Irl v erl tio~les

Invent. math. 88, 405-422 (1987)

matbematicae 9 Springer-Verlag 1987

Local units, elliptic units, Heegner points and elliptic curves Karl Rubin* Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

Introduction

Suppose E is an elliptic curve defined over Q with complex multiplication by the ring of integers C of an imaginary quadratic field K. Let p be a prime of K lying over the rational prime p, and form the extension K(Ep~)/K(Ep) with Galois group Z 2. In this tower one can study, following Coates and Wiles [5], U~:/%~ - the inverse limit of the local units at p modulo the inverse limit of the elliptic units attached to E, restricted to a certain eigenspace under the action of Gal(K(Ep)/K). When p splits in K, Yager [23] has shown that as a module over the Iwasawa algebra A2=Zp[[Gal(K(Ep,~)/K(Ep))~ ~-Zp~S, T~, Uoo/~'oo ~- A 2 / F A 2

(1)

where F is a 2-variable p-adic L-function attached to E. This isomorphism also gives evidence for the " M a i n Conjecture" 1-3] which identifies F as the characteristic power series of a Selmer group. When p does not split in K the picture is not nearly so satisfactorily developed. One difficulty is that U~ has A2-rank 2 while cg~ has A2-rank 1, so U~/Cg~ is not a torsion A2-module. In this paper we begin with the observation (see w that the "restriction" V~ of U~ to the anticyclotomic Zp-extension of Kp has 2 natural A-submodules V + and V - of rank 1 over A, the Iwasawa algebra attached to the anticyclotomic Zp-extension of Kp. These modules depend only on p, not on E or on K (see w3). A further remarkable fact is that the "restriction" of cgo~ always lies in V + or V , depending on the sign in the functional equation of L(E, s). Conjecture 2.2 (w states that Vo~- V + | V-. Most of this paper is devoted to proving a criterion (Theorem 8.4) under which Conjecture 2.2 is true. This criterion (which seems likely to be true for all primes) can be checked for many primes p; see w It is interesting that although Conjecture 2.2 is purely a statement about local fields, the methods used in this paper to attack it are very global: we make crucial use of elliptic *

Partially supported by NSF grant DMS-8501937

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units and Heegner points on elliptic curves with complex multiplication. It would be interesting to have a completely local proof. The last two sections give applications to elliptic curves. In w9 we prove the counterpart for supersingular primes of a theorem of Greenberg [7]. Our result states that if E is an elliptic curve defined over Q with complex multiplication, and the sign in the functional equation of L(E, s) is - 1 then either E(Q) is infinite or the p-part of the Tate-Safarevic group of E is infinite for every prime p > 5 where E has good, supersingular reduction. (With the word "supersingular" replaced by "ordinary" this is Greenberg's theorem.) In w10 we discuss the analogue of (1), showing that if e = _+ 1 is the sign in the functional equation of L(E, s) then V ~ / ~ ~-A/FA where F has some properties of a p-adic L -function.

Acknowledgement. I would like to thank David Rohrlich for several helpful conversations. w 1. Preliminaries Fix a prime p > 5 . Denote by q~ the unique unramified quadratic extension of Qp, O its ring of integers, and let E denote a fixed Lubin-Tate formal group over O for the uniformizing parameter - p . Write q~n= cb(Ep,,+ ~), the extension of c/, generated by the p"+l-torsion of E, and q ~ = w ~ b , . Then we have a natural isomorphism Gal (@~/~b) -~ Aut (Ep~) ~ O* ~ A x O

(2)

where A =Gal(~o/~b)~-(O/pO) * is a cyclic group of order p 2 1. Write tr for the isomorphism of Gal(~b~/~) into O* given by (2), and let ~o denote the restriction of ~ to A. F r o m (2) we see that G a l ( q ~ / ~ b 0 ) ~ Z 2. The natural action of Galffb/Qp) on this group gives a decomposition Gal(q~ /~bo)~ G + | G + ~ G - ~ Z e, where G + (resp. G - ) is the maximal subgroup on which Gal(q~/Qp) acts via the trivial (resp. nontrivial) character. Fix topological generators a of G + and r of G-. Define the Iwasawa algebras

Az=O[[Gal(~b~/~bo)]]=lim O[Gal(q~j~bo) ] (

and

A=O[G-~.

As usual we will identify A 2 with O[[S, T]] and A with O[[T~ by sending a to I + S and ~ to I + T . For every n let U, denote the units of q~, which are congruent to 1 modulo the maximal ideal and define

V =(limm Uo|

O) ~

where the inverse limit is taken with respect to the norm maps, and for any O[A]-module A, A ~' denotes the submodule of A on which A acts via co. Also define

= This quotient does not depend on the choice of or.

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407

Proposition 1.1. U~ ~- A22, and V~, ~ A 2. Proof The first assertion can be found in [22] and the second follows immediately.

w2. Logarithmic derivatives Fix a generator {G} for the Tate module Tp(E). For any u = { u , } e l i m U. Coleman [6] has shown the existence of a unique power series f.eO[[X~* with the property that Ju(G)=u. for every n. For every n we define the Coates-Wiles logarithmic derivatives 6: U~-~ O, 6.: U~--, q~. by

s

1 L'(v.) ~"(u)=2(v~ L(v.)'

~(u)-f.(o)

where 2 denotes the logarithm map of E. (Note that we have extended these maps O-linearly to limmU,| ). For any character z : G a I ( q ~ / ~ ) ~ Q * of finite order, choose n large enough so that Z factors through Gal (~b,/~) (i.e. the conductor of Z divides p,+l) and define an O-linear homomorphism 6z: U~--* ~, by

~(u)=p-"- 1

~

z(~) ~.(u) ~'.

7s Gal (q~n/q))

The properties of f , [6] show that the right hand side of this definition does not depend on the choice of n. Lemma 2.1. (i) U" ueU~, and 7eGal(~,/~0) then cS,(uT)=tc(7)6,(u)L (ii) I f Z is a character of G- then 6z((a-~c(a))U~)=0. (iii) I f )~ is a character of Gal(~b~/q~0), ueU~ and F(S, T ) e A 2 then 6z(F. u)=F(~c z l ( a ) - 1, ~cZ-l(r) - 1) (Sz(u). (iv) 6 = - p / ( p +1)(3zo, where Zo denotes the trivial character. (v) I f ;( is a character of G-, u~U~ and 6z(u)=0 then 6p(U)=0 Jor all characters p of G- with the same conductor as )~. Proof Statements (i) and (iv) follow from the basic properties of f,; see for example [6]. Then for any fl~Gal (q~/~b),

G(u ~) = p - " - ' ~ z(~) ,~~

= p - " - ' y ~(~) z(~) ~.(u) ~ = ~z- l(fl) G(u)

If X is a character of G- then Z(e)= I, so applying this with f l = a proves (ii). Taking fi= ~" z" proves (iii) for F(S, T)=(1 +s)m(1 + T)", and the general case follows since these power series generate A 2 topologically. To prove (v), observe that since G - is cyclic, any such p can be written p = z p with some fl~G + and then (~p(U) = p - n--1 E )~'(~J fl) (~n (u)7B = )~[l([J) [ ( ~ z ( U ) ] / / = 0.

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We call the characters of G - anticyclotomic characters. N o t e that (ii) of the l e m m a shows that if ) is an anticyclotomic character then 6 x induces a welldefined h o m o m o r p h i s m of V~. Define ~ + = {anticyclotomic characters ): the conductor of )~ is an even power of p} S-=

{anticyclotomic characters Z: the conductor of Z is an odd power of p}

and V + = { v ~ V ~ : 6z(v)=0 for every ) ~ Z - } V - = {re V~: 5z(v)=0 for every Z e 3 + } .

Conjecture 2.2. V + _~ V - ~- A and V~ = V + 9 V _

w3. Elliptic curves and elliptic units F o r the basic facts a b o u t elliptic curves with complex multiplication, formal groups and elliptic units stated in this section see [9], [4], [15]. Let g denote the collection of elliptic curves E with the following properties : (i) E has complex multiplication by the ring of integers of some imaginary quadratic field K in which p does not split. (ii) E is a Q-curve in the sense of Gross [9] (i.e. E is defined over the Hilbert class field H of K and E is isogenous over H to all of its G a l ( H / Q ) conjugates) and further there is a grossencharacter q~=q~E of K such that the grossencharacter ~/,= ~bE of H attached to E is given by ~ = q~ o NH/K. (iii) E has good reduction at all primes of H above p. If E is any element of g condition (ii) above implies that for all ideals a of K, (3) We have the L-function identities L ( E m , s ) = L(~, s) 2 = H L(~Pz, s) z

(4)

where the product is taken over characters Z of Gal (H/K), and by (3), L(~, s ) = L(~p, s), so the constant w = wE in the functional equation of L(~p, s) is + 1. Thus we can write ~' = g + w g - where ~ + = {E ~ ~: w~ = + 1}. Note that condition (ii) above is automatically satisfied if E is defined over Q. Also, we will make use of the Q-curve A(q) defined by Gross in [9] for primes q - 3 (mod 4): A (q) has complex multiplication by the ring of integers of Q(lf~-q), good reduction outside q, and the sign in the functional equation of L(~pA(q~, s) is (--1) (~ If --q is not a square m o d p then A ( q ) ~ g .

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L e m m a 3.1. I f E ~ g then q~((p))= - p .

Proof F o r m the Dirichlet character e(n)= ~p~((n))/n and let f be its conductor. If q is a rational prime not dividing f and q--qO in K, then e(q)=l~oE(q)lZ/q= 1 by (3). Since e ( - 1 ) = - 1, e is not the trivial character. Thus on integers prime to f, e must be the quadratic character attached to K/Q, so e(p)= - 1. Now suppose E e ~ and let H, K be as above. For any n set H,=H(Ep,,+I), and H ~ = u H , . The principal prime ideal (p) of K splits completely in H/K, so H has [H: K] primes above p, all with absolute norm p2. Fix one of these primes and denote it by p; then H ~ = K p = q ~ . The theory of complex multiplication shows that the formal group E over O giving the kernel of reduction modulo p on E is a Lubin-Tate group for the uniformizing parameter O(p)--cp((p))= - p , so /~ is isomorphic over O to the formal group E of w1. It follows that for every n, p is totally ramified in Hn/H, and the completion of H, at the unique prime above p is

eb(Ep,+,) = qs(/~p. +i) = ~b(Ev, +1)= ~b. This gives a canonical embedding (once p is chosen) of H , into cb, for each n, and we can identify Gal(q~/~b) with Gal (H~/H) by restriction. Let O=(~ E be a period of E, i.e. O~C* satisfies

~2- 1 L = C

(5)

where L is the period lattice of a Weierstrass model of E with discriminant prime to p and (9 is the ring of integers of K. As in [4] we can construct from E elliptic units in each of the fields H,. Viewing these as units of ~b we obtain elements of U~j. Fix once and for all an embedding of 0 into 0p, extending the map above from H into ~b, so that we can identify (among other things) complex and p-adic valued characters. With this identification, we have the following result due to Coates and Wiles. Theorem 3.2. Suppose E~Co. There is an elliptic unit ?,= ~(E, f2E)~ U~ such that 6(~) =f2~-' L(q~E, 1)

and for all characters Z of Gal (q~/q~) C~Z(~) = _ _ ~ e ~ E l ( l

_ _ -rpEZ(p) p - 2

) L (~o~Z , 1).

Proof [4] when H = K , or [15] in general. Corollary 3.3. Suppose E ~

+- and let ~ denote the projection of ~(E, f2E) to V~.

Then ~6 V +-. Proof Let w denote the sign in the functional equation of L(rp, s). Greenberg [7] showed that if Z is a character of G - of conductor p", then L e m m a 3.1 implies that the sign in the functional equation of L(c~)~,s) is ( - 1 ) " w . Thus L(CoZ, 1)= 0 if ( - 1 ) " ~ w, and the theorem follows from Theorem 3.2. Corollary 3.4. rank A V + > 1.

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Proof Choose a rational prime q - 7

(mod8) such that - q is not a square modp. Rohrlich and Montgomery [13] have shown that the L-function of the Q-curve A(q) does not vanish at 1. Therefore if ~ is the unit of Theorem 3.2 for A(q), 6(~)~0 so ~ is a nonzero element of V +. The corollary follows since V~ is torsion-free (Proposition 1.1).

Remark. Using theorems of Rohrlich [14] or Waldspurger [20] one can find elliptic units ~ e V - with 6~(~)#:0 for some Z, so rank A V - > I obtain an alternate proof of this fact in w7 (Corollary 7.5).

w

as well. We will

Background: A-modules, formal groups

In this section we gather the results we will need concerning formal groups and Iwasawa theory. We let A=O[[F]], where F is a subgroup of Gal(q~/q~) isomorphic to Zp, and let ? be a topological generator of F.

Lemma 4.1. Suppose X'~A 2, Y is a rank-1 submodule of X and X / Y is torsionfree. Then Y is free. Proof Fix a A-basis c~, fl of X, and let Z = A ( F e + G [ d ) be a maximal free submodule of Y. Then F, G~A have no common factor. Since A is a UFD, X / Z is easily seen to be torsion-free, so Y/Z, which is a torsion submodule of X / Z , must be 0.

Lemma 4.2. Suppose X is a free, rank-i A-module, and d~ is a continuous homomorphism from X to O. I f x e X is such that (~(x)~O* then x generates X over A. Proof Writing m for the maximal ideal of A, the continuity of ~b implies that ~b maps m X i n t o pO. Thus x r so by Nakayama's Lemma x generates X.

Lemma 4.3. I f X is a rank-1 A-module and H o m o ( X , Ep~)r~-cb/O, then X is free. Proof Homo(X, Ep~)r~- Homo(X~( 7 - ~c(7))X, Epic), so we conclude x / ( 7 - K(r)) x ~- o. By Nakayama's Lemma it follows that X is generated by a single element, hence free.

Lemma 4.4. Let A be an O[F]-module with no p-torsion and write Dp= Qp/Zp. Then (A | D p)r/(Ar @ D p) ~- H a(F, A). Proof From the exact sequence O--~ A ---~A @ Q p ---~A @ D p --~O we get a F-cohomology exact sequence

Ar ~ Ar | which proves the Lemma.

| D flr--* HI(F, A)-* HI (F, A |

=0

Local units, elliptic units, Heegner points and elliptic curves

411

Proposition 4.5. Let G be a Lubin-Tate .formal group of height 2 defined over O, and let Fo~/cb be a ramified extension with Galois group Z~, d => 1. Write F, Jor the subfield of F~ with Gal (F,/F) ~_(Z/p ~Z) d and write N, Jbr the norm map from G(F,) to G(q)). Then (i) ~ N. G (F.) = 0. n

(ii) G(F~) is torsion-free. Proof The first statement is due to Hazewinkel [12]. The second follows from the fact that since G is a Lubin-Tate group, Gal(cl)(Gp)/Cb)~-(O/pO)*, which has order prime to p,

w5. Kummer theory and the dual situation For all n =< 0o define O. = q~a, the subfield of q). fixed by A, and 7in= O.G+. Then we have restriction isomorphisms Gal(Ojq))~-Gal(q),/Cl)o)~-O/p"O

Gal(O~/q))~-G + x G-

a a (7~q))~G /p G ~-Z/p"Z l

--

n

aal(!/'~/q))~G-.

--

Then ~ is the anticyclotomic Zp-extension of 4) in O~,, the maximal subfield on which Gal (q~/Qv) acts via the nontrivial character. For any algebraic extension ,~- of ~b write E(Y) for the maximal ideal of o~ endowed with the group law of E. For each n the exact sequence pn+

1

0-~ Ep.+,-~ E ( q ) ) - - 4

E(~)~ 0

gives rise to a Ga1(45/O.) cohomology exact sequence O ~ E(O.)/p "+' E(O.)-~ HI(Gal(@O.), Ev.+d H ' ( G a l (@O.), E(~))vo +1 ~ 0. Lemma 5.1. (i) ~

(6)

HI(Gal(@O.), E(~))v.+ ~=0. n

(ii) limmHI(Gal (@70, E(qS))v.+, =0. tt

Proof. If E is any elliptic curve over q~ whose formal group /~ is isomorphic to E (E~$, for example) then H 1(Gal (@O.), E(qS))p. +~~ limmH 1(Gal (45/O.), E(~)) v. . . . n

n

By Tate duality [19] applied to E, this direct limit is dual to li~mE(O.), the inverse limit with respect to the norm maps, and since E is a height 2 formal group Proposition 4.50) shows that this inverse limit is 0. The proof of (ii) is identical. Recall D p = Qp/Zp.

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K. Rubin

Proposition 5.2. The exact sequences (6) induce an isomorphism E(O ~) | Dp ~ Hom o (Us, Ep~).

Equivalently, we have a nondegenerate, O-linear, Galois equivariant Kummer pairing ( , >: E(Oo~)| • U~ ~Ep~. Proof Taking the direct limit over n of the exact sequences (6) and using Lemma 5.1 (i) shows E(O ~) | Dp _~H1 (Gal (@O ~), Epic) Hom (Gal (~/r ~), Ep~) ~ Horn o ((Gal (45/4,) @ O)~ Ep~)

~ Homo (Uo~,, Ep~), where the last isomorphism is given by local class field theory. We will need the following formula for the pairing ( , > due to Wiles. Recall that 2 is the logarithm map of E and {v,} is the generator of the Tate module Tp(E) used to define the logarithmic derivative maps. Write Tr, for the trace map from 4', to 4~. Theorem 5.3 (Explicit Reciprocity Law). I f y~E(O,) and ur U~, then

= [ p - " - 1 Tr,(~,(u) 2(y))] vk. Proof Wiles [21]. Proposition 5.4. The Kummer pairing of Proposition 5.2 induces a nondegenerate pairing <, >: E ( ~ ) | v • V~ ~ Ep-~.

Proof From Proposition 5.2 we get (E(O o~)|

De) G+ ~

(Homo (U~,, Ep~))z + = Hom o (U~/(a- so(a)) U~, Ep~)

= Homo(V~, Ep~). By Lemmas 4.4 and 4.5(ii), (E(Oo~)| G+/(E(To~)@Dp)~-Ha(G +, E(O~)), so to prove the Proposition we need only show that this cohomology group vanishes. The inflation-restriction sequence of Galois cohomology identifies HI(G +, E(O~)) with a subgroup of HI(Gal (@l/'~), E(q5)). But H l ( a a l (~/~t,), E (~)) ~ lim H ~(aal (~/7J,), E (q5)) n

which is 0 by Lemma 5.1(ii). For any character Z of Gal(7~oJT~)=G - define a map 2 x from E ( ~ ) t o ,/,~ by

~(y)=p-"

~ 7 e G a l (~n/q~)

z-~(7);4#

Local units, elliptic units, Heegner points and elliptic curves

413

where n is any integer large enough so that yeE(7-',) and the conductor of Z divides p'+ 1. Now define A + = {yeE(TJ~): 2x0,)=0 for all Ze~ + } and A- = { y e E ( ~ ) : Zx(y)=0 for all ZeS }. The next Lemma follows immediately from elementary properties of characters, and we omit the proof. In particular (iii) shows that G preserves A + and A-.

Lemma 5.5. (i) / f yeE(tP,) and the conductor of Z is greater than p'+ 1 then 2z(y) =0.

(ii) (iii) (iv) = pro-.

I f yeE(hU~) then 2 ( y ) = ~ 2z(y), summing over all characters Z of G-. If yeE(~o~), Z is a character of G and 7eG then ,~z(yT)=Z(7)2z(y). / f m>n, y e E ( ~ . ) and Z is a character of Gal(hU,/q)) then 2x(N,,/,y ) ,~ (y).

(v) A+c~A- =0. (vi) For every n, A+ c~E(~P.)+ A-c~E(iP,)~_p"E(tP,). There are natural injections A + | 1 7 4

and by (vi) of the Lem-

mR,

E(TJ~I|

= A+|

Proposition 5.6. The annihilator of A +-|

|

v.

(7)

r in V~o is V +.

Proof. If y e E ( 7 ~ ) and veV,, then using Theorem 5.3 and Lemma 5.5 yields = [ p - n -

1

Tr, 6n(V) ,~(y)] Ok 1'

Z

= [ ~ a~(v) ,~(y)] ~

(s)

Z

where the sums are taken over 7eGal((bjq)) and all characters Z of G_ Now it follows from the definitions that V • annihilates A+- | vNow suppose that veV~ and v annihilates A+- | and Ze~=-v. Choose any y e A +- such that 2z(y)4=0 (such a y must exist because E(tP,) has a submodule of finite index isomorphic to O[Gal(~,/ch)]). Then using (8) for every k shows that 0 = ~ 6p(v) ,~, 0") P

for every 7eGal (TJ,/cb). Thus

P

7

Since 2z(y)4=0 this shows 6z(v)=0. Therefore v e V +- as claimed.

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K. Rubin

Corollary 5.7. The Kummer pairing of Proposition 5.2 induces a nondegenerate

pairing <, 5" A+ |215

VojV + ~Ev-~.

Corollary 5.8. V+~ V- =0 and rank a V • __<1. Proof The first assertion follows from Propositions 5.4 and 5.6 and (7). For the second, since rank o A • is infinite, Corollary 5.7 implies that rank A V • < rank A Vo~= 2.

w6. Heegner points For the facts we will use concerning Heegner points see [10]. For this section we fix an elliptic curve E~d ~, and let K be its field of complex multiplication, D the discriminant of K, H the Hilbert class field of K, and ~ the conductor of the grossencharacter ~0E defined in w Define N=DNK/of. It is shown in [17] that there is a weight 2 newform f on Xo(N ) whose associated L-function L(f, s) is equal to L(q~e, s). It follows that As., the factor of the Jacobian Jo(N) associated to f by Shimura [17], is isogenous (over Q) to the restriction of scalars of E from F = Q ( j ( E ) ) to Q (see [9], w18), so there are nontrivial maps defined over H from Az to E, and therefore from Jo(N) to E as well. Using the fixed embedding of H into q' (see w we can view any such map as being defined over ~b. We will use Heegner points obtained in this way to obtain information about A • | v. Choose an auxiliary quadratic field 3U such that (i) all primes dividing N split into two distinct primes in ~ . (ii) p does not split in J(i

(9)

There will be infinitely many such J(" because N is prime to p. Let (9 denote the ring of integers of Of; and for every n let (9, denote the order of conductor p.+l in (9. Also, fix an ideal tt of (9 such that (9/n~Z/NZ; the existence of such an ideal is assured by (9)(i). Following" the construction in [10], we define the Heegner point x, eXo(N)(C ) to be the point corresponding to the moduli data (C/nc~(9,, (9,/n c~(9,), and x to be the point corresponding to (C/n, (9/n). Then in fact xEXo(N)(Jg ~) and x, eXo(N)(g/f,) where ~ is the Hilbert class field of Y and ~f.=~/'(j((9,)) is the ring class field of 3 f of conductor p,+l. Class field theory shows that Gal ( ~ , / ~ ) ~ Pic ((9,) ~ Pic ((9) x Z/(p + 1) Z x Z/p" Z and G a l ( Y / Q ) acts on G a l ( ~ , ] ~ ( ' ) via the nontrivial character. It follows that ~ , = ~f~o~ , , where :,f, is the extension of ~f" of degree p" inside o~o~, the anticyclotomic Zv-extension of Y. Thus (now using the fact that p does not split in og(,, and therefore splits completely in ~ / ~ f ' ) any embedding of ~ into q~ extends uniquely to an embedding of ~ , into ~u, and we can identify Gal(7~,/~) with Gal(Jt~X/~/~). Let N' denote the norm map from Jo(N)(~+) to Jo(N)(~6f'~), which we can identify with the norm map from Jo(N)(~) to Jo(N)(J~f~,) for every n.

L o c a l units, e l l i p t i c units, H e e g n e r p o i n t s a n d elliptic c u r v e s

415

Fixing a m a p 0 from Jo(N) to E defined over 9 as above, we define y, to be the image of the divisor [ x , ] - [ov] under the composition

Jo(N)(3~,) N, ) Jo(N)(~)~f~ir)_+ Jo(N)(~p) ~

(~+')~, E~(~',)~ s

E((p)

E(~U).

(10)

Here E 1 denotes the kernel of the reduction map: (p+I)2E(tP,)cEI(~P,) because (4), L e m m a 3.1 and the fact that 7~,/q) is totally ramified shows that E(7~,)/EI(~,,) has order ( p + 1) 2. The fifth m a p is the canonical m a p from the kernel of reduction to the formal group, and the final m a p is any isomorphism from /~ to E (see w3). Similarly we define y to be the image of I x ] - [oo]. The points y, y, depend on the choices of ~ , 0, and the embedding of o~' into (b but this will not concern us. Fix some such choice for the remainder of this section. The following proposition is crucial in using these Heegner points to study A [email protected] that N,,/, denotes the n o r m m a p from E(gJ,,) to E(~U). Proposition 6.1. (i) N . + 1/. Y.+ 1 = - Y . -

1 /f n>__ 1.

(ii) N1/o Yl = - Y (iii) y 0 = 0 .

(iv) Nn+2/nyn+:=-py,if n>O. Proof Let Tp denote the pth Hecke o p e r a t o r acting on Xo(N ) or Jo(N). By L e m m a 3.1, the Euler factor at p in L(~o,s)=L(f,s) is (1 + p l - 2 s ) - l . Therefore ap, the eigenvalue of Tp acting on f, is 0. It follows that OTp=O on Jo(N). In terms of the moduli data,

T(~.) = Z (c/n n L, L/n~C) where the sum ranges over the p + 1 lattices L of index p in (9,. If n__> 1 one of these lattices is (9,+ 1, one is pC,_ ~, and the other p - l represent the nontrivial classes in Pic((gn+l) which, when multiplied by (9,, become trivial in Pic((9,). Thus using (4.2) of [10], Tp(x.) = x._ 1 +

~

(x.+ 1)~.

(11)

?~Gal (3f'n + 1/',;r

N o w applying the m a p s (10) to (11) we conclude 0 = y , _ 1 + N,+ 1/,(Y,+ 1) since OTp=O. Applying Tp to x o yields in exactly the same way 0 = y + N1/0(yl). The p + 1 sublattices of (9 of index p represent the p + 1 elements of Pic((90) which b e c o m e trivial in Pic((9) when multiplied by (9. Thus, just as in (11) we get

Tp(x)=

F, ~E Gal (.~(o/Jt~)

(x0)'-

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K. Rubin

Now since yo~E(q)) applying the maps (10) gives 0 = ~ (y0)~= (p + 1) To. This completes the proof of (i), (ii) and (iii), and (iv) is immediate from (i). Corollary 6.2. y , ~ A + if n is even; y, e A - if n is odd. Proof Suppose k > l and Z is a character of Gal(gJk/Cb) which is not trivial on Gal(~Pk/g~k_l), i.e. Z is a character of conductor pk+l. If n < k then 2z(y,)=0. If n> _ n - k ,,tz(y,). If n - k is odd then applying (i) and (iv) of _ k then )~z(N,/k y,)--p Proposition 6.1, N,/k y, = _ ( __p)(n-k- 1)/2 Yk- l SO 2z(N,/ky,)=O. Similarly if Z is the trivial character and n is even, p. 2x(y,)= 2x(N,/o y,)=2x( ( _p),/2 To)=0. This proves the Corollary.

Proposition 6.3. For every E e g and map 0 as above there is an auxiliary quadratic field Jir satisfying (9) such that the corresponding Heegner point yeE(q~) is nonzero. Proof Write 0' for the map O'(z)=O([z]-Joel). Then 0' is a nonconstant map (defined over H) from X o ( N ) to E, and therefore finite-to-one. For any field .~f satisfying (9), x = x ( ~ , U ) e X o ( N ) ( ~ ) so the map N' of (10) on [ x ] - [ o e ] is multiplication by p + l . Thus the only way y = y ( • ) can be zero is if O'(x) is killed by ( p + l ) 2. But O'-l(E(p+l)2) is finite, so there are at most finitely many J{ for which y ( ~ ) can be zero. Since there are infinitely many ~" satisfying (9), this proves the Proposition. Remark. One could also prove this proposition by appealing to deep results of Gross-Zagier [11], Rohrlich [unpublished], and Waldspurger [20]. We did not need to do this because our Heegner point y~E(q)) comes directly from a global point in A y ( ~ ) without taking the norm to Ay(sC).

w7. Cohomology computations A reference for the Galois cohomology used below is [1]. Write G . = G a l ( ~ . / ~ ) . For any 0 [ G . ] - m o d u l e Z write h.(Z) for the Herbrand quotient h,( Z) = I/~~ Z)I/IH 1( G ,, Z)l. Standard results show that h,(Z1/ZE)=h,(Z1)/h,(Zz) and that h , ( Z ) = l if Z is finite. Fix E, off,, 0, and a collection y, Yl, Yz.... of Heegner points as in w Define A2 =A-nE(~U,), and write N, for the norm map N,/o. In this section we will use these Heegner points to study H I ( G -, A - ) = ~ H I ( G , , A2).

Local units, elliptic units, Heegner points and elliptic curves

417

L e m m a 7.1. Suppose y4=O. Then for all odd n, h,(A.-- ) w- p

n-

i.

Proof Since E(~u,) contains a submodule of finite index isomorphic to O[G,], A~- contains a submodule C of finite index,

C~-O[G~] @ O[Gk]/O[Gk_,]. k odd 3<_k<_n

An easy computation shows that /4~ O [Gk] ) =O/p "-k 0 and H I(G., O [Gk]) = 0. Therefore h,(A-,)=~,~2(n-1) H p2(n-k)/p2(n-k+l)=pn 1. k odd

3<_k<_n

Lemma 7.2. For all odd n, I/~~

,4,,)1 __
Proof /~~

A;)=(A~)G"/N,(A;); ( A d ) ~ = E ( ~ ) and by Proposition 6.1 and Corollary 6.2, N,(A~)D_ON,(y,)=p ("-1)/2 0y. Remark. This L e m m a is actually the only use we will make of Heegner points. The only fact we need is the existence of one sequence of points z,~A~- such that [p(~-l)/2E(~):ON,(z~)] is 1 (to prove Conjecture 2.2) or is bounded independent of n (to prove Theorem 9.1). Proposition 7.3. IH 1 (G,, A2)I __<[E(q0" Oy].

Proof We have IHI(G,,A2)I-=II~~

so the statement follows

from Lemmas 7.1 and 7.2.

Proposition 7.4. I(A-|

~ /E(q~)| (A- |

is finite. If Oy=E(q~) then G- =E(q~)|

Proof Since Proposition 7.3 holds for all choices of Heegner point y, this is immediate from Propositions 7.3 and 6.3 and L e m m a 4.4.

Corollary 7.5. rank A V - > 1.

Proof A |

If

not,

since

V~o

is

free

of

rank

2

we

would

have

Ep~) by Corollary 5.7. But then (A-@Dp) G ~(q)/O) 2,

which would contradict Proposition 7.4.

w8. Conclusions Proposition 8.1.

V +

~- V - ~- A and V +~ V - =0.

Proof Define W+={veVoo: for some nonzero FeA, F . v e V + } , and similarly for W-. By Corollaries 3.4, 5.8 and 7.5, V + c ~ V - = 0 and r a n k a V + = r a n k a V= l , so by L e m m a 4.1 W + and W are both free of r a n k l over A. Now if w+ is a generator of W -+, by L e m m a 2.1(iii) we conclude V + =F• W +, where F+ is a power series with simple zeroes at K Z - l ( r ) - 1 for all Z~E -v such that 6~(w+) 4=0 and no other zeroes. Thus V + ~ V - ~ A as well.

418

K. Rubin Define two sets of primes

,~+ = {p: for some E ~ g and period ~2~, ~2{1L (~oE, 1)~0 (rood p)} = {p: for some choice of Eeg, auxiliary field J{" satisfying (9) and map O: Jo(N)-~ E, the corresponding Heegner point y~pE(q~)}. Proposition 8.2. ;~+ has density 1.

Proof For any prime q = 7 (mod 8) fix a period •q of the Q-curve A(q) which satisfies (5) for all p. Rohrlich and Montgomery [13] have shown that for all of these q, L(~OA~ql,1)+ 0. If p > 5 does not split in Q(1//- q) then A(q)~g. Let S denote the complement of ~ , and for each q - 7 (mod 8) write Sq= {p: p splits in Q(]/--q) or p divides ~q-i L((PA~q) ' 1)}. Then each Sq has density 1/2, and furthermore if ql,q2, ...,q, are distinct primes then ~ Sq, has density 2-". Therefore S, which is contained in (~ Sq, has density 0. i= 1 q

Remark. Unfortunately the analogue for ~ of Proposition 8.2 is not obviously true: the question of for how many primes p a given point y~E(H~(f) lies in pE(~) seems very difficult. However, given y it is very easy (with a computer) to check this condition. Using computations of Stephens (unpublished, but for similar calculations see [2]) involving curves with CM by Q ( l f - I ) and Q(lf~- 3) I have verified the following: Computation. I f 5 < p < 1 0 0 0 and p ~ 1 (mod 12) then p ~ + n ~ . Proposition 8.3. IJpc?~ rank-1 A-module.

then V~,/V-~-A. l f p~.~+ then V~/V + is a torsionffree,

Proof If p ~ @ then by Proposition 7.4 and Corollary 5.7 HOmo(V~/V-,E;~.)G-'~q)/O. Therefore V~/V ~-A by Proposition 8.1 and Lemma 4.3. Now suppose p e ~ + . Recall from the proof of Proposition 8.1 that W+ ={v~V<: for some nonzero FeA, F . v e V +} is a free, rank 1 A-module. Choose an Ee~ ~ such that f2-~ L(~0, 1)~0 (mod p) and let { = {(E, f2) be the elliptic unit of Theorem 3.2 corresponding to E. Since L((p, 1)4=0, w~= 1 so ~ e V + c W + by Corollary 3.3. Since 3(~)=~ -~ L(~o, l)eO*, Lemma 4.2 shows that ~ generates W +, so V + = W +, and the second assertion follows. Theorem 8.4. Suppose peaA+ c~?~_. Then Conjecture 2.2 is true:

V+~-V ~-A

and

Vao=V+~V_

Proof By Propositions 8.1 and 8.3, Voo/V-, V + and V- are free of rank 1. Since pe~+, Theorem 3.2 and Corollary 3.3 show that there is a unit {eV + such

Local units, elliptic units, Heegner points and elliptic curves

419

that c~(~)~O*. Since 6 factors through V ~ / V , Lemma 4.2 shows that ~ generates V~,/V, so V~: = V + | V_ The next theorem is needed for the strengthening of Greenberg's theorem in w Recall W - = { v e V ~ : for some nonzero FeA, F . v e V }.

Theorem 8.5. For all p, 6lw =0.

Proof This can be deduced from Proposition 7.4 and Corollary 5.7 using standard methods of Iwasawa theory, but we can also argue directly as follows. By Proposition 7.4 and Corollary 5.7, Homo(VojV-, Epic)~ has a unique divisible subgroup D~-cl)/O. By L e m m a 2.1 (i) we can identify this subgroup as

D= {v--* iJ(v) z: zoEp.~ }. But since V~/W- has rank 1, H o m o ( V ~ ] W , Ep.,) c has a nontrivial divisible subgroup. Thus D must lie in the subgroup Homo(VojW-,Ep~J; of H o m o (V~,, Epic)a . i.e. 6 is 0 on W_

w9. Application : Greenberg's theorem With the word "supersingular" replaced by "ordinary", the following theorem was proved by Greenberg in [7]. In this section we use Theorem 8.5 to prove the supersingular version. It is worth noting that the methods of this paper do not yield a proof for ordinary primes.

Theorem 9.1. Let E be an elliptic curve defined over Q with complex multiplica-

tion, and suppose that the sign in the functional equation of L(E, s) is - 1. Then either E(Q) is infinite or the p-part of the Tate-Safarevic group of E over Q is infinite fi)r every prime p > 5 where E has good, supersingular reduction. Proof Fix a prime p where E has supersingular reduction (i.e. p does not split in the field K of complex multiplication). If F is any number field, Galois cohomology gives an exact sequence

0--~ E ( F ) | U ~(G(F/F), Ep,~) HI (G(F/F), E(F))p.. --, 0

(12)

which is a global version of (6). The Tate-Safarevic group IH(F) of E over F is the kernel of the localization maps

H' (G(F/F), E(F))-~ @ H 1( G(Fq/F,), E(F,)) q

the Selmer group Sp,~(F) is the HI(G(F/F), Epic) under (12). Thus and

0~-~ E(F)|

inverse

image

S~,~(F)--~ III(F)p, ~ 0

of

Ill(F)p~

in

(13)

and Galois cohomology shows that Sp~o(Q)~-Sp~(K) 6aI~K/QI. We first show that Sp~(K) is infinite. Set K~=K(Ep~). It is shown in [16], w1 that Sp~(K)= ker ( H o m o ( X ~, Ep~) Cal~K~jK)-~ Hl(G(~/cb), E(~5))) (14)

420

K. Rubin

where X ~ is the Galois group of the maximal abelian p-extension of K ~ which is unramified outside the prime above p. Consider the diagram Hom o (X

0-~ E(q~)|

~

~ ,

Ep~)GaI(K~

Homo(U~~ , Ep,~)C~l(~/e~--~

H~(G(@~), E(~)).

The bottom row is obtained from (6) using the isomorphism H 1(Gal (@~), Ep~) ~ H o m o (U~, Ep~) Gal(q'~/q'l which is deduced immediately from (6) and (7) of [16] by class field theory. Thus the bottom row is exact. By (14), Sp~(K)=ker(h), so f(Sp~(K)) =image(f)c~image(g). We will show that image(g) is contained in image(./'), so that Sp~(K) must be infinite. For each n let ~', denote the global units of the field K(Ep.§ ) which are congruent to 1 modulo the unique prime above p, and define

~u~=Oim % |

~ u~

where d//, denotes the closure of ~i, in }P*. Let {e~//~ he an elliptic unit attached to E as in Theorem 3.2. As a Az-module, ~//~ has rank 1 (see [-8]), so ~ll~/A2 ~ is Az-torsion. Projecting into V~ this implies (since ~ maps into V-) ~//~/~//~ c~((~-ir ~ W-, so by Theorem 8.5 6[ou =0. It then follows from the Explicit Reciprocity Law (Theorem 5.3) and Lemma 2.1(iv) that the image of the map g above is contained in H o m o (Uo~/~?/~,Ep~) Gal(q%/Oi, which by class field theory is precisely the image off. Since S_~(K) is infinite it contains a divisible submodule D ~ q~/O. Therefore Sp~(K)6"~(~Q~contains a copy of Qp/Zp, so Sp~(Q) is infinite as well. Now the theorem follows from (13).

w10. Application : p-adic L-functions For this section fix e = + 1, and fix a generator v of the free, rank-1 A-module V~. Suppose E s g ~ and let ~ V ~ be an elliptic unit as in Theorem 3.2. Write ~ = A ~ for the A-module of elliptic units attached to E. There is a unique power series FE(T)eA such that Fe.v=~, and then V~/~g~-A/F~A. F r o m Theorem 3.2 and L e m m a 2.1(iii) we see that for any anticyclotomic character ) ( ~

6~(v) F~(~z-'(r)- 1)=Q -1C(0z, 1). In other

words, the power series F E interpolates p-adically the values with the normalization factor 6z(v). Unfortunately we do not

g2-1L(CoZ, 1),

Local units, elliptic units, Heegner points and elliptic curves

421

know what the values fix(v) are, or even what their p-adic valuations are. Ideally one would like to show that 6x(V) is some sort of p-adic period. However at least these normalization constants are independent of E e ~ ~, and one can view F E as a sort of p-adic L-function attached to E in the anticyclotomic Zp-extension. By a theorem of Rohrlich [14] we can conclude that if E is defined over Q, then FE#:0. We also have the following: L e m m a 10.1. I f Conjecture 2.2 is true for p and v is a generator of V ~ then for all Z ~ ~, 6x(v)~=O.

Proof Fix such a Z. If 6z(v)=0 then 6 x must be 0 on all of V+@V - =V~, and by L e m m a 2.1(v) the same must be true for all characters Z' with the same conductor as X. Choose a nonzero yeE(~U~) such that 2 p ( y ) = 0 if the conductor of p is different from the conductor of )~. (If the conductor of Z is p"+ 1 this just amounts to choosing y~E(T,) such that N,/,_ l(y)=0.) Then by (8) (y| w) = 0 for every k and every weV~, which contradicts Proposition 5.4. Thus if Conjecture 2.2 is true for p (for example if p e ~ + r ) then we can write FE(t
Then the question raised

Question. W h a t is the relation between V ~ / ~ and A~/Ay? This is the analogue of a question asked by M a z u r when the curve E has ordinary reduction. F o r some evidence that there is a relation, see [10].

References 1. Atiyah, M., Wall, C.: Cohomology of groups. In: Algebraic number theory. Cassels, J.W.S., FrShlich, A. (eds.), pp. 94-I 15. London: Academic Press 1967 2. Birch, B., Stephens, N.: Computation of Heegner points. In: Modular forms. Rankin, R.A. (ed.), pp. 13-41. Chichester: Ellis Horwood Ltd (1984) 3. Coates, J.: Elliptic curves and lwasawa theory. In: Modular forms. Rankin, R.A. (ed.), pp. 51-73. Chichester: Ellis Horwood Ltd (1984) 4. Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39, 223251 (1977)

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5. Coates, J., Wiles, A.: On p-adic L-functions and elliptic units. J. Austr. Math. Soc. 26, 1-25 (1978) 6. Coleman, R.: Division values in local fields. Invent. Math. 53, 91-116 (1979) 7. Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72, 241 265 (1983) 8. Greenberg, R.: On the structure of certain Galois groups. Invent. Math. 47, 85-99 (1978) 9. Gross, B.: Arithmetic on elliptic curves with complex multiplication. Lect. Notes Math. vol. 776. Berlin-Heidelberg-New York: Springer 1980 10. Gross, B.: Heegner points on Xo(N ). In: Modular forms. Rankin, R.A. (ed.), pp. 87-105. Chichester: Ellis Horwood Ltd (1984) 11. Gross, B., Zagier, D.: Points de Heegner et d6riv6es de fonctions L. C.R. Acad. Sci. Paris 297, 85-87 (1983) 12. Hazewinkel, M.: On norm maps for one-dimensional formal groups III. Duke Math. J. 44, 305-314 (1977) 13. Montgomery, H., Rohrlich, D.: On the L-functions of canonical Hecke characters of imaginary quadratic fields II. Duke Math. J. 49, 937-942 (1982) 14. Rohrlich, D.: On L-functions of elliptic curves and anticyclotomic towers. Invent. Math. 75, 383-408 (1984) 15. Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. invent. Math. 64, 455-470 (1981) 16. Rubin, K.: Elliptic curves and Zp-extensions. Compos. Math. 56, 237 250 (1985) 17. Shimura, G.: On elliptic curves with complex multiplication as factors of the jacobians of modular function fields. Nagoya Math. J. 43, 199-208 (1971) 18. Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Princeton: Princeton Univ. Press 1971 19. Tate, J.: WC-groups over p-adic fields. S6minaire Bourbaki (1957/1958) expos6 156 20. Waldspurger, J-L.: Sur les valeurs de certaines fonctions L automorphes en leur centre de sym6trie. Compos. Math. 54, 173-242 (1985) 21. Wiles, A.: Higher explicit reciprocity laws. Ann. Math. 107, 235-254 (1978) 22. Wintenberger, J-P.: Structure galoisienne de limites projectives d'unit6s locales. Compos. Math. 42, 89-103 (1981) 23. Yager, R.: On two variable p-adic L-functions. Ann. Math. 115, 411 449 (1982)

Oblatum 18-IV-1986 & 6-X-1986