Iwasawa Theory of Elliptic Curves
with Complex Multiplication
p-adic L Functions
Ehud de Shallt
Mathematical Sciences Research Institute
Berkeley, California
@
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Library of Congress Cataloging-in-Publication Data
De Shalit, Ehud.
Iwasawa theory of elliptic curves with complex
multiplication.
(Perspectives in mathematics; vol. 3)
Bibliography: p.
1. L-functions. 2. Curves, Elliptic 3. Class
field theory. I. Title II. Series.
QA247.D36 1987 512'.74 87-1435
ISBN 0-12-210255-X (alk. paper)
87 88 89 90 9 8 7 6 5 4 3 2 1
Printed in the United States of America

TO THE MEMORY OF MY FATHER

Table of Contents
Introduction 1
I Formal groups, local units, and measures 5
§1 Relative Lubin Tate groups 5
§2 Coleman's power series 11
§3 Measures from units 15
§4 The explicit reciprocity law 28
II p-adic L functions 35
§1 Background 36
§2 Elliptic units 47
§3 Eisenstein numbers 56
§4 p-adic L functions (construction) 63
§5 A p-adic analogue of Kronecker's limit formula 87
§6 The functional equation 90
III Applications to class field theory 97
§1 The main conjecture 98
§2 The Iwasawa invariants 108
§3 Further topics 116
IV Applications to the arithmetic of elliptic curves
with complex multiplication 119
§1 Descent, the conjecture of Birch and Swinnerton-Dyer,
and Iwasawa theory 120
§2 The theorem of Coates-Wiles 126
§3 Greenberg's theorem 131
Bibliography 145
Index of Notation 151
vn

ACKNOWLEDGEMENTS
This book, based in part on the author's Ph.D. thesis, summarizes the work of
many people, and benefited from diverse sources. I tried to give appropriate credits
and references in the text, but there are undoubtedly omissions of which I remain
unaware.
Like everybody else in the field, I was heavily influenced by the beautiful ideas
of J. Coates and A. Wiles, and by the analogy with Iwasawa's work on cyclotomic
fields. In addition, Iwasawa's approach to local class field theory, and the work of
R. Coleman, formed the basis of chapter I. Chapter II grew out of, and is a natural
continuation to the work of R. Yager. Part of it was planned as a joint paper with
him, but due to geographical difficulties, this plan never materialized. To all these
people I would like to express my deepest gratitude.
Special thanks go to K. Iwasawa for guiding me in my first steps, to Andrew
Wiles for his encouragement and friendship, and to Robert Coleman who read parts
of the manuscript and made valuable comments.
I would like to thank all the other teachers and friends from whom I learned the
subject, in particular E. Friedman, R. Greenberg, B. Gross, B. Mazur and K. Rubin.
I am also grateful to Harvard University, the Mathematical Sciences Research
Institute, and the National Science Foundation for their support during the
preparation of this book, and to Faye Yeager for the excellent typing.
Ehud de Shalit
Berkeley, December 8, 1986
IX

INTRODUCTION
p-adic L functions are analytical functions of p-adic characters that, one way
or another, interpolate special values of classical (complex) L functions. The first
such examples were the p-adic L functions of Kubota and Leopoldt [K-Le],
interpolating Dirichlet L series. Manin and Vishik [M-V] and Katz [Kl] constructed p-adic
L functions which interpolate special values of Hecke L series associated with a
quadratic imaginary field K, in which p splits. (To fix notation write p = pp).
Their work gave p-adic interpolation of the Hasse-Weil zeta function of certain
elliptic curves with complex multiplication and good ordinary reduction at p (those
whose division points generate abelian extensions of K). The p-adic L function of
Manin-Vishik and Katz is the first object studied in this work.
Our point of view is nevertheless different, and goes back to the two
fundamental papers [C-Wl] and [C-W2] by Coates and Wiles. The program, pursued
by various authors since (see the introduction to chapter II), and which is brought
here to its fullest generality (so we hope), may be summarized in two main steps.
Fix an abelian extension Fi of K, and let Kqq be the unique Zp extension of
K unramified outside p (one of the two factors of p in K). If we assume that Fi is
the ray class field modulo fp, where f is an integral ideal relatively prime to p, we
do not lose any generality, and some notation is simplified. We therefore make this
assumption. The p-adic L function, then, is essentially a p-adic integral measure on
9 = Gal^K^/K).
Now in the first step we are given a norm-coherent sequence /? of semi-local
units in the completion of the tower Fqq = FiKqq at p. Out of each such sequence
we construct a certain measure fip on Q. We describe this construction in chapter I.
In the second step, carried out in chapter II, we introduce special global units, the
elliptic units. They come in norm coherent sequences, so we can view them inside
the local units. When the procedure from chapter I is applied to them we obtain
the p-adic L function.
1

Chapters I and II are carried out in full generality and are also attempted
to be self contained. This results in long tedious computations. The reader who
approaches the subject for the first time is advised to make two simplifying
assumptions: that K is of class number 1, and that the grossencharacters in question are
unramified at p. These eliminate most of the technical difficulties, yet very little is
lost conceptually. If still confused, one may restrict attention to grossencharacters
of infinity type (fc,0). This will only give the interpolation formula for the "one
variable" p-adic L function. We have actually treated this case separately in II.4,
despite some repetition, to facilitate the reading.
Other results obtained in chapters I and II include a new proof of Wiles' explicit
reciprocity law, a p-adic analogue of Kronecker's limit formula, and a functional
equation for the p-adic L function.
The immense interest in Katz' p-adic L functions arises from their significance
to class field theory (abelian extensions of K) and the arithmetic of elliptic curves
with complex multiplication. In the last two chapters we give a sample of results
in these two directions. Although largely self-contained, these chapters are not
intended to be exhaustive, and several topics are omitted. The selection of material,
and sometimes the method of proof, were influenced by our desire to show how the
results of chapter II are put to use.
Chapter III is mainly concerned with the "main conjecture" in the style of
cyclotomic Iwasawa theory. The fundamental idea is that the zeroes of the p-adic
L function ought to be those p-adic characters of Q whose reciprocals appear in the
representation of Q on a certain free Zp-module of finite rank. More precisely, this
module X is the Galois group of the maximal abelian p-extension of Foo which is
unramified outside p. See the introduction to chapter III for more details. We prove
that the Iwasawa invariants of X and the Iwasawa invariants of the p-adic L function
are equal, but we do not go into the recent evidence for this conjecture discovered
by K. Rubin, nor do we give Gillard's proof of the vanishing of the //-invariant.
While elliptic curves are deliberately kept behind the scene in chapter III, their
arithmetic, and in particular the conjecture of Birch and Swinnerton-Dyer, is the
2

main topic of chapter IV. First we show how Kummer theory and descent are used
to relate the Galois group previously denoted by X to the Selmer group over Foq.
Then we give a complete proof of two beautiful theorems of Coates-Wiles and of
R. Greenberg. These theorems are generalized here to treat elliptic curves with
CM by an arbitrary quadratic imaginary field, not necessarily of class number 1.
The crucial hypothesis that must be kept is that the division points of the curve in
question generate an abelian extension of K.
Of the topics not considered here, let us mention p-adic heights and p-adic sigma
functions, the work of Perrin-Riou on the algebraic analogue of the conjecture of
Birch and Swinnerton-Dyer [PRl], and her "Gross-Zagier-type" result [PR2]. As
this book goes to press, K. Rubin has announced important new results concerning
the conjecture of Birch and Swinnerton-Dyer. He kindly allowed me to report on
them here, and we refer the reader to his forthcoming papers for details.
The author is well aware of the lack of numerical examples in chapters III
and IV. These would illustrate the theory magnificently, but due to lack of skill in
computing, I was unable to produce any new examples. There is much relevant
numerical data in the paper of Bernardi, Goldstein and Stephens [B-G-S].
3

CHAPTER I
FORMAL GROUPS, LOCAL UNITS, AND MEASURES
Much of the first half of this book is devoted to the construction of p-adic L
functions associated with quadratic imaginary fields. This construction is "formal"
and "local" in the beginning. Only at a later stage results from the theory of
complex multiplication are incorporated. In chapter I we gather those results which
do not deal with elliptic curves. Our tools are formal groups and p-adic measures.
The key result is theorem 3.7, which describes the structure of a certain module
of local units. This module plays a central role in the following three chapters. In
section 4 we prove a version of the explicit reciprocity law in local class field theory,
that will be needed in chapter IV.
l. Relative Lubin-Tate Groups
1.1 Let R be a commutative ring with identity. For our purpose a (commutative)
one dimensional formal group law over R is a power series F € R[[X, Y]], satisfying
the following axioms.
(i) F{X, Y) = X + Y mod deg 2
(ii) F(X,0) = X = F(0,X)
(iii) F{X,F{Y,Z)) = F{F{X,Y),Z) (associativity)
(iv) F{X,Y) = F(Y,X) (commutativity).
We use the notation f = g mod deg n to mean that / — g involves only
monomials of total degree not less than n. It can be shown ([Haz] 1.1.4) that there
exists a unique power series l(X) € R[[X]] such that F(X,l(X)) = 0.
Let A be an .R-algebra and o an ideal such that A is complete and separated
in its o-adic topology (i.e. A = lim A/an). Then if f,g e o, F(f,g) and
5

t(/) converge to elements of o. We denote them by f[+]g and [—]/ respectively,
and observe that with [+] as addition o becomes an abelian group, written F(a)
("the o-valued points of F"), to distinguish from the ordinary addition on o. These
remarks apply in particular to A = R[[X}] and o = (X), and to the case where
A = R is a complete local ring and o is its maximal ideal. Almost everything
we shall need about formal groups can be found in the book of Hazewinkel [Haz].
Henceforth we let "formal group" stand for "commutative one-dimensional formal
group law", unless otherwise specified.
A homomorphism f between two formal groups F and F' over R is a power
series without constant term such that F'{f{X), f(Y)) = f{F{X, Y)). The collection
Hom(F, F') of such homomorphisms forms a group with respect to the addition law
of F' : (f+g)(X) = f(X)[+]'g(X), and End (F) becomes a ring under composition
as product.
Let R be a domain of characteristic 0, and / 6 Hom{F,F'). Then f(X) =
aX + (higher terms) and the map / h-> a = /'(0) is an injective group
homomorphism of Hom{F,F') into R ([Haz] 20.1). When F = F' this is a ring
homomorphism. We shall write [oj^jr/ or [o]f, or simply [a] instead of / in such
a case. Over the field of fractions K of R all formal groups are isomorphic. Any
isomorphism A : F c± Ga over K (Ga(X,Y) = X + Y is the additive
formal group) is called a logarithm of F. If A is normalized so that A'(0) = 1,
then A'(X) € -R[[X]]X has coefficients in R ([Haz] 5.8). All these statements are
blatantly false (or void) in non-zero characteristic.
Let F be a formal group over a field of characteristic p > 0. Then [p]jr(X) =
X[+]... [+]X (p times) is a power series in Xq with q = ph for some h > 1. The
largest possible h is called the height of F ([Haz] 18.3). If [p]f — 0 F is of infinite
height.
Finally, we shall need the concept of a translation-invariant derivation on F.
This is a continuous derivation D of i2[[X]] (over R) satisfying D(f(X[+]Y)) =
Df(X[+]Y), where Y is treated here as a constant for D (i.e. D is extended to
6

iE[[X, y]] via DY = 0). If R is a domain of characteristic 0, then D = ——
A'(X) dX
where c € R and A is the logarithm of F, normalized to A'(0) = 1.
The multiplicative formal group Gm is given by Gm(X, Y) = X + Y + XY =
(1 + X)(l + Y) - 1.
1.2 Let A; be a finite extension of Qp, the field of p-adic numbers. Let 0 and p be its
valuation ring and maximal ideal. Let the residue field 0/p have q elements. Lubin
and Tate introduced an extremely useful class of formal groups defined over 0 [L-T].
Their handiness stems from the fact that they each possess a special endomorphism
which "lifts" the Frobenius substitution X i-> Xq in characteristic p. Here we
generalize a little (see [dSl]), and as usual in this theory, focus first on the lifting
of Frobenius, and web the formal group around it.
Let d be a positive integer and k' the unique unramified extension of k of
degree d. Let kur be the maximal unramified extension of k, and K its completion.
The Frobenius automorphism (relative to k) <p generates Gal{kur/k) topologically,
and extends by continuity to K. It is characterized by <p(x) = xq mod pur for
all x € 0ur. We let 0',p',<p' denote the corresponding objects for k', so that
<p' = <pd. Finally let v : Kx —► Z be the normalized valuation (normalized in
the sense that v(Kx) = Z).
Fix f € kx, v(£) = d, and consider
Jt = {/ e 0'[[X}} | / = 7r'Xmoddeg2, Nk>,k{^') = £ and / = Xq mod p'}.
Any / in 7^ is going to play the role of an endomorphism lifting Frobenius. Its
differential is /'(0) = n', and its reduction is Xq.
1.3 Theorem. For every f € 7^ there exists a unique one-dimensional
commutative formal group law Ff defined over 0' satisfying Ff o f = f o Ff.
In other words, / € Hom(Ff,Ff). Here, and elsewhere, the superscript tp
means that we apply <p to the coefficients of the power series. Note that Ff € 7$
too, and Ff = -F^/) (apply <P to the equation defining Ff). When d = 1 we
7

are in the situation studied by Lubin and Tate. When d > 1 we call F/ a relative
Lubin Tate group (relative to the extension k'/k). For the proof we need a lemma.
1.4 Lemma. Let f,g E 7$ and let Fi(Xi,...,Xn) be a linear form in
0'[Xi,...,Xn}. Suppose f o Fi = Ff o (<jr,..., <jr) mod deg 2. Then there
exists a unique F 6 0'[[Xi,... ,Xn}} satisfying (i) F = Fi mod deg 2, (ii)
f o F = F* o (g,...,g).
PROOF: (Compare [Se] p. 149). Let / = 7TiX + ..., g = n2X + Set
FM = Fx and define successive approximations F(m) satisfying (ii) mod deg m + 1
through (m > 2) F<m> = F^"1) + Fm where Fm is homogeneous of degree
m. For this we need
/ o (f^-1) + Fm) = (f^-1) + Fmy o g mod deg m + 1
or
/ 0 F{m-l) + ^pm = F{m-l)<p Q g + „mF^
Let t be the homogeneous part of degree m of F^"*-1^ o g — / o F^m-1). Since
F(m-i)<p 0 g = F^-^iXl^.^Xl) = (F^-1))' = / o F(m_1) mod p',
t = 0 mod g>'. We have to find Fm satisfying
Fm - Tr^TT^ = Trf1*.
This is possible because m > 2 and 0' is complete (proceed by induction mod p'r).
Setting F = X)m=i -**"* concludes the proof.
PROOF OF THEOREM 1.3: In the lemma, let / = g and Fi = Xx + X2. We
have to show that F/ = the resulting F, is a formal group law. This is done by
repeated application of lemma 1.4 and is left as an exercise (or look it up in [Se]
p. 150).
Let F be the reduction of F/, i.e. the formal group over O'/p' obtained by
"reading F/ modulo p'n. It is easily verified that F is of height [A: : Qp]. By abuse
of language we refer to it as the height of F/ too.
8

1.5 Proposition. Let f = niX + •••, g = tt2X + •■• be in 7$. Let a G 0'
satisfy a^-1 = n2/^1 (such an a exists by Hilbert's theorem 90). Then there
exists a unique power series [a]/)9 G 0'[[-X]] for which
(0 [a\f,g = aX mod deg 2,
00 [«]/,, ° / = 9 ° [a]f,g.
Ako[a\ftg G Hom(Ff,Fg). If h(X) = n^X + ■•• is another element of 7$ and
b<p~1 — 7T3/7r2, then [afc]/^ = \b\g>h ° [a]f,g- Moreover, the map a i-» [a]/)9 is
a group isomorphism from {a G 0' \ a^-1 = ifilxi} onto Hom(Ff,Fg), and if
f = g it is a ring isomorphism 0 a End (Ff).
Corollary. If f,g G J^, Ff and Fg are isomorphic over 0'.
PROOF: In lemma 1.4 let Fx = aX, so that g o Fi = Ff o / mod deg 2.
Then call the resulting F [a]f>g. Transitivity [afc]/^ = [b]g}h ° [a]f,g is clear from
the uniqueness. To show F9 o [a] = [a] o Ff it is enough to check (in fc'ffX]],
where the inverse is with respect to composition) that g o [a] o Ff o [a]-1 =
[a]'p o Fj o ([a]^)-1 o g, which follows from (ii) and the defining property of Ff.
The only non-obvious point is that every homomorphism A : Ff —► Fg is of this
form. But let a = A'(0). By the general remarks in 1.1 concerning the injectivity of
the map A i—► A'(0) in characteristic 0, it is enough to show that a'p~l = Kil^x,
which is obvious. Notice that even if we allow homomorphisms defined over the
integers of an extension field of A;', we do not get anything new.
1.6 It follows from the last section that the various Ff, for / G 7$, can be viewed
as different models of the same formal group. Over Ok (recall K = kur) one
need not even distinguish between the various £. In other language, Ok is strictly
Henselian so formal groups over it are canonically classified by their reduction (over
Fp), which is determined up to isomorphism by its height. In down to earth terms,
we have the following proposition.
Proposition. Suppose v(£) — v{£') = d, set v = £'/£, and let u be a unit of
k' such that Nk'/kiu) = v (the norm map on units is surjective in an unramified
9

extension). Pick Q G 0K such that fi^-1 = u, and f G 7$. Then there exists a
unique power series 0(X) G Ok[[X\\ satisfying
(i) <pd{0) = 0 o [v]f,
(ii) 0(X) = QX + (higher terms).
Put f = 0f o f o 0-1. Then f G 7? and 6 is an isomorphism of Ff onto Ff,
over Ok-
The proof is left to the reader. Note that Q exists because Ok is complete and
its residue field is algebraically closed.
1.7 Fort > 0 and / G % let /W = v?'_1(/) ° ■■• o <p{f) ° /.Then
/W G fTom(F/,FV9.(/)), and if i/(f) = d, /(«9 = [£]/ G £nd (F/) (we write
[a]/ for [a]/,/). Note also that p>(/(0) ° /(;) = /('+;)-
Let Cp denote the completion of an algebraic closure of Qp, M its valuation
ideal, and Mf = Ff(M), the M-valued points of Ff.
DEFINITION: Let 7r be any prime element in 0, n > 0. Define
Wf = {u G Mf | [o]/(w) = 0 for all a G g>n}
= {w G M/ | [***]/(w) = 0}
= Ker{fW : Mf -> Mv.(/)).
These are the division points of level nm Ff. The equality of the first two expressions
is obvious. They are also equal to the third one because, quite generally, if a G 0',
b G 0 andi/(o) = i/(ft), [a]/,, G #om(F,,F9), then [a]Lg = [afc"1]/^ o [b]f
and [o6-1]/,9 is an isomorphism.
Proposition, (i) W? is a finite sub-0-module of Mf of qn elements. W? C
(ii)Ifu G Wf = \Vf\Wf~1, a i-> [a] f(u) gives an isomorphism 0/pn c±
Wf.
(Hi) Wf = UW? = k/0 (noncanonically) and is the set of all 0 —torsion in
Mf.
The proof is the same as in [Se], §3.6, prop. 6, and we omit it.
10

1.8 Proposition. The Geld k^W?) = k? does not depend on which f we choose,
as long as f 6 J^. It is a totally ramiGed extension ofk' = A;2 of degree (q— l)qn~1
(n > 1), which is abelian over k. Any u 6 W? generates k? over k' and is in fact
a prime element in k?. There is a canonical isomorphism (0/pn)x ^ Gal(k^/kf)
given by u i—► aUi ou(u) = [u-1]/(a;) (u G W?), and this isomorphism is again
independent of f.
PROOF: See [Se] §3.6 and 3.7. The only point that deserves special attention in
the "relative" situation is that A;> is actually abelian over A;, and not only over k'.
First, since it is independent of /, it is clearly Galois over A;. Let r be an extension
of <p to fc£ and a 6 Gal(k^/k'), say a = au. Then tou(u) = r([u-1]/(u;)) =
\u~X\ip{f){T<jJ) = ou{tu) since the isomorphism u i—► au is independent of /,
which proves that a and r commute.
See §4 for more about these fields and local class field theory. The map u i—► au
is the well known local Artin symbol. A fact to bear in mind is that fc? is class
field to the subgroup < £ > • (1 + pn) of kx (< f > • 0X if n = 0). If k" is an
unramified extension of k' of degree e, k"k^ = A;?,, for f = fe. In particular, for
any n, f i and £2, there exists k" for which fc"A;? = k"k? . See also the forthcoming
book of Iwasawa [Iw 3].
2. Coleman's Power Series
Robert Coleman introduced in [Coll] a method of dealing with norm-coherent
sequences in the tower of local fields k1}. The Coleman power series is a simple but
ingenious device which associates with any such sequence a power series over 0'.
2.1 THE NORM OPERATOR: Notation as in §1 let R = 0'[[X]], £ € kx,
z/(f) = d, and / G J$.
Proposition. There exists a unique multiplicative operator M : R —► R (M =
11

Mf when we want to emphasize the dependence on /), such that
(1) Mh o f = JJ h(X[+]u) Vh G R.
(addition is on the formal group Ff, of course). It enjoys the additional properties:
(i) Mh = h* mod p',
(ii) Mv(f) = <p o Mf o tp-1,
(iii) Let My — M^i-i^ o ••• o .A/^/) o A//. Then
(M}*>h) o /W(X) = J] h(X[+]u).
uEWJ.
(iv) Ifh <E R,h = 1 mod pH (i > 1), then Mh = 1 mod p,i+1.
PROOF: Clearly (1) characterizes the power series Mh uniquely, and from this
N(h\h,2) — Nhi ■ Mh>2, if we only show how to construct Nh. Let go(X) be the right
hand side in (1). Then g0(X[+]u) = g0{X) {u <E W}) so
go(X) — <7o(0) = gi{X) • f{X), by Weierstrass preparation theorem. Now
gi(X) is Wj-translation-invariant, so similarly gi(X) — <7i(0) = 02(^0 • f[X),
etc. Hence
go(X) = 9o(0) + 9l(0)f(X) + g2(0)f(X)2 + ■•■
and this infinite series converges in the topology of R. Mh = <fo(0) + <7i(0)X +
g2(0)X2 + • • • therefore satisfies (l). To prove (i) note that
Mh{Xq) = Mh o f = h{X)q = ^(X9) mod p'.
Point (ii) is obvious, applying <p to (1). The case i = 1 of (iii) is (1) and to prove
12

the general case we proceed by induction, assuming it for i — 1. Then
= II Mfh{f{X)[+]v{f)0) (induction)
I] MfHf{X[+]fa)) {f-.W} -* Wij}))
aEWJ. mod Wj
II II KX[+}M+}fl) (by(l))
aEWJ. mod Wj iEW}
= JJ h(X[+]fu) (grouping together).
Finally let pn be the valuation ideal in A;> (so po = &')• We prove (iv) by induction.
What we need from the t — 1 step is that if h = 1 mod p'1, then Mh = 1 mod pH.
For i = 1 this is easy to check directly. Now let i > 1 and consider the congruence
AM(**) = Mh o f = h{X)q = 1 mod g>'Vi
which holds because h = Mh = 1 mod p'x. Since .A/h G 0'[[X]], actually
Mh = 1 mod g>'i+1.
Call .A/ (Coleman's) worm operator. Notice that if h G XXRX (i > 0) then
Mh G X*i2z as well, so M extends to 0'{{X))X (0'((X)) is the ring of Laurent
series over 0') and the same remark holds there too. We shall be applying M
primarily to power series in 0'((X))X. If a is in 0', Ma = aq.
2.2 Theorem. Let f3 = (/?n) be a norm-coherent sequence in (k'g) (i.e. (3n G
(fc?)z for n > 0, and Nm>n(l3m) — fin where Nm>n is the norm from km to h^ for
m > n). Let v((3) be the normalized valuation of ft (i.e. fin0n = pn" for any
n > 0, where 0n and pn are the valuation ring and ideal in k?). Fix f G J<t, and
ui G W^i{f)\W^l}{f) = ^;_i(/) such that (p-7)(w<) = w."-i(l < » < <*>)•
Then there exists a unique gB G XVW • 0'[[X]]Z such that {(p-'gp)^) = ft for
alii > 1.
By abuse of language we shall call (wi)t>o a generator of the Tate module of
Ff (but notice that Ui is a division point on ■F,y)-«(/) and not on Ff). We shall call
13

gp(X) Coleman's power series of /?. It converts /-compatible division points into
norm-compatible elements. A stronger result holds "at finite levels n" but we shall
not need it, and its proof is slightly more complicated (see [Coll] for the "absolute"
case k' = A;).
PROOF: Fix to > 1. Since um is a prime of k1?, which is a totally ramified
extension of A;', there exists h G XV^RX such that h(um) = /?m.Ifl < n < to,
W^f^M = u£z?h o (P-"7)<"-»>(Wm)
(2) = II M«m[+]oO
v~mf
= Nmtn(h{wm)) = pn.
On the other hand,
The first equality follows from multiplicativity and prop. 2.1(H). Then from the fact
that h G XV^RX and from 2.1(i) the quantity in parenthesis is in 1 + p'R. The
congruence now follows by successive application of 2.1(iv). Let gm = yiK_jn,h.
Together (2) and (3) imply for all 1 < n < to that (p"njm)H/|8n =
1 mod p'm-n+1. Since XV^RX is compact, we may choose in it a limit point g
of {gm}. Then by continuity (<p~ng)(<jJn) = Pn for all 1 < n. The Weierstrass
preparation theorem shows that g = gp is unique, so actually g = lim gm.
This concludes the proof of the theorem.
2.3 Corollary.
(i) gpp> = gp • gp<,
(ii) Mfgp = g%,
(iii) gpio)1-*-1 = fa (HP) = 0).
(iv) If a G Gal(k^/k') and k(g) G 0X is the unique unit for which o(u) =
[k(«7)]/(w) for any u G Wf (and any f G 7$—see 1.8; k^ = Ukn, Wf =
UWf), then ga(p) = gp o [k(<t)]/.
14

PROOF: Point (i) is clear from the defining property of gp. For (ii) notice that
= V/(^"'w) ° (p"V)(w.-)
= ^~*(^/^)K-i),
while we also have /?t_i = (v1-,0/9)(wt-_i). Hence flrj — A//^ has infinitely many
zeroes, so must be identically 0. Point (iii) follows from (ii) because when v(0) = 0,
V3"1^^) * -Ni,o/?i = V3"1^^) • A)- Note that (iii) is in accordance with
the known fact that Nk'/k{@o) S < f >, and if /? is a unit, Nk>/k{Po) = 1-
Finally (iv) follows from <p~%{gp o [/c(cr)]/)(wt-) = {<p~{gp) o [K(ff)]v-</(w,-) =
(p~*0j9)(<Wt) = ^((V3-*^)^)) = ffjS,-.
For a generalization of 2.3(iv) to cr G Gal(k^/k) see 3.7(15).
3. Measures from Units
It is instructive to think of (classical, abelian) L-functions as functions of
characters of the Galois group, or quasi-characters of the idele group. With this in mind,
Np_* and x{p) appear on equal footing, because both arise from characters on the
idele group.
p-adic (abelian) L-functions, like those of Kubota-Leopoldt, Deligne-Ribet, or
Katz, should be conceived in the same light. This time the situation is even
better, because grossencharacters of type Aq (and in particular N) which could not
be interpreted as C-valued characters of any Galois group, become just so when
considered p-adically, i.e. as Cp-valued (Cp is the completion of an algebraic
closure of Qp). Needless to say, the Galois group is now profinite and not necessarily
finite. This point of view is due to Weil [We]. As functions of p-adic characters on
a group G, the p-adic L functions are more than just locally analytic. They belong
to the Iwasawa algebra. This means that their value at a character x is obtained by
15

integrating x against a p-adic measure on G. For all practical purposes, the p-adic
L function may be identified with the measure. See [Mazl] and [Se2]. The
translation into the language of power series, or functions of "5", is routine, as will be
explained, and not always beneficial.
In this section we first review some general definitions. We then go back to
the situation of §2, but assume that the formal group is of height 1 (k = Qp).
The method of Coleman's power series is used to turn a coherent sequence of units
into a measure on the local Galois group. This important procedure underlines the
construction of chapter II.
3.1 Let M be an abelian group and G aprofinite group (usually a Galois group). An
M-valued distribution on G is a finitely additive function from the Boolean algebra
of compact-open subsets of G to M. We denote the abelian group of M-valued
distributions on G by A(G, M). If M is an A-algebra for a commutative ring A, so is
A(G, M), and A.(G,A) is even a ring with convolution as product. The convolution
of A and // is defined as follows. If U is open and compact, so is the subgroup
H = {7 e G I Ui = U}, so G = up=1 OiH, a disjoint union. Set
(A • H){U) = £?=i \{UaZx) • n{°iH) (= fG \(Uo-l)dn(a)).
Now suppose M C Cp. If M is bounded (\x\ < R < 00 for all x G M
and some R) we call an M-valued distribution a p-adic measure. If M C Dp =
{x G Cp I \x\ < 1} we talk about integral measures.
If G is finite A(G,M) a M[G] under A h+ £*£<? A({cr})cr. If M = A
is a commutative ring, convolution corresponds to the usual product. In general,
A(G,M) = lim k{G/H,M) a lim M[G/H] = M[[G]] (notation), where the
inverse limit is over the family of normal subgroups H of finite index in G. M[G] is
dense in M[[G}}.
The (standard) Iwasawa algebra is obtained when M = Zp and G = T, a
group isomorphic to Zp (usually 1 + pZp for odd p, 1 + 4Z2 when p = 2).
It is well known that in this case A ~ Zp[[5]] non-canonically. The isomorphism
depends on a choice of a topological generator u of T, and maps ua to (1 + S)a.
16

When T = Zp however (written additively), we naturally take u = 1. The power
series Pn(S) corresponding to fi is then
(1) P^S) = f (1 + S)a dn{a).
Jzp
If x '• G -* Cp is any continuous function, and A is a p-adic measure, then the
Riemann integral fG x{a)d\(o) exists. Simply approximate x uniformly by locally
constant functions. In particular, if x S Hom(G, Cp), then
(2) f X d(Aji) = [ Xd\ • [ xdft.
J G J G J G
The augmentation ideal in A(G, A) is the kernel of fi ►-»■ /^(G).
Finally, assume that G is commutative, and let S C A(G, A) be the
multiplicative set of non-zero-divisors. A pseudo-measure is an element X/s of 5_1A(G, A).
If X S Hom(G, Cp) and f x ds ^ 0, set / x d(\/s) = / X ^A/J x ds. In view
of (2), this is well defined.
REMARK: If G admits subgroups whose indices are divisible by arbitrarily high
powers of p, then there is no non-trivial p-adic measure on G which is translation
invariant. In our framework, therefore, there are no Haar measures.
3.2 A FORMAL CONSTRUCTION OF MEASURES: We return to the situation of
§§1,2 and assume from now on that Ff is a relative Lubin-Tate group of height one.
Thus A; = Qp, 0 = Zp, and k' is an unramified extension of Qp. By proposition
1.6 there exists an isomorphism
(3) 6: Gm * Ff, T = 6{S) = U • S + ••• G 0K[[S}],
over the ring of integers of the completion of A;ur. Let f(T) = n'T + •••be
the special endomorphism of Ff. The special endomorphism of Gm is of course
[p](S) = pS + •••. Proposition 1.6 implies then
(4) n*"-1 = tt'/p, / o e = ** o [p],
17

where (p is, as usual, the Frobenius automorphism.
Fix once and for all primitive pn roots of unity fn (n > 0) such that f£ = fn_ \.
Letting
(5) un = O^ (fn - 1)
we see that un G W"n, and (<p~nf)(un) = un-i. Thus (un) is a generator of
the Tate module of Ff in the sense of 2.2.
Let A = Xf be the logarithm of Ff, normalized so that A'(0) = 1. Then
\'{T) G 0'[[T]}X (cf. 1.1).
3.3 For any local field k let U(k) be the subgroup of principal units (units congruent
to 1 modulo p). Let
(6) P = {pn) € U = lim I7(*jf)
be a norm coherent sequence of principal units in the tower A;? = k'(Wf).
In theorem 2.2 we attached to /? a power series gp{T) G 0'[[T\[X such that
{(P~n9p){(jJn) — Pn {n > 1). Since fin are principal units gp = 1 mod (p',T) and
we can take its logarithm using the power series expansion of log (1 + h). Thus
log gp G k'[[T\\.
Lemma. The power series
(7) \oTgp{T) = log gp{T) - - Y, lQg 9f>{T\+)w)
has integral coefficients.
PROOF: Let g = gp. Since gp = g^ o f mod p' we get from corollary 2.3(ii)
that gp = Yl flf(^1[+]w) mo^ &'• Taking logarithms, and noting that np \ pn for
n > 1, we see that p times the right hand side of (7) is congruent to 0 mod p',
which proves the lemma.
We remark that (7) is also equal to log gp — - log^J o /).
18

Let ap(S) = log gp o 0[S) £ Ok[[S]] and let \ip be the Ojf-valued measure
on Zp for which PM = a p. In other words,
(8) afi{S) = /" (1 + 5)" dfip{a).
Jzp
The measure /ti^ is actually supported on Z* This is a consequence of (7)—
removing the Euler factor at p—and without it we would get a distribution, but
not a measure. Indeed, if ji. = f<i\Z*, extended by 0 to pZp, then Pp, = PM, where
(7') PM(5) = PM(5) - - Y. PM* + s) ~ !)•
" ?p=i
But with PM = ap, (7) implies PM = PM, hence p, = \i.
We may now use the isomorphism
G = Gal{kt/k')
'(u) u € Wf
to pull back fxp to G (cf. 2.3(iv). Note that when we follow k by the local Artin
symbol—see 1.8—a goes to a~l). We still denote our measure by fxp.
(») (K: g * z;
1' U(«) = [«W]/(
3.4 DEFINITION. For any ft £ U let up be the Ok — valued measure on G —
Gal(kz/k') satisfying
(10) loTJ/9 o 6(S) = / (1 + S)K& dnp{o).
Jg
Some elementary properties of the map ft *-* ftp are summarized below.
Lemma.
(i) »pp' = Up + »p-
(ii) Let 7 G G. Then n^p^U) = Hp{U).
(Hi) fxp depends only on the choice of (fn). If $„ = f« (7 G G), then the
resulting measure is given by fx'p(U) = fXpfaU).
PROOF: (i) and (ii) are consequences of 2.3(i) and (iv) respectively, and (10). Part
(iii) is proved similarly, and is left to the reader.
19

Let Q = Gal(k^/k), so that Q/G = Gal(k'/k) is cyclic of order d. Suppose
U is an open subset of Q contained in a coset of G, If 7O" C G define (i>p{U) =
U»i(p){lU) (now 7 € £). Part (ii) of the lemma shows that this is independent of
•7. We can now extend \ip to a measure on Q, so we get a map » : U —»• A((), Ok),
i((3) = \ip. Since gp is non-constant for /? ^ 1, (10) shows that i is injective. The
lemma implies the following.
Corollary. The map % is an injective homomorphism of Zp{[§]} — modules.
3.5 THE COATES-WILES HOMOMORPHISMS: These variants of fxp were first
considered by Kummer ([Kum] p. 493) and are also known as Kummer's logarithmic
fl d
derivatives. Let D = w. r —- be the translation invariant derivation of Ff over
\'{T) dT J
0K (fl as in (3)). Letting T = 6{S) and using A o 6{S) = fl • log(l + S)
we see that in terms of S, D = (1 + S) —, the standard translation invariant
derivation of Gm. The moments of \ip are given by the formula
(11) / K{a)kdn0{a) = Dk log^O) (k > 0).
Jg
When Ff is an (absolute) Lubin Tate group, so (p acts trivially on gp{T), this
can also be written I 1 — — J Dk log gp(0), w = /'(0). The map <pk{P) =
jG K(a)kdfj,p(a) is called the kth Coates-Wiles homomorphism. It satisfies
(i) <ph{pp) = <pk(/3) + <pk[F)
(») MliP)) = <l)k ' <Pk{fi) (l € G).
Together <pk for k > 0 determine \ip uniquely on G (but not on Q).
3.6 Another useful formula is for Hp{Gn), where Gn = Gal(k^fk^)
= /c-1(l + pnZp), n > 1. From (8) we deduce at once
1 pn~1
(12) p0{Gn) = ^ E Mrf - !) ' (nJ-
v j=o
Here fn can be any primitive pn root of 1. The formula remains valid with ap instead
of a,p.
20

3.7 Let N > 0 be the largest integer such that ftv G k^. If [k1: Qp] — d, local
class field theory shows that N is the largest integer for which pd£-1 = 1 mod pN
(cf. the discussion at the end of 1.8). N — oo if and only if Ff = Gm over 0'. We
call N the anomality index of JF/ (over A;f), and call Ff itself anomalous if iV > 0.
Since Q acts on (mpn, Ok ® A*p* = 0k/pN{1) (Tate twist) is a A.(G, 0jr)-module.
Let N : Q -> (Z/p^Z)* be the character giving the action on /ttpw. Observe
that for-7 G G, N(i) = /c(7) mod p^. Define a map j : A.(G,0k) —¥ 0k/pN{1)
by
j{fi) = / N{o)dp{p).
Jq
Then j is a surjective homomorphism of Q modules. In the following key result i
was extended by linearity to the completed tensor product U ®zp Ok- Its proof
will take up the rest of this section.
Theorem. Tlie sequence
(13) 0 -* U ®Zp 0K -*- H9,0K) -*-* (0k/pn)(1) -» 0
is exact (If N — 00 it ends with Ok (1)).
PROOF: We have already mentioned that j is surjective. We shall show now that
j o j = 0. Let 0 G U, A calculation based on (11) gives
Jg NW4.,w . /c .(cWM = n g - (n gg)' -p".
Now /c can be extended to £ in the following way. If a G Q, there is a unique
isomorphism h : Ff m .F^/) such that ^(w) = o(u) for all u; £ VK/. This & is
of the form [/c(a)]/>(T(/) for a unique/c(a) G 0,a:, and/cfa)^-1 = /'(O)'-1. Then
k: Q —*• (0r)x is a. 1-cocycle
(14) k(to) = /c(a)r • /c(r),
and the generalization of corollary 2.3(iv) is
(15) gff{P) = ffj o [«(a)]/^(/).
21

From the definition of up one gets now
J N{o)dvfi{o) ^ £ ^(i - <p) (n ^)
This proves jot = 0. The rest is more difficult. Although not absolutely necessary,
we shall first reduce the proof to the case of Gm (sections 3.8-3.9).
3.8 We want to change perspective somewhat. The local Artin map identifies
U = lim U(k%) with Gal(kurM(k^)/kab) where M(L), for any L, is the maximal
abelian p — extension of L. Viewing Fj over an unramified extension k" of k', and
letting Q' = Gal(k"kz/k), W = lim U(k"k%) (note that A"AJ = *J„ with
£' — flfc :fc 1), we obtain an exact sequence like (13), but over k". The two exact
sequences are related through a commutative diagram.
0 _ u'&Ok - HP, 0K) ± (0k/pn')(1) -> 0
(16) | I I
o-+u®oK -u ng,oK) ± (oK/PN)(i) ->o.
The first vertical arrow is Nk"/k' ® 1> the middle one is the map induced from the
projection Q' —*• Q and the third is the canonical map. All three are surjective.
Taking the projective limit over k' C k" C kur we arive at
(17) 0 -> Gal{M{kab)/kab) ®0K + A{ga,0K) ± 0*(1) -> 0
where Qa = Gal(kab/k), and where we used the Artin map to identity lim W with
the Galois group.
Lemma. (17) is exact if and only if (13) is (for every k').
PROOF: If (13) is exact for every kf, then so is (17) because the vertical arrows in
(16) are surjective. Suppose (17) is exact. Let H = Gal{kab/k^) =* Gal{kur/k')
and let a be the generator of H restricting to <pd on kur. Then (13) is the sequence
of a-coinvariants of (17) (the a-coinvariants of a module E are E/{a — l)E). If
22

(a — 1) | Ok (1) is injective, the snake lemma concludes the proof. This is always
A A
the case, unless Ff cz Gm over 0 . Since in the case of Gm we shall prove the
theorem directly, we assume that Ff ^ Gm over 0'.
The advantage of (17) is that the particular formal group disappears from the
scene. The following proposition is interesting in its own right.
3.9 Proposition. The homomorphisms i and j in (17) are independent of k', f
or f G J<t. In other words, they are canonically associated to k = Qp.
PROOF: Let Fi and F^ be two relative Lubin Tate groups, both defined over 0'.
Let t G Gal(M(kab)/kab) and let ^(r) = m and i2{r) = n2 be the measures
associated to t on Qa (relative to F\ and F<i). Let 6X : Gm ^ F\ and r\ : F\ c* F2
be isomorphisms over Ok, 02 = V ° 0i, and set Wi)n = {(p~n0i){^n ~ 1) and
similarly W2,n, as in (5) above. Write k% = k'(ui>n) and similarly k% for the fields
denoted fe> before. For each n there exists an unramified extension k" of k' for
which k"ki = Wk^ (see 1.8) and we consider the collection S of all subgroups
of Qa of the form H = Gal(kab/k"ki) for such pairs (n,k"). S is a basis of
neighborhoods at zero, and we shall prove Hi(H) = H2(H) for H G S. This, for
all r, will prove our claim, because
(18) *{T){<rH) = i{c-lTo){H)
where a G Qa and a is any extension to M(kab), and where i is ii or i% (see lemma
3.4(H)).
So fix n and k" as above and consider F\ and F% over k". Let 0i = (0i,m) £
Ui = lim U[k"k™) and (similarly) 02 G i/2 correspond to r. Then 0iim — 02,m
for 0 < m < n. The field diagram is as below.
23

.ur
k"
k'
Kl — K2
.ab
h
H = Gal{kab/k?)
kx = Ufcf*,
k2 = Uk?.
Let gi and g2 be the Coleman power series of 0i and /?2 (relative to F% and F2).
From 3.6 (12) we get
Pn-i
(19)
"*(*) = ^El0S^° Mrf-1) • £"'"
y=o
and similarly for /tt2(.ff). Now for 0 < m < n
£>
'(<7l O $i){sm-l) = £l,r
#»,r
^"m(ff2 O «2)(fm-l),
so extending <p to feab so as to fix all fm, and applying <pm to the last equality,
we see that g\ o 8i($m — l) = jj ° ^2(fm — 1) for all 0 < m < n.
Conjugating by Ga/^^m)/^) it follows that gi o ^i(^-l) = g2 o 02(f'-l)
for all 0 < j < pn — 1. It remains to show gi(0) = 02(0). Note first that
X-*{(1 + X)Pn - 1} divides gi o 6X - g2 o 82, so gi(0) = g2{0)modpn.
Consider now Fi and F2 over k', let 61 = Nk"/k'Pi, &2 = Nk"/k'P2 be the
inverse systems of local units corresponding to r over k', and /ii = Nk"/k'9i,
ti2 = Nk"/k'92 their Coleman power series. It follows that hi(6) = /i2(0)modpn.
With k' held fixed, 6t- and hi are fixed, and n can be chosen arbitrarily large.
Hence /h(0) = /i2(0). Repeating this argument with k" instead of k', we deduce
01(0) = fl'2(0)> and the proof of the proposition is concluded.
24

3.10 Proposition 3.9 shows that it is enough to prove our theorem in the case
Ff = Gm (over arbitrary k'). In this case we shall actually show that
(20) 0 - U ®zp 0' ± 0'[[Q)\ ± 0'(1) - 0
is exact (notice that i now maps U into 0'[[i?]]). Exactness of (13) will follow from
the (topological) flatness of Ok over 0'. There are two points to check. Injectively
of i (this is a-priori clear only when k = k'), and that any \i £ Ker(j) is \ip for
a suitable /?. We begin with a series of three lemmas due to Coleman [Col2].
Lemma. Let M be the norm operator for Gm over 0' (see 2.1). Let O'/p' — Fq.
Then for any g e F, [[£]]* there exists g e O'ffS1]]35, Mg = g* and
g = g mod p'.
PROOF: Let go be an arbitrary lifting of g~. Let grt- = <p~lM^go- It follows from
2.1(i) and (iv) that g = lim grt- exists and satisfies Mg = g^, g = go mod p'
(compare the proof of theorem 2.2).
3.11 Lemma. Consider the map d : F9[[X]]Z -> Fg[[X]], dg = X • g'/g. The
image of d consists of all the power series h = X)^Li cnXn with cpn = c^.
PROOF: Modulo F*, which belong to Ker(d), Fg[[X]]z is generated (topologically)
by 1 — aXk for a £ F*. and k > 1. Since
oo
(21) d{l - aXk) = -k - J^ ajXjk
satisfies the condition cpn = c£, it remains to show that this is the only
restriction. Let h = Y^Li cnXn be as above. If p \ k, then d(l - aXk) =
—kaXk mod Xk+1, so adding up successively power series of the form (21), for k
relatively prime to p, we find in the image of d a power series that agrees with h
at least at cn for p \ n. But since cpn = c£ holds for h, as well as for (21), this
power series automatically coincides with h.
25

3.12 Lemma. Let £ be the subgroup of all h G 0'[[X]] satisfying
(22) fc*((l + IF-1) = ^ fc(f(l + X) - 1).
p fp=l
Then X(l + X)-1 • £ mod p' = Imoge(d).
PROOF: For any h the right side of (22) is an integral power series, and an argument
parallel to 2.1 shows that there exists a unique Sh £ 0'[[X]] such that
(23) sh((i + xy - i) = - J2 h(c(i + x) - i).
" fp=l
We are interested in £ = {h \ Sh = h*>}. Let 6 : 0'[[X]]X ~> 0'[[X]],
6g — (1 + X)g'/g = D log g. (D=(l + X) —— is the translation invariant
dX
derivation on Gm). Thus d = X(l + X)-x6 mod p'. If Mg = g* then S{6g) =
{6g)'p so in view of lemma 3.10 X(l + X)-1 • £ mod p' D Image(d). Suppose
this was a strict inclusion. From the description of Image(d) given in 3.11, would
then follow that there is some h G £ with X(l + X)_1/i = ££Li cnXpn mod p',
and not all cn = 0 mod p'. Equivalently, h = YZ?=\ cn{Xpn~l + Xpn) mod p'.
However, S(XP") = Xn and S(Xpn_1) = Xn_1 mod p', as a simple calculation
reveals. This contradicts Sh = h^, and the lemma is proved.
Corollary. The map 6g = (1 + X)g'/g maps {g G 0'{[X]]X | Mg = g^}
onto {h G 0'[[X]] | Sh = h^}. Its kernel consists of the roots of unity in 0'.
PROOF: Let H be the first group and £ the second. Clearly 6 maps M. into £, and
the lemma shows 6(H) mod p' = £/p£, so 6(M) = £. The statement about
the kernel is obvious. The corollary is all we shall need to finish the proof.
3.13 Lemma. Let v be an 0' — valued measure on Z* and assume that
Trk,/k(u(ZD) = 0. Then there exists h G 0'{[X}] satisfying (22) (i.e. Sh = h*>)
such thai, with
h{X) = h{X) - - Y, %(1 + X) - 1) = h{X) - hT{{l + X)p - 1)
^ fp=i
26

(see (7')), h(X) = JZx (1 + X)*dv{a).
p
PROOF: Extend i/toa measure on Zp by (1) u(p{U) = <p{v{U) (U c Zp5),
(2) u(ptdZp) = 0 [d = \k' : k}). These two conditions are compatible because
Trk./k{u{ZD) = 0. Let h = Pv, i.e.
h{X) = f (1 + X)adu{a).
Jzp
Then fc"((l + X)p - 1) = JZp (1 + X)P-^(a) = JpZp (1 + X)««fc/(a) =
/i(X) — h[X), so h^ = S/i and the lemma is proved.
3.14 We are now in a position to conclude the proof of (20). Recall our notation:
[*' : Ik] = d, kp = Uk{(n)t £ = pd, k^ = kpk', and G = Galfe/h'). Let
H = Gal(k^/kp) and denote by <p the unique element of H inducing <p on k'. Then
g = GxH = Gx<ip>. Note that in the notation of 3.7 (14) and (15),
n((p) — 1, and g<p(p) = g%. Let C be the subgroup of 0'[[^]] consisting of all \l
satisfying
(a) j{fi) = 0 (6) n{<p-xU) = <p(»(U)).
Since k'/k is unramified, it is easily seen that Ker(j) = 0' ®zp C It is therefore
enough to prove that i maps U isomorphically onto £. That i(U) C t follows at
once from g^p) = g'a. Let \i € £, and view its restriction to G as a measure on
Zp through the cyclotomic character /c (= N). Because of (6), condition (a) reads
Trw/kilnx adfi(a)) = 0. Let dv(a) = adfx(a), and let h be the power series of
p
lemma 3.13. Let g be as in corollary 3.12, normalized (dividing by a root of unity
from 0') so that g(0) = 1 mod p'. Then /?„ = {<p~ng){$n ~ 1) € I7(fcj?).
.Vtf = ^ implies Nm,nPm = Pn, so /? G U, g = gp, h = D log gp, and
h = D log^ = / (1 + X)aad/x(a).
It follows that a • df/,p(a) = a • d/^a), which proves /it = \ip on G. Comparing (6)
with the way /it/? was extended to Q (3.4), fj, = fi,p = i(/?). Therefore i(i/) = £.,
and the proof of theorem 3.7 is complete.
27

REMARKS: It was not necessary to reduce to the case of Gm. The analysis of
3.10-3.14 could be carried out with the original Ff, mutatis mutandis. However,
proposition 3.9 is independently interesting, so we chose to prove it first.
Iwasawa (and later Wintenberger [Win] in a more general context) determined
the A-structure of U as a Zp[[£]] module ([Iw] 12.2). The significance of our theorem
is that U <§> Ok is canonically embedded in Ok-[[,P]] with the right cokernel.
Theorem 3.7 will be used later on (III.l). Special cases of it are equivalent
to lemmas 23-26 and theorem 27 of [Ya], but we found the proofs there somewhat
ad-hoc. In particular, they rely on the general structure theorems of Wintenberger,
while the approach taken here does not. In fact, in our case we can deduce Win-
tenberger's results from theorem 3.7.
4. The Explicit Reciprocity Law
Our aim is to prove a version of the explicit reciprocity law in local class field
theory, that was discovered by Wiles [Wi], generalizing earlier work of Artin-Hasse
[A-H] and Iwasawa [Iw4]. For further generalizations, as well as discussion of the
literature, see [dS2] and the bibliography there. This section will be needed only in
chapter IV, so the reader may skip it and proceed directly to the next chapter.
Notation and assumptions are as in §§1,2.
4.1 KUMMER THEORY ON LUBIN TATE GROUPS: Let Ff be a relative Lubin Tate
group defined over 0', and A;? = k'iWJ) the field of division points of level n. Let
an € pn (the valuation ideal of k£). Recall that f^ = <pn~xf o ... o <pf o f e
Hom{Ff,F<pTif). Let an be any root (in the open unit disk of A;) of
(1) (^"V(n))(an) = On.
The extension k^(an) = L of k% is abelian. Indeed, Gal(Lfk^) is embedded in
W™ni via the map a ►-»■ a{an) [—]an. This map is a group homomorphism
independent of an because W£_nf C jfej. The Hummer pairing Ff{pn) x Gal(k^'ab/k^) ->
28

W"n. is the pairing
(2) < e*n,a> = <j(an)[-]an
(Lab, for any L, is its maximal abelian extension). It is bilinear, because in addition
to the linearity in a mentioned above, < an[+]a'n,a > = < ocn,(j > [+] < a'n,cr > .
The pairing is even 0-linear in the first variable: < [e]an,a > = [e] < an,a >
for any e E 0. Clearly the kernel on the left is f^n\pn)i but it is not at all clear
what the kernel on the right is, i.e., what is the field obtained by "extracting f^
roots" of all a E pn. We only know so far that it is an abelian extension of A;? of
exponent qn (however, see [dS2], theorem 2).
Using the local Artin map, we define a pairing Ff(pn) x (A;>)z —»• W£_n* by
(3) {ocn,fin) = («„,/?„)„,/ = <an,apn>
where op is the Artin symbol of fi. Our aim is to compute (3) by means of some
analytic formula.
4.2 For simplicity of notation and exposition we shall assume from now on that
k' = A;, i.e. Ff is an absolute Lubin Tate group. The "relative" case is treated
similarly, but because of the need to work with all F^-if at once, the notation
becomes cumbersome.
Assume, therefore, that f(X) = nX + • ■ •, -k a prime of A;, and fix a generator
of the Tate module (un) as in 2.2, un E WJ, [n](<Un) = <^n-i-
Theorem (WILES [Wl]). Let j3n E (A£)z satisfy Nn(/3n) E < tt > (Nn is
the norm from k% to k; this assumption means that fin sits in a norm-coherent
sequence j3 E B = lim {k%)x). Choose a norm coherent sequence 0 with /?„ as
its nth coordinate, and let gp be the Coleman power series of 0 (2.2). Let X(X) be
a logarithm of Ff, so that X'(X) E 0[[X]]X. Put 6g = — ■ - E X~x ■ 0[[X]].
A g
Let
(4) [otn,l3n} = [otn,Pn]nJ = [7r-nTrkn/k{\{ctn)6gp{un))}f{un).
29

(The fact that the quantity in square brackets is in 0, and modulo 7rn depends only
on an,/?n, is part of the theorem). Then
(5) {<Xn,Pn)n,f = [ocn, (3n}n,f.
The analytical pairing (4) is (in principle and in practice) computable. Indeed,
one only needs to compute a certain element of 0/7rn, and this requires a finite
amount of computation.
Our proof is different from Wiles' in one important way: It is based on the
observation that it is easier to prove (5) for all Lubin-Tate groups at once, adjusting
Ff "to fit" /? (see the "reduction lemma" below).
4.3 Let B = lim (A;£)z, the inverse limit taken with respect to the norm maps.
Thus B is an extension of Z by the profinite group F* x U, where U is the inverse
system of principal units.
Let Af = lim Ff(pn), where Ff(pn) maps to Ff{pn+1) by / = [jt].
The first observation is that the Kummer pairings (3) combine to give a single
pairing
(6) ( , ), : Af x B -> Wf.
To check this, let m > n, am = fm~n{an), f3n = iVm>n/?n, and observe
that (an,/?n)n = (am,/?m)m. Similarly, denoting the trace from A;£ to k by Trn,
Trm{\{ocm)6gp{um)) = rrn(7rm-"A(an)rrm,n%(u;m))
= Trn(nm~n\(an)irm~n6gp(u)n)) because property 2.l(iii), asserting gp o /* =
ELew"' 9p{X[+]u), gives, upon applying D log = 6 to it,
(7) r'Sgp o P{X) = £ Sg0{X[+]u).
It follows that [ocn,fin]n,f = [o=m, fim]m,f, hence the symbol [ , ]/ is also defined
on Af x B (provided it makes sense!).
30

The second observation is that both pairings ( , )/ and [ , ]y are bilinear, so
it is enough to prove (5) for prime /? (i.e. (3n is prime for every n), as these generate
B.
The third observation is that if—with a given /? £ B—(5) holds for one f £ 7%
and every a £ Af, then it holds—with the same/?—for all f £ Jn and a £ Af.
Indeed, if r\ : Ff ~ Ff> is an isomorphism over 0 (where /,/' £ 7„), a £ Af
and/3 £ B, then r)((oc,P)f) = (77(a),p)f, and also rj([«,/?]/) = [77(a),/?]//.
Now, given /? for which /3n is a prime of A;", gp £ XO[[X]\x. Let go1
denote the inverse of gg with respect to composition of power series. Then /' =
9p ° / ° 9pl € Ix too, and gp : Ff =* Fy. Furthermore gg{un) = fin so (/3n) is
a generator of the Tate module of Ffi. By the last remark, we may prove (5) for /'
instead of /. In other words, without loss of generality we may assume un = (3n.
REDUCTION LEMMA. To prove theorem 4.2 we may assume that Nm,nWm = wn,
and that (3n = wn for each n. The formula to be proved is then
K),
/
and its validity should only be established for high enough n.
PROOF: Everything follows from the discussion above, except for formula (8), but
with/3 = w, g0 = X, so 8gp = x x,,xy
Note that in general wn is not a norm coherent sequence. It is easy to check
that this is the case if and only if MX = X.
4.4 PROOF OF (8): Throughout the proof, it will be convenient to use
= mod 7ran+c to mean "there exists an integer c, independent of n, such that
the congruence holds for large enough ri". The letter c will stand for different
constants in different places.
STEP 1: If a £ pn, then (a,a:)n = 0.
PROOF: As noted above, MX = X, so if [?rn](a) = a, a = rLeW" (a[+]w)-
Let L — k%{a), and let V C W? be the subgroup isomorphic to Gal(L/k%)
(8)
{<Xn,Un)n =
*-nTrn
AK)
w,
A'K)
31

under the map (2). Let R be a system of representatives for W^ modulo V. Then
a = Iluefl ILev (a[+M+]«) = Iluefl NL/ks{a[+]u). By class field theory, the
Artin symbol of a on L is trivial.
STEP 2: Fix a € Af. For large n, an[+]wn is a prime of A;", because an —> 0
rapidly (recall [«r](aB) = an+1 so i/(a»+i) > min(qv(an),v{an) + v{n))).
Write
(9) an[+]wn = w„ • (1 + 7n).
By step 1, and bilinearity of ( , )n,
(10) 0 = (a„[+]w„,w„(l + 7n)) = (a»,w„)[+](a„,l + 7»)[+](w„,l + 7n).
STEP 3: For large n, (an,l + 7n)n = 0. Indeed, if m > n, (am,l + 7m)m =
(an,iVm)n(l + 7m))n- When m -> oo, am -* 0, so 7m -> 0 and iVm>n(l + 7m)
tends to 1. By continuity of the Artin symbol, (am, 1 + 7m)m tends to 0. But with
fixed n, (am,l + 7m)m is always in W?, which is discrete. Hence for large m,
(am,l + 7m)m = 0.
STEP 4: For large n
(11) (aB)u4i = H(w»,l + 1n)n (steps 2 and 3)
= W2»[-][W»(1 + 7n)_1]/(W2n)
= [1 - Nn{l + 7n)_1](w2n).
The second equality shows the advantage of dealing with un. Its a7rn-root" u2n
generates a field L = k%n abelian over k, so for the Artin map (1 + 7n,L/A;£) =
(iVn(l + 7n),L/A;). The effect of this symbol on u2n is known by the theory of
Lubin and Tate!
STEP 5: 1 - Nn{l + 7n)_1 = Trn(7n) mod ir3n~c.
PROOF: Note that an = 0 mod 7rn-c, hence the same congruence holds for 7n
(cf. (9)). Since the different of k^/k is "of size" tt""0 [Se3, IV §1], Trn{in) =
32

0 mod 7r2n_c, and Trn(Yn) = 0 mod 7r3n-c. Next
(12)
1 - Nn(l + 7,)"1 = 1 - J] (1 - 7n + llr mod n3n~c
l7n
2
Trn(ln) - \ Tr^l) - \{Trn{ln)f
= ^rn(7n) mod tt
Zn—c
STEP 6: Take A of both sides of (9) and Taylor-expand in 7„. We find
\{an) = 7n • un • A'(u;n) mod 7r
2n—c
hence
In
AK)
Thus Trn(7n) = Trn
A(o»)
mod 7r
2n—c
STEP 7: Combining steps 4, 5, and 6 we get
mod 7r3n c.
i \ I\fi / Man) M t \
{<Xn,Un)n = Trn -f—rr (w2n)
(for large n). Since the left hand side lies in W^, the quantity in square brackets is
divisible by 7rn, and this concludes the proof of (8).
33

CHAPTER II
p-ADIC L FUNCTIONS
The method of chapter I, 3.1-3.6 will be applied here to construct p-adic L
functions over quadratic imaginary fields. These p-adic L functions appeared first
in a paper of Manin and ViSik from 1974 [M-V], and almost simultaneously Katz
gave another construction [Kl]. Coates and Wiles [C-W2] introduced the point of
view taken here when they showed how to produce the "one variable" functions (see
below) from norm-coherent sequences of elliptic units. They related the p-adic L
functions to class field theory, and formulated a "main conjecture". Their approach
was developed further mainly by R. Yager [Yal], [Ya2], and also by P. Cassou-
Nogues [CN], R. Gillard [Gil], J. Tilouine [Ti] and the author, hand in hand with
the emergence of Coleman's power series as a powerful tool. Our aim here is to give
a general and self contained exposition, free from any restrictive assumptions.
The extension of these results to arbitrary CM fields poses some tantalizing
problems. Out of the three approaches (Manin-Vi&ik, Katz, and Coates-Wiles),
only Katz' method of p-adic interpolation of real analytic Eisenstein series seems to
work [K2]. The approach taken in this book, for example, would require a supply of
special units in abelian extensions of the ground field, similar to the elliptic units.
Exhibiting such units is a major open problem in number theory. Moreover, a
consequence of a Coates-Wiles type construction (when compared with [K2]) would
be a deep theorem on monomial relations between p-adic periods of abelian varieties
with complex multiplication, analogous to Shimura's period relations between the
corresponding complex periods. We have managed to demonstrate these p-adic
period relations in some cases [dS3].
The p-adic L functions constructed in this chapter belong to a long array of
functions bearing this title. Among the abelian ones, there are first and foremost
those of Kubota and Leopoldt [K-Le], and their generalizations by Deligne and
35

Ribet [D-R], as well as the ones mentioned above. Then there are "non-abelian"
p-adic L functions arising from modular forms. Of fundamental importance here
are the works of Mazur and Swinnerton-Dyer [M-SD], and of Manin [Man], but they
all lie outside the scope of this book. See also the paper by Haran [Har].
An outline of chapter II follows. In §1 we summarize various results from
the theory of elliptic curves with complex multiplication, needed later on. Section 2
introduces special global units, the elliptic units of Siegel, Ramachandra and Robert.
They come in norm-coherent sequences, and when viewed inside the local units in
the right tower of fields, the procedure of 1.3.1-3.6 may be applied to get measures
out of them. This simple idea translates into elaborate computations in §4. Th,e
measures so obtained art the p-adic L functions, up to simple twisting factors (see
the discussion of measures and L-functions in 1.3). The relations that exist between
elliptic units, special values of Eisenstein series, and L-series, worked out in §3,
allow us to conclude that the p-adic L functions indeed interpolate special values
of classical Hecke L series, so they deserve their name. The rest of chapter II is
devoted to the exploration of some analytical properties of these p-adic L functions.
Algebraic and arithmetic aspects are left out for the next chapters.
l. Background
1.1 GROSSENCHARACTERS: Throughout this chapter, K denotes a quadratic
imaginary field of discriminant — die [d-K is a positive integer congruent to 0 or 3
modulo 4). The number of roots of unity in K, wk, is 2, 4 or 6. Fractional ideals
of K are denoted by gothic letters o, f, p etc. If f is an integral ideal let Wf be
the number of roots of unity congruent to 1 mod f. The ray class field modulo f is
denoted K{f), and K{fg°°) is the union of K{fgn) for all n. K{1) is the Hilbert
class field, and [K{f) : K{1)} = |(0*/f)x| ' «>f/«>tf•
We view all our number fields as subfields of a fixed algebraic closure Q of the
rationals, and choose, once and for all, a rational prime p and embeddings i'oq and
36

ip of Q in C and Cp. When there is no danger of confusion, we shall drop i'oq and
ip from the notation.
Let F be a number field containing K, and x a grossencharacter of type Aq of F.
Thus x is a homomorphism from the group of all fractional ideals relatively prime
to some f into Q . There exist an integral ideal f and an element w = ^ n(o)o £
Z{Emb{F,Q)\ such that x((«)) = aw for any o E Fx, a = 1 mod* f. The smallest
f with this property is called the conductor of x, denoted fx, and w is its infinity type.
Put x(a) = 0 if o is integral and not relatively prime to fx. The L — series of x
{with modulus m) is the complex analytic function Lm{x,5) = X) x(a)No_a, where
o runs over all integral ideals relatively prime to m. When m = (1), we simply
write L{x,s) and refer to it as a primitive L function. Note that in our notation
we do not insist that fx|m. The analytic continuation and functional equation of
L{x, s) are well known. When F = K we shall also say that x has infinity type
(fc>i) if x((a)) = afcflJ for o = 1 mod* fx. We then let
m r (s) - r(*-m*"(M)
(2) £oo(x,s) = Tx{s)L{X,s), R{X,s) = (djcNy'/'Loofoa).
With these definitions, the functional equation takes the form
(3) R{X,s) = W • R{x,l + k+j-s).
W is a constant of absolute value 1, the Artin root number.
Following Deligne, call x critical if the T factors in the functional equation,
namely Tx{s) and Tx (1 + k + j — 5), are finite at 5 = 0. This holds if and
only if k < 0 and 0 < j\ or 0 < k and j < 0. Schematically, the critical infinity
types {k,j) are depicted as follows.
37

3
(4)
If the infinity type of x corresponds to a point P, x and x^~k~:>~1 are represented
by the reflections of P in the lines k = j and k + j = — 1 respectively. The
integer k + j is called the weight of x- For reasons that will become clear later, we
call the dotted lines k = j and k + j = —1 the cyclotomic and anticyclotomic
lines respectively. Note that the cyclotomic line lies outside the critical range, while
all the lattice points on the anticyclotomic line are within it.
Finally, let x be a grossencharacter of K of type Ao and conductor f. It is well
known ([We]) that the values of x lie in a number field of finite degree, and that,
when viewed p-adically via the embedding ip : Q c-> Cp, x extends continuously
to a Galois character (also denoted x)
(5)
| x : 9 = Gal(l
Ix(ffa) = X(a),
cx
X : 9 = Gal(K(fp°°)/K)
(a,fp) = 1, aa = (a,X(fp00)/ir).
If furthermore p splits in if, p = pp, and p is the place induced by the inclusion
K C Q c-> Cp, and if the infinity type of x is {k,0) for some k, then the Galois
character (5) factors through Gal(K(fp°°)/K). If the infinity type is (0,j), then (5)
factors through Gal(K(fp°°)/K).
1.2 ELLIPTIC CURVES: General references for elliptic curves are the book of
Silverman [Sil] and the survey article by J. Tate [Ta]. The purpose of this section is
to recall some terminology.
38

If jP is a number field and E an elliptic curve over F, a Weierstrass model for
E/F is any plane model
(6) y2 = 4x3 - g2x - g3, g2,g3 E F, A = g\ - 21g\ ± 0.
Such a model is unique up to {gi^Qz) •-»■ («4<72ju6<73)j u £ Fx. The standard
differential associated with (6) is the differential
7 uE = —.
y
This is a basis for the (one dimensional) space of differentials of the first kind,
rational over F.
The pair (E,ue) determines a lattice L C C of periods, L = {f ue \
7 6 Hi(E(C), Z)}. Conversely, the lattice L determines the Weierstrass model (6)
via 02 = 92{L), gz = gz{L) (we then write A = A(L)), where
(8) g2{L) = 60 • Yl ""4> ML) = 140 • ^ a;"6.
The complex points of E are uniformized by C/L through Weierstrass' p—function
and its derivative
The map z i-> £(z,L) = (p(z,L),p'(z,L)) is an analytic isomorphism of C/L
onto £(C).
The j — invariant of E is given by
(10) jE = J(L) = 1728 gl/A.
Two elliptic curves defined over F axe isomorphic over F if and only if they have
the same j-invaxiant.
39

1.3 COMPLEX MULTIPLICATION: General references are [Sh], [Gr], and the more
classical expositions in [Bo] and [Deu]. The older monographs of Weber, Fricke and
Fueter contain much valuable information not to be found in modern literature.
Let E be an elliptic curve defined over a number field F. Then E admits complex
multiplications if End(E), the ring of endomorphisms of E as an algebraic group
over F, is strictly larger than Z. In such a case End(E) is isomorphic to an order
0 in a quadratic imaginry field K, F contains K(0), the ring-class-field (ringk-
lassenkorper) of 0, and E is isomorphic over F to an elliptic curve defined over
K(0). We always identify a € 0 with the endomorphism t(o) whose differential is
o, i.e. l(o)*ue = au)E- Write E[a] for the kernel of t(o), and for any ideal o C 0,
let E[a] = naG„ E[a], E{a°°} = Ui<n E[a% Etor = Ua E[a}.
The main theorem of complex multiplication ([Sh] 5.3) asserts first that the
extension F(E[a])/F obtained by adjoining to F the coordinates of points of E[a]
in one (hence any) projective model of E over F, is abelian. Second, there exists a
unique grossencharacter of type Aq, V = ^e/f °f ^> with values in K, and with
the following property. If 21 is an ideal of F, relatively prime to the conductor of if),
then V>(21) € 0, and
(11) cra(u) = t(V>(2l))(u)
for any u € E[c], (c,Nf/k%) — 1. The infinity type of rj) is the "half norm type"
J2 o~, the sum extending over all the embeddings of F which restrict to the identity
on K. The conductor of E over F is by definition the conductor of if). This agrees
with other, more general definitions of conductors (see [Se-Ta]).
1.4 A SPECIAL CLASS OF ELLIPTIC CURVES: It has been known for almost a
century that the arithmetic of abelian extensions of K is related to elliptic curves
with complex multiplication by K, in the same sense that the arithmetic of abelian
fields is related to the multiplicative group. We shall now introduce a special class
of curves, particularly well-suited for that purpose.
From now on, until the end of §1, fix an abelian extension F of K, of conductor
40

fF/Kj and let E be an elliptic curve over F satisfying
( (i) E has complex multiplications by Ok ,
\ (ii) F(Etor) is an abelian extension of K.
It follows from (i) that F D K(l). To understand (ii), recall the theorem ([Sh]
7.44) saying that it is equivalent to the existence of a grossencharacter <p of K, of
type (1,0), satisfying
(13) %1)E/f = <p o Nf/k-
Fix once and for all such a <p. The other candidates are <px, X € Gal(F/K). Since
fF/K = Lc.m. {fx},
(14) f = Lc.m. (f„, fF/K)
is equal to Lc.m. {f^x}, and therefore depends only on F and i}). Note that f^, is a
proper ideal.
Another consequence of (ii) is that for any a £ Gal(F/K), i^e" /f = i*E/F-
Since the grossencharacter of E/F completely determines its F-isogeny class ([Gr]
9.2), all the Galois conjugates of E are F-isogenous.
Lemma, (i) Let <p be a grossencharacter of K of type (1,0), and F = K(f<p).
Let j be the j — invariant of an elliptic curve with complex multiplication by Ok-
Then there exists a unique elliptic curve E over F with j'e = j, satisfying (12),
for which ij>e/f = <P ° Nf/k-
(ii) Let j be an integral ideal of K. Then there exists a grossencharacter <p of
type («>f,0) and conductor f.
PROOF: (i) Let i\) = <p o Np/K- If 21 is an ideal of F relatively prime to f^,,
then Nf/k(21) = (a) for some a = 1 modx f<p, so ^(21) = a. Let Eq be
any elliptic curve defined over F with CM by Ok and ./-invariant j. Let V'o be its
grossencharacter. Then e = V'/V'o is a character of finite order with values in 0K.
Let E = E^ be the twist of £0 by e ([Gr] 3.3). Then ([Gr] 9.2) r/}E/F = *!>, and
since E and Eq are isomorphic over Q, Je — 3- The uniqueness is a consequence
41

of the fact that the isomorphism class over Q and the isogeny class over F together
determine the isomorphism class over F ([Gr] §9).
(ii) This is obvious, and left to the reader.
1.5 Notation and assumption as above, let o be an integral ideal relatively prime to
f. By the main theorem on complex multiplication ([Sh] 5.3) we deduce the following
strengthening of (11).
Proposition. There exists a unique isogeny A (a) : E —> Effa over F, of degree
No, characterized by
(15) <ra(tt) = A(o)(u)
for any u G E[c], (c, a) = 1.
Let u be an F-rational differential of the first kind on E. Define the quantity
A(o) g Fby
(16) ui<Ta o A(o) = A(a)w.
It is easily verified that A(-) satisfies the cocycle condition (cf. 1.3.7 (14))
(17) A(ob) = A(o)CTbA(b) = A(b)CTaA(o).
This enables us to extend the definition of A to any fractional ideal relatively prime
to f so that (17) remains valid. A different choice of u results in modifying A by a
coboundaxy. When a Weierstrass model (6) is given, we tacitly assume that A(-) is
associated with the standard differential <jJe-
If(o,F/X) = 1, E<Ta = E, soA(o) G Kx and A(o) = t(A(o)). Moreover
(18) A(o) = <p(a), (a,F/K) = 1.
Note that both A(a) and <p(a) are now independent of any choice. To prove (18)
observe that A is multiplicative on the group of ideals for which (o, F/K) = 1. If
<p' is any extension of it to a grossencharacter, <p' o Nf/k = A o Np/K = ip
42

so <p' is one of the <p satisfying (13). To sum up, <p and A are uniquely determined
by E and F on {a | (a,F/K) = 1}. From there on they can either be extended
as a cocycle A, unique modulo coboundaries, or as a character <p, unique modulo
Gal{F~/K).
For an interpretation of <p by means of the abelian variety Resp/icE, see [Go-
Sch].
Finally, let L be the period lattice of u, and
(19) £: C/L -> E(C) £(z,L) = (p(z,L),p'(z,L))
the corresponding analytic uniformization. Then
(20) La = A(o)o_1X
is the period lattice of u^0 on E°a, and the following diagram commutes (o integral,
(«.f) = 1):
vrxL <-+ C/L ^ C/La
(21) I l£(;L) lt(',La)
KerX{a) <-+ E{C) ^ E°*.
It follows that gk{LYa = gk{La) {k = 2,3) and A(Lya = A(La).
EXERCISE: Show that a0F = A(a)0F.
1.6 Proposition. Let 0 be an integral ideal divisible by f. Then K(q) = F(E[q\).
PROOF: Let 21 be an integral ideal of F prime to 0 and suppose o%(u) = u for
every u G E[q\. Since a = NF/K{W) = (a) with a = <p(d) = ^(B), (11)
implies that o = 1 mod 0. If B is an ideal of .F(2?[0]) and 21 its norm down to F,
then the argument above applies to 21, so Np(E[g])/KB = (a), a = 1 mod 0.
By class field theory, K(q) C F(E[q}). Conversely, F C K(q), and if o = (a),
a = 1 mod 0, then A(o) = t(^(a)) = t(o), so by (15) oa fixes £[fl] pointwise,
showing F(E[g]) C tf(fl).
43

1.7 Corollary. Let g and m be any two relatively prime integral ideals, (f, m) =
1. Then F(E[m\) and F(E[q\) are linearly disjoint over F, and Gal(F(E[m])/F) =*
(0K/m)*.
PROOF: Without loss of generality, we may assume f|g, because the hypotheses
are not changed if we replace g by fg. Clearly Gal(F(E[m])/F) C (O^/m)*.
Let $(m) = ${OkI™)x (Euler's $ function). By class field theory, $(m) =
[lf(mg) : lf(g)], since (m, g) = 1, and ws = 1. Tta corollary now follows from
this and from proposition 1.6. The field diagram is described below.
F(E[q\) = K(S) X(mg)
F F{E[™})
K
1.8 GOOD AND BAD REDUCTION: Let ty be a prime of F. E has good reduction
at ^JJ if there exists an elliptic curve £ over the localization R of Of at ^J, whose
generic fiber is isomorphic to E over F. The special fiber £ x Spec Of/V is then
denoted by E, the reduction of E mod ^JJ. A basic theorem asserts that £, hence
E, depend only on E. It is also known that if E has good reduction at ^}, there
exists a Weierstrass model over R with A 6 Rx (if the residual characteristic of
^J is 2 or 3 we must allow for a generalized Weierstrass model; see 1.11 below).
The corresponding differential of the first kind w^ is then a basis for H°(£,fl\iR).
Every F-rational point of E extends uniquely to an .R-rational point of £ (since
£/R is smooth and proper) and when we read it modulo ty we get the reduction
map E(F) -> E{OfI^)- The kernel of reduction mod %$ will be written -Ei.qj.
In a Weierstrass model (6) with A € Rx, ^i,(p consists of all the points with
non-integral affine coordinates.
The following fundamental theorem is proved in [Se-Ta] as a consequence of
the theory of Neron models.
44

Theorem, (i) The primes of bad reduction for E/F are precisely the primes
dividing its conductor.
(ii) (Criterion of Ogg-Neron-Shafarevitch). Let m be an integral ideal of K,
relatively prime to ^JJ. Then ty is a prime of good reduction if and only if F(E[m°°])/F
is unramified at ^JJ.
1.9 We shall now determine the decomposition patterns of primes of F in the
extensions obtained by adjoining division points of E.
Proposition. Let p be a prime of K, (p,f) = 1, and let Fn = F{E[pn]),
0 < n < oo.
(i) All the primes above p are totally ramified in Foo/F.
(ii) Primes not above p are finitely ramified (i.e. their inertia group in
Gal(Foo/F) is finite), unramified if they are primes of good reduction.
(Hi) If p is split in K/Q, every prime not above p is finitely decomposed in
Foo/F. Ifp is inert (resp. ramified) and £2 is a prime of F not above p, the order
of the decomposition group of Q. in Gal(Fn/F) is asymptotic to cpn (resp. cpn/2)
for large n, c being a constant.
PROOF: We shall deduce (i) in 1.10 from formal group considerations. To prove
(ii) and (iii) we may replace F by K(g), f|g, so that over K(q) V = <P ° ^k(s)/k
is unramified, and E has good reduction everywhere. Then (ii) follows from 1.8.
(Note that it is not enough to take g = f in general).
To prove (iii) let Q be a prime of F not above p. Since we may assume now that
Q is unramified, its decomposition group in Fn/F is cyclic, generated by Frobenius
at £2. It follows from (11) that its order is the least positive integer / such that
(22) r/){£l)f = 1 modpn0p.
The assertions in (iii) all follow from this.
1.10 THE FORMAL GROUP: Let (p, f) = 1, and denote by ^J a prime of F above
p. Fix a Weierstrass model of E over R (the localization of Of at ^J) with A £ Rx.
45

Let E be the one-parameter formal group law of E with respect to the
parameter
(23) t = - —
y
(see 1.11 for modifications in residual characteristic 2 or 3) ([Ta] §3). E is defined
over R but we consider it over the completion of R, 0<p.
Lemma. E is a relative Lubin-Tate group with respect to the unramiGed extension
F<$/Kp. It is of height 1 ifp is split in K/Q, and of height 2 ifp is inert or ramified.
PROOF: Let 4> — o~p be the Frobenius automorphism. The isogeny A(p) : E —►
E^ induces a homomorphism of formal groups \{p) : E —> E^ which is of the
form
m \HP)(t) = Hp)t + ... e ov[[t]\
I = t« mod qJO^j
with q = N^J. See 1.1.2. The rest is obvious in view of the results of chapter I.
Corollary. <P is totally ramiGed in F^ = F{E[p°°}) (see 1.9(i)).
PROOF: Comparing the representation of Ga/(Foo/F) in 0pz = Aut(E[p°°\) (see
corollary 1.7) and the corresponding representation of the local Galois group in
Op = Aut(E[p°°}) {E[p°°} was denoted Wj in chapter I), we see that the two
images coincide. Therefore, ^J does not decompose in Fqq. But the local tower
(k^/k' in the notation of chapter I) is totally ramified, so the corollary follows.
Another consequence that will be needed later is the following. Consider
n'*(Q) where the product is over non-zero Q in E[p]. Since F(E[p])/F is totally
ramified at ^J, and the Np — 1 points Q are conjugate to each other, the order of
this product at ^J is 1. This applies to each ty\p.
1.11 CHARACTERISTIC 2 AND 3: Some formulas concerning Weierstrass models
have to be modified when the residual characteristic is 2 or 3. See [Ta] for details.
A generalized Weierstrass model is given by
(6') y2 + aixy + a3y = x3 + a2x2 + a4x + a6.
46

The A function is given by [Ta](2). With these, the assertions made in 1.8 about
good reduction and Weierstrass models remain valid. The differential w# becomes
>„~,\ dx dy
(10') uE y
2y + aix + a,3 3x2 + 2a2X +04 — aiy'
and the lattice L is still its period lattice. The uniformization £ : C/L —► E(C)
is accomplished through
(x = p{z,L) - {a\ + 4a2)/12
\y = (p'(z,L) - axx - a3)/2.
The parameter t for the formal group is now given by
(23') t = --
y
and with it E and W£ are defined over Z[a1}..., a6]. The kernel of reduction mod ^J,
-E'l.qSj is still the subgroup of points where x and y are non-integral. Restricted to
Eity t is finite and |t|<p < 1.
2. Elliptic Units
Elliptic units are units in abelian extensions of quadratic imaginary fields,
obtained as special values of elliptic modular functions. They play a role analogous
to that of circular units in abelian number fields. See the works of C.L. Siegel
[Sie], Ramachandra [Ra], Robert [R], Gillard and Robert [Gi-R], and Kubert and
Lang [K-La]. Except for nineteenth-century results on elliptic functions (out of the
many references here we mention Whittaker and Watson [W-W], Chandrasekharan
[Chand], and Weil's beautiful little book [We2]), our exposition is self contained,
and some of the proofs, I believe, are new.
2.1 THETA FUNCTIONS: Let L = Zu>\ + Zw2 be a lattice in C, whose basis is
ordered so that r = ijJ\/ijJ2 belongs to the upper half plane. Recall that Weierstrass'
47

a function and Ramanujan's A function have the absolutely convergent product
expansions
d) „(„.£).„. n'(1-i)«p(i +1(£)')
oo
(2) A(L)= (2ni/u2)12qT]l(l ~ ^)24> * = <?™■
v=\
cr(z, L) satisfies the transformation law
(3) a{z + u,L) = ±a{z,L)exp{r){w,L){z + <^)), w 6 L,
where r\ is an R-linear form on C, explicitly given by the following set of formulas.
(4)
/ M w1r?2-W2^i _ . w2r?i-wir?2
"(*'L) = 2*iA{L) * + 2WA(L) *
A(L) = (27ri)-1(wia72 — W1W2) = 7r_1 Area(C/L)
^1 = WlEnE'm(mwl +nw2)~2, *?2 = W2 Em E'nHl + nw2)~"2.
The order of summation matters. Half way in between r\\jw\ and 772/^2 we find
(5) s2(L) = BmQ £ Owl-2'.
With these definitions,
Wi7?2 - WlW2 ' S2[L) = —r—■
A[L)
W2r7l - WlW2 • 52(-^J = -7777
from which one recovers Legendre's relation W1772 — W2*?i = 27ri, and also (using
(4)) f,(z,L) = A(L)-1^ + a2(L)*.
The fundamental theta function
(7) 0{z,L) = A{L) • e-6"(*.L)* • a(z,L)
12
48

is non-holomorphic, but its arithmetical usefulness makes up for this blemish. If
c 7^ 0, 6(cz,cL) = 6(z,L) (the reason for A(L) in (7)), and the exponential is
chosen so that \0(z,L)\ is JL-periodic. In addition, 6 posseses an important product
expansion. Normalize L so that u»i = r, u>2 = 1, and let qz = e2%tz. Then
([We2], IV §3(15))
(8) 6(z,L) = «W'^ • gr(rf/a-fcl/a)ia ' II {(l"«rfa)(l-«rfc 'J}12.
Many identities, and in particular (ll) below, are consequences of (8).
2.2 SIEGEL UNITS IN THE HILBERT CLASS FIELD: We shall have almost no use
for these units, but since their construction is so simple, we describe it briefly.
Let if be a quadratic imaginary field. For any ideal a of K, consider
Proposition, (i) u(a) € K(l),
(ii) u{ab) = u(o)'b • u(b),
(in) («(o)) = a-120*(i).
PROOF: See [Sie], chapter II, §2.
If h is the class number of K, then ah = (a), a € ifz, and although a is
not unique, a12 is, so 6(a) = u(a)ha12 is a well defined unit in K(l). The group
of Siegel units in K(l) is generated by the S(a). It is of finite index in the group of
all units, and stable under Galois.
2.3 From now on assume that L admits complex multiplications by Ok- Let a be
an integral ideal of K. The function
(Q(z;L,a) = e(z,L)Na/e(z, a^L)
(10) = A(L) , A(L)
is an elliptic function with respect to L (to obtain the second expression in (10)
compare divisors and leading terms in the Taylor expansions).
49

Proposition (THE DISTRIBUTION RELATION). Let a and b be integral ideals of
K, relatively prime to each other. Then
(11) J] ®{z + v,L,a) = G^b-^o).
v£b~1L/L
PROOF: Both sides of (11) are jL-elliptic functions with the same divisor, so their
ratio is a constant. Comparing the Taylor expansions at z = 0 shows that this
constant is
(12)
K'} \*{a-iL)J \±(b-iL)J 11 11 (p(u,L)-p(v,L)r
Here u runs over a~1L/L — {0}, v over b~lL/L — {0}, and the denominator never
vanishes since (a,b) = 1. Despite the apparent asymmetry, e(a, b) = er(b,a),
because (A(L)/A(b-1L))<T(>-1 = (A(JL)/A(a"1L))<Tb-1 (2.2(H)). We shall prove
that e(a, b) = 1.
Let H = K(l) be the Hilbert class field of K, and wh the number of roots
of unity in it. A short analysis of ramification patterns reveals that iujy|l2. In fact
4\vjh if and only if djc = 4 mod 8, and 6\wh if and only if 6k = 0 mod 3.
Now e(a,b) evidently lies in H, because we may choose L so that the particular
Weierstrass model is defined over H. Next we claim that e(a, b) is a root of unity.
The proof of this fact is a tedious but straightforward computation, based on (8).
It can be found in [K-La] chapter 2, theorem 4.1 (i), from where it actually follows
that e(a, b)Nb = 1. Thanks to the symmetry between a and b, e(a, b)No = 1.
So far we know that e(a, b)m = 1 where m = 0.c.d.(u;jy,Na,Nb). To
conclude we need the following lemma of Robert ([K-La] chapter 11, 5.7).
Lemma. Suppose gk{L) € H, k = 2,3. Then for any a relatively prime to wk,
A(L)No/A(a-1JL) € {Hx)12.
Consider the expression (12) for e(a, b), where L is chosen so that gk{L) G H.
Since p(u) = p{—u), Yl'Y['{p{u) — p{v))e is a twelfth power in Hx, unless both
50

a and b axe even (not relatively prime to 2). But this can only happen if 2 splits in
K, and then wjj |6. Therefore, in any case
vNo\Nb_1 / Afr\ \ No-<To
M <-'>-(«) (*$a -o
%
Choose 0 E Kx so that b' = (/?)b is integral and (b',6a) = 1. Then (13) implies
kNo x Nb-Nb'
(14) e(a, b) ee (x^Zy) W (^X)
WH
because e(a, b') = 1. From Robert's lemma we conclude that e(a, b) = 1 if
(a,iyK) = 1.
Suppose K is not Q(t) or Q(y/~3~). If a or b is odd, e(a, b) = 1. If both
a and b axe even, 2 splits in K, so tu# = 2 or 6. Pick b' = (/?)b such that
(b',2a) = 1 and Nb = Nb' mod 3. Then if iuff = 6, (14) shows that e(a,b)
is a cube. If we also show that e(a, b) is a square, it would belong to (HX)WH. But
then e(a,b) = 1. Formula (14) implies e(a, b) = A(L) mod (iP)2. Now, with
the model y2 = 4x3 — hx — h, h = 27j/(j — 1728), for the elliptic curve with
j-invariant j, A = j - 1728 mod (Hx)2. A theorem of Weber ([Web] §§134-135)
asserts that j — 1728 is a square in H, and concludes the proof whenever K is not
Q(») or Q(VC3).
Finally, in the two exceptional cases K = H, and a and b are principal. If
K = Q(t) either a or b must be odd, so e(a, b) = 1 is a consequence of (14) and
Robert's lemma, as above. If K = Q(y/—S), either a or b is odd, and one of them
is also prime to 3. From (13) we get
e{a,b) = £(L)(F*-m™-i) mod{Hx)e
and 6|(Na — l)(Nb — 1), so again e(a, b) is a sixth power, hence 1. The proof of
the proposition is now complete.
2.4 Proposition. Let m be a non-trivial integral ideal of K, and v a primitive
m-division point of L (i.e. v € m-1L, but v £ n~1L for any proper divisor n of
m). Then, if (a, m) = 1,
51

(i)Q(v;L,a) <E K{m).
(ii) 9(w;L,o)'e = 9(w; c"1^ a) = 0(t>;£, ac)0(w;L,c)-No fc integral,
relatively prime to m).
(Hi) Q(v;L, a) is a unit if m is not a prime power. If m = pn, it is a unit
outside p.
PROOF: (i) We may assume that L corresponds to a Weierstrass model E defined
over the Hilbert class field K(l). Part (i) follows from (10) and standard results from
the theory of complex multiplication ([Sh], theorem 5.5). It is also a consequence
of (ii).
(ii) Once more we are free to choose a model E for C/L, changing L by a
homothety, because Q(v\L, a) depends only on the complex isomorphism class of
the elliptic curve. Choose an integral ideal f with Wf = 1, relatively prime to c, and
assume that the model E corresponding to L is defined over K(j) = F, and satisfies
^E/F — <P ° Np/K for a grossencharacter <p whose conductor divides f. Then
(10) and 1.5(21) yield e(w;L,o)'« = 0(A(c)tr, A(c)c~1JL, a) = Q{v; t~lL, a). The
rest of (ii) follows from this, and from 0(t>; L, a) = 6(v,L)Na/6(v, a-1L).
(iii) We shall give two different proofs of this important result.
FIRST PROOF: We make the same assumption as in (ii) on the model E, and in
addition we assume (f, m) = 1. Suppose p|m and let Mn = F(E[mpn}), n > 0.
Corollary 1.7 implies that the conjugates of a primitive mpn-division point v over
A^n-i (» > 1) are t; + u for u € E[p}. Let en = Q(v;pnL, a). Although
en = Q(A(p~n)v, A.(p~n)pnL, a) refers to a^n(E), and not to E, it lies in Mn,
because all the conjugates of E are F-isogenous, so the fields ^(^^[c]) coincide
for all a £ Gal(F/K). Since v is a primitive mpn-division point of pnL, the
distribution relation (ll) implies (with b = p) Nn>n-ien = en_i. Here Nm,n is
the norm from Mm to Mn.
Suppose first that p is a split prime. According to 1.9, only primes above p
may ramify in Moo/Mo, and all the others are finitely decomposed in that tower.
Let q be a prime of K different from p, and choose n large enough so that all the
52

prime divisors of q are inert in Moo/Mn. Since en € NminM^, m > n, en is a
unit above q, hence so is eo = Nn>oen.
If p is inert or ramified this argument needs a slight modification. First,
replacing m by mpn, c0 by cn, we may assume that Gal(Moo/Mo) = Zp (a redundant
step if (p,6) = l!). Proposition 1.9 implies that the decomposition group D of
any prime of Mo above q (they all coincide) is Zp. Once again replacing Mo by
some Mn, we shall assume that Gal(Moo/Mo)/D = Zp too. It follows that Moo
contains a Zp extension iVoo of Mo in which every prime above q is inert. The rest
is identical to the split case, because Co sits in a norm-coherent sequence in the
tower iVoo-
We conclude that Q(v;L, a) is a unit outside p, therefore if m is divisible by
two distinct primes, a unit.
REMARK: If m = pn, 0(w;L, a) is not a unit. The reader should keep in mind
the cyclotomic analogy: exp(27ri/m) — 1 is a unit if and only if m is divisible by
two distinct primes.
SECOND PROOF: This proof does not use the distribution relation (11). It does not
even use complex multiplication, and works whenever j(L) is an algebraic integer,
or even more generally, if one is willing to use some results about Tate curves. This
interesting observation, that the elliptic units "are units" even in the absence of
complex multiplication, does not seem to have found yet far reaching applications.
We may assume that the lattice L corresponds to an elliptic curve E defined
over a number field F, with good reduction everywhere, and that £[mo] are rational
over F. Fix a prime O. of F, and a Weierstrass model over R, the localization of Of
at £2, with A £ Rx. We shall use ~ to mean "both sides have the same valuation
at O". Let P = £(u,L) be the point corresponding to v, and E\ the kernel of
reduction modulo Q. Thus Q € E\ if and only if |x(Q)|Q > 1. Suppose first that
a is prime and recall (1.10) that
(15) fj t(Q) ~ a ifjQ|a.
QeE[a]
53

Since A(L) ~ 1, (10) implies
(16) Q(v,L,a) ~ a"12 J] W) ~ *(p))~6-
ges[a]
Now suppose that either m is divisible by two distinct primes, or else Q { m.
In both cases P, which is a primitive m torsion point, does not belong to E\.
If O f a5 then any non-zero Q €i E[a] lies outside E\, and so do Q ± P.
Therefore, x(P) and x(Q) are O-integral and non-congruent modulo 0, whereby
(16) is a unit at D. If, on the other hand, 0|a, then for any Q £ E[a] — {0},
x(Q) - x(P) ~ x{Q) ~ t(Q)~2, and (16) ~ 1 because of (15). We conclude
that as long as a is prime and (a, m) = 1, (iii) holds. The general case follows at
once, since for integral a, b, Q(v;L, ab) = Q(v;L, a)Nb • Q(v;a~1L,b).
EXERCISE: If m = pn, and ty\p, show that the ^J-adic valuation of ®{v;L, a) is
(No - l) X (a constant depending on m).
2.5 Proposition, (i) Let f be a non-trivial integral ideal of K, and g = fl, with
a prime I. Let e = iuf/iu0. Then if v is a primitive ^-division point of L, and
Kfl) = 1,
at r^t ri- ^e / 0(t;;L, a)1"^1 if! f f
{Q(v;L,a) if I | f.
(ii) Let I be prime, (a, I) = 1 and e = wk/wu Then if v is a primitive
l-division point of L, and L represents the trivial class in Pic (£«•),
NK(i)/K(i)&(v,L,ay = u(l)"«-Na
(see 2.2).
PROOF: (i) We have seen that [K(q] : K{j)] = iy0$(g)/iyf$(f), where $(f) =
|(0K"/f)z|- Consider first the case I|f. The conjugates of Q(v;lL, a) over K(j) are
Q(v + u; IL, a), u £ L/IL, and when u ranges over the NI possible points, each of
them is counted e times. Our formula follows from the distribution relation (10).
54

If I f f, then the conjugates of Q(v; IL, a) are Q(v + u; IL, a) for the Nl — 1
points u £ L/IL for which v + u is of level Q. Again each conjugate appears e
times. If uo is the unique point such that v + uo is an f-division point of IL, then
from 2.4 (iii)
Q(v + u0; IL, a)*7! = Q(v+u0;L, a) = Q(v;L, a).
As before, (10) gives the desired formula
(ii) This is proved in a similar way, and is left to the reader.
2.6 ROBERT UNITS: We want to briefly indicate the relation between Q(v;L, a)
and Robert's units.
Let f be a non-trivial ideal of K, and / the least positive integer in f f~l Z.
Let C/(f) be the ray class group mod f, identified with G(f) = Gal(K(f)/K). Let
J(f) C Z[G(f)] be the ideal generated by aa - No, (a,6f) = 1. J(f) is the
annihilator of MK'(f)- For any a € <2(f), Robert's invariant is defined as
(17) tpfa) = 0(1, fc-1)', a = (c,K(f)/K), c integral.
This is well defined, and the symbol tpf extends linearly to Z[G(f)].
Proposition ([R], [Gl-R]). (i) <pf(u) <E jRT(f).
(ii) a<pf(u) = <pf{o-u).
(iii) If f is divisible by two distinct primes, or if u belongs to the augmentation
ideal, <Pj(u) is a unit. If f = pn, it is a p-unit.
(iv) Assume tha.t f is the conductor of Ktf). Ifu € «^(f), <Pf{u) is a (I2fwf)th
power in K(f) ([Gi-R] proposition A-2).
Now the relation with 2.5 is given by
(18) Pf(No-«r.) = 0(1; f, a)', f ? (1).
The moral is that to get well defined units from 0(z,L) evaluated at a primitive
f-division point, we may either raise to power /, or twist by <r0 — No, and the two
operations are related by (18).
55

2.7 The following result is similar to 2.6(iv), and will be used in 4.12 to deal with
p-adic L functions for p = 2,3.
Proposition. Let v be a primitive ^-division point of L (f ^ (1)). Then for
(a,6f) = 1, 0(v;L, a) is a 12twf power in K(f)x.
We omit the proof. The proposition allows us finally to define the group of
primitive Robert units of conductor f, denoted Cf.
DEFINITION: Let f be a non-trivial integral ideal. Let 0f be the subgroup of K{f]x
generated by 0(1; f, a), (a,6f) = 1. Define Cf to be the group of all units in K(f)1
whose 12iWf power lies in /x^(f)0f-
Notice that Cf is Galois stable, that 0f C Cf only if f is divisible by two
distinct primes, and that from Robert's point of view, Cf consists of all the units
in if (f) whose 12/iWf power belongs to the group generated by £>f(w), u € «J(f),
and by roots of unity.
3. ElSENSTEIN NUMBERS
The bridge between special values of Hecke L series associated with K and
elliptic units is provided by Eisenstein series. We use the term "Eisenstein numbers"
for their special values at CM points on the modular curve, because their role
parallels the role Bernoulli numbers play in the cyclotomic theory. This section
owes a great deal to Weil's book [We2], and also to the paper of Goldstein and
Schappacher [Go-Sch].
3.1 Throughout, let f be an integral ideal of K with twf = 1, relatively prime to
the prime p. Let E be an elliptic curve with complex multiplication by Ok-, defined
over F = K(f). We assume that E satisfies (1.4) i^e/f = ¥> ° NFfK, and that
the conductor of <p divides f. Fix also a (generalized) Weierstrass model of E over
F, with good reduction at the primes above p. Let uje be the standard differential
56

on that model, and L its period lattice. Recall that A(-) is the cocycle associated
to L (or oje) as in 1.5.
Letting $(z,L) = a'(z,L)/a(z,L) stand for Weierstrass' zeta function, we
observe first of all that
(1) Ex{z,L) = s{z,L) - r){z,L)
is L-periodic. From 2.1(7) and the formula for r](z,L) preceding it, another
expression is derived:
(2) E^L) = -L ± log 6(z,L) - \zA{L)-K
Following Weil, put L = Zu>i + Zu>2> im(u>i/u>2) > 0, and treat z,~z,oj\,uj\,u}2
and ZJ2 as six independent variables. Formulas 2.1(4) express rj(z,L) in terms of
these six variables, and for f (z, L) there is the well-known expansion
(3) f(z,L) = - + Yll " + 1 + 7 T «}•
z ^—' \ z — mux — nut mwi + nwo mwi + nu>2) J
Let us introduce two differential operators
(4) 9 = -±, p = -^)-(^+<+^).
It is easy to check that V(A(L)) = 0, so A(L) is a constant with respect to the
two operators.
If 0 < —j < k, and a is an integral ideal, define
(EJ)k(z,L) = D-idW-iEfaL),
(5) j E3>k{z;L,a) = No • Ej>k{z,L) - E^a^L),
{ Ek = E0)k-
The next two formulas, easily verified from the definition, are the basis for the
connection between Ejtk and L functions, on one side, and elliptic units, on the
other.
57

(6) Ejtk{z,L) = {k-l)lA{L)j • Y^{Z + oj)-k{z + ZJ)->, k + j > 3,
(7) -12 • Ek{z;L,a) = dk log ®{z;L,a), k > 1.
3.2 A crucial step in the study of the special values of EJ}k{z, L) is provided by the
lemma below. It will allow us in 4.14 to replace the EJtk by expressions involving
Ek only.
Lemma. There exists a unique polynomial $Jtk in Z[Xi,...,Xk-j], of degree
1 — j, isobaric of weight k — j (Xi is assigned weight i), such that
Furthermore,
$jtk = (—2Xi)~3Xk + (terms in which Xi appears to degree < —j).
PROOF: The first statement is proved by Weil in [We2], VI §4. The fact that $J)fc
has integral coefficients and the second assertion are not explicitly mentioned there,
but follow easily from the proof.
3.3 Let us summarize some properties of Ejtk that will be needed later. In view
of (7), it is not surprising that they look very much like the facts proven about
0(u;L,a) in §2.
Let m be any non-trivial integral ideal, and v a primitive m division point of
L. Recall that L was the period lattice of a good Weierstrass model (at p), defined
over K{f) = F, and 0 < -j < k.
Proposition, (i) Ejtk{cz,cL) = c^~kEj)k(z,L).
(ii) (Rationality). EJ>k{v,L) € K(l.c.m. (f,m)).
(Hi) (Galois action). If c is integral, (c, fm) = 1, then
(8) Ej>k(v,Lr = Ej>k(Hc)v,A(c)c-lL) = A(cy-kEjtk(v, c~lL).
58

(iv) (Integrality). Suppose that m is not a power of p, and that p is split in
K/Q. Then vHEx(v,L) and (2m)-3'Ej>k(vtL) for 1 < Jfc, 0 < -j < k, are
p -integral.
PROOF: (i) This is a consequence of (5).
(ii) Let 0 = /.c.m.(f, m). We have already seen (1.6) that K(q) = .F(i?[m]),
where E is the Weierstrass model corresponding to L. By the lemma, it is enough
to prove (ii) for Ek, k > 1. By (7), Ek(z;Lt a) is an .F-rational elliptic function,
so Ek{v\L, a) e K{q). Choose a = (a), a = 1 mod m. Then ^(t;, o_1L) =
akEk{ocv,L) = akEk{v, L), because Ek is L-periodic. Since we may arrange Na ^
ak, (5) shows that Ek{vtL) 6 K(q).
(iii) This may be deduced from 2.4(ii) and (7) by the same reasoning as above.
(iv) We shall first prove that Tff^(v,L) is p-integral for A; > 1. It is enough
to show that when we fix the embedding of Q into Cp it is a local integer there.
We shall prove later on (4.9) that if (a, mp) = 1, 0(v — z;L, a) has a p-integral t
expansion G(t) at 0. Thus
a* log e(.-.;£,.) = (-jpj!) it CM
is also a p-integral power series. Furthermore, up to a constant, G(t) is a 12*^ power
of another power series with p-integral coefficients. By (7), Ek(v;Lt a) is p-integral.
Choose a = (a), a = 1 mod mpr for some large r, such that the same power of
p divides a — 1 and m. Then, if r is large enough, a — ak~l and iff are divisible
by the same power of p. Now
Ek{v;L,{a)) = a{a - a*"1) • Ek{v,L)t
somEk(vtL) is p-integral.
To prove the rest of part (iv) we may assume, again by lemma 3.2, that j — 0.
First observe that with a generalized Weierstrass model 1.11(6'), and with 62 =
a\ + 4ct2, 64 = a\az + 2ct4, as in [Ta], one has
(9) p" = e{p - 62/12)2 + b2{p - 62/12) + 64.
59

Supppose that A; > 3. Then Ek(v,L) = (-l)kp(k~2>>(v,L). Since v is a primitive
m-division point and m is not a power of p, £(v,L) does not belong to the kernel
of reduction, hence its x and y coordinates are integral. It follows from 1.11(19'),
(9) and easy induction, that p(k~2\v,L) is integral.
There remains the case k = 2. Now, as follows for instance from (1) and 2.1,
E2{v,L) = p{v,L) + s2{L), so by 1.11(19') we only have to show that
(10) s2(L) + ±(a2 + Aa2)
is a p-adic integer. To establish this, let n be an auxiliary ideal, (n,p) = 1, and
w a primitive n torsion point. By the discussion above, r\E2(w,L), hence E2(w,L)
as well, is integral. Applying the previous argument in reverse,
s2{L) + ^{a2 + 4a2) = E2{w,L) - x{£{w,L))
is integral. The proof of (iv) is complete.
REMARK: Any change of coordinates in 1.11(6') transforms b2 = a\ + Aa2 to
62 = u2b2 -f 12r with some unit u and an integer r. Choosing r appropriately, we
may therefore assume that s2(L) = —b2/12.
3.4 CONGRUENCES: The integrality results of proposition 3.3 lead to some very
useful congruences between Eisenstein numbers. As in 3.3(iv) we continue to assume
that p = pp is split, and ip : Q ^-> Cp induces p on K. We continue to denote
by v a primitive m division point of L. As already implied by the proposition,
Ek(v,L) is in general p-integral only if A: > 2. At places above p Ei(v,L) has a
denominator whose order is at most, and as we shall see later, precisely, the power
of p in m.
Lemma. Assume that m is not a power of p. In the following statements, all
congruences should be read locally, in Cp.
(i) Ex{v,L)a< = NcA(c)-1E1(v,L)modl, (c,fm) = 1.
In (ii) and (Hi) assume f|m, so that F(I?[m]) = K(m).
60

(ii) Let q be a prime, (q,mp) = 1. Then for any u € q 1L,
(Nq - l)Ex{v,L) = (Nq - l)Ex{v + u,L) mod 1.
(in) Let g be integral, (fl,p) = 1, and let pr be the power of p in n(Nq — 1),
the product taken over all the primes q dividing Q, but not m. Then
prE1(v,L) = prTStQEi(v,QL) mod 1.
REMARK: Parts (ii) and (iii) perhaps hold without the factors Nq — 1 and pr.
However, the weaker congruences recorded here will be sufficient for all our needs
in §4. The reader should keep in mind the typical case, where m = fp*1^)™, n
fixed, and m very large. It is in this context that the lemma will be used later on.
PROOF: (i) We begin by showing that we may assume (c,p) = 1. If not, let
a € Kx be such that b = (a)c is integral, a = 1 mod vnfpn for some
large n, and (b,p) = 1. The left hand side of (i) is unchanged if we replace c
by b. Also, NbA(b)-1 = NcA(c)-1a, so if n is large enough, the right hand
side of (i) is unchanged too. Assuming therefore (c, fmp) = 1, we know that
Ei(v;L,c) = NcEi(v,L) - Ex(v, t~xL) is p-integral (see the proof of 3.3(iv)).
To derive (i) divide by A(c), now a unit, and use (8).
(ii) Observ° first that
(11) E1{ziq~1L) = Y, M* + »>L)-
This distribution relation may be deduced from proposition 2.3, or more easily, from
the fact that Ex (z, L) equals the value at 5 = 0 of the analytic continuation of
£ {z+u)-x\z+u\-2a ([We2],VIII§14). By 1.7, since f|m, the numbers E^v+^L),
for u E q~xL — L, are all conjugate under Gal(K(mq)/K(m)). If c is an ideal
whose Artin symbol belongs to this Galois group, NcA(c)-1 = v?(c) = 1 mod m,
so if uo £ q~xL — L,
(Nq - l)Ex{v + uo,L) + Ex{v,L) = Y NcA(c)~1^i(v + u0tL) + Ex{v,L)
c
= Y E^v + uoiLy + Ex{v,L) = Y, E^v + ^L) = Ex{v,q~xL)
= Nq • Ex(v,L) mod 1,
61

by part (i) and 3.3(iv).
(iii) An easy induction argument shows that to prove (iii) we may assume that
0 is a prime q. We distinguish between two cases, according to whether q divides
m, or not. In the former case, (11) gives
Ex{v,L) = ^2 Ex{v + u,qL) = Nq • Ex{v, qL) mod 1
uSL/qL
by the same argument as above, since now all Nq points £{v + u, qL) are conjugate
under Gal(K(mq)/K(m)). If q \ m, on the other hand,
(Nq - l)Ex{v,L) = (Nq - 1) • £ Ex{v + u,qL)
ueL/<\L
= (Nq - l)Nq • Ei(v, qL) mod 1
by part (ii). This concludes the proof of the lemma.
For future reference, we mention here another (local) congruence that follows
from lemma 3.2 and 3.3(iv). Let m be divisible by some prime different from p, and
let v be an m-primitive division point of L, as before. Then, for 0 < — j < k, if
m|c,
(12) (2c)-'Ejlk(v,L) = (-^(u.ijj-^t^Ljmodm.
3.5 The results obtained so far may be of some independent interest, but their
significance stems mainly from the relation between the numbers Ejtk(v,L) and
special values of L functions. Recall that the partial L function L ( x>s;
is defined to be £) x(fl)Na-8, the sum extending over all integral ideals a such that
(Mm/x) = l,and(a,M/tf) = {c,M/K).
Proposition. Let <p be a grossencha.racter of type (1,0) whose conductor divides
m. Then for any integral ideal c, (c,m) = 1, and H € Cx,
Nm-J.Eyfc(n,c-1mn) =
(13)
(*-!,« (^V-.p(c,W.L(^i4!(S^)).
M/K
62

PROOF: This is a straightforward computation, which, at least when A; + j > 3,
boils down to (6). In the remaining cases one has to use Hecke's trick of introducing
\z + u\~2s into (6) and analytically continuing in s. The relevant fact here is that
Ejtk is the value of this analytic continuation at s = 0 ([We2], VIII §14). See also
[Go-Sch], corollary 5.7.
Note that if H is chosen so that L = mH is the lattice of a Weierstrass
model E as in 3.1, the left hand side of (13) belongs to K(m), hence so does the
right hand side. This was originally discovered by Damerell [Da], In this case
Ej)k{n,c-1mn) = A(c)*-J'£ylfc(n>mn)"«.
4. p-ADIC L FUNCTIONS
We have now developed all the ingredients needed to construct the p-adic L
functions in full generality. Chapter I 3.1-3.6 contains a recipe for turning a norm-
coherent system of local units into a measure. Section 2 of this chapter supplies
interesting examples of such norm-coherent sequences, namely the elliptic units.
Section 3 relates, as we shall see below, their Coleman power series to special values
of classical L functions. We now combine everything.
From now on p will denote a prime that splits in K
(1) P = PP
and we assume that p is the prime induced from the fixed embedding of Q in Cp.
The splitting restriction is imposed upon us because of the fundamental assumption
made in 1.3.2, that the formal group is of height 1.
Traditionally, the p-adic L functions were labeled either one-variable or two-
variables. Since our construction employs measures rather than power series, this
distinction is not so prominent. See 4.16 for comparison with power series
terminology.
The main theorems are formulated in 4.12 and 4.14, and the reader is advised
to look them up before he is deluged with computations. It is perhaps worth noting
63

from the outset, that to understand the statements alone, nothing about elliptic
curves is needed. The construction requires the elliptic curves studied in 1.4, whose
torsion is abelian over K.
4.1 Fix an integral ideal f of K, with Wf = 1, and relatively prime to p. Let
F = if(f) and Fn = K(jpn). We emphasize that f need not be relatively prime
to p. In fact, our chief interest in 4.14 will be in what happens when one lets fp™
stand for the original f.
Fix a grossencharacter (p of type (1,0), whose conductor divides f, and an
elliptic curve E over F, satisfying 1.4 (12), for which i>E/F = ¥> ° Nf/k- Such
<p and E exist by lemma 1.4. For any q divisible by f, K(q) = F(^[g]), and this
in particular holds with g = jpn. Furthermore, Gal(Fn/F) = {0K/pn)x via its
action on E[pn] (corollary 1.7).
It is easy, nevertheless essential to what follows, to give a semi-local version of
the local results from chapter I, considering all places above p simultaneously. This
we shall now carry out. Let us put
(2) $ = F ®K Kp = e?p|p Fv> R = 0F ®0k °p = ®<V\p 0y*
and let $n and Rn be defined similarly, with Fn instead of F. Recall that above
each prime ^3 of F dividing p, lies a unique prime ^Jn of Fn.
Let Q = GaUFoo/K), G = GaliF^/F) = T x A, where r S 1 + pZp,
A = Fp. Needless to say, $ acts on $n via its action on Fn.
K
64

Let R% = Un x Vn be the decomposition of the semi-local units into a pro-p
part Un (principal units), and a finite group Vn of order prime to p. Obviously, Vn
is independent of n. If [F : K] = fg, and p decomposes into g ^J's, each of norm
q = pf, then Vn S (F*)«.
Finally define inverse systems with respect to the norm map
(3) U = lim £7n, V = lim Vn.
4.2 From now on let E stand for a particular Weierstrass model defined over the
localization of Of at p, with good reduction above p, i.e. A is a unit at each place
$ dividing p. The models Ea, a 6 Gal(F/K), are of the same type, and what is
said here about E, applies equally well to them.
The formal group E, with its parameter t corresponding to the specific Weier-
strass model fixed above, is defined over R. At each ^3, E projects to a Lubin-Tate
formal group of height 1, relative to the unramified extension Fy/Kp, which is
defined over 0«p. We denote it by Ey (1.10).
If (a, f) = 1, the isogeny A (a) induces a homomorphism of formal groups over
R
(4) A(o) : E -► EP~*.
If a = aa belongs to the decomposition group of p in Gal(F/K), this homomor-
phism breaks down as the product of E<$ —»• Em for ^3|p. In the notation of 1.1
these arrows are [■A-(fl)]/)(7(/), and indeed A(a) 6 Fy satisfies A(a)^-1 = A(p)'7-1
(^ = ffp), which must hold in order for [A(a)]y>(7(y) to make sense (proposition
1.1.5). Furthermore, if (a, fp) = 1 and (a,F/K) = 1, then A(o) 6 End(E), and
comparing 1.5 (15) to 1.3.3 (9), we obtain
(5) A(a) = «(fffl).
With the convention of 1.3.7 (14), regarding the extension of /c from G to §, (5)
remains valid even without the restriction (a, F/K) = 1. Now, of course, A(a) 6 F,
and /c(ct0) 6 $.
65

We let L, as before, be the period lattice of E, and La = A(o)o lL that of
Eaa. Replacing E by one of its conjugates, if necessary, we assume
(6) L = fif, n 6 Cz.
Fix a choice of fi once and for all, and note that <p and (6) determine E up to
F-isomorphism, so n mod Fx is independent of the specific Weierstrass model.
4.3 p-ADIC PERIODS: Let F' = F{E\p°°]), and
(7) $' = F' ®K Kp R' = 0F> ®0k 0V.
Since p is finitely decomposed and unramified in F', $' is a finite direct sum of
A A
fields, each unramified over Kp. Let $ and R be the completions of $ and R .
Clearly Gal(F'/K) acts on <&' continuously, via its action on F', and the action
extends to <&.
^^ A
Proposition. There exists an isomorphism of formal groups defined over R,
(8) 0: Gm k E, t = 0{S) = QPS + ••• 6 E[[5]],
satisfying
(9) if) c » = f o [Nc]6m, (c,fp) = 1.
f2p G Rx is uniquely determined by (9) modulo 0px.
PROOF: This is the semi-local version of 1.1.6 and 1.3.2 (3). First observe that by
Tate's theorem on invariants ([Ta2], p. 176) the fixed subring of $ under Gal(F'/K)
is Kp. Since (9) implies
(io) nj'-1 = aconc-1, (c,fp) = i,
and these ac are dense in Gal(F'/K), the last assertion follows.
To construct 0 we first find an np as in (10). If at fixes F and ^[p"*] pointwise,
then (1.5 (18)) A(c) = <p(c), and by Weil's pairing, or from (p(c)<p(c) = Nc,
66

A(c)Nc_1 = 1 mod pm. It follows that the map ac !->• A(c)Nc_1 extends to
a continuous 1-cocycle Gal{F'lK) -»■ Rx C Rx (see 1.5 (17)). Since F'/K
is unramified at p, Hilbert's theorem 90 implies H1(Gal(F'/K),Rx) = 1, so Qp
exists (compare 1.1.6).
Proceed as in chapter I. By the semi-local analogue of lemma 1.1.4, there exists
a power series 9 with A(J) o 0 = 0°> o [p]6 , 0(S) = QPS + • • •, and this
A A
implies that 0 : Gm ~ E. Finally (9) holds because both sides are homomorphisms
A A^
from Gm to Eat with the same derivative at 0.
4.4 CHOOSING Qp: Fix once and for all a generator (fn) of the Tate module of
A
Gm, as in 1.3.2. Such a choice should be regarded as orienting Cp. Now, it is clear
from proposition 4.3 that the following are equivalent:
(i) a choice of np as in (10),
(ii) a choice of 0 as in (8).
Let
(11) ^ = **""(£» - 1) e Rn
(see 1.3 (5), <f> = Gp). Then there exists a unique point un 6 L/pnL such that
(12) un = *(£(A(p->n,A(p->"L)).
Recall that £(•, A(p~n)pnL) : C/A(p-n)pnL cz ^""(C). Furthermore, since
(<£-nA(p))(wn) = wn_i, un mod pn~1L = un_x. See 1.5 (21). It is now clear
that (i) and (ii) are also equivalent to either of the following:
(iii) a choice of (wn), for which <£-nA(p)(u;n) = un-i, and <£-nA(c)(wn) =
c«K), (c,fa) = i,
(iv) a choice of un 6 L/pnL, primitive of level pn, such that
un = un-i mod pn~1L, for each n > 1.
We are now going to show how, with a fixed orientation of Cp, Q
determines Qp canoniccdly. Let wn be the unique f-division point of pnL for which
(t>n{Z{k{p-n)wn,A{p-n)pnL)) = £{n,L). Diagram 1.5 (21) implies that
67

wn = wn-i mod pn lL, and in particular wn = wo = ft mod L. If we
define un = wn — ft mod pnL, we obtain a sequence (un) as in (iv).
DEFINITION. With L = ffi (6), let wn and un be the unique f- and pn-division
points ofpnL for which wn — un = ft mod pnL. Let un be determined by (12),
0 by (11), and ftp (the p — adic period corresponding to ft) by (8).
It is easy to verify that < n,np > G (Cz x Cp)/Q is independent of ft; it
only depends on the fixed choice of (fn). See also remark (iv) to theorem 4.11. In
fact, more is true; the pair < n,np > is well defined modulo Fx, and even modulo
those elements of Fx which are units at p.
4.5 The following proposition is the semi-local version of 1.2.2.
Proposition. Let (3 = ((3n) 6 U. There exists a unique power series gp{T) €
R[[T]]X for which
(13) !3n = (<Tn<7/?)K), n > 1.
Furthermore the following properties hold:
(i)Po = MO)1"*"1,
(H) 9/3/3' = 9/3 • 9/3',
(ui)g* o X5)(T) = UueE[p) ^M").
(*y) g<rt&) = otto) ° M^). (c,fp) = i.
PROOF: See chapter I, 2.2 and 2.3. For point (iv) see also 1.3.7, especially (15).
4.6 Similarly, the following proposition summarizes the results of 1.3.2-3.4 in a
semi-local framework.
Proposition. There exists a unique Q — homomorphism i : U —► A((),R)
(= R — valued measures on Q) t(/?) = (ip, satisfying
(14) \0T9p o 8{S) = / (1 + 5)«W dnp(a),
Jg
where k : G ~ Zp is the character giving the action on E[p°°], and where log g
is defined by 1.3.3 (7), with Wj = E[p]. The measure (ip depends on the choice of
(fn), but not on 0 (or equivalent data as in 4.4).
68

4.7 Recall that we have fixed' an embedding of Q into Cp. This means that $ is
mapped into Cp in a way that sends one of its field components isomorphically onto
a subfield of Cp, and the rest to 0. The image of y,p under this map will be denoted
/zjj; it is an integral p-adic measure on Q. The reason for introducing /z£ in lieu of
\Lp is that we wish to integrate certain Cp-valued characters of Q against it, and
these characters are not ^-valued in general. The next two lemmas complement
1.3.5 and 1.3.6.
Lemma. Let x be a character of GaI(F/K), and <p the grossencharacter of type
(1,0) fixed in 4.1. Fork > 0 and 0 G U let
(15) 6k(/3) = Dk logfo o 9)(0), D = (1 + S)±,
be Rummer's logarithmic derivatives (1.3.5), and let 8k{P)° be their projection from
R to Cp. Choose ideals c ofK, relatively prime to fp, whose Artin symbols (c, F/K)
represent Gal(F/K). Then, for any k > 0 the following equality holds:
(16) (l - ^^j • J2 X¥>*(0 • MM/*))0 = / X<Pk{o)dn%o).
Here we regard <p also as a p-adic character of Q (cf. remark at the end of 1.1). In
particular (p{o) — k[o) for a in G.
PROOF: If 6k(fi) is defined as in (15), but with log(^ o 9) (1.3.3 (7')), then
(17) 6k{0) = 6k{0) - Pk-lap{6k{fi)).
Also, if (c, fp) = 1, it follows from proposition 4.5(iv) and from (9), that
(18) **("«(£)) = Nc* * °*W))-
69

Substituting in the left hand side of (16),
(i _ *eM\ J2 xvH'-1) • Ncfc • *«(w))°
= E X^Cc"1) • &(a«(0))° (from (17) and (18))
c
= E X^Cc"1) • / p{*)k dp^wic) (1.3.5(11))
= E X^Cc"1) • / <p{o)kdn%oa^)
c *'G!
J9
4.8 Lemma. Let x be a character ofGal(Fn/K), n > 1, and suppose that n is
the exact power of J) in its conductor. Define the Gauss sum.
(19) r(X) = i E X(T)f»«W
P ieGal(Fn/F)
(this is well defined because i determines k(i) modulo pn). Fork > 0 and (3 € U
let
d
(20) fc,n(0) = Z>* logfc, o 0)($n - 1), £> = (1 + 5) —
fin $ ® iCp(fn)), and let £&,«(/?) ° be its projection to Cp. Choose ideals c, relatively
prime to fp, whose Artin symbols (c,Fn/K) represent Gal(Fn/K). Then
(21) r(X) • E X'p'ic-1) • SkAMP))0 = / X^(<^M».
c J9
PROOF: This time we begin the computation on the right hand side. Just as in
1.3.6, it is easy to see that
/ X<Pk{°)dn%o)
J9
= E X^c-1) • / <Pk{°)d&tip){°) Gn = Gal(Foo/Fn)
= E x^^c-1) • JL P£ Dk iog^((/J) o e)(d -1) • f-'.
j=0
70

If 7 G G and k(^) = j mod pn, then
(22) Dk log(^cW o *)($£-1) = /c(7)"fc • Dfc logfo,e7(/9) o *)(fn-l),
as follows from proposition 4.5(iv). Now for each c, and each j relatively prime to
p, there exists a unique i 6 Gas above such that acq = aci is again one of the
representatives chosen for Gal(Fn/K). It follows that the part of the double sum
corresponding to (j,p) = 1, is equal to the left hand side of (21). On the other
hand, the terms with p\j disappear after summing over the c's, because n is the
exact power of p in fx. To see this, let b be such that o\, 6 Gal(Foo/Fn-i). Then
D" log(^b(/3) O *)(<£_! -1) = <pk(b) • Dk logfa, O *)(<£_! -1)
(a special case of (22)). If j = pa, break the collection of c into cosets modulo
Gal(Foo/Fn^i), and use this last formula together with
<reGal(Fn/Fn-1)
to verify our claim.
4.9 Lemmas 4.7 and 4.8 (which easily admit a unified formulation) allow us to
compute the integrals of x<Pk against (ip, if we only know gp explicitly. It is time
to bring forth the elliptic units.
Let o be an integral ideal, relatively prime to fp, and define, for each n > 0,
(23) en(o) = 0(fi;pnL,a).
Then en(o), n > 1, is a unit in Fn (prop. 2A) and iVm>nem(o) = en(o) for
m > n > 1 (prop. 2.3; see also the first proof of prop. 2A(iii)). We let
(24) e(o) = lim en(o) (w.r.t. Nm,n)
and denote by /3(a) the projection of e(o) to the pro-p part U of lim R%. Our first
task is to compute the Coleman power series ge(a)-
A
Let A^ 6 $[[T]] be the logarithm of the formal group E, normalized to
71

Proposition. Let P{z) 6 F[[z\] be the Taylor series expansion of 0(n — z\ L, a).
Let Q{T) = P{\${T)) (operating formally with power series in *[[T\]). Then
(i) Q(T) e R[[T\]',
(ii) en(o) = {<f>-nQ){un), n > 0.
Thus 0e(a) = Q{T), and gp(a)lge(o) is a constant.
PROOF: First observe that since wg = dz is an F-rational differential, z is an
F-rational local parameter on E at the origin. Since E[j] C F, the elliptic function
0(n — z; L, o) is defined over F, hence P(z) indeed belongs to ^[[z]]. This allows
us to move from the complex domain to the p-adics, and Q{T) G $[[T]] is well
defined.
We claim that as local parameters at the origin, z and t are related via z =
A^(£). Indeed, since the formal group law in the z variable is the additive law, z is
some logarithm of t. Since dt/dz(0) = 1, z is the normalized logarithm of t. Thus
Q(T) is nothing but the expansion of 0(n — z; L, o) in terms of t at 0.
To verify (i) one can argue by "pure thought" that the function 0(H — z; L, o)
has good reduction at p, as may be read off 2.3 (10). But it is instructive to compute
its ^-expansions, using [Ta] (14) (caution: Tate's z is our t), and the addition law
on a cubic. In a generalized Weierstrass form 1.11 (6'), the sum of two points
-Pi = (zi5yi) and P2 = (z25y2) has x-coordinate
fy2-yi\2 . fy2-yi\
+ ai - a2 - xx - x2.
\X2-XxJ \X2-XiJ
Now A(L) and A(o-1L) 6 Rx, so it is enough to find the ^-expansion of
p(Q — z, L) — p(v, L) for v G a~1L/L — {0}. The formula above shows that this is
a power series in t, with p-integral coefficients and constant term p(Q,L) — p(v,L).
But this is a p-adic unit since (f, o) = 1 and both ideals are relatively prime to p
(see the second proof of 2.4(iii) for similar arguments).
To prove (ii) we must use the choice of (un) made in 4.4. Recall that wn was
defined to be the unique f-division point of pnL for which
(25) <t>n(Z(Hp-n>n,Hp-n)PnL)) = £(n,L).
72

It follows that (<f>-nP)(z) = 0(A(p-n)wn - z\ A(p_n)pnL, o). In the ^-expansion
of this function (on the elliptic curve E* ) substitute t = un. In view of 4.4 (12)
we obtain the value
8(A(p->B - A(p->n; A(p~n)pnL, a),
which is precisely en(o), thanks to the choice of un. This concludes the proof of the
proposition.
4.10 To get an idea what iip(a) looks like, lemmas 4.7 and 4.8 tell us, we have to
compute 6k,n{P{o))- Proposition 4.9 implies, substituting t = 9(S),
**,»(/*("))= [yJAJt) l0S 3e(a){t)\t=»(Sn-l)
(26) = Qk • (£\ log e(Q-z;L,a)\z=Vn
= -12 • nj • Ek{n-vn;L,a) (c/.3.1(7))
where vn is the pn division point of L for which t(£(vn,L)) = 9($n — 1).
Lemma. Let k > 1, n > 0, and choose a prime q, (q,fp) = 1, such that
Nq = lmodpn, {q,F/K) = (pn,F/K). Then
(27) n~k • 6k,n(0(a)) = n~k • (-12)(fc-l)! • <pk(pn) •
Here L I x>s; I —-— 1 1 = ]£ x(°)No_a, the sum extending over all integral o
such that (o,fM/K) = 1, {a,M/K) = (c,M/if). Both sides of (27) lie in Fn.
PROOF: Step 1. Ek(Q - vn; L, a) = A(pn)fc • Ek{n;pnL, a)**. Indeed,
A(p")fc- Ek(Q;pnL,ay« = (A(p-n)-kEk(n-pnL,a))°«
= Ek(K(p-n)Q;A{p-n)pnL,a)^ (proposition 3.3)
= Ek{A{qp-n)Q;L,a), since A(qp-n)pnq"1 = 0K-
73

However, from (12) and (11) we obtain vn = A(qp~n)un mod L, so from the
definition of un and wn (4.4), A(qp~n)ft = A(qp~n)wn — vn modulo L. Finally,
since wn = ft mod L and A(qp-ri) = <p{qp~n) = 1 mod f, we conclude that
A(qp~n)ft = ft — vn mod L, as desired.
Step 2. From the first step and (26) we see that it remains to prove
A{pn)kEk{ft;pnL,a)^ = ft~k{k - l)\<pk{pn) -
Writing A(pn) = A(q) • <p{pnq~1), and using the relation
Ek(ft;pnL,a) = Na • Ek(ft,pnL) - A(a)fc • Ek(ft,pnLya
(derived from (5) and (8) of §3), we see that it is enough to prove
(28) A(c)fc • Ek(ft,pnL)°< = ft-k(k-l)\<pk(c) • L(vk,k;(^^)y
for any integral ideal c, (c, fp) — 1. This was done in proposition 3.5.
4.11 We can now formulate the first main theorem. Let f be an ideal of K with
Wf = 1, and p a split prime, (p,f) = 1. Let en(a) = 0(l;fpri,o), (a, fp) = 1,
n > 1, be the elliptic units (23) in Fn = K(fpn). Define e(o) and /?(a) as in 4.9
and recall that Q = Gal(Foo/K). Let fj,a = V°p(a) be the p-adic integral measure
on Q corresponding to (3(a), as in 4.6-4.7.
Theorem. There exist complex and p — adic "periods" ft £ Cxandftp 6 Cp,
for which the following interpolation formula, both sides of which lie in Q, holds.
The grossencharacter e is assumed to be of type (k,0), k > 1, and of conductor
dividing fp°°.
(29) ft~k • / e{o)dtia{o) =
h
ft-k\1{k - 1)! • G{e) (l - ^-Y{e{a) - Noj-Lf^1^).
74

Here the complex L function is taken with modulus f. The "like Gauss sum" G(e)
is deGned as follows. Let F' = K(fp°°), so that F'Fn = K{jpnp°°). Write
e = x<pk with a grossencharacter <p of type (1,0) whose conductor divides f. Let
S = {7 6 Gal{F'Fn/K) | i\F' = {pn, F'/K)} where n is the exact power of
p dividing the conductor of e. Then
(30) G(£) = 5^£l . £ xW(£)-.
REMARKS: (i) G(e) is well defined because fn G F'Fn, and it depends only on e as
a whole (but not on <p or x)- G{e) = 1 if er is unramified at p, and it is an ordinary
Gauss sum if fc = 0. It lies in a CM field ([Go-Sch] §4), and G{e)13{eJ = p^*""1).
(ii) Note that 0 intervenes in the right hand side of (29) only through the
twisting factor er(o) — No.
(iii) The proof shows that we may take fip to be a unit in the completion of
Qur.
(iv) Both the period-pair-class < Q,Qp >6 (Cz x Cp)/Q and fia are
uniquely determined by (29). Indeed, suppose 17, Hp and p, also satisfy it. Let
d\x(cr) = <p(a)dtia(a) and ^2(0") = <p(a)dp(a). Then there exists a constant
c € Q such that for any x °f finite order J x^i = cf X^2- Hence Ai = cA2,
and fia = cp. But cHfi-1 = Hpfi"1, so if (29) is supposed to hold for all <pkx,
k > 1, then c = 1.
This is perhaps the point to remark that the dependence of fia, fip and G(e)
on (fn) is compatible with (29).
PROOF: Write e = x^* ^ above. Fix E as in 4.1-4.2, n and np as in 4.2 (6)
and 4.3-4.4, and compute (29) from formulas (16), (21), and (27). Use (2.4(H))
ac{P(a)) = /?(oc)/?(c)-Na. We arrive at (29), with p*(Pn)x(qMx) for G{e) (see
(19)). Now let F'n = F{E[pn}). Then FnF'n = K(fpn)sofn € FnF'n (this is also a
consequence of Weil's pairing). We may choose q so that (q, F'n/K) = (pn, F^/K),
andNq = 1 mod pn. Then if 7 6 Gal{FnFjK) and ^ = {q,FjK),
.7 _ .^q1 _ "(f'q1)
Sn — S« — S«
75

because 7<rq * fixes E[p ] pointwise, so its action on E[pn] and (ipn. is given by the
same character. With this in mind, (30) follows.
4.12 Theorem, (i) Let f be any non-trivial integral ideal ofK, and p a split prime
(p,f) = 1. Let G(e) be defined as above (30). Then there exist periods Q 6 Cz and
ftp £ Cp, and a. unique p-adic integral measure fj,(f) on £(f) = Gal(K(fp°°)/K),
such that for any grossencharacter e of conductor dividing fp°° and type {k,0),
k > 1,
(31) n;k J e(a)dfi(j;a) = Cl~k • G(e) (l - ^-) • W^O).
J9V) \ P )
(ii) Ifj\Q and JI(q) is the measure induced from (i(q) on ^(f), then
(32) Ji(Q) = Hi1 ~ T1) ' Mf)
where the product is over all I dividing Q but not f.
(Hi) If f = (1) the same conclusion holds, except that now fi(l) is just a
pseudo-measure, but for any o £ £(1) = Gal(K(p°°)/K), (1 — <7)m(1) is a
p — adic integral measure.
REMARKS: (i) In view of (ii) the theorem remains valid if f is replaced by f Q°° with
f, Q prime to p. An important case occurs when p\g, because then grossencharacters
of any infinity type (k,j) may be integrated, and when (k,j) is critical, a formula
similar to (31) holds. See 4.14.
(ii) The grossencharacters for which (31) applies are exactly those that can be
interpreted as p-adic characters of £(f), and are also critical.
(iii) £00,f refers to the complex L function with the Euler factor at 00 (1.1 (2))
but without the Euler factors at the primes dividing f.
(iv) Just as in 4.11, fi(j) and < fi,fip >£ (Cz x Cp)/QZ are uniquely
determined by (31). In view of (i), the period pair class < Q,np > is even independent
of f, although the individual periods depend on f and on the Weierstrass model of
E chosen in the construction.
76

(v) The (possible) pole at the trivial character—this is the meaning of part
(iii)—is a common feature of p-adic L functions. Compare [Iw2], p. 29 for the
Kubota-Leopoldt L function. We shall prove later (5.3) that (i(l) indeed has a pole
at the trivial character.
PROOF: We note first that (ii) follows from (31), and implies (iii) as well. Simply
choose any prime I, and let fi(l) = Ji(l)/(1 — <rf*). We may also assume that
itff = 1, because if f fails to satisfy it, some power of it will, and we may let (i(f)
be the measure induced from j*(fm).
So fix f as in 4.11, and let 6a = aa — No be the "twisting measure" associated
with o. By (29),
(33) na • 6b = Aib • £«, (ob,fp) = 1,
because the integrals of the two against any admissible e are equal, and there are
enough admissible e to separate measures apart (grossencharacters <px with a fixed
<p, and x ranging over all the characters of finite order, already accomplish it). We
shall prove, roughly speaking, that the greatest common divisor of all the 6a is 1,
hence the pseudo-measures fj,a/6a, which are all equal, are actually measures.
Let Koo be the maximal Zp extensions of K inside K(jp°°), and G' —
GaliKtfp00)/^), a finite group, |G'| = m. Let I" = GaliK^/K), and fix
an isomorphism Q = T' x G'. Let D be the ring generated over the integers
of the completion of the maximal unramified extension of Kp, by the mth roots of
unity. Then^ 6 £>[[£]] « £>[[F]][G;], and Q <g> £>[[£]] = Q <g> D[[T'}}m. The
last isomorphism is through A t-* (... ,0(A),...) where 6 runs over the characters
of G', extended to homomorphisms -D[[,p]] —► J9[[r']]. It is also well known that
Z)[[r']] = J9[[X]], and this is a unique factorization domain, since D is a discrete
valuation ring.
Now for any 6, 6(6a) = 6{aa\G') • (<70|r') — No is non-zero since <r0|r' =^ 1.
Thus 6a is a non-zero-divisor in -D[[,p]]. Furthermore, for a fixed 0, the greatest
common divisor of 0(6a) is 1. To see this, observe first that 7T, the uniformizer of
D, does not divide 0(6a). Secondly, if £n e K{jp°°), but £n+i £ K{jp°°), then
77

for any txgEY'xG' = § fixing $n, and any u 6 1 + pnZp, we can
find o's such that 0(6a) converge to 6(o)t — u. A common divisor of all the 0(6a)
must therefore divide (in D[[X]]) 6(o)(l + X)a - u for all u 6 1 + pnZp,
a £ pnZp, for some n. This being impossible, our assertion is proved.
Applying 0 to (33) we conclude that there exists \i$ 6 -D[[r']] such that
0(^a) • He = 0(/*a) for any o. Let t% = m-1 • ^2aeG, 0(a)a~1 be the idempotent
corresponding to 6. Then \l = Yl V-6^6 is a p-adic measure, 6a • \l = /xQ for any
o, and m • // is integral.
Even if m is divisible by p, we conclude from this that // is integral, as follows.
Suppose not, and let D° be the maximal ideal of D. Write = to denote congruence
modulo -D°[[p]]. We can find a scalar multiple u of //, v 6 -D[[P]], v ^ 0, but
6a • v = 0 for all o. Let u = Ylaec v°a-> u<* e -^[PHls an<* assume without
loss of generality that V\ ^ 0. Then
I* • v = Y, (»V«|r' - Nw*)o, P = {oa\G')-\
Thusi/pff = ^NaKlr')-1 for alia 6 G'.If/ = 1,^(1 - (Na(<70|r/)_1)d) =
0. Since i/x ^ 0, Nod = {aa\T')d in D[[T% which is a contradiction.
We conclude that /xQ/^o = /x is an integral measure independent of o. We
claim that //(f) = /x/12 is also integral. When (p, 6) = 1 there is nothing to
prove. Otherwise we appeal to the results of Robert and Gillard (prop. 2.7). It is
easy to deduce from them that if (a,6fp) = 1, then /3(a) is a 12*fc power in U,
hence /xn is divisible by 12. Replace /xQ by /xQ/12 and repeat the arguments above.
The claim follows. Comparing (29) with (31) concludes the proof.
4.13 As mentioned above, of special interest is the case when f is replaced by
fp°°. Then Q = Gal(K(jp°°)/K) contains the unique Z^-extension of K, and
any grossencharacter of type Aq and conductor dividing fp°° may be regarded as
a p-adic character of Q. Such grossencharacters can now be integrated against the
measure produced in 4.12 on Q, and theorem 4.14 will give the interpolation
property extending (31).
78

One point is nevertheless different. A glance at (36) below reveals that its
validity is not insensitive anymore to the choice of (?n). If 7 6 Gal{K{fp°°)/K{flp))
and $n = $%, then the new //, fip and G(e), for £ = x<Pk¥>3 with [j<p,p) — 1,
are given by
(34) H(U) = n^U), fip = N7-1 • np, G{e) = X(7)_1 ■ G(s).
Note that ^(7) = 1, since 7 fixes E\p ] pointwise, so <p(i) = N7. Formula (36) is
insensitive to such a change only if j = 0. We shall therefore begin by prescribing
the (fw) for which the theorem holds. However, a few words of explanation are
probably needed.
There are actually two p-adic periods associated to <p, fi' and fi'', which
intrinsically lie in Cp(l) and Cp, and satisfy Wp • Hp = 1 <g> 2ni. Both are units
in the completion of the maximal unramified extension of Qp (with the right Tate
twist), and Galois acts on them like
(10') <7C(np) = A(c)np,
(10") <7C(np') = NcA(c)-1n;/.
Here, by choosing an appropriate orientation (fw)5 we shall identify Cp(l) with Cp
so that 1 <g> 2iti corresponds to 1, Q'p = Qp <8> 2%i, and Hp = fi"1. The
transcendental part of f tp^^dfi, intrinsically given by QpkQp3, becomes Clp~i,
which accounts for (36) below. See [dS3] for generalizations to CM fields, and for
the relation with Hodge theory of abelian varieties.
CONVENTION. From now on, let $n be the primitive pn root of unity given by the
Weil pairing (see [La] ch. 18, [Mum] §20)
(35) $n = epn(wn,un)
with respect to the lattice pu0kH, where wn and un are the unique p — and pn—
division points of that lattice for which wn — un = fi mod pw0iffi.
The following facts are easy to establish:
79

(i) & = £»-i.
(ii) If f = p and 0 < n < m, and if wn and uw are defined as in 4.4,
then fw = epm(wn,un) with respect to the lattice pnffl. For n = m this agrees,
of course, with the definition.
REMARK: fw can be given in a closed form. See lemma 6.2.
4.14 Theorem. Let Q be an integral ideal of K, and p a split rational prime,
(p>fl) = 1- Let // be the measure /x(gp°°) on Q = Gal{K{$p°°)/K) (see remark
(i) to theorem 4.12), and fix < n,flp > 6 (Cs x C£)/Qs as above. Then the
following formula, both sides of which lie in Q, holds for any grossencharacter e of
conductor dividing Qp°°, and of type (k,j), 0 < — j < k :
(36) IJJ-* J «(„)«,(„) = 11*-* (jgy • G(.) (l - ^) • L^il'-l,0).
Here to define G(e), write e = ipkfp3X with a grossencharacter tp of conductor
prime to p and type (1,0), and x a character of finite order, and let (see (30))
(37) G(.) = ?^m • £ X(1) . W)->.
PROOF: This is similar to 4.11, except that the computations are more
complicated. We outline the main steps, and leave out some routine verifications to the
reader.
Step 1. Fix an auxiliary ideal o, (o, Qp) = 1. With the notation of 4.11, and
with f = Qp , m > 1, //„ = 12(ct0 — No)/x(f). Since the measures
//(f), for various m, are compatible (4.12(H)), so are /xn, and their inverse limit is
a measure //„ on Q. This //„ is associated to the double inverse system of units
©(l,0pmpn;o) = e»,m(o) {n,m > 1).
Now fix some large m > 1, let f = Qp , and assign F, E, L, fi, fip, <p,
X and A their previous meaning, e.g. L = ffi etc. Let Fn = K(fpn) and
Gn = Gal(Foo/Fn). Let c run over integral ideals of K whose Artin symbols
represent Gal(Fn/K). Write = to indicate congruences in Cp (not necessarily between
80

integers). Now, if n > 0 is the power of p in fx,
nj-*- / x^V'W4.W
J9
(38)=nj-fc-X;X^V(c-1)-/ <Pk{o-)-dnlt{m){o) modp™
= ni-k-r(x)- (l - X^(P)) •J^X<Pk^(c-1)-6k,n(ac((3(a))0 mod
as in lemmas 4.7 and 4.8. If we write ac(/?(a)) = /?(ac) • /?(c)-No, we deduce at
once from (26) that (38) is congruent modulo pm to
12 • flj - r(X) • (l - &&M) • £ X^V(c-) •
• {No • Ek{n-vn-L,t) - Ek{n-vn;L,ac)}.
Let q be an ideal as in lemma 4.10. As in the first step of that lemma, we conclude
that (38) is congruent modulo pm to
12 • flj ■ r(X) • (1 - *^M) . A^q-1)* ■ E^V^1)-
(39) \ P J j
•{A(ocq)fcJEJfc(n,pnL)c7oC(i -Na-A(cq)fc£Jfc(n,p,*L)<7«i}.
Step 2. Recall that //„ = 12(ct„ — No)/x, and e = XV*^'- We shall prove that
for(c,fp) = 1,
nj.A(p»q-1c)*.tffc(n,p»L)*« =
(40) (fc-i).^^ ^.pVrt-pVlc)-
L (<pk-j,k; (!^IE\\ mod tfp
xn—ttiq
for some mo independent of m. Simple algebra shows that (39) and (40) together
imply a congruence modulo pm-m<> between the two sides of (36). Our mo will
depend on g and e, but not on m. Since m may be arbitrarily large, the two sides
of (36) are in fact equal.
81

The key to (40) is the congruence
(41) n-1 = -N(fpn)JE1(n,pnfn) modPm-mo
(mo independent of m), to be treated in 4.15. Assume (41) is proven, and consider
the fundamental relation 3.5(13):
(41) (*-1)'(#yn^.,(^.i(^>;(^f)) =
N(fp»)-'A(c)fc-'" • Eilk{ntpnL)"t
of which (28) is a special case. We would like to replace Ej>k by an expression
involving Ek only, possibly weakening the equality in (42) and replacing it by a
congruence. This can be done, according to 3.4 (12), but with EJ)k(A(p-n)n, A(p~n)pnL),
because the lattice A(p~n)pnL (unlike pnL) is the period lattice of a Weierstrass
model with good reduction. Indeed, A(p~n)pnL is the period lattice of E^ ", where
E refers to the specific model fixed in the beginning, which had good reduction at
every place of F = K(f) above p. Therefore express the right hand side of (42) as
N(ft>»)->A(c*r»)*-'" . £;)fc(A(p-»)n,A(p-")p»Lrs
and use 3.4 (12) to conclude that (42) is congruent modulo 23'pm~nk to
A(c)fc^'(-N(fpn)Je;1(n,pnL)c7c)-^fc(n,pnL)c7c
= <rcn£ • A(c)fc"J' • Ek{U,pnL)at mod 2jpm-mo~nk by (41)
= nJNc-'A(c)* • ^fc(n,pnL)ac mod 2jpm~m°-nk by (10).
To deduce (40), multiply the left hand side of (42) and the last line by
NcJA(pnq-1)fc, and recall that A(pnq-1) = <p{pnc\~1) because pnq~1 isaprincipal
ideal congruent to 1 modulo f.
REMARK: It is plausible that mo = 0 satisfies (41) and (40). See the remark
following lemma 3.4. On the other hand, the extra powers of 2 (when p — 2) seem
to be necessary. One way or another, the final result is unaffected by this, because
our concern is with the limit when m approaches oo.
82

4.15 STEP 3. PROOF OF THE CONGRUENCE (41): The idea behind the proof
is simple. Specifying an isomorphism 0 : Gm ~ E is tantamount to giving
an isomorphism between the corresponding p-divisible groups, and modulo pm,
O'(0) is determined by the restriction of that isomorphism to the group schemes
\ifm. ~ ^[pm]. However, by the Weil pairing, this is the same as giving a primitive
p division point on E. We are therefore led to compute a formula for the Weil
pairing in terms of the parameter t on E.
First, by lemma 3.4(iii), and at the expense of introducing mo, we may assume
that Q = (1) and prove (41) with f = p . Let un and xvn be, as in 4.4, the unique
pn- and f- division points of pnfQ for which wn — un = Q mod pnffi. Let Pn —
£{A{p-n)un,A{p-n)pnfn), Qn = Z{A(p-n)wn,A{p-n)pnfQ) be the corresponding
points of <j>~nE. We consider only those n lying between 0 and m. As mentioned
at the end of 4.13, $n = epm(Qn,Pn). Put, for brevity, Ln = A{p~n)pnfn,
and let us write a(z) for the corresponding a(z,Ln). Put also rn = A(p~n)un,
vn = A(p~n)wn, and choose auxiliary torsion points Xn of Ln, primitive of order
£> (Afp) = 1, in such a way that £(AW,LW) = <f>~n£(\0,L), 0 < n < m.
If / and g are .Lw-elliptic functions with
div(f) = pm((un + Xn) - (Xn)), div(g) = pm((rn) - (0)),
then the Weil pairing of Qn and Pn is computed by
/(fo.) - (Q))
(43) ^(Q.,P.) = 9(K + K) _ {K)].
Let hn be the Lw-elliptic function
Ka ) nn{z) a^mz _ A^ a^ + ^ ^K + pm^ _ z).
Expressing / and g in terms of sigma functions we find out that
(45) epm(Qn,Pn) = hn{rn).
Now let us expand hn in terms of the parameter t on <f>'~nE. Since dz/dt(0) = 1,
one gets
(46) hn = 1 + {{pm - l)£(An) - pm<T(An + un) + f(An + pmun)}t + • • •
83

where f = a'jo is Weierstrass' zeta function. However, since pmvn € Ln,
${Xn + pmVn) = ${Xn) + Pmil{i/n), so the coefficient of t becomes
Pm(<T(A)-r?(A))- pm($(X + v)-r,(X + v)) = pm{Ei{\n,Ln) - Ei{Xn + i>n,Ln))
= pm • <f>-n(E1(X0,L) - EtiXo+votL))
= _p™ . ^-nE^^L) modpm-mo
= ^-"(-Nf-^^n.fn)) modpm-mo,
where mo is independent of m. The congruence is a consequence of 3.4(ii), and with
an appropriate Xn we could reach mo = 0, but this is irrelevant to our purpose.
It is clear that hn is rational over ir(£'[£]). We claim that its f-expansion (46)
has p-integral coefficients. To prove the claim, let
/
(47) *„(*) = J]
lE(p-mLn/Ln)/±l
P \—^- -*)- P(l)
P (yZ ~ Z) ~ PM
n
0<t<p3n»
M*^-)-^)!
'P^-')-^)).
Here p(z) = p(z,Ln), u = i/n, X = Xn, and the first product is taken over
a full set of representatives of non-zero pm-division points of Ln modulo ±1. This
has to be modified when p = 2 as follows. If 71, 725 73 axe *ne three non-zero
2-division points, one should only take p'(-)/2 = {n?=i(*>(') ~ p{li))} M
their contribution to the first product. Now it is easy to see that hn and kn have the
same divisors. Furthermore, the same arguments as in the proof of proposition 4.9,
or proposition 2.4(iii) ("second proof" there), show that the ^-expansion of kn(z)
around 0 has p-integral coefficients. It also shows that kn(0) is a p-adic unit. Hence
hn(z) = kn(0)~1kn(z) has a p-integral ^-expansion, and our claim is verified.
The choice of Xn and un makes the power series
h = <f> hn
84

independent of n. It has p-integral coefficients, and satisfies
( h = 1 + (-Nf-^njn))* + ••• modpm-mo
\$n = (<f>-nh){un) 0 < n < m.
The last equality is a consequence of (45) and the specific choice of (fw) outlined in
4.13. For m large enough (48) already guarantees that —Nf • Ei(Q, fn) is a p-adic
unit, and the power series h — 1 has an inverse with respect to composition, which
we denote by 0. Recall (11) that 0 was the unique isomorphism Gm ~ E for which
un = {<f>~n0){$n — !)• Hence 0 and 0 are two power series with coefficients in an
unramified extension of Qp, for which
(*""*)(£» - 1) = {<t>-nQ){Sn - 1), 0 < n < m.
This implies that [(1 + X)p" - l]/[(l + X)p"~l - 1] divides <j>-n(0 - 0), hence
0 - 0, fori < n < m, so 0'(O) = 0'(0) mod pm. Since 0'(0) = Qp, we finally
obtain (41). The proof of theorem 4.14 is complete.
REMARK: Step 3 settles affirmatively a question left open in [Ya2], and also shows
how to choose (fw) so that the periods denoted by fij, in [Ya], and by 7j, in [Ya2],
become equal.
4.16 p-ADIC L FUNCTIONS: For convenience and comparison with results in the
literature, we give the translation of our main theorems to power series language.
As usual, the quadratic imaginary field K and the embeddings of Q in C and
Cp are fixed once and for all, and p is assumed to split in K as pp, where p is
the place induced from Cp. Also fixed throughout the discussion is the generator
(fw) for the Tate module of /zpoo, as in 4.13, and the period-pair-class < fi,fip > 6
(Cs X Cp)/Q , as in the preceding sections. We denote by e a grossencharacter
of K of type Ao, or more generally, a p-adic character of the Galois group of an
abelian extension of K.
DEFINITION. Let f bean integral ideal for a "pseudo-ideaF of the form ob°°j reia-
tively prime to p. The p-adic L function of K with modulus f is the function whose
85

domain is the set of all p-adic continuous characters on §(f) = Gal(K(fp°°)/K),
and which assigns to every e (e / 1 if f = (1)) the value
(49) Lptj{e) = f e-\c)d^c).
J 9(f)
Here /x(f) is the integral measure constructed in theorem 4.12 (see also remark (i)
there).
With this definition, the interpolation formulas (31) and (36) say the following.
Let e be a grossencharacter of type {k,j) with 0 < j < —k. Define G(e) by (37).
Let q be relatively prime to p, and divisible by the non-p part of the conductor of e.
Let f = Qp , and Loq^{s,s) the classical L function with the right gamma factor
(1.1) but without the Euler factors at the primes dividing f. Then
(50) nJ-'Lp.Ke) = nk~* (-^=)' -GCe"1) (l - ^M) .LeOt1{et0).
Furthermore, if jf = 0 and e is unramified at p, the same formula holds with fl
instead of f.
4.17 ONE AND TWO-VARIABLE p-ADIC L FUNCTIONS: Let Koo be the unique
7ip extension of K unramified outside p. Fix an isomorphism /cx : Ga/(.K'00/.K') c±
1 + pZp (1 + 4Z2 if p = 2). For example, if p does not divide the class number
of K, p is totally ramified in Kqq, and /Ci may be taken to be the inverse of the
local Artin map, once we identify the Galois group with the inertia group at p. Let
F be an abelian extension of K, linearly disjoint from K^, and i^ = FKoq (this
notation is somewhat different from the one adopted until now). Fix an isomorphism
Gol^oo/K) a Gal{F/K) x GaliK^/K). If x is a character of finite order of
Ga^Foo/K), then we may define
(51) Lp,i{x,*) = LPti(x*r) V5 € Zp
(f must be divisible by the non-p part of the conductor of x) • Let 70 be a topological
generator of Gal(Koo/K), /ci('yo) = «, and decompose x as x = XoXi? where xo
86

is trivial on Ga^K^/K) (a character "of the first kind", at most tamely ramified
at p), and Xi is trivial on Gal(F/K) (in general wildly ramified at p). Then the
fact that Lp^{xKT") is given by an integral like (49) translates to the following
statement. There exists a power series G[xo',T) £ D[[T]] such that
(52) £p,f(x,«) = G(xolXi(7o)-V - 1).
If f = (1) and xo = 1, G[xo\T) G r_1D[[T]]. Indeed, we may assume
F C K{f), so that Foo C K{fp°°) and GaliF^/K) is a quotient of ${f) =
Gal(Ktfp°°)/K). Fix an isomorphism £(f) ~ Gcd^^/K) x GaliKtfp00)/K^) =
T' x H', and put
(53) G{Xo;T) = ]T xolW / (1 + r)0d/x(f;r7§).
One often refers to G(xol T) as the p-adic L function of xoj and more generally
to G(x\T) = G(xo;Xi(7o)_1(l + T) - 1) as the p-adic L function of x (with
modulus f). Note however that as a power series in T it depends on the choice of u
(or 7o)j and only Lp^(x,s) has an intrinsic meaning.
The two-variable functions are defined by the same procedure. Let K'^ be the
unique Zp extension unramified outside p, and assume for simplicity that K'^ n
Koq — K. Fix an isomorphism /C2 : Ga^K'^/K) ~ 1 + pZp (1 + 4Z2 if
p = 2). Then we let
(54) LP)f(x;si,s2) = lpAxkT81 K2°2) V 81,82 € Zp.
We leave it to the reader to define G{x\ Ti, T2) as a power series in two variables for
any character of finite order x of K, and to derive the analogues of (52) and (53).
5. A p-ADIG ANALOGUE OF KRONECKER'S LIMIT FORMULA
5.1 In the cyclotomic case, Leopoldt has shown that the value of the (Kubota-
Leopoldt) p-adic L function at 1 is given by a formula which resembles the classical
87

formula for L[x,l). The interest in his result is that the point 5 = 1 is the first
integral point outside the range of interpolation, yet when the complex logarithm is
formally replaced by the p-adic one, the same identities hold with Lp(x, 1) instead
of L(x, 1). See [Iw2] theorem 3, p. 61.
When K is quadratic imaginary, there is a similar classical result derived from
Kronecker's second limit formula. See [Sie] theorem 9, p. 110. Since the functional
equation relates the points 5 = 0 and 5 = 1, the result can be restated as follows
([Ta3] p. 97):
Theorem. Let g be a non-trivial integral ideal of K, and x a character of finite
order whose conductor divides g. Let <pB(C), C £ Cl(g), be Robert's invariants
associated with classes in the ray class group modulo g (2.6 (17)). Let g be the
least positive rational integer in g, and wB the number of roots of unity congruent
to 1 mod g in K. Let Loo.otXj5) = (2n)~aT(s)L(x,s), as in 1.1 (2). Then
(1) Loo,,(x,0) = -=i- £ X(C) • log|^fl(C)|2.
In this section we prove a p-adic analogue of this classical result. See also [Kl],
10.4.
5.2 Theorem. Notation and assumptions as above, let p be a prime that splits in
K, and write g = fpn, p f f. Define -Lp,f(x) as m 4.16 (49), and let LP)B(x) =
LP,f(x)(l - X(P)) ifn > 0. IfX = 1, assume f / (1). Then
(2) ^.»M = t^t • G(*-1) f1 - *-^) • £ x(c) •log ^(C)-
12gw* \ pJ ci£(B)
Here "log" denotes any branch of the p — adic logarithm ([Iw2] 4.1), and G[x) is
the Gauss sum 4.11 (30).
Compare 4.12 (31). Forgetting for the moment the difference between complex
and p-adic logarithms, we may say that (31) "extends" to A; = 0. Recall that
LPif(x) is the integral of x_1 (and not of x) against /x(f).
88

PROOF: All the references below are to formulas from §4. Assume first that tuf = 1
and n > 0 is the exact power of p in fx (but f may be divisible by primes where
X is unramified). Then formulas (16) and (21) imply
J r'WW = r(X"1)(l - £±M) • £ x(c)So,n(°M»)))°
(3) = rix-1) (l - ^51) £ *(C> M^.</»<«)) o «)(f» " 1)
= r(x-X) (l - *^1) £ X(c) • log a«(en(a))'«
where en(a) = 0(ft;pnfft, a) (see (23)), and q is as in lemma 4.10: (q,K(f)/K) =
<f>n, Nq = 1 mod pn. Note that G(x_1) = X-1(qMx-1)-
Combining (3) with the definition of LPtB(x) and the fact that /x0 =
fM(f)12(aa - No), we get
(4)
Wx) ■ 12 • (x-\a) - Na) = Gfo"1) (l - ^1) £ *(c) ' lo8 *<(«»(«))•
Formula (2) follows from the last one, because by 2.4 (18)
en(a)» = 0(1; 0, a)" = p8(l)NB-'",
and we have assumed wg = 1.
The general case follows easily from the special one above, because if we denote
the right hand side of (2) temporarily by Mfl(x), the distribution relation 2.3 (see
also [K-La] p. 242) implies that for any prime I,
I (1 - x(l))Mg(x) otherwise.
The same relation holds between LPtB(x) and ipJi(x)i as already mentioned in
theorem 4.12(H). We may therefore assume that Wf = 1 without any loss of
generality.
89

5.3 Corollary. -kPl(i), the p — adic L function with conductor 1, has a true pole
at the trivial character.
PROOF: Fix an auxiliary prime ideal f. According to 4.12(iii) it is enough to show
-kp.f(l) ~ /^(^(f)) / 0 (note that if f is divisible by two distinct primes this is
zero). But (4) implies
(5) 12 • /x(£(f)) • (1 - Na) = (l - i) . log NK{f)/K0(l;f,a)
and iVx(f)/jcQ(l; f> o) is a p-adic unit which is not a global unit, since its valuation
at f is non-zero. In particular its p-adic logarithm is non-zero, as desired.
6. The functional equation
The "two-variable" p-adic L function satisfies a functional equation similar to
the classical functional equation. In this respect the elliptic theory differs
significantly from the cyclotomic theory of Kubota-Leopoldt L functions. The reason is,
vaguely speaking, that complex conjugation "acts on everything" and enables us
to define an involution e i-» £ on characters of type Ao of K, that preserves the
critical ones (see also 6.7 below).
6.1 For any grossencharacter e of K, let
(1) £{a) = r^No"1.
Clearly fg = fe, and if the infinity type of e is (k, j), that of £ is (—j — 1, —k — 1). In
diagram 1.1(4) the infinity types of e and £ are symmetrical about the line k + j =
—1, so if is critical if and only if e is. Now let f be an integral ideal, (f, p) = 1, and
suppose fe | fp°°. Let Q = Gal{K{fp°°)/K) and $ = Gal{K(fp°°)/K). If we
consider e as a p-adic character of Q (see 1.1(5)), and denote complex conjugation
by p, £ may be considered a character of $, and (1) reads
(2) £{c) = e^N-^pap-1).
90

As ee = Nfc+J', we also have L{i,0) = L(e-1N-1yO) = L{e,l + k +j), so
the complex functional equation 1.1(3) becomes
1+k+j
(3) Ax,M) = W ■ (dKNfe)~T~ . LoofrO).
Artin's root number W = W(e) is a product of local factors W = \[ Wv (over
all places of K). Here W^ = *lJ'-fcl, and the finite factors Wq = VTq(e) are given
as follows. Let Eq be the quasi-character on K\ associated to e as in Tate's thesis.
Thus eo is trivial on principal ideles, and £o(*q) = e(q) if (ijfe) = 1> and *q is
any idele whose q-component is a uniformizer of Kq, and which is 1 at the other
coordinates. Let qe||(\/—djf) and qm||fe, and denote by tr the map
tr : Kq *-**£* Qq -* Qg/Zg — Q/Z.
In the following sum, u ranges over ideles trivial outside q, whose q-components
represent 0* mod 1 + qm0q. We then have ([La2] XIV, §8)
m(k+j-l)+e(k+j)
(4) W^e)-1 = Nq 2 . £ e0{ut-m-)exp [2jr«tr(trf-m-)] ,
u
and this expression is independent of the choice of £q and {u}.
6.2 For the functional equation we shall need to know explicitly the fn defined by
4.13(35). Write sj^d^ for iy/d^.
Lemma. Let 6 £ Z* be the image of yj—dx under K "-^ Kp = Qp. Then
(5) d = *-2*i/pn-
PROOF: Write 0K = Z + Zr, t = y/-dK/2 or (1 + y/-dK)/2, according to
whether d^ = 0, 3 mod 4. On the pn-division points of the lattice pn0K we then
have the following formula, giving the Weil pairing:
(6)
pn(x,y) = exp I 27U——=== ) , x,y G 0K/pn0K-
\ pnyJ-dK)
91

Now suppose wn, un are as in 4.13(15), i.e. wn € pn, un £ p"", wn — un =
1 mod pn. Then (6) gives us
fn = epn[wn,un) = expl 2m ——===) .
\ pn\J~dK)
Now wn — wn = (wn — un) + (un — wn) = 1 mod pn, so also wn — wn =
— 1 mod pn. It follows that in Kp (wn — wn)/y/—dK = —6-1 mod pn. Since
(wn — wn)/y/—dic € Q, the congruence holds mod pn if we consider 6 in Z*.
This proves the lemma.
6.3 The key to the p-adic functional equation is a comparison between the local
root numbers Wp and Wp-, and the quantities denoted by G(e) in 4.14(37).
In the following, let e be a grossencharacter of type (k,j), and fe = fpnp ,
(f,p) = 1. Let
(7) vs € GaliKifp^/Kifp00))
be defined by as{$) — $6 for all p-power roots of unity f. Thus p(os) = 1,
<p(cs) = N(a$) = 6.
Lemma, (ij Wp(e)"1 = p^k+i+^6-ke{as) ■ Gfe"1)
PROOF: (i) When q = p in formula (4), we have e = 0, and for tp we take
p = (...l,p,l,...) € K%. Then
Wpie)-1 = pT^+J'-1)^£o1(Pn")^P(27ri"_1P"n)
u
= p^(^-i)^-v-i(Pn)-E^1(pn«)^p(2^u-1p-n),
u
where we have written, as usual, s — <pk~<p3x with <p of type (1,0) and conductor
dividing fp . Replace exp{2mp~n) by $~s (see (5)), and use local class field theory
to identify the ideles pnu, u € {0p/pn0p)x, with their Artin symbols in Q =
92

Gal(K{fp°°)/K). FutF' = K{fp°°) and F'n = K{fp°°pn). Then pnu correspond
to S = {a € Gal{F^K) \ a\F' = {pn,F'/K)} (cf. 4.11). We arrive at
wye)-1 = p^^'-^^-v^p^.^x-1^)?;
= pT(*+i+i).x(a5).G(e-i).
To obtain the expression in (1) note that <p{os) = 6 and <p{os) = 1.
(ii) It is easy to see, by "transport of structure", that
w-{e) = wy^N-1).
The character £-1N-1 is of type (j, A;), and the power of p in its conductor is m.
Applying part (i) to it we get
Wpie)'1 = p^k+3'+1H-j-1i-1{asyG{iN)
where, to be perfectly honest, as is defined as in (7), but with f instead of f. Let
<r_i be defined in the same way, with —1 replacing 6 : C-i(f) = f-1. Now
G{ZN) = pmG{i), and from G{e)G{e~1) = p-m£(a_1)(-lp'+1 we easily deduce
the formula given in (ii).
6.4 THE p-ADIG FUNCTIONAL EQUATION: Let e be a p-adic continuous character
of Q = Gal(K(jp°°)JK), (f,p) = 1, and assume that e is ramified at all the
primes dividing f. As mentioned above, £ (given by (2)) is a character of ^ =
Gal(K(fp°°)/K). Let us write simply Lp(e) for the p-adic L function of e, with
modulus fp (see 4.16), and similarly Lp(£). Thus
(Lp{e) = fg e-\c)d^r°\o)
( \lp(£) = f$ i-'WdnCfp00;*).
Theorem (CF [K2] 5.3.7). (i) There exists a p-adic unit W^adic{e) (the p-adic
root number) for which
(9) Lp(e) = wrpa*«(e).^).Lp(g).
93

(ii) Suppose e is a grossencharacter of type Ao, of conductor fpnp . Then
(10) W^adic(e) = -i J! W,{e)
q
where the product is over all finite primes q, (q,p) = 1.
PROOF: Let us assume first that e is a grossencharacter of type (k, —k — 1), k < 0,
so its infinity type lies on the anticyclotomic line. Then £ has the same infinity
type, and both e and £ lie within the interpolation range of theorem 4.14. Since
(1 — —) = (1 — £(p)), and f is the exact non-p part of fe, formula 4.14(36)
P
(with e~ * instead of e) yields
(11) nlk^Lp(e) = n2fc+1(^) + G(e-l)(l - £(p))(l - e^-L^O).
Similarly,
(12) Uf^Lp{£) = n2^1 (^j + G(£-l)(l - e(p))(l - £(p)yLoQ(£,0).
Dividing (11) by (12), we deduce from the complex functional equation (3) that
Lp{e) _ G{e~lyW{e)
Lp(£) ~ G(g-i)
= S^-'Mi-Sr^ia.syW^eyHw^e),
where we have used lemma 6.3. Finally, Woo(e) = i~1-2k gives (9) and (10) in
this case, and it is well known that (10) is a p-adic unit.
To treat the general case, let Koq be the maximal Z2 extension of K, G —
GaliK^/K), H = GaliKtfp00)/^), and & = GaliKtfp^/K^). Fix
splittings § = GxH,$ = Gx]i, which are compatible with the isomorphism
Q = $,o- h-> pop~l (p is complex conjugation). Any character e of Q can be
written uniquely ase = eqShi where eq is trivial on H, and vice versa.
Furthermore eg is ramified only at p and p, and if e is a grossencharacter of type (k,j), so
is eg, because eh is of finite order.
94

Consider the two measures on Q defined by
idu{a) = <f/x(fp°°,<™$)
It is easy to see that (9) is equivalent to the statement that if ejj is ramified at all
the primes dividing f,
(14) / E-\a)dv[a) = WPadic{e)- f E~l{a)di>{o).
Jq Jg
Now fix £h and vary eg- Observe that Wpadtc(£) depends only on eh, so it is
unchanged. Thus (14) amounts to equality between measures on G:
(15) e-H\v) = W^{eH).e-H\i>),
where we used e^1 to project D[[,p]] to D[[G]]. We have verified before that for
any grossencharacter eg of type (k,—k — l), k < 0, the integrals of egx against
both sides of (15) coincide. Since we may twist eg by any character of finite order
without affecting its infinity type, there are enough admissible eq to separate points
in D[[G]]. Thus (15) is proven, and with it (14) and (9). Formula (10) also follows
from the above, as well as the fact that Wpadtc(e) is a unit.
REMARK: The involutive nature of e i—► £ together with (9) imply
(16) Wpadic{E)-Wpadic{£)-E£{a-1) = 1.
This may be verified directly. For example, let us check it when f = (1).
Then (10) gives Wpadic(E)Wpadic(£) = -1. On the other hand er£(<7_!) =
—E(a-ipaz\p~1) = —1, because <T_ipaZiP_1 is the Artin symbol of the idele
a = (ctv) defined by ctp = ccr = —1, av = 1 otherwise, and we may change the
remaining l's to —1 because e is unramified outside p, thus obtaining a principal
idele, whose Artin symbol is trivial.
6.5 Of special interest are the anticyclotomic characters of K. These are, by
definition, the p-adic characters satisfying
(17) e = e.
95

For such an e the sign in the functional equation (9) is
(18) sgn{e) = WK'^yeip-x) = ±1,
because of (16). If sgn(e) = —1, e is a zero of the p-adic L function.
REMARK: An anticyclotomic character e must be ramified at some prime not above
p, and, of course, its conductor is stable under complex conjugation. To see this
compare (18) with (10). Alternatively, suppose a grossencharacter e satisfies e = i.
Then if "/„ denotes the idele which is —1 at v and 1 elsewhere, e(7p) = £("7p) =
e('Yp), hence ^('yoo'ypTp-) = e(~ioo) = —1, because the infinity type of e is (A;, —fc—1).
But if e is unramified outside p, e{~loo1py$) = e(—15 —15 • • •) = 1- Our argument
actually proves that e must be ramified at a non-split prime.
6.6 The functional equation has the following obvious, but important, corollary.
Corollary. Lp{e) = 0 -O- LP{S) = 0.
6.7 Another corollary of the functional equation is the following.
Corollary. Notation as in theorem 4.14, the conclusion of that theorem (the
interpolation formula (36)) is valid for any grossencharacter e of type (k,j), where
k > 0 and j < 0.
In other words, the interpolation range is extended to half of the critical values.
We cannot extend this result to the other half. The tautological functional equation
(19) L{e,s) = L{eop,s),
where p is complex conjugation, does not have a p-adic analogue. Note that the
p-adic functional equation was the counterpart of (3), which is the "composition"
of (19) with the classical functional equation 1.1(3), and not of either of these two
alone.
96

CHAPTER in
APPLICATIONS TO CLASS FIELD THEORY
There are two main themes to Iwasawa theory of elliptic curves with complex
multiplication, the interplay between which enriches our understanding of both.
One deals with abelian extensions of quadratic imaginary fields, especially their
units and ideal class groups. The other is concerned with the arithmetic of CM
elliptic curves, and problems having their origin in diophantine equations. Although
these two themes are intricately woven, we shall try to concentrate on the first here,
and postpone the striking applications to the conjecture of Birch and Swinnerton-
Dyer to the fourth and last chapter.
To put things in perspective, let us regress a little and discuss cyclotomic
Iwasawa theory. Motivated by Weil's work on curves over finite fields, Iwasawa
sought an analogue for the Jacobian variety in the case of number fields. Unable to
find a satisfactory answer in the large, he restricted his attention to the p-primary
part of the Jacobian (p not equal to the characteristic). Consider Q as a base
field. The proposed analogue of the "geometric" curve (over the algebraic closure
of the finite field) is Fqq = Q(p°°), the maximal abelian extension of Q unramified
outside p (the real subfield of Q(A4p~)). The analogue of the p-adic Tate module of
the Jacobian would be the Iwasawa module X = Ga^Moo/Foo), where Mqq is the
maximal pro-p abelian extension of F^ unramified outside p. The study of X as a
module over A = Zp[[£]], Q = Gal(Foo/Q), is the focal point of Iwasawa theory.
In Weil's theory, the zeta function of the curve is essentially given by the
characteristic polynomial of Frobenius on the p-adic Tate module of the Jacobian.
Its zeroes are the reciprocals of the eigenvalues of Frobenius in its action on the
Tate module. The celebrated theorem of Mazur and Wiles (formerly the "main
conjecture" of Iwasawa) asserts that the eigencharacters by which Q acts on X (a
free Zp module of finite rank) are precisely the reciprocals of the zeroes of the
97

Kubota-Leopoldt p-adic L function. Note how elegant this formulation becomes
when we view the p-adic L function as a function of characters on Q (see the
discussion preceding 1.3). Although the semi-simplicity of X as a A-module remains
unresolved, the theorem of Mazur and Wiles supplies the link between the p-adic
analytic side and the algebro-arithmetic theory of Zp extensions.
Our main aim here is to formulate a corresponding main conjecture in the
elliptic case. See 1.9-1.11. We shall give some evidence in favour of it, proving
the equality of the Iwasawa invariants of the two modules which are supposed to
have the same characteristic polynomial. Thus instead of matching the roots of two
polynomials, we show that their degrees are equal. We also discuss without proofs
Gillard's theorem on the vanishing of the ^-invariant, the important recent work
of K. Rubin on relations in ideal class groups of abelian extensions of K, and its
relation to the main conjecture.
Throughout chapter III we assume p > 2.
l. The Main Conjecture
1.1 We keep the notation of chapter II. Thus K is our quadratic imaginary field.
Fix an integral ideal f of K, and a split prime p, (p, f) = 1. Let F = K(j) and
Fn = K(jpn) be the corresponding ray class fields. Every prime ty of F lying
above p is totally ramified in Fqq.
Denote the integers of the completion of Qpr by D, let Q = £(f) =
Gal(K(fp°°)/K), and A = A(£,D), the convolution algebra of D-valued
measures on Q. Recall (II.4.1) that
(1) U = U(j) = lim Un
denoted the inverse limit of the groups Un of (semi-local) principal units in the
completion of Fn at p. Unlike II.4.1, we do not assume anymore tuf = 1. In particular,
f may be trivial. U is a torsion-free pro-p group, and also a Zp[[^]]-module.
98

Let Z be the decomposition group of p in Q, G = Gal(Foo/F) and T =
GaliFeo/Fj so that
K - • -F-Fx-Foo
$dZdGdTd{1)
is the field diagram. Let $J be any prime of F above p, and consider the groups
fJ>p<*>{Foo) and Aip°°(^oo,!p)- Let their annihilators in Zp[[£]] and Zp[[Z]] be denoted
by Jo and J\ respectively, and define three ideals in A:
■A-o = Jo-A-
Ax = JiA
Aoo = Ao fl (the augmentation ideal of A).
EXERCISE: (i) Show that A0 is generated by aa - No, (o,6fp) = 1, where
°* = {a,Foo/K).
(ii) Let $ be a prime of F dividing p and pN the number of p-power roots
of unity in jFoo,<p- If «>f = 1, £ is the unique generator of Np/K^P satisfying
£ = 1 mod f, and / = /(^J/p) the relative degree, show that N is the precise
power of p in p^£-1 — 1.
(iii) Show that A/Ax = T>/{pN)[g/Z].
1.2 THE HOMOMORPHISM i: When tuf (the number of roots of unity in K
congruent to 1 modulo f) is 1,1/ was embedded in A by means of the ^-module homo-
morphism t : fi h-> [iQ (II.4.6 and 4.7).
Lemma, (i) The map i is independent of the elliptic curve E used in its
construction. It is canonically associated to the extension Fqq/K, and the choice of
(ii) Suppose f|g and Wf = 1. Set A(g) = A(£(a.),D), and let
*Blf : A(a) -> A(f)
nU9 ■ A(f) - a(b)
99

be the maps corresponding to restriction and corestriction on the Galois groups.
Then the following diagrams commute (iV0jf is the norm from K(gp°°) to K(fp°°))
U{B) ^ A(fl) U{2) ^ A(fl)
(2) NBtfl |7r0>f inc/.T T»7f,0
U(f) ^ A(f) U(j) *M A(f).
(We remind the reader that for r £ 5(f)> *7f,e(r) = Hcn-n-47)-
PROOF: (i) Obvious in view of II.4.6 (14).
(ii) Both parts follow from the definitions and are left to the reader.
1.3 Proposition. There is a unique way to extend the definition of i(f) to all f so
that (2) remains commutative. Furthermore, i induces an isomorphism
(3) i: U <8>zp D a Ax.
PROOF: Assume first that tuf = 1, so that i is defined (II.4.6). Let Uz be
the inverse limit of the groups of principal units in Fn>%, and Ai)Z the ideal of
Az = D[[Z]] generated by J\ (see 1.1 for notation). Then U = Ind^ Uz and
Ai = Indz A1)Z. In 1.3.7 we proved that t maps Uz ®zp D isomorphically onto
A-i,z = Ker(j). Our claim is the semi-local version of this and follows from it since
A is free over Az.
In the general case one may use (2) to define t(f). Pick Q with ws = 1, f|g.
It is easy to check that 7re,f(&'(fl)(/?)) = 0 if and only if NStffi = 1. On the other
hand iV0)f is surjective. To see this consider the norm from K(gpn) to K(fpn) in
two steps: K(gpn) -> K(g)K(fpn) -> K(jpn). The first step is ramified at p, but
only tamely, since the degree (for large n) is tUf, and (tUf,p) = 1. The second step
is unramified at p. In any case the norm on principal units is surjective. Thus iVflif
is surjective and there is a unique way to define i(f) so that (2) is still commutative.
It also follows from what was said above that (3) remains valid.
100

1.4 THE GROUP Cfi Let f be any integral ideal of K and Cn = Cfpn the group
of primitive Robert units of conductor fpn, n > 1. Recall their definition (II.2.7).
If f ^ (1), then each 0(l;fpn, a), (a,6fp) = 1 is a 12*h power of a unit in
K(fpn) = Fn. Denote by ©12(1; fPn, 0) one such root. The group Cn is generated
by 0i2(l;fPn,a), (a,6fp) = 1, and by the roots of unity in Fn. If f = 1,
0(1; pn, 0) still admits the twelfth root ©12(1; pn, 0) in Fn, but these are not units.
A product of the form n ©12(1; Pn, a)m(o) (K6p) = 1) is a unit if and only if
^2 m(o)(No — 1) = 0 (exercise II.2.4). We let Cn be the group generated by all
such products and by the roots of unity in Fn. In any case Cn is Galois-stable, and
Nm,nCm = Cn modulo roots of unity (proposition II.2.5(i)). Notice that since we
assume p > 2, and since p is split in K, wv = 1. Let Cn be the closure of Cn in
Un x Vn, < Cn > its projection to Un, and
(4) Cf = lim <Cn> C U{f).
Proposition. The map i : ii(f) —+ A. induces an isomorphism
t5) i:C'®*'D^Woo f-(l)
wAere fi{f) is the measure constructed in II.4.12.
REMARKS: (i) If f ^ (l), Cf may be described as the group generated by the
12"1 roots of /3(a) in tt(f), for (o,6fp) = 1. If (p,6) = 1 there is no need to extract
12t;i roots. If Q = fI with I prime, and f ^ (1), proposition II.2.5 (i) implies that
\c, if 1 I f.
Compare II.4.12 (32) and (5). If f = (l), iV[,(i)C[ C C(i).
(ii) jLt(l) is only a pseudo-measure, but for any A in the augmentation ideal,
H{1) • A E A (II.4.12 (iii)).
PROOF: Recall the notation of II.4 in case twf = 1 : i(0(o)) = fxa =
12 • fx(f) • (<j„ — No). Our assertion follows from this, and from the definitions of Cf
101

and Ao. In the rare event that f ^ (1) but ttff > 1, replace f by g = fm for some
large m and "push down" (5) from level a. to level f using (6) and II.4.12 (32).
If f = (1) the situation is truly different because now en(a) = 0(1; pn, a) ^
Cn. Using exercise II.2.4 we deduce that IIeri(a)m(0) is a unit if and only if
J2 m(a)(aa — No) belongs to the augmentation ideal, i.e. ]T ra(a)(l — No) = 0.
This explains why in (5) we substituted Aoo for Ao. The rest of the argument is
identical to the previous case (f ^ (l)).
1.5 Corollary. The map i induces an isomorphism
In particular ii(f)/Cf is a torsion noetherian Zp[[^]]-modu/e (i.e. killed by a
nonzero-divisor from the ring).
A remarkable observation is that Ai C Ao, so unless Ai = Ao, yu(f) is not
a unit. In particular, if there are no p-power roots of unity in i*oo» but there are
some in jPoo.qjj ^(f) is n°t invertible. Notice that in this case, in the language of
1.3.7, any E as in II.4.1 is anomalous above p (meaning that E is anomalous).
Note also that when f = (1) there are no non-trivial p-power roots of unity
in jPoo.qj) so Ai = Ao = A, and Aoo is the augmentation ideal.
1.6 The measure #(f) is deprived of the Euler factors at the primes dividing f
(II.4.12 (32)). While it is impossible to restore them to At(f), we may remedy that
matter on the left hand side of (7), by considering a different module.
For any divisor a. of f, C0 C U{q) C U{j), and iVfjBCf C C0, where N^iB
denotes the norm from K(jp°°) to K(Qp°°), as before. In fact, equality prevails
here if f and a. are divisible by the same primes (6).
DEFINITION. The Iwasawa module of elliptic units in K(jp°°)/K is
C(f) = II Co-
elf
If f is indeed the conductor of K(f), we refer to Cf as the primitive (or new) part
of C(f).
102

1.7 THE FUNDAMENTAL EXACT SEQUENCE: Let En be the global units in Fn,
En their closure in Un X Vn, and < En > the projection of En to Un. Let
(8) <?(f) = lim <En>C U{f).
We thus have a chain of inclusions
(Q) (Cn C ... C En C Un x Vn
U lCf C C(f) C «?(f) C l/(f).
Let Mn be the maximal abelian pro-p extension of Fn unramified at (the primes
above) p. Let us put (1 < n)
(10) Xn = Gal(Mn/Fn), X = r(f) = GaliM^/Feo).
The first is a Zp[Ga/(irri/i;C)]-module, and the second a topological Zp[[^]]-module.
Clearly M^ = UMn.
Class field theory provides an exact sequence
(11) 0 - UJ < En > ± Xn - K - 0
where U^ = Gal(Ln/Fn) is the Galois group of the maximal unramified abelian
p-extension of Fn (the Hilbert p-class field). The injection a is the (idelic) Artin
symbol. Taking projective limits over n, (11) yields
(12) 0 - U(j)/£(f) - X(f) - V(f) - 0.
We shall re-write (12) as a four-term exact sequence
(13) 0 - <?(f)/C(f) - W(f)/C(f) - r(f) - V(f) - 0.
The idea is that ii/C is somewhat better understood, at least in terms of L functions,
than U/£.
The "main conjecture" compares U(f)/C(f) with Z(f) as representation spaces
for £. Considered as Zp[[^]]-modules, both of them are torsion and noetherian.
This was already verified for U/C, and it is a well known fact about T^(f), which
holds in any Zp extension ([Wa] 13.3). The exactness of (13) implies it for X(f)
too.
103

1.8 Let Kqo be the unique Zp extension of K unramified outside p,
I" = Ga^Koo/K), H' = GaJiFoo/Koo), and fix, once and for all, an
identification
Q c* H' x T'
so that characters of H' will be naturally considered as characters of Q too. If
K(l) D Koo = Kt {[Kt : K] = p*), then the image of T in V, under restriction
toKco/isT'P*.
Let D' be the finite extension of D generated by the values of all x from H'. We
want to decompose the modules in (13) according to x £ ■" > an(i study each piece
as a D'[[r']]-module. If p f [F : K], then H = H', T = T, D = D' and any
D[[£]] module M automatically breaks up as the direct sum of its x-components
Mx. In general, let us agree to consider D'[[r']] as a A = D[[£]]-module via x>
extended to a homomorphism of the group algebras. Put
(14) Mx = M ®A,X D'[[r']],
the largest quotient of M on which H' acts through x (scalars extended to D').
Recall the structure theory of D'[[r']]-modules ([Wa] 13.2). Such a module is
pseudo-null if it is of finite length. M and M' are pseudo-isomorphic if there exists
an exact sequence
0->e->M->M/->e/-+0
where e and e' are pseudo-null. This is an equivalence relation if M and M' axe
noetherian and torsion. Every noetherian torsion module M is pseudo isomorphic
104

to an elementary module
(15) E = J] D'[[T']]/(/.-).
The ideal (JJ /,) is an invariant of M, called its characteristic ideal. Any generator
of it is called a characteristic power series of M (the terminology stems from the
fact that D'[[r']] S D'[[X]], cf. 1.3.1). We denote the characteristic ideal by
char.(M).
If the p-torsion submodule of M is pseudo-null, then Mx is pseudo-isomorphic
to Mx = the largest submodule of M on which H' acts via x- We shall see later
that this is the case with X{f).
1.9 THE MAIN CONJECTURE. Let f be any integral ideal and fix a decomposition
5(f) ~ H' x r as above. Then for any x e ff',
(16) char. (H(f)/C(f))x = c/mr. Z(f)x.
1.10 The following lemma is more or less obvious, in light of 1.5. Let A*(f;x) =
xW)) e D'[[r']].
Lemma, Suppose fx is g or gp (fl|f). Then
(17) char.[U[f)/C[f))x = If*'*], h„ * * )
[ (70 - i)a*(1; i) x = i,
where 70 & a topological generator ofT'.
PROOF: Consider first the map x ° «(f) : U{f) <§> D' -> D'[[r']]. We claim that
X o t(f)(C(f) <§> D') C x ° ,(fl)(Cfl <§> D') (here we consider x as a character on
5(f) and 5(fl) simultaneously), and that the cokernel of this inclusion is pseudo-
null. Indeed, x ° *(fl)(Cfl) = X ° *(fl)(C(s)) because x has conductor fl or flp.
Also, proposition II.2.5 implies JVfiflC(f) C C(fl) (compare (6)—here we have to
deal with NUiCh for any h|f). Thus, x ° »(f)(C(f)) = X ° *f>8 ° »'(f)(C(f)) =
X ° t'(fl) o iVf)flC(f) C x ° t(fl)(C(fl)), proving half of our claim. To see that the
105

quotient is pseudo-null observe that it is annihilated by [K(fy°°) : K(gp°°)] — m
(since C(fl) C C(f)), and also by JJ (1 — xV)~1^i\t^)^ where the product is over all
I dividing f but not fl (because of (6)). These two elements of D'[[r']] are relatively
prime, and any noetherian module annihilated by two relatively prime elements is
pseudo-null. So the claim is established. Now apply corollary 1.5. If {j ^ (1) we
are done, because A/Ao and A/Ai are pseudo-null. If {j = (1) but x ¥" 1 then
x(Aoo) contains x(r) — 1 for some r e H', x{T) ¥" 1, and also 70 — 1. These two
are again relatively prime, hence x(A)/x(Aoo) is pseudo-null. Incidently we have
shown that a*(1;x) £ D'[[r']]. Finally if x = 1> x(Aoo) is the augmentation ideal
of D'[[r']], which is generated by 70 — 1-
1.11 Corollary. The main conjecture is equivalent to the statement that
m(0;x)> if X ¥" 1> or (7o — 1)m(1> 1)> if X — l» is a characteristic power series for
X(f)x. Here g is the prime-to-p part of fx.
While (16) makes the connection with the fundamental exact sequence (13)
evident, this last formulation is more practical, because ^(fljx) is nothing but the
p-adic L function of x~X» restricted to characters of GallKoo/K). In the notation
of II.4.16, for any p-adic character p of T' (a character of "the second kind")
(18) Wx~V-1) = / XPdfi{g) = I pdn{g;x).
J9(s) Jr>
As mentioned above, we shall see in the next section that X (f) is free of finite
rank as a Zp-module, and on the other hand fi[g; x) (or (70 — 1)m(1> 1) if X = 1) is
not divisible by the uniformizer of D'. From the general theory of Iwasawa modules,
the main conjecture then reduces to the following statement:
The eigencharacters of £(f) in its action on the finite dimensional vector
space X(f) <S> Q are precisely those e = XP f°r which (18) vanishes.
I.e. they are the reciprocals of the zeroes of the (primitive) p-adic L
functions of conductors dividing f.
106

Compare with the discussion at the beginning of this chapter!
In one direction the main conjecture says that if we know a zero of the p-adic
L function, we should be able to construct an abelian p-extension of Fqq on which
H' X r" acts in a prescribed way, and which is unramified outside p. How do we
construct such an extension? A natural approach is through Kummer theory on
elliptic curves with complex multiplication by K. The main conjecture is thereby
intimately related to the arithmetic of such curves, a topic that will make up the
next chapter.
1.12 THE TRIVIAL CHARACTER: With scant information as we have about X (f),
any partial result is welcome. We shall show that at least at the trivial character
(x — P = 1 in (18)) the main conjecture holds.
Proposition, (i) fi(l) has a true pole at the trivial character,
(ii) p = 1 is not a zero of char. £(l)i.
PROOF: (i) This is a restatement of corollary II.5.3. (ii) If p = 1 is a zero of X(i)i,
there is a Zp extension of Fqq = K(p°°) on which Gal(Foo/K) acts trivially, and
which is unramified outside p. Therefore there exists already a Zp extension N of
this type over Kqq. But such an N would be abelian over K, because it is a trivial
extension of a pro-cyclic group by an abelian one, and this is impossible.
Similarly, one can use the fact that Leopoldt's conjecture is known for abelian
extensions of K [Br] to prove the following.
Proposition. Let e be a character of finite order of K, e ^ 1, and f the prime-
to-p part of fe. Then
(i) J5(f) e «fc(f) * 0.
(ii) e does not occur in the representation of £(f) on X(f).
1.13 The following result of Greenberg will be used later to show that the p-torsion
of X(j) is trivial.
Theorem ([GREl], END OF §4). The Zp[[()]]-module X{f) does not have any
non-trivial finite submodules. (Finite means of finite cardinality).
107

1.14 THE "TWO-VARIABLE" MAIN CONJECTURE: When f is replaced by fp°°
one can write
(19) 9(fp°°) ^ H' x Tx x T2
where Ti (resp. T2) is the Galois group of the unique Zp extension of K unramified
outside p (resp. :p). Taking the inverse limit of the measures ft(fpn) we obtain a
measure i/(f) = A*(fp°°) (see II.4.14) on 9{fp°°), and for any x £ H'
(20) x(Hf)) = ^(f;x) € D'[[rx x r2]].
The structure theory for T-modules generalizes easily to Ti X IVmodules ([Cu]
§1 and IV.3.7) and we may make the conjecture that for any x £ ■"
(21) i/(f;X) = char.X{fF°)x,
provided f is the prime-to-p part of fx.
2. The Iwasawa Invariants
2.1 We retain the notation and assumptions of the previous section. In particular
we fix a decomposition £(f) ~ H' x T' as in 1.8. Recall that any h € D'[[r']]
can be written uniquely in a Weierstrass form
(1) h = irmhwu
where 7r is a uniformizer of D', m a non-negative integer, hw a monic polynomial,
all of whose coefficients except for the leading one are divisible by 7r, and u a unit.
Let e be the absolute ramification index of D'. The invariants
(2) fj, — inv(h) = —, A — inv(h) = deg hw
are called its Iwasawa invariants. If M is a noetherian torsion D'[[r']]-module, we
refer to the invariants of char.(M) as the invariants of M. If M = Mq <E> D' is
108

obtained from a Zp[[r']]-module by extension of scalars, its invariants coincide with
those of Mo (which we leave to the reader to define).
Let fx — char. X{f)x, 9X = char. {U(f)/C(f))x be the two power series
figuring in the main conjecture (1.9), for x £ H'. We shall prove the following
corollary of 1.9:
Theorem. The total (i— and X-invariants of X(f) and U(f)/C(f) are equal:
,z, Ex A* - inv{fx) = Ex M - inv{gx)
T,x X ~ invtfx) = Ex A - inv(gx).
In fact, the /^-invariants vanish (2.12), but this is proven for the gx, and (3) is
needed to get it for the fx-
Apart from giving support for conjecture 1.9, (3) reduces its proof to showing
gx\fx f°r a^ X (or the other way around). In the cyclotomic theory, the same
phenomenon enabled Mazur and Wiles to deduce the cyclotomic main conjecture
from a afifx|/x"-type result.
The proof of (3) goes as follows. Let / be a characteristic power series for X (f),
considered as a T'-module, and flf = fl 9x- ^e snaU compute the invariants of /
and g separately, using class field theory in the case of the former, and the analytic
class number formula for the latter, and conclude that they are equal. Quoting the
important result of Gillard [Gi2] on the vanishing of the /^-invariant of g, we deduce
that the p-torsion of X(f) is finite, hence trivial by 1.13. But then the /^-invariant
of each X(f)x vanishes too, and f = Yl fx- This gives the identities in (3).
2.2 For any group G, and any G-module X, let Xq be the module of G-coinvariants,
i.e. the largest quotient of X on which G acts trivially. With Tn = Tp =
lV+re = GaliFoo/Fn+J (n > 0),
(4) Go/(Mn+1/JP00) = Z(f)r„,
109

because Mn is the largest abelian extension of Fn inside Mqq. To find the A and
(i invariants of X(j) we shall compute in 2.7 (11) below, following Coates-Wiles
[C-W3], the order of (4). Then we use the following fundamental lemma of Iwasawa
([Wa] 13.3):
Lemma. Let X be a noetherian torsion Zp[[T']]-module, and f its characteristic
power series. Assume that Xrn {Tn = T't+n) is finite for all n. Then there exists
a constant v such that for large enough n
(5) length(XTn) = \i - inv(f) ■ pt+n + A - inv(f) • n + v.
(The invariants are as T'-modules. When we replace T' by T = T't, the A-invariant
is unchanged and the ^-invariant is multiplied by p*. The length, in our case, is
simply ordp jJ(Xrre), because we have not yet extended scalars to D').
2.3 p-ADIC REGULATORS: For the moment, let F be any abelian extension of K
of degree d. Let <7i,..., Od be those embeddings of F in Cp that induce p on K. Let
E be a subgroup of finite index in the full group of units of F, and choose generators
ci,..., ed-i for E/tor(E). Let log denote the p-adic logarithm.
DEFINITION. The p-adic regulator of E is
(6) Rp(E) = det{\og <7t-(ey))i<f|y<<i_i.
It is well defined up to sign, because log Np/K{e) = 0 for e in E. If we let
ed = 1 + p we get (add the first d — 1 rows to the last one):
(6') Rp{E) = {d log ed)"1 • det{log <7,-(ey))i<tV<d-
Since we assumed that F/K is abelian, Leopoldt's conjecture holds, namely,
Theorem (BAKER-BRUMER [Br]). Rp(E) ^ 0.
We shall denote by discp(F/K) the p-part of the relative discriminant. It is
an ideal in K, which is a certain power of p.
110

2.4 For each prime *$ of F lying above p let win be the number of p-power roots of
unity in the local field Fin. Let $ = F <g> Kp = J~J Fin, and let U be the group of
principal units in $ (compare II.4.1). The p-adic logarithm gives a homomorphism
log :£/—>$ whose kernel has order fj win (the win, for ^3|p, are actually
equal). The image log(J7) is an open subgroup of $.
Let E be a subgroup of finite index in 0f> as above, and D — E ■ <l + p>.
Let D be its closure in the units of $, and < D > the projection of D to £/.
Lemma. The index of log(< D >) in \og(U) is finite, and given by
(7) ordp[\og(U) : log(< D >)] = ordp ( _^£^— . ft (^N^)"1
yy/discp{F/K) JjJ
PROOF: See [C-W3], lemma 8.
2.5 Corollary. Let w(E) be the number of roots of unity in E. Then
(8) ordp\U:<D>\ = ordp I , d***W . TT Nqr1
K} P[ J " [w(E)y/discp{F/K) JA
PROOF: An immediate consequence of (7) ([C-W3] lemma 9).
2.6 We return to the situation considered in theorem 2.1. With the notation of
1.7, the Artin symbol induces an isomorphism
(9) Unl <'En>= Gal{Mn/Ln).
Recall that Ln (resp. Mn) is the maximal abelian unramified (resp. unramified
outside p) p-extension of Fn. Clearly Fqq C Mn and Ln n Fqq = Fn.
Let Yn C Un be the subgroup for which
(10) Ynl <^n>= GaliMn/LnFn).
Lemma. Yn = Ker NFn/K\Un-
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PROOF: The norm is of course from $n = Fn <g> Kp to Kp. If u G i7n and
(u,Foo/Fn) = 1, then {NFn/KU,F00/K) = 1, so the idele Npn/K^, which is 1
outside p, is necessarily 1. This argument can be reversed to prove the converse.
2.7 Proposition ([C-W3] THEOREM 11). For any held F containing K let
h(F) be its class number and Rp(F) the p-adic regulator of 0f>. Let wp be the
number of roots of unity in F. Then in the situation considered above Mn/Foo is a
finite extension (n > 1) and
(11) ordp[Mn : Foo] = ordp <
p»HF)Rt{Fn) n _
(the product is over the primes *$ of Fn).
PROOF: Letp6||[F: JK'],sopn+5-1||[Fn : if],for n > 1. Let Dn = En • <l+p>,
and note that since NFn/K(En) = 1, NFn/K(< ~Dn >) = 1 + pn+s0p. Prom
local class field theory, NFn/K{Un) — 1 + Pn0p. Consider the diagram
0 -►<£„> -►<!)„> -► l+pn+6Op -> 0
i i i
o -> yn -> c/n ^£ i + pnop -> o
with exact rows and injective columns. Since d = [Fn : if] is exactly divisible
by pn+5_1, (11) is a consequence of (8), (10), and the fact that [LnFoo : Foo] =
[Ln : Fn] = the p-part of h(Fn).
2.8 Corollary. Let f be a characteristic power series for X(f) as a Y'-module.
Then for n » 0
li - inv{f) ■p*+n-1(p-l) + A - inv{f) =
(12) , f hRp /T, x / ^p ,„ x
1 + ordp \ *- (Fn+i) / *— (Fn)
112

PROOF: By the previous proposition, the right hand side is equal to
ordp([Mn+i : Foo]/[Mn : Foo])- Now apply (4) and lemma 2.2 (5).
2.9 We now begin the computation of the Iwasawa invariants of the p-adic L
functions gx. Let g = JJ gx, the product taken over all x £ H'. We have the
following useful result.
Lemma. For any character of finite order p of T', let level (p) — s if
p(r'P') = 1, but p{T*'~l) ^ 1. Then for n » 0
(j, - inv(g) •pt+n~1(p~l) + A - inv(g) =
ordp I fj p{g)
\level(p)=t+n
Here ordp is the valuation of Cp normalized by ordp[p) = 1.
PROOF: We may enlarge D' to assume that g has all its zeroes there. This reduces
the proof of (13) to the two special cases g = n (a uniformizer of D') or g —
7o — 1 — <* (7o a topological generator of V, |a| < 1). Both are easy exercises.
2.10 Proposition. For any ramified character e of Gal(Foo/K) let g = fc,
(g) = fl n Z, and
(14) Sp(e) = --i- • £ e~\C) log <pg(C)
9 ° ceci(g)
(compare II.5.2 (2)). Define G(e) as in II.4.11 (30). Let An be the collection of all
e for which pn||fe. Then, for large n,
(15)
ordp( J! G(e)Sp(e)) =
ordp
±^- (F„) / -±fy= (J..,)
w-\/discp I Wy/dfsct
The proof of the proposition will be given in 2.11. Before embarking upon it,
we conclude the proof of the equality of the invariants of / and g.
113

Every e G An+i can be written asc = x/>> where x is a character of H', and
p of T'/T't+n but not of r,/r;+n_1. Theorem II.5.2 shows that
p{9x) = I Xpdfi{Qo) = G{e)Sp{s), if X ^ 1
JQ(Bo)
(fl = fe = 9oPn)- Ifx = 1, we similarly get
P(fc) = (PM ~ l)G(s)Sp(s)
where 70 is a topological generator of I". When we take the product over all
e G An+i, we take all x £ ir, and all p G V of level £ + n. Therefore
(15) yields
tie) n 'M ~ " ■ -J%= e»«) / -t£=» «o
P{9) ~ P ' —
level(p)=t+n
where ~ means "up to a p-adic unit". Here we used the fact that n(p(7o) — 1) =
rj(f — 1) ~ P if f runs over all primitive roots of unity of order pn+t.
Comparing (16) with (13), and then with (12), finally shows that / and g have
the same invariants.
2.11 To prove (15), observe first that
(17) J] G{e) ~ {discp(Fn^/K) / disc^F^K)}1'2 .
seAn
This is nothing but the "conductor-discriminant formula" localized at the prime p.
It remains to show, therefore, that
(18) JJ Sp(E) „ *& (,„) / *& (,^).
To this end, introduce, for a ramified er, the sum
(19) S^e) = -—j— ■ £ £_1(C) log|^(C)|2
(the logarithm here is the ordinary logarithm, and g and g are as in (14)). If e is
unramified, but non-trivial, let
114

(14'' s^ = m^ ' £ e"<c> log *<c»
* K CGC/(1)
K K ceci{i)
where 5(C) is Siegel's unit (II.2.2).
Let Hn = Gal(Fn/K), I{Hn) = the augmentation ideal in Z[ifn], and choose
an embedding 77 of I{Hn) into the units of Fn, as Galois modules. The image of 77
is a subgroup E'n of finite index in En. If we put
Sp(e) = S £"V) lo8 tfM (p-adic log)
Soo(e) = ]£ e_1(a) log|i7(a)|2 (ordinary log)
(e: ^ 1) then there is a non-zero algebraic number re such that
(20) 5p(e) = rs ■ Ep(e), ^(e) = rs • £«,(£).
Now, the analytic class number formula together with Kronecker's theorem (as
formulated by Siegel) II.5.1, gives
(21) n s~& = <* • ^ (^)
s^l,sGH„
where a is a constant independent of n. Here Rqq is the usual (complex) regulator, h
the class number and w the number of roots of unity. In the following computation
e ranges over all non-trivial characters of Gal(Fn/K). Use has been made of the
fact that Rp{E'n)/Rp{En) = i2oo(^)/J2oo(£») = \En • E>fJ- We-find out
II 5p(s) = J] Sp(£) ' (f^f ) (from (2°))
= Rp{E'n) (Q-%tt- ) (Frobenius determinant)
V Roo[E'n) J
-«™ • tm)
= a • ^ (^) (from (21)).
115

Dividing these expressions for n — 1 and for n we arrive at (18), as desired.
2.12 In [Gi2] R. Gillard uses ideas of W. Sinnott to prove the following. (L. Schneps
proved the same result independently too.)
Theorem ([Gl2] 2.9). Fix a decomposition £(f) = H' x I" as above, and let
gx be defined by the right hand side of 1.10 (17), as before. Then ft — inv(gx) = 0
(Vx € H').
As explained at the beginning, this concludes the proof of theorem 2.1. It also
has the following
Corollary. p. — inv[fx) = 0 for every x £ H'. The §(f)-module X(f) is a free
Zp-module of finite rank.
Gillard's theorem is another example, so common in number theory, where a
purely algebraic statement (as the last corollary) is proved by analytical means.
3. Further Topics
In this short section we make a few more comments concerning the main
conjecture, without proofs.
3.1 RELATIONS IN IDEAL CLASS GROUPS: Building upon ideas of F. Thaine,
K. Rubin proved recently a remarkable theorem in the direction of the main
conjecture. Among other things he showed that in the fundamental exact sequence
1.7(13), the module £ /C "governs" "W. So far the results depend on various
simplifying assumptions, which we proceed to describe, but these seem to be removable.
Assume that K has class number 1, and that E is an elliptic curve denned over
K with complex multiplication by Ok, and conductor f. Let p be a split prime of
good reduction, and assume p > 2. Consider the fields Fn = K(E[pn]). Our
notation is different from that employed previously, but note (II.1.6) that Fn C
K{E[fpn})) = K(fpn). Write, as usual, Ga^F^/K) = T x A, and let
116

X € A. The main conjecture for x asserts that {U{f)/C(f))x and X(f)x have the
same characteristic ideal as Zp[[r]]-modules. By 1.7(13), this is equivalent with
(1) char.{£{f)/C{f))x = char.1V{f)x.
Theorem ([RU3]). Let hx = char.{£(f)/C{f))x. Then hx • IV {f)x is pseudo-null
(i.e. finite).
This comes close to (1). Since we know that the total A-invariants (summed
over all x) of the two modules in (1) are the same, and their /^-invariants vanish, (l)
would follow from the theorem if all the zeroes of char. W{f)x are simple, for all x-
More generally, it would be enough to know that W{j)x is a cyclic Zp[[r]]-module,
for all x>
3.2 THE FUNCTIONAL EQUATION: We have seen in II.6 that the p-adic L function
satisfies a functional equation with respect toe i—*■ i, and in particular Lp(e) =
0 -O- Lp(i) = 0, where the modulus of the p-adic L function consists of the
ramified primes for e, as well as p and p. In view of the discussion in 1.11 we would
like to say that e occurs in the representation of Qtfp ) on X(fp ) precisely when
i occurs in the representation of ^(fp ) on X(fp ). This is indeed true, but to
prove it one has to await a different description of the module X (cf. IV. 1.5). Even
then the "algebraic functional equation" is a deep fact. See [PRl] V, §1, or [Maz2]
§7.
3.3 A KUMMER CRITERION FOR QUADRATIC IMAGINARY
FIELDS: In analogy with the well known notion of a regular prime (in Q), one
may define a split prime p of K to be regular for F = K(f) ((f, p) = 1 as
before) if the module jf(f) is trivial. Since by Nakayama's lemma X(f) = (0) if
and only if Z(f)r = (0) (r = Gal(Foo/^i)), and since X(f)r = Go/(Mi/foo)
(see 2.2(4)), p is regular for F if and only if the only cyclic extension of degree p of
K(fp), unramified outside p, is Ktfp2). Compare with the cyclotomic theory. There
117

p is regular (for Q) if the only cyclic extension of degree p of Q(p) = Q(cos —),
P
unramified outside p, is Q(p2).
Theorem 2.1, in conjunction with 1.13, shows that a necessary and sufficient
condition for p to be regular for K(f), is that the power series gx, X £ H', are
units. (See 1.8 for notation, and recall that gx is defined by 1.9(17)). It is now
easy to obtain a criterion for regularity in terms of divisibility of special values of L
functions by p. When K has class number 1, this was done by Coates and Wiles (if
f = 1, [C-W3]), and by Yager, in a more general context ([Ya3], theorem 3). Note
that there is no reason to stick to F which is a ray class field. The definition of
"regular for Fn, and the corresponding criterion, carry over to any abelian extension
of K.
118

CHAPTER IV
APPLICATIONS TO THE ARITHMETIC OF ELLIPTIC
CURVES WITH COMPLEX MULTIPLICATIONS
This chapter deals with the conjecture of Birch and Swinnerton-Dyer. The
conjecture relates arithmetical invariants associated to the group of F-rational points
on an elliptic curve E (defined over a number field F), to analytic invariants derived
from its Hasse-Weil zeta function L(E/F, s). It sprung from extensive computations
done in the early sixties on the curves y2 = x3 — Dx [B-SD] and x3 + yz — D
[St]. Both families of curves have complex multiplication, by Q(i) and Q(^—S)
respectively. Tate has studied far-reaching generalizations in [Ta4], but until 1976
very little was known.
The last decade has witnessed a few remarkable breakthroughs. Coates and
Wiles [C-Wl] and R. Greenberg [Gre2] used p-adic techniques and Iwasawa theory
to obtain results about elliptic curves with complex multiplication. Gross and
Zagier [G-Z] used special points on modular curves to obtain a theorem pertaining
to any elliptic curve defined over Q, and uniformized by a modular curve. These
include the curves with potential complex multiplication which are defined over
Q. Recently, K. Rubin combined new p-adic results with the three works cited
above, together with important work of B. Perrin-Riou on p-adic heights [PR2], to
obtain stronger theorems in the CM case. He also gave the first examples of finite
Tate-Shafarevitch groups, thereby verifying the full conjecture for the first time.
In this chapter we shall give a complete account on the theorems of Coates-
Wiles and Greenberg. These theorems were originally proven for curves defined
over their field of complex multiplication. Here we give generalizations to quadratic
imaginary fields of higher class number, and curves satisfying 11.1.4(12). The
generalizations are due to N. Arthuand [Art] and K. Rubin [Rul] for the Coates-Wiles
theorem, and to the author for Greenberg's theorem, but they are mainly variations
on the original themes.
119

The work of Gross and Zagier uses an entirely different circle of ideas. We refer
the interested reader to their paper.
The latest developments mentioned above, although a natural continuation of
this chapter, are not treated here. As this book goes to press, the work of K. Rubin
is in preparation, and will appear shortly.
l. Descent and the Conjecture of
Birch and Swinnerton-Dyer
In this section we describe the problem and the method of attack. Sections
1.1-1.3 are general, and are intended as background material. Starting in 1.4 we
specialize to elliptic curves with complex multiplication. The key result is Coates'
theorem relating the Selmer group to the Iwasawa module denoted by X in III.l.
Our treatment of descent is far from complete, and we give the bare minimum
required for §3. For a comprehensive study of descent and Iwasawa theory see [C]
and [C-Go].
1.1 Let F be a number field, and E an elliptic curve defined over F. The group
E(F) of P-rational points of E is a finitely generated abelian group ([Sil] VIII, 6.7).
It is called the Mordell-Weil group, and its rank is the first important invariant
attached to E/F.
The Hasse- Weil zeta function of E over F is defined by the Euler product
(1) L(E/F,s) = J] M*)
p
where P runs over all the non-archimedean primes of F. If P is a prime of good
reduction (II.1.8) Lp(s) is the reciprocal of the numerator of the zeta function of
the reduced curve:
(2) LP{s) = {(1 - aNP"a)(l - a'NP-*)}"1
where a and a' are the two roots of the characteristic polynomial of the Frobe-
nius automorphism (relative to Of/P)- If P is a prime of bad additive reduction
120

we set Lp(s) = 1. If P is a prime of bad multiplicative reduction, Lp(s) —
(1 ± NP-a)-1 where the — sign is chosen if the tangents at the node of E(mod P)
are rational over Of/P, and the + sign otherwise.
3
The product (1) converges absolutely in Re(s) > —. This is a consequence
of the Riemann hypothesis (|a| = VNP in (2)), proven by Hasse in his 1934
dissertation. It is conjectured that (1) admits analytic continuation to the whole
complex plane, and satisfies a functional equation with respect to s t-+ 2 — s (see
[Ta]).
If E has complex multiplication by a quadratic imaginary field, then all the bad
places are places of additive reduction, so in (1) we may simply ignore them. In that
case it is also known that L(E/F,s) admits analytic continuation and functional
equation. Indeed, if F D K and tjj = ijje/f is the grossencharacter of E/F (see
II. 1.3), then
(3) L(E/F,s) = L(tl>,s)-L$,s),
so the zeta function is expressed as a product of two Hecke X-series. Note that if,
in addition, F/K is abelian, and E satisfies (11.1.4(12)), so that ij) = <poNp/K f°r
some grossencharacter <p of K, then
(4) L(^,s) = J] Hx<P,s).
XeGal(F/K)
1.2 In general, we have the following famous conjecture.
Conjecture of Birch and Swinnerton-Dyer (part l):
(5) rk E{F) = orda=zlL{E/F,s).
Thus the order of the zero at s = 1 (where in general L(E/F, s) is not known to
exist) should yield the most interesting arithmetical invariant associated to E over
F, namely its rank.
If E has complex multiplication by Ok, and F D K, then E(F) is an Ok-
module, and clearly (5) is equivalent to
(5') rk0K E{F) = orda=1 Lty,s).
121

1.3 There is a second part to the conjecture of Birch and Swinnerton-Dyer, giving
the leading coefficient of the Taylor expansion of L(E/F, s) at s = 1 in terms of
other arithmetical invariants associated to E over F. See [Ta4] for details. Here we
are solely concerned with (5). To study the rank of E(F) one usually uses descent,
a method that we shall now briefly sketch. See [Sil], chapter X, for a fuller account
and examples.
Let a G End{E/F). The Kummer exact sequence
(6) 0 -> E[a] -> E(F) -^ E(F) -> 0
gives an exact sequence in Galois cohomology
(7) 0 -> E{F)/aE{F) -> Hl{GF,E\a\) -> Hx{Gf,E(F))\cl] -> 0.
HereGF = Gal(F/F). The group Hl{GF,E{F)), called the Weil-Chatelet group,
classifies principal homogeneous spaces for E over F.
For each prime P of F choose once and for all a prime P of F lying above
it. Let Gp C Gp be the decomposition group of P/P and identify the algebraic
closure of Fp with the corresponding subfield of the completion of F at P. In this
discussion we have to include the archimedean primes, but if F is totally imaginary
they have no effect on what follows.
For each P there is a sequence like (7) with Fp replacing F. In
particular we may consider the local Weil-Chatelet groups H1(Gp,E(Fp)). The Tate-
Shafarevitch group U1(E/F) is by definition
(8) U1(E/F) = KeriH^Gp^iF))^ UP Hl{Gp,E{FP))),
where the direct sum is over all P. It parametrizes locally trivial principal
homogeneous spaces for E over F.
The pre-image of UI(E/F)[a] in Hl{GF,E[a\) is called the a-Selmer group,
Sa(E/F). We therefore obtain the a-descent exact sequence
(9) 0 -> E{F)/aE{F) -> Sa{E/F) -> U1{E/F)[a] -> 0.
122

The Tate-Shafarevitch group is conjectured to be finite, and today is even known to
be so in some cases, for example for many curves of the form Y2 = X3 — DX [Ru3].
The advantage of (9) over (7) is that if we show U1(E/F)[a] = 0, we may compute
the rank of E(F) from Sa(E/F) (see Silverman's book and [Ru2] for examples).
The full Weil-Chatelet group, on the other hand, is huge and uncontrollable.
1.4 From now on assume that F contains K, our quadratic imaginary field, and
that E has complex multiplication by Ok- Comparing the exact sequences (9) for
an, n > 1, with a fixed a G Ok, we get in the limit
(10) 0 -> E{F) ®oK lim <x~n0K/0K -> Saoo(E/F) -> U1{E/F)[a°°] -> 0.
The group Sa<*>(E/F) is a subgroup of fT1(GJp,^[a00]). The first map sends
u ® a~n{i (u G E(F), /? G Ok) to the cohomology class of the cocycle
{a (-* av — v}, where t; G E(F) satisfies an(v) = P{u).
Before we state the next theorem let us make a general remark about Galois
cohomology. Suppose L is an algebraic extension of F, but not necessarily finite,
and R any Gir-module. Then Gl is a closed subgroup of Gp and
(11) Hl{GL,R) = lim Hl{GM,R),
where M runs over all intermediate fields F C M C L, [M : F] < oo, and the
limit is taken with respect to restriction maps. This is clear if we keep in mind that
all the cocycles intervening are continuous for the discrete topology on R. We agree
to define U1{E/L) and Sa{E/L) as the inductive limits of UI{E/M) and Sa{E/M),
F C M C L, [M : F] < oo. (The subtle point here is that if P is a place of L,
the completion Lp is not necessarily UMp).
1.5 Let p be a split prime of K, unramified in F, such that E has good (necessarily
ordinary) reduction at every place of F above p. Pick any a G Ok with (a) = ph.
Then (10) becomes
(12) 0 -> E(F) ®0k KP/0P -> S(E/F)(p) -> UI(E/F)(p) -> 0,
where D(p) denotes the p-primary component of the divisible group D. Let Fn =
F(E[pn}), n > 0.
123

Theorem (COATES). There is a canonical isomorphism of Galois-modules
SiE/F^ip) ~ HomiXiF^^lp00}),
where X (-Poo) is the Galois group of the maximal abelian p-extension ofFoo
unramified outside p, over Fqq.
PROOF: S'(^/F00)(p) is by definition a subgroup of ^(G^^p00])
= Hom{U, E[p°°]) where U is the Galois group of the maximal abelian p-extension
of Fqo over F^. We therefore have to identify the homomorphisms that belong to
the Selmer group with those which are unramified outside p.
First observe that over F\, E has good reduction everywhere. Indeed, Fqq is
the compositum of Fi with Kqq, the unique Zp-extension of K unramified outside
p, as follows easily from the main theorem of complex multiplication. Since Fqo/Fx
is unramified outside p, the criterion of Ogg-N£ron-Shafarevitch (II. 1.8) shows that
E has good reduction outside p over F\. By our assumption on p, E/Fi has good
reduction everywhere.
Suppose now that P is a place of Fqq not above p, and / e S(E/Foo)(p). Then
there exists a point v € E(Fp) such that for any a in the decomposition group
GP C GalfFoo/Foo), f{a) = a{v) - v e E[p°°}. The p°°-torsion points of E
map injectively under reduction mod P, because P \ p, and there is good reduction.
If Ip denotes the inertia group in Gp, and a £ Ip, f{a) = a{v) — v = 0,
so f{a) = 0 (t> is the image of v under the reduction map). It follows that / is
unramified at P.
The converse is more difficult. Suppose / is a homomorphism which is
unramified outside p. Let P be a prime of F^ not above p. If we now denote by
Gp and Ip its decomposition and inertia groups in M, Gp = Ip, because X is a
pro-p extension, and the residue field of P, while not algebraically closed, has no p
extensions. (P is inert in Foo/Fm for some large m, and a finite field has a unique
extension of degree pr for each r). Thus f\Gp = 0, so the local condition at P for
/ to be in the Selmer group, is met.
124

There remain the primes P of F^ dividing p. We shall show that for such P,
H1 (Gp, E(Fp)) (p) = 0, so the local condition at P is trivially satisfied, completing
the proof that / € S(E/Foo)(p). We need the following.
1.6 Theorem (TATE'S LOCAL DUALITY [Ba]). Let k be a locai Geld, E an
elliptic curve over k, a E End(E/k), a the dual isogeny.
(i) E(k) and H1(Gk,E(k)) are Pontrijagin dual of each other (the first is
compact, the second discrete).
(ii) The duality induces duality between E{k)fblE(k) and i?"1 (£*,£(£)) [a].
(in) Ifk' is a finite extension ofk, then the transpose of iVV/fc : E(k') —> E(k)
is Res: H^G^E^)) -> ^(G*/, £(£)).
To pair u e E(k)/mE(k) with v e JT1 (£*,£(&))[»?»], one lets u' be the
image of u in H1(Gk,E[m]), and v' any lifting of v there (cf. (7)). Then take
u' U v' e H2(Gk,fJ.m) ^ Br(k) s Q/Z, using the Weil pairing, and the
canonical isomorphism of the Brauer group with Q/Z.
1.7 To conclude the proof of 1.5 let a € Ok be such that (a) = ph. Then,
setting fcn = the completion of Fn at P, fT1(Gp,£?(i;ip))[a] is dual, in view of
1.6(ii) and (iii), to
(13) lim E{kn)/aE{kn),
inverse limit with respect to the norm maps. However, since (a, P) = 1, and
P is a place of good reduction, a induces an isomorphism on Ei(kn), the
kernel of reduction mod P. Thus (13) is equal to lim E(Kn) j'diE{Kn), where E is
the reduced curve and Kn the residue field of kn. Now P is totally ramified in
Foo/F (cf. II.1.9(i)), so Kn = «0 for all n, and E(ko)/'<xE(ko) is a fixed finite p-
group. The reductions of the norm maps become multiplication by pm~n (for Nm,n,
m> n> 1), so the inverse limit vanishes. By duality, Hl{Gp,E (Fp)) [a] = 0.
1.8 Here is an example of the usefulness of theorem 1.5.
125

Corollary. Suppose F(Etor) is an abelian extension of K (cf. 11.1.4(12)). Then
E(Foo) is finitely generated modulo its torsion.
PROOF: In this case X{F00) is a finitely generated free Zp module (III 1.7, 1.13
and 2.12). From the theorem S(E/Foo)(p) =* (Qp/Zp)r for some r. The corollary
follows from the descent sequence (12), with Fqq replacing F, because it is easy to
see that E(FQO)/torsion is a free abelian group.
2. The Theorem of Coates-Wiles
In this section we shall prove the following.
2.1 Theorem. Let K be a quadratic imaginary Geld, F a finite abelian extension
of K, and E an elliptic curve over F with complex multiplication by K. Assume
that F(Etor) is an abelian extension of K. IfE(F) is infinite, then L(E/F, 1) = 0.
The method of proof is p-adic. The idea is to show that the "algebraic part"
of the special value L(ip, l) (ip = tpE/F is the grossencharacter of E over F) is
divisible by arbitrarily high powers of a chosen prime. This is a variation (due to
K. Rubin) on the original proof, which showed that the value in question is divisible
by infinitely many primes, another property peculiar to zero!
2.2 We may assume, to begin with, that E is given in Weierstrass form. Let
<p be a grossencharacter of K satisfying tpE/F = tP°^F/K (H.1.4), and f =
Lc.m.(f<p, Jf/k)- Changing E by isogeny, if necessary, we may assume that it has
dx
complex multiplication by Ok, and that the lattice of periods of ue — — 1S
L — fCl for some Q E Cx. All this is standard notation from chapter II.
Choose an auxiliary prime p which splits completely in F, and is relatively
prime to 6fA(L). Let p be one of the factors of p in K, and embed Q once and
for all in Cp so that the place induced on K is p. For any prime P of F above p,
7r = fp(P) = <p{p) is a generator of p, which is therefore principal.
126

2.3 Suppose a e E(F) is a point of infinite order. Replacing it by some multiple,
if necessary, we may assume that a is in the kernel of reduction mod P for each
2x
P\p, hence \t(a)\p < 1 (t = as usual).
y
Let Fn = F(E[pn}). K{f) and Fn are linearly disjoint over F, and K{f)Fn =
K(fpn) (cf. II.1.7). Each of the [F : K] primes P above p is totally ramified in
Fqo (II.1.9), and we denote by the same letter P the unique prime of Fn above it,
0 < n < oo.
Let an £ E(F) be a solution of
(1) 7r>n) = a,
and Ln = Fn(an). The Kummtr map
(2) Gal{Ln/Fn) -+ E[pn], a >-> o{an) - an,
exhibits Gal(Ln/Fn) as a subgroup of the pn torsion points. Since E has good
reduction everywhere over Fn (proof of theorem 1.5, n > 1), Ln/Fn is unramified
outside p.
2.4 Fix a generator of the Tate module of E[p°°] : un e E[p% un £
E[pn~1], 7r(un) = un-i. Let ojn = t(un) be the value of the local param-
2x
eter t = at un. Then un E Fn is of valuation 1 at each P\p.
y
Let e = (en) be a norm-compatible sequence of units in the tower (Fn). Let
R be the completion of Of at P (11.4.1(2)), and
(3) 9e(T) E R[[T})X
the Coleman power series of e. (See II.4.5).
The idelic Artin symbol (en,Ln/Fn) is trivial, since tn is a principal idele. On
the other hand, it is equal to the sum of the local Artin symbols, and there only
primes above p contribute, since LnjFn is unramified outside p, tn is a unit, and
Fn is totally complex. Therefore
(4) 0 = £ {( /FnjP){an) - an}.
P\P
127

2.5 Let A be the logarithm of the formal group E of E, with respect to the
parameter t. It is a power series with coefficients in F, which we also consider in
Id
F ®/c Kp- Let Do = . . — be the translation invariant derivation on E.
A'(Z J dl
Recall that the explicit reciptocity law 1.4.2 computes for us the local contributions
to (4) as follows.
(5) t((en,Ln>P/Fn)P){an) - an) =
[*-nTrn{\{t{a)yDo\ogge{un)}}p{un).
Here Trn is the trace from Fn to F, which is the same as the trace from Fn,p to Fp.
The subscript P is meant to remind us that we deal with the formal group E over
Op. Now A(t(a)) can be pulled out of the trace because a € E(F). When we do
so, the formalism of ge{T), and in particular property L2.1(iii), (see also II.4.5(iii))
give for (5) the value
(6)
(1 - i)-A(t(a)).£>olog0e(O)
(wn).
P
As Gal(F/K) acts transitively on the P's, (4), (5) and (6) give
(7) (l - i) TrF/K{X{t{a)yD0\ogge{0)} = 0 mod Tn
for each n > 0. Here we view ge{T) and A(T) as power series over F ®k Kp = $,
and Trp/K 1S the trace from $ to Kp. Since n is arbitrary,
(8) £{A(t(a))-i?o log 0.(0)}*' = 0,
c
where c runs over ideals of K whose Artin symbols represent Gal(F/K).
2.6 Let rip be the p-adic period associated with Q as in chapter II.4. Recall
(11.4.3(10)) that n^-1 = A(c)Nc_1. Recall also (11.4.7(18)) that if we put D =
npD0
(9) c({D\ogge{0)) = ffc(M«)) = Nc_1 M*c(«))>
128

where 6t is the first Kummer logarithmic derivative. "Removing the Euler factor
at p" gives (11.4.7(17))
1
(10)
• ce(Dlogge{0)) = Nr'-^a^))
= Nc"1 • / <p(a)dfxac(e)(a) (G = Ga/^/F))
JG
= Nc-1 • / <p(aca)dfj.e(a).
J<T7lG
Here we have used various elementary properties of Si(0) and up. See 1.3.4-3.5. To
sum up, (8) is equivalent to
(11)
£
\{t{a))
o-c
A(c)
[/.,
ip(a)dfxe(a)
(TT'G
= 0.
In the last sum, the two quantities in square brackets depend only on (c, F/K). We
denote them, for brevity,
(12)
W(c) = X(t(a)r
A(c)'
(13)
M(c) = / <p{tx)dfxe{a).
2.7 We now claim that, in fact, for any (0, f) = 1,
(14)
^2 ^(c)M(cO-1) = 0.
Alternatively, £c W{rt)M{t) = 0. Indeed,
W{tV) = (\{t{a)Y<> ■ ^
°*J ' A(0)'
so up to a constant <p(d)/A.(d), independent of c, W(c0) is the same as W(c),
provided we replace the elliptic curve E with which we started by its conjugate
Eav, and a by aj,(a). Note that the measure /xe is independent of E.
129

Consider the cyclic matrix (M(c9-1)) = M whose rows and columns are
labeled by Gal(F/K). Since a is of infinite order, W(c) / 0, hence M is singular,
and det(M) — 0. The Frobenius determinant relation ([La3], p. 89) yields
(15) I]^ (£x(c)M(c)) = 0.
XGGal(F/K)
2.8 We now specify e. Let e = {en{a)), where
(16) e'n{a) = 0(l;fpn,a) (see 11.4.9(23)),
(17) en{a) = NK{fpn)/Fn{e'n{a)).
The measures fie on Q = Gal(Foo/K) are then given by (II.4.12)
(18) iie = 12(a0 - Na)/x
where \i is the measure induced on Q from (i(f) via the natural projection
D[[£(f)]] -*■ D([5]]- We easily deduce from (15) that
(19) J] / <px{c)iv{c) = 0.
The interpolation formula 11.4.2(31) gives, on the complex side,
(20) n w.j) = °-
x
Theorem 2.1 follows from this since f] -^(x^.l) = L{^,1), and L(E/F,1) =
L(if), l)L(ij), 1). The proof is now complete.
2.9 REMARKS: When F — K the proof is shortened considerably, because in
(8) there is only one term and X(t(a)) can be dropped. The use of the Frobenius
determinant in 2.6-2.8 to complete the proof in the general case seems to be new. In
[Rul] a result of Bertrand in transcendental p-adic number theory is used instead.
However, Rubin succeeds in saying which of the L(ipx, 1) vanishes. For this we
130

need to be able to distinguish between the various <px, X € Gal(F/K), and this is
possible if we introduce the abelian variety ResF/K^, which is lurking behind many
of our arguments. To keep the exposition more elementary, and to avoid Bertrand's
theorem, we contented ourselves with theorem 2.1.
Our proof is slick in the sense that the use of the explicit reciprocity law
obscures the role played by descent. Indeed, we did not even have to know the
definition of the Selmer group, let alone a result like theorem 1.5, or its analogues
"at finite levels." However, this is not the way the theorem was discovered in the
first place. Coates and Wiles resorted to techniques involving the explicit reciprocity
law, only because in the descent sequence they could not tell whether a point of
infinite order gave rise to an abelian extension of -K"(J5/[p]) which was truly ramified
at p. The recent ideas of F. Thaine and K. Rubin enable one to revive the original
approach, and free the proof from the explicit reciprocity law, at least when F = K.
3. Greenberg's Theorem
Theorem 2.1 concludes the vanishing of the zeta function of J? at 1, from the
existence of a point of infinite order. In the converse direction we have a theorem
of R. Greenberg. Not as strong as the Coates-Wiles theorem, this result relies
on the finiteness of the p-primary part of the Tate Shafarevitch group, for some
prime p of good ordinary reduction. It only applies to a more restricted class of
elliptic curves. A much more definite result is contained in the work of Gross and
Zagier [G-Z], at least when the field of definition is Q. Nevertheless, Greenberg's
theorem is a beautiful application of the ideas developed in chapters II and III,
and was recently put to use, together with [G-Z] and other ingredients, to yield the
following strengthening of theorem 2.1 (about which we say nothing more here).
Theorem (K. RUBIN [RU3]). Suppose that E is an elliptic curve de&ned over
Q, with complex multiplication by K. Then
(i) L{E/Q, 1) / 0 =>• E{Q) is finite.
131

(ii) L{E/Q,l) = 0, L'{E/Q,l) # 0 =► rk E{Q) = 1.
3.1 We begin by describing the class of elliptic curves with which we shall be
dealing. Let K be a quadratic imaginary field, and F an abelian extension of K. F
is called an anti-cyclotomic extension of K if it is Galois over Q, and Gal(K/Q) acts
on Gal(F/K) by —1. Thus if p denotes complex conjugation and a & Gal(F/K),
pop~l = a~l. A typical example is the Hilbert class field of K. Fix such an
extension F, and set F' = F n R.
Let E be an elliptic curve defined over F', which, over F, has complex
multiplications by Ok- Assume in addition:
(i) F(Etor) is an abelian extension of K.
(ii) If y? is a grossencharacter of K for which <poNp/K = fpE/F (see II.1.4),
then
(1) <p{a) = p(a).
Since x(o) = x(a) f°r any X € Gal(F/K), if (1) holds with one choice of <p, it
holds with any.
EXAMPLES: (a) If rfjc = q is a prime congruent to 3 mod 4, and F is the Hilbert
class field of K, the curves A(q)d (d = a square free integer), introduced by Gross
in [Gr] p. 35, satisfy (i) and (ii). In this case genus theory easily implies that the
class number of K is odd, so (ii) is superfluous, as the next example shows.
(b) If [F : K] is odd, (ii) is a consequence of (i). Indeed, since E is defined over
F'y ^(21) = i>{W). Letting c(tp) = potpop-1, c(ip) = tp, hence c((p)oNp/K =
(poNF/K, and c(<p) = <p-x for some x € Gal(F/K). Since c(x) = X> <P =
c(c(<p)) = (px2, showing x2 = 1> and eventually x = 1, because [F : K] is odd.
(c) If K has class number 1, and F = K, (i) and (ii) are automatically
satisfied.
3.2 Theorem (CF. [GRE2]). Let F/K be an anticyclotomic extension, F' =
F n R, and E an elliptic curve over F', satisfying conditions (i) and (ii) above.
132

Suppose that for some x € Gal(F/K) L(lpx, s) has a zero of odd order at s = 1.
Then either E(F') is infinite, or U1(E/F') has an in&nite p-primary component for
every odd prime p that splits in K, is unramified in F, is relatively prime to [F : K],
and above which E has good reduction (in F').
Of course, the second alternative is believed not to occur. The hypothesis is
clearly satisfied if L(tp, s) itself has a zero of odd order at 5 = 1. If F is the Hilbert
class field of K, then, according to a theorem of Shimura ([Sh2]), the vanishing of
L(jpx, 1) is a property independent of x- In that case, it is also known that rk E(F')
is a multiple of [F : K] = hK ([Gr] theorem 16.1.3).
The proof of theorem 3.2 is carried out in two main steps. In 3.3-3.9 we use
descent and theorem 1.5 (relating the Selmer group to the Iwasawa module X), to
reduce it to a statement about the characteristic power series of X (10).
In 3.10-3.13 we prove this statement. The key ingredients are the theory of
p-adic L functions as developed in chapters II and III, and a non-vanishing theorem
for complex L functions.
3.3 Pick a prime p as in the statement of the theorem, In light of the descent exact
sequence (see 1.3)
(2) 0 - E(Ff) ® Qp/Zp - S(E/F')(p) - U1(E/F')(p) -+ 0,
we actually have to prove that the p-primary part of the Selmer group over F' is
infinite.
We now describe the various fields that we shall encounter, and give them
names. The notation is different from the one used in chapters II and III, because
we need to consider more fields.
Let 7oo = F(E[p°°}), /oo(p) = F(E[p°°}), and ^(p) = F(E[p°°}).
Let Q, §\, Qi be the Galois groups of these fields over K, and G, G\, G2 the
subgroups fixing F. Then, since p / 2 and p is unramified in F, there are
canonical isomorphisms
J ki : Gx = Ai x rx c*. Of c*. Z£,
^ \k2: G2 = A2 x T2 c* Of cz Z*.
133

As usual, IAjI = p - 1, I\ = icf^l + pZp), and we write A = Ax x A2,
r = rx x r2.
£»(p)
\ A2 .
Foo(P)
H
/
K
We let Foo(p) (resp. Foo(p)) be the fixed field of Ai (resp. A2) in 7oo(p) (resp.
^oo(p))» and Fqo = FooCPJ-PooCp)- The field Fqq contains the cyclotomic Zp
extension F<+ of F, as well as the maximal anticyclotomic extension of K in Foo, which
we denote F^. The latter is the field fixed by pap-1 a, for all a G Gal(Foo/K) (p is
complex conjugation), and it contains F by assumption. Since p ^ 2, F^ and F^
are linearly disjoint over F, and their compositum is Fqo- If we now let k+ = /ci/c2,
1 (/c» is viewed as a character of G after projecting G to G{), k+ is the
cyclotomic character, and factors through 7£ = F(/jp<»). This is a consequence of
the Weil pairing. On the other hand k~ cuts the maximal anticyclotomic subexten-
sion 7^ of Joq. Note that [7£ n 7^ : F] = 2 = \7oo : 7£7^], but restricted to T,
k~ factors through F^, and yields an isomorphism Gal(F^/F) = T~ ~ 1 + pZp.
It is useful to fix a generator u of 1 + pZp, and to define 71, 'y2, 7+ and 7"
£0 (P)
134

in T by
{/ci(-7i) = u, /ci(-72) = 1,
«2(-7i) = 1, k2(i2) = «,
1+ = TfiTf2. 'V- = Tfi Aft-
Then 71, 72, 7+ and ^~ project to generators of Ti, T2, T+ = Gal(F^/F) and
T~ = Gal(F^/F), and axe trivial on T2, Ti, T~ and T+ respectively.
3.4 Lemma. Notation as above, the following are equivalent.
(a) The conclusion of theorem 3.2.
(b) S(E/F')(p) is infinite.
(c) S{E/F)(p) is infinite.
PROOF: (6) -#■ (a): Already observed in 3.3.
(c) -O- (6): The inflation-restriction sequence shows that S(E/F')(p)
= S{E/F){p)Gal(F/F') {p jk 2). Since S(E/F)(p) = S(E/F)(p) 0 S(E/F)(p),
and complex conjugation interchanges these two factors, (c) is equivalent to (b).
3.5 Lemma ([PR 1]). The restriction homomorphism
(5) S(E/F)(p) ± S(E/F-)(pf-
is injective, with finite cokernel.
PROOF: Since £^(i^)[p] = 0, the inflation-restriction sequence yields an
isomorphism
(6) Hl{Gal{FlF),E[p°°)) » H1(Gal(F/F-),E[p°°})r-,
hence (5) is clearly injective. To deal with Coker(r), let us examine the local
restriction maps (for each place v of F)
(7) B^Efa)) * H^F-^EiFv)),
135

where we write Hx{k,—) for Hx{Gd{klk), -). We shall prove: (a) If v \ p, rv is
injective. (b) If v\p, Ker(rv) is finite. The lemma follows from these two assertions,
because by (6) the cokernel of r maps injectively into IIW Ker(rv).
PROOF OF (a) : Assume first that v is a place of good reduction. Since F^ V/Fv
is unramified, every point in E(FV) is a norm from each E(F~V), so by Tate's local
duality Ker(rv) = H1{F^)tV/FVlE{F^)tV)) = 0 (see [Maz2] corollary 4.4). If v
is a place of bad reduction, the same proof works, because we can "translate" the
tower F^/F by the extension 7\{p), which is of degree p — 1, and over which E
acquires good reduction everywhere. Having proven the same result there, we may
descend to F taking Ai-invariants.
PROOF OF (B): A theorem of Mazur ([Maz2] proposition 4.39) implies that
Ker{rv) = H^F^jF^EiF-J) is finite.
3.6 Lemma. The restriction homomorphism induces an isomorphism
(8) S(E/FZ)(P) * S(tf/*oo)(p)r+
(r+ = GaliFoo/F-)).
PROOF: As in the previous section, since ^(Foojfp] = 0, there is an isomorphism
(9) H^GaliF/F-^Eip00}) ~ tf1(GW(F/*0o),£[p00])r+,
and (8) is injective. To prove the surjectivity, note that Fqq/F^ is everywhere
unramified, so like claim (a) in 3.5, the local restriction maps H1(F^}V,E(FV))
^ H1(FOQ)V,E(Fv)) are injective, as desired.
3.7 We briefly recall the structure theory of Zp[[r]]-modules (recall T
= Ti x Tj = Zp), to which we alluded in III. 1.14. Let M be a noetherian
torsion Zp[[r]]-module. M is pseudo-null if its localization at any principal ideal
(= ideal of height 1) of Zp[[r]] is trivial. Geometrically, the associated coherent
sheaf is supported in codimension 2. The notion of pseudo-isomorphism, and the
136

characteristic ideal char.(M) are defined precisely as in IILl.8. Thus there exists
an exact sequence
0 _> N - M - JJ Zp[fr]]/(/*) - ^' "> °
l<t<r
with iV and iV7 pseudo-null, and c/iar.(M) = (JJ /,-). For details see [Cu] and
Bourbaki's Commutative Algebra.
Proposition. Let X = Gal(M(?oo)/?oo) be the Galois group of the maximal
abelian p-extension of Jqq = F(£^[p°°]) unramiRed outside p, over Jqq. Let u =
/ci|A : A —► Zp (recall A = Galftoo/Foo)) be the character giving the action
of A on E[p\, and Xu the corresponding eigenspace of X. Let fu € Zp[[r]] be a
characteristic power series for Xu : char.(Xu) = (/w). Let ^+ be a generator of
T+ = GallFoo/F^) and suppose that
(10) (7+ ~ /ci(V))|/w.
Then the Pontrijagin dual ofS{E/F^)(p) is not Zp[[T~]]-toi8ion.
3.8 Before proving the proposition, let us draw the consequence that we need.
Corollary. Suppose that (10) holds. Then S(E/F)(p) is infinite, hence by lemma
3.4 the conclusion of theorem 3.2 holds too.
PROOF: Let Y be the Pontrijagin dual of S{E/F^)(p). Then the dual of the T--
invariants is Y/(i~ — 1)F, where 'y- is a generator of T~. If this last group is
finite, say of order pm, then pm(i~ — 1) annihilates Y, contrary to proposition 3.7.
Thus Y/(i~ - 1)Y, as well as S(E/F^)(p)r , are infinite. Lemma 3.5 concludes
the proof.
3.9 PROOF OF PROPOSITION 3.7: By theorem 1.5
SiE/FooKp) ~ Hom(X,E[p°°])A.
137

Combining this with lemma 3.6,
(11) S(E/F~)(p) c Hom(X,E[p°°})A * r+
= Hom(X"/(1+ - K1(7+))rw,JB[p00]).
The module X is a noetherian torsion Zp[[r]]-module, as is implicitly included in
the statement of proposition 3.7. This is proved by the same argument used for the
"one-variable" module at the end of III. 1.7. By the structure theory for Zp[[r]]-
modules, there exists an exact sequence
(12) o - N - X" - J] Zp[[r]]/(/,-) - N' - 0
l<t'<r
where N and N' are pseudo-null, and /w = II /*'• By our assumption 7+ — «i (7+)
divides one of the /,-, say f\. Now, if N is pseudo-null, it is annihilated by two
relatively prime elements of Zp[[r]], hence N/(~/+ — Ki(~/+))N is still Zp[[r-]]-
torsion. Since Zp[[r]]/(/!,7+ - k1(7+)) = Zp[[r"]], Zw/(7+ - aCi(7+))Xw
can not be Zp[[r~]]-torsion, and the proposition follows from (11).
3.10 ROOT NUMBERS: We begin the second part of the proof of theorem 3.2 with
a discussion of root numbers. Without loss of generality, we may assume that <p is
chosen such that L(jp, s) has a zero of odd order at 5 = 1. Let
(13) ek = ^2fc+1N-fc~1 = Tpkip-k~l, k > 0.
Since <p(a) = ¥?(a) by assumption, ik = £k (i-e. ejt is an anticyclotomic character
(II.6.5)). Since the functional equation relates L(ek,s) to L(ek,— s), W^(^fc) = ±1
(11.6.1(3)). By assumption, W(e0) = -1.
Lemma. Let m = (p — l)[F : K\. Then
(i) If k,j > 0, k = j mod m, then W(ek) = W{ej).
(n)W{em-X) = 1.
PROOF: Notice first that ((p/lp)™ — <p2mN-m is an unramified character. Indeed,
letting d = [F : K], (pd{a) = (poNF/K{aOF) = i>{aOF) e Kx, so (pdw* is
138

unramified, and wk\p — 1. Now quite in general, if A is a Hecke character of K of
infinity type (k,j), put i/(A) = \k — j\. Then if Ai and A2 have relatively prime
conductors fi, f2, we have for A = A1A2
(14) W(X) = ^(A1)^(A2)A1(f2)A2(f1) i"^)+"M-"W,
where A = A/|A|. This follows from Tate's thesis, and may be found for example
in [We3] p. 161, or deduced directly from II.6.1. Since we have seen that <pm is
unramined, applying (14) to Ai = A2 = <pm we get W((<p/lp)m) = W(<p2m) =
W{(pm)2 = (±1)2 = 1. Applying it with Ai = ek+m and A2 = (<pf<p)m, and
using the fact that fx = fi and that y?(a) = <p(a), we see that
W{ek) = W(ejfc+m).t-|»+am+l|+|2m|-|2*+l| = ^(£jfc+m),
which proves (i). For (ii),
W(em_i) = W^T1 (£>/<£>) m) = ^(^-l),'l+2m-(2m-l)
= -W{lp"1) = -W{<p-X) = 1.
3.11 THE NON-VANISHING THEOREM: The key to the rest of the proof is a result
saying that generically, the numbers Z»(efc,0) do not vanish, unless they are forced
to vanish by the sign in the functional equation. Recall that m = [p — 1) [F : K].
Theorem ([GRE3] THEOREM 1, [Roh] P. 384). If W(ek) = 1, then
L(ej,0) = 0 for only finitely many j > 0, j = k mod m.
Indeed, by the previous lemma W(sj) = 1 for all those j. There are two proofs
of the theorem. Rohrlich's proof (although stated only for K with class number 1, it
works in general) is purely complex-analytic, but uses a non-archimedean version of
Roth's theorem at a crucial point. Greenberg uses a mixture of p-adic and complex
arguments, accompanied by the classical version of Roth's theorem! Both yield
stronger results than what is quoted above. Since their methods are different from
the spirit of this book, and since the two papers are easily accessible, we omit the
proof.
139

3.12 Recall that p did not divide [F : K], hence
Q = Gal{Too/K) = T x H,
where H = Gal(7i/K) is of order prime to p, and T = Galtf^/J^) = 1\. We
may decompose any p-adic character e of Q as er^H, where ep is trivial on H, and
vice versa. In particular we have <p = <pr<PH, and <Ph\A = «i|A = u.
Complex conjugation p acts on Q, and we let
(15) c(a) = poo-op-1, a € Q.
If e is a p-adic character of Q, so is eoc. If e is a grossencharacter, eoc(a) = e(o),
so by our assumptions <poc = Tp.
If X is any Zp^]] module, and x £ -ff, we write Xx for (D <g>zp X)x, where
D is the ring of integers in the completion of Qpr. This slight abuse of notation is
justified since we are only interested in char.(Xx), and the characteristic ideal of a
T-module behaves well under extension of scalars.
Proposition. Fix a congruence class ko mod m. Let x € H be defined by
(16) X = <PkH°+1VHk°-
Let g be the prime-to-p part of fx, and
(17) 9X = x{n{SP°°)) € A = D[[r]]
the primitive, "two-variable", p-adic L function of x (see II.4.14, III. 1.10, 1.14).
Then the following are equivalent:
(a) L(efc,0) = 0 for infinitely many k > 0, k = k0 mod m.
(b) L(ek,0) = 0 for every k > 0, k = k0 mod m.
(c) {l+ — Ki(i+))\gx, where 7+ is a generator of T+ = Ga^Foo/F^).
PROOF: Let T = 7+ - /ci(7+), S = ^~ - 1. It is well known that
Zp[[r]] = Zp[[5",r]], the power series ring in two variables (although a more
140

standard choice for T is ^+ — l). Furthermore, e^l{T) = Kk+1/cJa'{/1+) —
Ki{l+) = Ki{l+)iK~{l+)k — 1) = 0> because <pr = /ci|I\ lpr = k^\T (see
(4)). Similarly, er^S") = Kk+1K2k(l~) - 1 = u2k+1 - 1, where u is a generator
of 1 + pZp. It follows that if we write
gx = a(S) + Tb(S,T),
then for k = ko mod m (ej^jy = x> an<i
(18) / e^{a)dn{Qp°°-a) = f e^(a)dgx(a)
J 9 Jr
= gx{u2k+1 - 1,0) = a{u2k+1 - 1).
We have identified the measure gx with the corresponding power series gx(S,T).
Therefore T divides gx if and only if (18) vanishes for infinitely many k, and in that
case, it vanishes for all k = ko mod m.
It remains to relate (18) to the special value of the complex L function.
However, this is precisely the contents of theorem II.4.14. Formula (36) there gives for
(18)
(19) SI-™-* I e^ d„{ti°°) =
The proof of the proposition is now complete.
3.13 We can now conclude the proof of theorem 3.2. Suppose that p is as in the
statement of that theorem, and consider the fundamental exact sequence 111.1.7(13)
(20) 0 -> £/C -> U/C -> X -> IV -> 0
for the field J^ = F(E[p°°]). Recall that X is the Galois group of the maximal
abelian p extension of Jqq unramified outside p, W its (absolutely) unramified
quotient, and U, £ and C the Iwasawa modules of local (at p), global, and elliptic units,
141

respectively. More precisely, to get (20) from 111.1.7(13) let f = Lc.m.(fi?/K,f<p),
so that F C K{f). Take the inverse limit of 111.1.7(13), with f replaced by fpm,
as m —► oo. Finally take Gal(K(j)/F) invariants to descend from K(fp°°) to Jqq.
We decompose the modules in (20) with respect to x € H, and call the
characteristic power series of the resulting A = D[[r]]-modules
hx = char.(£ / C)x
(21) gx = char.{U/C)x
fx = char.X*.
This notation is in accordance with (17), because of lemma IIL1.10, which gives
the p-adic L function as the characteristic power series of U/C. Notice that the
exceptional case x = 1 does not occur in the "two-variable" theory, because the
p-adic L function always has the p-Euler factor removed from it. The notation also
agrees with that of proposition 3.7 if we observe that for any 6 6 A
(22) X9 = J] Xx,
hence /w = Y[ /x' *he product taken over all the characters of H extending w.
Recall that A is a unique factorization domain. Clearly hx\gx, and <7xl^x/x-
Now consider the special eigenspaces corresponding to x = Phi and to xoc = ¥?jy-
The module of global units modulo elliptic units in the field K(jp°°) can be defined
as the inverse limit of the p-Sylow subgroups of groups of global units modulo elliptic
units in the fields K(jpn). These fields are Galois over Q (f = Lc.m.(JF/K, f<p) =
f), and complex conjugation preserves global or elliptic units in them. When we
take Gfa/(K(fp00)/^)0)-invariants this is still true (note that 7^ is also Galois over
Q). It follows that complex conjugation induces a natural isomorphism of {£ /C)x
with (£ /C)x°c as groups. As T-modules we have, by "transport of structure", that
(23) hxoc = c(hx).
This is a crucial observation. It is false for gx or /x, but, as expected from the main
conjecture, true for char.(W) at the other end of (20) (we do not need this fact).
142

Now c(i+ — Ki{l+)) = 7+ — /ci^+jjso
(24) (-7+ - Kifr+))\hx O h+ - /ci(-7+))|/iXoc
According to our assumptions on <p and lemma 3.10(i), W(ek) = —1 if A; > 0,
k = 0 mod m, hence -L(efc,0) = 0. By proposition 3.12 (ko = 0),
(25) (i+ - Ki(i+))\gx.
On the other hand, by lemma 3.10(H), W(ek) = 1 if A; > 0, k = -1 mod m,
hence by the non-vanishing theorem (3.11), L(sk,0) = 0 for only finitely many of
those k. By proposition 3.12 (ko = —1),
(26) (-7+ - /ci(<7+)) f <7xoc
Since 9x\^xfx an<^ ^xoclfl^oo (24) implies that
(27) (<y+ - /ci(-7+))|/x.
The restriction of x to A is w, hence /w is divisible by 7+ — /ci('y+). Corollary
3.8 completes the proof of the main theorem.
143

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150

Index of notation
Page numbers refer to first occurence(s). If the meaning of a symbol slightly
varies, this should be clearly indicated in the text.
0. General.
pi
R[[T]]
k
Zp,Qp
Cp
FP
Nf/k,Ttf/k
Mm
A
G
N
M[n]
MG,MG
M*,MX
ir
rk M
E{F)
a
1. Fields, rings and ideals.
units of R
power series ring over R
algebraic closure of k
p-adic integers and p-adic numbers
completion of Qp
completion of F at P
norm, trace
mth roots of unity
character group of G
absolute norm
Ker(n : M —► M)
G-invariants, G-coinvariants
X-eigenspace, x-coeigenspace
the prime means that an obvious exception
should be made to the index set
rank of a finitely generated abelian group
.F-valued points of E
complex conjugate of a
Local
k,0,p
v (valuation)
kur,K (inch. I only)
<p(<f>)
D,D'
Semi-local
$,$n,$',$
it, Rny R j R
7
7
7
11
98,104
64,66
64,66
151

Global _
WP: Q «-> C,Cp 36-37
K,dK,WK 36
f,f0°°,^f 36
p = pp
2. Galois groups.
fx>9x
3. Units.
38
K(f),F 36,64
i^oo 64
F^,F' 66,75
#oo 77
Mn,M(Joo) 103,137
In IV.3 only:
?nJn{p),Tn(p) 134
Fn,Fn{p),Fn{p) 134
F+ F~ 7+ 7~ 134
•ioo'-ioo>,/oo''/oo -L°^t
F' 132
G = r X A 19,64
£,£(f) 20,64,76
r',i?' 77,104
aa = (*,-/*) 38
p (complex conjugation) 90,132
l0 86,105
TV f 100
In VI.3 only:
S,9u92,GuG2 134
r = ri x r2,r+,r- 134
Iwasawa modules
r(f),v(f) 103
cfcor.(M) 105,137
A-inv.,/it-inv. 108
109
Local
y yn 65
^£/(f),V,V(f) 18,65
/? = (Pn) 13'68
152

Global_ _
En,En,<En> 103
£ = £{f) 103
in 18,67,79,91
RP(E) 110
Elliptic
Cf _ _ 56
Cn,Cn,<Cn> 101
Cf,C(f) 101,102
en(o),e(o) 71
Ma),0{a) 71
<pg(a) 55
u(o) 49
4. Measures and power series.
A(£,D) = D[[£]],A 16,98
A0,Ai,A0o 99
A*/?,A*J,Ma 19,68-69,74
«,«(f) 20,68,99
0£ 13,68
log gp = ap 18,68
**,**(= Wkinch. I),4,n 20,69,70
A*(f),A*(ffl°°) 76
A*(f;x) = xMf)) 105
5. Characters and L functions.
X, fx 37
ipE/F = V> ,40
93(c), 93(a) 41
/c(«r) 18,65
A(c) 42
e = xfk^P3 80
£ 90
N 38
w 137
r(x) 70
G(e) 75,80
-£'m(£)S))I'oo,m(£) s) 37
L{E/F,s) 120
LP)m(er),LP|m(x;si,S2) 86,87
W,W(er) 37,91
153

In IV.3 only:
Ki,K2,K+,K~ 134
6. Formal groups and related symbols (especially ch. I).
Ga,Gm
A,A^
E
wy,wy,Wf
0: Gm ~ E
QP,Q (only in ch. I)
D
wn
X
S
6,7
7
6,9
6,71
46
10
11,14
17,66
10,17
6,20,
13,18
11
26
7. Elliptic curves and special functions.
92,93,A
we
L,La,Wi,U}2
p(z,L),£(z,L)
JE
E[a],Etor
A(o),X(S)
E, Ei
t
A
E
a(z,L)
r,(z,L),A(L)
0(z,L)
s2(L)
e{z;L,a)
S{*,L)
Ej>k{z,L),Ek{z,L)
d,D
n
Un,Wn
U1{E/F)
Sa(E/F)
39,48
39,47
39,43
39,47
39
40
42,65
44
46,47
46
48
48
48
48
49
57
57
57
66
67,79
122
123
154

Perspectives in Mathematics
Vol. 1 J.S. Milne, Arithmetic Duality Theorems
Vol. 2 A. Borel et al., Algebraic D-Modules
Vol. 3 Ehud de Shalit, Iwasawa Theory of Elliptic Curves with Complex
Multiplication