The F ourier-Analytic Proof
of Quadratic Reciprocity




PURE AND APPLIED MATHEMATICS
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The Fourier-Analytic
Proof of Quadratic
Reciprocity
Michael C. Berg
Loyola Marymount University
Pure and Applied Mathematics
A Wiley-Interscience Series of Texts, Monographs, and Tracts

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Contents
PREFACE vii
ACKNOWLEDGMENTS xi
INTRODUCTION xiii
1. Heeke's Proof of Quadratic Reciprocity 1
1.1 Hecke it-functions and Their Functional Equation
/ 3
1.2 Gauss (-Hecke) Sums / 5
1.3 Relative Quadratic Reciprocity / 11
1.4 Endnotes to Chapter 1 / 14
2. Two Equivalent Forms of Quadratic Reciprocity 16
3. The Stone-V on Neumann Theorem 20
3.1 The Finite Case: A Paradigm / 21
3.2 The Locally Compact Abelian Case: Some Remarks
/ 24
3.3 The Form of the Stone-Yon Neumann Theorem Used
in  4.1 / 25
v

vi CONTENTS
4. Weil's" Acta" Paper
4.1 Heisenberg Groups / 28
4.2 A Heisenberg Group and A Group of Unitary
Operators / 32
4.3 The Kernel of 7T / 35
4.4 Second-Degree Characters / 44
4.5 The Splitting of 7T on a Distinguished Subgroup of
B(G) / 52
4.6 Vector Spaces Over Local Fields / 57
4. 7  Quaternions Over a Local Field / 63
4.8 Hilbert Reciprocity / 70
4.9 The Stone-Von Neumann Theorem Revisited / 73
4.10 The Double Cover of the Symplectic Group / 77
4.11 Endnotes to Chapter 4 / 79
26
5. Kubota and Cohomology
5.1 Weil Revisited / 84
5.2 Kubota's Cocycle / 86
5.3 The Splitting of lX A Over SL(2, k) / 92
5.4 2-Hilbert Reciprocity Once Again / 96
82
6. The Algebraic Agreement Between the Formalisms of Weil
and Kubota 98
6.1 The Gruesome Diagram / 99
6.2 The Even More Gruesome Diagram / 102
7. Heeke's Challenge: General Reciprocity and Fourier
Analysis on the March
103
BIBLIOGRAPHY
109
INDEX
113

Preface
This is not a book for experts. This is not a book for raw beginners. It
is, instead, an exposition of and commentary on a handful of sources,
most of them classical by now and at least one of them notoriously
austere. The material presented has been compiled as an aid to
number theorists seeking to work on the analytic proof of reciprocity
laws. This is a notorious affair, of course. The quadratic case is
completely settled by Hecke [He23], and resettled by Weil [We64], but
for n > 2 the matter is still open and ranks as one of the hardest open
problems in the field. This book is written for those reckless few who
are predisposed to enter this area of research at an early (but not too
early) stage of their career, when they don't yet know any better and
don't know a lot about the indicated specialized techniques either.
The goal is to make entry into this field a little easier by explicitly
delineating and comparing the three existing approaches to the
(Fourier-) analytic proof of quadratic reciprocity which qualifies as a
paradigm for the general case.
Of course, there is really only one Fourier-analytic proof of
quadratic reciprocity, traced back to Cauchy's treatment of the classi-
cal absolute case and Hecke's treatment of the relative case. Hecke
then posed the generalization problem mentioned above, which still
remains open today. Not until some four decades after Hecke's work
was the quadratic case cast into a form amenable to modern tech-
niques, that is, unitary representation theory. This breakthrough, of
vii

vii; PREFACE
paramount importance in the subject, was due to Weil, in a famous
(and famously difficult) paper whose objective went considerably
beyond a reformulation of the analytic proof of quadratic reciprocity.
These two formulations, Hecke's and Weil's, stand in contrast one to
the other in that Hecke's approach is classical while Weil's is anything
but. But the book goes one step beyond even Weil's formulation and
includes a treatment of Kubota's reformulation (and "algebraization")
of Weil's work. This reformulation resulted in the context in which
some dramatic subsequent achievements of the 1980s and the 1990s
(by, e.g., Matsumoto, Kazhdan-Patterson) were carried out-while the
proof of the general case remained, and remains, out of reach!
The goal of this book is to make it possible (if not easy) for the
reader to proceed from here to the papers by Weil and by Kubota, as
well as some of the aforementioned papers of very recent vintage.
Beyond this I intend that this book provide incentive to the reader to
tie up some loose ends which I leave untied, by going to the indicated
supplementary sources to which I refer rather copiously. So, from this
perspective, this book is something of a guide to the more peripheral
literature, too.
Here are a few examples of these loose ends. In the discussion of
Hecke's proof I take a lot of Fourier analysis for granted; the reader
should read Hecke or the cited books by Lang [Lang70] and by
Garrett [Ga90]. I also leave out consideration of the full general case,
treating only the case of a totally real algebraic number field. While
nothing dramatic happens in the former case as compared to the
latter case, the reader, if he is a Fourier-analysis-in-number-theory
rookie, should study the general case also.
And then, in the chapter on the Stone-Von Neumann Theorem, I
give only the proof of the finite group case, while I quickly go on to
use the theorem as it applies to locally compact groups. The reader is
urged to carry out one (or both) of the following projects: learn the
proof for the locally compact case; figure out without looking at  4.9
how Weil's very intricate arguments succeed in achieving what the
Stone-Von Neumann Theorem brings about without ever bringing the
theorem in explicitly. In any case,  4.9 gives a relatively detailed
sketch of what goes into designing the proof for the more general
case.
Rather than present an exhaustive treatment of the full quadratic
case, then, I have purposely left the reader some tasks which should
go a very long way toward providing him with a feeling for how this
kind of number theory interacts with, for example, harmonic analysis

PREFACE ix
on suitable topological groups, unitary representation theory, and so
on. I presuppose, therefore, that the reader has a solid foundation
along the indicated lines, that he is able to navigate easily through the
indicated analysis, topology, and algebra (possibly with standard
sources nearby), and that he is reasonably patient. This is dense
material, but it is concerned with beautiful and deep mathematics.
Daring to look ahead and prophesy-which is in itself very reckless,
of course-I claim that the resolution of the open problem Hecke
posed in 1923 will indeed come about by means of the discovery of a
generalization of Poisson's formula applicable to an as yet elusive
generalization of Weil's @-functional fitted into the (algebraic) setting
laid out by Kubota. This prophecy is proposed at the end of the book,
in the context of some algebraic architecture of my own design
(actually nothing more than a careful comparison between Weil's
formalism and Kubota's). It is my hope that this will resonate with the
reader as he proceeds onward.

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Acknowledg11lents
I should like to thank an anonymous referee who corrected a number
of mistakes in an early version of this work. My wife dissected my
prose with a sharp and sure scalpel; the extent to which this book is
readable is largely credited to her. (Of course, the extent to which it is
not is to be blamed only on me.) I wish also to express my gratitude to
Cathy Herrera, my department's world-class secretary, who prepared
the entire manuscript from my hand-written pages. Finally, thanks are
due to my research assistants, Ryan Brown and Donna Morano, who
slaved away arduously on the thankless job of indexing the book.
xi

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Introduction
Aside from its unquestionable novelty, leading to its inclusion in most
if not all introductory courses in number theory, the law of quadratic
reciprocity stands out as one of the deepest facts of the theory of
algebraic number fields. This was certainly already understood by
Gauss, who in his lifetime gave six proofs of this beautiful theorem
first conjectured by Euler. There are a number of good sources
available treating this central theme of Gauss' arithmetical work,
among which we recommend Variationen ilber ein Zahlentheoretisches
Thema von Carl Friedrich Gauss [Pi78], and the indicated section of
Scharlau-Opolka [S084].
Gauss' work laid bare deep connections between at first glance
rather disparate aspects of the behavior of rings of integers of alge-
braic number fields. Presently it became clear that the splitting of
primes in quadratic extensions is completely governed by the fine
structure of the Legendre symbol, that is, by quadratic reciprocity,
and this set the stage for Gauss' work on the genera of quadratic
forms.
If there is a tool par excellence in Gauss' armory for these arithmeti-
cal investigations it is surely the method of Gauss sums. Their relation
to the Legendre symbol is fundamental: it is an easy exercise to show
that Gauss sums transform very nicely under the Legendre symbol's
natural action. It is a quick step from there to the formulation of
quadratic reciprocity as an identity between so-called reciprocal Gauss
sums. But where are the quadratic forms?
xiii

xiv INTRODUCTION
The answer to this question takes us to what is properly called the
analytic approach to reciprocity laws. Interestingly enough, although
these interconnections between different well-delineated (if young)
parts of nineteenth-century algebraic number theory were doubtless
known to Gauss, the analytic approach has its genesis not with Gauss
but with Cauchy. This is certainly the attribution made by Hecke (see
below), but he neglects to give chapter and verse. I leave it as an
exercise for the historically minded reader to unearth Cauchy's proof
of quadratic reciprocity in his voluminous Oeuvres Completes.
The main idea behind the analytic approach to reciprocity, whether
it be for the Legendre symbol or, equivalently, for Gauss sums, is to
determine a suitable analytic object which associates to these arith-
metic objects in the right way and then to derive the right kind of
arithmetic identity from a functional equation. For Gauss sums the
corresponding analytic object is a {t-function and its very definition
depends on a quadratic form.
The way one gets from a {t-function, that is, qua data, a quadratic
form, to a Gauss sum is by means of a limiting process involving the
so-called it-constants or Thetanullwerte. The all-important it-func-
tional equation is usually established by methods of Fourier analysis
with Poisson summation in a starring role. It is important to note,
however, that a {t-function is a half-integral weight modular form and
from this modern perspective a it-functional equation arises as an
instance of automorphy.
In the 1920s this classical strategy was adapted by Hecke to provide
the analytic proof of relative quadratic reciprocity. The Legendre
symbol can easily be extended to an arbitrary algebraic number field
and this more general symbol also satisfies a reciprocity relation;
hence: relative quadratic reciprocity. In the nineteenth century, with
the flurry of activity surrounding reciprocity laws, relative quadratic
reciprocity had already been dealt with to some extent, but the
analytic proof did not appear until the last chapter of Hecke's famous
1923 treatise, Vorlesungen ilber die Theolie der algebraischen Zahlen
[He23]. Hecke's strategy corresponds exactly to the preceding.
Of course, with the emergence of class-field theory in the late
nineteen th century and, most importantly, Hilbert's Zahlbericht of
1897 [Hi32], it had become clear that reciprocity laws occupy a central
position in algebraic number theory, in keeping with Gauss' pre-
science. It follows that an analytic approach, not just to quadratic
reciprocity but also to reciprocity laws in general, would be of great
value.

INTRODUCTION XV
In the last paragraph of the Vorlesungen, Hecke says it this way:
In der Art, wie wir die Theone des relativquadratischen Korpers begrUndet
haben, erscheint das Reziprozitatsgesetz als das erste, die Existenz der
Klassenkorper als eine Folge desselben. In den klassischen Begrilndungen von
Hilbert und Furtwiingler (bei der Untersuchung auch hoherer Potenzreste)
verlauft der Gedankengang umgekehrt. Es wird erst die Existenz der
Klassenkorper auf anderem, ubrigens sehr kompliziertem Wege bewiesen,
ihr Zusammenhang mit den Idealklassen diskutiert und daraus dann
das Reziprozitiitsgesetz algeleitet. Dabei ist das sog. Eisensteinsche
Reziprozitatsgesetz ein bisher unentbehrliches Hilfsmittel. Auf diesen Weg ist
man in alien Fallen angewiesen, wo es sich um Korper von hoherem Relativ-
grad als 2 handelt. Man hat bisher noch nicht solche transzendenten Funktio-
nen entdeckt, welche, wie die Thetafunktionen unserer Theone, eine
Reziprozitatsbeziehung zwischen den Summen ergeben, die fUr hohere
Potenzreste an Stelle der Gau{3schen Summen treten . .. .
This provides a greater purpose for the enterprise of looking at
quadratic reciprocity's analytic resolution as something of a prototype.
ffhe huge question in the background is how to generalize Hecke's
exploitation of -functions so as to get at an analytic approach to
higher reciprocity laws.
Surprisingly little attention was paid to Hecke's open problem until
some forty years later. In 1964 Weil published his famous paper, "Sur
certains groupes d'operateurs unitaires," whose aim it was to lay bare
the secrets of Siegel's avant garde analytic theory of quadratic forms
[Si35]. Weil shows that this theory is very closely related to the
behavior of square-integrable representations of unitary groups associ-
ated to locally compact abelian (LCA) groups. The focus of the
subject has changed rather dramatically, falling now on representation
theory. And it has been so ever since.
In this same paper, sometimes called "Weil's Great 'Acta' Paper,"
Weil also shows that the reciprocity law for the quadratic Hilbert
symbol is tantamount to the existence and behavior of a certain
double cover (or a two-fold central extension) of Siegel's symplectic
group. Then, almost in passing, he observes that his proof must indeed
be quite the same as the classical Cauchy-Hecke proof of the relative
quadratic reciprocity law using the functional equation satisfied by
certain it-functions. To be sure, one of Weil's central arguments
concerns an automorphic representation in a Hilbert space of general-
ized it-functions and a beautiful application of a generalized form of
Poisson summation. This quickly brings us back to the last chapter of
Hecke's Vorlesungen. In no uncertain terms, in Weil's hands the

xvi INTRODUCTION
analytic proof of quadratic reciprocity emerged as property situated
within unitary representation theory.
Moreover, Poisson summation's role was expanded to show ulti-
mately that the so-called projective Weil representation of the adelic
symplectic group reduces to a true representation when restricted to
the subgroup of rational points. In turn, this provides that the afore-
mentioned double cover of the symplectic group (adelized) splits on
the rational points, which, in turn, yields 2-Hilbert reciprocity. Noting
that 2-Hilbert reciprocity is quite the same thing as relative quadratic
reciprocity, Weil's en passant observation is seen to be exactly on
target.
Only three years after Weil's paper appeared, Kubota demonstrated
in "Topological covering of SL(2) over a local field" [Ku67] that over a
totally imaginary number field the general higher degree reciprocity
law is equivalent to the existence and certain splitting properties
(relative to subgroups of rational points) of a metaplectic cover of a
linear group. .
Kubota introduced metaplectic covers first of special, then of gen-
eral two-dimensional linear groups, both locally and adelically, with
the attendant 2-cocycles constructed directly from 2-Hilbert symbols.
As a result of this explicit formulation, 2- Hilbert reciprocity can be
derived directly from the fact that the adelic double metaplectic cover
splits over the subgroup of rational points imbedded in the adelization
of the (special or general) linear group (cf. [Ge76]). Due to a famous I
cohomological result of C. C. Moore [M068], in the quadratic setting,
this construction must yield the same object as Weil's construction,
because the lowest dimensional symplectic group can be identified
with the special linear group.
-Then, in 1969, in "On automorphic functions and the reciprocity
law in a number field" [Ku69], Kubota went on to address the question
of constructing higher metaplectic covers. Unfortunately, while his
work shows that the splitting of an n-fold adelic metaplectic cover on
the rational points is indeed enough to give, say, n-Hilbert reciprocity,
he had to presuppose this reciprocity for his construction. The prob-
lem, then, is how to do Kubota's construction without presupposing
any reciprocity laws.
Finally, this splitting of an n-fold metaplectic cover on the rational
points is in itself a very difficult affair. Proving this without assuming
reciprocity would involve, for example, the manufacture of a higher
degree Weil representation, a higher degree Weil 8-functional, and
even a higher degree counterpart to Poisson summation. Indeed, it is
evident that a successful attack on Hecke's challenge issued along the
indicated lines requires a fundamental advance in abstract Fourier

INTRODUCTION xvii
analysis. At the date of this writing, with the recent work by Laumon
[Lau84] and others on Deligne's Fourier transform, there is reason to
be hopeful.
Having appraised the analytical approach to reciprocity laws for
number fields in the preceding terms, the reasons for this book
become apparent. It grew out of a long set of notes prepared in order
to explain Weil's pivotal 1964 paper, dotting the i's and crossing the
t's. As conveyed above, this paper marks the advent of the modern
representation-theoretic approach which has breathed new life into
the attack on Hecke's old generalization problem and is responsible
for a wealth of mathematical innovation. On Weil's recommendation
[We 89] I came to this paper from another part of number theory-
modular forms and classical Fourier-theoretic methods a la Hecke.
While the transition was natural, particularly as it was driven by an
established open problem, it was not a trivial one. This is already
evident by comparing the structure of Hecke's proof of relative
quadratic reciprocity as outlined above to the more involved, or
certainly more sophisticated, tactics employed by Weil. But it cannot
be otherwise, of course, if one is to get at a deeper understanding of
quadratic reciprocity-let alone the higher-degree case. Although
more machinery is called for, the prevailing viewpoint has to be that
of unitary group representations, and the Fourier transform has to be
fitted into that framework.
So there is no doubt that the central chapter of this book is, in fact,
the middle chapter, Chapter 4, which is entirely devoted to a careful
and complete explication of those parts of Weil's "Acta" paper which
are directly concerned with quadratic reciprocity. Accordingly the
book is arranged in a chiastic pattern. In Chapter 1 we address
Hecke's groundbreaking work and in Chapter 7 we discuss the status
of the challenge he posed at the end (indeed in the last paragraph) of
the Vorlesungen. In Chapter 2 we give an account of the equivalence of
the formalisms of Gauss-Euler and Hilbert for quadratic reciprocity
itself. In Chapter 6 we compare the formalisms of Weil and Kubota,
concerning the transformation groups that are involved, and their
all-important covering groups. Finally, in Chapter 3 we discuss the
theorem of Stone, Yon Neumann and Mackey, which provides a
welcome streamlining of the representation theory Weil develops in
his treatment of quadratic reciprocity, while in Chapter 5 we treat
Kubota's work of 1967, which provides an explicit algebraic develop-
ment of Weil's double symplectic cover whose splitting properties
precipitate quadratic reciprocity. One must bear in mind that the
entire edifice rests on this adelic splitting: the first half of the book
leads up to it; the last half expands on it. In what follows we discuss

xv;;; INTRODUCTION
the specifics of the individual chapters and some of their prerequi-
sites.
Chapter 1 deals with Hecke's work, as we have indicated' already.
The techniques involved come from classical Fourier analysis in R n
rather than R 1. A good understanding of the one-dimensional case is
more than adequate, as we do not require anything that fails to
translate directly to the higher-dimensional case. Not surprisingly, the
viewpoint of Poisson summation as a prevailing tool for proving
functional equations is of paramount importance.
For convenience, we confine attention to the case where the under-
lying algebraic nUlp.ber field is totally real, but everything carries over
to the general case. It should be an easy matter for the reader to go
on to Hecke's Vorlesungen to examine the general case in detail, but
this is not necessary for our purposes.
Chapter 2 deals with something that a number of sources tend to
leave unaddressed, even as it is referred to unabashedly: the equiva-
lence between the various forms of the law of quadratic reciprocity
that one encounters in the literature. On the one hand, there is
quadratic reciprocity in the style of Gauss and Euler, involving just the
Legendre symbol. On the other hand, there is the product formula for
the quadratic Hilbert symbol. These formulations allow for generaliza-
tions, as we have instances of these for number fields beyond the
rationals and for higher-degree generalizations of the Legendre and
Hilbert symbols. In Chapter 2 we consider the equivalence between
classical (absolute) quadratic reciprocity in the style of Gauss-Euler I
and the progenitor of Hilbert reciprocity, the case of the 2-Hilbert
symbol. The corresponding equivalence between (Hecke's) relative
quadratic reciprocity and 2-Hilbert reciprocity for an arbitrary number
field works the same way, and we have left the details as an exercise.
The goal is to provide the reader with some opportunity to "get his
hands dirty" as he progresses through the first half of the book,
preparing for Weil. It is for this reason that Chapters 1 and 2 do not
quite concern themselves with the indicated general cases. Investigat-
ing these on his own should provide the reader with an increasingly
comfortable feel for this still classical material. The cases we deal with
will provide a more than adequate Gestalt for our particular needs.
We also leave the general case as an exercise in Chapter 3, which,
however, we do not classify as classical for it is here that representa-
tion theory rears its head. A sufficiently detailed discussion of the
Stone-Yon Neumann Theorem is presented to serve our purposes in
Chapter 4, but we only give the proof of the case where the underlying
group is finite. The important case is where this group is locally
compact. We have opted simply to refer the reader to Mumford's

INTRODUCTION xix
elegant discussion in part one of his Tata Lectures [Mu91], giving
hints on what steps must be taken to effect the transition to the more
general case. Of course, this is much more of an ambitious exercise
than the ones in the previous chapters, but Mumford is always well
worth reading.
It bears mentioning that the general case of Stone-Yon Neumann
we deal with is due to Mackey, and his papers of the middle-to-Iate
1950s are classics and come highly recommended. Moreover, it is
possible to go through Weil's "Acta" paper blissfully ignorant of the
Stone-Yon Neumann Theorem, as Weil himself makes no use of it.
But when I made mention of this to V. S. Varadarajan in 1995 he
replied immediately, "But surely it's implicit there!"
This takes us to Chapter 4. We are now concerned only with
quadratic reciprocity for an arbitrary algebraic number field, ex-
pressed by a Hilbert product formula, and unitary representation
theory takes center stage. We progress from a distinguished Heisen-
berg group to a projective representation of its isotropy group and
then to splitting properties. Thereafter it is a question of playing the
local cases against the adelic case in order to get at the climactic
product formula. We finish this long chapter with a section on the
specific application of Stone-Yon Neumann as a simplifying device to
Weil's difficult argument and a discussion of how Weil shows that his
symplectic cover is, in fact, a double cover. The latter is a prelude of
sorts to the next chapter, on Kubota's work.
The prerequisites for Chapter 4 are a bit more severe than those
for the first three chapters, which are, of course, considerably less
ambitious. We presuppose a fair amount of familiarity with modern
expositions of algebraic number theory, that is, the behavior of num-
ber fields both locally and adelically. We also assume the reader has
some knowledge of unitary representation theory and the theory of
integration over locally compact abelian groups (not that much, actu-
ally). There are marvelous sources available for this material: Weil's
own books, Basic Number Theory [We74] and L'!ntegration dans les
Groupes Topologiques et ses Applications [We51]. Toward the end of
the chapter we need some results on quadratic forms. For this
material we recommend Lam's The Algebraic Theory of Quadratic
Forms [La73]. Still, we have tried very hard to make this discussion
self-contained and claim that it is possible to go through this fourth
chapter profitably even without consultation of outside sources.
With Weil's work taken care of we proceed to Chapter 5 and
Kubota's manoeuvres with 2-cocycles on SL(2) over a local field
expressible in terms of 2-Hilbert symbols. This chapter has two main
points. First, we give Kubota's proof that the object he proposes as a

xx INTRODUCTION
2-cocycle for the indicated cover actually does the job. Second, we give
his proof that the cover defined in this way is non-trivial. The reader
needs only the rudiments of the theory of group extensions, that is,
second-degree cohomology, as found in many books on homological
algebra. For the historically minded we suggest Eilenberg-MacLane's
Collected Works [EMacL86]. Naturally, also presupposed is a modicum
of topology, and some familiarity with linear groups, locally and
adelically, as given in, for instance, Humphreys' Arithmetic Groups
[Hu80], would be useful too.
Chapter 6 is a comparison of the formalisms developed by Weil and
Kubota, at least as far as the evolution of covering groups is con-
cerned. It is all a question of comparing diagrams of exact sequences,
and we offer along these lines two such comparisons, the "gruesome
diagram" and the "even more gruesome diagram." Beyond their enter-
tainment value these diagrams serve the purpose of bringing the
reader to a vantage point where he can consider all at once the
representation-theoretic context for the entire development from Weil
to Kubota. This vantage point is of importance as regards the last
chapter, where we go into the matter of Hecke's open question per see
Finally, Chapter 7 is an evaluation of the status of Hecke's venera-
ble challenge as of the date of this writing. It is a personal perspective
on this notorious matter, on the part of one who has little right to call
himself an authority on this matter. Mter all, as indicated in the
Preface, more than anything else this book aims at providing an entry
into the field by dissecting the quadratic case. It does not attempt to
say anything too radical or prophetic as far as research proper is
concerned. Still, as it is altogether impossible not to speculate, we
present in this last chapter some preliminary glimpses at what looks to
be a very beautiful approach. This line was suggested to me by V. S.
Varadarajan in 1995.
Chapter 7 also contains an extremely concise formulation of the
analytic proof of quadratic reciprocity, giving only the architecture
and leaving out the bricks and mortar. Mter working through Chap-
ters 4 and 5, where all the brick-laying is done, this kind of sketch may
serve to provide an overview of the whole affair and thereby convey
the beauty of the structure all at once. It is perhaps apposite to recall
Weil's own words by way of introduction to his beautiful book, Elliptic
Functions According to Einstein and Kronecker [We76]: "When kings go
building, carters have work to do."

Hecke's Proof of
Quadratic Reciprocity
With (; a a rational prime, define the Legendre symbol in the usual
way: - ) = 1 or - 1 according as a is or is not a square modulo p.
p p-l
Equivalently, we can say that ( ; ) = (a mod .p)) , p -r a. Then, with
p and q both odd primes, we have quadratic reciprocity in the familiar
form of Euler and Gauss, as follows:
p-l q-l
(  H ; ) = (-1)';
moreover,
p2-1
( ; )=(-1)S-,
and
( -1 ) 
P = (-1) 2 .
The last two assertions are traditionally referred to as the
Ergiinzungssiitze of the quadratic reciprocity law. It is important to
1

2 HECKE'S PROOF OF QUADRATIC RECIPROCITY
note that the Legendre symbol is a multiplicative homomorphism
modulo p, mapping to {1, - 1}.
Gauss gave six proofs of this reciprocity law for the Legendre
symbol, and all of us know one or two of them; they are easily found
in the literature in any case. Beyond this, perhaps the best-known
nineteenth century non-Gauss contribution of a proof of quadratic
reciprocity would be Eisenstein's (cf. [S085]). Some of us saw his
proof as undergraduates. It is comparatively less known, however, that
Cauchy gave a proof also [Ca82]. It is Cauchy's approach that matters
to us, as it forms the prototype for what Hecke does over an arbitrary
number field.
Such a multiplicity of different proofs of a single theorem unques-
tionably indicates a wealth of deep connections between different
parts (and methods) of number theory. Cauchy's proof is particularly
remarkable for its use of the functional equation of Jacobi if-functions
and the deep relationship it reveals between "reciprocal" it-constants
(Thetanullwerte) and therefore reciprocal Gauss sums. This is alto-
gether fitting for we know (with the benefit of hindsight) that quadratic
reciprocity is not only integrally tied to the theory of genera, to the
splitting of primes in quadratic extensions, and so on, but, most
important for our purposes, also to quadratic forms. And quadratic
forms, in turn, define it-functions.
As we just indicated, Cauchy's proof was beautifully generalized by
Hecke in the 1920s [He23], yielding the so-called relative quadratic I
reciprocity law (over an arbitrary algebraic number field). This ap- i
proach brings out the salient point in this analytic proof of quadratic
reciprocity very clearly-the fact that a it-function is automorphic of
half-integral weight and is amenable to Fourier analysis. We will
recognize the profound importance of this viewpoint later on, in
connection with the Weil representation.
Hecke's generalization, a tour de force of Fourier analysis, occurs in
Kapitel VIII, the final chapter of his Vorlesungen [He27]. Briefly, the
Cauchy-Hecke strategy is as follows:
Step 1. Start with an appropriate it-function (the Jacobi it-function in
the absolute case, and what is today called a Hecke f)-function in
the relative case), realizing that such a half-integral weight auto-
morphic function is associated to a suitable quadratic form occur-
ring in the exponents of its summands. By means of manipulating
the Fourier coefficients of this f)-function, establish a functional
equation of sorts whose primary feature is that it relates a pair of

HECKE f)-FUNCTIONS AND THEIR FUNCTIONAL EQUATION 3
it-functions associated to "reciprocal" ideals (modulo the inverse
different). Of course, this all boils down to Poisson summation.
Step 2. Evaluate a certain limit at a boundary point of the region of
analyticity of the it-function in order to change the aforementioned
it-functional equation to an identity involving similarly "reciprocal"
Gauss sums. This also entails that these resultant Gauss (or Gauss-
Hecke) sums are essentially it-constants [Fr91].
Step 3. Proceed from this reciprocity relation between Gauss sums to
the desired reciprocity between extended Legendre symbols by
arithmetical manoeuvres. This is possible because Legendre sym-
bols are expressible as quotients of suitable Gauss sums or, what is
quite the same, Gauss sums transform nicely with respect to the
Legendre symbol. In [Be95] this important connection between a
quadratic symbol and a Gauss-type sum is generalized to higher
powers along classical lines.
We showed in [Be93] that a faithful mimicry of Hecke's methods
leads to a functional equation (for generalized if-functions, associated
to forms of higher even degree). This functional equation is not
suitable for Steps 2 and 3; to find out the specifics behind this failure,
the reader should look at [Be93].
Now on to the proof of relative quadratic reciprocity. This is vintage
Hecke: analytic number theory with a vengeance.
1.1 HECKE i)-FUNCTIONS AND THEIR FUNCTIONAL
EQUATION .
Let k be an algebraic number field of (absolute) degree n; <.Ok is its
ring of integers. To begin with, we assume, with Hecke, that k is
totally real. The central result, (1.3.7), however, holds for arbitrary k
-the motivating discussion just happens to be less cumbersome (and
remains sufficiently general) in case k's conjugate fields are real.
We also have that if Q(x 1 ,..., x n ) is a quadratic form (over k, i.e.,
in <'ok[X 1 ,..., x n ]) with positive definite real part, then the associated
it-series is, by definition,
- <l eu U ) " e - 7TQ (ml+ul,...,m l1 +u l1 )
'U l' · · ., n = L.J
(1.1.1)
mEZ n

4 HECKE'S PROOF OF QUADRATIC RECIPROCITY
where U 1 '..., un are real variables and m = (m 1 ,..., m n ) E zn. Let a
be an integral ideal of k (i.e., in <Ok)' having Z-module basis
{aI' · · · , an}. Under the assumption of k being totally real, setting
n
zp = E aP)uq' for 1 < p, q < n, with ap) denoting the pth conju-
q=1
gate of u q , we have that z = (Zl'. . . , zn) ranges over R n as u =
(U 1 ,..., un) does. Then it is easy to prove that, with t = (t 1 ,..., t n ),
tt(t, z; Q) = E e- 7TE ;_1(P(P.(P)+Zp)2
/-LEa
(1.1.2)
defines a it-series in the preceding sense, that is, (1.1.2) becomes
(1.1.1) upon substituting for the Z p in the (two-variable) Hecke it-
function attached (or belonging) to Q.
By means of manipulating the Fourier expansion of (1.1.2) (which
exists because (1.1.1) is invariant under the substitutions UqUq + 1,
with 1 < q < n) one obtains, with N{a) denoting a's ideal norm,
Proposition 1.1.1. If b (resp. d) is the different (resp. discriminant) of
k/Q, then
A(P )2
"n 2 '"n dp)
e-7Ti...p-l-- 7Tli... p "'ll\ zp
t p ,
1
it(t Z. a) =
, , N(a)la h/ t 1 ... t n
E
1
AE-
ab
(1.1.3 )
1
where ab is the inverse (reciprocal) ideal to ab in k's ideal group.
Next, writing it{t; a) (a single-variable it-function) for the it-con-
stant it(t, 0; a), it is possible to derive the following all-important
resul t:
Corollary 1.1.2. The Hecke it-function it(t, a) obeys the functional
equation
1 ( 1 1 )
it t. Q - it ·
(, ) - N(a)lldhjt 1 ... t n t' ab '
(1.1.4)

GAUSS (-HECKE) SUMS 5
1 ( 1 1 )
where - = -, . . . , - . Moreover, this relation obtains that 1}(t, a)
t t 1 t n
can be viewed as an analytic function in (R > 0 X i R) n .
The proofs of these two assertions are wonderful exercises in
classical Fourier analysis; Poisson summation figures in an obvious
way. The reader who is new to this kind of application of Fourier
analysis should look up Hecke's original arguments in Vorlesungen
[He27]. In addition, Garrett [Ga90] and Lang [Lang70] provide good
coverage of this material.
1.2 GAUSS (-HECKE) SUMS
-
I
If the fractional ideal 1/ ab has Z-module basis {{31'...' {3n}, so that
A E l/ab is uniquely expressible as a sum A = m 1 {31 + ... +mn {3n,
m E zn, then A runs through l/ab simultaneously as m runs through
zn. Moreover, A = 0 if and only if m = o. Let r = min Re (  ) > O.
15.p5.n t p
Viewing L r{p)2 as a positive definite quadratic form in its own right,
p
it follows from a classical theorem of Minkowski [Mill] that there
exists an absolute real constant C > 0 such that
ex p ( - 7T L J\.(d/ tp ) < ex p ( - 7Tr L J\.(P)2)
P P
< exp{ -7TrC( m + ... +m)}.
Now we get the following proposition:
Proposition 1.2.1. Letting t p  0 (1 < p < n) in such a way that
Re( t )  00 (i.e., r  00), we obtain
1
lim V tl." t n iJ(t, z; a) = ()IJdf
t--.o N a d
(1.2.1)

6 HECKE'S PROOF OF QUADRATIC RECIPROCITY
Proof. Going to (1.1.3) we have that with A E l/ab,
( A(p)2 )
L exp - 7T L - 27Ti L A(P)Zp
A+O p t p p
( A (p)2 ) n
= L: exp -1T L: < { L: exp( -1TrCm 2 )} - 1
A+O p  mEZ
= {1+2 L: eX P (-1Trcm 2 )f -1«1+2 L: e-'lTrcmf-1
mO mO
( 2 e - 7T rC ) n
= 1 + 1 _ e - 'IT rC - 1 r --+ ,,,,> O.
B ut this im plies that as t ---+ 0 in the indicated way (i.e., r ---+ 00), then
y t1,...,t n it(t,z,a) must, by means of (1.1.3), tend to the term corre-
1
sponding to A = 0, that is, I r- I ' and we are done.
N( a ) v ex
.
Next, if w E k such that, in k's ideal group, b w = b /a with
(a, b) = 1, that is, the integral ideals a and b are relatively prime,
define the Gauss-Hecke sum [Sh64]
C( w) = L e27TiSp{WJL2),
J..L E lJ k / a
(1.2.2)
where Sp denotes the absolute trace from k to Q.
Proposition 1.2.2. Write w for the vector of conjugates of w, that is,
w = (w(l),..., w{n» and, with x E R>o, write x - 2iw for (x - 2iw(l),
. . . , x - 2iw(n»). Then
I . c:rl _Q. ( 2 . - In ) C ( W )
1m vX.-'u X - lW; uk = ( I C I ·
xO+ N a) vex
Proof. Setting t p =x - 2iw(p), 1 < p < n, z = 0, and a = Dk in (1.1.2)
yields that t}(x - 2iw; <'ok) = L exp{ - 7T L (x - 2iw(p») J.L Cp )2}. De-
J..L E lJ k P
composing Dk as D k ::: a EB <.Ok/a allows us to write the preceding sum

GAUSS (-HECKE) SUMS 7
over J.L as a double sum over v E a and Q E Dk/a:
-& (x - 2iw; <!:l)
- L exp{ - 'TT LP( x - 2iu}P»)( v + (} )(p)2}
J..t=V+(}
- JL=+1l [ex p  {2'TTiw(P)( v + (} )Cp)2} exp  { - 'TTx( v + (} )(p)2} ]
- L { exP2'TTiL(}CP)2wCP)}exP { -7TLX(V+ Q)(P)2 } ,
J..t=V+(} P P
because L w{p)( V(p)2 + 2 v{P)Q{P») E Z. (This follows from the fact that
P
w( v 2 + 2 vQ) E wa = b /b since v E a; now we simply apply a funda-
mental property of the inverse different.) But this means that
-& (x - 2iw; lOk) = L exp 2'TTi Sp( (}2w) L ex p { - 'TT LX( v Cp ) + (} CP»)2 }
Q v P
= Lexp27TiSp(Q2 w )it(x,{j;a),
(}
with {j being Q's vector of conjugates. Thus, by means of (1.2.2) and
Proposition 1.2.1:
lim Vxn -&(x - 2iw; lOk) = lim Lexp2'TTiSp( (}2w)vxn -&(x, Q; a)
x-+O + x-+O +
(}
=C(w) lim ..Jxn -&(x,Q,a) = C(11 '.
x-+O+ N(a) d
Clearly, Proposition 1.2.2 is a large part of Step 2, namely, the
characterization of a Gauss-Hecke sum in terms of the right {}-
constant. The next step is to use (1.1.4) and (1.2.1) to derive what
Hecke calls "a reciprocity relation between Gauss sums."
In addition to the assumptions already in place, write b 1 for the
reduced denominator of a/40 in k's ideal group and let 'Y E k such
that by E Dk and (y, 0 1 ) = 1. Write Sp(sgn w) for Lsgn w{p) where
w p
sgn w = I wi = + 1 (as k is totally real). Then we have the following;

B HECKE'S PROOF OF QUADRA TIC RECIPROCITY
Proposition 1.2.3.
C(w)
V N(a)
yN(2b) :i Sp(Sgnw) ( _y2 )
- N(b 1 ) e C 4w ·
Proof. From (1.1.4) we get that
tt ( X _1 2iw ;  ) = Iv'dl v (x - 2iw(1)) ... (x - 2iw(n)) tt(X - 2iw; (,l),
1 ( 1 1 ) I
writing . _ for 2' (1) "'" 2' (n) · Next, an easy calcu-
x -lW x - lW X - lW
lation yields that
7Ti
lim vi (x - 2 i W( 1 )) ... (x - 2 i w( n) ) = V N (2 W ) e - 4"" Sp (sgn w) ,
xo
and so
( 1 1 )
lim xn/2it .-
xo + X - 2iw ' b
7Tl
I I -Sp(sgn w) ( )
= lim xn/2 v'd V N (2w) e 4 iJ x - 2iw; Dk
xo+
7Ti C(w)
= 1 v'd l y N ( 2w ) e- -Sp(sgn w)
4 N(a)IVdI'
by Proposition 1.2.2. By an argument very much like the derivation of
Proposition 1.2.2 from Proposition 1.2.1 (consult [He27] for details) it
is possible to prove that
( 1 1 ) N(2w8)
lim Xl'l/2iJ . - = A
xo+ x-2iw'b N(v1c)Iv'dI'
where c b = 8 in <.Ok and A is the sum
A = Eex p ( -27riSp (i 2 ) '
Q 4w8

GAUSS (-HECKE) SUMS 9
with (l running through the residue classes modulo v, which are
ult!plies of c. But then, since N( : ) = N( b) = Idl, we get the simpli-
fIcatIon
( 1 1 ) N(2w)
lim x n 12it . _ ; - b = (u) I v'cl IA ,
xo+ x-2zw N I
whence
C( w) V _ :i Sp(sgn w) N(2 w) 1 _0 1
N(a) N(2w) -e N(u 1 ) vd A.
Therefore,
C(w) =e :i Sp (sgnw q/N(2w) 1v'clIA
!V(a) N(v l )
I
7Tl
-Sp(sgn w)
=e 4
( V ) 7Tl
-Sp(sgn w)
N 2b"a 1v'clIA = e 4 1v'clIVN(2u)
N(v l ) N(vl)Iv'clI y' N(a)
where
C(w)
y' N( a)
l
V N(2 fJ) 7Ti Sp(sgn w)
- N(u 1 ) e 4 A.
a
It only remains for us to note that if a is divisible by c and IS
c
prime to VI' then, in the definition of A, (l can be replaced by (la,
which quickly yields that A coincides with C ( - y2 ) , writing 'Y = a .
4w D
.
Finally, we collect a few intrinsic results about Gauss-Hecke sums
which will be required in the upcoming derivation of the relative
quadratic reciprocity law from Proposition 1.2.3. Again, we follow
Hecke's treatment very closely.
Proposition 1.2.4. Let Xl' X 2 , a E <'ok' each relatively prime to a. If
Xl = x 2 a 2 (mod a) then C(xlw) = C(x 2 w).

10 HECKE'S PROOF OF QUADRATIC RECIPROCITY
Proof. Because J.L and J.La run through Dk/a (in (1.2.2)) simultane-
ously, C(xzw) = C(xzaZw). Next, Sp(J.LZXlw) - Sp(J.LzxzaZw) =
Sp( J.LZ(Xt -x 2 a Z )w) E Z, since Xl -xza Z Ea. (This is, once again, an
immediate consequence of the definition of the different b, noting
that w has reduced denominator ab.) The result follows readily. .
Proposition 1.2.5. If a l = cla l , a z = cza z , (aI' a z ) = 1, and
(a l a z , c 1 C Z ) = 1, then
C ( f3 ) =c ( azf3 ) c ( alf3 ) .
a l a z a l a z
Proof. Note first that as QI (resp. Qz) runs through Dk/a, (resp.
Dk/aZ)' J.L = Ql a z + Qz a l runs through Dk/a l a z . Then we have that
c( a: Jc( a: )
" . ( Q a z {3 ) " . ( Q a z f3 )
= '-' exp 27Tl Sp '-' exp 27Tl Sp
Q1 a l Q2 az
= 1: exp 27Ti Sp( Qi a + Q an f3
Q,Q2 ala Z
_" 2 . s ( z Z z Z ) f3
- '-' exp 7T l P Q I a Z + Q Z a I + 2 Q I Q Z a z a Z ,
a a
Q1,Q2 1 2
because Sp(2QI Qz {3) E Z. Consequently, with J.L as above,
C ( a z f3 ) c ( a l f3 ) = l:ex P 27TiS P ( t-t 2 f3 ) =c ( f3 ) ,
a l a z J.L a l a z a l a z
as required.
.
Proposition 1.2.6. If a = , an odd prime ideal, and if X E D k , (x, )
= 1, then
C(xw) = ( ; )C(W).

RELATIVE QUADRATIC RECIPROCITY 11
Proof. It is a straightforward exercise to prove that, with J.L runnIng
through Dk/tJ, L exp 27Ti Sp( J.Lw) = O. Consequently,
J..L
 ( ; )ex p 21Ti Sp( JLw)
=  ( ( ; ) + 1 }ex p 21TiSp( JLw)
= 1 + 2 L exp 27Ti Sp( J.Lw) = L exp 27Ti Sp( J.L 2 w) = C( w),
J..L square JJ..
noting that for any g, if J.L satisfies J.L 2 = g (mod tJ), the same is true of
- J.L. Now, replacing J.L by x J.L,
C( w) =  ( x; )ex p 21Ti Sp( JLxw)
= ( ; )  ( ; )ex P 21TiS P ( JL(xw)) = ( ; )C(XW).
Since ( ; ) = + 1 =1= 0, the theorem follows.
.
Corollary 1.2.7. If a is merely an odd ideal and x E Dk with (x, a) = 1,
then
C(XW)=( : )C(W).
Proof. See [He27].
.
1.3 RELATIVE QUADRATIC RECIPROCITY
For convenience we suppose that k's different is principal, say, b = (D),
D E Dk. An odd algebraic integer a E Dk is said to be primary if it is
congruent modulo 4 (i.e., 4 D k ) to a square in Dk.
Proposition 1.3.1. Suppose that one of a, f3( E D k ) is p rimary. Then
( a ) ( (3 ) = e :i Sp(sgn a{38-sgn a8-sgn (38) V N() .
f3 a C(-4"8)

12 HECKE'S PROOF OF QUADRATIC RECIPROCITY
1
Proof. In Proposition 1.2.3, take w = , which is to say, (af3) =
af35 1
a, Dk  b, where b 1 = 4 D k , and we may take 'Y = 8' This yields
c( aB ) 1 7Ti Spgn a{3li) ( af3 )
= e 4 C --
v N(af3) V N (8) 45 .
(1.3.1 )
By Proposition 1.2.5, C ( 1 ) =C ( 1/D ) =C (  ) C (  ) ; by
aD a 8 aD
Corollary 1.2.7, C( ;a ) . ( ; )c( ;a ) and c( :a ) = ( : )c( :a ).
Consequently, c( aa ) = ( ; )( : )c( ;a )c( :a )' Next, again in
1
Proposition 1.2.2, set w = -, that is, a = (a), b = D k , b 1 = 4 D k ,
1 a8
'Y = 8' Then
( 1 ) _ Fm a ) :i Sp(sgn ali) ( a )
C - - N - e C --
a8 8 48 '
(1.3.2)
and, mutatis matandis,
C( ;a ) = V N( ) e :i SPgn{3li)C( -  ).
Of course, (1.3.2) and (1.3.3) also follow from (1.3.1) just by setting
f3 = 1, a = 1, respectively. Substituting everything into (1.3.1) yields
(1.3.3)
( a ) ( f3 ) _ :i Spgn a{3li-sgn ali-sgn (3li) ,; N( 8 ) C( - "*)
- e ( ) ( a. ) . (1.3.4)
f3 a C - C -
48 48
Finally, by Proposition 1.2.4, taking a to be primary (without loss of
generality) C ( - a ) = C ( -  ) and C ( -  ) = C ( -  ) . This
, 48 48 45 48
( af3 )
C -- 1 1
. I . h 48 h b .
Implest at C ( - ) C (  ) = C ( - ) = ( 1 ) ,w ere,su Stl-
48 48 48 C --
45

RELA TIVE QUADRATIC RECIPROCITY 13
tuting into (1.3.4), the desired result follows. .
Under the preceding assumption that b = (5), c(  ) i defined on
the ideal Dk and it is easy to derive the fact that C ( - ) = 1. This
allows us to prove the following proposition. 8
Proposition 1.3.2.
,; N(8)
C ( - :8 )
7Tl
-Sp(sgn S)
= e 4 .
7Tl
- Sp(sgn S)
Proof. By (1.3.1), this time with a{3 = 1,C(  ) = e N(8) c( -  ) =
1, and simplifying. .
7Ti
Corollary 1.3.3. (  ) ( : ) = e """4 Sp(sgn afi8-sgn a8-sgn fi8+sgn 8), provided
that one of a, f3 is primary.
Proof. The proof is obvious.
.
Since sgn a{38 - sgn a8 - sgn {38 + sgn 8 = sgn 8(sgn a - 1)(sgn {3 - 1)
= (sgn a - 1)(sgn f3 - 1)(mod 4), we may strike sgn 8 in Corollary
1.3.3:
7Tl
( ; )( : ) = e """4 Sp {(sgn a-l)(sgn fi-l)).
(1.3.5)
Because
n
Sp{(sgn a - 1)(sgn {3 - 1)} = E (sgn a(p) -1)(sgn {3(p) -1),
p=l
we get, at long last,
( a ) ( f3 ) n sgn a (p) - 1 sgn (3 (p) - 1
{3 a = ( -1 )Lp-l 2 2,
(still) provided that one of af3 is primary, and this is indeed the
relative quadratic reciprocity law. .
(1.3.6)

14 HECKE'S PROOF OF QUADRATIC RECIPROCITY
It remains for us to observe that several of the hypotheses we
stipulated earlier can be omitted: (1.3.6) remains valid even if k is not
totally real and even if k's different is not principal. Note that in
general, if k has '1 (resp. '2) real (resp. complex) imbeddings, so that
(k: Q) ='1 + 2'2' and if we stipulate that for any wE k, w(l),..., w(rl)
are the real conjugates, then (1.3.6) becomes
( a ) ( f3 ) "1 sgn a (p) - 1 . sgn f3 (p) - 1
- - =(_1)L p .l 2 2.
{3 a
(1.3.7)
In the Vo,lesungen Hecke does full justice to this most general case
but this is not necessary for our purposes. Suffice it to say that with
(1.3.6) and (1.3.7) we have reached Hecke's formulation of relative
quadratic reciprocity and that, in light of our earlier discussion (in the
Introduction) this will be equivalent to Weil's formulation of Hilbert
reciprocity.
1.4 ENDNOTES TO CHAPTER 1
The material in this chapter adds up to the following points. First,
working over an arbitrary algebraic number field, k, so that we get
relative quadratic reciprocity in the form (1.3.7), the analytic proof of
quadratic reciprocity consists mainly in establishing the Hecke it-func-
tional equation (1.1.4). There are two ways of viewing this functional
equation, either as manifesting that Hecke it-functions are automor-
phic functions of half-integral weight or as a consequence of a suitable
form of Poisson summation. Both of these features prove important in
what follows, with Poisson summation being critical.
Next, once (1.1.4) is in place, the transition to Gauss sums and
finally to the right kind of identity between generalized Legendre
symbols is, so to speak, just the work of gravity.
Finally, as far as the dream of generalizing this strategy to higher
degrees is concerned, it is part and parcel of [Be93] that a straightfor-
ward approach is doomed to failure because of the forbidding appear-
ances of functional equations obtained for the generalizations of
Hecke's it-functions. It is this obstacle that suggests that we look
toward a different context in which to situate the analytic proof of
quadratic reciprocity, and we are led to Weil's 1964 classic [We64]
which is the focus of Chapter 4.
Weil's strategy is to recast it-functional equations as instances of
invariances of half-integral weight automorphic functions under the

ENDNOTES TO CHAPTER 1 15
action of the underlying transformation group whicn he connects to
the symplectic group. This shifts the focus to projective representa-
tions of this group, given that it-multipliers can be interpreted in
terms of low-dimensional cohomology. Specifically, it turns out that at
the local level (i.e., working over k p with  a place of k) the effect of
it-multipliers is captured by the so-called Weil indices (cf. 4.4), which
owe their existence to the action of the Weil representation evaluated
at the involution ( -  ); here we have tacitly passed from the
symplectic group to SL(2), which is harmless (see 4.4). The Weil
representation is, in fact, the only irreducible projective representa-
tion of the required type and it is obtained in the context of the
Stone-Von Neumann Theorem.
Because the Stone-Von Neumann Theorem has not enjoyed a lot of
recent air-play in proportion to its importance, we devote Chapter 3
to it, in preparation for the chapter on Weil's 1964 paper. In part III
of his Tata Lectures on Theta [Mu91] Mumford notes that "the only
treatment of the Stone-Von Neumann Theorem I have been able to
understand" is that given by Varadarajan in [Va68]. Our position
draws heavily on these sources. Beyond this we refer the reader to
Mackey's papers of the 1950s.
Next, before we get to the Stone-Yon Neumann Theorem, that is,
to unitary representation theory, in Chapter 3, we give a brief discus-
sion of the equivalence between quadratic reciprocity in the form
(1.3.7) and 2-Hilbert reciprocity.

Two Equivalent Farms
of Quadratic Reciprocity
In Chapter 4 our focus will shift entirely to the famous product
formula for the 2-Hilbert symbol, first proposed in Hilbert's Zahlbericht
[Hi32] for the case k = Q, the so-called absolute case. We will concern
ourselves with the general case where an arbitrary number-field k
takes the place of Q; to wit, if  ranges over the places of k, finite as
well as infinite, then, for all a, b E k X , ,
IJ( ab )=1.
(2.0.1)
As always, we characterize ( a  b ) by stipulating that it equals 1 or
-1 according as b is or is not a norm in k (fa); this definition is
symmetric in a and b (cf. [Has80]).
On the other hand, in Chapter 1 we sketched Hecke's 1923 proof of
relative quadratic reciprocity (over k), which is to say the reciprocity
law for the extended Legendre symbol, in the form of Gauss and
Euler:
( a ) ( b ) 1"1 sgn a(p)-l sgn b(p)-l
b a = ( - 1 )Ep-l 2 2,
(2.0.2)
where k has r 1 real imbeddings (and r 2 complex ones: (k: Q) =
r 1 + 2r 2) and, for any x E k, x(l), X(2), . . . , x(rl) is a full list of real
conjugates of x (cf. (1.3.7)).
16

TWO EQUIVALENT FORMS OF QUADRA TIC RECIPROC/7Y 17
As indicated in the Introduction, in his" Acta" paper W eil observes
explicitly that his derivation of what amounts to 2-Hilbert reciprocity
is, at a rather deep level, quite the same thing as Hecke's proof of
relative quadratic reciprocity. Specifically, Weil says:
. . . il fait servir les theoremes 2 et 5 du Chapitre I [our Corollary 4.5.5 and
Proposition 4.5.5] a une demonstration de la loi de reeiproeite quadra-
tique, apparentee a eelle qui figure au dernier ehapitre du livre elassique
de Heeke. . .
Thus there should be no distinction between (2.0.1) and (1.3.7).
Certainly, from the bird's eye view of, say, Artin's reciprocity law
[Ar65], one expects a priori that even such disparate-looking identities
should be equivalent; but this is a far more austere approach than one
needs. Indeed, let us begin by indicating how one gets one's hands on
this equivalence between 2-Hilbert reciprocity and quadratic reciproc-
ity for the absolute case, k = Q, by brute force. Subsequently, the
ge,neral case, that (2.0.1) implies (1.3.7), is handled in exactly the same
way. We leave this as an exercise for the enthusiast.
We follow Tate's derivation [Ta76] of the absolute quadratic re-
ciprocity law,
p-l q-l
(  )( ; ) =(_l)Z-z
(2.0.3)
for p, q odd primes in Z, from Hilbert's product formula,
IJ ( ab ) = 1,
(2.0.4)
where a, b are non-zero integers and p runs over all places of Q
(including 00). It is clear that the 2-Hilbert symbol ( ab ) can be
defined to be 1 or -1 according as the norm from ax 2 + by2 = 1 is or
is not solvable over the field of p-adic numbers, Qp. The following
properties of the 2-Hilbert symbol are very well known and, in any
case, are rather easy to prove. First, ( a  b ) is symmetric and bimulti-

18 TWO EQUIVALENT FORMS OF QUADRATIC RECIPROCITY
plicative in a and b; second, if p -4= 2, 00 and a E Z, P X a, then
( a  b ) = ( ; r/ b )
(2.0.5)
where, as usual, vp(b) is the exponent of p in b's prime factorization.
Now we have the following proposition:
Proposition 2.0.1. (2.0.4) implies (2.0.3).
Proof. Take a = p, b = q in (2.0.4) with p, q distinct odd primes. Note
that if I is another odd (finite) prime, that is, I =1= 2,00, p, q, then
( P q ) = 1 by (2.0.5), since v/(q) = O. So, (2.0.4) simplifies to
( P; q )( P q )( P q )( P: q ) = 1.
(2.0.6)
Next, since p, q > 0, the quadratic form px 2 + qy2 = 1 is solvable in
1 1 . . ( p, q )
R: x = _, y = . , for Instance. AccordIngly, - = 1. So,
v2p v 2 q 00
since each factor in (2.0.6) is + 1 it can be rewritten as
( P q )( P q ) = ( P; q ).
It therefore suffices for us to show that the left-hand side is ( p ) ( q ) ,
p-l q-l q P
--
while the right-hand side is (-1) 2 2
As regards the former matter, by the symmetry of the 2-Hilbert
( P,q )( P,q ) ( q,P )( P,q )
symbol we get that P q = P q; by (2.0.5) we then
obtain that
( qP )( pq ) = ( ; r/ p )( : rqCq) = ( : )( ; ),
as required. Finally, as for the latter identity, recall from elementary
number theory (e.g., [Ser73]) that if x = 1 (mod 8) then x is a

TWO EQUIVALENT FORMS OF QUADRA TIC RECIPROCITY 19
square in Q2' whence we get immediately that ( P; q ) =
( p(mod 8), q(mod 8) ) ( p, q )
. 2 · Now we obtain that 2 agrees with
p-l q-l
--
( - 1) 2 2 simply by substituting a finite number of values for p
and q. This completes the proof. .
It is easy to see how this proof generalizes to arbitrary number
fields k, as we already suggested.
We have made no mention of the Ergiinzungssiitze of the quadratic
reciprocity law:
( -1 ) 
P =(-1) 2 (2.0.7)
and
p2-1
( ; )=(_1)S-. (2.0.8)
Suffice it to say that their derivations, which can be structured along
the same lines as that of Proposition 2.0.1, constitute good exercises
for the reader. Hecke also proved a corresponding pair of such
Ergiinzungssiitze attached to (1.3.7), for which we simply refer to
[He27]. They are quite a bit more cumbersome. In view of our
dominant emphasis on 2-Hilbert reciprocity (over k), and since these
Ergiinzungssiitze of Hecke are certainly derivable from 2-Hilbert re-
ciprocity, there is no need to explore them any further.
With this admittedly paltry sketch in place and modulo the sug-
gested exercises for the enthusiastic reader, we will from now on make
no distinction between these two forms of quadratic reciprocity. It is
the case, of course, that since we will presently embark on local versus
adelic considerations, we will work almost exclusively with (2.0.1).

The Stone-Von Neumann
Theorem
It is possible to read Weil's "Acta" paper with no foreknowledge of
the theorem of Stone and Von Neumann (and Mackey), but it is
hardly advisable. Although Weil makes no reference to the fact, this
powerful theorem's influence pervades the paper. The very existence I
of the Weil representation is indeed intimately tied to the Stone-Von
Neumann Theorem. Perhaps it is not improper to say that this
absence of mention of the Stone-Yon Neumann Theorem typifies the
notorious austerity of [We64].
In any event, it is clearly judicious to discuss the Stone- Von
Neumann Th.eorem in some detail now, prior to embarking on a
dissection of [We64] in Chapter 4. We follow the eminently readable
presentation of the theorem and its proof given by Mumford in
[Mu91], who, in turn, refers to Varadarajan [Va68]. With Mumford we
first carry out the proof of the theorem in the case of a finite
underlying group. However, while Mumford then goes on to general-
ize the proof to the case of a locally compact abelian group, we merely
touch on what is needed to do this (in  3.2); the reader is referred to
Mumford for the details (or, if particularly enthusiastic, the reader
can try to fill them in). Of course, Varadarajan's treatment is elegant
and complete, and Mackey's original papers of the 1950s are still
worth reading.
Our ostensibly cavalier tactic of bothering only with proofs for the
finite case can perhaps be excused on the grounds that so much of
20

THE FINITE CASE: A PARADIGM 21
finite group representation theory goes through almost unchanged for
compact or even locally compact groups, modulo the right measure
theoretic manoeuvres. But the measure theory that enters in is very
important, of course, and, beyond this, it occurs elsewhere in investi-
gations which have bearing on the general questions motivating this
book. We refer to Mackey's original papers for this material.
3.1 THE FINITE CASE: A PARADIGM
Let G be a finite abelian group, written additively, and say that we
have a central extension
1  C  c:H  G  0,
meaning that there is a 2-cocycle (class) a E HZ(G, C) so that c:H
can be identified with G X C equipped with the group law:
(gl' l)(gZ' z) = (gl + gz, a(gl' gZ)l z).
Naturally, that a is a 2-cocycle is nothing else than the associative law
imposed on c:H. Here 7T is just the projection map (g, g)g.
Generally, if g E G, write g for an element of c:H which maps to g.
Moreover, if gl,gzEG, define e(gl,g2)=glg2g1 1 g2 1 . It is an easy
exercise to check that e is independent of the choice of gl' g2' given
gl, gz. This gives that e is well defined as a mapping from G X G to
C (since the first coordinate of glgZg1 1 g2 1 is 0). Evidently e is
linear in both coordinates, anti-symmetric, and
a(gl,g2)
€(gl' g2) = ( ) ·
a gZ,gl
In other words e gives a skew-multiplicative pairing into C .
Now, with G being G's character group (which will be replaced by
G's Pontryagin dual if G is a locally compact abelian group), set
A
cp: G  G
gt-+[g  C]
g't-+e(g, g').

22 THE STONE-VON NEUMANN THEOREM
By definition, df is a Heisenberg group if and only if 'P is an
isomorphism.
With df = G X C, a Heisenberg group, a subgroup M < G is
maximal isotropic if € restricts to 1 on M X M, with M maximal for
t his pr operty. Suppose too that 'P gives rise to an isomorphism, M =
(G /M), and that M = M 1-:= {g E G with €(g, h) = 1, Vh EM}.
Proposition 3.1.1. € restricts to 1 on M X M if and only if there exists
a homomorphism (Le., a section) r: M  df such that
7T
1 C  df GO,
r:.. I
r .. 
M
that is, for all hEM, 7T(r(h)) = h. (In other words, M is isotropic if
and only 7T (or df) splits on M.)
Proof. Suppose E is 1 on M X M. Infer that 7T- 1 (G) is an Abelian
subgroup of df. Dualizing we get the exact sequence
"
o M  7T-l(M)  Z  0,
and 1 E Z can be associated to a pre-image  E 7T- 1 (M), that is, a
character of 7T- 1 (M) which (by exactness of the original sequence)
fixes C element-wise. So, 7T restricts to an isomorphism between
ker( ) and M. The inverse isomorphism gives r.
Conversely, the existence of r defines in r(M) an Abelian subgroup
of df so that a is symmetric in its coordinates when restricted to M;
in other words E equals 1 on M X M. .
In light of the preceding, we can write
r: M  df
h(h, f(h))
with f: M  C. Of c9urse, f is an emphatically co homological
object:
Proposition 3.1.2.
[(hI +h 2 )
f(h1)f(h z ) = a(h1,h z ), Vh1,h z EM.

THE FINITE CASE: A PARADIGM 23
Proof. Because r is a homomorphism,
r(hi +h 2 ) = (hI +h 2 ,f(h l +h 2 ))
= r(hl)r(h2) = (hI' r(h l ))(h 2 , r(h2))
= (hI + h 2 , r(hl)r(h2). a(hl' h 2 )). .
With the notion of a maximal isotropic subgroup fixed, we now get
Proposition 3.1.3 (Stone-V on Neumann-Mackey). If df is a Heisen-
berg group as above, with M maximal isotropic in G, then there is a
unique irreducible unitary representation
Q: dfGL(SM)
with Q( g) = g.id on C; here SM may be taken as SM = {<I>: G  C
(Borel measurable) so that Vh EM, <I>(f + h) = f(h)-Ia(h, g)-I<l>(g)
and 1 1<1>1 2 < oo}.
I G/M
Furthermore, Q(go, )<I>(g) = a(g,go).<I>(g+go)' and going from
one maximal isotropic M to another amounts to conjugating in
GL(SM) (we say Q is "essentially unique").
Proof. First, suppose Q is just a unitary representation of df acting in
a representation space V. With r(M) an Abelian subgroup of f),
decompose V as
v = $, EM VC - $:=0 L2( M n ; On),
where AVC = {x E  with Q(x(h))(x) = (h).x for all hEM}, as always,
and M n = {E M so that dim  = n}, fl n any reasonable n-dimen-
sional Hilbert space, and L 2 (M n ; fl n ) just the space of fln-valued
A
functions <l>n on M n . The indicated isomorphism consists in associat-
ing Q(r(h)) to multiplication by (h) for all hEM. (This is a standard
ploy; see Serre [Ser77].) Next, if we just identify 7T- I (M) with M X Cr
in df = G X Cr (viar), and if we set 7T(g) = g (where g E df) and we
identify hEM X C with (hI' g) (i.e., 7T(h) = h, etc.), then
ghg - I = (h, E ( g , h ) ) = (h, cp ( g ) ( h ) ·  ) .
Consequently, if x E V,
{ghg-l)(X) = qJ(g)(h){J(h)(x),

24 THE STONE-VON NEUMANN THEOREM
- A _ A
and if Q(h) acts on x by 'EM then Q(ghg- 1 ) acts by Q(Y) + 'EM:
Q(g)( h) · Q( h)( x) = 'P(g)( h) ·  (h) · x = ( 'P(g) + )( h)( x),
(as we are working in the dual group). Infer immediately that dim  =
dim Vcp(g) + , for all g E G, whence, because 'P is an epjmorphism, it
follows that dim  is altogether independnt of 'E M. So there is
some fixed dimension n so that V = L 2 (H; an) and then, \/<1> E V,
A
Vh EM, EM,
Q(h,!(h»<I>(C) = '(h).<I>().
Now define for g E G,
2 ( A ) A
W:L H;On --+SMOn
A
by W<I>(g) = Q(g, 1)<I>(id), with  indicating the topological tensor
product. Then, for all hEM, a tedious but straightforward calculation
gives that
W<I>(g + h) = a(h, g)-lf(h)-lW<I>(g)
(for the details, see [Mu91], p. 7). An equally tedious but straightfor-
ward calculation provides that W is a unitary map. Finally, we note
that, evidently, WQ(g', 1)<I>(g) = a(g, g ')W<I>(g + g ').
With these ingredients in the mix it remains only to note that I W
is surjective and, in fact, defines a unitary isomorphism between
A A
V = L 2 (H; On) and S M  an. Adding in the requirement that Q be
irreducible forces n = 1 and realizes V as an isomorphic copy of S M
itself. The proof is complete. .
3.2 THE LOCALLY COMPACT ABELIAN CASE:
SOME REMARKS
Again, the details that pertain to this short section can be found in
[Mu91].
It turns out that the first part of the proof, concerning writing the
A
representation space V as EB:=l L 2 (H n ; an)' involves the theory of
spectral multiplicity and the fact that  is independent of 'E M fits
in rather easily. Of course, we need to exploit translation invariance,
and Haar measure enters on the scene, which is no problem since G is
locally compact.

THE FORM OF THE STONE-VON NEUMANN THEOREM USED IN  4.1 25
The most difficult aspect of the generalization has. to do with the
last part of the proof. With Mumford, we refer to Mackey and leave
the detective work to the reader.
It is also the case that Weil's development of the unitary represen-
tation theory of the symplectic group possesses many features that
correspond to several of the constructions and arguments set forth in
the preceding. Weil's circumvention of the Stone-Von Neumann
Theorem actually reveals some subtle facets of the way this theorem is
handled in the locally compact case. We discuss some aspects of the
way the Stone-Von Neumann Theorem can be brought to bear on
[We64] in  4.9.
3.3 THE FORM OF THE STONE-YON NEUMANN THEOREM
USEDIN4.1
Having treated the Stone-Von Neumann Theorem in detail in the
finite case, we now have sufficient motivation to state the precise form
of the theorem that we will employ in  4.1. There the Heisenberg
group that underlies everything is (k p EB k) X C, where k is our
number field localized at p, and the 2-cocycle used to twist the
attendant group law covers from a skew-symmetric bilinear form. In
this connection, see (4.1.1)-(4.1.5).
Proposition 3.3.1. There is an essentially unique, smooth and irre-
ducible representation of the given Heisenberg group with a(ny)
preassigned central character.

Weil's "Acta" Paper
We pick up the thread in the 1960s: by his own description, Weil's
derivation, in [We64], of 2-Hilbert reciprocity (i.e., the product for-
mula for the 2-Hilbert symbol) is fundamentally quite the same as
( a, b )
Heeke's. We reeall from Chapter 2 that the 2-Hilbert symbol 1)'
for p a prime of an algebraic number field k, is 1 or -1, according as
ax 2 + by2 is or is not a -adic square (i.e., a square in k), and then
2-Hilbert reciprocity asserts that
1] ( a/ ) = 1.
Weil's characterization certainly .stands to reason if we just observe
that the most fundamental objects in Hecke's approach are the
quadratic forms which arise in the exponents of summands of it-series.
Beyond this, Mackey, in his review [Ma64] of [We64], observes that
" . . . the author proposes to throw light on the role played by the
symplectic group in the celebrated work of C. L. Siegel on quadratic
forms." The review then goes on to identify the primary features of
Weil's approach to this subject, which in fact constitutes a profoundly
important departure from the classical Fourier-analytic approach
identified with Hecke: "... To a large extent the result... may be
formulated and proved as theorems about a certain group of unitary
26

WElL'S "ACTA" PAPER 27
operators invariantly associated to a locally compact commutative
group."
So we discern a critical shift in focus: while Hecke exploited
classical Fourier analytic methods in order to get to generalized
if-functional equations, Weil's perspective is clearly that of represen-
tation theory. And, as regards the symplectic group's role in all this, as
alluded to by Mackey, suffice it to say by way of an early introduction
that Weil's group of unitary operators can be realized as a semi-direct
product of the symplectic group and the complex unit circle: truly the
symplectic group occupies center stage.
Two more remarks are in order at this point. First it should be
noted that the symplectic group is, of course, defined by the property.
that its canonical action leaves a skew-symmetric bilinear form invari-
ant (cf. [Ig72]). This suggests a priori that there should be a close
connection with quadratic forms. Second, in most of what follows-
mainly because our focus is the quadratic reciprocity law in and of
itself-it eventually suffices to work in the lowest non-trivial dimen-
sion, in which case the according symplectic group coincides with the
special linear group over the attendant base field. (This is a straight-
forward exercise.) However, insofar as we seek to convey aspects of
Weil's original ground-breaking work, we often assume a correspond-
ingly general perspective in the earlier part of our discussion.
With the special linear group now in the game we are perched to
shift perspectives once more, going over to the sparser algebraic
setting of Kubota [Ku67]. Anticipating Chapter 5 a little, we note that
the principal aspect of Weil's work in this connection, the theory of
the Weil representation or, essentially equivalently, the behavior of
the double cover of the symplectic group, translates to some explicit
low-dimensional cohomology courtesy of Kubota (abetted by a pretty
result of C. C. Moore [M068]). Quadratic reciprocity then emerges as
a direct algebraic consequence of the phenomenon of adelic splitting
of the aforementioned double cover-or central extension -over the
subgroup of rational points. As we shall see this amounts to having the
so-called Weil @-functional be invariant under the natural action of
the rational points. Indeed, quadratic reciprocity comes out of an
invariance property of a (8- )functional which, as we shall show later,
ultimately devolves to little more than Poisson summation.
We now proceed with an exposition of the theory of the Weil
representation and the double cover of the symplectic group from
basically Weil's own perspective, although we do deviate from [We64]
in various places. In addition to Mackey's review [Ma64], some good
supplementary references for this material are [lg72] and certainly

28 WElL'S "ACTA" PAPER
[Ra93]. Gelbart's Springer Lecture Notes [Ge76] are also very helpful,
despite an ostensibly different objective.
Beyond the references given in the Introduction we suggest as
prerequisites for this chapter a smattering of elementary representa-
tion theory, say, as in Curtis-Reiner [CR81]; low-dimensional coho-
mology ([CR81] is already more than enough); and the standard
material on local fields (Serre [Ser79] or Weil [We74]) and adelization
(ibid). Otherwise the material presented is rather self-contained and,
where it is not, many other references are given below for the reader's
convenIence.
4.1 HEISENBERG GROUPS
On the one hand, Andre Weil's great Acta Mathematica paper of 1964
[We64], in which the Heisenberg group plays a role of fundamental
importance, had its genesis, at least in part, in a paper by I. E. Segal
[Se63] having to do with quantum mechanics. On the other hand, in
his marvelous exposition [H080], Roger Howe credits Hermann Weyl
as "one of the pioneers in introducing the Heisenberg group into
Quantum Mechanics" and observes that "many physicists still call the
Heisenberg group the 'Weyl group.'" It is very striking that this object,
the Heisenberg group, occupies a crucial role not only in physics but
also in algebraic number theory, and this would, of course, be particu-
larly consonant with Weyl's sweeping philosophy of mathematics.
Weil, however, concerns himself not at all with physics, and neither
do we.
For OUf purposes down the line it behooves us to work with a
particular Heisenberg group, namely, that attached merely to a local
field. So let k be an algebraic number field and let  denote a place
(or prime) of k. With k),J the according local field, let k be the
Pontryagin dual of k'p. If
[ , ] : (k),J X k) X (k),J X k)  k),J
(4.1.1)
is a skew-symmetric bilinear form (into the base field) then, by
definition, the object
Heis (k),J EB k) = (k),J EB k;) X k),J' (4.1.2)
equipped with the group law
(WI' ZI)(W 2 , Z2) = (WI + W 2 , ZI +Z2 + [Wl' w 2 ]), (4.1.3)

HEISENBERG GROUPS 29
for WI' W 2 E kp EB k, ZI' Zz E kp, is the linearized Heisenberg group
attached to kp. Let Xp be any fixed character on the additive group of
kp (so that X p : kp  CT), and denote Xp 0 [,] by < , >. Now
<,>:(kpG)k;)X(kpG)k;)-kpC (4.1.4)
effects a bilinear form from kp EB k into the circle group (ct), and
we characterize
Heis(k p EBk) = (kp EBk) X ct
( 4.1.5)
with (twisted) group law
(WI' t 1 )(W 2 , t 2 ) = (WI + W 2 ' t 1 t 2 < WI' W 2 »
(4.1.6)
as "the" Heisenberg group of kp. As will become immediately clear
upon reflection, the ensuing theory of the double cover of the sym-
plectic (or, rather, special linear) group can be developed in the same
way, starting either from (4.1.2) or from (4.1.5), so we can abuse
notation with impunity.
For the Heisenberg group as given by (4.1.5) we have the following
version of the Stone-Yon Neumann Theorem (discussed at length in
the previous chapter).
Proposition 4.1.1. Up to unitary equivalence there exists a unique
smooth (cf. [JL70]) irreducible unitary representation of Heis (k p EB k)
in the space of Schwartz-Bruhat functions on kp, which has Xp as its
central character.
Explicitly, then, we obtain an "essentially unique" (which is to say,
unique up to unitary equivalence or conjugations in the unitary group)
irreducible representation
r xp : Heis(k p G)k) -GL( .d(k p )),
( 4.1.7)
where (kp) is the aforementioned Schwartz-Bruhat space, for which
'x (0, t) = Xp(t) (since the center of Heis(k p EB k) is k p ). Now it is
esy to see that SL(2, k p) acts on Heis (k p G) k) by
SL(2, kp): Heis (kp EB k)
u: (w,t)(WU,t)
(4.1.8)

30 WElL'S "ACTA" PAPER
and, as an immediate consequence, each a E SL(2, k p ) acts on r xp by
(j
rxp(w, t)rxp(wU, t) =: r (w, t).
(4.1.9)
Proposition 4.1.2. r x and r x u are unitarily equivalent (for any a E
p p
SL(2, k p )).
Proof. Just note that these two representations have the same central
character, namely, X p , and invoke Proposition 4.1.1. .
Thus, for any aE SL(2, k p ) there exits r*(a) E GL( c£(k p )) such that
r;: ((T, t) = r*( (T )rxp(w, t)r*( (T) -1.
( 4.1.10)
Proposition 4.1.3. There is a unique homomorphism
r*: SL(2,k p ) GL( -S(kp))/C
(4.1.11)
characterized by (4.1.10).
Proof. Use Schur's Lemma.
.
Corollary 4.1.4. r* gives rise to a multiplier representation, r, in
c£(k p ), with multiplier (i.e., 2-cocycle) a E H 2 (SL(2, k p ), Cr) charac-
terized by
r(a 1 )r(a 2 ) = a(a 1 a 2 )r(0"10"2).
(4.1.12)
Proof. This is a general result from elementary representation theory
and low-dimensional cohomology (cf. [CR81]). Indeed, it all amounts
to little more than a calculation, starting with the definition of r as
f 0 r* for f any section, GL( -S(kp))/C L GL( -S(k p )), mapping the
scalars to 1 (in fact, we may simply take f: r*«(T)Crr*«(T)). .
We take the liberty to refer to r also as the projective representa-
tion arising from r* as per (4.1.10). Indeed, (4.1.12) serves as the
defining property for r as a projective representation in (k),J).
Furthermore, there is a somewhat subtle distinction between the
multiplier, a, of r, and a as realized in H 2 (SL(2, k p )' C): in the
latter sense a is unique only up to coboundaries. This turns out to be
rather immaterial, however, as far as our covering groups go, because
of a result of C. C. Moore [M068] to be discussed later.

HEISENBERG GROUPS 31
Corollary 4.1.5. In turn, r gives rise (i.e., lifts) to an ordinary repre-
sentation
r: (o-,A)Ar(o-)=A(for*)(o-)
( 4.1.13)
of the group SL(2, k p ) x C (see below), which is just the set of
a
ordered pairs (0-, A) E SL(2, k p ) X Cr subject to the combination rule:
(0- 1 , A 1 )( o-z, Az) = (0- 1 o-z, Al Az a( 0- 1 , o-z)).
Proof. Immediate. .
Parenthetically, the notation" x " will be used a great deal in these
a
notes: in general we posit that if E E HZ( G, T), which is to say that we
have a group extension,
1TGxTG1
,
where G X T has the implied ("twisted") group law (gl' t 1 )(gz, t z ) =
(glgZ' t 1 t Z E(gl' gz)), then we uniformly denote this group by G x T.
Accordingly, (4.1.13) defines a representation of SL(2, k p ) x Cr. €
. a
With SL(2, k p )  C we have reached one of the most important
players in the game, the prototypical metaplectic group of Weil, often
denoted by Mp(k p ). Nowadays (following Kubota [Ku69]), the adjec-
tive "metaplectic" is used to describe any of a family of group
extensions of SL(2) or GL(2) (or Sp, the symplectic group), but
Corollary 4.1.5 describes the archetype. We say more about other
metaplectic groups later, in connection with Kubota's work.
We close this section with two remarks. First, note that Mp(k p )
constitutes a covering of SL(2, k p ) by the full circle, so one is apt to
ask, right off, how it is that double covers of SL(2, k p ) enter the scene.
It turns out, however (cf. [We64]) that a, descendent as it is from the
skew-symmetric bilinear pairing (4.1.1) (defining Heis (k p €a k)), is an
inalienably quadratic object in the sense that it can be proved that a
defines an element of order 2 in H Z (SL(2, k p ), Cr), and so we
actually get an element taking values in JLz = {1, - 1}, that is, in
H Z (SL(2, k p ), JLz). In  4.10 we give Weil's own proof of this remark-
able fact.
Second, it is a standard tactic (from [JL70]), aimed at introducing
square integrability, to expand the representation space in Proposi-
tions 4.1.1-4.1.5 from S(k p ) to LZ(k p ). Ultimately this is nothing
more than the'recognition that the former space is dense in the latter.

32 WElL'S "ACTA" PAPER
Our next objective is to contrast the preceding construction of
Mp(k'p) with Weil's original construction in order to prove a set of
critical results for r x ' r*, as well as for rand r, each of which has
been referred to at sme time or other as "the Weil representation."
Of course, in truth these are projective representations.
4.2 A HEISENBERG GROUP AND A GROUP OF UNITARY
OPERATORS
For a while we work with a far more general object than kp, namely,
an arbitrary locally compact abelian (LCA) group, G, written addi-
tively. G's Pontryagin dual is denoted by G* and we write ( ,) for the
canonical dual pairing from G X G* to C. We also suppose we are
given a Haar measure on G, suitably normalized, typically denoted by
dg (for g E G), dx, or du-whatever is appropriate. We write LZ(G)
for the space of square-integrable (complex valued) functions on
G and d(G) for the dense subspace of Schwartz-Bruhat functions
on G.
If w = (u, u*) E G X G*, let
ru (w): LZ( G)  LZ( G)
<p(x)<p(x + u)(x, u*).
(4.2.1) \
Define dfE-l(G) to be the set of operators tCZ1.(w), wE G X G*,
t E C, subject to the group law
tICU(WI)tZCU(W Z ) = CU(w I + WZ)tItZ(UI'U)
(4.2.2)
for WI = (U I , ui), W z = (u z , u). Obviously df£l(G) is a group of uni-
tary operators because each CU (w) is, in fact, unitary (an easy exer-
cise) .
Now, if we take kp for G and match <,), with (4.1.4) then
the mapping tCU(w)(w, t) effects an isomorphism, df£l(k'p) --
Heis(k p EB k;). This is, of course, very readily generalized to the case
of an arbitrary LCA group G, in exactly the same way, and accord-
ingly we define
Heis(G) = G X G* X C,
equipped with the group law
(w l , t l)( W z , t z ) = (w l + W z , t 1 t z < U l ' ui > ),
( 4.2.3)
(4.2.4)

A HEISENBERG GROUP AND A GROUP OF UNITARY OPERA TORS 33
where w l = (u l , ui), w 2 = (u 2 , ui). The content of the preceding re-
mark is that we have a canonical identification, dfE.i( G) ::: Heis (G).
Finally, using the notation of 9 4.1, we can present Heis(G) as
(G X G*) x Cr (or, for that matter, as (G E9 G*) x Cr), simply by
w w
virtue of the fact that the mapping «ul,ui),(U2,ui))<Ul,ui) =
ui(u l ) defines an element of H 2 (G X G*,C).
Our present objective is to construct the covering group
SL(2, k p ) x C, i.e., Mp(k p ), by approximately the same route that
a
Weil took in [We64]. It will turn out in due course that this metaplec-
tic group can be realized as a subgroup (or even a direct factor) of the
normalizer of dfE.i( G) in the supergroup consisting of all unitary
operators in L 2 (G). We now develop this approach, in contrast with
the tactics in 9 4.1, in order to underscore the fact that the theory of
Weil's metaplectic group or, more accurately, of the Weil representa-
tion, is fundamentally a theme belonging to abstract Fourier analysis.
Throughout what follows in this chapter, the reader should bear in
mind that even through Weil's formalism in [We64] is self-contained
and Weil makes no explicit mention of the Stone-Yon Neumann
Theorem, this theorem is certainly implicit there, as mentioned before
and as we shall explore in 9 4.9. It will turn out that Corollary 4.1.4,
that is, the relation (4.1.12), is essentially equivalent to Proposition
4.5.4; and for both of these the all-important consequence is Proposi-
tion 5.3.3, "adelic splitting."
To wit: Weil's formalism:
N(dfd(G)):= {s, unitary on LZ(G), with Sdfd(G)S-l =dft:l(G)},
( 4.2.5)
the normalizer of the Heisenberg group in the unitary group.
Proposition 4.2.1. Each s E N(dfE.i( G)) induces an automorphism,
(u,g), of dfE.l(G), as follows: uEAut(GXG*),g: GXG*Cr,
and
(U , g): dfE.i( G) ---+ dfE.i( G)
(w, t)(WU, tg(w))
(4.2.6)
with g characterized by
g(W l + w z )
w(wf,w2)=w(w 1 ,w Z ) ( ) ( )'
g w l g W z
(4.2.7)

34 WElL'S "ACTA" PAPER
Proof. The only reasonable way to define a pair (u, g), as indicated,
from s E N(cJl-E.i(G)), is to exploit the condition scJI-E.i(G)S-l =
cJl-E.i(G), that is, to posit that for all (w, t) E G X G* X C we should-
have stf'11(W)S-l = tg(w)f'11(W U ). It now only remains for us to show
that this association, s( u, g), is well defined. But this simply
reduces to (4.2.7) by means of the following computation:
St 1 CU(W 1 )t 2 '1l(W 2 )S-1 = St1CU(Wl)S-lst2CU(W2)S-1 and so, by (4.2.2),
st 1 t 2 W(W 1 , W 2 )'ll(W l +W 2 )S-1 =t 1 t 2 W(W 1 , W 2 )g(W 1 +w 2 )'ll«W 1 +w 2 )U)
=t 19( w 1 )f'11( W f )t 2 g( w 2 )'ll( w 2 ) = t 1 t 2g( WI) g( w 2 )f'11( u 1 u + w) W( WI' w 2 );
simplifying we get w(W 1 , W 2 )g(W l + W 2 ) = g(w1)g(wz)w(wf, w 2 ),
which amounts to (4.2.7). .
We can take this one step further by establishing that the associa-
tion, s( u, g), actually defines a homomorphism:
Proposition 4.2.2. The mapping
7T: N(cJI-£i(G») -Aut(cJI-£i(G») (4.2.8)
s(u,g),
given by st CU (W)s -1 = tg( w) CU (w 0'), with (4.2.6) and (4.2.7) in effect,
is multiplicative.
Proof. Regarding 7T(Sl S2) we have (Sl S2) -1 =tg 1,2( W) CU (w 0'1,2), say (i.e."
7T(SlS2) = U 1 ,2, gl,2))' and so tg 1 ,2(W)CU(W U1 ,2) = S1(S2 tCU (W)S2 1 )Sl =
Sl(tg 2 (W)'1l(w u 2)s1 1 = tg2(W)gl(WO'2)CU(WO'2O'1). But, in GL(cJl-E.ij.(G))
= GL«(G X G*) x X (C(), we have (u 1 , gl)( u 2 , g2) = (u 1 (T2' g3) where
a
g 3( w) = g 2( W) g 1( W 0'2). (This is a standard fact; see [EMacL86], e.g.)
Accordingly, since (u 1 , gl) = 7T(Sl)'(U 2 , g2) = 7T(S2)' we have that
7T(Sl S2) = 7T(Sl)7T(S2)' as desired. .
Writing B(G) for the image of 7T (so that B(G) < Aut(cJl-E.Lj.(G»)
we get that 7T is an epimorphism.
Prom a slightly different point of view, the group B(G) can be
presented (courtesy of its group law) as a semi-direct product, one of
whose factors is Sp( G), the symplectic group for G. Obviously this is
very important for our purposes and we say more about this later. For
now, merely note that the rule
( u l' g 1) ( u 2' g 2) = ( U 1 U 2 , g 3 = g 2 · (g IOU 2) ) ( 4.2.9)
actually defines a semi-direct product just by itself (cf. [Ha76]).
As regards 7T, the next question to ask is, what is its kernel?

THE KERNEL OF 7T 35
4.3 THE KERNEL OF 7T
We will prove that ker 7T ::: C, which will allow us to associate to a
suitably chosen G, or rather, B(G), a local symbol (the Weil index)
which is intimately related to the 2-Hilbert symbo1. Following Weil's
own tactics, the quadratic reciprocity law-for this 2- Hilbert symbol
-will then be obtained in  5.8 from a predictable product formula
for the Weil index.
The derivation that ker 7T is nothing else than the group of scalar
multiples of the identity is a sequence of manoeuvres in harmonic and
functional analysis and amounts to something of a tour de force on
Weil's part. But this development also serves to underscore the extent
to which this aspect of the theory of Weil's metaplectic cover is a
chapter in abstract Fourier analysis. Although we will eventually
proceed to a more algebraic perspective, it is very instructive to go
through this part of Weil's argument with some care. A good sub-
sidiary reference is [Ig72], but it is important to note that Weil's
development is entirely autonomous (as we shall see). So, what follows
is essentially a direct derivation of (4.1.11).
Without question, the pivotal observation in  4.1 was that (4.1.8)
defined an action of SL(2, k-p) (also a symplectic group, of course) on
the Heisenberg group, Heis (k -p E9 k). Subsequently, an application of
what we might call "the Stone-Von Neumann-Schur game" resulted in
the desired projective representation, r*, as per (4.1.11). The main
idea behind Weil's proof that ker 7T ::: C consists in unraveling the
action of B( G) on dfgij,( G) given by
B( G): dfgij,( G)  dfgij,( G)
(u,g): tru(w)tru(WU)g(w).
(4.3.1 )
(We shall argue a little later that (4.3.1) is, in fact, algebraically
equivalent to (4.1.8), in a very straightforward manner.)
Now, just as the Stone-Von Neumann theorem guaranteed in  4.1
that the attendant representation of SL(2, k'p)' lifted to a square-
integrable representation by methods from [JL70], is "essentially
unique," so it turns out that (4.3.1) also gives rise to a square-integra-
ble representation with something of a uniqueness property attached.
Weil uses transfer of Hilbert space structure as opposed to essential
uniqueness per se, but ultimately these ploys amount to quite the
same thing.

36 WElL'S "ACTA" PAPER
We begin with some functional analysis. The extremely energetic
reader can simply read the definitions that follow, skip to Corollary
4.3.9 and devise his own proof. The lazy reader can just skip to
Corollary 4.3.9. For everyone else we have:
Proposition 4.3.1. Let K(x, y) be the (kernel) function
K(x, y) = f cp(y -x, u*)(x, u*) du*
G*
(4.3.2)
for cp(w) = cp(u, u*) E L 2 (G X G*). Then
G
W: L2( G X G*) --+ L2( G X G*)
cp(w)t--+K(x,y)
(4.3.3)
gives a Hilbert space isomorphism.
Proof. Of course, (4.3.2) effects a generalized Fourier transform, and
it is easy to see that the inverse transform obtains via
<p(w) = <p(u, u*) = f K(x, x + u)(x, - u*) dx.
G
In other words, our theorem reduces to nothing else than Plancherel's
Theorem: cp and K have the same L 2 -norms and we have the desired
Hilbert space isomorphism. . '
Now, given tru(w) E c:H-£lj.(G), that is, given ru(w), associate to it
a mapping from L 2 (G X G*) to L 2 (G)*, as follows:
U: L2(G x G*) L2(G)*
'P(w)[U('P): L2(G) C]
<I> ( x )  1 {CU ( w) <I> ( x) } 'P ( w) dw. ( 4.3.4 )
GxG*
And, in tandem with (4.3.1), we define the actions of B(G) on each
U by the rule (for S E B(G)):
s: U(cp) [ U(CP)s: <I> (x) f {CU(W)s<I>(X)}cp(W)dW ] , (4.3.5)
GxG*
which is to say, U(cp)S(<I>(x) = j {ru(wlT(<I>(x))}g(w)cp(w)dw.
GxG*

THE KERNEL OF 7T 37
Proposition 4.3.2. For s=(u,g)EB(G), and 'P(W) ELZ(G xG*),
define
-} -}
'PS(w) = 'P(wO" )g(wO" ).
(4.3.6)
Then U('P S ) = U('P)s.
Proof. An easy exercise using the translation invariance property of
the Haar product. .
Next, given 'PI' 'Pz E LZ(G X G*), define
'PIX'PZ(W)= ! 'PI(wz)'Pz(z)w(w-z,z)dz (4.3.7)
GxG*
for w as before ( 4.2), namely,
w((U I , ui), (u z , ui)) = ui(u l ) = (u l , ui).
(4.3.8)
Proposition 4.3.3. U( 'PI X 'Pz) = U( 'PI)U( 'Pz); consequently, W( 'PI X
'P2)(X, y) = ! W('Pl)(X, Z)W('P2)(Z, y) dz.
G
Proof. The first equality follows by a brute force calculation using
(4.3.4), (4.3.3), and a straightforward change of variables. The second
equality then follows from Fubini's Theorem. .
Soon we will approach ker 7T (which, as a normal subgroup of
dfE,i(G) is certainly a group of unitary operators on L 2 (G)) by means
of a transfer of Hilbert space structure from L 2( G) X L 2( G) to
L2(G X G). Toward this purpose we define, with Weil, the following
binary operations:
X: L2(G X G) XL 2 (G X G) LZ(G X G)
(4.3.9)
(Kl' K2) ! Kl(X, z)Kiz, y) dz,
G
and
@: LZ(G) XLZ(G) L2(G X G)
(P, Q)P(x)Q(y),
(4.3.10)
so that we get an embedding of LZ(G) X LZ(G) in LZ(G X G).

38 WElL'S "ACTA" PAPER
"
A propos, I grant that (4.3.9) is notationally ambiguous with regard
to (4.3.7), but plead by way of an excuse that in this difficult section
we are staying as faithful as possible to Weil's original notation in
[We64].
Proposition 4.3.4. Let K E L 2 (G X G). Then K = P  Q (i.e.,
K E im ) if and only if, for each K' E L 2 (G X G), K and K X K' X
K( E L 2 (G X G)) differ by a scalar factor.
Proof. If K = P @ Q, then
KxK' xK= f f P(x)Q(z)K'(z,y)P(y)Q(g)dzdy
G G
= f {K'(z, y)Q(z)P(y) dzdy}P(x')Q(g)
GxG
= a(K, K')P  Q,
where a(K, K') = f K'(z, y )Q(z)P(y) dzdy, a scalar. Conversely,
GxG
let K'(x, y) be the characteristic function of the x, y-trace of the
compact set in (GxG)2 on which K(x,z)K(y,g) is supported.
1 ,
Define a(K, K') as before and let P(x):= V f K(x, z) dz,
1 a(K, K') G
Q(y):= V f K(y, g) dg. Then K(x, g) = P  Q follows by a
a(K,K') G
trivial calculation. .
In a similar vein (but considerably more labored):
Proposition 4.3.5. With P,P',Q,Q' EL 2 (G), let K=PQ, K' =
P'  Q' then P, P' .(resp. Q, Q') differ by a scalar factor if and only if
for all K" = P"  Q" , with P", Q" E L 2 (G), we have that K X K" and
K' X K" Crespo K" X K and K" X K') differ by a scalar factor.
Proof. Suppose P = EP'. Then
KXK"-=(P0Q)X(P" 0Q")= fP(x)Q(z)P"(z)Q"(y)dz
G
1
=p(x) f Q(z)P"(z)dzQ"(y) = -P'(X)E'Q"(y),
G E

THE KERNEL OF 17 39
for E ' = 1 Q(z)P"(z)dz. Let E" = 1 Q'(z)P"(z)dz. Then
G G
E' E'
K X K" = 7 P' (x )Q" (y) = EE" P' (x) f G Q' (z )P" (z) dzQ" (y)
E
= _ I P'(x)Q'(z)P" (z)Q" (y) dz
EE" G
E' E'
= -(P' @ Q') X (P" @ Q") = -K' XK",
EE" E
as required. The converse is equally straightforward and the second
assertion is (naturally) proved in the same somewhat annoying way. .
Proposition 4.3.6. Suppose s: L Z( G X G)  L Z( G X G) is a Hilbert
space automorphism which commutes with x: (K I X Kz)S = Kf X K
for all K I , Kz E LZ(G X G). Then s gives rise to another Hilbert space
automorphism, t: LZ(G)LZ(G), such that for all P,QELZ(G),
(P @ P)S = p t @ Qt, where t: L 2 (G)  L 2 (G) is defined by Q t = Qt.
Proof. By Proposition 4.3.4, K = P @ Q if and only if, for all K' there
exists a = a(K') with K X K' X K = aK; consequently, aK s = K S X
(K')S X K S . Since (K')S is unrestricted we can apply Proposition 4.3.4
again to get K S = p' @ Q' for some P', Q' E L 2 (G); so, (P @ Q)S =
P'@Q'. Define t,u: LZ(G)LZ(G) by pt=P',Qu=Q' (and the
reader should verify that t and u are well-defined maps). Because all
this goes through mutatis mutandis with S-I in place of s, both t and
u are, in fact, automorphisms of LZ(G). Finally, since (P @ Q) X
(P @ Q) = (P, Q )P @ Q, where (,) is the underlying inner product on
LZ(G) (this amounts to a trivial calculation), {(P @ Q) X (P @ Q)}S =
(P, Q )p t @ QU; on the other hand, (P @ Q)S X (P @ Q)S = (P t @ QU) X
(P t @ QU) = (P t , Q U)pt X QU. Since s commutes with X this implies
that (P, Q ) = (P t , Q U), and the desired result follows from the fact
that t and u are unitary. .
We now come to the main result of this section (whose corollaries
are crucial), namely,
Proposition 4.3.7. With 'PI X 'Pz as in (4.3.7), the mapping 'P'Ps,
s E B(G), obtains a unitary operator on LZ(G X G*) which satisfies
the relation ('PI X 'Pz)S = 'P: X 'P.

40 WElL'S "ACTA" PAPER
Proof. Using (4.3.6) and (4.3.7),
S
( (,01 X (,02) (w)
= ('PI X 'P2)(u,g)( w) = g( W U - 1 ) ( 'PI X 'P2)( W U - 1 )
= g( W U - 1 ) 1 epl( w u - 1 - z) 'P2( z) w( w u - 1 - z, z) dz
GxG*
=g(W U - 1 ) j (jOl(W -z)U- I )(j02(ZU- I ) w(w _z)U- J , Zu- I ) d(zU- I ).
GxG*
On the other hand,
'Pf x 'P (w)
= {g( W U - 1 ) 'PI( W U - 1 )} X {g( W U - 1 ) (,02 ( W U - 1 )}
= 1 g(w -z)U-1)'PI(W _z)U-1)g(ZU- 1 ) w(w -z, z) dz.
GxG*
Since, by (4.2.7), for any WI' W 2 E G X G*, g(W I )g(W 2 ) = g(W I + W 2 )
W(W t , W 2 ) .
( au ) , we obtaIn that
W WI' W 2
( a-I a-I )
-1 -1 -1 W (W-Z) ,Z
g(W -z)U )g(ZU ) =g(W U ) ;
W(W-Z,Z)
Therefore, substituting,
'Pf x 'P ( W)
= j (jO 1 ( ( W - Z ) U -I ) (jO 2 ( Z U -I ) g ( W U -I ) W ( ( W _ Z ) u- I , Z U -I ) dz.
GxG*
Because of the invariance of Haar measure, that is, dz = d(za- 1 ),
this latter expression agrees with the former expression for (I X
2)S( W ). .
Corollary 4.3.8. {W( I) X W( 'P2)}S = W( 'Pl)S X W( 2)S, for all S E
B(G) (where W is as in (4.3.3)).
Proof. Use the preceding theorem together with the fact that W
effects a transfer of structure (in that (4.3.3) defines a Hilbert space
automorphism-see Proposition 4.3.1). .

THE KERNEL OF 7T 41
Corollary 4.3.9. S E B(G) gives rise to an automorphism, t, of L 2 (G),
as follows:
L2(G X G)
1
L2(G) XL 2 (G)
S ) L2(G X G)
1
(ri!)(
) L2(G) X L 2 (G),
(4.3.11)
where t  t: p  Q  pt  Ql as in Proposition 4.3.6.
Proof. By Proposition 4.3.7, S E B(G) yields a unitary operator (and
an automorphism) on L2(G X G*). By the preceding corollary s acts
on L 2 (G X G), so the assertion reduces to the trivial observation that
Proposition 4.3.6 can be applied to the present situation by means of
the transfer of structure afforded by W (and W- 1 ). .
For reasons which will become clear soon, write s -1 for t In
Corollary 4.3.9.
I
Proposition 4.3.10. S-1 E N(cJf£ij,(G)) (cf. (4.2.5) and Proposition
(4.2.1)).
Proof. With arbitrary P, 'Ll(w)Q EL 2 (G), note that (by a straightfor-
ward calculation) (P, f11(w)Q) = W- 1 (P  Q )(w) and so (P, f11(w)Q)S
= W- 1 (P  Q )S because W- 1 and s commute. Since (P  Q )S =
pt  Ql (by Proposition 4.3.6), this yields that (P, 'Ll(w)Q)S =
W-1(pt  Qi) = W-1(S-lp  S-lQ) = (S-lp, CU(W)S-IQ). On the
other hand, by (4.3.6), (P, 'Ll(w)Q)S = g(WO'-I)(p, 'Ll(WO'-I)Q) =
( p, 1 -I '11 C WU - 1 )Q ) . Accordingly, noting also that s is unitary,
g(wO' )
( 1 -1 )
P, -1 CU(wO' )Q = (P,sCU(W)S-IQ). Therefore, because P,Q
g(wO' )
1
are entirely arbitrary, -1 'Ll(WO'-l) = s'Ll(W)S-I, or, equivalently
1 g(wO' )
'Ll (w) = s'Ll (w O')s -1. But this, in turn, implies the relation
g(w)
S-ICU(W)=g(w)CU(wO')=CU(w)S (via (4.3.1), with (u,g)=s, of
course), meaning that the action 'Ll(w)'1l(w)S is nothing else than
conjugation by s( E GL(L 2 (G)), unitary). This proves our claim. .
-4

42 WElL'S "ACTA" PAPER
And now, finally,
Proposition 4.3.11. ker 7T ::: C .
Proof. s E ker 7T implies, with 7T(S) = S E B( G), that ru (w)S = ru (w ),
that is (by the preceding proposition), that s  1 '1l (w)S = '1l (w ).
By means of (4.3.4), (4.3.2), and Fubini's Theorem, '1l (w) associ-
ates to U(cp): <I>(x)jK(x,y)<I>(y)dy, where K(x,y)=
G
j cp(y -x, u*)(x, u*) du* (just write U(cp)(<I>(x)) as j <I>(x + u) X
G* GxG*
< X, u*) P(u, u*) du du* and change variables). So, writing K( = W( cp),
via (4.3.3)) as P@ Q , we obtain U(cp)(<I>(x)) = jP@ Q( x,y)<I>(y)dy
j _ G
= P(x) <I>(y )Q(y) dy = P( <1>, Q). Thus, seeing that the commutation
of s wifh ru ( w) translates to the commutation of t with U( cp):
<I>( <1>, Q)P, it follows that (<I>t, Q)P = (<I>, Q)P t , valid for all <1>,
Q E LZ(G). In other words, for all <1>, Q E LZ(G), we have that pt
= i :t: ; P = €p, € a scalar (because, given P, p t , this relation holds
for all (<I>, Q)). But this just amounts to saying that the effect of
s E ker 7T is simply to multiply each PEL Z( G) by a scalar, E E C,
d'epending only on s (i.e., on t); to wit: s(P) = EP. Finally, since s is,
unitary, (Pt, Pz) = (sP t , sPz) for all Pt, Pz E LZ(G), where (Pt, Pz) =
(EP t , EPZ) = E€(P t , Pz), so that E€ = 1 and E E C. .
Corollary 4.3.12. The following sequence is exact:
1 C N(dfEl(G)) B(G)  1.
(4.3.12)
Equivalently, B( G) ::: N(df£i( G)) /Cr.
Proof. This is just a reformulation of the preceding.
.
The exact sequence (4.3.12), which is to say, the central extension
N(df£i(G)), already captures the metaplectic group of Weil in the
sense that we will presently be able to demonstrate that B(G) con-
tains a copy of the symplectic group whose pre-image (under 7T) in
N(df£i(G)) will turn out to coincide with Mp(G) (Weil's metaplectic
group for G, generalizing Mp(k p ) as per  4.1). More precisely, we

THE KERNEL OF 7T 43
shall see that (4.3.12) can be completed to the commutative diagram
1
) c x
1
) Mp(G)
1
) N(dti( G))
) Sp( G)
1
7T ) B(G)
) 1
(4.3.13)
1
) c x
1
) 1
,
where Sp(G) denotes the symplectic group of G. In the simplest case,
G = kp, the top row of (4.3.13) obviously coincides exactly with
SL(2, k p )  C (cf. (4.1.13)) simply because Sp(k p ) can be identified
with SL(2, k p ).
Thus, once we have described in adequate terms the way Sp( G) sits
inside B(G) (again a cohomological affair), we need have no qualms
about observing that Weil's development of (4.3.12) is mathematically
indistinguishable from the culminating facts in  4.1, namely, Proposi-
tion 4.1.3 and its two corollaries.
But there is a subtle difference which we stress from now on.
Whereas Corollary 4.1.4 introduces U E H 2 (SL(2, k p ), C) in order to
give the explicit group law for Mp(k p ), Weil's approach pushes such
explicit low-dimensional cohomology into the background. To be sure,
(4.3.13) entails the existence of cocycles U G , (3G' respectively, in
H 2 (B(G), C) and H 2 (Sp(G), C), such that N(dtt:ij, (G)) =
B( G) x C and Mp( G) = Sp( G) x Cr, but they are not given explic-
aG G
itly. In due course we will get an explicit local cocycle, courtesy of
Kubota (cf.  5.2), but it behooves us to carry out that development for
yet another subgroup of N(dtt:ij,(G)), very closely related to Mp(G),
namely, the double cover of the symplectic group. For in  4.10 we
shall be able to prove (with Weil) that (3G actually determines an
involution in H 2 (Sp(G), Cr) so that (4.3.13) can, in turn, be extended
to
1
-
) Sp( G)
1
) Sp( G)
) 1
) J.L2
1
1
) C x
1
) Mp(G)
1
) N( dtd( G))
) Sp(G)
1
7T ) B(G)
) 1 (4.3.14)
1
) C x
1
) 1
,

44 WElL'S "ACTA" PAPER
where (generally) J..L n is the group of nth roots of 1. The establishment
of (4.3.14) constitutes one major objective of the remainder of this
chapter. The other major goal is the development of attendant ma-
chinery in order to derive quadratic reciprocity within this formalism.
In the former instance the crux of the matter is the location of Sp(G)
inside B( G), while in the latter instance everything devolves to the
proper identification of a unimodular index (in C :: ker 7T). This is
the so-called Weil index [Sc85], which turns out to be intimately
related to the 2-Hilbert symbol. In both instances, concerning the
symplectic group as well as the Weil index, the common thread
consists in the fact that both are defined relative to a bilinear (or the
associated quadratic) form and, therefore, we now turn to Weil's
notion of second-degree character. The pending process of lineariza-
tion (already hinted at in  4.1, in connection with Heisenberg groups)
will bring out the important connection with quadratic forms very
explici tly.
4.4 SECOND-DEGREE CHARACTERS
f(u l + u z )
A mapping, f: G -- Cr, is a second-degree character if f(ul)!(u Z ) lS
an ordinary character in u l (resp. u z ) for every U z (resp. u I ) in G.
Proposition 4.4.1. The map (J: G  G* defined by the rule
f(u l +u z )
(ul,uf) = !(ul)!(U Z )
(4.4.1 )
is a homomorphism.
Proof. This follows quickly from the fact that
f(u l + U z + u) f(u l + u z ) f(u l + u)
-
f(ul)f(u z + u'z) f(ul)!(u Z ) f(uz)f(u'z)
(as a character in u z ).
.
If {J is, in fact, an isomorphism then ! is said to be non-degenerate.
For the time being we restrict attention to such non-degenerate
second-degree characters. In view of [Ra93], however, it is now known
that this restriction may.be dispensed with; nonetheless, we make this
stipulation, in keeping with [We64].

SECOND-DEGREE CHARACTERS 45
Next, for [ a second-degree character as above, define the following
unitary operator on L 2 (G):
Uj: L2(G) L2(G)
<1>( x )C .[( x )<1>( x Q),
(4.4.2)
"
where <I> is <I>'s Fourier transform and C is chosen in such a way as to
render Uj unitary.
Proposition 4.4.2. Uj E B( G); Uj3 E ker 7T.
Proof. This amounts to a tedious exercise in Fourier analysis, after a
fashion, but due to the centrality of this result we carry this out in full
"
detail. We have that <I> I ) <I>(x + u)( x, u*) =: '¥(x)C[(x)'¥(x Q).
ru(u, u*) w f
" (-u,u*) 1; 1
Also, <I>(x)C[(x)<I>(xQ) =: 8(x) ( -1 ) 8(x)(u*)P- X
W f [ (u*)Q
(x, ( - u)Q - u*), with the last mapping being effected by the operator
(-u,u*) -1
( -1 ) ru«u*)Q , (-u)P - u*). If these two mappings on <I>(x)
[ (u*)P
are the same then eo ipso
(-u,u*) ( -1 Q )
W f ru (u, U*)Uj-l = ( -1 ) ru (u*)P , (-u) - u* ,
[ (u*)P
fr.om which it follows immediately that Uj E B( G). So we must show
that
" (-u,u*) ( Q ) ( -1 )
C[(x)'I1(xQ) = ( -1 ) x, (-u) - u* E x + (u*)Q ;
[ (u*)Q
but this is equivalent to
" ( - u, u*) ( Q ) ( -1 )
C'I1(x Q ) = ( -1 ) x,(-u) -u* E x+(u*)Q ,
.[(x)! (u*)Q
that is,
C{<I>(x + u)(x, u*)} /\ Ix*=x p
(-u,u*)(x,(-u)Q-u*) ( -1 ) " (( -1 ) Q )
= -1 C[ x + (u*)Q <I> x + (u*)Q .
f( x )f( (u*)11 )

46 WElL'S "ACTA" PAPER
In turn, this is equivalent to
{ <p (X + u ) < x , u * > } 1\ Ix. =x lJ
e 
= < - U, U*><X, (-U) - U*><X, U*><P(X e + U*),
( -1 )
I x + (U*)e _
in light of the fact that ( _I) = (X,(U*)I? 'I?) = (x,u*). In
I(x)! (u*)e
order to handle the left-hand side, note that since, by definition,
(x*) = f '¥(g)( g, x*) dg, whence (xl?) = f '¥(g)( g, x I?) dg, we get
G G
that {<I>(x + u)(x, u*)} /\= f <I>(g + u)(g, u*)(g, x*) dg and so
G
{<I>(x+u)(x,u*)}/\Ix*=x p = f <I>(g+u)(g,u*)(g,xl?) dg.
G
Also,
<1>( x*) = f <I>(g)( g, x*) dg,
G
which gives
<I>(xl? + u*) = f <I>(g)(g, xl? + u*) dg.
G
Accordingly it remains only for us to prove that
f <I>(g+u)(g,u*)(g,xl?) dg
G
= ( - u, u*)(x, u*)(x, (-u)1? - u*) f <I>(g)(g, xl? + u*) dg
G
= (-u,u*)(x,(-u)l?) f <I>(g)(g,xl?+u*) dg.
G
If we replace g + u by g' then the left-hand side changes to
f <I> ( g ') ( g I - u, u * ) ( g I - u, X I?) dg I
G
= f <I> (g ') ( g I , U * ) ( - u, u * ) ( g I , X I? ) ( - u , X I?) dg I ,
G

SECOND-DEGREE CHARACTERS 47
whereas the right-hand side easily simplifies to
f <I> (g ) ( g , x P ) ( g , u *) dg ( - u, u * ) ( x , ( - u ) P ) ;
G
so it all comes down to showing that ( - u, xf') = (x, (-u)Q). But this
.. h b . h ( II f ( u 1 + u 2) ) . .
IS Just tea servatlOn t at (} or, actua y, f( u1)f( u z ) IS symmetnc.
So, indeed, Uj E B( G), and we single out the fact that
(-u,u*) Q
W f CU(u,U*)W f - 1 = -1 CU ( (-u) -u*).
f( u*)Q
Next, writing Uj 3 CU(u, U*)Uj-3 in the form
Uj[ Uj{UjU( U, U*)Uj-l}Uj-l] Uj-l,
d . · d . h < Q ) f ( U 1 + u 2 ) f . d h
an usmg agam an agam t at u 1 , U z = f( u1)f( u z ) , we m t at
-1
( - U u*)(u + (u*)Q u Q )
Uj 3 'l1(u,U*)Uj-3= ' f(-u)f(u) , U(u,u*).
-1
. < - u, u*)<u + (u*)Q , u Q ) 3
So, If we can show that f( _ u )f( u) = 1, so. that Uj
commutes with all CU(w) = CU(U, u*), then by Proposition 4.3.11 or by
Corollary 4.3.12, Uj3 E ker'iT ::: C, seeing that (4.3.12) describes a
central extension. We are left with a straightforward computation:
-1
< - u, u*)(u + (u*)Q , uf')
f( -u)f(u)
1 ( -1 ) Q
= f( -u)f(u) ( - u, u*)(u, uP)(u, (u*)P )
1 (-u u Q ) (0 u Q )
- f( -u)f(u) (0, u*)(u, uP) = feu  u) (u, uP) = ;(0)
1 f(u') f(O+u') ,Q
- f(O) - f(u)f(u') = f(O)f(u') = (0, (u ) ) = 1,
for any u' at all.
.

48 WElL'S "ACTA" PAPER
The upshot of Proposition 4.4.2 is that for every second-degree
character, f, there exists a unimodular constant, '}'(f) E C, so that
3 = '}'(f) 0 ide This all-important object, '}'(f), is called the Weil
index or the Weil constant. As Mackey says in [Ma64]: "... this
constant, . . . y(f), plays a central role in the proof of the quadratic
reciprocity law."
Before we go on to Weil's derivation of this law, however, we give
Weil's own definition of '}'(f) which differs prima facie from the
definition above, y(f) 0 id = 3 (which is given by Mackey (cf.
[Ma64])). Weil's formulation turns out to be better suited to the
upcoming discussion of the product formula for the Weil index and
has the additional benefit of affording an explicit depiction of the
Weil representation, r*, as given in (4.1.11). (We address the latter
theme in  5.3.)
Subsequent to giving Weil's characterization of y(f) we will prove
that y(f)Mackey = y(f)Weil' so to speak.
For convenience, write X 2 (G) for the group of all non-degenerate
second-degree characters on G. Let Iso ( G*, G) denote the set of all
isomorphisms y: G*  G. (This marginally ambiguous notation makes
sense in view of the fact that a typical (a, g) E B( G) is expressible as
(( ),g), where a: GG, f3: GG*, '}': G*,
and 5: G*  G) Also, if a: G  G (resp. 5: G*  G*), denote by
ex * (resp. 8 *) the unique homomorphism associated to ex (resp. 8) by
Pontryagin duality. Finally, if f E X 2 ( G), write (} for the symmetric
homomorphism associated to f by (4.4.1), and define f(x, x*) = f(x),
if (x, x*) E G X G*.
Consider the following mappings:
do: Aut(G) B(G)
(4.4.3)
a(( a-l),l);
d: Iso ( G*, G)  B( G) (4.4.4)
'}'(( _*-l),<U'_U*»);
to: X 2 ( G)  B( G) (4.4.5)
f(( ),;.).
i ,1t'

SECOND-DEGREE CHARACTERS 49
By virtue of the surjectivity of 7T (i.e., by Corollary (4.3.12) or
actually by the definition of B( G) given in  4.2) we can pull the maps
do, d, to back to 7T- 1 (B(G)) = N(cJf£Lj,(G)) so as to get
do: Aut(G) N(cJfEl.1(G))
a[L2(G) -)L 2 (G)]
<I> ( x )  I a 1 1 / 2 <I> ( x a ) ;
d / o : Iso ( G*, G)  N(dfEl.1( G))
y[L2(G) -)L 2 (G)]
<I> ( X )  I I' 1- I/Z <i> ( ( - X yr* - 1 ) ;
to: IsoXz(G) N(dfEl.1(G))
f [L2( G) -) L2( G)]
<1>( x )<I>( x )f( x),
where 1 a I, 1 y 1 denote the indicated Haar moduli.
(4.4.6)
(4.4.7)
(4.4.8)
Proposition 4.4.3. do = 7T 0 do, d = 7T 0 d ' o , to = 7T 0 to.
Proof. We treat the case of do vs. do; the other two cases are simi-
lar. From the proof of Proposition 4.3.11 we know that 7T(S) = s
translates to s- 1 '1l(w)s = '1l(W U )g(w), if S = (iT, g). So, we need
only show that d o (a)-1'1l(w)d o (a) = '1l(u a ,(u*)a- 1 ), seeing that
(u, u*)( (: *_lx-I)) = (u a , (u*)a- 1 ). But on *_t1he one hand,
'1l(u a ,(u*)a ) maps <I>(x) to <I>(x + ua)(x,(u*)a ), while, on the
other hand, do(a)-lCU(w)do(a) maps <I>(x) to <I>(x + ua)(x a -\ u*)
(a straightforward calculation). It just remains for us to observe that,
by Pontryagin duality, <x,(u*)a*-l) =x a -\ u*). .
Caveat. As hinted above, (4.4.3)-(4.4.8) conspire to describe a
projective representation, in L 2 (G), of the free group generated inside
B(G) by the elements (a, 1), a of the form ( a -1 ),
( 0 - *-1 ) ( 1 (3 )
I' 1'0 '0 1. Indeed, the subgroup cut out in B( G) by
these elements is easily enlarged to Sp( G) (in fact, B( G) is realized as

50 WElL'S "ACTA" PAPER
the semi-direct product X 2 (G) X Sp(G)), and the automorphisms of
L 2 (G) defined via (4.4.6), (4.4.7), and (4.4.8) characterize a projective
representation of Sp(G) in L 2 (G), which -a forteriori must coincid'e
with r*. In other words, we shall see (in  4.9) that this explicit
presentation of distinguished automorphisms of L2( G) gives rise to
what is usually called the Schrodinger model of the metaplectic (or
Weil) .representation (see, e.g., [Ge93]).
It also bears mentioning that, following Jacquet-Langlands [JL70], it
is not too difficult to proceed directly from Proposition 4.1.3 to an
explicit presentation of r* by giving its action on generators.
Back to the Weil index.
Proposition 4.4.4. For f E X 2 ( G), with symmetric morphism Q, define
f-(x) = f( -x); then there exists y(f)Weil E C, so that
d'o( - Q-1 )to(f)d'o( Q-1 )to(f-) = y(f)Weii to(r 1 )d'o( - Q.
Proof. Write s (resp. s') for d'o(-Q-l)to(f)d(Q-l)to(f-) (resp.
t o (P-l)d'o( _Q-l)), so that we have to establish that s = y(f)Weil S '.
Using Proposition 4.3.11 this reduces to showing that 7T(S) = 7T(S'),
that is, given that 7T is a homomorphism (Proposition 4.2.2), we need
only show that d( - Q-l )to(f)d( Q-l )to(f-) = t o (f-l )d( - Q-l). This
amounts to a "matrix" calculation in B(G) which we leave as an
exercise. .
I
Corollary 4.4.5. The unimodular constant, y(f)Weil' IS equivalently
characterized by
1/2" 1
<P * f = y(f)Weiil Qr <p. f' (4.4.9)
where <I> E S( G) and <I> * f denotes the convolution of <I> with f.
Proof. With s, s' as in t he p receding proof it evidently suffices to
prove that s'(<p)(x) = I QI(<P * j)(x(J) and s'(<p)(x) = I Q1 1 / 2 f() <I>(x(J);
for convenience, since f is non-degenerate, we use Q as a means of
identifying G and G*. The action of S on <I> is this
.f
<P ( x ) I ) <P ( x ) f ( - x ) I ) I Q 1 1 / 2 ( <P f- ) 1\ ( X Q ) I )
to(f-) d'O(Q-1) to(f)
_I/)
I Q ,- 1 / ( <P r )" ( x (J ) f ( x) I ) I - Q - 11 1 / 2 1 Q 1 1 / 2 { ( <P r ) /\ f} /\ (x (J )
d' ( - n -1 )
, 0 
=IQI(<I>*f)(x Q ),

SECOND-DEGREE CHARACTERS 51
where the last simplification is achieved as follows: by definition,
(<I> * f)(x) = f <I>(g )f(x - g) dg, whence
G
(<I> * f)(xl?) = f f <I>(g)f(x - g)(x, xi?) dgdx;
G G
on the other hand,
{  } A
{( <I> f-) 1\ f} 1\ (xi?) = f( x) fc <I>(g )f( - g)( g, xi?) dg
= ff(x)f <I>(g)f(-g)(g,xl?)dg(x,xl?)dx
G G
f( X - g )
. fcfc <I>(g)f(x)f( -g) f(x)f( _g) (x, xl?) dgdx,
and the desired equality follows upon canceling f(x)f( -g). It now
only remains for us to compute the action of s' on <1>: <I>(x) 
d'o( l? -1 )
1-' Q- 1 1-l/2<i>(( _x)(-i?*)-l) = I QI 1 / 2 <i>(Xi?)  I QI 1 / 2 <i>(Xi?). 1 .
t'o( f- 1 ) - f ( x )
The proof is complete. .
Finally,
roposition 4.4.6. y(f)Mackey = y(f)weu. (Here y(f)Mackey is y(f) as
given in Proposition 4.4.2: Uf3 = y(f) 0 id.)
Proof. The identity d( - (}-1 )to(f)d( (}-1 )to(f-) = t o (f-1 )d (_ (}-1)
in B(G) (cf. the proof of Proposition 4.4.4) can be shown to be
equivalent (as a "matrix"identity) to the relation {to(f)d(-(}-1)}3=
ide Using Proposition 4.3.11 and Proposition 4.4.4 this yields that in
N(dfE,i( G)) we must have that {to(f)d'o( - (}-l )}3 = y(f)Weil 0 id (just
pull back along 7T). It is also evident, however, that to(f)d'o( - (}-l)
can be identified with Uf so that the preceding relation becomes
Uf3 = y(f)Weil 0 ide But then, courtesy of Proposition 4.4.2, and so on,
we have also that Uf3 = y(f)Mackey 0 ide So, y(f)Weil = y(f)MaCkey. .
From now on we will use the notation, y(f) unambiguously, for any
and all of the unimodular constants characterized in Propositions
4.4.2, 4.4.4, and (most importantly) Corollary 4.4.5. Each and all of
these go by the name "Weil index" (cf. [Sc85]).

52 WElL'S "ACTA" PAPER
4.5 THE SPLITTING OF 7T ON A DISTINGUISHED SUBGROUP
OF B(G)
Let f be a closed subgroup of G and, via Pontryagin duality, identify
f's dual, f*, with AnnG.(f), f's annihilator subgroup in G*. As
further analogues of the Fourier transform and its inverse define the
operators
z: <I>(x)e(x, x*):= 1 <I> (x + O( g, x*) dg (4.5.1)
r
Z-1: e(x, x*)<I>*(x*):= j e(x, x*)(x, x*) dx, (4.5.2)
G/r
where dx is the induced Haar measure on G If. We obtain immedi-
ately
Proposition 4.5.1. If ( g, g *) E f X f* then
@(x + g, x* + g*) = E>(x, x*)< g,-x*),
(4.5.3)
that is, writing z = (x, x*) E G X G*, and C = (g, g*) E f X f*,
@(z + C) = @(z)w(, Z)-l.
(4.5.4)
Proof. e(x + g, x* + g*) = 1 e(x + g+ 1])( 1], x* + g*) d1] = 1 <I>(x +
r r
g+ 1])( g+ 1] - g, x* + g*) d(g+ 1]) = 1 <I>(x + O( C, x*)( c, g*) dC'
(-C,x*+g*). However, g*Ef*, CJf=>(C,g*)=l and, since
g E r, < - g, x* + g *) = < - g, x*) = < g, - x*). .
Obviously, (4.5.4) qualifies as an automorphy condition (in which
the 2-cocycle w, as given in 9 4.2, is realized as a it-multiplier, once
@(z) is suitably explicated). Indeed, the functions @(z) are in the
obvious sense automorphic under r X f* with automorphy factor
w(, Z)-l. These functions are WeB's famous "fonctions theta
generalisees," about which he says: "si fest discret, l'integrale
[(4.5.1)]. . . se reduit a une somme... pour des choix con venables de E>
[cettes integrales] se reduisent a des series theta au sens classique . . . " (" if
f is discrete, the integral [(4.5.1)]... reduces to a sum... for proper
choices of E> [these integrals] reduce to theta series in the classical
" )
sense. . . .

THE SPLITTING OF 1T ON A DISTINGUISHED SUBGROUP OF B(G) 53
We now impose a few obvious analytic conditions in order to get a
Hilbert space structure:
df (G, f) = {0( z) as in (4.5.4), locally integrable on G X G* ,
square integrable on Gjf X G* jf*}. (4.5.5)
Proposition 4.5.2. Z: L2( G)  df( G, f) is an isomorphism.
Proof. The Hahn-Banach Theorem permits us to restrict our atten-
tion to Z's action on compactly supported continuous functions <1>,
and in this setting we need only verify that Z is well defined (for,
subsequently, Pontryagin duality in the presence of Z-l (as per
(4.5.2)), suffices to ensure that Z is invertible on all of df(G, f)).
Thus it all reduces to showing that Z(<I>(x)) E df(G, f). However,
tha,t Z( <I>(x)) should be locally integrable on G x G* follows from
Fuhini's Theorem and the square-integrability of Z( <I>(x)) on G jf X
G* If* is, following Weil, tantamount to an application of Plancherel's
Theorem. .
The upshot of this result is that for all ru (w) E N(dft:i1.( G)) the
diagram
L2(g) ru(w) ) L2( G)
Z- l l 1 z (4.5.6)
df(G, f) zcu.(w)Z- \ df(G, f)
is commutative; abusing notation only mildly we may identify
zru(W)Z-l and ru(w) and realize N(dft:ij,(G)) as well as dft:ij,(G) as
subgroups of GL( df(G, f)). Indeed, we have from Corollary 4.3.12
that the sequence (4.3.12)
1  Cr N(dfgi(G)) B(G)  1
is exact, and certainly N(dft:i1.(G)) < GL(L 2 (G)) by (4.2.5); thus, given
that (4.5.6) allows us to identify GL(L 2 (G)) with GL( df(G, f)), we
can view (4.3.12) as describing the behavior of 7T relative to represen-
tations in df(G, f).

54 WElL'S "ACTA" PAPER
As in  4.4, present a typical element of B( G) as (u, g)
= ( (  ) , g ), where a : G ---'; G , {3 : G ---'; G *, '}' : G * ---'; G, 8 :
G* ---+ G*, and g: G X G*  Cr (cf. (4.2.6), (4.2.7)). Define the
following subgroup of B( G):
B(G, f) = {( u, g) with u E GL(f X f*), g = 1 on f X f*}. (4.5.7)
Additionally, cut out the subsets
fl(G) = {(u,g) EB(G) with I'EIso(G*,G)}, (4.5.8)
fl ( G , f) = {( u , g) E B ( G , f) with 'Y E Iso ( G * , G)} . ( 4.5.9)
Note that this arrangement of subsets of B(G) mimics the classical
setting for developing a theory of automorphic functions-B( G, f) is
substantially a congruence subgroup attached to f < G. Again mim-
icking the classical case we have the following decomposition theorem:
Proposition 4.5.3. Each S E O(G) is (uniquely) expressible as s =
to(/l)d( l' )t o (/2) for It, 12 E X 2 ( G) and 'Y E Iso ( G*, G). In other words,
O( G) = (X 2 ( G) )10 Iso ( G*, G)( X 2 ( G) )10.
Proof. A straightforward calculation in B( G).
(4.5.10)
I
.
Weare now in a position to exploit the Weil representation.
Referring back to  4.4, that is, (4.4.3)-(4.4.8), define
'0: !leG) ---,;N(dft:i(G»)
to(/l)d( l' )t o (/2) to(/t )d'o( l' )t o (!2)
( 4.5.11)
and
'r: B(G, f) ---+ GL( df(G, f)) (4.5.12)
s = (u, g) [e(z)e(za)g(z)].
The latter map is a group representation. Furthermore, (4.5.6) permits
us to view 'r as a mapping into the unitary operators on L 2 (G) and,
most importantly, we have the following result:
Proposition 4.5.4. '0 and 'r agree on O(G, f). This yields that 7T
splits on fl( G, f).

THE SPLITTING OF 7T' ON A DISTINGUISHED SUBGROUP OF B(G) 55
Proof. In view of (4.5.10) it is enough to prove that if f E X 2 (G) and
f = 1 on f then ro(to(f)) = rr(to(f)) and, similarly, if y E Iso(G*, G)
n !so(f*, f), then ro(d( y)) = rr(d( y)). However, for the first asser-
tion we have that ro(to(f)) = to(f): <I>(x)<I>(x)f(x) (by (4.4.8))
while rr(to(f»=rr(( i),f): 8(x,x*)8(x,xl?+x*)f(x) (by
(4.4.5) and (4.5.12)), so we need only verify that Z( <I>(x)f(x)) =
8(x, xl?+x*)f(x). However, this is an easy matter: L.H.S.= 1 <I>(x+ g)
f
f( X + g) ( g, x *) d g = 1 <I> (x + 0 f( x) f( g ) ( g, X I? ) ( g, x *) d g =
f
f(x) 1 <I>(x + g)( g, xl? +x*) dg= f(x)8(x, xl? +x*), using (4.4.1), the
f
fact that f = 1 on f, and (4.5.1). This takes care of the first assertion.
As regards the second claim, Weil states in [W e64], "... !es deux
operateurs en question ne different !'un de !'autre que par un facteur reel
> 0; comme ils sont unitaires, i!s sont done egaux. . . " ("the two opera-
tors in question only differ by a real factor > 0; as they are unitary,
they ,are therefore equal... "). Specifically, by (4.4.7) we have that
I
. roe d' (1')) = d'o( 1'): <1>( x )I'Yr 1/2 f 8( x, - Xy'-l)( x, - Xy'-l) tU,
G If Y Y d{
because, by (4.5.2), <I>*(x*) = f 8(x, x*)(x, x*) tU. On the other
G/f
hand, by (4.5.12),
rr(d(y)): 8(X'X*)8((X'X*)( -'Y O *-l))(X,-x*)
( Y *-1 )(
=0 (x*) ,-x Y x,-x*).
But this yields that the effect of Z-lrr(d( y))Z on <I>(x) is this:
<I>(X) I
( y *-1 ) )
) 0(x,x*) ' ) 0 (x*) ,-x Y (x,..x*
Z 'r(d o ( y ))
I ) f e ( ( x *) y , - x y* -1 ) ( x, - x *) dx,
Z-1 G*/f*
where dx* denotes Haar measure on G* /f*. In light of (4.5.9) we get
from this that (x*)Y = x, X y *-1 = x*, whence the latter integral becomes

56 WEIL'5 "ACTA" PAPER
j <3(X,-X'Y*-l)(X,-X'Y*-l> dx*. Therefore, the agreement be-
G*jr* .
tween 'o(d( 1')) and 'r(d( l' )), up to a real constant factor, amounts
to changing variables via 1': G*  G (and f*  f). The proof is
completed by showing that this constant must equal 1, which in fact
follows from a stronger result (as Weil indicates): if rut and ru 2 are
unitary operators and, for some real € > 0, f1.4 = € ru 2 , then € = 1. For
since flhru:;l is also unitary, we get, for all ,'I',(,'I')=
(f1.4ru:;l, f1.4ru;l'1') = (€, €'I') = €€(, '1') = €2(, '1'), so that €2
= 1 and so € = 1. .
With Proposition 4.5.4 in place we extend diagram (4.3.12) as
follows, with the vertical arrow entailing the canonical projection:
1
) c x
1
) N( df£i5{ G)) Tr) B( G)
. 1
. . . . . ;. . . . . . . . . . .
O(G, f)
) 1
(4.5.13)
Finally,
Proposition 4.5.5. Suppose f E X 2 (G) so that f = 1 on r (and we
employ the convention that f(x, x*) = f(x), f = 1 on f X f*), and I
() E Iso(G*, G) n Iso(f*, f). Then yet) = 1 (where this l' denotes the:
Weil index as per (4.4.9), say).
Proof. From the proof of Proposition 4.4.6 we have that y(f) can be
characterized by the identity d'o( - (}-l )to(f)d'o( - (}-l )to(f-) =
y(f)t o (f-l )d'o( - (}-l) (indeed this is nothing else than a restatement
of (4.4.9)). In other words, writing s=d(_(}-l)to(d(-(}-l)to(f-) in
B( G), and also s' = t o (f-1 )d( - (}-1), the unimodular constant y(f)
comes about by lifting the identity s = s' (cf. the proof of Proposition
4.4.6) to N(dfE-i1.(G)) by means of what amounts to an inverse to 7T
(cf. Proposition 4.4.3); of course 'r affords such an inverse and our
result now follows because 'r is a group representation (which is
immediate from (4.5.12)). Specifically, writing s = 11' 0 s, s' = 11' 0 s', for
s, s' as above, we get that the aforementioned restatement of (4.4.9)
boils down to s = y(f)s'. On the other hand, since s, s' involve only
d and to, we use the fact that '0 and 'r agree on O(G, f) to get that
s = 'o(s) = 'r(s) = y(f)s' = y(f),o(s') = y(f)'r(s). But because 'r is
a group representation and so a homomorphism (acting on B( G, f) 
O(G, f)), s = s' yields 'res) = 'res'). Thus, y(f) = 1, as desired. .

VECTOR SPACES OVER LOCAL FIELDS 57
Proposition 4.5.5 is very important indeed, because the eventual
derivation of the product formula for the 2-Hilbert symbol, that is, the
fact that an infinite product of 2-Hilbert symbols reduces to 1, will
follow almost immediately from it.
4.6 VECTOR SPACES OVER LOCAL FIELDS
Having obtained in  4.4 the Weil index, y(f), attached to any
non-degenerate second-degree character f on any LCA group G, we
now proceed to map out the specific setting useful to algebraic
number theory. In Gelbart's phrase (cf. [Ge93]) this means that we
"linearize" the theory surrounding 7T, that is, the material developed
in  4.2-4.5, specializing to finite dimensional k p-vector spaces. In
the lowest dimensional case (G = k p ), what results from this lineariza-
tion is nothing other than the content of  4.1, where the according
Weil representation was derived in the context of applying the Stone-
Von Neumann Theorem to the Heisenberg group Heis (k p €a k).
Indeed, the tactics of  4.1, along the lines of our Stone-Von Neumann-
Schur game, so to speak, apply mutatis mutandis to the case with
which we now concern ourselves, namely, G ::: kg. But this line is
arguably inferior as far as the exhibition of Weil indices is concerned.
For our purposes it is far more expeditious to follow the same route
Weil followed in [We64], which is to say that we seek to interpret
Corollary 4.4.5 in the right fashion. Our final goal is to get. some
appropriate Weil index expressed in terms of 2-Hilbert symbols.
With kp and k as in  4.1 let X be a d-dimensional vector space
over kp (so X ::: kg) and let X* be X's Pontryagin (or linear) dual.
Generalizing (4.1.1), let
[,]: (XXX*) X (XXX*) kp
(4.6.1)
be a skew-symmetric bilinear pairing into kp. Because of the agree-
ment of Pontryagin duality and linear duality in this (restrictive) case
we get that if q is any quadratic form on X then the canonical
homomorphism Q: X  X* associated to q by the rule
[x,ylJ] =q(x+y) -q(x) -q(y)
(4.6.2)
is manifestly the natural counterpart to (or linearization of) (4.4.1).
Indeed, if X p is any fixed character on k p (just as in  4.1) then
X p 0 [,] plays the role of ( ,) in (4.4.1).

58 WElL'S "ACTA" PAPER
Proposition 4.6.1. Xv 0 q is a second degree character on (the additive
abelian group of) X.
Proof. Trivial.
.
Moreover, as regards (4.4.1) the analogy is complete since we may,
of course, identify X with k so that (with the product topology) we
get an LCA group. But in the present linearized setting we encounter
second-degree characters as images of quadratic forms under Xp. In
other words, just as Corollary 4.4.5 presents the Weil index as belong-
ing to a generic second-degree character, linearization underscores
the fact that in the setting of finite-dimensional vector spaces over a
local field the Weil index belongs to a quadratic form. We can
immediately state the following proposition.
Proposition 4.6.2. If q is any non-degenerate quadratic form on
X -- kg (i.e., q is a d-dimensional quadratic form over k p ), then there
exists a unimodular object y( Xp 0 q) E C such that
{ } 1\ 1/2 " 1
cD * ( X p 0 q) = l' ( X'1 0 q ) 1 Q 1- cD . ( 4.6 .3)
+ Xp 0 q
for Q associated to q by (4.6.2) and for all <I> E t::S(kg)( :::: cS'(X)).
I
Proof. Corollary 4.4.5. .
Wit.h (4.6.3) in place we are now in a position to prove what turns
out to be the most important characterization result concerning
y( Xv 0 q), which, in essence, sets the stage for the introduction of
Gauss sums. We first make a few trivial observations which will prove
to be very useful in the upcoming proof. First, by virtue of (4.6.2) it is
the case that [x, yQ] = [y, xQ]. Second, by virtue of the current agree-
ment between Pontryagin duality and linear duality it is the case that
if L is a lattice (i.e., an open and compact Dp-submodule) in X then
L * is once again nothing else than Ann x*(L).
Lemma 4.6.3. Given any quadratic form, q, on X, there exists a
lattice L c (L*)Q-I such that Xp 0 q = 1 on L.
Proof. By homogeneity q(O) = 0, that is, Xp 0 q(O) = 1. Since ker X p is
open in X, X p 0 q = 1 in some neighborhood of 0; pick L to be any
subset of this neighborhood which is also a lattice. Because L is an
lOp-module, x, y E L yields x + y ELand therefore Xp 0 q(x) =
Xp 0 q(y) = Xp 0 q(x +y) = 1, that is, Xp([x,yQ]) = 1 and LQ c L*. .

VECTOR SPACES OVER LOCAL FIELDS 59
Proposition 4.6.4. For convenience, write L' for (L*)(}-I. Then
. y( Xp 0 q) = I l?1 1 / 2 ,( Xp 0 q)(x) dx.
(4.6.4)
Proof. A straightforward change of variables in (4.6.3) gives the rela-
tion
f (cI> * ( Xp 0 q»)( x) Xp( [x, x*]) dx
X
-1/2 A 1
= y( Xp 0 q)\ Q\ <p(x*) ( -1 ) .
X p 0 q (x*)Q
Setting x* = 0 we obtain (since X p 0 q(O) = 1) that
f ( f cI>(x - Y)Xp 0 q(y) d Y ) dx = y( X p 0 q)1 l?r 1 / 2 f cI>(x) dx.
x x x
Let L be as in Lemma 4.6.3 and set <I> = CPL' the characteristic
function of L. So
f f (1)£( x - y) Xp 0 q(y) dydx = y( X p 0 q)1 l?1-l/2JL( L),
xx
writing J.L(L) for L's Haar measure. Changing variables, y' =x - y, in
the left-hand side, yields immediately that
f f 4>L(Y')XP 0 q(x - y) dy' dx = f 4>L * (Xp 0 q)(x) dx.
xx x
We now claim that 4>L *( Xp 0 q) = J.L(L)4>L'( Xp 0 q), from which the
desired result follows very easily: writing L.H.S. (resp. R.H.S.) for
"left-hand side" (resp. "right-hand side"), we obtain that
L.H.S. = f J.L(L)( 4>L'( X p 0 q))(x) dx = f J.L(L)( Xp 0 q)(x) dx, whence
x L'
JLtL) .L.H.S.= ,(Xpoq)(x)dx= JLtL) R.H.S. = y(X p oq)Il?I-1/2.

60 WElL'S" ACTA" PAPER
It remains for us to prove the claim. To wit:
cPL*(Xpoq)= fXpoq(x-y)dy
L
1. Xp(q(X-Y))
=Xp(q(X)) (()) ((_ )) Xp(q(-y))d Y
 Xp q x Xp q Y
= Xp(q(X)) f X p ([ x, (-y)l?]) dy,
L
because by Lemma 4.5.3, Xp 0 q = 1 on L. Now distinguish cases:
If x" E L * then Xp([x, ( - y) "]) = X p ([ - y, x"]) = 1, where
f Xp([x, (-y )I?] dy = p.,(L). If, on the other hand, xl? $ L*, then for
L
any Yo E L,
f x p ([ -y, xl?]) dy = f xp([y + Yo, - xl?]) d(y + Yo)
L L
= jxp([y,-xl?])xp([yo,-xl?])dy
L
= xp([yo,-xl?]) f X p ([ -y, xl?]) dy;
L
so, since we can take Yo EL with Xp([Yo,-x"]) =1= 1 (because
-xl?$L*), it follows that jxp([-y,xl?])dy= jxp([x,-yl?])dy=O.
L L
Therefore, since x"EL* if and only if xE(L*),,-l =L', we get that
f Xp([x, - yl?]) dy = JL(L)cPL" whence (substituting) cPL *( X p 0 q) =
L
( X p 0 q) /-L(L)cPL'. .
We now make the transition to Gauss sums. The characterization
(4.6.4) of the Weil index for q suggests the following definition: If
Xp 0 q = 1 on Land M -:J L (so, in a sense, M is "large enough") let
g(q, M) = f (X p 0 q)(x) dx.
M
( 4.6.5)
Proposition 4.6.5. g(q, M) = /-L(L)'E xE Mn L'/L( Xp 0 q)(x).

VECTOR SPACES OVER LOCAL FIELDS 61
Proof. Decomposing M as L EB MIL and using the definition of L
we get that
q}(q,M) = fXtJoq= f !(XtJ°q)(y-x)dydx
M MIL L
= f f (XtJ°q)(y)(XtJ°q)(-x)XtJ([y,-x!?])dydx
xEM/L yEL
= f (XtJ°q)(x)f XtJ([y,-x!?])dydx
xEM/L yEL
- L (XtJ°q)(x)fxtJ([y,-x!?])dy
xEM/L L
Here we have made use of Fubini's Theorem, of the evident fact that
(Xpoq)(y-x) _ ([ P ]) d f L ' d . .
( )( )( )( ) - X p y, - x , an 0 s Iscreteness In
X p 0 q y Xp 0 q -x
M (an easy exercise using the topology of k p ). Now, as in the proof of
Prop:osition 4.6.4, the latter integral reduces to plL) or to 0 according
as x E L' or x $; L'. The result follows. .
Corollary 4.6.6. If M::> L' then q}( q, M) is independent of M.
Proof. Obvious: M n L' = L' .
.
Thus we may now define the object
q}(q)=JL(L) L (Xpoq)(x),
xEL' /L
where Land L' = L*)e- 1 as before.
(4.6.6)
Corollary 4.6.7. 1'( Xp 0 q) = q}(q)/I q}(q)l.
Proof. With L' eM, certainly q}(q) = q}(q, M) = f XtJ 0 q, using
Proposition 4.6.5 and (4.6.5). At the same time, by Proposition 4.5.4,
Y(XtJ 0 q)= IQI 1/2 !,xtJ 0 q. Therefore, taking M = L', Y( XtJ 0 q) =
IQI 1/2 q}(q); accrdingly, because Iy( XtJ 0 q)1 = 1 by Proposition 4.6.2,
the result follows gratis. .
Now, because L is a lattice in X ::: k we have in (4.6.6) an
analogue to the classical notion of Gauss sum. The principal differ-
ence between the object q}(q) and a classical Gauss sum consists in

62 WElL'S "ACTA" PAPER
the fact that g(q) exists in the context of the local field kp, equipped
with its incomparably useful ultrametric topology, while classical Gauss
sums, properly speaking, are so-called global objects. On the other
hand, as we shall see presently ( 4.8), local data such as the charac-
terization of the Weil index via g (q) as afforded by Corollary 4.6.7
yields global data almost automatically as a result of adelization.
Loosely put (and we make this precise below), the fact that Corollary
4.6.7 holds for each place p of k gives rise to a similar relation
holding for k itself, and this could be nothing else than a characteriza-
tion of something like a classical Gauss sum in terms of a global (or
adelic) Weil index. Prom this point of view, Corollary 4.6.7 presages
nothing short of Gauss' famous result concerning the sign of the
classical Gauss sum. So, at this point it is in order to add a few
historical comments; here we refer to [S085] and [Sc85].
2m ( k2 )
Writing G(2m) for the Gauss sum E exp 27Ti- , the central
k=l m
result in the entire affair is that G(2m) = E f2m , where E is a certain
1 + i
eighth root of 1 (specifically, € = fi ). It turns out that the determi-
nationpf ( \S ) uffice ( s to kac ) ilitate the determination of the sign of the
sum E - exp 27Ti- , which is conspicuous by virtue of the
k=l P P
appearance of the Legendre symbol. This is also a Gauss sum, of
course, and we are rescued from ambiguity because one can show
p -1 ( k2 )
rather easily that this sum agrees with E exp 27Ti- . In any case,
k=l P
the upshot is that the determination of E is tantamount to the
solution of a problem whose solution taxed Gauss considerably [S085]:
The determination of the sign of the root has vexed me for many years.
This deficiency overshadowed everything that I found; over the last four
years there was rarely a week that I did not make one or another attempt,
unsuccessfully, to untie the knot. Finally, a few days ago, I succeeded-but
not as a result of my search but rather, I should say, through the mercy of
God. As lightning strikes, the riddle has solved itself [S08S].
And the denouement is Gauss' famous theorem:
P-l ( k ) ( k ) { fP,
E - exp 27Ti- = . C
k=l P P lVP'
P = 1(mod4)
P = 3(mod4).
Of course, from our point of view the most significant feature of
this beautiful formula is the suggestion that the reciprocity law for the

QUATERNIONS OVER A LOCAL FIELD 63
Legendre symbol should be derivable from a suitable relation between
Gauss sums. Indeed, such a relation, a Reziprozitiitsgesetz zwischen
Gaussschen Summen, in Hecke's phrase [He27], holds true for what
are sometimes called Gauss-Hecke sums ([Sh64], [Be95]), generalizing
G(2m) to arbitrary number fields.
As we saw in  1.2, Hecke derived this Reziprozitatsgesetz as a
relation between so-called Thetanullwerte (it-constants or holomorphic
it-series) coming directly out of the functional equation obeyed by
suitable Hecke it-functions (cf. (1.1.4)). It is precisely this set of tactics
which constitutes Hecke's generalizations of Cauchy's treatment [Ca82]
of the reciprocity laws for the Legendre symbol by means of Jacobi
it-functions [K084], as we explained in Chapter 1.
As regards the all-important Weil index, y( Xp 0 q), characterized by
Corollary 4.6.7, we stress that it actually carries the germ of the
determination of the sign of the Gauss sum (after adelization) and,
interpreting the Weil index in Hecke's formalism, this is tantamount
to setting the stage for deriving quadratic reciprocity as in [He27].
However, in the presence of Corollary 4.6.7 the quickest (and most
transparent) way to get to quadratic reciprocity is to exploit the
aforementioned process of adelization in order to get a product
formula for the y( Xp 0 q), indexed on the places, +>, of k. This, in
turn, precipitates Hilbert's form of quadratic reciprocity because suit-
able choices of X and q permit one to realize y( Xp 0 q) in terms of
the 2-Hilbert symbol. This is the line Weil himself takes in [We64] and
it is manifest that the equivalence between Weil's proof of quadratic
reciprocity and Hecke's ultimately reduces to the fact that in 2-Hilbert
reciprocity all other forms of quadratic reciprocity are subsumed in
one fell swoop. (See Chapter 6 in this connection.)
Finally, as regards our goal for this section, namely, the proper
interpretation of Corollary 4.4.5, suffice it to say that Corollary 4.6.7
does the job:
y( X p 0 q) = g(q)/I g(q)l.
(4.6.7)
Next, we specialize to the right X and q to yield 2-Hilbert reci-
procity.
4.7 QUATERNIONS OVER A LOCAL FIELD
The proximate objective is to realize the Weil index, y( Xp 0 q), as a
2-Hilbert symbol in kp by adroitly choosing X  k and q. There are
at least two ways of achieving this aim, one of which (Weil's) we

64 WElL'S II ACTA II PAPER
discuss in this section. It turns out, however, that the other approach
fits far better into the formalism of Kubota and we accordingly discuss
this alternative in Chapter 5.
In fact, it is very instructive to contrast Weil's route to 2-Hilbert
reciprocity with Kubota's, if only because the critical issue of splitting
of cocycles on the subgroup of rational points appears in essentially
the same way in both settings due to the fact that the corresponding
formalisms, Weil's and Kubota's, are in fact "algebraically" equivalent
(cf. Chapter 6). On the other hand, deriving 2-Hilbert reciprocity in
Kubota's setting is a far more explicit affair than it is in Weil's
approach, so the correspondence of these two routes to quadratic
reciprocity suggests a deep interconnection. It is indeed possible (and
natural) to capture both of these approaches to the construction of
metaplectic covers under a single umbrella. This is discussed in future
chapters.
Proceeding, then, toward Weil's realization of y( Xp 0 q) as a 2- Hil-
bert symbol, suppose kp =F C and let X be the algebra of kp-quater-
nions, H(k p ) = H(k)  kp, expressed in the variables x, y, z, t. Let n:
(H(kp))X k denote the norm map, a surjective homomorphism
with compact kernel (see, e.g., [We74]). First we wish to prove that
Yp( X p 0 n) = -1 (the reason for this will appear later; see Proposition
4.7.6), and in order to do this-by exploiting (4.6.7)-we require a few I
basic facts. The reader is referred to Weil's own Basic Number Theory
[We74] for details.
We actually need comparatively little. Let 7T I tJ, 7T E Dp, which is to
say that we have  = 7T Dp, 117T1I = 1, -adically. Now, for all v E Z, the
set Mv:= {z E H(k p ) with n(z) E 7T- V D p } is a lattice in H(k p ) and,
furthermore, obviously Mv C MJL if v < JL. Moreover, we may suppose
that the measure dx on k p is normalized in such a way as to restrict to
dx
a probability measure on tOp; we take Ix I to be dx x the (multiplica-
tive) measure on k. Then, writing dz for the induced additive
dz
measure on H(k p ) --- k we write dz x for 2 ' the corresponding
n(z)
multiplicative measure on (H(kp))x.
Proposition 4.7.1. Y( Xp 0 n) = -1.
Proof. By Corollaries 4.6.6 and 4.6.7, it suffices to show that if Mv is
large enough (Le., v is large enough), then g(m, Mv) < O. By (4.6.5),
g(n, Mv) = f Xp 0 n = f ( Xp 0 n). c/>M' where for convenience we
M x v
v

QUATERNIONS OVER A LOCAL FIELD 65
write X instead of H(k p ), and cPMv denotes the characteristic function
of Mv. If cP is the characteristic function of lOp itself, then cPMv(X) =
cP( 7T' Vn(x)) and so, writing o/v(x) for Xp(x)<I>( 7T' V x )lIxIl 2 , we get that
d(i)
g(n, M) = f I/J,,(n(z)) dz x = f. f f I/J,,(n(i)) (_) dz x , where i
X"" ker n X/ker n n z
E X/ker n, of course. By n's surjectivity we may simplify this to
g(n, Mv) = f dyX 1 o/v(X) dx X = A 1 Xp(X)cP( 7T' Vx)lIxlldx, where A
ker n k x k x
 
= ,u(ker n). Next, it is a relatively straightforward exercise to show
that cP( 7T' Vx)lIxll = E IIx JLII{ cP( 7T'- JL x ) - cP( 7T'- JL-l x)} and therefore
J..L-v
g(n, Mv) = A E 117T' JLII{ f Xp(x) dx - f Xp(x) dx}. We now go
'Tr IL ,n 'Tr IL+ 1
JL  - v \.:...l
on to show that if m = min{ ,u with 7T' JL lOp c ker Xp} then g(n, Mv) =
- AII7T' 11 2m -1 (1 + 117T II) -1, which will do the trick. So, note that if ,u < m
then ?T JL Dp i ker Xj:P whence X/y) "* 1 for some y E ?T JL Dp. But then
f Xp(x) dx = f Xp(x + y) d(x + y) = Xp(y) f X/x) dx,
'TrILlO 'TrILlO 'TrILlO
I p  
whence f X/x) dx = O. On the other hand, if /-L > m then certainly
nIL
p
f Xp(x)dx= f Xp(?TJLx)d(?TJLx)=I!?TJLII (since ?TJLDp c kerx p ).
'TrILlO lO
p 
Consequently, with ker Xp :) 7T' JL lOp :) 7T' JL+ 1 lOp :) ..., the only integrals
remaining in g(n, Mv)'s expansion have ,u > m, allowing the simplif
cation g(n, Mv) = E 117T JLII<II7T' JLII -117T JL+ 111). Now all that remains
J..Lm
is to sum a geometric series.
.
The next step in the characterization of Y(Xp oq) on X=H(k p ) is
to establish that, up to equivalence (isometry) there exist only two
distinct relevant quadratic forms on X, so that y( Xp 0 q) takes on
exactly two values, 1 and -1. It then follows, almost automatically,
that y( Xp 0 q) can be identified with the quadratic Hilbert symbol for
suitably chosen q (cf. Proposition 4.7.6).
We begin by recalling certain facts from the algebraic theory of
quadratic forms; the reader is referred to [La73], [Sc85], and [DD93].
Two n-ary quadratic forms f, g, on k p' are equivalent, f -- g, if and
only if there is some C E GL(n, k p ) such that f(x) = g(Cx) for all
(column vectors) x E k;. Equivalently, if f (resp. g) has associated
matrix M f (resp. M g ), then f :: g entails M f = CtMgC. As expected,

66 WElL'S "ACTA" PAPER
we get an equivalence relation and any class of forms is canonically
identified with the data (V, B), a so-called quadratic space, where V is
an n-dimensional kp-vector space and B: V X V  kp is a symmetric
bilinear form with the property that any assignment of coordinates to
V produces a quadratic form in the given equivalence class (by
computing R(x, x)). Thus, since we only have to change bases,
equivalence of quadratic forms translates to similarity of the attend-
ant matrices. Furthermore, f(x)f(x + y) - f(x) - f(y) and
B(x, y) B(x, x) set up a bijective correspondence between quadratic
forms and bilinear forms so that we may use the notation (V, f)
instead of (V, B), for Bf(x, y) := f(x + y) - f(x) - f(y) and fB(X):=
B(x, x), wherever desired.
Next, two quadratic spaces, (V, B) and (V', B'), are said to be
isometric, written (V, B) :::. (V', R '), if there is an isomorphism 'T:
V  V' such that B '(x'T, y'T) = B(x, y) for all x, y E V.
Proposition 4.7.2. (V, B) :::. (V', B') if and only if fB  fB'.
Proof. An easy exercise.
.
The upshot of Proposition 4.7.2 is this (in the words of Lam [La73]):
" . . . we . . . often prefer to argue geometrically with quadratic spaces,
and then back freely with quadratic forms, viewing the above one-one
correspondence as an identification. . . ."
The Witt-Grothendieck group, which has the Witt group as a factor
group, is obtained by defining orthogonal addition to be the group law
on the set of all quadratic spaces or quadratic forms over kp.
..L : (( VI , B I) , (V 2 , B 2) ) I ) (VI EB V 2 , B I + B 2 )
n 11 n (4.7.1)
( E aijxix j , E 13kl X k X l)  E aijxix j + E 13 k1 X n +k Xn+l ,
where (B I + B 2 )«X I , x 2 ), (YI' Y2)) = B1(X 1 , Yl) + B 2 (X 2 , Y2). It is easy
to see that, modulo the equivalence (or isotropy) defined above, ..L
defines a semi-group structure on the set of quadratic forms or,
equivalently, quadratic spaces.
In keeping with the example of the construction of, say,
Grothendieck groups of representation modules (cf. [Ser77], it now
appears more or less natural to introduce Grothendieck's notion of a
formal difference so as to gain a full group structure for a certain
canonical factor group. For our situation, of quadratic spaces rather
than representation spaces, what results is the Witt-Grothendieck

QUATERNIONS OVER A LOCAL FIELD 67
'"
group, W(k), and this is precisely the algebraic object we need in
order to continue with our investigation of 1'( X 0 q). However, before
going on with this we make a few observations about the full algebraic
structure enjoyed by the set of quadratic spaces: we have in fact a
ring, the Witt-Grothendieck ring [La73].
As regards multiplicative structure, note first that, by definition, the
matrix of the bilinear form B: V X V  ktJ is given by (B(v i , Vj))i,j'
where V = ffi i kpv i ; then this matrix is given with respect to a basis for
V, but, since we are dealing with identification under isotropy we
regain well-definition by associating B with the conjugacy class of
(B(v i , Vj))i,j: By abuse of notation call this conjugacy class of (n X n - )
matrices, again, simply B. It is a standard result from multilinear
algebra that if B: Vx Vkp (resp. B': Vx V' kp) then the
Kronecker product B @ B' maps (V @ V') X (V @ V') to k p. In other
words, we have a binary operation given by (V, B) @ (V', B') =
(V  V', B  B '); this operation gives us the multiplicative structure
'"
on W(k), yielding the Witt-Grothendieck ring.
Specifically, then, the set of isometry classes of quadratic spaces
(bi-uniquely associated with quadratic forms) over kp is dealt the
structure of a semi-group under .1.. and is closed and associative
under  (and we have a multiplicative identity). Following
Grothendieck (cf. [Ser77], [La73]) we say that, qua pairs of quadratic
forms, (f, g)  (f', g') if and only if f .1.. g' = f' ...L g (again, modulo
equivalence). So, setting ([I' gl) + ([z, gz) = ([I J..[z, gl .1.. gz), we get
a semi-group structure on the pairs of quadratic forms (spaces)
divided out by the aforementioned equivalence. Moreover, since (f, g)
.1.. (g, f) = (f .1.. g, g .1.. f)  (0,0) because f...L g ...L 0 - 0 ...L g J..f, we get
inverses, anq so this factor semi-group is, in fact, a grop. This is, by
definition, W(k); with this additive structure being compatible with
@ as introduced above we again have a ring, the Witt-Grothendieck
ring of k. As we indicated above, we are exclusively concerned with
this object's structure, however.
Having in this way defined the Witt-Grothendieck group as the
group of isometry classes of quadratic spaces (or forms) over k, we
now go on to the Witt group itself by dividing out by the so-called
hyperbolic spaces. Indeed, the binary form x; - x and its equivalents
produce the hyperbolic spaces and we write Z 1) for the free group
'"
they generate. Obviously Z1) < W(k) and we can form
W(k):= W(k)/Z1),
(4.7.2)
k'p's Witt group.

68 WElL'S "ACTA" PAPER
The seminal structural result in this connection is the Witt decom-
position theorem (cf. [La73], [Sc85]), which generalizes Sylvester's Law
of Inertia:
Proposition 4.7.3. Any (V, B) decomposes as (V, B) = (VI' B I ) 1- (€a f))
1- (V 2 , B 2 ), where (VI' B I ) has Bl = 0 ("totally isotropic"), the direct
sum is finite, and fB 2 does not represent 0 (i.e., (V 2 , B 2 ) is "aniso-
tropic").
Proof. [La73].
.
Corollary 4.7.4. W( k p) = {totally isotropic forms} 1- {anisotropic
forms} .
Proof. Obvious from (4.7.2). .
It is understood in what follows that we work with representatives
of equivalence or isometry classes and we "add" modulo hyperbolic
planes.
Proposition 4.7.5. y 0 Xp is a character of W(k p ).
Proof. Says Weil: " ... comme ly(f)1 = 1, il ne depend pas du choix des
mesures de Haar dans les groupes qui intervennient dans sa definition,
puisque, lorsqu' on change ces measures, cela ne modife les formules
du [Corollary 4.4.5]... que par des facteurs reels> 0" (... "because
ly(f)1 = 1, it does not depend on the choices of Haar measures on the
groups which figure in its definition, seeing that when one changes
measures this does not modify the formulas in [Corollary 4.4.5]...
by other than real factors> 0.") The upshot is that if f ::: g then
y( Xp 0 f) = y( Dp 0 g), for if g = f 0 a, say, with a a suitable auto-
morphism, then, by Proposition 4.6.4,
'Y ( lq, 0 g ) = 'Y ( lO 0 loa ) = 1 () ( X  0 g) 1 1 / 2 f L , X pol 0 a
= I ()( lOp 0 g ) 1 1 / 2 1 Dp 0 f( ax) d( ax)
(L')U
I () ( lO 0 g ) 1 1 / 2
- 1 ()( X 0 1)1 1 / 2 'Y( X p 0 I),
and we need only observe that because f ::: g we have that Q( Xp 0 g)
= Q( Xp 0 f). Accordingly, y 0 X p is well defined on W(kl). Next, by

QUATERNIONS OVER A LOCAL FIELD 69
Corollaries 4.6.6 and 4.6.7,
0( -f)
Y( X p o( -f») = I 0( -f)1 -
J.L( L )Exp 0 f
1 0( -f)1
0(f)
10(f)1
1
=y(xpof)= y(xpof) '
and so it just remains for us to establish that y 0 Xp is multiplicative;
not surprisingly (in view of Proposition 4.6.4) this boils down to
Fubini's Theorem. Accordingly,
y( xp(f -Lg)) = 1 Q( Xp 0 (f -Lg))1 1 / 2
,( X p 0(1 1-g»)
= 1 Q( xp(f 1- g) )1 1 / 2 '=L'l mL'2 ( X p ° (f( Xl) + g( X 2 »)) dx 1 dx 2
= I Q( X p ° f)Q( X p ° g)1 1 / 2 {I {'2( X p ° g)(X2) dx 2 }( X p ° f)(x 1 ) dx 1
= (I Q( X p ° f)1 1 / 2 'lXP ° f) (I Q( X p ° g)11/2 ,/p ° g)
= Y( Xp 0 f)y( Xp 0 g).
(For the definitions of L', [;1 EB [;2( ::: L '), see Proposition 4.6.4.) This
completes the proof. .
We can now prove the central result of this section, modulo a few
facts from the classical theory of quadratic forms (for which the
historically-minded reader can actually go to Dirichlet-Dedekind
[DD93] itself).
( a,b )
Proposition 4.7.6. (Xp(x 2 - ay2 - bz 2 + abt 2 )) = - , the quadratic
Hilbert symbol on k p . p
Proof. The classical result referred to above holds that if k p =1= C,
char. (k) =1= 2, and as before, the coordinates of H(k p ) are x, y, z, t,
then there are exactly two classes of forms in these variables (with
square discriminant), namely, the class of the norm form, n, and the

70 WElL'S "ACTA" PAPER
class of the form xy + zt. But this implies that the form of our
proposition, x 2 - ay2 - bz 2 + abt 2 , is equivalent to n according as a is
the norm of an element in k p ( fb) (or, equivalently, b is a norm from
k'p({(i)). Thus, with both y( X/x 2 - ay2 - bz 2 + abt 2 )) and ( ab )
characters on the indicated Witt group it now follows from the
definition of ( a  b ) that these characters coincide. .
4.8 HILBERT RECIPROCITY
From the profusion of results obtained in the preceding sections we
single out two-Proposition 4.5.5, stating that under certain stipulated
hypotheses on f, ,,(f) = 1, and Proposition 4.7.6, singling out a
quadratic form in four variables, say, q, so that ,,( Xp 0 q) coincides
with the Hilbert symbol. The former result is a consequence of the
fact that if '0 and 'r are defined as in (4.5.11) and (4.5.12), respec-
tively, then they agree on a subgroup of B(G) (cf. Proposition 4.5.4),
which then, by virtue of 'r being a representation, entails that 7T:
N(cHE-l(G))  B(G) splits on this subgroup. Indeed, as per (4.3.11)
we get (by restriction) a corresponding subgroup of Sp(G)( < B(G)), :
which splits on Mp(G) E H 2 (Sp(G), C) as well as on Sp(G) E
H2( G), JLz).
Now, if a suitable global f can be constructed from the local data
Xp 0 q, so that we get a relation of the form 1'(f) = TIp1'( Xp 0 q), then
the two propositions cited above will conspire to give us 2-Hilbert
reciprocity immediately, that is, 1 = Q ( a  b ). In this section we do
exactly that, which is to say that we give Weil's argument that
n ,,( Xp 0 q) defines the right cohomological object on the adelization
p
of (4.3.9) for G = kp (if q E W( D k ) < W(k p )).
What exactly is the cohomological status of this Weil index? The
proof of Proposition 4.4.4 yields a characterization of ,,(f) in terms of
certain elements s, s' E N(cHE-l(kp))-we will take G = kp for the
time being-specifically, s = ,,(f)s'; however in [We64] Weil goes
on to prove something stronger. By means of brute force calcula-
tions in the group N(cHdj.(k'p)) one shows that if s = ( ( ), g ),
s' = ((: : ), g,), and ss' = (( :: :: ), g") in B(k'p)' then, with

HILBERT RECIPROCITY 71
f the second-degree character associated to e' e" e' -1, we have
ro(s)ro(s') = y(f)ro(ss'). (To be proper, in light of (4.5.8) and (4.5.11)
we should take s, s' E ll(k p ) < B(k p )). On the other hand, while
(4.5.11) gives '0 as a mapping from B(k p ) to N(dft:ij.(k p )), evidently a
multiplier representation in view of the preceding remark, we get
from (4.1.12) (or, rather from Proposition 4.1.3 and Corollary 4.1.14)
that r maps 5L(2, k p ) to GL( (kp)) as a multiplier representation.
It is the case, however, that (i) , can be taken to have GL(L 2 (k p )) as
range (cf. [JL70]); (ii) N(dft:ij.(k p ))  GL(L 2 (k p )) (as a result of (i));
and (iii) '0 can be restricted to a map from 5L(2, k p ) to M(dft:ij.(k p )).
Here (iii) follows directly from (4.3.13), with 5p(k p ) identified with
5L(2, k p ). But now (ii) and (iii) permit us to view ro: 5L(2, k p ) 
GL(L 2 (k p )) as a multiplier representation, so by means of the Stone-
Yon Neumann Theorem (to be precise, Proposition 4.1.3) it must be
that, and '0 define the same mapping. So, inasmuch as (4.1.12) gives
the characterization r( (71)'( (72) = a( (71' (72)r( (71' (72)' a E
H 2 (5L(2, k p ), C), it follows that a '" y(f), that is, the Weil index
may in fact be viewed as a 2-cocycle. So much for its cohomological
status.
Now for the adelization, n y( Xp 0 q), with G = k p' q E W( D k ).
I P
We view the k-adeles, k A , as the set of all valuation vectors (a.k.a.
adeles, of course) x = (xp)p,  ranging over the places of k, with
x p E Dp a.e. ; operating coordinate-wise, k A is a ring (containing k
via the obvious embedding). If, for each tJ, X p : kp  Cf is a charac-
ter, then the object X:= n Xp defines an ad61ic character, provided

it is well defined in the sense that for all x = (x) E k A , X),J(x p ) =
1 a.e. .
In [Ta50], for instance, we find that any X p E k, that is, any
additive character in kp, is of the form X p = X 0 Tp, where Tp is
multiplication by a constant and X is the canonical character
texp(27TiA(Trp(t))), where Trp is the trace from kp to the underly-
ing p-adic base field Qp (with Ip) and, for any a E Qp' A(a) = n/pll
or -{a}, according as p is or is not non-Archimedian; n, v are defined
via the requirement n - plla Ep 1J Z p ' and {a} is a's fractional part.
Right off we get the following:
Proposition 4.8.1. If t E tOp then x:(t) = 1.
Proof. t E Dp yields Trp(t) E Zp and so, in the non-Archimedian case,
o - pllTr p (t) E pllZp, while, in the Archimedian case, {Trp(t)} = 0; in
any case A(5p p (t)) = O. .

72 WElL'S "ACTA" PAPER
Corollary 4.8.2. n Xp 0 q is well defined.
p
Proof. Write x = (xp)p for a typical element. With q defined. over
tO k (  tO1) we certainly have that (X p 0 q)(x p ) = 1 for , seeing that
x p E <Op a.e. , where q(x p ) E <Op also. .
Corollary 4.8.3. n Y( X p 0 q) is well defined.
p
Proof. The proof is clear.
It now remains for us merely to demonstrate that n y( Xp 0 q)
p
defines an element in the group H 2 (B(k A ), C() as follows:
Proposition 4.8.4. n Y( Xp 0 q) = Y( X 0 q) E H 2 (B(k A ), C().
p
Proof. Corollary 4.8.3 guarantees that the infinite product exists;
Y( X p 0 q) is defined via Corollary 4.4.5, for instance, although Propo-
sition 4.6.2 (adelized) is more directly to the point. In any case, the
above identity is ultimately nothing else than an infinitary Fubini's
Theorem, as it were. Indeed, in an unpublished work we showed that
the theorem of Brge Jessen [HS65] can be invoked to yield th
desired result. However, following [We64] one can also give a deriva-
tion ab 000 and we do so now. Evidently (in view of the nature of the
integrals that occur in, for example, Corollary 4.4.5) it is enough to
demonstrate that if, for each place , <l>p E (H(kp)), with <l>p =
ljJH(kp)O a.e. , where ljJ denotes the characteristic function and
H(kp)O is the collection of those points of H(k p ) -- k with coordi-
nates in <Op' then with <P:= n <Pp, we have that 1. <P * ( X 0 q) =
P H(k)A
n 1. <p;( X p 0 q), where H(k)A (which is not itself an algebra in
p H(k)
general) is the adelization of the local data provided by the H(k p ).
So, let J-Lp denote Haar measure on H(k p ), normalized so as to
yield J-Lp (H(kp)O) = 1. It is a standard argument from topo-
logical measure theory that these data (for each place, ) yield a
product measure (or, rather, a projective measure), J-L, on H(k)A'
and it is in this sense that a priori, we get a formal agreement,
1. <P * ( X 0 q) = n 1. <Pp * ( X p 0 q). Accordingly, it only remains
H(k)A p H(k p )
for us to prove that n 1. <Pp * ( X p 0 q) is finite. However, by (the
p H(k,)
proof of) Proposition 4.6.4 we have that a.e. , <l>p * ( Xp 0 q) =

THE STONE-VON NEUMANN THEOREM REVISITED 73
4>H(k p )O *( X p 0 q) = JL p (H(k p )O)4>H(k p )O( X p 0 q). Here we need to
observe that since q: <0 p 4 - <0 p ' H(kp)O is "small enough" in the
sense of Lemma 4.6.3, and also that, because kp-spaces and their
maximal lOp -submodules are self-dual, if (} is as in (4.6.2), then
(H(kp)O)P-t = H(kp)o. Accordingly, for such ,f. <I>p *( Xp 0 q) =
H(k p )
J.Lp(H(k p)O) f. 4>H(k )o( Xp 0 q) = { J.L p H(k p )O}2, and this reduces to 1
H(k,) p
for a.e. . In other words, a.e. factor in the infinite product reduces to
1 (depending, of course, on the adelic function <f) = n <f)p). The
infinite product therefore converges. p.
At last, then, we obtain the following:
Proposition 4.8.5 (the 2-Hilbert reciprocity law):
Q ( ab ) = 1.
Proof. For q the quadratic form x 2 - ay2 - bz 2 + abt 2 we have, from
Proposition 4.7.6, that ')'( Xp 0 q) = ( ab ). From Proposition 4.8.4,
( a, b )
n Y( X p 0 q) = y( X 0 q), that is, n - = y( X 0 q). Finally, from
P P V
Proposition 4.5.5, with f=xoq, G=H(k)A and r= nH(kp)O
p
(closed in H(k)A)' it follows that y( X 0 q) = 1. .
4.9 THE STONE-VON NEUMANN THEOREM REVISITED
We now return to the constructions given in  4.1-4.5 and discuss, in
the context of a general LCA group, G, how the two approaches to
the Weil representation (via the Stone-Von Neumann Theorem (Pro-
position 4.1.1) and by means of Weil's original constructs) are, in some
sense, complements. In most of the recent literature on the Weil
representation the point of departure for discussing metaplectic groups
is either the explicit Kubota formalism (which we will address in
Chapter 5, below) or the Stone-Von Neumann Theorem for Heisen-
berg groups, applied with respect to the so-called Schr6dinger model.
It is relatively rare that a motivation for this approach is given along
the lines of fitting the material into Weil's unitary representation
theoretic context; in this connection Gelbart [Ge93] and Igusa [Ig72]
can be mentioned, but very few others.

74 WElL'S "ACTA" PAPER
Thus, in order better to motivate the (quicker) approach to the
theory of metaplectic groups, specifically double covers (cf.  4.10),
centered on applying the Stone-Yon Neumann Theorem, we proceed
to establish that there is something of a dictionary correspondence
between the two routes described above. We shall also see in Chapter
8 that Kubota's approach fits perfectly into this scheme.
As is apparent from  4.1 and 4.2 it all starts with N(dfgi(G)) as
per (4.2.5). Weil's presentation of N(dfgi(G)) consists in interpreting
any (w, t) E dfgi(G) as the linear operator tf11.(w) (cf. 4.2.1) in the
sense that
L2(G) L2(G)
<I>(x) t<l>(x + u)(x, u*),
(4.9.1)
where, as always, w = (u, u*). In the context of the Schr6dinger model
(see also, e.g., [Ge93] or [Ig72]) of the Heisenberg group, the Stone-Von
Neumann Theorem, generalizing Proposition 4.1.1 all but verbatim to
G in place of kp, yields that for X a fixed character on G there is, up
to unitary equivalence, a unique smooth irreducible representation
r x : (w,t)[<I>(x)t<l>(x+u)<x,u*)],
(4.9.2)
having X as central character.
So,
r x : dfgi( G)  GL( L2( G))
(w, t)tru (w).
(4.9.3)
With B( G) as in  4.3, realized in Aut(dfgi( G)) according to
(4.3.1), and with 7T the canonical projection (4.2.8), we quickly dis-
cover the salient point:
N(dfd(G)) Aut(dfEi(G)) =Aut(rAdfEi(G))) =B(G)
s(a,g),
where B(G) is, of course, identified with (G E9 G*) X Sp(G). That is,
via Proposition 4.2.2,
sf ru ( w )s -1 = tg( w) ru (w (T).

THE STONE-VON NEUMANN THEOREM REVISITED 75
Group actions next enter the picture, as follows.
B(G): dft:ij.(G)
(0-, g): (W, t)(WO", t)g(W)
(4.9.4)
or, equivalently,
B(G): rAdfEi(G))
( 0- , g): t CU (W) tg( W )U( w.O").
And, "canceling" G E9 G* from B(G), as it were (cf. (4.3.10) below),
(4.9.5)
Sp(G): df£i(G) :::: rAdfEi(G))
0-: (w,t)tCU(wO").
(4.9.6)
Indeed (4.9.6) is tantamount to the formulation chosen most often in
recent presentations of the Weil representation, for instance, in [Ge93]
and (Ig72]-it is manifest that (4.9.4), (4.9.5), and (4.9.6) ultimately
amount to the same thing. Suffice it to say for now that the effect of
"canceling" G E9 G* from B(G) - (G E9 G*) X Sp(G) is captured by
the fact that in (4.3.14) the commutativity of the diagram entails the
cohomological equivalence between the rows. In a sense, to be made
precise in  4.10, the construction of Mp(G) is quite the same thing as
the construction of N(dft:ij.(G)) and, at the same time, the group
action (4.9.6) is quite the same thing as the group action (4.9.5). In
order to strip the theory of the Weil representation of some of its
complexity and to effect a transition froni N(dft:ij,(G)) to Mp(G),
anticipating  4.10, we now proceed to exploit the essential unique-
ness part of the Stone-Von,Neumann Theorem as it pertains to the
symplectic group, Sp(G). A propos, as is well known, in case G =
kp, Sp(G)  SL(2, k p ) (cf. (4.1.8)); we say much more about this in the
next section.
By means of (4.9.6) we get an action of Sp(G) on r x in that
0- E Sp( G) maps r x to r;- defined as follows:
r;: dfEi(G)  GL(L2(G)) 
t\.
( w, t)  [ <1>( x) t<l>( x + u 0")< x, u xu > ] ,
(4.9.7)
where (uO", u*O") =: (u, u*)O". Evidently, then, in light of (4.9.1) and
(4.9.6) and the uniqueness part of the Stone-Yon Neumann theorem,

76 WElL'S "ACTA" PAPER
r;(w, t) = t'U(W U ) must be unitary equivalent to rx(w, t) = tfl1(w),
whence, in particular, 'U (w u) and 'U (w) must be conjugates in the
unitary group. In other words, given (J' there exists s, unitary, with
stCZ1(W)S-l = t'U(W U ).
As regards Proposition 4.2.2, where (0", g) E B( G) pulls back (via
7T) to S EN(dfE.Lj.(G)) with st'U(W)S-l = tg(w)'U(W U ), note that, of
course, Sp( G) embeds into B( G) by means of the explicit monomor-
phism (J't--+( (J' , 1), so taking g = 1 in the latter identity in order to get
the former is entirely licit.
Next, writing s = r*(O"), expanding on the material in  4.1 in the
obvious way, we find that
r; (w, t) = r*( (J' )r x ( w, t)r*( (J') -1,
(4.9.8)
just as in (4.1.10). Mutatis mutandis the map r*, as a projective
representation, yields-by means of Schur's Lemma-a multiplier
representation r, with 2-cocycle {3, so that (cf. (4.1.12))
r( O"l)r( (J'2) = f3( (J'1' (J'z)r( (J'1 (J'z),
(4.9.9)
and the sequence
1 C Sp(G) x C Sp(G)  1
{3
(4.9.10) I
is exact. In view of the relatively transparent fact that «(J' , g) E B( G) if
and only if (J' E Sp(G) (Aufgabe fUr den Leser: use the definition of the
symplectic group relative to the underlying skew-symmetric pairing
from G E9 G* into kp as per (4.1.1) and (4.6.1) and realize that B(G)
is, in fact, realized as the semidirect product (G E9 G*) X Sp( G) via
(4.2.9)), it now follows immediately that (4.5.10) is valid.
Furthermore, note that since r, as a multiplier representation, maps
0" to the coset sC, where s is characterized by stCU(W)S-l = t'U(W U ),
it also follows trivially that if p is the restriction of 7T to Mp( G) in
(4.3.10) then ker p :::: C. We have the pure counterpart to Proposition
4.3.11 which, in the context of 7T itself, Weil derived essentially by
brute force. It is in this sense, then, that we propose that Weil's
approach and the approach through the Stone-Von Neumann Theo-
rem stand together as complements if not mutual converses.
Now, at last, we proceed to the most dramatic algebraic simplifica-
tion of (4.3.12).

THE DOUBLE COVER OF THE SYMPLECTIC GROUP 77
4.10 THE DOUBLE COVER OF THE SYMPLECTIC GROUP
Realize Mp(G) as Sp(G) x C (as in (4.9.7)), and consider an epimor-
(3
phism cp: Mp(G)  C with the property that cp(id, t) = t n for some
fixed n. If such a cp exists then, by the first isomorphism theorem, we
can also realize Mp( G) as ker cpC , so that we get an identification
Mp(G)/C  ker cpC /C -.; Sp(G).
By means of the second isomorphism theorem we then obtain
Sp( G) -.; ker cp/ker cp n C .
(4.10.1)
On the other hand, by virtue of the definition of cp we may identify
ker cp n C with J.L n , being, as always, the nth roots of 1. Therefore we
have proved the following:
Proposition 4.10.1. If the map cp: Mp(G)  C, as above, exists, then
1  J.L n  ker cp  Sp ( G)  1
( 4.10.2)
.
is exact.
The remainder of this section is devoted to proving that with G a
finite-dimensional kp-space, if kp = C (i.e.,  is archimedian) then
such a cp exists with n = 1, while if  is non-archimedian there exists
such a cp with n = 2. In particular, for  non-archimedian (4.10.2)
defines a double cover, we write sp(G) for ker cp, and (4.3.10) expands
to the following commutative diagram:
1
) J.Lz
1
-
) Sp( G)
1
) Mp(G)
1
) N( df£i( G»)
) Sp( G)
) 1
1
) c x
1
1
) C X
1
) Sp( G)
1
) B(G)
) 1 (4.10.3)
) 1.
The construction of cp proceeds from a few results from the theory
of groups for which the reader may consult [We64]. The upshot is
simply that Sp( G) is, in fact, generated by elements of the form
t o (/1) d( 'Y )t o (/2) as in Proposition (4.5.3).

78 WElL'S "ACTA" PAPER
Proposition 4.10.2. Sp(G) = < fl(G), for fl(G) as in (4.5.10).
Corollary 4.10.3. cp: Mp(G) --+ Cr is determined by the values of
'P 0 ro on O(G), for ro as in (4.5.11).
Proof. The proof is trivial.
.
Proposition 4.10.4. If  is non-Archimedian (resp. archimedian), then
we get a homomorphism cp(id, t) = t 2 (resp. t) and, accordingly, Mp(G)
'. defines a nonsplit 2-fold central extension of Sp(G) (resp. Mp(G)
defines a split extension of Sp( G)).
Proof. As we argued in  4.8 (and, to some extent in the proof of
Proposition 4.5.5), the 2-cocycle defining Mp(G) can be taken to be
cohomologous to the Weil index whence, with s, s' E O(G), we get,
for some quadratic form q, that r(s)r(s') = y( Xp 0 q)r(ss'). As we
already stated in  4.7 (cf. Proposition 4.7.6), the classical theory of
quadratic forms provides that over kp non-archimedian (resp. archi-
median-i.e., kp = C) there exists exactly two classes (resp. there
exists exactly one class) of forms whence, via Proposition 4.7.5,
Y( Xp 0 q) = + 1 or 1, according as k'p is or is not non-archimedian.
Now, with cp as above, that is, cp(id, t) = t n for some fixed n, obtain
immediately that cp 0 r(s) cp 0 r(s') = y( Xp 0 q )ncp 0 r(ss '), so that
cpor(s)cpor(s') =( + l)ncp o r(ss') or cpor(ss') according as kp is or is I
not non-archimedian. However, with cp and rln(G) being homomor-
phisms, we had better arrange that ( + l)n = 1 in the non-archimedian
case, and so we may take n = 2. In the archimedian case n = 1
obviously suffices. .
With Proposition 4.10.4 we obtain that (4.10.3) is valid for G =:. k,
say, and, in particular we get that
-
1 ) JL2 ) SL(2, kp) ) SL(2, kp) ) 1
1 1
1 ) C x ) Mp( kp) ) SL(2,k p ) ) 1
1
1 1
1 ) C x ) N( dld(kp)) ) B( kp) ) 1.
1
(4.10.4)

END-NOTES TO CHAPTER 4 79
4.11 END-NOTES TO CHAPTER 4
The principal result of this long chapter is Proposition 4.8.5, of course:
the 2-Hilbert reciprocity law. Given our discussion in Chapter 2 we
know that this is quite the same as (1.3.7), even as the contexts are
dramatically different. While Hecke's work shows no sign of local
analysis whatsoever, Weil's approach is almost exclusively locaL In
fact, Weil's method is an exemplar of the according modern approach
which in recent decades, indeed throughout this ce!ltury, has proved
so fruitful: the hard work is done locally, that is -adically, and then
adelization is brought to bear on the matter in order to get results
which have global meaning. Such global (or adelic) formulations
correspond to classical facts in Number Theory.
But beyond their utilitarian benefits, local methods afford a pro-
found insight into, for lack of a better word, the deep structure of
certain parts of Number Theory, and so it is with Weil's development
of quadratic reciprocity. The facile observation that it is ultimately all
about iJ--functions is true as far as it goes, but one must look beyond
them, beyond their beautiful and important functional equations, to
the representation theory of the isotropy groups attached to the
bilinear forms whose quadratic forms define these it-functions. If this
tactic at first appears a bit byzantine, it quickly becomes very reason-
able since, after all, functional equations are fundamentally concerned
with invariance under a transformation, which is to sayan element of
a distinguished group. In Weil's approach this distinguished group is
already mentioned in the title of his paper: the focus falls on "certain
groups of unitary operators." And the representation theoretic aspect
of this matter is explicitly evoked by the requirement that these
operators be unitary. We are located squarely within Fourier analysis
and harmonic analysis. Without doubt this is the right perspective.
It is particularly telling, of course, that in doing local analysis on the
aforementioned isotropy groups which one is led to, the local Fourier
analysis done by Weil is in fact adelized into what amounts to
Hecke's iJ--functional equation (1.1.4) in Proposition 4.5.5, a candidate
for the deepest result of Chapter 4. The point is that the global Weil
index, 'Y, for now, trivialized to 1 on a certain isotropy subgroup
because the mapping 7T (as a covering map) splits on that group
since the Weil' representation becomes a homomorphism there (cf.
(4.5.12) and Proposition 4.5.4). Let us say it differently, in more
modern language: the global Weil index restricts to 1 on the so-called
subgroup of rational points of the adelic symplectic group because the

80 WElL'S "ACTA" PAPER
double cover of the symplectic group splits on this subgroup; this
splitting follows from the fact that the Weil (projective) representation
restricts to a homomorphism on the rational points.
In terms of (4.10.3), which will figure more and more prominently
from here on out, we have simply that it is amenable to adelization
(i.e., G is allowed to be ), and then Weil's arguments translated to
the top row yield that SL(2, k A ) splits on SL(2, k) < SL(2, k A ) as a
consequence of nothing less than Proposition 4.5.4, mutatis mutandis.
Presently we turn to Kubota's reformation of these lines of argument
in more algebraic terms, with SL(2) taking central stage. But it
behooves us now to inquire after a few of the more subtle aspects of
the proof of Proposition 4.5.4.
Upon analyzing this proof we note that the principal tactical fea-
tures are that the Weil representation, restricted as per (4.5.12), is
identifiable with a discrete group action on a space of generalized
it-functions and, more importantly, that the required invariance under
the action of ((  i), f) comes down to the fact that (4.4.7) and
(4.5.12) conspire to recast this as the familiar fact that even a general-
ized it-function is automorphic of half-integral weight. But this fact
about it-functions is, of course, the most dramatic dividend Fourier
analysis pays in the setting of automorphic functions: it follows from
Poisson summation.
It is for this reason that in the Preface and Introduction we have
seen fit to characterize Poisson summation as the key to the general-
ization problem. There is no doubt that it is Poisson summation that
makes adelic splitting on the rational points come about. In  5.3 we
return to this theme in order to fit this critical part of the analytic
proof of quadratic reciprocity in a more transparent context by bring-
ing in the Weil E)-functional. Poisson summation then comes in
explicitly to yield this functional's invariance under the action of the
rational points via the adelic Weil representation, and this is essen-
tially the same as Proposition 4.5.4.
So, by way of a synopsis of what has transpired so far, we have that
whether it be implicit or explicit, the Stone-Von Neumann Theorem
applied to Weil's group of unitary loal operators gives rise to a
projective representation of this group. And one should always view
this group as canonically attached to an underlying quadratic form: it
is the isotropy group of the associated bilinear form. With the Weil
representation being projective, one gets gratis a covering of the group
projectively represented, and Weil shows (cf 9 4.10) that this cover has
order 2. Obtain therefore that the associated 2-cocycle belongs to

END-NOTES TO CHAPTER 4 81
cohomology with coefficients in J.Lz. (This will be critical in Chapter 5.)
It is also the case that, in a canonical fashion, the projectivity of the
Weil representation gives a cohomological meaning to the local Weil
index Yp(:= Y( Xp 0 q) in  4.8). Since Yp gives the 2-Hilbert symbol
once the right assignments have been made, the goal is to prove a
product formula: n Yp = 1.
p
Given the cohomological interpretation of yp' this product formula
(and therefore 2-Hilbert reciprocity) will follow if we show that the
adelization of the double symplectic cover mentioned above is split on
the rational points, since." the right assignments" invoke nothing worse
than applying Proposition 4.5.5, that is, restricting attention to the
rational points. This feature of the argument becomes incomparably
clearer in Kubota's formalism, in the next chapter.
Finally, the manoeuvre proving that the double symplectic cover is
split on the rational points is ultimately an application of a somewhat
subtle form of Poisson summation, as we discussed at length above.

Kubota and Cohomology
In light of what transpired in Chapter 4 it is not inaccurate to
characterize Weil's derivation of quadratic reciprocity as a conse-
quence of the following three facts:
First, N(cHE.l(G)) is a central extension of B(G) (containing Sp(G)
as a semi-direct factor) by the unit circle; locally the map 'r (cf. I
(4.5.12)) yields a projective representation of Sp(G) while globally-
adelically-the map '0 (cf: (4.5.11)) brings about a splitting of the
natural projection 7To on a distinguished subset on which '0 agrees
with , r.
Second, as discussed in 9 4.9, when applied to G =X/k p , a finite-
dimensional kp-space, this line of argument takes us to the lineariza-
tion of the theory surrounding the projection mapping 7T, and then to
the important fact ( 4.10) that the symplectic group admits an
essentially unique double cover, realized as a subgroup of N(cHE.l(G)).
Third, d(.ta about 7T, which is to say information about the coho-
mology groups H 2 (B(X), C) and H 2 (Sp(X), C), lead to a non-split
element of H 2 (Sp(X), J.L2)' as stipulated in Proposition 4.10.4.
So, reversing the field, as it were, it ought to be possible to start out
with a nontrivial element of the group H 2 (Sp(X), J.L2) and proceed
from there to that part of Weil's formalism where 2-Hilbert reciprocity
is proved as a consequence of an adelic splitting property ( 4.5).
Unquestionably, however, details of Weil's derivation of reciprocity
in this way are somewhat cumbersome and opaque, at least in the
sense that the all-important result, Proposition 4.5.5, is not as trans-
parent as one would like. With the adelic splitting of 7T, via the partial
82

KUBOTA AND COHOMOLOGY 83
coincidence of '0 and 'r for suitable r, and this being the single most
important ingredient in the analytic proof of 2-Hilbert reciprocity, it
should certainly serve us well if this property might be lifted to the
level of Sp(X A ), and if everything might be phrased in more explicit
terms by virtue of the accessibility of H 2 (Sp(X A ), J.Lz). Happily, this is
precisely what Tomio Kubota achieved in [Ku67], only a few years
after the appearance of Weil's "Acta" paper.
While Weil's derivation of 2-Hilbert reciprocity in  4.8 proceeds
from the assignment X = H(k)  kp, we may now simplify matters
even further by setting X = k p' no more, no less: as we shall see, it is
indeed possible to derive 2-Hilbert reciprocity in this leanest of
environments (and 9 4.1, together with 9 4.10, can be taken as a
prelude). Under these circumstances we find that Sp(k p ) can be
identified with SL(2, k p ) (an easy exercise, actually), so that Sp(k A ) is
also identified with SL(2, k A ). We then obtain for free that (4.10.2),
with n = 2, is nothing else than
1  J.L2 sr(2, kp)  SL(2, kp)  1,
where we write SL (2, k p ) for ker cp in Corollary 4.10.1. In other words,
Weil's (local) formalism regarding 7T leads, for G = X = k p' to a
nonsplit element of H 2 (SL(2, k p ), J.Lz) if kp =1= C, while we get a split
exact sequence if k p = C.
Next, we invoke a marvelous result due to C. C. Moore [M068]: if
kp =1= C then H 2 (SL(2, k p ), J.Lz) has only two elements (otherwise it is a
singleton), that is, any two non-split elements of H 2 ( SL(2 , k p ), J.Lz) are
cohomologous. As a result, any presentation of SL(2, k p ) (resp.
SL (2, k A )) with its group law given in terms of a nontrivial local (resp.
adelic) 2-cocycle, is entirely equivalent to Weil's. Therefore, there is
only one metaplectic group of order 2. It should accordingly be
possible to go directly from group extensions of SL(2, k p ) and
SL(2, k A ), defined by explicit 2-cocycles, to 2-Hilbert reciprocity.
This explicit presentation of a non-split 2-cocycle in
H 2 (SL(2, k p ), J.L2)' built up from 2-Hilbert symbols, constitutes the
modus vivendi of [Ku67].
After discussing briefly how 2-Hilbert reciprocity is gotten from
Weil's formalism applied to G = X = kp (and then k A ), we present
Kubota's beautiful result in detail and explore how Weil's approach is
related to Kubota's on a more fundamental level.
One final point: as mentioned above, as far as Weil's derivation of
2-Hilbert reciprocity is concerned, the single most important tool is
the aforementioned partial coincidence of '0 and 'r, resulting in
Proposition 4.5.5. It is no exaggeration to say that the analytic proof of

84 KUBOTA AND COHOMOLOGY
quadratic reciprocity ultimately devolves to nothing else than proving
that the adelic symplectic cover of degree 2 splits in the according
fashion. But this fact is amenable to a far more transparent approach
than we presented in  4.5: there is no real difference between 7T'S
splitting on fl( G, f), as per Proposition 4.5.4, and the splitting of the
adelization of Kubota's 2-cocycle on the subgroup of rational points,
SL(2, k). Rather than mimicking Weil's approach through auto-
morphic representations and transfer of structure as regards repre-
sentation spaces (cf. Proposition 4.5.2), we will now derive this
all-important adelic splitting properly by exploiting the behavior of the
so-called Weil @-functional as hinted at at the end of the preceding
chapter. This is the standard approach taken by most contemporary
authors on this subject (e.g., [Ge76]).
5.1 WElL REVISITED
We know from the first part of Proposition 4.8.4 that if G = H(k) @ k,
adelizing to H(k) @ k A , and if X:= nXp is an adelic character
p
presented in terms of the local factors X\J' additive characters on kp,
then, for f a suitable quadratic form in the right number of variables, I
n y( X 0 f) = y( X 0 f). Of course, all this goes through mutatis
p
mutandis for G = k (or k A ), in which case f involves a single
variable. We can phrase things thus (see [Ge76]): for a E k; define
Xa: kp  Cr by x  Xp(ax) and take f to be the form x  x 2 . An
elementary but nontrivial calculation (cf. [Ge76], [Ra93]) yields that if
we define y( a) to be the quotient y( Xa 0 f) I y( Xp 0 f), then
y ( ab ) ( a, b )
y(a)y(b) = 1) · (5.1.1)
Since Proposition 4.8.4 is obviously valid for k and k A in place of
H(k) @ kp and H(k) @ k A , because Xp(ax) is as good a generator of
kp's character group as X is ([Ta50]), we obtain that n y(a)
p
= n y(b) = n y(ab) = 1. In light of (5.1.1) this readily yields 2-
p p
Hilbert reciprocity.
As we indicated above, from this point onward we concern our-
selves with the one-dimensional case, G = k (and then k A ), or, what
amounts to the same thing, with Weil's formulation as applied to
1  J.L2  SP(k p )  Sp(kp)  1
(5.1.2)

WElL REVISITED 85
and
-
1  ILz Sp(kA) Sp(kA)  1.
(5.1.3)
Furthermore, from generalities concerning the low-dimensional coho-
mology of groups ([EMacL86], [CR81]) we also know that there is a
bijective correspondence between exact sequences
ITEG1
(5.1.4)
attached to the pre-given data (G, T), G any (topological) group and T
a trivial G-module, and 2-cocycles in HZ(G, T), factoring out by
cohomological equivalence. Indeed, HZ(G, T) can be viewed simply as
the set of all sequences (5.1.4) equipped with the Baer product as
group law. As in Chapter 4 we have that any E E HZ(G, T) corre-
sponds to E = G x T, by definition the set G X T with the (twisted)
€
group law
(gl' t 1 )(gz, t z ) = (gl gz, t 1 t z E(gl' gz)).
(5.1.5)
Usng this sort of convention the diagrams (4.10.3) and (4.10.4),
I
capturing much of the Weil formalism, can be fleshed out in the
following way:
1
) ILz
) 5P(k p )
) Sp(k p )
) 1
Sp( kp) x ILz
'Yp
1 ) C x ) Mp( kp) ) Sp ( k p ) ) 1
1
(5.1.6)
Sp( kp) x ILz
(3p
1 ) C x ) N( df£l( kp) ) ) B( kp) ) 1
1
B(kp) x C.
Qp

86 KUBOTA AND COHOMOLOGY
5.2 KUBOTA'S COCYCLE
Kubota characterizes the substance of [Ku67] as follows. "We shall
obtain a topological covering of SL(2, F) by proving in an elementary
way that an expression containing Hilbert's symbols is actually a factor
set of SL(2, F)." Here Kubota's F is a non-archimedean local field
containing the nth roots of unity, with the Hilbert symbols mentioned
being m-Hilbert symbols. So Kubota actually constructs a factor set,
that is, a 2-cocycle, taking values in the group of nth roots of unity,
J.L m , which implies that for our purposes we need only restrict atten-
tion to the case m = 2. The one exception to this convention is our
Chapter 7, for which we need to take note of the fact that [Ku67]
yields a nontrivial m-fold cover of SL(2, F) for any m > 2.
When m = 2 the above requirement on F is vacuous, of course, so
what follows applies to any F = kp,  < 00.
Write a . for the factor set or 2-cocycle we are concerned with, that
is, a p E H 2 (SL(2, k p ), J.L2). By forming a A := nap one obtains an
p
element of H 2 (SL(2, k A ), J.L2); in this connection consult also Gelbart
[Ge76] and Kazhdan-.Patterson [KP83]. (It should be noted that we
still need to take care of the archimedean cases.) Identifying the
symplectic group of Weil (and Siegel, actually) with SL(2), we need
only invoke the result of [M068] already mentioned in  4.1 to get that
the adelization of Weil's double symplectic cover coincides with the
cover of SL(2, k A ) defined by a A . By taking this route, that is, by going
with Kubota's algebraic construction, we can in effect dispense with
 4.10. And, as we indicated in  5.1, the goal now is to get to the
exact sequences (5.1.2) and (5.1.3) by Kubota's quicker algebraic route.
First, define
x: SL(2, kp)kp
(5.2.1)
(]"= ( a f3 )  ( y,
y 8 8,
if y =/:= 0
if y = o.
Then, with () the 2-Hilbert symbol as before, set

a p : SL(2,kp) XSL(2,k p ) -+J.L2
( X(lT),X(T) )
((T,T)

(5.2.2)
X(T)
x ( IT) , X ( lTT )

Kubota's main result is this:

KUBOTA'SCOCYCLE 87
Proposition 5.2.1. For all 0-, T, Q E SL(2, k p),
a p ( 0- , T ) a p ( o-T , Q) = a p ( 0- , TQ ) a p ( T , Q ) .
(5.2.3)
And, therefore, a p E H 2 (SL(2, k p ), J.L2).
Proof. First, some conventions. If a = (  ), then '}'( a ) = '}', that
is, Y( ) just picks off the subdiagonal entry. Also, in the calculations
which follow we often write r- I when r E J.L2' making the exponent,
- 1, superfluous. But keeping this exponent in place makes the
various tactics easier to follow; note, too, that these go through also
when m > 2, when we cannot dispense with the exponent. Finally, we
take for granted a number of standard elementary properties of the
2-Hilbert symbol for which the reader is referred to Hasse's classic
[Ha80].
If b p ( 0-, T, Q) : = a p ( 0-, T) a p ( o-T, Q) a p ( 0-, TQ) - I a p ( T, Q) -1, it all
comes down to roving that b./ a , T, (J) = 1 for all a, T, (J E SL(2, k p)'
Say, E = (   ), through this proof. Note that if J..L is close to °
(tJ-dically) then E is close to (, ). Write J..L '" 0, E '" (  ) for
"J..L is sufficiently close to 0,"" E is sufficiently close to (  )." Since
'}'( ES) = 0  J..L = - '}' (if a *- 0), there is at most one solution to
a
Y( ED) = 0; likewise for Y( DE) = O. Bearing this in mind, we claim
that there exist E, E I (both of the indicated form (  )) so
that bp(Eo-,T,Q)=bp(o-,T,Q), with Y(ED)*O, Y(EDT) *0, and
bp(Eo-, QE') = bp(Eo-, T, Q), with Y(QE') * 0, Y(TQE') * 0,
Y( Eo-TQE') * o.
Proof of the claim: It clearly suffices to show that if E is as above
and J.L"'O then bp(Eo-,T,Q)=bp(a,T,QE)=bp(o-,T,Q). In other
words, if E"'( ) then we have ap(E,a)ap(Ea,T)=
ape E, o-T )a p ( 0-, T) and ape 0-, T )a p ( o-T, E) = ape 0-, TE )a p ( T, E). SO, re-
place T by 0- -IT to get that the first assertion is equivalent to
ap(E, a)ap(Ea, a-IT) = ap(E, T)ap(a, a-IT ( )' !lTt e hav ) e
( x( Eo-), x( o--IT) ) - x(eu) ,x( ET)
a (E 0- 0-- IT ) a (0- 0- - IT ) - 1 =
p' p'  

88 KUBOTA AND COHOMOLOGY
( x( U-1T) ] -1
X(U),X(U- 1 T) -1 - x(u) ,X(T),
(  )  ' a product of Hilbert sym-
bols via (5.2.2) and (5.2.1). We now distinguish four cases. If 1'( U) =1= 0,
Y(T)*O then, as /-LO and so E( ) (-adically), X(EU)
x( a-) and x( ET)  x( T) so that, by the continuity of the Hilbert
symbol, a(Ea-, a-- 1 T)a p (a-, a-- 1 T)-1  1. If y(a-) =1= 0, yeT) = 0
( JL,X(U) )
we get that a p ( EU, u- 1 T)a p( U, U- 1 T) -1 =  if /-L"'" O. ;
( JL, X(a-) )
Y( U ) = 0, Y( T) * 0 then a p ( EU, U- 1 T )a p ( u, U- 1 T )-1 = 
if JL  O. An d if y ( a-) = y ( T) = 0, a p ( E a-, a- - IT ) a p ( a-, a- - IT ) - I
( JL, X(U)-I X (T) )
=  if /-L"'" O. Now a straightforward calculation with
Hilbert symbols (using their continuity) gives that if E"'" (  ), then
1,
if y ( a- ) =1= 0
ap(E, a-) =ap(u, E) =
( x ( E ) ' x ( u) ) ,
(5.2.4)
if y ( a- ) = 0,
and this is enough to give the claim. For example, in the first case, te
assertion that a( E, U )a p ( Ea-, a-- 1 T) = a p ( E, T )a p ( u, a-- 1 T) simplifies
to the relation ap(E, T)ap(E, U)-1 = 1, and with y(u) =1= 0, y(r) =1= 0 we
get a p (E,a-)=l,a p (E,T)=l. And in the third case, ?e indicated
assertion simplifies to a p ( E, T )a\J( E, U ) -1 = ( /-L, X( U » ) ; with Y( U )
( X(E),X(a-) ) -I
= 0, Y( T) * 0 we have a pC E, U ) -1 =  and a \J ( E, T ) = 1,
by (5.2.4), and we need only note that JL = x( E) in this case. The
second and fourth cases proceed in the same way. This gives the proof
of the claim ape E, a- )a1( Ea-, a-- 1 T) = a p ( E, T )a p ( a-, iT-IT) and so of the
equivalent assertion a p ( E, U )a p ( Ea-, T) = a p ( E, a-T )a p ( a-, T), if E is
sufficiently close to ( ) -adically. One shows by a similar argu-
ment that if, again, E"'" ( ), then a\J(u, r)a\J(UT, E) =
apea-, TE)ap(T, E).
But we are now in a position to show that b p ( Ea-, T, Q) =
b p ( a-, T, QE) = b p ( a-, T, Q). For instance, by virtue of a p ( E, a- )a p ( Ea-, T) =

KUBOTA'SCOCYCLE 89
ap(E, O"T)ap(O",T)weobtain that bp(EO",T,Q) =ap(EO",T)ap(EO"T,Q) X
a p ( EO", TQ) - 1 a p CT, Q) - 1 = a p ( E, 0" ) - 1 a p ( E, O"T ) a p ( 0" , T ) a p ( E, O"T ) - 1 X
a p ( E, UTQ) a p ( UT, Q) a p ( E, U) a p ( E, UTQ) - I a p ( U, TQ) - I a p ( T, Q) - I =
a p ( E, U) - I a p ( E, U) a p ( E, UT) a p ( E, O"T) - I a p ( E, UTQ) a p ( E, UTQ) - I X
ap(u, T)ap(O"T, Q)ap(O", TQ)-lap(T, Q)-I = bp(u, T, Q). The equality
b p ( 0", T, QE) = b p ( 0", T, Q) is handled in the same way. The claim is
proved.
As a consequence of this claim, which should be seen as license to
perturb btJ( cr , T, Q) by a factor of € '" (  ) in the first or third
coordinate, we may restrict our attention to 0", T, Q so that y( 0" ) =/:= 0,
Y( Q) =1= 0, Y( O"T) =1= 0, Y( TQ) =1= 0, Y( O"TQ) =1= 0 (just compute). We need
to show that b p ( 0" , T, Q) = 1.
Let N < SL(2, k p ) be the subgroup of upper triangular matrices. It
is a straightforward if cumbersome calculation to show that if
VI' V 2 , V 3 , V 4 EN then b'p(O"' T, Q) = b'p( VI O"V 2 , v 2 1 TV 3 , V3tJV4)' which
allows us to restrict ourselves to the two cases, cr = ( -  -1 ),
T =  -  - 1 ), Q = ( -  -1 ), if '}'( T) 4= 0, and cr = ( -  - 1 ),
T = b  1  ), Q = ( -  -1 ), if '}'( T) = o. In either case, x( cr ) =
a, x( T) = b, x( Q) = c, of course.
Consider the former case, Y( T) =1= o. We have that x( O"T) = bd,
ac
x( TQ) = be, x( crTQ) = bde - b. Compute away:
btJ ( cr , T , Q) = ( a  b ) ( -; bd ) ( b: C ) ( -  ' b;e -  )
X ( a, :e ) - 1 ( -  ' b;e -  ) -1 ( b  C r 1 ( - t' be ) -1
= ( a e r 1 ( d C)( T ) -1 ( -' d )( Cb r 1
c - I ( ac bd _ ac ]
X ( -  ' e ) (bde)b' p e b .

90 KUBOTA AND COHOMOLOGY
But by properties of the Hilbert symbol, we can replace the last factor
with
( aCb )( 'd )( Cd )( ae )( -:,e ),
and this yields that
_ ( a, b ) - 1 ( c, b ) - 1 ( ac, b )
b p ( CT, T, Q) - - - .
p  
The multiplicativity of the Hilbert symbol finally yields that
b p ( 0" , T, (}) = 1. Again, the other case, Y( T) = 0, proceeds along similar
lines. This completes the proof. .
Proposition 5.2.2. The cover defined by a p is nontrivial.
Proof. Since a p is a factor-set mapping into the group J.L2' we get
that if 7T: SL(2, k p) x J..L2  SL(2, k p) is the canonical projection
a p
on the first coordinate, then 'VCT, T, E SL(2, k p ), a p ( CT, T) = I
7T(CT)7T(T)7T(CTT)-l. So, the nontriviality of a p will follow if we show I
that there is no mapping s into J..L2 for which ape CT, T) =
s( 0" )s( T )s( O"T) -1. It is not difficult to show, and we leave it as an
exercise for the reader, that if this is true then VO" E SL(2, k p ),
Vv E N,s( VO" ) = s( 0" ) = s( O"v) (see [Ku67], p. 119, e.g.). Set CT =
( 0 -1 ) ( a /3 )
1 0' T = 'Y 8 with a -4= 0, 'Y -4=  W e he from the preced-
ing proof that 3v, v' EN so that VTV' = (  ) , and 3v", v'" EN
so that Y
V /I lTTV'" = V /I ( -a 'Y  8 ) V", = (  -  - 1 ) .
This gives that
a  ( CT , T) = S ( CT ) S ( T ) S ( O"T ) - 1 = S ( (]" ) S ( VTV ' ) s ( V" CTTV"') - 1
. =s(( l))s(( _-l))S(( _-1)).

KUBOTA'SCOCYCLE 91
( -y a )
At the same time, however, by (5.2.1) and (5.2.2), a p ( a, T) = p' .
Set a = 1 and get that for all '}' E k; , ( -;' 1 ) = 1 = s ( (  -  -1 ) ) .
So, s(( l))s(( _-1))S(( -.-1))=1.1.1=1, so
( -y a )
that a p (a,T)=1, that is, p' =1 for all a,'}'Ek;. This, how-
ever, is false. So, s cannot exist and we are done. .
And now, in view of the remarks at the beginning of this section, we
can state the following result, which hugely simplifies the entire
situation.
Corollary 5.2.3. We may identify Kubota's a p , as per (5.2.2), with the
a given in Corollary 4.1.4. The according double cover, SL (2, k p ), of
SL(2, k p ), can also be identified with Weil's double cover of the local
symplectic group.
Proof. The upshot of Proposition 4.10.4 is that Mp(k p ) = Sp(k p )  C
can be identified with SP(k p ) = Sp(k p ) x JL2 (cf. (4.10.3)), identifying
a p
Sp(k p ) with SL(2, k p ). Our claim is exactly this result. .
Finally, it behooves us to note first that because of  4.1 we know
that a is also realized as the 2-cocycle canonically attached to the
projective Weil representation, r, in that (4.1.12) holds. Invoking
Corollary 5.2.2 we can now characterize Kubota's a p in the same way:
r( O"l)r( O"z) = ape (T1' lTz)r( 0"1 O"z)
(5.2.4)
for all 0"1' U z E SL(2, k p ). The arithmetical benefit derived from this
formulation is that via (5.2.2) the relation (5.2.4) directly involves the
Hilbert symbol. As we shall see presently, this fact considerably
shortens the analytic proof of quadratic reciprocity.
Beyond this, we note that by virtue of the fact that, by definition,
the 2-Hilbert symbol takes values in JLz, the groups SL (2, k p ) and
SL(2, k A ) are automatically double covers of SL(2, k p ) and SL(2, k A ),
provided that a p and nap are non-split. In other words, we have in
p
Kubota's construction a hugely simplified route to the majority of the

92 KUBOTA AND COHOMOLOGY
results of Chapter 4. The remaInIng feature to be dealt with in
Kubota's formalism is the splitting of nap = a A on the rational
p
points. 8L(2, k), of 8L(2, k A ). This will be enough to give 2-Hilbert
reciprocity directly. We cover this critical manouvre in the next
section.
5.3 THE SPLITTING OF a A OVER 8L(2, k)
The splitting of a A on SL(2, k) is equivalent to the splitting of the
adelized projection 7T, as per  4.5, on the rational points of the
symplectic group. We know by Proposition 4.5.5 that this ultimately
devolves to quadratic reciprocity, but to get there one needs the full
force of Proposition 4.5.1-4.5.5. In the present, sparser, context
afforded by Kubota's approach we seize the opportunity to go at
quadratic reciprocity more effectively, by working directly and explic-
itly with Weil's famolis E)-functional.
With a A = nap as above, it is easy to prove:
p
Proposition 5.3.1. a A E H 2 (SL(2, k A ), J.L2).
As already indicated at length, a A defines SL (2, k A ) := 8L(2, k A ) x J..Lz
aA
as an adelic (or global) metaplectic group. Playing fast and loose with
the obvious identification of adelic cocycles (as per the adelization of
(5.1.6)) and the agreeme betweehe adelizations of the SL(2, k p )
d Sp(k p ), we identify SL(2, k A ), Sp(k A ), and Mp(k A ) and just write
SL(2, k A ) for "the" adelic metaplectic cover (of SL(2, k A )) of degree 2.
What is today most often referred to as the Weil @-functional is now
defined as follows:
':1: Si(2, k A )  C d(k A )
(<T,t) [ <P E fA(<T,t)(<P(X)) ] ,
XEk
(5.3.1)
where fA is as in (4.1.13) with k A in place of kp, and C.$(k A ) is the
space of all functions from (kA) to C. Equivalently, we can just
define fA as @p r p' of course. By virtue of the denseness of  (k p) in
L 2 (k p ) for every place  and using the transfer of structure (4.5.6)

THE SPLITTING OF a,l\ over SL(2, k) 93
with G = k A and r = k, we can recast this functional as
'J: SL(2, k A )  Cdf(kA,k)
(o-,t)r-+ [ @ E fA(o-,t)(@(X)) ] .
xEk
(5.3.2)
Here cH(k A , k) is the Hilbert space defined in (4.5.5), entailing in
particular that an element E>(z) E cH(k A , k) is characterized by (4.5.4)
for  E k X k*:
8(z +) = 8(z).w(, Z)-l.
As we noted before, one recognizes in cH(k A , k) a space of "fonctions
theta generalisees," in Weil's own phrase. He notes in Theoreme 6 of
[W e64]: " . . . [ 'J] est un fonction continue. . ., invariant par les transala-
tions a gauche determinees par les elements... de la forme [(k,l)] avec
[k E SL(2, k)]." In other words, we have the following theorem:
Proposition 5.3.2. 'J is invariant under the left natural action of
SL(2, k).
Proof. Evidently (cf. (5.3.1)) we have to show that if 0- 0 E SL(2, k) or,
- .
more precisely, (0- 0 , 1) E SL(2, k)  SL(2, k A ), then, for all <t> E (kA)'
L rA(uOu,t)(<I>(x)) = L rA(u,t)(<I>(x)).
xEk xEk
(5.3.3)
Corollary 4.1.5 yields easily that fA is a true (ordinary) representa-
tion and, seeing that rr = r k agrees with fA on SL(2, k) (by Proposi-
tion 4.5.4) we get that f A ( 0- 0 u, t) = r k ( o-o)r A ( 0-, t). Therefore (5.3.3)
will result if we can show that for U o ranging over the generators of
SL(2, k), we have that, for every <I> E (kA)'
E <t>(x) = E rk(o-o)(<I>(x)).
xEk xEk
(5.3.4)
Using (4.4.3)-(4.4.8), with the suitable assignments having been
made, we find that for generators of the form (  a  1 ) or (  ),
the identity (5.3.4) follows by direct calculation, taking into account
the fact that an adelic character is necessarily trivial on k (this is
immediate from Proposition 4.8.1). Finally, for the remaining genera-
tor ( _   ), it is easy to see that (5.3.4) reduces to a variant of
Poisson summation. That does it. .

94 KUBOTA AND COHOMOLOGY
For completeness we note that locally, that is, over kp, we get from
(4.4.3)-( 4.4.8) that
rk((
r k ( ( 
a1 ) ) (<I>( x)) = I a 1 1 / 2 <1>( ax),
) )(<I>(X)) = Xp([ f3x, x])<I>(x),
(5.3.5)
(5.3.6)
and
r k ( ( _  ) ) (<I>(x)) = y( Xp 0 q)<I>(x),
(5.3.7)
for the appropriate quadratic form, q. Now just adelize. Note also that
in our earlier treatment of splitting ( 4.5), the central issue was the
substance of Proposition 4.5.4, whose critical consequence is the fact
that adelically the Weil index, y, reduces to 1. Evidently, this is
equally critical for demonstrating that the adelization of (5.3.7), when
substituted into (5.3.4), presents us with nothing else than Poisson
summation. Ultimately the two approaches are quite the same at a
deeper level.
We should observe that (5.3.5)-(5.3.7) are frequently proposed as
the de facto definition of the Weil representation (see, e.g., [JL70],
[Ge76]), especially when explicit calculations are indicated. The
definitive treatment of these three relations, in the setting of  4.1
(especially Proposition 4.1.3 and its corollaries), is that of Jacquet-
Langlands.
Now for the main result of this chapter:
Proposition 5.3.3. a A splits on SL(2, k).
Proof. Adelizing (4.1.12) we have that if lT 1 , lT z E SL(2, k A ), then
'A( lT 1 )'A( lT z ) = a A ( lT 1 , lTZ)'A( lT 1 lT z ).
Here we have replaced U A by a A as per Corollary 5.2.2 (adelized).
Therefore, certainly,
E 'A( (Tl)'A( lTz)(<I>(x)) = a A ( lT 1 , lT z ) E 'A( lT 1 lTz)(<I>(x)).
xEk xEk
Now, if lT 1 , lT z E SL(2, k), then both sums reduce to E <I>(x) as a
xEk
result of (5.3.4), that is, a consequence of Proposition 5.3.2's proof.

THE SPLITTING OF aA over SL(2, k) 95
Here, again, we have used the fact that on SL(2, k) the rational
points, r A and fA' agree, that is, Proposition 4.5.4 suitably interpreted.
The result follows easily: for all 0"1' 0"2 E SL(2, k),
a A ( 0"1,0"2) = Oa p ( 0"1,0"2) = 1.
p
(5.3.8)
.
And we have the immediate (in fact, tautological) corollary:
Corollary 5.3.4. SL (2, k A ) splits on SL(2, k), its subgroup of rational
points.
1
) IL2
- 7T
) SL(2, k A ) ) SL(2, k A )
<:.......p.......... 1
SL(2, k)
) 1
,
(5.3.9)
where p is the according splitting homomorphism whose existence is
I
guaranteed by Corollary 5.3.4:
-
p: SL(2, k)  SL(2, k A ) = SL(2, k A ) x ILz
a A
O"( 0", Ep(O")).
Thus Ep maps into IL2 and we have that
p( 0"0"') = ( 0"0" ' , E p ( 0"0"'))
= P ( 0" ) P ( 0" ') = ( 0"0" ' , E P ( 0" ) E P ( 0" ') a A ( 0" , 0" ') ) ,
where E p ( 0"0"') = E p ( 0") E p ( 0" ')a A ( 0", 0"') for all 0", 0" ' E SL(2, k). But
then by (5.3.8), E p ( 0"0"') = E p ( 0") E p ( 0" '). In other words,
p = id @ Ep
with id = idl sL (2,k) and Ep: SL(2, k)  J.L2' a homomorphism.
(5.3.10)
We add, for the sake of completeness, that, possibly as a response
to the seminal influence exerted by Jacquet-Langlands [JL70], the
preceding considerations are most often developed for GL(2) rather

96 KUBOTA AND COHOMOLOGY
than SL(2). Of course, there are sound technical reasons for this, and
one obvious reason is that qua theories of automorphic forms or
automorphic representations, GL(2) is the group of choice. Under
these circumstances the Weil representation is defined on GL(2, k p )
and GL(2, k A ) by adding to the list «5.3.5)-(5.3.7)) the effect this map
should have on (  ), say, thus exhausting the set of generators of
GL(2). For our purposes we state only the following result, without
proof:
Proposition 5.3.5. GL (2, k A ) splits on GL(2, k). Indeed (cf. (4.10.4)):
1
) J..L 2
) Si(Z,k A ) ) SL(2,k A )
......... \
. .
. .
. . .
......
SL(2,k A ) X 2 SL(2,k)
aA
1
) J..L 2
) GL (2,k A ) ) GL(2,k A ) ) 1
.............. \
GL(2, k A ) x 2 ......GL ( 2 k )
 ' ,
) 1
(5.3.11 )
where b A E H 2 (GL(2, k A ), J..L2); see more on this in Chapter 6.
5.4 2-HILBERT RECIPROCITY ONCE AGAIN
With Proposition 5.3.3 and Corollary 5.3.4 in place it is now a very
straightforward matter to derive 2-Hilbert reciprocity directly.
Proposition 5.4.1. If a, f3 E k x, then (as in Proposition 4.8.5)
Q( aJ3 )=1.

2-HILBERT RECIPROCITY ONCE AGAIN 97
Proof. By (5.2.2) and an easy property of the 2-Hilbert symbol
(see, e.g., [Ha80]), a(( ),( ))*( al3 )( -lal3 )=
( a  13 ). So, by the definition of a A' we get that I] ( a  13 ) =
aA(( ),( )). Since ( ),( )ESL(2,k), we now
need only invoke (5.3.8). .

The Algebraic Agreelnent
Between the Forlnalisms
of Weil and Kubota
The purpose of this short chapter is to delineate the precise manner
in which the arrangement of central extensions of symplectic groups
presented in [We64] corresponds to the explicit cohomological formu-
lation introduced in [Ku67]. Specifically, the groups N(dfE-i(k))
(4.2.5), Mp(k) ( 4.1), and SP(k p ) ( 4.10) that appear in Weil's "local
formalism" (the term is self-explanatory) are naturally located in the
commutative diagrams (4.10.4) and (5.1.6). As we first noted in  5.1,
identifying Sp(k'p) and SL(2, k'p) we find that the top row of (4.10.4) is
cohomologically equivalent to (5.1.2), which begins Kubota's develop-
ment of these quadratic metaplectic groups in the context of explicit
2-cocycles for linear groups. We now go to ask how the bottom rows of
(4.10.4) might be transported into Kubota's formalism.
One reason for asking this question is the profound practical
benefit one derives from being able to carry out Weil's arguments
concerning the representation and the E)-functional that bear his
name in the setting of SL(2) and, as we shall see, GL(2), locally as
well as adelically. Indeed, work in progress by the present author
suggests that we can achieve this in completely algebraic terms, that is,
qua mappings between second cohomology groups and, most impor-
tantly, we thereby provide a means of transporting Weil's treatment of
splitting of (adelized) covering groups over subgroups of rational
points directly into Kubota's explicit set-up.
98

THE GRUESOME DIAGRAM 99
In Chapter 7 we shall discuss (in very brief terms) the extrinsic
motivation behind this algebraic exercise: this "dictionary," Weil-to-
Kubota, generalized from quadratic to higher metaplectic covers,
would constitute a necessary feature in the attempt to prove the
according splitting result for the higher metaplectic covers introduced
in [Ku69] by generalizing or mimicking Weil's approach.
6.1 THE GRUESOME DIAGRAM
In order to keep from interrupting the flow of the discussion and to
avoid lengthening our treatment by virtue of the inclusion of a heavy
dose of low-dimensional cohomology, we simply "cut to the chase," so
to speak. The overly scrupulous reader, already mentioned earlier in
the book, might try to provide the sundry computational details for
himself. Suffice it to say that there appear to be only two possibilities
(a priori) for the Weil-to-Kubota dictionary, and the choice of one
(i.e., (6.1.1), below) over the other is suggested by means of the
Rmak-Klein-Fricke Theorem (cf. [R078]). This might make for a nice
bit of algebra. First, some additional players and a redundancy.
In order to conform with the notation of [Be97], from which (6.1.1)
is lifted verbatim, rewrite  as v (so kp is now k JJ ) and N(cJfE.i(k'p))
as B(k p ). Also, given Kubota's a JJ (i.e., a'p) as defined in ( 5.2), set
p: GL(2, k JJ )  SL(2, k JJ )
u( detOu- 1 )u,
(6.1.1)
and for y E k: ,
( )y: GL(2, kJJ)  GL(2, k JJ )
u( lY)U( );
(6.1.2)

100 THE ALGEBRAIC AGREEMENT BETWEEN THE FORMALISMS OF WElL AND KUBOTA
,..,..-I ,..,..-I ,..,..-I ,..,..-I
I 1 r I
  
  
... 
  
.. ) N N ' ) 
N  
  c
 t5 t5 ..c
I N 1 N I x_
::1, ::t U x_
  x:' X )(:. U
 X r:::J:'   
    X 0)(:.
  
..  ... 
N  ) N =..:::c:" ---+   ---...  
  l::{ -   &q0 ;:.
l ... 
N r  I 
  I 0
r  t5 ..c
N N x_ x_
::t ::1, ' ) u U
1 1 1 I
,..,..-I ,..,..-I ,..,..-I
,..,..-I
,..,..-I
1 ,..,..-I 
1 1
,-,.
  
  
'-'  ) 
  
 CQ
1 N r x_ I x_
::1, U U
 x   X :.
  Xed: 
  )
  ' )    
lr5}    = 
..:::c  
  I 
1  1 r5} 0
CQ
N x_ x_
::1, ' ) u u
r 1 r
 ,..,..-I ,..,..-I
(6.1.1)

THE EVEN MORE GRUESOME DIAGRAM 101
finally, with ()" = (  ), let
v(,): k: X GL(2, k,,)  J.L2
(6.1.3)
{ I,
(y, (}")r- ( Yd ),
if c =1= 0
if c = 0,
reintroducing the 2-Hilbert symbol. Under these circumstances one
obtains [Ku69] that
b,,: GL(2, k,,) X GL(2, k,,)  J.L2
(6.1.4)
(()", T )r-a.,( (p( ()" )detT, p( T)) v( det t, p( (}"))
defines an element in H 2 (GL(2, k,,), J.L2). We naturally write GL (2, k,,)
for the corresponding double cover, that is, GL (2, k,,) =
GL(2, k,,) x J.L2.
b v
Now define bo(k,,) to be the semi-direct product k; X SL(2, k,,).
The principal result of the author's preliminary investigations holds
that there exist (canonical) 2-cocycles K" E H 2 (GL(2, k,,), C() and
K E H 2 (b o (k,,), C(), which suffice to complete Kubota's formalism in
such a way as to produce a complete "match" to Weil's: locally, the
dictionary is complete. Thus with !B(k,,) corresponding to K" and with
(kv) corresponding to K (as the indicated central extensions, of
course), we obtain the following gruesome diagram, delineating the
agreement between Weil's left half and Kubota's right half.
6.2 THE EVEN MORE GRUESOME DIAGRAM
Adelizing everything in sight and writing flo(k A , TI" lOll)' O(k A , TI"lO,,),
SL(2, k), and GL(2, k) for the various subgroups of rational points, we
get from (4.10.4) and from the indicated splitting theorems (e.g.,
Corollary 5.3.4),

102 THE ALGEBRAIC AGREEMENT BETWEEN THE FORMALISMS OF WElL AND KUBOTA
,.....j
,.....j
,.....j
,.....j
r r<
< ,-..
,-..
 
N ) N
 
 \5------l \ 5
I I
</ I
too'} I  N I rloo\ N
I  I 
I < I
ot X t:J< ,-..ot X 
 <  <
N ,-.. ) N ,-..
   
l  l 
r  r tS
N N
  '
r r
r
<
,-..

..
N


r
,-..
<


) ::::;

r
x_
U
X:
<
,-..

..
N


,-..
<
) 

o
r
,-..
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) 

&q
,.....j
,.....j
r
x_
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r
,.....j
x_
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,.....j



r
,-..
<

"'-"
l
1
N
 '
1
N

X
1 1
,-.. ,-..
< <
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 \ -----________J
.. l ..
I < I <
I  x_ I 
I "-'" U I o
,-.. I c: ,-.. I &... c:
<ot X  < ot
 
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i  i
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( 6.2.1 )
"The rest is silence." (Hamlet, Act V, Scene II).

Hecke's Challenge:
General Reciprocity
and Fourier Analysis
on the March
The famous last paragraph of Hecke's Vorlesungen, already quoted
verbatim in the Introduction, contains his critical conjecture that there
should exist "transzendenten Funktionen" which will bring about for
higher reciprocity what his it-functions achieve for quadratic reciproc-
ity. Indeed, this is precisely what would constitute a full-fledged
Fourier analytic proof of general reciprocity in the classical sense.
And, while the formulation of this reciprocity law would be in the
style of Gauss-Euler rather than that of Hilbert, this would not be
objectionable at all since an agreement along the lines of that given in
Chapter 2 is also available for higher degrees. Accordingly, there
would be a perfect correspondence between such a classically Fourier
analytic formulation and the general framework designed by Kubota,
already discussed somewhat in Chapter 5 and about which a little
more is to follow.
Regarding Hecke's specific request, however, there are some very
severe obstacles to be dealt with. My paper, "On a naive generaliza-
tion of Hecke it-functions and the analytic proof of higher reciprocity
laws" [Be93], goes at the central problem of the elusive "tranzendenten
Funktionen" directly by considering, with Hecke's arsenal, generalized
it-series obtained simply by replacing the quadratic orm defining a
103

104 HECKE'S CHALLENGE: GENERAL RECIPROCITY AND FOURIER ANAL YSIS
Hecke it-function by a form of higher but still even degree. This parity
for the degree is needed to bring about suitable convergence of the
series and, indeed, one obtains an a priori reasonable facsimile of a
Hecke it-function. Doing the Fourier analysis, however, one is led to a
bizarre if independently interesting state of affairs concerning the
functional equation of such an object. While these functional equa-
tions turn out to bear a fascinating relation to a Cauchy-Euler system
of partial differential equations, making for a closed-form presenta-
tion of these relations, they involve multipliers and generalized r -fac-
tors which preclude going to the next step of Hecke's program.
One notes, of course, that the basic ploy of this naive approach
consists in replacing a Gaussian kernel (or Gaussian density), consti-
tuting a typical it-series' summand, by an admittedly simplistic higher-
degree generalization. Of course this move is part and parcel of
setting up a sweeping generalization of the Fourier transform itself,
and this objective, although obviously ambitious, is actually more
reasonable than it appears. We turn to this matter at the very end of
this chapter.
Before we leave the naive approach behind altogether we briefly
men!ion another attempt at generalizing a part of Hecke's tactics to
higher degrees, but one which succeeded. The fact that Gauss sums
transform so nicely with respect to the natural action of the Legendre
symbol generalizes entirely to higher symbols. This is shown in
[Be9S].
Now we address what might perhaps be called "the Weil-Kubota
formalism," as synopsized in the two gruesome diagrams of the pre-
ceding chapter. As we have taken pains to indicate throughout, it is all
a question of fitting the appropriate Fourier-analytic phenomenon
into what is really an algebraic arrangement. Specifically, the grue-
some diagrams are a set of roadmaps of central extensions of transfor-
mation groups (or even isotropy groups for bilinear forms) which are
unitarily represented. The critical Fourier analytic phenomenon is the
fact that the according adelized short exact sequences allow splitting
on the rational points by means of the invariance of the Weil @-func-
tional under the action of the rational points via the projective Weil
representation. We have also taken pains to indicate that this invari-
ance property of the adelic @-functional is ultimately identified with a
suitable form of generalized Poisson summation (also due to Weil).
There is no question that with this Weil-Kubota formalism properly
delineated, and taking into account that the Weil representation is
fundamentally attached to a Heisenberg group's Schr6dinger repre-

HECKE'S CHALLENGE: GENERAL RECIPROCITY AND FOURIER ANALYSIS 105
sentation, the analytic proof of quadratic reciprocity acquires a partic-
ularly compact formulation. Thus:
- .
1  J.Lz SL(2, k)A = SL(2, k)A x J.Lz  SL(2, k)A  1,
'YA
with 'YA-llp 'Yp E H 2 (SL(2, k)A' J.Lz), and
7T A : SL(2, k)A Aut( .sA)'
projective, with A a Schwartz-Bruhat space, are given; 7T A defines a
-
representation of SL(2, k)A. We get that
7T A (GG') = 'YA(g, g')7TA(g)7TA(g')
for all g, g' E SL(2, k)A. Next, define the Weil 8-functional (which we
now denote by 8 instead of 'J):
8( <1»:= E <1>( x),
xEk
here <I> E A. From Poisson summation and properties of 7T A =  7T p
I 
we get that
8(7T A (gO)(<I») = 8(<1»
for all go E SL(2, k). Therefore, iterating, we find that for go, g' E
SL(2, k),
8 ( <1» = 8 ( 7T A (g 0) 7T A (g ) ( <I> ) ) ,
while, at the same time,
8(<1» = 8(7TA(gog')(<I»).
Infer that 7TA(gOg) = 7TA(go)7TA(g), which implies that 'YA restricts to
1 on SL(2, k) X SL(2, k). Now just note that if go = ( ), g =
( 1 b ) ( a, b )
o l' a, b E k X , then 'Yp(go, g) = T ' the quadratic Hilbert
symbol. We end up with
1 = 'YA( ( 
a ) ( 1 b )) = n ( a,b )
1 ' 0 1  P ·
(Q.E.D.)

106 HECKE'S CHALLENGE: GENERAL RECIPROCITY AND FOURIER ANAL YSIS
Mimicking this argument for the sequence
- (n)
1  ILn  SL(2, k)A := SL(2, k)A x ILn  SL(2, k)A  1,
"llt)
where "i n ) = II p ,,In) is the generalization to (5.2.2) which occurs in
Kubota's 1967 paper [Ku67] and is used by him to great advantage in
[Ku69], would give the analytic proof of general reciprocity in the
modern sense. We have to stress two points, however. In the first
place, Kubota's treatment of the splitting properties of the sequence
vis a vis 5L(2, k) requires nth order reciprocity beforehand, making
for circularity from the outset. In the second place, and this is of
course very closely related to the preceding point, the problem of
determining a counterpart to the Weil representation is of huge
difficulty and significance in its own right. Indeed, it is really the
central problem.
From a broader perspective, it is the case that the generalization
problem rests on the process of going from bilinear forms and
quadratic forms, Gauss kernels and Hecke if-functions, and how they
pertain to double symplectic covers, to higher-dimensional counter-
parts of all these, fitted into Kubota's algebraic framework surround-
ing 5L(2, k)i n ) or GL(2, k)i n ), its meaning being clear by analogy (cf. \
Jacquet-Langlands [JL80], Kazhdan-Patterson [KP83]). And then the
final quarry is adelic splitting on the rational points, SL(2, k) or I
GL(2,k).
In the final analysis the crux of the matter is the problem of
generalizing a number of mainstays of Fourier analysis itself. Combin-
ing the viewpoints of Hecke and of Weil, and generalizing accordingly,
it appears that we require, first, the notion of a higher-degree coun-
terpart of a Gaussian kernel, second, a generalization of the Weil
representation and the Weil @-functional, and third, a suitable gener-
alization of Poisson summation.
As already indicated above, the first attempt at generalizing a
Gaussian kernel in the setting of Hecke if-functions simply leads to
apparently untenable functional equations. Therefore the preferred
viewpoint is that of Weil and reciprocity laws in the sense of Hilbert.
Indeed, the formalism of Kubota's 1969 paper [Ku69] already presents
us with much of the low-dimensional cohomology invoked. We need,
however, to fit a generalized Weil representation and E)-functional
into this formalism and this means a full frontal attack on the Fourier
transform itself. The following remarks (of a very preliminary nature)
suggest a possible approach.

HECKE'S CHALLENGE: GENERAL RECIPROCITY AND FOURIER ANAL YSIS 107
Returning to Hecke once more we bring in a strategy which has
proven itself incomparably fecund in modern number theory, namely,
the correspondence between modular forms and L-functions by means
of the Mellin transform and its inverse. In case the modular forms in
question have half-integral weight, as in the case for it-functions, we
have at our disposal Shimura's well-known 1972 generalization, which
is usually referred to as the Shimura Correspondence (cf. [Sh72]). In
the classical case of the standard Jacobi it-function,
it(z) = Ee- nz2 ,
n
the Shimura Correspondence maps this half-integral weight form to
the Riemann -function. Similar things occur for Hecke's it-functions,
of course.
From the standpoint of Hecke's and Shimura's correspondences, it
is a question of delieating a dictionary of sorts from, for our purpose,
it-functions to L-functions, and back again. In a recent work [Be98]
we apply this dictionary to the case of Dedekind l"-functions in order
to bring out an analytic connection (one integrates twice) between
local constants or Artin root numbers and Weil indices. While this
work is very immature as yet, in that only a relatively easy special case
has been considered, it does suggest that it is possible to approach
problems concerning Weil indices by looking at local constants.
Local constants have their historical roots in the work of Artin (cf.
[Ar65]) and, most importantly, in Tate's thesis [Ta50]. Roughly speak-
ing, a -function, indeed an adelic -function, is decomposable as a
product of local factors, each of which satisfies a local functional
equation. Tellingly, the latter is obtained by means of applying a
Fourier transform, and the result is an identity equipped with a
multiplier, a local constant. Going over to the global or adelic l"-func-
tion, one obtains a product formula for these local constants involving
a classical Artin root number.
We claim that it is in the context of the study of local constants for
L-functions that the problem of generalizing Gauss kernels, indeed,
the Fourier transform itself, should be tackled. Subsequently there
should be two ways available to return to the setting of the analytic
proof of higher reciprocity laws: either by working directly with the
purported new-found Fourier transform or by expanding and exploit-
ing the correspondences of Hecke and Shimura.
The idea of seeking a higher-dimensional Gaussian kernel within
the context of the study of local constants was suggested to me in 1995

108 HECKE'S CHALLENGE: GENERAL RECIPROCITY AND FOURIER ANALYSIS
by Varadarajan. He suggested that the direction in which to proceed
leads through Deligne's 1972 work [De72] on local constants to rela-
tively recent work by Laumon ([Lau84], e.g.) which inter alia recasts
Deligne's central result in this connection in terms of the l-adic
Fourier transform (also due to Deligne) and what Laumon calls "the
method of stationary phase." There is a profound similarity between
Laumon's co homological version of stationary phase and the classical
notion -of the same name.
We should note that these considerations by Deligne and Laumon
deal with local constants in the context not of number fields but of
function fields with prime residual characteristic. This makes for an
obvious additional difficulty, of course. On the other hand, function
fields have reciprocity laws, too.
Perhaps it is proper to end this chapter (and the main body of this
book) with a bit of flagrant optimism. The problem of the analytic
proof of higher reciprocity laws is generally viewed with grim pes-
simism, so it is not meant for the timid. A number of mandarins whom
I consulted when I first embarked on these investigations some ten
years ago expressed little hope for a solution to Hecke's challenge.
However, Weil himself was a notable counterexample to this rule.

[Ar65]
[Be93]
[Be95]
[Be98]
[ Ca82]
[CR81]
[De72]
[DD93]
[EMacL86]
[Fr91 ]
Bibliography
Artin, E., Collected Papers, Springer-Verlag, Berlin (1965).
Berg, M., "On a generalization of Hecke 1J--functions and the
analytic proof of higher reciprocity laws," J. Number Theory, vol.
44, no. 1 (1993), pp. 66-83.
Berg, M., "On generalized Gauss-Heeke sums and theta con-
stants," Integral Transforms and Special Functions, vol. 3, no. 1
(1995), pp. 1-20.
Berg, M., "A relationship between Weil indices and local con-
stants," presentation at the 1998 Western Number Theory con-
ference, San Francisco.
Cauchy, A. L., Oeuvres Completes: Gauthier-Villars (Publ.), 55
Quai des Grands Augustines, Paris (1882).
Curtis, C. W., and Reiner, I., Methods of Representation Theory,
John Wiley & Sons, New York (1981).
,
Deligne, P., Les Constantes des Equations Fonctionelles des Fonc-
tions L, in Modular Functions of One Variable II, Springer
Lecture Notes, Vol. 349 (1973).
Dirichlet, P. G. J., and Dedekind, R., Vorlesungr uber Zahlen-
theorie, Braunschweig (1893), and Chelsea, New York (1968).
Eilenberg, S., and MacLane, S., Collected Works, Academic
Press, Orlando, FL (1986).
Freitag, E., Singular Modular Forms and Theta Relations, Springer
Lecture Notes, vol. 1487, Springer-Verlag, Berlin (1991).
109

110
[Ga90]
[GauOl]
[Ge76]
[Ge93]
[Ha76]
[Has80]
[He23]
[HS65]
[Hi32]
[H080]
[Hu80]
[Ig72]
[JL70]
[KP83]
[K084]
[Ku67]
[Ku69]
[Ku90]
[La73]
[Lan27]
BIBLIOGRAPHY
Garrett, P. B., Holomorphic Hilbert Modular Forms, Wadsworth
& Brooks/Cole, Pacific Grove, CA (1990).
Gauss, C. F., Disquitiones Arithmeticae, Yale University Press,
New Haven, CT (1966), Latin original: 1801.
Gelbart, S., Weil's Representation and the Spectrum of the Meta-
plectic Group, Springer Lecture Notes, vol. 530, Berlin (1976).
Gelbart, S., "On theta-series liftings for unitary groups," in
Theta Functions, M. Ram Murty, Ed. (1993), pp. 129-174.
Hall, M., The Theory of Groups, 2nd ed., Chelsea, New York
(1976).
Hasse, H., Number Theory, Grund!. Math. Wiss., Bd. 229,
Springer-Verlag, Berlin (1980).
Hecke, E., Vorlesungen ilber die Theorie der Algebraischen Zahlen,
Leipzig (1923).
Hewitt, E., and Stromberg, K., Real and Abstract Analysis, Grad.
Texts in Math. 25, Springer-Verlag, Berlin (1965).
Hilbert, D., Gesammelte Abhandlungen, Springer-Verlag, Berlin
(1932), and Chelsea, New York (1981).
Howe, R., "On the role of the Heisenberg group in harmonic
analysis," Bull. AMS, vol. 3, no. 2 (1980), pp. 821-843.
Humphreys, J. E., Arithmetic Groups, Springer Lecture Notes,
vol. 789, Berlin (1980).
Igusa, J. I., Theta Functions, Grund!. Math. Wiss., Bd. 194,
Springer-Verlag (1972).
Jacquet, H., and Langlands, R., Automorphic Forms on GL(2),
. Springer Lecture Notes, vol. 114, Berlin (1970).
Kazhdan, D. A., and Patterson, S. J., Metaplectic Forms, Publ.
Math. I.H.E.S. (1983), pp. 35-142.
Koblitz, N., Introduction to Elliptic Curves and Modular Forms,
Grad. Texts in Math. 97, Springer-Verlag, Berlin (1984).
Kubota, T., "Topological covering of SL(2) over a local field,"
J. Math. Soc. Japan, vol. 19 (1967), pp. 114-121.
Kubota, T., "On automorphic functions and the reciprocity law
in a number field," Lect. in Math., Kyoto Univ., Kinokuniya
Bookstore Co. Ltd., Tokyo (1969).
Kubota, T., Letter to the author (1990).
Lam, T. Y., The Algebraic Theory of Quadratic Forms,
Benjamin/Cummings, Reading, MA (1973).
Landau, E., Elementary Number Theory, Chelsea, New York
(1958), original German edition: 1927.

[Lang70]
[Lau84]
[Ma64 ]
[Mill]
[M068]
[Mu91]
[Pi78]
[Ra93]
[Re89]
[R078]
[Sc85]
[S08S]
[Se63]
[Ser73 ]
[Ser77]
[Ser79]
[Sh64 ]
[Si35]
BIBLIOGRAPHY 111
Lang, S., Algebraic Number Theory, Addison-Wesley, Reading,
MA (1970).
,
Laumon, G., Transformation de Fourier, Constantes d'Equations
Fonctionelles, et Conjecture de Weil, Publ. Math. I.H.E.S. (1984),
pp. 131-210.
Mackey, G. W., Review of [We64] in Math. Reviews, no. 2324
(1964).
Minkowski, H., Gesammelte Abhandlungen, Leipzig (1911), and
Chelsea, New York (1967).
Moore, C. C., "Groups extensions of p-adic and adelic linear
groups," Publ. Math. I.H.E.S. (1968), pp. 157-222. .
Mumford, D., Tata Lectures on Theta III, Birkhauser, Boston
(1991).
Pieper, H., Variationen ilber ein Zahlentheoretisches Thema von
Carl Friedrich Gauss, Birkhauser Verlag, Basel (1978).
Rao, R. R., "On some explicit formulas in the theory of Weil
representation," Pac. J. Math., vol. 157, no. 2 (1993), pp.
335- 371.
Reiter, H., Metaplectic Groups and Segal Algebras, Springer
Lecture Notes, vol. 1382, Berlin (1989).
Rose, J. S., A Course on Group Theory, Cambridge University
Press, Cambridge (1978).
Scharlau, W., Quadratic and Hermitian Forms, Grundl. Math.
Wiss., Bd. 270, Springer-Verlag (1985).
Scharlau, W., and Opolka, H., From Fermat to Minkowski,
Lectures on the Theory of Numbers and its Historical Development,
Undergrade Texts in Math., Springer-Verlag, Berlin (1985).
Segal, I. E., "Transforms for operators and symplectic automor-
phisms over a locally compact abelian group," Math. Scand. 13
(1963), pp. 31-43.
Serre, J. P., A Course in Arithmetic, Grad. Texts in Math. 7,
Springer-Verlag, Berlin (1973).
Serre, J. P., Linear Representations of Finite Groups, Grad. Texts
in Math. 42, Springer-Verlag, Berlin (1977).
Serre, J. P., Local Fields, Grad. Texts in Math. 67, Springer-
Verlag, Berlin (1979).
Shirantani, K., "On the Gauss-Hecke sum," J. Math. Soc. Japan,
vol. 16, no. 1 (1964), pp. 32-38.
Siegel, C. L., "Ueber die analytische theorie der quadratischen
formen, I, II, III," Annals Math., vol. 36 (1935), pp. 527-606; vol.
37 (1936) pp. 230-263; vol. 38 (1937), pp. 212-291.

112 BIBLIOGRAPHY
[Ta50] Tat.e, J. T., "Fourier analysis in number fields and Heeke's zeta
functions," thesis (1950) (Princeton), in Cassels, J. W. S., and
Frolich, A. (Eds.), Algebraic Number Theory, Thompson (1967),
pp. 305-347.
[Ta76] Tate, J. T., "[Hilbert's] Problem 9: The general reciprocity law,"
Proc. Symposia Pure Math., vol. 28 (1976), pp. 311-322.
[Va68] Varadarajan, V. S., Geometry of Quantum Theory, Springer-
Verlag, Berlin (1985).
[We51] Weil, A., l'Integration dans les Groupes Topologiques et ses Appli-
cations, Hermann, Paris (1951).
[We54] Weil, A., "The mathematics curriculum (a guide for students)"
in Oeuvres Scientifiques, vol. II (1951-1964), pp. 121-126.
[We64] Weil, A., "Sur certains groups d'operateurs unitaires," Acta.
Math. 111 (1964), pp. 143-211.
[We74] Weil, A., Basic Number Theory, Grund!. Math. Wiss., Bd. 144,
Springer- Verlag, Berlin (1974).
[We76] Weil, A., Elliptic Functions According to Eisenstein and Kro-
necker, Ergebnisse der Mathematik 88, Springer-Verlag, Berlin
(1976).
[We89] Weil, A., letter to the author (1989).

Index
'Some entries in the index below are simply dealt a reference of the form [pg] &c so as
to indicate that the item in question occurs almost everywhere in the book, so to
!speak. For example, we have for "Hecke, Erich" the entry vii ff, 1 &c."
Absolute quadratic reciprocity, xviii
Absolute quadratic reciprocity law, 17
Adelic character, 71, 93
Adelic cocycles, 92 ff
Adelic splitting over the rational points,
33, 84
Adelization, 71 ff, 92
Analytic proof of higher reciprocity,
103 ff
Analytic proof of quadratic reciprocity,
i &c.
Anisotropic forms, 68
Artin root number, 107
Artin's reciprocity law, 17
Automorphic form of half-integral
weight, xiv, 2
Automorphic functions, xvi, 14, 54, 80,
112
Automorphic representation, xv
Automorphy, xiv, 52
Bilinear form, 25 ff, 66 ff, 80
Brge Jessen's Theorem, 72
Cauchy, Augustin-Louis, vii, xiv ff, 2 &c
Central extension, xv, 21, 27, 42 ff, 78
Cocycle, xvi, 14, 21
Cocycle, Kubota's, 86 ff
Cover, double, xv, 27 ff, 43, 77 ff, 91, 101
Covers, metaplectic, xvi ff, 64, 99
Covers, symplectic, 106
Deligne, Pierre, 108
Double cover, xv, 27 ff, 43, 77 ff, 91, 101
Dual pairing, 32
Eisenstein, Gotthold, 2
Ergiinzungssatze, 1, 19
Euler, Leonhard, xiii, xviii, 1 &c
Even more gruesome diagram, xx, 102
Fourier transform, generalized, 36, 104 ff
Gauss, Carl Friedrich, xii, 1 &c
Gauss sums, xiv ff, 2 &c
Gauss-Euler (quadratic) reciprocity,
xvii ff, 1, 103
113

114 INDEX
Gauss-Heeke sums, 6 &c
Gaussian kernel (or density), 104, 107
Gelbart, Steve, 28, 57
Genera, xiii, 2
Generalized Fourier Transform, 36,
104 ff
Generalized theta functions, xv, 27, 93
Grothendieck group, 66 ff
Group extensions, xx, 31, 83
Gruesome diagram, xx, 99-102
Hahn-Banach Theorem, 53
Hamlet, 102
Harmonic analysis, viii, 79, 110
Heeke, Erich, vii ff, 1 &c
Heisenberg group, xix ff, 28 ff
Heisenberg group, linearized, 29 ff
Higher Kubota formalism, 106
Higher metaplectic groups, 106
Higher Weil formalism, 106
Hilbert, David, xiv ff
Hilbert reciprocity, xiv ff, 16, 17 ff, 103 ff
Hilbert symbol, quadratic, 16, 17 ff
Howe, Roger, 28
Hyperbolic spaces (planes), 67
Isometry (of quadratic spaces), 65 ff
Isotropy, xix, 66 ff, 79 ff, 104
Kronecker product, 67
Kubota formalism, higher, 106
Kubota, Tomio, xvi &c
Lam, T. Y., xix, 66
Laumon, Gerard, xvii, 108
Legendre symbol, xii, xviii, 1, 3, 16, 62 ff,
105
Legendre symbol, generalized, 14
Linearized Heisenberg group, 29 ff
Locally compact abelian ( = LCA) groups,
XIX
Mackey, George, xix, 26 ff, 75
Metaplectic covers, xvi, 64, 99
Metaplectic groups, 31, 73, 98, 111
Metaplectic groups, higher, 105
Moore, C. C., xvi, 27, 30, 83
Multiplier representation, 30, 71, 76
Normalizer, 33
Plancherel's Theorem, 5, 36, 53
Poisson summation, xv ff, 27, 80 ff, 105
Pontryagin dual, 21, 28, 32
Pontryagin duality, 48 ff, 57 ff
Primary algebraic integer, 11 ff
Projective representation, 15, 30 ff, 49,
76 ff
Quadratic forms, xiv, 3 ff
Quadratic forms, analytic theory of, xv
Quadratic Hilbert symbol, xv, xviii, 65,
69, 105
Quadratic reciprocity, i &c "
Quadratic reciprocity (absolute), xviii
Quadratic reciprocity (Gauss-Euler),
xvii ff, 101, 108
Quadratic reciprocity (Hilbert), xvi ff, 16,
17 ff
Quadratic reciprocity law, xv, 2 ff
Quadratic reciprocity law (absolute), 17
Quadratic reciprocity law (relative), xv, 2,
9, 14
Quadratic spaces, 66 ff
Quaternions over a local field, 63 ff
Reciprocity, quadratic See quadratic
reciprocity (1ots of entries)
Reciprocity law, Artin's, 17
Reciprocity law, higher, vii, xvi, 104 ff
Remak-Klein-Fricke Theorem, 99
Representation, unitary, 32 ff
Representation, Weil [projective], xv &c
Schrodinger model, 50
Schur's lemma, 30, 76
Schwartz-Bruhat functions, 29 ff, 105
Second-degree characters, 44, 48, 58
Segal, I. E., 28
Semi-direct product, 27, 35, 50, 102
Shimura, Goro, 108
Siegel, Carl Ludwig, xv
Skew-symmetric bilinear form, 25 ff
Splitting of primes, xiii, 2
Stone-Yon Neumann Theorem, xviii ff,
20 ff, 73 ff
Symplectic covers, 106

INDEX 115
Theta constants, 3 ff, 109
Theta functions, generalized,
xv, 27, 93
Theta functions, Heeke, 3 ff
Theta-functional (WeiI), 80, 92 ff
Theta series, 52, 111
Totally imaginary number field, xvi
Totally isotropic forms, 68
Transfer of Hilbert space structure,
35, 37
Vector spaces over local fields, 57 ff
Unitary groups, xv, 110
Unitary representations, 32 ff
Weil, Andre, vii &c
Weil formalism, higher, 107
Weil index, 61, 63 ff
Weil-Kubota formalism, 103 ff
Weil [projective] representation, xv &c
Weil theta functional, 80, 92 ff
Weyl group (physics), 28
Weyl, Hermann, 28
Witt group, 66, 67, 70
Witt-Grothendieck group, 66, 67
Witt-Grothendieck ring, 66, 67
Varadarajan, V. S., xix, 20, 107
Zahlbericht , xiv, 16

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Founded by RICHARD COURANT
Editors Emeriti: PETER HILTON and HARRY HOCHST ADT
Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX,
JOHN TOLAND
ADAMEK, HERRLICH, and STRECKER-Abstract and Concrete Catetories
ADAMOWICZ and ZBIERSKI-Logic of Mathematics
AKIVIS and GOLDBERG-Confonnal Differential Geometry and Its Generalizations
ALLEN and ISAACSON-Numerical Analysis for Applied Science
* AR TIN-Geometric Algebra
AZIZOV and IOKHVIDOV-Linear Operators in Spaces with an Indefinite Metric
BERG-The Fourier-Analytic Proof of Quadratic Reciprocity
BERMAN, NEUMANN, and STERN-Nonnegative Matrices in Dynamic Systems
BOY ARINTSEV-Methods of Solving Singular Systems of Ordinary Differential
Equations
BURK-Lebesgue Measure and Integration: An Introduction
*CARTER-Finite Groups of Lie Type
CASTILLO, COBO, JUBETE and PRUNEDA-Orthogonal Sets and Polar Methods in
Linear Algebra: Applications to Matrix Calculations, Systems of Equations,
Inequalities, and Linear Programming
CHA TELIN-Eigenvalues of Matrices
I CLARK-Mathematical Bioeconomics: The Optimal Management of Renewable
Resources, Second Edition
I COX-Primes of the Formr + ny2: Fermat, Class Field Theory, and Complex
Multiplication
*CURTIS and REINER-Representation Theory of Finite Groups and Associative Algebras
*CURTIS and REINER-Methods of Representation Theory: With Applications to Finite
Groups and Orders, Volume I
CURTIS and REINER-Methods of Representation Theory: With Applications to Finite
Groups and Orders, Volume II
*DUNFORD and SCHWARTZ-Linear Operators
Part I-General Theory
Part 2-Spectral Theory, Self Adjoint Operators in
Hilbert Space
Part 3-Spectral Operators
FOLLAND-Real Analysis: Modern Techniques and Their Applications
FROLICHER and KRIEGL-Linear Spaces and Differentiation Theory
GARDINER-Teichmiiller Theory and Quadratic Differentials
GREENE and KRANTZ-Function Theory of One Complex Variable
*GRIFFITHS and HARRIS-Principles of Algebraic Geometry
GRILLET -Algebra
GROVE-Groups and Characters
GUSTAFSSON, KREISS and OLIGER-Time Dependent Problems and Difference
Methods
HANNA and ROWLAND-Fourier Series, Transforms, and Boundary Value Problems,
Second Edition
*HENRICI-Applied and Computational Complex Analysis
Volume 1, Power Series-Integration-Conformal Mapping-Location
of Zeros
Volume 2, Special Functions-Integral Transforms-Asymptotics-
Continued Fractions

Volume 3, Discrete Fourier Analysis, Cauchy Integrals, Construction
of Conformal Maps, Univalent Functions
*HIL TON and WU-A Course in Modem Algebra
*HOCHST ADT -Integral Equations
JOST-Two-Dimensional Geometric Variational Procedures
*KOBA Y ASHI and NOMIZU-Foundations of Differential Geometry, Volume I
*KOBA Y ASHI and NOMIZU-Foundations of Differential Geometry, Volume II
LAX-Linear Algebra
LOGAN-An Introduction to Nonlinear Partial Differential Equations
McCONNELL and ROBSON-Noncommutative Noetherian Rings
NA YFEH-Perturbation Methods
NA YFEH and MOOK-Nonlinear Oscillations
PANDEY-The Hilbert Transfonn of Schwartz Distributions and Applications
PETKOV-Geometry of Reflecting Rays and Inverse Spectral Problems
*PRENTER-Splines and Variational Methods
RAO-Measure Theory and Integration
RASSIAS and SIMSA-Finite Sums Decompositions in Mathematical Analysis
RENEL T -Elliptic Systems and Quasiconformal Mappings
RIVLIN--chebyshev Polynomials: From Approximation Theory to Algebra and Number
Theory, Second Edition
ROCKAFELLAR-Network Flows and Monotropic Optimization
ROITMAN-Introduction to Modem Set Theory
*RUDIN-Fourier Analysis on Groups
SENDOV-The Averaged Moduli of Smoothness: Applications in Numerical Methods
and Approximations
SENDOV and POPOV-The Averaged Moduli of Smoothness
*SIEGEL-Topics in Complex Function Theory
Volume I-Elliptic Functions and Unifonnization Theory
Volume 2-Automorphic Functions anQ Abelian Integrals
Volume 3-Abelian Functions and Modular Functions of Several Variables
SMITH and ROMANOWSKA-Post-Modem Algebra
STAKGOLD-Green's Functions and Boundary Value Problems, Second Editon
*STOKER-Differential Geometry
*STOKER-Nonlinear Vibrations in Mechanical and Electrical Systems
*STOKER-Water Waves: The Mathematical Theory with Applications
WESSELING-An Introduction to Multigrid Methods
tWHITHAM-Linear and Nonlinear Waves
tZAUDERER-Partial Differential Equations of Applied Mathematics, Second Edition


*Now available in a lower priced paperback edition in the Wiley Classics Library.
tNow available in paperback.