J. Coates R. Greenberg K. A. Ribet K. Rubin Arithmetic Theory of Elliptic ~urves- Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, July 12-19, 1997 Editor: C. Viola Fonduiione C.I.M.E. Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Springer Paris Singapore Tokyo Authors John H. Coates Ralph Greenberg Preface Department of Pure Mathematics Department of Mathematics and Mathematical Statistics University of Washington University of Cambridge Seattle, WA 98195, USA 16 Mill Lane Cambridge CB2 1 SB, UK The C.I.M.E. Session "Arithmetic Theory of Elliptic Curves" was held at Kenneth A. Ribet Karl Rubin Cetraro (Cosenza, Italy) from July 12 to July 19, 1997. Department of Mathematics Department of Mathematics The arithmetic of elliptic curves is a rapidly developing branch of University of California Stanford University Berkeley CA 94720, USA Stanford CA 94305, USA mathematics, at the boundary of number theory, algebra, arithmetic alge- braic geometry and complex analysis. ~ftetrh e pioneering research in this Editor field in the early twentieth century, mainly due to H. Poincar6 and B. Levi, Carlo Viola the origin of the modern arithmetic theory of elliptic curves goes back to Dipartimento di Matematica L. J. Mordell's theorem (1922) stating that the group of rational points on Universiti di Pisa an elliptic curve is finitely generated. Many authors obtained in more re- Via Buonarroti 2 cent years crucial results on the arithmetic of elliptic curves, with important 56127 Pisa, Italy connections to the theories of modular forms and L-functions. Among the main problems in the field one should mention the Taniyama-Shimura con- jecture, which states that every elliptic curve over Q is modular, and the Birch and Swinnerton-Dyer conjecture, which, in its simplest form, asserts that the rank of the Mordell-Weil group of an elliptic curve equals the order of Cataloging-in-Publication Data applied for - vanishing of the L-function of the curve at 1. New impetus to the arithmetic Die Deutsche Bibliothek CIP-Einheitsaufnahme of elliptic curves was recently given by the celebrated theorem of A. Wiles (1995), which proves the Taniyama-Shimura conjecture for semistable ellip- Arithmetic theory of elliptic curves : held in Cetraro, Italy, July 12 - 19, 1997 / Fondazione CIME. J. Coates ... Ed.: C. Viola. - Berlin tic curves. Wiles' theorem, combined with previous results by K. A. Ribet, ;H eidelberg ; New York ; Barcelona ;H ong Kong ; London ; Milan ; J.-P. Serre and G. Frey, yields a proof of Fermat's Last Theorem. The most Paris ; Singapore ; Tokyo : Springer, 1999 recent results by Wiles, R. Taylor and others represent a crucial progress (Lectures given at the ... session of the Centro Internazionale towards a complete proof of the Taniyama-Shimura conjecture. In contrast ... Matematico Estivo (CIME) ; 1997,3) (Ixcture notes in mathematics to this, only partial results have been obtained so far about the Birch and ; Vol. 1716 : Subseries: Fondazione CIME) Swinnerton-Dyer conjecture. ISBN 3-540-66546-3 The fine papers by J. Coates, R. Greenberg, K. A. Ribet and K. Rubin Mathematics Subject Classification (1991): collected in this volume are expanded versions of the courses given by the l 1605, 11607, 31615, 11618, 11640, 11R18, llR23, 11R34, 14G10, 14635 authors during the C.I.M.E. session at Cetraro, and are broad and up-to-date contributions to the research in all the main branches of the arithmetic theory ISSN 0075-8434 of elliptic curves. A common feature of these papers is their great clarity and ISBN 3-540-66546-3 Springer-Verlag Berlin Heidelberg New York elegance of exposition. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is Much of the recent research in the arithmetic of elliptic curves consists concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, bl-oad- casting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this in the study of modularity properties of elliptic curves over Q, or of the publication or parts thereof is permitted only under the provisions of the German Copyright Law of structure of the Mordell-Weil group E(K) of K-rational points on an elliptic September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. curve E defined over a number field K. Also, in the general framework of 0 Springer-Verlag Berlin Heidelberg 1999 Iwasawa theory, the study of E(K) and of its rank employs algebraic as well Printed in Germany as analytic approaches. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply. even in the absence of a specific statement, that such names are exempt from the relevant protective laws Various algebraic aspects of Iwasawa theory are deeply treated in and regulations and therefore free for general use. Greenberg's paper. In particular, Greenberg examines the structure of Typesetting: Camera-ready TEX output by the authors the pprimary Selmer group of an elliptic curve E over a Z,-extension of SPIN: 10700270 4113143-543210 - Printed on acid-free papel the field K, and gives a new proof of Mazur's control theorem. Rubin gives a detailed and thorough description of recent results related to the Birch and Swinnerton-Dyer conjecture for an elliptic curve defined over an imaginary quadratic field K. with complex multiplication by K . Coates' contribution is mainly concerned with the construction of an analogue of Iwasawa theory for elliptic curves without complex multiplication. and several new results are Table of Contents . included in his paper Ribet's article focuses on modularity properties. and contains new results concerning the points on a modular curve whose images in the Jacobian of the curve have finite order . The great success of the C.I.M.E. session on the arithmetic of elliptic curves was very rewarding to me . I am pleased to express my warmest thanks Fragments of the GL2 Iwasawa Theory of Elliptic Curves to Coates. Greenberg. Ribet and Rubin for their enthusiasm in giving their without Complex Multiplication fine lectures and for agreeing to write the beautiful papers presented here . Special thanks are also due to all the participants. who contributed. with John Coates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 their knowledge and variety of mathematical interests. to the success of the 1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . 2 . session in a very co-operative and friendly atmosphere 2 Basic properties of the Selmer group . . . . . . . . . . . . . . . . 14 3 Local cohomology calculations . . . . . . . . . . . . . . . . . . . . 23 Carlo Viola 4 Global calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Iwasawa Theory for Elliptic Curves Ralph Greenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 Kummer theory for E . . . . . . . . . . . . . . . . . . . . . . . . 62 3 Control theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4 Calculation of an Euler characteristic . . . . . . . . . . . . . . . . 85 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 05 Torsion Points on Jo(N)a nd Galois Representations Kenneth A . Ribet . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 45 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 45 2 A local study at N . . . . . . . . . . . . . . . . . . . . . . . . . . 1 48 3 The kernel of the Eisenstein ideal . . . . . . . . . . . . . . . . . .1 51 4 Lenstra's input . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 54 5 Proof of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . .1 56 6 Adelic representations . . . . . . . . . . . . . . . . . . . . . . . .1 57 7 Proof of Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . .1 63 Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer Karl Rubin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 1 Quick review of elliptic curves . . . . . . . . . . . . . . . . . . . . 1 68 . . . . . . . . . . . . . . . . . . . . . . . . Elliptic curves over C 170 Fragments of the GL2 Iwasawa Theory . . . . . . . . . . . . . . . . . . . . 172 Elliptic curves over local fields . . . . . . . . . . . . . . . . . . of Elliptic Curves 178 Elliptic curves over number fields Elliptic curves with complex multiplication . . . . . . . . . . . . 181 without Complex Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 88 Elliptic units 193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler systems 203 John Coates . . . . . . . . . . . . . . . . . . . . . Bounding ideal class groups 209 . . . . . . . . . . . . . . . . . . The theorem of Coates and Wiles 213 . . . . . . . . . . . . . 2 16 Iwasawa theory and the "main conjecture" . . . . . . . . . . . . . . . . . . . . "Fearing the blast Computing the Selmer gmup 227 Of the wind of impermanence, I have gathered together The leaflike words of former mathematicians And set them down for you." Thanks to the work of many past and present mathematicians, we now know a very complete and beautiful Iwasawa theory for the field obtained by ad- joining all ppower roots of unity to Q, where p is any prime number. Granted the ubiquitous nature of elliptic curves, it seems natural to expect a precise analogue of this theory to exist for the field obtained by adjoining to Q all the ppower division points on an elliptic curve E defined over Q. When E admits complex multiplication, this is known to be true, and Rubin's lectures in this volume provide an introduction to a fairly complete theory. However, when E does not admit complex multiplication, all is shrouded in mystery and very little is known. These lecture notes are aimed at providing some fragmentary evidence that a beautiful and precise Iwasawa theory also exists in the non complex multiplication case. The bulk of the lectures only touch on one initial question, namely the study of the cohomology of the Selmer group of E over the field of all ppower division points, and the calculation of its Euler characteristic when these cohomology groups are finite. But a host of other questions arise immediately, about which we know essentially nothing at present. Rather than tempt uncertain fate by making premature conjectures, let me illustrate two key questions by one concrete example. Let E be the elliptic curve XI (1I ), given by the equation Take p to be the prime 5, let K be the field obtained by adjoining the 5-division points on E to Q, and let F, be the field obtained by adjoin- ing all 5-power division points to Q. We write R for the Galois group of F, over K. The action of R on the group of all 5-power division points allows us to identify R with a subgroup of GL2(iZ5), and a celebrated theorem of Serre tells us that R is an open subgroup. Now it is known that the Iwasawa Elliptic curves without complex multiplication 3 2 John Coates Write algebra A(R) (see (14)) is left and right Noetherian and has no divisors of zero. Let C(E/F,) denote the compact dual of the Selmer group of E over F, (see (12)), endowed with its natural structure as a left A(R)-module. We prove in these lectures that C(E/F,) is large in the sense that for the Galois groups of F, over Fn,a nd F , over F, respectively. Now the action of C on E,- defines a canonical injection But we also prove that every element of C(E/Fw) has a non-zero annihi- lator in A(R). We strongly suspect that C(E/F,) has a deep and interest- When there is no danger of confusion, we shall drop the homomorphism i ing arithmetic structure as a representation of A(R). For example, can one from the notation, and identify C with a subgroup of GL2(Zp). Note that i say anything about the irreducible representations of A(R) which occur in maps En into the subgroup of GL2(Zp) consisting of all matrices which are C(E/F,)? Is there some analogue of Iwasawa's celebrated main conjecture congruent to the identity modulo pn+'. In particular, it follows that & is on cyclotomic fields, which, in this case, should relate the A(R)-structure of always a pro-pgroup. However, it is not in general true that C is a pro-p C(E/F,) to a 5-adic L-function formed by interpolating the values at s = 1 group. The following fundamental result about the size of C is due to Serre of the twists of the complex L-function of E by all Artin characters of R? [261. I would be delighted if these lectures could stimulate others to work on these fascinating non-abelian problems. Theorem 1.1. In conclusion, I want to warmly thank R. Greenberg, S. Howson and (i) C is open in GL2(Zp) for all primes p, and Sujatha for their constant help and advice throughout the time that these lectures were being prepared and written. Most of the material in Chapters (ii) C = GL2(Zp) for all but a finite number of primes p. 3 and 4 is joint work with S. Howson. I also want to thank Y. Hachimori, K. Matsuno, Y. Ochi, J.-P. Serre, R. Taylor, and B. Totaro for making im- Serre's method of proof in [26] of Theorem 1.1 is effective, and he gives portant observations to us while this work was evolving. Finally, it is a great many beautiful examples of the calculations of C for specific elliptic curves pleasure to thank Carlo Viola and C.I.M.E. for arranging for these lectures and specific primes p. We shall use some of these examples to illustrate the to take place at an incomparably beautiful site in Cetraro, Italy. theory developed in these lectures. For convenience, we shall always give the name of the relevant curves in Cremona's tables [9]. 1 Statement of Results Example. Consider the curves of conductor 11 1.1 Serre's theorem Throughout these notes, F will denote a finite extension of the rational field Q, and E will denote an elliptic curve defined over F, which will always be The first curve corresponds to the modular group ro(ll) and is often de- assumed to satisfy the hypothesis: noted by Xo(ll), and the second curve corresponds to the group (ll), and is often denoted by X1(ll). Neither curve admits complex multiplication (for Hypothesis. The endomorphism ring of E overQ is equal to Z, i.e. E does example, their j-invariants are non-integral). Both curves have a Q-rational not admit complex multiplication. point of order 5, and they are linked by a Q-isogeny of degree 5. For both curves, Serre [26] has shown that C = GL2(Zp) for all primes p 2 7. Subse- Let p be a prime number. For all integers n 2 0, we define quently, Lang and Trotter [21] determined C for the curve ll(A1) and the primes p = 2,3,5. We now briefly discuss C-Euler characteristics, since this will play an important role in our subsequent work. By virtue of Theorem 1.1, C is a We define the corresponding Galois extensions of F padic Lie group of dimension 4. By results of Serre [28] and Lazard [22], C will have pcohomological dimension equal to 4 provided C has no ptorsion. 4 John Coates Elliptic curves without complex multiplication 5 Since C is a subgroup of GL2(Zp),i t will certainly have no ptorsion provided whence the assertion of the Corollary is clear from (i) of Lemma 1.3. The p 2 5. Whenever we talk about C-Euler characteristics in these notes, we shall corollary is useful because it does not seem easy to compute h2(E) in a always assume that p 2 5. Let W be a discrete pprimary C-module. We shall direct manner. say that W has finite C-Euler characteristic if all of the cohomology groups We now turn to the proof of (i) of Lemma 1.3. Let K, denote the cyclo- Hi(C, W) (i = 0,. . . ,4) are finite. When W has finite C-Euler characteristic, tomic Zp-extension of F, and let E,m (K,) be the subgroup of Epm which is we define its Euler characteristic x(C, W) by the usual formula rational over K,. We claim that Epm (K,) is finite. Granted this claim, it follows that r where denotes the Galois group of K, over F. But H1(r,E pm (K,)) is a Example. Take W = Epm. Serre 1291 proved that Epm has finite C-Euler subgroup of H1( E,E pm) under the inflation map, and so (i) is clear. To show characteristic, and recently he determined its value in [30]. that Epm(K,) is finite, let us note that it suffices to show that Epm(Hm) is finite, where H, = F(p,-). Let R = G(F,/H,). By virtue of the Weil Theorem 1.2. If p 2 5, then x(C, Epm)= 1 and H4(C,E ,~)= 0. pairing, we have R = C n SL2(Zp),f or any embedding i : C v GL2(Zp) given by choosing any %,-basis el, e2 of T,(E). If Epm (H,) was infinite, we This result will play an important role in our later calculations of the Euler could choose el so that it is fixed by 0. But then the embedding i would inject characteristics of Selmer groups. Put (: 1). R into the subgroup of SL2( Z,) consisting of all matrices of the form (i,) where z runs over Z,. But this is impossible since 0 must be open in S L ~ as 27 is open in GL2(Zp).T o prove assertion (ii) of Lemma 1.3, we need the We now give a lemma which is often useful for calculating the hi(E). Let pp- fact that 27 is a Poincar6 group of dimension 4 (see Corollary 4.8, 1251, p. 75). denote the group of pn-th roots of unity, and put u Moreover, as was pointed out to us by B. Totaro, the dualizing module for 27 is isomorphic to Q/Z, with the trivial action for C (see Lazard [22], Ppm -- Pp-, T,(p) = ltim p pn . (8) Theorem 2.5.8, p. 184 when C is pro-p, and the same proof works in general n>,l for any open subgroup of GL2(Zp) which has no ptorsion). Moreover, the By the Weil pairing, F(ppm) c F(Epm) and so we can view C as acting in Weil pairing gives a C-isomorphism the natural fashion on the two modules (8). As usual, define Using that C is a Poincar6 group of dimension 4, it follows that H3(C,E pn) is dual to H1(C,E pn (-1)) for all integers n 2 1. As usual, let T,(E) = lim Epn here 27 acts on both groups again in the natural fashion. e Passing to the limit as n + oo, we conclude that Lemma 1.3. Let p be any prime number. Then (i) ho (E) divides hl (E). H3(E,E pm) = l--i+m H~(zE~p,n ) (ii) If C has no p-torsion, we have h3( E) = #HO( C, Epm (- 1)). is dual to Corollary 1.4. If p 2 5, and h3 (E) > 1, then h2(E) > 1. Indeed, Theorem 1.2 shows that Write V,(E) = Tp(E) @ Q,. Then we have the exact sequence of E-modules 6 John Coates Elliptic curves without complex multiplication 7 Now V,(E)(-~)~ = 0 since Ep,(H,) is finite. Moreover, (10) is finite as Iwasawa was the first to observe in the case of the cyclotomic theory, it is by the above duality argument, and so it must certainly map to 0 in the more useful to view them as modules over a larger algebra, which we denote Qp-vector space H1(E, Vp(E)(-1)). Thus, taking E-cohomology of the above by A(R) and call the Iwasawa algebra of 0, and which is defined by exact sequence, we conclude that where W runs over all open normal subgroups of R. Now if A is any discrete As (11) is dual to H3(E,E pm),t his completes the proof of (ii) of Lemma 1.3. pprimary left 0-module and X = Hom(A, U&,/Z,) is its Pontrjagin dual, then we have Example. Take F = Q, E to be the curve Xo(ll) given by (4), and p =*5. A = UAW, X = limXw, The point (5,5) is a rational point of order 5 on E. As remarked earlier, t W Lang-Trotter [21] (see Theorem 8.1 on p. 55) have explicitly determined C where W again runs over all open normal subgroups of 0, and Xw denotes in this case. In particular, they show that the largest quotient of X on which W acts trivially. It is then clear how to extend the natural action of Z,[R] on A and X by continuity to an action of the whole Iwasawa algebra A(R). as C-modules. Moreover, although we do not give the details here, it is not In Greenberg's lectures in this volume, the extension H, is taken to difficult to deduce from their calculations that be the cyclotomic 23,-extension of F. In Rubin's lectures, H, is taken to be the field generated over F by all p-power division points on E, where ho(E) = h3(E) = 5, and hl (E) 2 52. p is now a prime ideal in the ring of endomorphisms of E (Rubin assumes that E admits complex multiplication). In these lectures, we shall be taking It also then follows from Theorem 1.2 that h2(E) = hl (E). H, = F, = F(Ep-), and recall our hypothesis that E does not admit complex multiplication. Thus, in our case, R = .E is an open subgroup of 1.2 The basic Iwasawa module GL2( 23,) by Theorem 1.1. The first question which arises is how big is S(E/Fw)? The following Iwasawa theory can be fruitfully applied in the following rather general set- result, whose proof will be omitted from these notes, was pointed out to me ting. Let H, denote a Galois extension of F whose Galois group R = by Greenberg. G(H,/F) is a padic Lie group of positive dimension. By analogy with the classical situation over F, we define the Selmer group S(E/H,) of E over Theorem 1.5. For all primes p, we have Hw by Example. Take F = Q, E = X1(ll), and p = 5. It was pointed out to me some years back by Greenberg that where w runs over all finite primes of H,, and, as usual for infinite extensions, H,,, denotes the union of the completions at w of all finite extensions of F contained in H,. Of course, the Galois group R has a natural left action on S(E/H,), and the central idea of the Iwasawa theory of elliptic curves (see his article in this volume, or [7], Chapter 4 for a detailed proof). On the is to exploit this R-action to obtain deep arithmetic information about E. other hand, we conclude from Theorem 1.5 that This R-action makes S(E/H,) into a discrete pprimary left R-module. It will often be convenient to study its compact dual This example is a particularly interesting one, and we make the following observations now. Since E has a non-trivial rational point of order 5, we have which is endowed with the left action of R given by (of)(%)= f (a-'x) for the exact sequence of G(Q/Q)-modules f in C(E/H,) and a in 0. Clearly S(E/H,) and C(E/H,) are continuous m6dules over the ordinary group ring Zp[R] of R with coefficients in Z,. But, 8 John Coates Elliptic curves without complex multiplication 9 This exact sequence is not split. Indeed, since the j-invariant of E has order It is natural to ask what is the A(C)-rank of the dual C(E/F,) of the -1 at 11, and the curve has split multiplicative reduction at 11, the 11-adic Selmer group of E over F,. The conjectural answer to this problem depends Tate period q~ of E has order 1 at 11. Hence on the nature of the reduction of E at the places v of F dividing p. We recall that E is said to have potential supersingular reduction at a prime v of F if there exists a finite extension L of the completion F, of F at v such that E has good supersingular reduction over L. We then define the integer and so we see that 5 must divide the absolute ramification index of every r,(E/F) to be 0 or [F, : Q,], according as E does not or does have potential prime dividing 11 in any global splitting field for the Galois module E5. It supersingular reduction at v. Put follows, in particular, that [Fo: Q(P~)=] 5, where Fo = Q(E5). Moreover, 11 splits completely in Q(p5), and then each of the primes of Q(p5) divid& 11 are totally ramified in the extension Fo/Q(p5). In view of (15) and the fact that Fo/Q(p5) is cyclic of degree 5, we can apply the work of Hachimori and Matsuno [15] (see Theorem 3.1) to it to conclude that the following where the sum on the right is taken over all primes v of F dividing p. Note < assertions are true for the A(r)-module C(E/FO(P~~w))h,e re r denotes the that rP(E/F) [F : Q]. Galois group of Fo (p5m) over Fo: (i) C(E/ Fo( ~ 5 ~is) A)( r)-torsion, (ii) the pinvariant of C(E/F0(p5m)) is 0, and (iii) we have Conjecture 1.7. For every prime p, the A(C)-rank of C(E/F,) is equal to 7, (EIF). It is interesting to note that Conjecture 1.7 is entirely analogous to the con- However, I do not know at present whether E has a point of infinite order jecture made in the cyclotomic case in Greenberg's lectures. Specifically, if which is rational over Fo.F inally, we remark that one can easily deduce (16) r K, denotes the cyclotomic Z,-extension of F, and if = G(K,/F), then from Theorem 3.1 of [15], on noting that Fn/Q(p5) is a Galois 5-extension it is conjectured that the A(r)-rank of C(E/K,) is equal to rP(E/F) for all for all integers n 3 0. primes p. We now return to the discussion of the size of C(E/F,) as a left A(C)- Example. Consider the curve of conductor 50 module. It is easy to see (Theorem 2.7) that C(E/F,) is a finitely generated left A(C)-module. Recall that F, = F(Epn+1),a nd that En = G(Fm/Fn). We define @ to be El if p = 2, and to be &, if p > 2. The following result is a well known special case of a theorem of Lazard (see [lo]). Take F = Q. This curve has multiplicative reduction at 2, so that 72 (EIQ) = 0. It has potential supersingular reduction at 5, since it can be shown to Theorem 1.6. The Iwasawa algebra A(@) is left and right Noetherian and achieve good supersingular reduction over the field Q5 ( p3, ?-). Hence has no divisors of 0. r5(E/Q) = 1. It has good ordinary reduction at 3,7,11,13,17,19,23,31,.. . , and so rp(E/Q) = 0 for a11 such primes p. It has good supersingular reduction Now it is known (see Goodearl and Warfield [ll],C hapter 9) that Theorem at 29,59,. . . , and rP(E/Q) = 1 for these primes. 1.6 implies that A(@) admits a skew field of fractions, which we denote by K(@).I f X is any left A(C)-module, we define the A(Z7)-rank of X by the Theorem 1.8. Let tp(E/F) denote the A(r1)-rank for C(E/F,). Then, for formula all primes p 2 5, we have This A(C)-rank will not in general be an integer. We remark that the lower bound for tp(E/F) given in (22) is entirely analo- It is not difficult to see that the A(C)-rank is additive with respect to gous to what is known in the cyclotomic case (see Greenberg's lectures [13]). short exact sequences of finitely generated left A(C)-modules. Also, we say However, the upper bound for tp(E/F) in (22) still has not been proven un- that X is A(E)-torsion if every element of X has a non-zero annihilator in conditionally in the cyclotomic theory. We also point out that we do not at A(@). Then X is A(C)-torsion if and only if X has A(C)-rank equal to 0. present know that tp(E/F) is an integer. 10 John Coates Elliptic curves without complex multiplication 11 1.3 The Euler characteristic formula Corollary 1.9. Conjecture 1.7 is true for all odd primes p such that E has potential supersingular reduction at all places v of F dividing p. Exact formulae play an important part in the Iwasawa theory of elliptic curves. For example, if the Selmer group S(E/Fw) is to eventually be use- This is clear since rp(E/F) = [F : Q] when E has potential supersingular ful for studying the arithmetic of E over the base field F, we must be able reduction at all places v of F dividing p. For example, if we take E to be the to recover the basic arithmetic invariants of E over F from some exact for- curve 50(A1) above and F = Q, we conclude that C(E/F,) has A(C)-rank mula related to the C-structure of S(E/F,). The natural means of obtaining equal to 1 for p = 5, and for all primes p = 29,59,. . . where E has good such an exact formula is via the calculation of the C-Euler characteristic of supersingular reduction. S(E/F,). When do we expect this C-Euler characteristic to be finite? We long tried unsuccessfully to prove examples of Conjecture 1.7 when rp(E/F) = 0, and we are very grateful to Greenberg for making a suggestion Conjecture 1.12. For each prime p 2 5, x(C, S(E/F,)) is finite if and which at last enables us to do this using recent work of Hachimori and Mat- only if both S(E/F) as finite and rP(E/F) = 0. suno [15]. As before, let K, denote the cyclotomic Zp-extension of F, and let T = G(K,/F). Let Y denote a finitely generated torsion A(r)-module. We shall show later that even the finiteness of HO(C,S (E/F,)) implies that We recall that Y is said to have p-invariant 0 if (Y)r is a finitely generated S(E/F) is finite and rP(E/F) = 0. However, the implication of the conjecture Zp-module, where (Y)r denotes the largest quotient of Y on which T acts in the other direction is difficult and unknown. The second natural question trivially. to ask is what is the value of x(C, S(E/F,)) when it is finite? We will now describe a conjectural answer to this question given by Susan Howson and Theorem 1.10. Let p be a prime such that (i) p 2 5, (ii) E = G(F,/F) myself (see [5], [6]). Let UI(E/F) denote the Tate-Shafarevich group of E is a pro-p-group, and (iii) E has good ordinary reduction at all places v of over F. For each finite prime v of F, let Eo(Fv) be the subgroup of E(F,) F dividing p. Assume that C(E/K,) is A(r)-torsion and has p-invariant 0. consisting of the points with non-singular reduction, and put Then C(E/F,) is A(C)-torsion. Example. Take E = XI(ll), F = Q(,u5), and p = 5. Then E has good ordi- 1, nary reduction at the unique prime of F above 5. The cyclotomic Z5-extension If A is any abelian group, A@) will denote its pprimary subgroup. Let ( of Q(p5) is the field Q(p5m ). AS was remarked earlier, F,/F is a 5-extension be the padic valuation of Q, normalized so that Iplp = p-l. We then define for all n 2 0, because Fo/Fi s a cyclic extension of degree 5, and F,/Fo is clearly a 5-extension. Hence C is pro-5 in this case. Hence (15) shows that the hypotheses of Theorem 1.10 hold in this case, and so it follows that C(E/F,) is A(C)-torsion. where it is assumed that III(E/F)(p) is finite. If v is a finite place of F, write The next result proves a rather surprising vanishing theorem for the coho- k, for the residue field of v and Ev for the reduction of E modulo v. Let j~ mology of S(E/F,). If p 2 5, we recall that both C and every open subgroup denote the classical j-invariant of our curve E. We define C' of C have pcohomological dimension equal to 4. !73 = (finite places v of F such that ord,(j~)< 0). (26) Theorem 1.11. Assume that (i) p 2 5, and (ii) C(E/F,) has A(C)-rank In other words, !lR is the set of places of F where E has potential multiplica- equal to rp(E/F). Then, for every open subgroup C' of C,w e have tive reduction. For each v E !73, let L,(E, s) be the Euler factor of E at v. + Thus L,(E,s) is equal to 1, (1 - (NV)-~)-' or (1 (NU)-*)-', according as E has additive, split multiplicative, or non-split multiplicative reduction at v. The following conjecture is made in [6]: for all i 2 2. Conjecture 1.13. Assume that p is a prime such that (i) p 2 5, (ii) E For example, the vanishing assertion (23) holds for E = 50(A1) and p = has good ordinary reduction at all places v of F dividing p, and (iii) S(E/F) 5,29,59,. . . , and for E = XI( 11) and p = 5, with F = Q in both cases. 12 John Coates Elliptic curves without complex multiplication 13 ES finite. Then HYE, S(E/F,)) is finite for i = 0,1, and equal to 0 for shows that (31) holds if we can find an imaginary quadratic field K, in which i = 2,3,4, and 11 splits, such that the Heegner point attached to K in E(K) is not divisible by P; here we are using Serre's result [26] that G(Fo/Q) = GL2(Fp) for all primes p # 5. The determination of such a field K is well known by computation, but unfortunately the details of such a computation do not seem to have been published anywhere. Granted (31), we deduce from (28) and (29) that Conjecture 1.13 predicts that We remark in passing that Conjecture 1 made in our earlier note [5] is not correct because it does not contain the term coming from the Euler factors in 17JZ. We are very grateful to Richard Taylor for pointing this out to us,. for all primes p 2 7 where E has good ordinary reduction. At present, we Example. Take F = Q and E to be one of the two curves Xo(l1) and cannot prove (32) for a single prime p 2 7. Xl(11) given by (4) and (5). The conjecture applies to the primes p = 5,7,13,17,23,31,. . . where these two isogenous curves admit good ordinary reduction. In these notes, we shall prove two results in the direction of Conjecture We shall simply denote either curve by E when there is no need to dis- 1.13, both of which are joint work with Susan Howson. tinguish between them. We have Theorem 1.14. In addition to the hypotheses of Conjecture 1.13, let p be such that C(E/F,) is A(C)-torsion. Then Conjecture 1.13 is valid for p. and Of course, Theorem 1.14 is difficult to apply in practice, since we only have rather weak results (see Theorem 1.10) for showing that C(E/F,) is A(C)- torsion. The next result avoids making this hypothesis, but only establishes This last statement is true because of Hasse's bound for the order of Ep(IFp) a partial result. Put and the fact that 5 must divide the order of Ep(IFp) for all primes p # 5,11. We also have c, = 1 for all q # 11, and Theorem 1.15. Let E be a modular elliptic curve over Q such that L(E, 1) # As is explained in Greenberg's article in this volume, a 5-descent on either 0. Let p be a prime 2 5 where E has good ordinary reduction. As before, curve shows that let F, = Q (Ep- ). Then (i) H1( 22, S(E/F,)) is finite and its order divides #(H3(C, ~ ~ r n )an)d, ( ii) HO(E,s (E/F~)) is finite of ezact order tP(E/Q)x #(H3(Z EPrn)). Hence we see that Conjecture 1.13 for p = 5 predicts that We recall that we conjecture that Hj(E, S(E/F,)) = 0 for j = 2,3,4 for all p 2 5, but we cannot prove at present that these cohomology groups are even finite under the hypotheses of Theorem 1.15. Note also that the order In Chapter 4 of these notes (see Proposition 4.10), we prove Conjecture 1.13 of H~(c,E p ) c an easily be calculated using Lemma 1.3. As an example of for both of the elliptic curves Xo(ll) and X1(ll) with F = Q and p = 5. Hence the values (30) are true. Now assume p is a prime 2 7. We claim that Theorem 1.15, we see that for E given either by Xo(l1) or X1(11), we have Indeed, the conjecture of Birch and Swinnerton-Dyer predicts that m(E/Q) for all primes p 2 7 where E has good ordinary reduction. Indeed, we have = 0, and Kolyvagin's theorem tells us that III(E/Q) is finite since L(E, 1) # H3(C, Ep-) = 0 for all primes p # 5 because of Lemma 1.3 and Serre's result 6. In fact, Kolyvagin's method (see Gross [14], in particular Proposition 2.1) that G(Fo/Q) = GL2(lFp) for all p # 5.