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Order Number 04209G0

On the M2-part” of the Birch and Swlnnerton-Dyer Conjecture for elliptic curves with complex multiplication

Gonzalez-Aviles, Cristian D., Ph.D.

The Ohio State University, 1004

UMI 300 N. Zeeb Rd. Ann Aibor, MI 48106

O n t h e “2 - p a r t ” o f t h e B ir c h a n d S w i n n e r t o n -D y e r c o n j e c t u r e f o r ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Cristian D. Gonzalez-Aviles,

* * * * *

The Ohio State University

1994

Dissertation Committee: Approved by

Professor Karl Rubin Professor Robert Gold Adviser Professor Warren Sinnott Department of Mathematics A cknowledgements

I thank my advisor Prof. Karl Rubin for his guidance and help; 1 have been very fortunate to be able to share in his wisdom and superb talent.

My thanks go to Prof. Warren Sinnott as well, who helped me overcome many of the technical difficulties involved in this work.

I also benefited from conversations with Professor Robert Gold and Mr. Cristian

Popcscu. I thank them heartily for their help.

Eslc trabajo csta dcdicado a mis queridos padres , quicncs me han apoyado sicmpre. V it a

June 25, 1961 ...... Born in Arica, Chile.

1985 ...... B. Sc., Univcrsidad dc Chile, Santiago, Chile.

1986 ...... M. Sc., Univcrsidad dc Chile, Santiago, Chile. Thesis: “Class numbers of quadratic function fields and continued fractions” (J. Number Theory 40, no. 1, January 1992, pp. 38*59). Thesis advisor: Prof. Ricardo Bacza.

1987-1994 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio.

F i e l d s o f S t u d y

Major Field: Mathematics T a b l e o f C o n t e n t s

ACKNOWLEDGEMENTS ...... ii

VITA ...... iii

CHAPTER PAGE

I Introduction ...... I

II The primes of bad reduction ...... 5

2.1 Preliminaries for Chapter II ...... 5 2.2 Sclmcr groups ...... 8 2.3 Sclmer groups, continued ...... 11

III The “main conjecture” of Iwasawa t h e o r y ...... 18

3.1 Preliminaries for Chapter III ...... 18 3.2 Elliptic units ...... 21 3.3 Euler systems of elliptic units ...... 26 3.4 A logarithm map ...... 29 3.5 Statement of the main conjecture ...... 33 3.6 Results from Iwasawa theory...... 35 3.7 Iwasawa modules of global and elliptic units ...... 39 3.8 A divisibility relation ...... 44 3.9 The Iwasawa invariants of X * ...... 54 3.10 The Iwasawa invariants of the p>adic L-function. Proof of the main conjecture ...... 60

iv IV The Birch and Swinnerton-Dyer conjecture ...... 67

4.1 The conjecture over A' ...... 67 4.2 The conjecture over Q ...... 71

BIBLIOGRAPHY ...... 79

v C H A PT E R I

Introduction

Let E be an elliptic curve defined over K — Q (\/—7) (one of the 9 imaginary quadratic fields of class number 1), with complex multiplication by the ring of integers Ok of K and with minimal period lattice generated by SI € Cx. Suppose that L (E fK , 1) ^ 0.

Then E(K) is finite (Coates and Wiles [8]) and the Talc-Shafarcvich group ]M{EjK} is finite (Rubin [22]). Further, for each prime q of J(t let Cq = [E(I

Eo(Kq) C E(K q) is the subgroup of points with non-singular reduction modulo q. In this dissertation we prove the following

Theorem A. If E has good reduction at 2 then

L(E/I<, 1) = m i. (# £ (/0 )-2 m• ( E / K ) • n^q. (1.1) q In other words, the full Birch and Swinnerton-Dyer conjecture is true for E.

Remarks, (a) Rubin [23] has shown that (1.1) is true up to multiplication by an element of K divisible only by primes dividing #0% (for all 9 imaginary quadratic fields K of class number 1). Thus when 0j[* = { ± 1} (as is the case for K — Q (\/—7)), the verification of (1.1) reduces, in a sense to be made precise shortly, to “checking

2-components”. The case K — Q (\/—7) treated in this paper is, for reasons that are explained below, the simplest of all. 2

(b) With more work it should be possible to remove the restriction on E in The­ orem A. We hope to be able to resolve this issue in the near future.

(c) Other authors have addressed the problem studied in this dissertation. Sec

[1] and [11]. The effectiveness of their methods is limited to those cases where the elliptic curve in question has few primes of bad reduction (at most 2). We hasten to remark, however, that the first of .the references given addresses this problem for other choices of K as well. This author is currently working on these remaining cases.

In this thesis wc also prove the following result. For any non-zero integer d let Ed denote the elliptic curve y2 = x3 -f-21dx3 + 112d3x, which has complex multiplication by Ok = Z[(l + V=7)/2).

Theorem B. Suppose d — l(m od4) and L(Ed/Q t 1) / 0. Then the full Birch and

Swinnerton-Dyer conjecture is true for Ed/Q.

The following is a brief outline of the paper.

Fix, once and for all, an embedding of K = Q(\/-7) in C. Assume that E has good reduction at 2 and L (E /K t 1) ^ 0, and let B denote the set of primes of K where E has a bad reduction. We will sec below (Proposition 2.1.7) that cq = 4 for each q € B. Further, let 4> denote the Heckc character of K attached to E and 2/(tf>,s) the corresponding Hecke //-function. Then the //-function of E over K factors as

L(E/K , s) = L{4>y s)L{$t s) (1.2) and L(4i, l)/Q € J<> so L{E/K, l)/fm = Nk/q(Z,(& l)/«) € Q+. Thus formula (1.1) is true up to multiplication by a power of 2 (cf. Remark (a) above), and therefore

Theorem A is equivalent to the statement that L (E /K }l)/SlO, and 46*(^£?(/C)3»)*'a* 3

# m 2°° differ by a 2-adic unit. Here b = #B.

Since 2 splits in K , say 2 = pp with p ^ p, one can show, using the factorization

(1,2) and the fact that = 4 (Corollary 2.1.2), that the result sought will follow from

l(0,l)/ft - 2t-2.#UIp» (1.3) where ~ means “equal up to a unit of /fp”.

To establish (1.3) we first show (in §2.3) that

#Hom(-Voo, Ep«>)** = 2h~2 • #£(/fp)p- • #UIP« (1.4) where A',*, is the Galois group of the maximal abelian 2-extension of = K(Ep«>) which is unramificd away from the prime above p, and *£ = Gal(KcafE). The essential ingredient in the derivation of this result is Theorem 2.2.6 below, which gives the index of the p°°-Sclmcr group of E in the larger group of 1-cohomology classes which arc locally trivial outside of B U {p}. The proof of Theorem 2.2.6 depends crucially on work of Bashmakov [2]. The rest of the argument leading to (1.4) is likely to seem familiar to those readers acquainted with Coates' paper [6] (but also a little more complicated, because here we deal with the troublesome prime 2).

Now Rubin ([23], §11) has shown how to relate the integer on the left-hand side of formula (1.4) to ordp(Xf($, l)/fl) when p^f 2, as an application of Theorem 4.1 of

[23], the “main conjecture" of Iwasawa theory for K . In Chapter III we prove a “main conjecture" for the extension KoojK (Theorem 3.5.1 below) which is well-suited for similar applications. The fact that 2 splits in I( means that we are in the setting of a one-variable main conjecture, which is the reason why the case K = Q (\/—7) is the 4 simplest of ail. Wc will then use the main conjecture to prove (in §4.1) that

# H o m (X „, E , . f ~ (1 -

Since 1 — 0(p)/N(p) ~ #E(Kp)p*> (Lemma 4.1.6), the above and formula (1.4) imply

(1.3), thus completing the verification of the Birch and Swinnerton-Dyer conjecture in the present case.

Theorem B will be proved in §4.2. C H A PT E R II

The primes of bad reduction

2.1 Preliminaries for Chapter II.

Let K denote the field Q (\/“ 7) and write O for its ring of integers. The rational prime 2 splits in K\ fix a factorization 2 = pp with p ^ p. Further, let E be an elliptic curve defined over K with complex multiplication by O , and assume that E has good reduction at both p and p.

For any C?-modulc M and ideal o C O, Ma will denote the a-torsion in M<, and

Ma» = Un» Farther, if F is any field, F denotes the algebraic closure of F and

G f = Gal(FfF).

Define the tower of fields Kn = I((Epn) and let Gn = Gal(KnfK ) for 1 < n < oo.

For any prime q of K we will write Oq (resp. Kq) for the completion of O (resp. K) at q. Define

Xn : G k -* (Op/p"O p)x by P° = Xtt(a)P for all P <= Epn and a* <= GK.

Proposition 2.1.1 (i) Xn ' Gn — * {OpfpnOp)* is an isomorphism .

(ii) The prime p is totally ramified in I(nfK.

(iii) If n > 2 then over Kn, E has good reduction at all primes not dividing p.

5 6

Proof. Since p is a prime of good reduction for E, statements (i) and (ii) follow from the theory of Lubin-Tatc formal groups. Sec for example [8] §3.

For (iii) also sec [8], Theorem 2. The proof is based on a variant of the criterion of

Neron-Ogg-Shafarcvich and the facts that E has potential good reduction everywhere and Gal(Kt»/Kn) cs 1 + pnOp is torsion-free for n > 2. □

Corollary 2.1.2 We have

E(K)2cc = E3.

Proof. Assertion (i) of the above proposition shows that K\ = K but K2 £ K, so

E(K)p» = Ep. Similarly, the analogue of Proposition 2.1.l(i) for the prime p implies that E(K)p<*> = Ep. □

Now let B denote the set of primes of K where E has a bad reduction.

Lemma 2.1.3 Suppose q € B, and let p denote either p or p. Then

(i) K{Epi)fK is ramified at q, and

(ii) £(/(,)„- =

Proof. For (i) sec [25], Theorem 2 and its corollaries. Clearly (i) implies Epi <£. E(Kq), and Corollary 2.1.2 shows that Ep C E(Kq)P», thereby proving (ii). □

Let r denote the element of Goo = Gal(KaojK) which maps to —1 under the isomorphism : Goo -* Op . This is the only non-trivial torsion element in Goo (of order 2); it acts as multiplication by -1 on Ep«>.

If q is a prime of K we will write lq|00 for the inertia group of q in KoafK. 7

Lemma 2.1.4 Let q € B. Then

Iq.oo = (7‘)>

Proof. It suffices to show that Iq)60 is finite and non-trivial. Since E fK i has good reduction at the primes above q (Proposition 2.1.1(iii)), K ^ /K j is unramified at these primes ([28] §VII.4). Thus Iq)00 may be identified with the inertia group of q in

K ifK , which is finite and non-trivial by Lemma 2.1.3(i). □

Remark 2.1.5 We point out that E cannot have good reduction everywhere over K i.e. 5^0. This follows from the fact that the Ilecke character of K attached to E has a non-trivial conductor.

For q 6 B we let JSo(A’q) C E{Kq) denote the subgroup of points with non-singular reduction modulo q.

Lemma 2.1.6 Let q G B. Then

Eq{K q) = 2Ea(/fq).

Proof. Write m for the maximal ideal of Oq and k for the corresponding residue field.

Fhrther, write E for the forma] group associated to f?, and let E\(Kq) C Eo(Kq) denote the kernel of reduction modulo q and Ent the non-singular part of the reduced A A curve. The groups J5i(/fq) and J3(m) are isomorphic, and J?(m) is uniquely 2-divisible

(because 2 € Ojf). Thus £?i(/{q) = 2Ei(A'q) C 2J3o(/fq), which shows that the reduction modulo q map induces an isomorphism between /Sof/fqJ^/sbf/fq) and a quotient of Ent{k). Now, since E has potential good reduction at q, the reduction 8 type of E over K q is additive i.e. Ent(k) ~ k+. Therefore ^ E M{k) is odd, which proves the lemma, □

Proposition 2.1.7 Let q g B, Then

E(Kq) = E2® Eo (IQ.

In particular, c, Is' [£(/g : £■„(/(•,)]4. =

Proof, The group £7(/Cq)/JSo(A'q) is annihilated by 2 ([17] Proposition 4.5). Now

Lemmas 2.1.6 and 2.1.3(ii) show th at E(Kq) = E2 + Eo{I(q) and E2 H Eo(Kq) C

Et n 2E(I

2.2 Selmer groups.

In this section wc compute a group index involving certain Selmer groups. Sec The­ orem 2.2.6 below. This result is a basic ingredient in the derivation of formula (1.4) of the introduction, which we carry out in §2.3.

Keep the notation and assumptions of §2.1. In addition, assume that our elliptic curve E satisfies L(E/K , 1) 0. In this case the finiteness of E(K) and of the Tate-

Shafarcvich group of E over K have been demonstrated by Coates and Wiles [8] and

Rubin [22], respectively.

If F is any field we will write H l(F, E) for H l(G f}E(F)),

Let III and S' denote, respectively, the Tate-Shafarevich group of E over K and the direct limit of the Selmer groups of E relative to powers of p. Thus

IE = ker\H\K,E) — * © q H \ K q,E)) 9 and

S =kcr[H1{I<,Ep-) — ♦ e qB 1{K

The q*component Aq : //'(/(, Ep») —♦ B x{I(^E)p» of the map appearing in the definition of S above is the restriction homomorphism to / / ’(/Cq, 15pw) followed by the canonical map from this group to IJl(KqtE)p«o.

Lemma 2.2.1 There is an isomorphism

UIp» ~ 5.

Proof ! Galois cohomology gives us an exact sequence

0 ■—+ E{K) 0 o (Kp/Op) — > S— * U I p . — ♦ 0

(see [22] §1).Since E(K) is finite, E(K) ®o (Kp/Op) = 0, hence the lemma. □

Recall that B denotes the set of primes of K where E has a bad reduction. Let

B f = B U {p}, and define a modified Selmer group S(B') D S by

S{B') = ker[U'(K%Er>) — © q,tB, H \ K ^ E )p»], (2 .2 )

We wish to compute the group index [S(J3'): 5].

Lemma 2.2.2 There is an Oq-module isomorphism

^(■^q) — ^(-^q)

Proof, This follows easily from the fact that E(Kq) has a subgroup of finite index which is free of rank one over 0 q. See [28] Proposition V1I.6.3. □ 10

Let A#* denote the restriction of ©qgg* Aq to S(B ’), Thus

Aflr: 5(5') — ©q€fl, H '{K ^E )P», (2.3)

and clearly ker(Ae») = S. Therefore

(S(fl'): 5] = n p-/#coker(Afl.). (2.4) q GB' Let b denote the number of primes of K where E has a bad reduction i.e. 6 = #B.

Proposition 2.2.3 With b as above,

I I #HHK„E)f. = 2t-#E(Kf)r. qS B» Proof. Let q 6 B 1. By Tate's local duality, B X(Kq, E)p«, is dual to Hm E(Kq)/pnE{Kq)t

inverse limit with respect to the natural maps. Now since q / p, Lemma 2.2.2 shows

that lim E (K q)/$nE(I

#y/I(/t'q,£;)p» = #£;(/

Finally if q p (i.e. if q € B ), Lemma 2.1.3(H) gives

#£(/Cq)p- = # £ , = 2, which completes the proof. □

Next we determine #cokcr(Afl»), where Ag> is the localization map (2.3).

Let Wb> denote the image of lim E(K)/pnE (K ) in © qeB» Ijm E{Kq)/pnE(Kq).

Theorem 2.2.4 (Bashmakov) /lssume that Ulp« is finite . Then there is an isomor­ phism

Hom(coker(Aflt), Q /Z) a Wb >. 11

Proof. See [2] (Corollary 2 of Theorem A). □

Corollary 2.2.5 We have

#cokcr(AB') ~ 2.

Proof Since E(K) is finite,

Ijm E(K)/pnE(K) sa E(K)Poo = Ep.

The result now follows from Theorem 2.2.4. □

Wc now combine Proposition 2.2.3 and Corollary 2.2.5 and use (2.4) to deduce the following

Theorem 2.2.6 Let S and S(B') be defined by (2.1) and (2.2), respectively. Then

[S(S '): 5] = 2*-1 • m > < p)|i“ . where b denotes the number of primes of K where E has a bad reduction. □

2.3 Selmer groups, continued.

In this section we establish formula (1.4) of the introduction. Wc do this by first comparing the Selmer group S(B') of the previous section with a Selmer group over the field Kao ((2.6) below) which is naturally isomorphic to Hom(A'oo» Ep«>)^ (notation as in the introduction). See Proposition 2.3.6 below. We will then combine this information with the index formula of §2.2 to obtain the desired result.

Recall Kn = K{Epn) for 1 < n < oo. At this point we deviate a little from the notation introduced in §2.1 and from now on write for Gal(KaofK) (this was 12

denoted Goo in §2.1). Also, let P = Gra/(/f00//iF2). Then under the isomorphism

& ~ Op of Proposition 2.1.l(i),

r ~ 1 + p2c?p ~ op ~ z2,

because p ^ p. Thus T is “cyclic”; a topological generator of T is given by the

automorphism of Iioo/E that acts as multiplication by 5 on Ep°°.

Lemma 2.3.1 (i) //'(r,^,® ) = 0 for all i > 1.

(ii) The restriction map induces an isomorphism

£ p») ~H X{I

Proof, (ii) follows from (i) and the usual inflation-rcstriction exact sequence. Now for

2 < n < oo let Tn = Gal(Kn{ K2), and observe that

H i(r,Epoo) = \m lP { rniEpn)>

direct limit with respect to the inflation maps.

Each r n is a cyclic group of order 2"-2, generated by the automorphism that

acts as multiplication by 5 on Ep«. A straightforward calculation then shows that

• W (Tni J5P«) = 0, thereby proving (i). □

Lemma 2.3,2 Wc have

# H l{tyEp») = 2.

Proof The inflation-restriction exact sequence together with Lemma 2.3.1(i) gives an

isomorphism

H x(&t E p - ) * H ' ( G 2tEp,) 13

where G2 = Gal(K 2 fK). This Galois group is cyclic of order 2, generated by the automorphism that acts as multiplication by —1 on Epi. Then

Ht(Gt,E9*)~Epl2Ep*2£Ep. □

Let 0 be a prime of K ^. For 2 < n < 00 wc will write KniQ for the completion of

Kn at the prime below 0 . Define K ^ q — Un>2 Kn,a- Wc note that Kao,a is contained in, but is not equal to, the completion of Koo at 0 .

Recall that B denotes the set of primes of K where E has a bad reduction and

B'-B U{p}.

Lemma 2.3.3 Let 0 be a prime of /{„>, and let q be the prime of K lying below 0 .

Then if q £ B't

H'(GaHK„,0/K,),E(K„,a)) = 0.

Proof This follows from the facts that the extension /C»,o//fq is unramificd ([28]

§VII.4) and E has good reduction at q. See Corollary 4.4 of [20]. □

Lemma 2.3.4 Let 0 be a prime of Koo, and let q be the prime of K lying below 0 .

Then ifq G B ,

H'(Gal(K„.a/l

Proof Since E has good reduction over (Proposition 2.1.1(iii)) and /foo.o/ffj.a is unramified (Lcmma2.1.4), H l(Gat(K03fQl /fa.o), E(Koo ,q )) = 0 (cf. Lemma2.3.3).

Thus the inflation-restriction exact sequence gives an isomorphism

II'(Gal(KM /K,),E(K„.a)) ~ H'(Gt,a,E(K7,o)) 14

where G^o = Gal(K 2 ,a/Kq). Note that #C?2,o = 2, by Lemma 2.1.3(i).

Now let k denote the residue field of 7fa,0i and write E\(K 2,q) for the kernel of reduction modulo 0 and E for the corresponding reduced curve. The exact sequence

0 — » Et(K2,Q) — * E(I<2 ,q ) — E(k) — . 0 gives rise to a cohomology exact sequence

H'(G,,0, £,(!<,,0)) -> H'(G,,aiE(K,,a)) //'(Gj.cCW) - H*(G,ia, E,(K,,a)).

Next, since # (72,0 = 2 and /?i(/?2,o) is uniquely 2-divisiblc, //'(Ga.o^iC^a.o)) = 0 for i = 1, 2, so the above cohomology exact sequence gives an isomorphism

//'(Ci.o, £(/(■„)) ~H'(G w ,E(k)).

Finally, Ga,o acts trivially on E(k) (because /(a.o/^q is totally ramified) and E(k)p is the reduction of Ep modulo 0 , so

= Hom(GJl0,£ P)^ E ,.D

Now for each prime q of K choose a prime of /? lying above q and let Iq denote the corresponding inertia group. Consider the inflation-restriction exact sequence

0 — ♦ //'(tf, Epoo) — ♦ H \K , Ep«) H om fG *., Ep~ f (2.5) and define

/(G/f„nI,) =0 for all q ^ p} ( 2.6) 15

(it is easy to see that this definition is independent of the choices made). Wc note that there is a canonical isomorphism

^(p )^ H o m (X o o ,£ p » )y (2.7) where Xoo is the Galois group of the maximal abelian 2-extension of K& which is unramificd away from the prime above p.

Lemma 2.3.5 ^ (p ) is contained in the image of the restriction map (2,5),

Proof. Let / G «^(p) and fix a prime q of K where E has bad reduction. As in

§2.1, let r denote the element of ^ that acts as multiplication by —1 on Ep*>, Since r generates the inertia group of q in K ^ fK (Lemma 2.1.4), wc can find an element r G lq whose restriction to is r. Then r2 G Gk„ fllq and consequently / ( t 2) = 0.

Also, Lemma 2.3.1(ii) shows that there is a 6 G H x(K^Ep<»)Ga^KifK^ whose image under the restriction map is / G //'(/foot /?p»)^. < * Now define c : G k —* £p» as follows: any a G G k can be written (uniquely) as

This can be easily verified using the facts: (a) (sf'^s1 fJ) = s(f's,r",)f,+J, (b) 6 is invariant under Ga/(/f’j//f'), and (c) £(f2) = /(f2) s= 0. Thus {c} G B x(K,Epoo) maps to / under the restriction homomorphism, and the proof is now complete. □

Recall the group S(B') C B X(K} Ep«>) defined in §2.2.

Proposition 2.3.6 The exact sequence (2.5) induces an exact sequence

0 — i Hx(&t Ep~) — > S(B') — ♦ y(p) — * 0. 16

Proof, That the inflation homomorphism maps Z/1^ , Ep<*>) into S(B') follows easily from Lemma 2.3.3. Now for each prime £1 of we have a commutative diagram

lB{I<,Ep«) ^ ------H l(Kv E)p-

res rcsQ

Horn (O * ., Ef. f ^ 3 S )p. where q is the prime of K lying below £}, Aq is the localization map defined in §2.2 and rcsa is the local restriction map, and where we have written A*J Q for the restriction of Aool0 to HomfGV,,,, Ep«o)^ (A<»(q = the analogue of Aq for the field /£«).

Note that the kernel of rcsQ is / / l(G'a/(/f0o1o//(q), E(K00ltJ))poot which is non-zero if q 6 B (Lemma 2.3.4). We have seen that //'(A ’q, ZJ)p» has order 2 for such q (see the proof of Proposition 2 .2 .3 ), and so it follows that rcsQ is the zero map if Q|q with q € B. If £2|q and q £B\ then rcsQ is injective (Lemma 2 .3 .3 ).

Next, we claim that

y ( p ) = n (2-8) OJP That Do|p M A ^ o ) is contained in S^(p) can be checked by a standard argument

([28] proof of Theorem X.4.2(b)), using the facts that if £ i/p then E has good reduc­ tion over Koo .q (by Proposition 2.1.1(iii)) and the p-torsion subgroup of E\(Koa,o) »s trivial. Conversely, suppose / € *^(p) and let £1 be a prime of /£«, not dividing p.

Then the restriction of / to factors through the Galois group of the maximal unramified extension of K ^ q . Therefore Aoo,q(/) = 0 (cf. Lemma 2.3.3), and (2.8) follows. 17

Now suppose c € S(B')i and let £2 be a prime of K » and q the prime of K below

£2. Then A0o,o(f,cs(c)) t= rcsa (Aq(c)) = 0 if q £ B’ (by definition of S(B')) or if q 6 B — B' \ {p} (since then rcsfl is the zero map). Thus, by (2.8), rca(c) € *^(p)*

Finally, let / 6 ^ (p ). By Lemma 2.3.5, wc can find an clement c € B l(K, Ep«>) such that / = rcs(c). Suppose £2|q with q £ B'. By (2.8),

res0(Aq(c)) = \oo,a(f) = °*

But rcsa is injective when q £ B\ so Aq(c) = 0. Thus c € 5(2?'), as required. □

Wc can now state formula (1.4) of the introduction as a

Theorem 2.3.7 Let b denote the number of primes of K where E has a bad reduction.

Then

#Hom(Afoo,i?p»)*= 2b-a • #£(/£p)r • #UIp«.

Proof. By Lemmas 2.2.1 and 2.3.2, Proposition 2,3.6 and (2.7), the integer on the left is equal to [5(2?') : 5]#IUp°°/2. Now use Theorem 2.2.6 to obtain the desired formula. □ CHAPTER III

The “main conjecture” of Iwasawa theory

3.1 Preliminaries for Chapter III.

Next we address the problem of relating #Hom(Xoo, Epoof* to L(0, l)/fl. As ex­ plained in the introduction, a first step towards establishing such a relation is to prove the main conjecture of Iwasawa theory for the extension A'00//<’; this is the subject of the present chapter.

Keep the notation introduced in Chapter II. Thus K denotes the field Q (\/—7) and O is its ring of integers. In addition, if m is an ideal of O we will write K(m) for the ray class field of K modulo m.

Recall that p|2 and Kn = K(Epn) for 1 < n < oo. Define /T(p°°) = Un>i

Lemma 3.1.1 (i) K(m) C K(Em) for any ideal m of O,

(ii) / / 2 < n < oo then [Kn : /T(p")) = 2 and I

Proof. Assertion (i) follows from the theory of complex multiplication. See [27], p.

124. Statement (ii) for n = oo follows without difficulty from the corresponding state­ ments at finite levels, so we assume n < oo. Since n > 2, we have Gal(K(pn)f K) o;

(0/pn)x/C?x. Thus [I<„ ; /f(pn)] = #O x = 2, by Proposition 2.1.l(i). The rest of assertion (ii) follows from this and from Lemma 2.1.3(i). □

18 19

Remark 3.1.2 Recall that G a ^K ^f K2) — % 2 (cf. discussion preceding Lemma

S.3.1). Thus, by assertion (ii) of the above lemma, Gal(K(p°°)/I() ~ %2.

For any field F we will write f i ^ F ) for the group of all roots of unity in F and

/*2«(F) for the subgroup of 2-power roots of unity.

Lemma 3.1.3 (i) /*«,(/£») = {±1}.

(ii) / f « n Q W = Q.

(iii) h 2«>(Koo ®k Kp) is finite.

Proof. Let ( € Koo he an m-th root of unity, and assume that A '(C ) ^ K. By

Proposition 2.1.1(ii), p ramifies in K(()/I(, so m is even and therefore p (the other prime factor of 2 in K) ramifies in K^fK. But this is impossible, because p is a prime of good reduction for E. Thus K(() = K, and (i) follows. A similar ramification argument shows that Ka, 0 K {p2«>) — K whence (ii) follows since K = Q(\/“ 7)

Q{p2«>)' A s regards assertion (iii), it is shown in [6] (Lemma4) that E(Koo®k Kp)p* is finite. Then a simple application of the Weil pairing gives (iii). □

Let r s= Too denote the automorphism of KmfK that acts as multiplication by —1 on JS^oo and let rn be the restriction of r to Kn (1 < n < 00). Write K+ for the fixed field of (rn) in Kn.

Proposition 3.1.4 Suppose 1 < n < 00. Then

K* = K( p”).

Proof. By Lemma 3.1.l(i), K(p) C K[Ep) = K, so we may assume n > 2, Now by

Lemma 2.1.4, K+ is the fixed field of the inertia group in KnfK of a prime q ^ p. 20

Therefore K+ D A'(pn), which proves the proposition since (by Lemma 3.1.1(ii))

[An : l<+) = [/fn : A-(P")] (= 2). □

Write An for the 2-primary part of the ideal class group of Kn, 1 < n < oo. The above proposition has the following

Corollary 3.1.5 Suppose 1 < n < oo.

(i) The class number of K+ is odd.

(ii) (1 + Tn)An ~ 0.

Proof, (ii) follows from (i) since (1 + r„)/t„ = N ^ b^+(/1„). Now Proposition 3.1.4 together with Remark 3.1.2 shows that K+fK is a Zj-cxtcnsion in which only p ramifies, and this prime is totally ramified. Then the fact that the class number of

K is odd (= 1) gives (i). Sec Theorem 13.22 of [30]. □

For every n < oo define 4>„ = Kn Kp. By Proposition 2.1.1(ii), 4>n/Ap is a totally ramified extension of local fields. Thus we may identify Gal(Kn/K ) with

Gal($n/Kp), Let fl** be the fixed field of (rn) in 4*n, and let i/„ denote the group of local units of 4*n. For convenience we will also write Up for U\, the local units of Kp.

Let Up ^ s= 1 + pnC?p C Up be the group of n-th principal local units of A'p. Note that

Up s Uil) = {±1} x U$2K

We have an isomorphism

/>. : Up/U^ — . Gal(Q„/Kr) given by pn(«)(A') = u~lP for all P 6 Ep»,

Proposition 3.1.6 Suppose 1 < n < oo.

(i) The map pn : Up -* Gal(

In particular, N

(ii) The kernel of the Artin map for $*/Kp is {±l}£/p”\

Proof. For (i) sec [24] §3.7. Assertion (ii) follows easily from (i), using the consistency property of the Artin symbol and the fact that pn(-l) = fn. □

3.2 Elliptic units.

In this section wc define the elliptic units of Kn for 1 < n < oo. The basic reference for the material presented here is [9].

Let xj) denote the Hcckc character of K attached to E and f its conductor. Fix a minimal model of E over K and let L denote the corresponding period lattice.

F\irthcr, write 7(6) for the set of integral ideals of K prime to C.

For a € 7(C) define the Klein function

K{z\a)= n (P(*» £) “ P(«; £))“l where p(s; L) is the Wcicrstrass p-function for the lattice L, and where the class (u)*, of u modulo L varies in a set T such that T U — T = a~'L/L \ (0)*,.

Robert [21] has constructed a 12-th root 7(a) of A (L )^^f A(a~l L) in K (here a 6 7(6) and A is the usual Ramanujan A-function) which is uniquely determined by the property that the function

Q0(s\a) = T}(a,)I<(z',a) satisfies the following distribution relation: if b is an integral ideal prime to of then

J J 0 o(z + u; a) - OoMb)*; a). (3.1) v£b-iL/L 22

This function 0q is the unique 12-th root of the function 6 (2; L, a) used in [9] (§11.2) which satisfies (3.1).

Proposition 3.2.1 Suppose 0 € /(C), m / 1 is prime to a and u € Cf L is a point of order exactly m.

(i) 0 o(o;a) € /('(m).

(ii) If c is an integral ideal prime to Gmf and at is the Artin symbol of c for K(m)/I(,

Oofuja)"* = 0 o(0 (c)u;a) = 0 o(o; ac)0 o(u; c)"N(o).

(iii) Suppose q j'af, and let tv € C/L be such that ^(q) ' w = v. Then if

NK(m<,)/K(in)(0o(u>; a)) = Oofuja)1-'

where s is the tnucrsc of the Frobcnius of q in Gal(K(m)/K) if q /m , and s = 0

otherwise.

(iv) 0o(u;a) is a global unit if m is not a prime power. If m = qn, it is a unit

at every prime of K(m) not lying above q.

(v) If q is a prime of good reduction for E which does not divide m and x 6 C/L

is a point of order exactly q,

0 o(u + x; a) 5 0q (o ; a) modulo all primes above q.

Proof. For (i) see [21], n°12. The first equality of assertion (ii) is valid because ac acts as multiplication by ij)(c) on torsion prime to c. The second can be derived from the corresponding identity for 0 = 0 j3 ([9] §11,2.4), using the distribution relation

(3.1). The proof of (iii) makes use of (ii) and (3.1), and is similar to the proof of the 23 analogous result for 0 in (9] (§11.2.5). Finally, statement (iv) follows from Proposition

II.2.4(iii) of [9], and (v) is proved in §12 of (22). □

Recall p|2, I(n = I<(Ep*) for 1 < n < oo and f is the conductor of r)>.

If 2 < n < co, the conductor of Kn over K is fpn ([8] Lemma 4). We also note that p/f, because E has good reduction at p, and f ^ 1. Thus fpn is not a prime power. Now choose a point v € C/L of order exactly fpn and write C„ for the group generated by

{N/f(fpn)/;^„(0o(ui a ) ) : (a,6f) = 1}. (3.2)

Statements (ii) and (iv) of Proposition 3.2.1 show that Cn is a Z[Gal(Kn/K)\- submodulc of the global units of Kn which is independent of the choice of u. We define the group of elliptic units of Kn to be

Vn = ti OQ{l

(the second equality by Lemma 3.1.3(i)).

Remark 3.2.2 This group

(defined in (a) below), which is why we have chosen not to include these in definition

(S.S). /Is regards the elliptic units of conductor gp" for gjf and g ^ 1 or f, see (b) below.

(a) (Definition of Cpn) Choose a point w € C/L of order exactly p", let

J = {p : 7(6) -♦ Z /p{a) = 0 for almost all a and Eae/(6)P (o )(N (fl)-l) = 0}, 24 and for ft in J set

0 o(«);/x)= JJ 0o(uj; oe/(o) Then every 0 o(u;; p) (ft € J) is a global unit of I((pn) ([9] §111.1.4). The group Cpn of elliptic units of conductor p" is by definition the group generated by {±0 o(to;/i): ft € J ). It follows from assertion (ii) of Proposition 3.2.1 that Cpn is Galois-stable and independent of the choice of w.

(b) Suppose g|f and g ^ 1 or f. The group Cg pn of elliptic units of conductor gpn may be defined by replacing /C(fpn) by h'(Qpn)Kn in (3.2) and so forth. Since KnfK has conductor exactly fpn, Lemma S.l.l(ii) shows that /f(gpn) H Kn = K(pn). Then a calculation based on (ii) and (iii) of Proposition 3.2.1 shows that Cg pn c Cpn,

(c) (Connection between Cpn and Gillard’s elliptic units for K(pn)).

In of [12] Gillard determined, for a finite abelian extension F of I(, the index in the global units of F of a certain subgroup fll(F) of elliptic units. Wc will now show that the group Cpn defined above coincides with Gillard’s group fll(F) for F — K( pn).

The case n < 2 is trivial, so we assume n > 3.

Put ft1 = & (K(pn)) and G = Gal(I((pn)f I(), and write 1(G) for the augmenta­ tion ideal ofZ[G\. The extension K(pn)/K is cyclic, of conductor exactly p" (see the proof of Lemma 3.1.1(H)). Using this and the fact that p oa(K(pn)) — {±1}, one can show that

fil = ± { w ( 1)/(° , }1/m where m = 12 • 2n and ^p«(l) is Robert's invariant, as defined in §11,2.6 of [9],

Now 1(G) is contained in the annihilator of /ioo(A’(pn)) — {± 1} in Z\G], which is 25 generated as a X-module by (N(a) — 6) — 1} (see [If] Lcmmc A-l; here aa denotes the Artin symbol of a for K(pn)/K ), Thus any a € 1(G) can be written as

a - ]C Ma)(N(a) -

Then, using formula (18) in §11.2.6 of [9],

W O)" = GoK/Om for some point w € C/L of order exactly pn. The above shows that fl1 = Cpn.

N ote. When F = K„, «'(F) C ?„£> but Sll(F) £ KnCpn, This is plausible since fll(F) is constructed by collecting together the elliptic units in all cyclic subcx- tensions of F fK and, in contrast to the situation of (c), F = Kn is not cyclic over

I<.

(d) Recall that /f{pn) = K* (Proposition S.l.f). Write <5*n+ for the group of global units of K+ and h(K +) for the class number of Ii*. By (c) wc may apply Theoreme

5 of [IS] with F = I(*. This yields

\K =

Since h(K*) is odd (Corollary 3.1.5(i))t we conclude that

Cpn 0 2 Z2 = 0 2 Z2.

This fact will be used in section 3.10, where we will compute (following [9]) the Iwa­ sawa invariants of the p-adic L-function. 26

3.3 Euler systems of elliptic units.

Wc now begin to work towards proving a divisibility relation similar to Theorem 8.3 of [23]. The methods wc use arc those of Rubin; this section in particular parallels

§1 of [23].

Fix n with 2 < n < oo and write F for the field Kn = K(Epn), Also fix a power

M of 2 and define St — Stptm to be the set of all primes q of K which satisfy

(i) q splits completely in Ff K, and

(ii) N(q) s 1 (mod2A/).

R em ark 3.3.1 Every q € St is prime to 2f, where f is the conductor of the Hccke character attached to E (sec the discussion following the proof of Proposition S.2.1).

In particular, every q € 12 is a prime of good reduction for E.

Recall that /f(tn) denotes the ray class field of K modulo the ideal m of O.

♦ * L em m a 3.3.2 Suppose q € St, There is a unique extension F( q) of F of degree

M in FI(( q). Further F(q)/F is cyclic, totally ramified at all primes above q and unramified everywhere else.

Proof. See [23] Lemma 1.1. □

Write St for the set of squarefree ideals of O which are divisible only by primes q € St. If o = rifei flf € ^ we define F{a) = P(qi)... P(qfc), and F(l) = F.

For every ideal g of O let

^ (g ) = { a e ^ : ( o tg) = l } c ^ , 27 and let ^(g) denote the set of functions a : &(g) —* F* such that for every a € @(g) and every prime q dividing a:

a(a) is a global unit of F(a), (3.4)

NF(fl)/f(a/q)(tt(o)) = ar(o/q),^ol,fl—1 (3.5) where Frobq denotes the FVobcnius of q in Gal(F(afq)/K), and

q (o ) = a(a/q)*N^ ”* ^ f modulo all primes above q. (3.6)

Define

& f - W f, m = U &(9), disjoint union over all ideals g of O. For a € ty/p we will write &(a) for the domain of a i.e. &(a) = &(g) if a € ^ ( 0).

Wc will see below (Proposition 3.3.4) that the elliptic units of F = J((Epn) con­ structed in §3.2 give rise to elements a € %f> Compare [23] §1.

Recall that f denotes the conductor of the Hcckc character attached to E, Put fn = fpn.

Lemma 3.3.3 Suppose q € .2.

(i) /<•(,(„) = F(E„).

(ii) |F (£ ,): F(q)) = (N(q) - 1 )/M.

Proof. For (i) see [9] §11.1.6. Asregards assertion (ii), note first that F(£?q) D

F /f(q) O F(q), by Lemma 3.1.l(i). Now since q is a prime of good reduction for E t q is totally ramified in I<(Eq)/K (cf. Proposition 2.1.1(H)), and unramified in F/K. 28

Thus, by Proposition 2.1.1(i), Gal(F(Eq)/F) ^ Gal(K(Eq)/K) ~ (O fq ) \ This proves (ii). □

Recall the group of elliptic units of F = Kn defined in §3.2.

Proposition 3.3.4 If u 6 ^fn then there is an element a € Wp such that a(l) = u.

Proof. Since V/p is closed under multiplication and inverses and the proposition is true when u is a root of unity (see [23] §1), we may replace u by

u' = Na*(|„)//K0 o(u; b)) where v € C/L is a point of order exactly fn and (b,6f) = 1.

For each q € B fix an element xq € C fL of order exactly q. Wc shall also view the coscls v and xq as points on E(C) via the usual isomorphism C/L a E(C) (given by Wcierstrass’ paramctrization). Now for every a € ^ (b ) define

q(q) = N/4'(aft))/F(a)(0o(u + £q|a ®q; b)).

Clearly a (l) = u'. That a satisfies axioms (3.4) and (3.5) follows easily from assertions

(ii)-(iv) of Proposition 3.2.1, We will show that a satisfies (3.G) as well. For every a €

3?(b) let 0(a) = Gal(K(tf„)/F(a)). Now fix a € ^(b), let qo be any prime dividing a and let // = Gal(K(tfn)/F(a)K(afn/qo)) C 0(a). Ramification considerations and

Lemma 3.3.3(i) show that Q(a)/H a 0(a/qo), H a Gal(F(E^)/F(qQ))^ and that every h € H fixes both v and £ q^qoxq. Then by Proposition 3.2.1(v) and Lemma

3,3,3(ii), modulo all primes above q0,

°(°) = + S *q + x5oi **)* s °(o/«1q)#// = o(a/qo)(N(,to)~1,/Af. □ tfea(q)//Mettn II qyiqo 29

For every prime q € ^ write Gq = Ga/(F(q)/F). Then Gq is cyclic of order A/, and wc fix a generator ffq of Gq. Define

Oq = £ iai e nOq) 1 = 0 and if a € @ define

Da^l[Dq^Z[Ga] q\a where G0 = Gal(F(a)/F) ~ flqla^q.

Proposition 3.3.5 For every a € there is a canonical map

K = Ka :&(a) — *F*/(F*)M suc/i that for every a € ^(a), «(a) = o(o)0fl (mod (F(a)*)M),

Proof. See [23] Proposition 2.2. □

3.4 A logarithm map.

Keep the notation of §3,3, so F = Kn with n > 2. Farther, recall that t is the automorphism of KoofK that acts as multiplication by —1 on Fp».

Write H = Gal(F/K?). For reasons that will become apparent later (cf. §3.5), we will often disregard the natural Z[Gal(F/K)]-module structure of various algebraic objects and view these only as modules over Z[//]. Thus we let ^denote the Z [H]~ module of fractional ideals of F, and for any prime £2 of F define

S Q = Z[//]H C S. 30

Further, let An = X2[Gal(Kn/F 2)] = Z2{//] and define a map

t)a :F* — > Z[H] C An by ijQ(y) = £ ordfl(y(1“T,fc) / r l , /.€// and put

{V)a = Va{y)& € (y € F x).

Also define

70 : F*KF*)» — . A„/A/A„ by >ioM = £ ordQ(U,<,- ’l,')/r' h£H and

H o = 7o (w)Q € ^ q /A /^ o («> G ^ /(F ’*)").

Lemma 3.4.1 Suppose Q is a prime of F and y € F*. Then

(i) (l/)Q». = (y)o for all h € //, and

(ii) 0/)o' = - r ( y )Q.

Proof. Easy exercise. □

Recall that An denotes the 2-primary part of the ideal class group of F = Kn.

For y € F* and a prime q of K we define cy,q 6 A n to be the projection of the class

°f (y)a into A„, where JQ is any prime of F lying above q. The above lemma and the fact that (1 + r)An — 0 (Corollary 3.1.5(ii)) show that cyiQ is in fact independent of the choice of Q.

Proposition 3,4.2 Suppose y € F*. Then

CVi« = 0* q 31

Proof. For each prime q of K fix a prime q of F lying above q, and let G =

Gal(F/K) = / / U tH. It is not difficult to sec, using Corollary 3.1.5(ii), that cv,„ is the projection of the class of E jeoord^(yff)^- ,q in An> The proposition follows from this since SqX ^gcord^y^-'q ^1C principal ideal generated by y. □

Recall the set ££ of primes of K defined in §3.3, and let £ be a prime of F lying above q € .2. Note that in this case S q is free over Z[//J. Wc will now define a logarithm modulo £ map similar to the logarithm map

Recall Gq s= Gal(F(q)fF) where /'’(q) is given by Lemma 3.3.2. Let C?r(q) (resp.

Op) denote the ring of integers of ^(q) (resp. F). The prime £ is totally ramified in F(q)/F\ let £ denote the prime of F(q) lying above £ . If x 6 F(q)x and a 6 Gq then x,_

Op/Q. cz Op (q)/£ . Wc have a pairing

< , >=< , >0 : F(q)xxGq — ► ((C>/i'/£)x)d, (ar,

So ■■ m * — (Or/ar/((oF/ay)u, z 'id where o-q is the generator of Gq fixed previously and < x,Oq > denotes the unique d-th root in (Qp/Q)* /({Op/Q)*)M of '< x,trq > =< x,oq > mod ({Op/Q)*)M.

Since £ is tamely ramified in F(q)/F1 the higher ramification groups of £ in this extension are trivial, so the above pairing is non-degenerate in the second variable.

This is easily seen to imply that the map f Q is surjective. Further, the pairing is 32 trivial on { ( x ,^ ) : ordg(x) = 0}, which shows that

kcr(/o) = {s € ir(q)x : orda(x) = 0 (mod M)}.

Thus wc obtain an isomorphism logQ : (Op/Q)x/((O f/£1)*)m- ^ Z /M Z by setting

togfl(u>) = ordfi(x) (mod M) if w = f Q(x).

Let JtfQ = (Bhen(&F/0.h)x/({0p/0.h)*)M. Wc will also write loga for the composi­ tion

J f Q 2 $ ( 0 F/ 0 ) X/{{Of/H) x)m ZfM Z where prQ is the natural projection map.

Now define a logarithm modulo 0 map

y^fM yQ by

)) = log0(tul"T) for every w G «#q. We will also writetpQ for the induced homomorphism

Vo : {u> G F x/(F x)m : [ui]0 = 0} — * S Q/M S Q.

Proposition 3.4.3 Suppose a € ^ f > k = Ka is the map defined in Proposition 3,3.5,

0 ^ 1 is in S?(a), and 0 is a prime of F lying above the prime q of K.

(>) / / then l*(«)]o - 0.

(ii) If q|a, //ten [«(a)]Q = y>o(K(a/q)).

Proof. Both assertions follow easily from Proposition 2.4 of [23], □, 33

3.5 Statement of the main conjecture.

Recall p(2, Kn = K(EP«) for 1 < n < oo, # = G a l ^ / I Q and T = G a l ^ f l U ) .

Observe that

Gal{l<2fJ<) x T ~ Z/2Z x Z2.

In the next two sections we will adapt the methods developed by Rubin in §§5-7 of

[23] to prove a number of results from Iwasawa theory for the extension K^ojK. These results will be used in §3.8 to prove an analogue of Theorem 8.3 of [23]. Establishing this analogue is an important first step towards proving the main conjecture for

K0ofI(\ the form of the main conjecture that wc shall prove is Theorem 3.5.1 below.

Recall that, for 1 < n < oo, Un is the group of local units of 4>n = Kn ®k

An denotes the 2-primary part of the ideal class group of I(n, and is the group of elliptic units of Kn defined in §3.2. It follows from Proposition 3.2.1 (iii) that

C Let denote the group of global units of A'„, and write

Uoo = Inn £/„, Aoo = lhn A„, If*, = and = Hm

M„ denote the maximal abelian 2-cxtension of Kn which is unramified away from the prime above p and write A'n = Gat(Mn/K n)<

Each of A *, A'oo, Cf<»»

0 — (foo/^oo — ♦ tW ^o o — ► — * Aoo — ♦ 0. (3.7) 34

In this work, however, wc will always view Ac,, X W) Voo, <£» and oo as modules over the standard Iwasawa algebra

A = Za|(r]] = lira Z2[Gal(Kn/K 2)), inverse limit over n > 2. Doing so will enable us to apply the well-known classification theorem for Z3j[!T]j-niodulcs (T an indeterminate).

Let Y be a finitely generated torsion A-modulc. It follows from the classification theorem alluded to above that wc can find elements /,• € A such that there is an exact sequence

0 — ♦ ©A/7iA — ► Y — ► Z — ► 0 with Z finite. We will write char(K) for (FI/i)A, the characteristic ideal of Y.

We will see below that /l n | A'*,, £/„>, ^ and Sfoo are alt finitely generated A- modulcs, and A',*, and £/oo/^oo arc torsion A-modules (hence so are /!«> and , by (3.7)). The main result of this chapter is the following theorem (recall that r denotes the clement of & that acts as multiplication by —1 on Epoc):

T heorem 3.5.1 (the “main conjecture” for Keo/K) Wc have

char(A'oo) = char((l - r)t/oo/(l - t) ^ ) and

char(Aoo) = char((l - r) £ o /( l - r ) ^ ) .

(Later we will see that these identities are in fact equivalent.) 35

Theorem 3.5.1 will be proved in §3.10, using the divisibility relation of §3.8

(Theorem 3.8.5 below) and the fact, to be proved in §§3.9 and 3.10, that Xa, and

(1 — t )Uoo /( 1 - rj'ifoo have the same Iwasawa invariants.

3.6 Results from Iwasawa theory.

Keep the notation of §3.5, so F = G?a/( A'oo//^) 2; Z3. For 2 < n < 00 define «/„ to be the ideal of A generated by {7 — 1 : 7 G Ga/fA’oo/A’n)}. We will also write S(V) for J?2 , the augmentation ideal of A. Further, if Y is a A-modulc define

KW = {y G Y : 71/ = V for all 7 G Gal(KmIKn)} C Y and

y (n ) = Y fS nY.

Now fix n with 2 < n < 00. In this section wc will study the A-modulc structure of Xoo, A 00 and £/«> and the kernels and cokcrncls of the natural projection maps

7TX : A'oo(n) — ► X n, 7rA : /!«(») — ►A„ and irv : Um(n) — ► t/„.

Lemma 3.6.1 Suppose we have an exact sequence of A-modules

0 — >Y — ► Z — ►IV — » 0.

Then there is an exact sequence

0 —♦ y(") —♦ z(«) _* _♦ !'(„) —* Z(n) — * W(n) — * 0.

Proof. Choose a topological generator 7 of Gal(Koo/Kn) and use the given exact sequence as top and bottom rows of a commutative diagram with vertical maps equal to multiplication by 7 — 1. Then apply the snake lemma. □ 36

Remark 3.6.2 For later use, we call attention to the fact that the p-adic analogue of Leopotdt’s conjecture is valid for Kn i.e.

Theorem 3.6.3 (i) ker(7r*) — 0 anti ./(r)coker(jrx) = 0.

(ii) Xoo(n) is finite.

(iii) Xoo is a finitely generated torsion A-modulc.

(iv) A'*, has no non-zero finite submodules.

Proof. Recall that Mn (resp. A/,») is the maximal abelian 2*cxtcnsion of Kn (resp.

Koo) which is unramificd outside of p, X n = Gal(M„/Kn) and X& = Gal(Moof Koo)'

Since n > 2, Lemma 2.1.4 shows that K00/K n is unramificd outside of p. Thus Mn is the maximal abelian extension of K n in so J ^ X ^ = Gal(Moo/Mn) (see [18]

§3.1). Thus we have an exact sequence

0 —. A'„(n) M . X n —. Gal(K„/Kn) —. 0.

Assertion (i) is now clear.

Class Held theory together with Lcopoldt’s conjecture gives

Z2-rank(A'„) = Z2-rank(t/n/

This and the above exact sequence prove (ii). Statement (iii) follows from (ii) and

Nakayama’s Lemma, and (iv) was proved by Greenberg ([15], Proposition 3 and the comments at the end of §4). □

Proposition 3.6.4 We have

(i) A '& U O , and

(ii) char( A'«>) is prime to S n. 37

Proof. Since Xoo is finitely generated over A and Xoo(n) is finite (Theorem 3.6.3),

is finite. Thus = 0, by Theorem 3.6.3(iv).

Applying Lemma 3.6.1 to an exact sequence

0 — ♦ ©A//,A — ♦ Xoo — + Z — ♦ 0, with Z finite, gives (since = 0)

0 — . Z<’ > m /i-fr. + /(A) — *„(n).

Thus © A /t^n + /,A) is finite, and (ii) follows. □

T heorem 3.6.5 (i) The projection map irA is an isomorphism.

(ii) Aoo is a finitely generated torsion A-modulc.

(iii) char{Aoo) is prime to y n.

(iv) A*£} is finite.

Proof. Let Ln (reap. Loo) denote the maximal abelian 2-cxtcnsion of A'„ (resp. K ^)

# which is unramificd everywhere. Since Lnlioo is the maximal abelian extension of

Kn in Loot we have J?nAoo ~ Gal(Loo/LnKoo). Thus ker(^) = 0 and cokcr(7T*) =

Gal(Koo n LnfK n). Now Kao H Ln = Kn% because Keo/Kn is totally ramified at p, and (i) follows.

Since Aoo is a quotient of A'oo» assertions (ii) and (iii) follow from Theorem

3.6.3(iii) and Proposition 3.6.4(ii), respectively. The proof of (iv) is identical to the proof of Proposition 3.6.4(i), if we use the fact that Aoo(n) ~ A n is finite. □

T h eo rem 3.6.6 (i) is a torsion-free A-module of rank 2.

(ii) */(r)ker(irv) = 0 and J?(T)coker(irv) = 0. 38

Proof. For all m > 2 write = Km ®k Kp and let Zm — Gal(Nm/$ m), where Nm is the maximal abelian 2-cxtension of $ m. Local class Held theory allows us to view Uoo as a submodule of Zm, and it is not difficult to sec that T acts trivially on Zoo/Uoo and the projection map nz,u : Zoo/Uoo -♦ Zn/Un is an isomorphism (cf. [18] §12.1).

The A-modulc structure of Zm was determined by Iwasawa: since /x3«($oo) is finite

(Lemma 3.1.3(iii)), Zm is a torsion-free A-modulc of rank [$2 : Q2] — 2. Further, kcr(rrz) = 0 and «^(r)cokcr(7rz) = 0. Sec [18] §12.2. Assertion (i) follows from the above. Now consider the diagram with exact rows

0 -Zoo/tfe* - Voc{n) -Zooin) -Zoo/t/oo -0

Kv nz KzfU

0 O n Z n/Un 0 in which the top row comes from the exact sequence 0 —i Uoo —+ Zoo Z o o /U 0 a n d L e m m a 3.6.1. Applying the snake lem m a to this diagram yields

ker(7Tu) ss Z oojU oo and coker(7Tu) = cokcr(7rz).

Assertion (ii) is now clear. □

We now let n v a ry , 2 < n < 0 0 , and recall

An = Zi[Gal(Kn/K 2 )]. (3.8)

Proposition 3.6.7 Fix etements /i,...,/* G A so that there is an injective map from

©*=1 A//,-A to with finite cokemel. Then there is an ideal 3S of finite index in A with the following property: for every n, 2 < n < oo, there are classes Ci,...,Cjt € A n 39 such that for every i the annihilaior s/i C An of c,- in /ln/(A„Ci + ...+ A„C,_i) satisfies

®sfi C fi

Proof. This is similar to Proposition 6.5 of [23]. It can be proved exactly as in [23], using the fact that the projection map nA is an isomorphism (Theorem 3.6.5(i)). □

3.7 Iwasawa modules of global and elliptic units.

Keep the notation of §3.6. Further, extend definition (3.8) above to n = oo by setting

A oo = A, and recall that r = t,*, is the clement of Gal(Koo/K) which acts as multiplication by —1 on Ep«> and r„ is the restriction of r to Kn (2 < n < oo). If Y is a An-modulc on which Gal(K„/K) acts (e.g. Sn ) wc define

Y~ = {!/ € Y : rny = -j/} and Y + = {y € Y : Tny = y).

Both Y~ and V'+ arc An-submodulcs of Y , and (1 ±r„)V' C Y ± (because = 1). In addition, Y ±/{ 1 i r ,,)}7 is anniliilatcd by 2.

Now fix n with 2 < n < oo. In this section we will study the projection maps

^d 7Ty:^(n)—►

Lemma 3.7.1 We have

(i) (£/«,/*.)<"> = 0, and

(I*) ^U&OQ ft ^gg —

Proof, (i) is immediate from Proposition 3.6.4(i) and the inclusion £/oo/«S» C A'«, of global class field theory. Assertion (ii) follows easily from the fact that

Theorem 3.7.2 (i) ^(r)ker(ff^) = 0.

(ii) There is an ideal SS of finite index in A, independent of n, such that

2^(r)3S cokcr(7i£) = 0.

Proof. Let TTj,: <£o(n) —» Sn be the usual projection map. By Lemma 3.7.1(ii) we may view

Consider the diagram with exact rows

0 ------*(/„(«)------'(*/oo/<£o)(n) 0

Xu/S

Un Un/#n ■0 in which the top row comes from the exact sequence 0 —»-♦£/«, -* t/oo/d*oot/co /4 —♦ 0 and Lemmas 3.6.1 and 3.7.1(i). Applying the snake lemma to this diagram gives an exact sequence

0 -+ kcr(7Tj.) -* kerfTTy) -+ ker(7ry^ ) -♦ cokcr(fy) -+ coker (tt^) —► cokerfTr^) -+ 0.

(3.9)

In particular ker(ji£) injects into ker(TTy), so assertion (i) follows from Theorem

3.6.6(ii). Now consider the diagram where the top row comes from the exact sequence 0 —* Uoal&oa —♦ Xao —» /loo 0,

Lemma 3.6.1 and Proposition 3.6.4(i). Since irx is injective (Theorem 3.6.3(i)), the above diagram gives

ker(7rtr/

It then follows from the snakc-lcmma exact sequence (3.9) and Theorems 3.6.6(ii) and 3.6.5(iv) that

^(rj^cokerfj^) = 0 where SB is the annihilator in A of the maximal finite submodule of /loo. Then, by the observation at the beginning of the proof, 2Jt{V)SB annihilates coker(a£). □

L em m a 3.7.3 (i) is finite.

(ii) There is an exact sequence of An~modulcs

o — . { ± 1} — > # ; — >y — .o where Y is a submodule of finite index in An.

Proof. Recall the group Cpn C <5*+ of elliptic units of conductor p" defined in Remark

3.2.2. As observed there, contains Gillard's group of elliptic units for Knt so

KCpn has finite index in

We now prove (ii). Put G = Gat(Kn/K) and write I(C?) for the augmentation ideal of Z[G], It is easy to see that 1(G) ® zZ2 = (1 — rn)A„ + ^ ( r ) A n* Also, there is a G-homomorphism log :

(fn ®zK - 1(G) 0z K* By a standard argument from the theory of representations of finite groups, we conclude that there is an isomorphism t^n ®z Q — 1(C») ®z Q and, consequently, an exact sequence of Z[G]-modulcs

0 — + {±1} Z — * 0 where Z is a submodule of finite index in 1(G). Tcnsoring the above exact sequence with Z2 and then taking “minus parts” yields an exact sequence of An-modulcs

0 — ♦ { ± 1} — > — *Y — ► 0 where Y is a submodule of finite index in (1(G) ® zZ2)“ = (1 — t„)A„. Assertion (ii) follows from this since (1 — r„)An — An (as An-modulcs). □

Theorem 3.7.4 (i) ranka^oo) = 2, rankaf^) = 1, and ^ /(l - r)%, is finite.

(ii) 2coker(?r^) = 0 and ker(7r£) = 0.

Proof. It follows from Proposition II 1.1.4 of (9] that there is an isomorphism of

Z2[[jy]]-modules

tfoo ^ J{n ) where *^(/i) is the annihilator in Z2[[C?)] of /i3«>(/,f00) (see also Proposition 3.10.1 below). Since /i2<»(/f,00) = {±1}, S(fi) = (1 - r)A + 2A + ^(T), so we have A- module isomorphisms

JT(n) s A © (2A + J^(r)), *^(/i)" = (1 — r)A c- A, and 43

(1 - t)^(/x) = (1 - r)(2A + J^(r)) ~ 2A + J{T).

Noting that A/(2A + *?{T)) m Z /2Z, assertion (i) follows from the above.

Recall the subgroup Cn of *ifn defined in §3.2 and let Coo = Hm Cn, inverse limit with respect to the norm maps. Then *<#„ = {±l}Cn and — Coo • Now Proposition

3.2.1(iii) shows that the projection map from Coo(n) to Cn is surjcctivc, and assertion

(ii) for coker(;r£) follows without difficulty.

Now, by (i), ^ ( n ) — A„. Further, Lemma 3.7.3 gives us an exact sequence

0 { ± 1} — * K 0 where Y is a submodule of finite index in An. Define a map tp : A„ -+ Y so that the following diagram commutes

0--- * V ~ (n ) ------A„---- - 0

%< s . 9

{±1} o. Clearly, kcr(7r£) C ker(^) and coker(y>) is a quotient of cokcr(7r£). Then 2cokcr(^) = 0 so cokcr(p) is finite, and it follows that ker(y>) is finite. Since A„ has no non-zero finite submodules, this completes the proof of the theorem. P

C orollary 3.7.5 (i) //«>/^oo is a torsion A-modufe.

(ii) is a torsion A-moduie, and rankA(<^) = 1.

(iii) char(d£/^) is prime to

Proof, Statements (i) and (ii) follow easily from Theorems 3.6.G(i) and 3.7.4(i). Now 44 since is torsion-free, Lemma 3.6.1 applied to the exact sequence

o — V - —.<£ —.<£/*£—. o shows that injects into k c r(^ ). Thus Theorem 3.7.4(ii) gives

( 4 ' / ' 0 <"1 = o.

Now we can proceed exactly as in the proof of Proposition 3.6.4 to obtain (iii). □

Corollary 3.7.6 Let 3d C A be the ideal satisfying Theorem 3.7.2(H). Then for every

A 6 2 JP(T)3d there is a map 0nt\ : 8 “ -+ A„ such that

AJchar(<£/tf-)An C O ^ V ') .

Proof. This is similar to Corollary 7.10 of [23]. In adapting the proof given in [23] one uses Theorem 3.7.2 and the fact that rankA(dj£) = 1 (Corollary 3.7.5(H)). □

3.8 A divisibility relation.

In this section we will use the methods of Kolyvagin (as developed by Rubin in [23]) to study the structure of the ideal class group An for 2 < n < oo.

Fix n with 2 < n < oo and let F = Kn and Ii = Gal(FlKi). Thus A„ = Zjf//].

Further, fix a power M of 2, M > 2.

Lemma 3.8.1 Suppose A 6 An and m > 1 is an integer. Then

AA„/AmAn is finite. 45

Proof, Write // for the character group of // = Gal(Ff Kf). Every x £ H extends

by linearity to a homomorphism from An = Z2(//] to Z2[/*2»-*]* Define, for every

A <= A„, Z(A) = {X 6 : x(A) = 0}. Then 2(A) = 2 (Am) and

Z2-rank(An/AA„) = #2(A). □

Lemma 3.8.2 Lei p : F*/(F*)M —♦ F(fi2M)* f(F (p 2M)*)M ^ie natural map.

Then

4 kcr(p) = 0.

Froo/, If we identify F xf(F*)M with //'(F /F , *iM) (resp. for F(/t2M)) then p cor­

responds to the restriction map on cohomology. Thus kcr(p) = H X{F(H2 M)( F,itM).

Now, by Lcmma3.1.3(ii), Gal{F{n 2 M)l F) ~ (Z/2A/Z)x.Thcn theinfiation-rcstriction exact sequence gives

where C7_j (resp. C^) is the subgroup of (Z/2A/Z)* generated by —1 mod2A/ (resp. * 5mod2Af). A straightforward computation shows that //,((7_i,/i4) ~ a

Z/2Z, and the lemma follows. □

Theorem 3.8.3 Suppose W is a finite A„-submodule o/(F*/(Fx)M)~, c 6 2«/(r)A„,

and ij) : W —♦ An/A/An is an H-homomorphism. Then there areinfinitely many primes 0 of F of degree one such that

(i) the projection of the ideal class of Q into An is c,

(ii) N(q) = 1 (mod2A/), where q is the prime of K below Q, and 46

(iii) [u>]Q = 0 for all w € W, and there is a 11 € (Z/M Z)X such that

¥>a(to) = 16u^t{to)0 for all w € W, where [•](, and tpQ arc as defined in %3.f.

Proof, Write F' = F{^,2m ) and ^ = W/IV fl ker(/?), where p is the map of Lemma

3.8.2. Further, fix a primitive A/-th root of unity Cm and define a map

1 : A„/A/An — ► fiM by i(l) = Cm and t(h) — 1 for h £ //, h £ 1.

Kummer theory gives an isomorphism

Gal{F\W'>M)/F’) a Hom(lV„ M„), (3.10) and Lemma 3.8.2 shows that / rf= 8(to ^ ) £ 2 Hom(WC>,/**#)• Let 7 denote the element of 2Gal(F'(WlfM)JF') which corresponds to / under the isomorphism (3.10). Thus

f(w) = for all tn € W,

Now let L be the maximal abelian 2-cxtcnsion of F which is unramified every­ where. Thus we may identify Gal(L/F) with An> Write L‘ = LC\ F ‘ and L" =

L H F\\V*Im). We have the following diagram; 47

f '

l ': F H

I<2

K

By (3.10) there is a subgroup V of Wp such that

Gal{L"/L’) = C a/(iV /F ') ^ Hom(K,/iAf).

Since L" is abelian over Fy Gal(F'fF) acts trivially on Gal(L"F'/ V'1’) , s o

liom(V',M„) = Horn {V,nu F * f,/F> = H on.(V >2)

(recall that fi2«>(F) = //2). Thus 2Gal(L"/L') = 0, so 7 is trivial on L”. Farther, since L' is abelian over I(, ^(T) annihilates Gal(L'/F), so c 6 Gal(L/L"), It follows that we can choose an element <7 € Gal(LF‘ F) such that

o'l/r'piM/M) = 7 and a\L = c.

Let Q be any prime of F of degree one, unramified in F,(W1fM)jK } whose FYobenius in Ga^LF'flV'^^/F) is the conjugacy class of a, (Since W is finite, the Chebotarev density theorem guarantees the existence of infinitely many such primes.) We will show that £2 satisfies (i), (ii) and (iii). 48

Under the identification of A n with Gal(L/F), the class of 0 corresponds to the

Frobenius of 0 , so (i) is immediate. Further, since ff is trivial on L' and 0 is a prime of degree one, q splits completely in F(h 2m )/K i which proves (ii). The first assertion of (iii) holds because all primes above q are unramificd in F'{WxlM)l F and every to € W is an M-tli power in F \W x^m).

Now observe that

ordo(160(tw)0) =0 <=> 16(t o 0 (u>)) = /(u>2) = 1 <£* 7({uj2}1/a/) = {\o2}x^M

to2 is an A/-th power modulo 0 .

Also, from the definition of

ordo(^o(tij)) = 0 to2 is an A/-th power modulo 0 .

Consequently, there is a u 6 {Z/MZ)K such that

ord0(y?o(u>)) = 16uordo(^(tu)0) for all to € W.

It follows that the map

to *-*

Recall the set J2 of primes of K defined in §3.3. Also recall the maps t]Q and ijQ and the ideal classes cy,q € An defined in §3.4, where 0 is a prime of F} q is any prime of K , and y € F*, 49

Lemma 3.8.4 Suppose w € F*/(F*)M, go € Oo is a prime of F above qo, S is a set of primes of I( not containing q0, and f, Aj, A2 € A. Write B for the subgroup of

An generated by the primes of F lying above primes in S , C„ € An for the projection of the class of jQo and W for the An~submodule of F*f(F*)M generated by wl~T.

Assume

(i) [uj]a = 0 if 0 does not lie above a prime in S U {qo},

(ii) the annihilator stf C An of c0 in A n/B satisfies Ajj/C /An,

(iii) # A n\M, and ^oo(U)) divides (A f/#A n)A2 in An/M Ant and

(iv) /A is prime to J ’n.

Then there is an H-equivarianl map tp :W —* A„/A/An such that

ftp(wx~T) = 2Aj A 2 7o0(ui).

Proof. Let y € F* be any representative of to. By Proposition 3.4.2

cv.qo = — S (niotl B). q({Su{qo}

Now by (i) and the first condition of (iii), cV|() € M An = 0 for q S U {qo}. It follows that = »7oo(l/)co G B t so by (ii) Aiq0o(y) € /A „. Fix a 6 € An such that

6 f = AiqO0(y). Define tp : W -+ An/M A n by

tp ^u/l-Tlc) = 2#A2£ for allq € Z[H],

This map has the desired property, but we must check that it is well-defined. Suppose u,(j-r)c _ i c y(i-r)e _ XM wjtli x e F*. Then grjao(w) — Uoo(we) = 50 by (iii) eh(M/#A„)A„ C M An. Therefore f>A2An = 0. 50

Now from the definition of cr,, and the fact that = y2(,-r)^

C*,q = 2M * = 2q M * Cy,q*

Then, using Proposition 3.4.2,

2A jA / fj0o(j/P)c„ = 2A jA / Cv£ltflo ~ ^2C*,q0 = “ Aj ^ C^q q«Su{q0) = ~2pA2 A/_,cVlq = 0 (mod ^), q(SSu{q0) because pA 3An = 0. By (ii) it follows that 2AjA3i7O0(yf) = 2pA 3Sf € A//A„, and then (iv) gives

0 (u>*,“T)ff) = 2pAaJ 6 A/An. □

Fix elements /i, ...,/* € A so that ©{is, A//,A injects into A«, as a submodule of finite index. In particular,

char(Aoo) = ^11 /*)*•

Theorem 3.8.5 IVilii k os above,

clmr(Aoo) divides 2n+7 c h a r ( < ^ / ^ ).

Proof. See Theorem 8.3 of [23]. Fix a generator g € A of char( ^ ) , an ideal

# o f finite index in A satisfying both Proposition 3.6.7 and Theorem 3.7.2(H), and a

A € 2 . / ( r)38. Fix an n with 2 < n < oo and keep the notation introduced at the beginning of this section.

By Lemma 3.8.1 and the fact that gA is prime to (Corollary 3.7.5(iii)),

AAn/A3fe_,ffArt is finite. Thus for some power A/ of 2

A/AA„ c (2n( # A „ ) 2 5J'+ ,A3fc" 1

Now by Corollary 3.7.6 there is a map 0 — 0n<\ —> A„ such that A 2g € 0 (^“).

Then 2A 2g € 0((1 — Tn)^fn), and we fix ( £ so that 0(£'“f) = 2A 2g. Also fix an elliptic unit (0 € such that £ = & (mod (‘iff,)*1)*

By Proposition 3.3.4 we can choose an clement o € such that a (l) = £0. Let k = Ka be the map defined in Proposition 3.3.5. Further, let Ci,...,c& € An be as given by Proposition 3.6.7 and put Cfc+i = 0. As in [23], wc will use Theorem 3.8.3 inductively to select primes Jl2i,...,Ojt+i of F lying above primes qi,...,q*+i of K satisfying:

the class of £3; in An is Ac,-, and q,- 6 ^?(a), for 1 < i < k + 1 (3.11)

2qo,(«(qi)) =ui2°Aa3i and 2/i_1q0i(«(ai)) = u ,-25A3t;o._,(«(a»-i))

for 2 < t < + 1 (3.12) where a, = rij<» ui € (Z/A/Z)x, and the are as defined before the statement of the theorem.

For the first step take c — Aci € 2Jr(r)An, W f&~ H {F*)M, and t/> = 20 :

W -+ AnfM An (note that 0 does not necessarily map fl (F*)M into A/A„, but 20 docs). Let jQi be a prime of F satisfying Theorem 3.8.3 with the above data, lying above a prime qi € Then (i) and (ii) of Theorem 3.8.3 give (3.11) for i = 1. By

Proposition 3.4.3(ii) and Theorem 3.8.3(iii), for some «j 6 (Z/A/Z)x,

2Nq»)]o, = 2v>q,(k ( 1)) =

= 32ultf(^,- T)jQi = 52

Thus from the definition of [*]0l and the fact that is free over A„/A/Ani

2^Q|(«:(qi)) = ui 26A2<7, which is the first equality of (3.12).

Now suppose that 1 < i < k and that we have chosen 0 i ,. .. ,0 t- satisfying (3.11) and (3.12). We will define 0 ,+ i. Let a,- = n j,•)) divides 2S,+1A3*"1^ in A„/A/An, so by definition of A/, ijat(/c(a,)) divides (M /# A n)\ in A„/A/An.

Let Wi be the An-submodulc of F X/(F X)M generated by /i(o,),“r. Using Propo­ sition 3.4.3(i), Proposition 3.6.7 and Theorem 3.6.5(iii), we can apply Lemma 3.8.4 with w - «(<»,■), 0o = 0 ,-,S = {qi,...,q,-i}, / = TV, Aj = A3, and A2 = A to obtain a map : Wi -+ A„/A/An such that

/it&i(/c(fli),_T) = 2A3^ (« (a ,)).

Now choose 0 (+I satisfying Theorem 3.8.3 with W — Wi, c = Acl+i, and tp = tpit lying above a prime q;+1 € &{ot). Then (i) and (ii) of Theorem 3.8.3 give (3.11) for t + 1. By Proposition 3.4.3(ii) and Theorem 3.8.3(iii), for some ul+J 6 (Z/M Z)X,

2/i[/i(a;+i)]0(+I = 2 /,-^ + , {«(«<)) =

= 16ui+i/i0«(«(a,)I_r)0,+i = ui+,25A3j/Oj(K(oi))0l+i.

This proves (3.12) for i + 1.

Continue this induction process k *f 1 steps. Combining all the relations (3.12) gives

2*+ l( n / < ) W K(»wi)) = «2SH“A3‘+Js in A„/A/A„, 53 for some u € (Z/A/Z)x. Since this holds for every n and 2 n\M,

char ((jlfijh divides 2Ak+6\ 3k+2gA = 24fe+sA3*+3c h a r ( C / 0

The above holds for every A 6 2

Corollary 3.8.6 There is an integer m > 0 such that

char(Aoo) divides 2mchar((l - t)«£»/(1 - t ) ^ ) .

Proof. Consider the exact sequences

0 —» - r)^ —♦ - r)^ — £-!<#- — 0 and

0 —* (1 — t)^ o /( 1 - r)SPoo — « £ /(l - r)5foo — <6/(1 -r ) ^ — ► 0.

Since - t)%„, is finite (Theorem 3.7.4(i)), < ^/( 1 - rjtfj*, is annihilated by 2, and the characteristic ideal is multiplicative on exact sequences, we have

c M C / t f j ) = 2 0. Thus, if k is the integer of Theorem 3.8.5, m = / + 7k + 7 satisfies the conclusion of the corollary. P 54

3.9 The Iwasawa invariants of X^.

Later in this section we will compute, using ideas of Coates and Wiles [7], the Iwasawa invariants of A'oo. But first we wish to show that the two equalities in the statement of the main conjecture (Theorem 3.5.1) arc equivalent, and also to prove a divisibility result similar to Corollary 3.8.6 (sec Proposition 3.9.2 below).

Recall that is the fixed field of (r) in KM. Let M(K+) be the maximal abelian 2-cxtcnsion of K + which is unramificd away from the prime above p, and write X(JT+) = Gal{M{K+)/I<+).

Lemma 3.9.1 We have

(i) X(K+) = 0, and

0 0 V* = C

Proof, (ii) is immediate from (i) and the inclusion C A'(A’i) of global class field theory. Now identify P = G a^K ^/K f) with Gal(K^/K). We have

X(K+)/S(r)X(I<+) = Gal(Mi/I<+) where A/i is the maximal abelian extension of K in Af (/££). But /<+ =K{p°°) is the maximal abelian 2*extcnsion of K which is unramificd outside of {p}, so M\ — 7f+ and therefore A '(7^)/./(r)A'(7f'+) = 0. This and Nakayama’s lemma prove (i). □

P rop o sitio n 3.9.2 (i) The two equalities in the statement of the main conjecture are equivalent. That is,

charf/loo) = char((l - r ^ / f l - r)^*,) 55 if and only if

charfXoo) — char((l - r)LT0o/(l — r)^ * ).

(ii) Let m be the integer satisfying Corollary 3.8.6. Then

char(-Voo) divides 2mchar((l - r)£/oo/(l — r)^*,).

Proof. It follows from statement (ii) of the above lemma that the natural map

Moo/^oo -+ (1 — t)Uoo /(1 — t)£0o is an isomorphism. Thus global class field the­ ory gives us an exact sequence

0 —► (1 — r)«?oo/(l - rj'tfoo^ (1 - t)U oo/fl - t)Voo -+ 0, and therefore

char(/lM)char((l - t)Uoo /{1 - t)^oo) = charfA'ooJcharffl - r)

Assertions (i) and (ii) arc now clear. □

For the rest of this section we fix a generator / € A of charfA'oo). We will now compute the Iwasawa invariants of /. The argument that follows is standard (at least when p/2; see [9] §§111.2.2 IT) and depends on (analogues of) various results from [7].

Remark 3.9.3 Later (cf. end of §8.10) we will compute the Iwasawa invariants of

(1 — r)t/oo/(l — r)Sfoo. It will be seen that these invariants agree with the invariants of A'oo that arc computed below. This fact together with the divisibility relation of

Proposition 3.9.2(H) will enable us to prove the main conjecture.

For every /i € A we will write p(h) and A(/i) for the usual Iwasawa invariants of h i.e. p[h) is the largest non-negative integer such that 2"^ divides h and X(h) 56 is the degree of the “distinguished polynomial” part of h given by the Weicrstrass preparation theorem.

Lemma 3.9.4 For all sufficiently large n,

ord,(#-Mn)) = + X(/)n + c where c is an integer independent of n.

Proof. Sec, for example, [30] §13.3. □

Now for every n > 1 let Cn denote the idclc class group of Kn. We will view t/n, the local units of 4>n = I(n Kp , as a subgroup of Cn in the usual way. Farther, if LfH is an abelian extension of local or global fields, we will denote the Artin map for LfH by (•,£///).

Define, for each n > 1,

IK, = P) N/,'n+i//,*B(C,„+i) C Cn. i> 0 Lemma 3.9.5 Suppose n > 2. Then

Yn nf/„ - ker(N^n//,'p|t;n).

Proof (notation as in §SJ). Suppose first that £ € Un and N*„//,'p(£) = 1. Since n > 2, we have /(«, = K+Kn t so

(f, KJK.) |K* = (N«„/k,K).K*/K) = 1.

Therefore (£, KtnfKn) = 1 whence ( € Yn. Conversely, let £ € Yn C\Un. Since /(„+,• =

K*+if

{N*n/*p(£)t K+"IK) = <£, Kn+i/K n)\KUt = 1. 57

It follows that (N*n/fi-„{£)* * tn /K p) = 1 so N#„//<-,,(£) 6 {±l}t/p"+,\ by Proposition

3.1.6(ii). Since this holds for all t > 0 we have 6 {±1). Also, N*n/jfp(f) €

£/pn\ by Proposition 3.1.6(i). Thus, since n > 2, N$„/k,,(£) = 1. □

Remark 3.9.6 //£€ 2 we actually have

Nk„/k(£) = 1. Thus, by Lemma 3.9.5, 2.

Write C2 for the completion of Q2 wM respect to the 2-adic absolute value, and let log : C2 *-+ C2 be the usual 2-adic logarithm. Farther, let On denote the ring of integers of 4>n, write cn = ord2{# M2a°(4’n))t and choose an integer tn > 0 such that

2“‘"On D log Un.

Lemma 3.9.7 For all n > 1,

ord2((2“‘"On : Iog£/nJ) = tn2n~l + cn + I.

Proof. See (7), Lemma 7. □

Let Rp(Kn) denote the p-adic regulator of the full group of units of Kn, as defined in §111.2.3 of (9), and let A (K„/K) denote the relative discriminant of KnjK. We note that Rp{Kn) ^ 0, because Leopoldt’s conjecture is valid for K„, Now recall that C/p3* is the subgroup of the local units of Kp which are = 1 (mod p1), and define D„ = Ujpf, c U„.

Lemma 3,9.8 ([7], Lemma 8) Suppose n > 1. Then the index of log Dn in log Un is finite, and

ordjfllogt/, : logfl„]) = ordp + n - e„. 58

Proof. Using arguments analogous to those of [7] (proof of Lemma 8), one can show

that

ord2([2",nO„ : log A ,)) - ordp -f- f„2n"1 + n - 1 + ord2(log(5)).

Noting that ord2(log(5)) = 2, Lemma 3.9.8 now follows from Lemma 3.9.7. □

Lemma 3.9.0 For all n > 1,

ord,([t/„ : D„|) = ord, + n - 1.

Proof. This follows easily from Lemma 3.9.8. See [7], Lemma 9. □

Recall that Sn C Yn H Un for each n > 2 (see Remark 3.9.6).

Lemma 3.9.10 Suppose n > 2. Then

ord,([K„ n V. : &1) = ordp (lt,{l<„)/jA(l<„/l<)) + '■ - 2-

Proof. Recall that Dn = Uj^Sn. Then kcrfN ^/a-ploJ = (cf. Remark 3.9.6) and

N*n/^p(D„) = ((/<’>)’ ” ^pn+1** Thus we have a commutative diagram

0 Sn ------Dn .0

0 ------Yn fl-Un------»Un f/W------.0 where the bottom row comes from Lemma 3.9.5 and Proposition 3.1.6(i), and the vertical maps are the inclusion maps. Applying the snake lemma to the above diagram and noting that [Up1^ : £/pn+^] 2, the assertion of the lemma follows from Lemma

3.9.9. □ 59

Let Mn denote, as before, the maximal abelian 2-extension of Kn which is un-

ramified away from the prime above p. Further, let h(I(n) denote the class number of

Kn.

Proposition 3.9.11 Suppose n > 2. Then

ord,([Af„ : /f„]) = ordp + n - 2.

Proof. Write Ln for the maximal unramificd extension of Kn in Mn♦ and let An denote, as usual, the 2-primary part of the ideal class group of Kn. Thus we may identify Gal{LnJKn) with An.

Since K ^fK ^ is totally ramified at the prime above p, we have Ln H K<» = I(n.

In addition, class field theory gives an isomorphism

Yn r\Unl# n ~Gal(M nILnKo0).

Thus we have an exact sequence

0 — *Ynn Unf

We can now compute the Iwasawa invariants of A'oo* For brevity, let

= h[Kn)Rf(K.)/y/A(K„/K).

Corollary 3.9.12 For all sufficiently large n,

M r - 1 + x u ) = i + ordp . 60

Proof. By Proposition 3.9.11 the right-hand side of the above formula is equal to ord 2([A/n+i •' Koo)/[Mn : /C ,)). Noting that G'a/(A/n//f OC)) = ^foo(n) (see the proof of Theorem 3.6.3), the proposition follows from Lemma 3.9.4. □

3.10 The Iwasawa invariants of the p~adic L-function. Proof of the main conjecture.

In this section we will adapt arguments from Chapter III of [9] to compute the Iwasawa invariants of (1 — r)t/oo/(l — ^)^oo- As we shall sec, these invariants agree with the corresponding invariants of A'<» computed above. This fact and the divisibility relation of Proposition 3.9.2(ii) show that

char(A'oo) = char((l - T)Um /( 1 - t ) ^ ) .

This together with statement (i) of Proposition 3.9,2 proves Theorem 3.5.1, the main conjecture for the extension Koo/K.

Recall that f denotes the conductor of the Hcckc character attached to E. For

1 < n < oo put Fn = /f(fpn) and let Un be the group of local units of Fn ®k ^p which arc congruent to 1 modulo the primes above p. Define

W(|) = limWnt inverse limit with respect to the norm maps. Further, let t?(f) = Gal(Fao/K) where

— Un>i F*n, write D for the ring of integers of the completion of the maximal unramified extension of Qa, and let i(f) : U{f) -* D[[£7(f)]] be the injective (7(f)- liomomorphism defined in §§11.4.6 and 4.7 of [9]. We note that D is flat over Zj. 61

Recall th at &= G a l^ fK ) and Fro D K Write res : D[[0(f)]] -> D[[#]] for the map induced by the restriction map

m D [|g(f)n

N r . /Ko res

t / »------!— D |[0]|

Clearly, t is an injective ^-homomorphism. We will also write i for the extension by linearity of i to the completed tensor product Uoo ®Zj D.

Let ^ (/i)p (resp. ./(/*)) denote the annihilator of fa-Vtoo&h'Kp) (resp. in Za[[iy]], and define m(f) = rcs(/i(f)) where /i(f) € D[[f?(f)]] is the 2-adic integral measure constructed in §11.4.12 of [9].

Proposition 3.10.1 The map i : Uoo®Zi D —» D[(^]] induces isomorphisms

(i) and

(ii) tfoo0z1D =;m (f)./(#i)D([tf]].

Proof. See [9] Propositions II1.1.3 and 1.4. □

Recall T = Gal{J

The above proposition has the following

Corollary 3,10.2 We have

diar((l - r)t/„/(l - r)'d„) D[(l']] = m(|)D|[r]|. 62

Proof. Define ideals Q Z2[[r]] by (1 — T)./(fi)p — (1 — r)^/and (1 — r)^(/i) =

(1 - r)$8. It is easy to see, using the fact that /i2»(A'oo ®K Kp) is finite (Lemma

3.1.3(iii)), that ^/D[(F]] and .5?D[[r]] arc ideals of height 2 in D([P]]. Now by Propo­ sition 3.10.1 and the flatness of D over Z2, there is an isomorphism of D[[r]]-moduIcs

{(1 - r)£W (l - r )*?«} ®Zj D cz ^D[[r]]/m(f)^D[[r]].

In particular, these modules have the same characteristic ideal in D[[P]], which is the assertion of the corollary. □

Now let = Gal(K^fK). Define a map j : (1 + t )Uoo -* D[[S?+]] so that the following diagram commutes:

t/„ S D[(»J|

1 + r res+

(1 + r)t/«—-—. D[[»+l]

Here rcs+ is the map induced by the restriction map —» ^ +. Clearly, j is an injective ^ +-homomorphism.

Proposition 3.10.3 The map j defined above induces an isomorphism

(1 + t )Uoo ®Zj D ~ •Jr(/i+)p D([(y+]] where S (fi+)p is the annihilator in Z2[[^+]] of /i2» { / ^ ®k Kp)>

Proof. This follows easily from Proposition 3.10.1(i). □

Recall that p|2 and p ^ p. Let W(p3), £7(p3), i(p3) and /i(p3) have the same meaning as W(f), 5(f), *(f) and p(f) (respectively), but for the modulus p3. Observe 63 that /<*+ = /^(p°°) C /f(p 2p°°). Let res*: D([(?(p2)]] —* D([Sf+]] be the map induced by the restriction map t?(p2) —»£f+, and define a “pseudo-measure’1 m(l) on by

m(l) = rcs‘(/i(p2))/(l - Frobp1) where Frobp is the Frobcnius of p in ^ +.

Now fix a topological generator 70 of S^+ ~ Z2. Then ( 70— l)m (l) € D [[^+]) (see

Theorem 11.4.12 of [9]). Moreover:

Lemma 3.10.4 (70 - l)m (l) is a unit of D[[5f+]].

Proof. Let N : W(p2) -+ be the norm map. Using Lemma III. 1.2(H) of [9] one can show that the following diagram commutes:

N - ‘((l + 7•){/„)■ ^ . D[[ff(pJ)]|

N res

(1 + t )Uoo ------*D[lSf+J)

Recall the group Cp» of elliptic units of K+ defined in Remark 3.2,2. Let Cp* denote the closure of Cp» in Un and define Cp<» = lim Cp*, inverse limit with respect to the norm maps. A computation based on part (iii) of Proposition 3.2.1 together with the above diagram shows that

j : {(1 + r)Uoo flCp-} 0 Z, D - (70 - l)m (l)D((«f+]|. (3.13)

On the other hand, by Remark 3.2.2(d), Lemma 3.9.1(H) and Lcopoldt’s conjecture,

Cp* = <££ = £/+, Thus by (3.13) and Proposition 3.10.3

(70 - l)m(l)DI(sr+]j D ✓ (/i+)pD[[<*+]], 64

But J r(/£+)p D[[^+]j is an idea) of ticight 2, so the above containment can hold only if (70 — l)m (l) is a unit of D[[Sf+]]. □

We now begin the computation of the Iwasawa invariants of (1 -r)t/oo/(l

The argument that follows is a straightforward adaptation of §§111.2.9-2.11 of [9].

For each n > 2 let I\, = Gal{Koof Fn) C P. A character of finite order p of F (with values in C£) is said to be of level n, written l(p) = n , if p(Fn) = 1 but p (rn_j) ^ 1.

We will also write p for the extension by linearity of p to D[[P]].

Fix a generator g 6 Za[[P)] of c h a r((l-r) f/«,/{l-r)'i!fO0), and recall that /x(y) and

A(<7) denote the Iwasawa invariants of g. Let orda be the valuation of C 3 normalized by orda{ 2) — 1.

Lemma 3.10.5 For all sufficiently large n,

orda ( I I P(s)) = /‘W n~3 + % )• W)=»+i /

Proof, See Lemma 111.2.9 of [9]. □

Proposition 3.10.6 For all sufficiently large n,

° rdi L s H =1 +or

Write ,4n+i for the set of characters of # = (t ) x T whose conductor is divisible by pn+l but not by any higher power of p. Every e € A n+j can be written as e — XP 65 where x is a character of (r) and p is a character of T of level n +1. Now by Theorem

11.5.2 of [9]

W (m(f)) = G[e)S2(£) if X £ 1 and

Pii.'to - l) m ( I ) ) = (r'('To) - l)G(£)Sa(£) if X = 1 where 70 is the topological generator of = P fixed above, and where G(e) and

Sj(e) arc as defined in §11.4.11 and §111.2.10 of [9], respectively. On the other hand, by Corollary 3.10.2,

Xp(m(f)) = />(m(f)) ~ p(ff) if x ^ 1, and Lemma 3.10.4 gives

p(( 7o - l)rn(l)) ~ 1.

Thus since n(p(7o) — 1) = I1(C — 1) 'v 2, where ( varies in the set of all primitive roots of unity of order 2n_l, we have

I I M ~ 2 • II G(e)S>(e). /(p)=n+l *6^H+| Now by an argument similar to that of §111.2.11 of [9] (using the analytic class number formula for the fields Kn+1 and Kn) we have, for sufficiently large n,

n G(e)Sa(£) ~ ^ |(/C +I)j

This completes the proof of Proposition 3.10.6. □

Corollary 3.10,7 The Iwasawa invariants of A'«, and (1 — r)£/oo/(l — are equal. 66

Proof. This follows immediately from Lcmma3.10.5, Proposition 3.10.6 and Corollary

3.9.12. □

As shown at the beginning of this section, Corollary 3.10.7 completes the proof of the main conjecture for the extension Kt»fK. C H A PTER IV

The Birch and Swinnerton-Dyer conjecture

4.1 The conjecture over K.

Keep the notation introduced in Chapter 111. In this section wc will use the results of Chapter III to relate #Hom(A«tl2t>«)^ lo 1)/^* We will then combine this information with the main result of Chapter II (Theorem 2.3.7) lo prove formula (1.3) of the introduction (sec Theorem 4.1.7 below). As explained in the introduction, this will complete the proof of the Birch and Swinnerton-Dyer conjecture for E over K.

Recall A'*, s= Gctl(MoolKoo), where Moo is the maximal abelian 2-cxtension of

Koo which is unramificd away from the prime above p.

Proposition 4.1.1 Wc have

x t = 0.

ProoJ\ We will prove that A'oo/(l — t)A,00 is finite. This will show that A'+ is finite, hence zero because Xoo baa no non-zero finite submodules (Theorem 3.6.3(iv)),

Since A'oo/(l - t )X oo is the largest quotient of Xoo on which r acts trivially,

A '»/(l — t)A'oo = Gal(L/Koo) where L is the maximal extension of I

67 68

2*extenslon of K+, which is unramificd outside of p. Therefore

o<./(£//c)= n h where the product extends over all primes v of not lying above p and /„ is the inertia group of v in Gal(L/K^). Since LjKoa is unramificd outside of p, the restriction-to-A'oo-homomorphism maps /„ injectivcly into the inertia group of v in

Ga/(/foo//f^)t for each u/p. This inertia group is trivial unless u lies above one of the finitely many primes of K that ramify in Koo/K> in which case its order is 2.

Finally, it is known ([9] Proposition II. 1.9) that every prime of K other than p is finitely decomposed in the tower K +, and this completes the proof. □

Recall T = G a/(/f«//f3) and G a l ^ j K ) = (r) x F.

Corollary 4.1.2 Wc have

Hom(A'w,,Z?p»)^= Hom(A'oo,Z?poo)r .

Proof. By the above proposition (1 + r)A'oo = 0, so if / € Hom(A'oot7?p») then f T(x) — t/(t~ 1x) = -f(-x) = f(x) for all x € A'*, i.e. f T - f. □

Remark 4.1.3 Interestingly, Proposition 4.1.1 implies that the 2-torsion subgroup of A'oo is trivial, which in turn shows that the Iwasawa /i-invariant of Xoo is zero.

This result partially complements work of Gillard [IS] and Schneps [26], who showed

(independently) that the fi-invariant of A'oo = Gal(Moo/X(E p«)) vanishes for all curves E defined over the imaginary quadratic field K, with complex multiplication by the ring of integers of K, and all split primes p/G such that E has good reduction at both p and p. 69

Let k : T —► Z j denote the character giving the action of F on Z3p«. We will also write k for the extension by linearity of k to the Iwasawa algebra A = Zj[[F]]. Recall that 70 is our fixed topological generator of F.

Lemma 4.1.4 Wc have

#(* 00/(70 - k(7o))A'oc) = # (Z 2/«(char(*0o))).

Proof. For any A-module K, let ipY : Y —♦ Y be the map induced by multiplication by 70 “ «(To) °n y* Find an exact sequence

0 — 4 © A //,A — + A'*, — » Z — > 0 (4.1) with Z finite, so cb ar(*oo) = (n/*)A* Arguing as in the proof of Lemma 3.6.1, wc obtain from (4.1) an exact sequence kcr(y>*) kcr(y>z) -4 ©A/(/,A + (70 - k(7o))A) -►

A '«/(7 o - k (7o ))* oo -* cokcr(

Now # © A /( /,A + (70 — «(7o))A) = # (Z 2/K(char(A',00))) is finite, so cokcr(^x) =

A'«/(7o ~ «(7o))A" 00 is finite. It follows that kcr(ipx) i® a finite submodule of A oo. therefore trivial. Noting that #ker(y»z) = #cokcr(^z), the lemma follows easily. □

Recall that rj> denotes the Hcckc character of K attached to E and O is the ring of integers of K. Fix an fl € C* such that flC? is the period lattice of a minimal model of E over K. If a, b € K* we will write a ~ b to mean that ordp(a/6) = 0.

Proposition 4.1.5 If £($, 1) ^ 0 then

- (1 - M )/N(p))I($,l)/n. 70

Proof. There is an isomorphism

Hom(Xoo,£T)»)r ^ Hom(A'00/(70 - K(7b)))t and by Lemma 4.1.4 and Theorem 3.5.1 (the main conjecture for Koo/K)

#(* 00/(70 - k(7o))*«) « #(Z,/*(char((l - ^ / ( I - r)**))).

Thus if g is a generator of char((l - r)f/oo/(l - r)^foo)*

#Hom{X ^E ^f ~n{g).

Finally, by Corollary 3.10.2 and Theorem 11.4.12 of [9], char((l — t)Uoo /(\ — r)^oo) has a generator g € A such that «(

Lemma 4.1.6 Wc have

1 -

Proof. Write Frobp for the FVobcnius of p in Gal{Kcaj I(), and note that Frobp acts as multiplication by ^>(p) 011 Ep». Then

1 - v>(p)/n(p) = 1 - m - ' ~ m - 1 ~

= m w , - = m K , ) r . □

Theorem 4.1.7 Let b denote the number of primes of K where E has a bad reduction, and let IU p» denote the p -primary part of the Tate-Shafarevich group of E over K .

Then if L($, 1) 0,

L(xj>, l)/fl ~ 2fc"3 • #UIp°°. 71

Proof. By Proposition 4.1.5 and Corollary 4.1.2

(1 - 0(p)/N(p))£(fc I)/fl ~ #Hom(X„, Ey.f, and Theorem 2.3.7 asserts that

#Hom{XtxnEP'* f= 2 h-2 • #E{I

Now Lemma 4.1.6 completes the proof of the theorem. □

» Corollary 4.1.8 Wc have

L $ , 1 )/tt = ± 2 ‘ • \J#U1(E/K)/#E(K).

In other words, Gross' refinement [16] of the Birch and Swinnerton-Dyer conjecture for E is true.

Proof. This follows easily from Theorem 11.1 of [23] and Theorem 4.1.7 above, using the fact that #Ulq<» — #IHq» for any prime q of K (see p. 228 of [16]). □

4.2 The coqjecture over Q.

There is a unique elliptic curve E t defined over Q, with complex multiplication by the ring of integers O of K = Q (\/—7), and with minimal discriminant ideal (—73).

FVom this curve, all the curves defined over Q and having complex multiplication by

O can be obtained as twists E x for some quadratic character \ °f @Q — C?a/(Q/Q).

See §12 of [16]. For convenience, we will write E d for E x if d is a squarefree, non-zero integer and Q(Vcf) is the quadratic extension of Q determined by y. 72

Assume that E d lias good reduction at 2 and L(EdfQ i l) ,4 0. In this section we will show that Birch and Swinnerton-Dyer’s conjectural formula for L(EdfQ , 1) is valid t.c. wc will show that

H & fQ , 1) = lV,- ( # E J(Q))-! • # m (B '/Q ) • n

First wc note that when d is divisible by 7 the curves Ed and E~d^ arc isogcnous over Q [16]. Thus Ed and E~d^ have the same L-function. Further, Casscls [5] has shown that the expression on the right in (4.2) is an isogcny invariant of Ed. From these facts it follows that wc need only consider values of d which arc prime to 7.

Now, if d is prime to 7, the sign in the functional equation for L(Ed/Q ts) is d/\d\ ([16]

§19), so L(Ed/Q x 1) - 0 if d is negative. We note also that E d has good reduction at

2 precisely when d = l(mod 4) (see [28], cxer. 10.15).

In accordance with the above remarks, we now fix (for the remainder of this section) a positive and squarcfree integer d, prime to 7 and = l(mod 4), such that i(V5',/Q ,l)5 £ 0 .

A global minimal model of E d over Q is given by the equation

y2 + xy = x3 + ^(21 d - 25)x3 + {7# - 21d + 12)x - (Md3 - 21d + 8), (4.3) which has discriminant —7 3d°. It follows that Cp > 1 exactly when p\7d.

The fundamental real period of Ed is

Jo y/x3 + 21 dx2 + 1 l2tPx 73

Lemma 4.2.1 Let b denote the number of primes of K where E d/K has a bad rc- duction. Then

Y[cp = 2h.

Proof. The Cp terms can be computed (quite easily) by applying Tate’s algorithm

[29] to the model E d: y1 = x3 + 21 dx1 + l^ t^ x . Wc find that cy = 2, and that if p ^ 7 divides d then Cp — 4 if = 1 and Cp = 2 if = -1, where is the

Legendre symbol. The assertion of the lemma is now clear. □

Let rpd denote the Heckc character of I( attached to E d.

Lemma 4.2.2 (i) L(Ed/Q,s) s= L(tpdys) = L(rj)d,s),

(ii) 0:Ed(Q) — 2 and #JSrf(/C) = 4.

(iii) L{rj>d} 1) > 0.

Proof. Statement (i) is due to Dcuring [10], and assertion (ii) is proved in §14 of [16].

Statement (iii) follows from Theorem 2 of [19]. □

Remark 4.2.3 The above observations and results (beginning wilh (4-3)) are also valid for the curve E~7d.

Lemma 4.2.4 (i) Wd = W„yd.

(ii) The period lattice of the minimal model (4.3) is Wd • O.

Proof. We have Wd — W \f\/d and Trf = W-\f\/Td. Thus assertion (i) follows from the equality W_j = \/7 Wj, which can be easily verified using formulas 241.00,

113.02 and 900.02 of [4]. Statement (ii) holds for d = 1 ([16], p. 82). That it holds for any d > 0, d = l(m od4), will follow from this particular case. 74

Let u>d denote the invariant differential associated to the model (4.3) and Ld the corresponding period lattice. The change of variables x — u2x'+ r, y = u3y'-fsu2x'+f, where r = 2 — 2d, s — (\/d — l)/2 , I = d — 1 and u = y/dt transforms (4.3) into the analogous equation with d = 1. It follows that wj = {y/d)~ ’u?! and Ld = (v/5)”1//1, so Wj • O = ° = Li n y d Proposition 4.2.5 IVc have

HE1/Q, 1) = Wi ■ (#EJ(Q))-J •y/#m(&/K) • n% .

Proof. Since d is prime to 7, equation (4.3) is a minima] model of Ed over K. Thus, by Lemma 4.2.4(H), we can take ft = Wd in the statement of Corollary 4.1.8. Now use Lemmas 4.2.1 and 4,2.2 to obtain the desired formula from Corollary 4,1.8. □

The above proposition shows that in order lo prove conjecture (4.2) it is sufficient to show that, for each rational prime p,

#UI(EJ/E)P. = (#UI(E j/Q)p»)1. (4.4)

To this end, consider the Selmcr groups

S(E'/Q. p“) = M E ’(Q,Ep.) —. and

S(E'/E,p“) = kcr|«’(E ,£pJ.) —. where the direct sums extend over all the places (including the archimedean ones) of the field involved. Then there are isomorphisms

mfJSVQ),- a S(Ed/Q tp°°) and III {Ed/I<)p« ~ S(E d/I<,p°°) (4.5) 75

(cf. Chapter II, Lemma 2.2.1).

Fix an automorphism c G G q whose restriction to K is the non-trivial element of

G = Gal{KfQ). For any (?Q-modulc M, define

M + = {m € M :m c = m) and M~ = {m 6 M : me = —m}.

If G/, acts trivially on M {e.g. M = JP{K%Ed00))l, then M is a G-modulc and

M + = A/°, the submodule of M of G-invariant elements.

The next rcsutt is an immediate consequence of Lemmas 4.2.8 and 4.2.9 below.

Proposition 4.2.0 The restriction map //*(Q, 2iy») —+ lV (K ,E dM)+ induces an isomorphism

S{Ed/Q,p°°) ~ S(EdfK }p°°)+.

Corollary 4,2.7 We have

#ui(£'7Q)p~ = # mI&IK)*..

Proof. Immediate from (4.5) and the above proposition. □

Lemma 4.2.8 The restriction map //’(Q, E d*>) —♦ H x{K,Ed»)+ is an isomorphism.

Proof, This follows from the fact that H'(G,Ed(K)p«>) = 0 (i — 1, 2), which in turn follows (easily) from Lemma 4.2.2(ii). □

Lemma 4.2.0 Suppose v is any place of Q and w is a place of K lying above v.

Then the local restriction map H l(Qv,E d) -♦ Hl(Kw,E d) is injective. 76

Proof. The kerne] of the restriction map is lV(Gal(Kw/Q v), Ed(Kw)). This group is zero if K w = Q„, or if v is a (finite) prime of good reduction for E djQ (since then v ^ 7, so KwfQ v is unramified; cf. Chapter II, Lemma 2.3.3). Suppose now that v is a prime of bad reduction for E d which docs not split in K/Q (hence v 2). We identify Gal(Kw/Q v) with G = Gal(KfQ). By Lemma 2.2.2, there is an isomorphism

E d(Kw) 2; E d(KJ) © /I, where A is a uniquely 2-divisiblc G-modulc. Then, since

H'(GtA) = 0 for all i > 1, the cohomology exact sequence gives an isomorphism

H'(G,Ed(Kw)) ~ H '(G ,Ed{Kw)2«).

Now, since ui is a prime of bad reduction for Ed/K } Lcmma2.1.3(i) gives E d{KJ)i<*> =

Ed{K)2. Therefore, as noted in the proof of Lemma 4.2.6, Hl(G,Ed(Kw)j») =

H x(G,Ed{I<)2) = 0.

Finally, suppose that v is the infinite place. The exact sequence

0 — ♦ E d{I<)2 — > E d(C) E d(C) 0 gives rise to a cohomology exact sequence

H '{G ,Ed{K)a) — ♦ H '{G ,Ed{C)) 7/‘(G, Ed{C)) — » lP{G,Ed{K)2).

But /P(G,EJ(I{)2) — 0 for i = 1, 2, and / / ’(G, JSd(C)) is annihilated by 2, so

/ f ‘(G, Ed(C)) = 0, as required. □

We can now prove formula (4.4) (and hence conjecture (4.2)).

Recall that c 6 Gq restricts to the non-trivial clement of Gal(K/Q).

Case I: p = 2. Recall that 2 splits in K as pp with p ^ p (p — pc). Then

W {Ed/K )2~ =* W (E d/I<)p~ © Ul{Ed/K)p », #UI(JS‘,/^)p- =# W {E d/I<)p~% and 77 multiplication by (1 + c) on 1XL{Ed f K)p«> induces an isomorphism

n i ( £ 7 / < v -

Therefore, using Corollary 4.2.7,

m i & I K ) , - = ( = (#m (£'/K )J.)3 =w m & I Q M ' -

Case II: p ^ 2. We need to consider the curve E~7d. Let x 5^ 1 be the quadratic character of Gq which is trivial on G/v-. Then E~7d = (Ed)x, the twist of Ed by x-

If we choose the equation y7 = x3 + 21dx2 + 112 d?x as a model for Ed, then E~7d is given by the equation: —7 y7 = x3 + 2ldx2 + 1 12d?x, and the map

JS-W{Q)+ ~ ^ ( Q ) “ .

The map l!l(E~7d/K) A Ul(Ed/I() induced by

Ul(E~7d/K )+ ss Ul(Ed/!<)-.

It now follows from Corollary 4.2.7 (applied to both E d and J3”7d), and the decom­ position IH(£d/A V = ni(JSJ/A’)+„ e that

# W (E d/I

Finally, the Q-isogenous curves Bd and E~rd have the same Tamagawa product, the same number of Q-rational points, and the same real period (Lemmas 4.2.1 and

4.2.2(ii), Remark 4.2.3, and Lemma 4.2.4(i)). Thus by the result of Cassels referred to above

#ni(£-«/Q),_ = #iii(Js7Q)p-, which completes the proof of formula (4.4), and of conjecture (4.2). B ibliography

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