On the Finiteness of X for Motives Associated to Modular Forms
Amnon Besser1 Received: February 1, 1996
Revised: January 15, 1997 Communicated by Don Blasius
Abstract. Letf be a modular form of even weight on 0(N) with asso- ciated motive Mf. Let K be a quadratic imaginary eld satisfying certain standard conditions. We improve a result of Nekovar and prove that if a rational primepis outside a nite set of primes depending only on the form f, and if the image of the Heegner cycle associated with K in the p-adic intermediate Jacobian of Mf is not divisible by p, then the p-part of the Tate-Safarevic group of Mf over K is trivial. An important ingredient of this work is an analysis of the behavior of \Kolyvagin test classes" at primes dividing the level N. In addition, certain complications, due to the possibility of f having a Galois conjugate self-twist, have to be dealt with.
1991 Mathematics Subject Classication: 11G18, 11F66, 11R34, 14C15.
1 Introduction
Letf be a new form of even weight 2rfor the group 0(N), letMf be ther-th Tate twist of the motive associated to f by Jannsen [Jan88b] and Scholl [Sch90]. For all but a nite number of primespthere is a canonical choice of freeZp-latticeTp(Mf) with a continuous action of Gal(Q=Q) such thatTp(Mf)Q is thep-adic realization ofMf. In [Nek92], Nekovar showed that under certain assumption one could apply the Kolyvagin method of Euler systems toMf and obtained, among other things, the following result:
Theorem 1.1. LetK be a quadratic imaginary eld of discriminant D in which all primes dividing N split, and let p be a prime not dividing 2N. LetTp(Mf) be the p-adic realization ofMf and letP(1) be the image inH1(K;Tp(Mf)) of the Heegner cycle associated with K under the p-adic Abel-Jacobi map. If P(1) is not torsion, then thep-part of the Tate-Safarevic group ofMf overK,Xp(Mf=K), is nite.
1Partially supported by an NSF grant
We remark that in [Nek92] there is a stronger condition onpfor the theorem to hold which is removed in a remark on the last paragraph of [Nek95].
The purpose of this note is to give the following renement of the above result:
Theorem 1.2. There is a nite set of primes (f), depending only on f, such that for a prime p not in (f) the following holds: for K as in theorem 1.1, if P(1) is not torsion, thenp2IpXp(Mf=K) = 0, where Ip is the smallest non-negative integer such that the reduction of P(1) toH1(K;Tp(Mf)=pIp+1) is not 0. In particular, if
Ip= 0, thenXp(Mf=K) is trivial
Remark 1.3. 1. The Tate-Safarevic group discussed here is not exactly the same as the one that appears in [Nek92]. The main dierence is in the local conditions at the primes of bad reduction. Nekovar makes no conditions at these primes, which is whyX comes out too big. The local condition that we use is the one dened by Bloch and Kato. The analysis of this local condition is one of the main ingredient of this work.
2. The nite set (f) contains the primes dividing 2N and primes with an excep- tional image of Gal(Q=Q) in Aut(Tp(Mf)) (see denition 6.1).
It is our hope that the methods used here allow a complete analysis of the struc- ture ofXp(Mf=K) in terms of various Kolyvagin classes following [Kol91, McC91].
Notice however that some diculties are already visible in the fact that the power of pannihilatingX is 2Ip whereas in the elliptic curves case one gets annihilation by pIp. This diculty is caused by the more complicated structure of the image of the Galois representation associated toMf (see remark 6.5).
A natural problem raised by theorem 1.2 is to bound the numbers Ip. In par- ticular, one would hope that Ip = 0 for all but a nite number of p's. This would show the niteness ofX(Mf=K) except for possible innite contribution at primes dividing 2N. It is useful to compare the situation to the case where the weight off is 2, where the triviality ofXp(Mf=K) for almost allphas been previously established in [KL90]. In that case, the class P(1) correspond to a point on the Jacobian of a modular curve, andIp = 0 for almost allpwheneverP(1) is of innite order. This last result uses essentially the injectivity of the Abel-Jacobi map (up to torsion) and the Mordell-Weil theorem, neither of which is known for greater than 1 codimension cycles. One possible way of getting some control over the indicesIpcould be to use the results of Nekovaron thep-adic heights of Heegner cycles: According to [Nek95, corol- lary to theorem A] one has the equalityh(P(1);P(1)) = fK;pL0p(f K;r) where h(; ) is thep-adic height pairing dened by Nekovar and Perrin-Riou,Lp(fK) is a p-adicL-function off overKdened by Nekovar and fK;pis somep-adic period.
Thep-adic height of elements ofHf1(K;T) has a bounded denominator (it is integral for universal norms from a Zpextension) and so the estimation of Ip is reduced to giving estimates on thep-divisibility ofL0p(fK;r).
Another problem is to handle primes dividing 2N. The diculty here is that we do not understand yet the image of the Abel-Jacobi map withQp coecients for varieties over an extension ofQp and with bad reduction. Recently there has been some progress on that problem [Lan96] but the results do not yet cover the cases we need.
Here is a short description of the contents. After a few preliminary remarks and denitions in section 2 we will recall in section 3 some of the main points of [Nek92].
For brevity this will be far from a full account. We merely attempt to indicate the main changes that need to be made and explain where the local conditions at the bad primes come into play. These conditions are then discussed in sections 4 and 5. We then give the proof of the main theorem in section 6. It would have been nice to skip this section or make it shorter and refer instead to the corresponding sections in [Nek92].
However, it turns out that to get the result we want under weaker conditions than the ones stated there (see the remark in loc. cit. page 121), the proof has to be modied somewhat. I have therefore chosen to give the full details of the proof. In the appendix we give a proof of a Hochschild-Serre spectral sequence for continuous group cohomology which is used in section 5.
As the reader will notice, this work is closely related to [Nek92]. Familiarity with that paper is helpful for reading this one but not necessary, as one may choose to trust the results quoted from there.
I would like to thank Wayne Raskind, Don Blasius, Haruzo Hida, Dinakar Ra- makrishnan and Jan Nekovar for helpful discussions and remarks. I would also like to thank Farshid Hajir for encouraging me to write down my ideas on this subject.
Finally, I would like to thank the referee for some useful corrections and remarks.
2 preliminaries
For this work, a motive is eectively equivalent to its set of realizations. We only need thep-adic realizations for the dierentp's and a brief mention of the Betti realization.
Thus, a motiveM has a Betti realization which is a Q-vector space VQ and p-adic realizations which are continuous representations of Gal(Q=Q) onVp=VQQpfor the dierentp's. By choosing a suitableZ-latticeTZinVQwe have in eachVpan invariant
Zp-lattice Tp =TZZp. The p-part of the Tate-Safarevic group of M depends on the choice ofTp but statements about thep-part for all but a nite number ofpare clearly independent of the choice of TZ. In the cases we will be considering there is a standard choice (a Tate twist of a piece of the etale cohomology of a suitable Kuga-Sato variety, see [Nek92,x3]) and the theorem will be proved for this choice.
To be more precise:
TpQp =f;pQp(r); (2.1) wheref;pis the standardp-adic representation associated tof.
To dene the p-part of X, we start with the free Zp-module of nite rank, T =Tp(M), on which Gal(Q=Q) acts continuously. LetV =TQp andA=V=T, so that there is a short exact sequence:
0!T !i V pr!A !0:
Let`be a prime, possibly1. LetF be a nite extension ofQ` and let F be an algebraic closure ofF. In [BK90, (3.7.1)] Bloch and Kato dene the nite partHf1of
the rst Galois cohomology ofF with values inV, T orAas follows:
Hf1(F;V) := KerH1(F;V) res!H1(Fur;V) when`6=p; Hf1(F;V) := KerH1(F;V)!H1(F;V Bcris) when`=p; Hf1(F;T) :=i 1Hf1(F;V);
Hf1(F;A) := ImHf1(F;V),!H1(F;V) pr!H1(F;A);
where Fur is the maximal unramied extension of F. The ring Bcris is dened by Fontaine. We will not need to use the denition directly in the case`=p.
Let nowKbe a number eld. WhenBis a Gal(Q=K)-module we have restriction maps for each placev of K: H1(K;B)!H1(Kv;B). When x2H1(K;B) we will denote its restriction toH1(Kv;B) byxv. Thep-part of the Selmer group ofMover Kis now dened as
Selp(M=K) := KerH1(K;A) !Y
v H1(Kv;A)=Hf1(Kv;A); where the product is over all placesv ofK. We also dene
Hf1(K;V) := KerH1(K;V) !Y
v H1(Kv;V)=Hf1(Kv;V):
Thep-part of the Tate-Safarevic group of M over K is the quotient of Selp(M=K) by the image of Hf1(K;V). Nekovar denes the same group as the quotient of the Selmer group by the image of an appropriate Abel-Jacobi map. It follows easily from his result that in the case of interest here his denition coincides with the one we are using.
LetApkbe thepk-torsion subgroup ofAand let redpk:T !Apkbe the reduction modpk. We will use the same notation for the reduction mapApn !Apk which is given by multiplication bypn k whenn > k and we notice that all reduction maps commute with each other. We will abuse the notation further to denote by redpk the maps induced by the reduction on Galois cohomology groups.
To simplify the notation slightly, we assume the following:
Assumption 2.1. There is a Galois invariant bilinear pairing T T ! Zp(1) such that the induced pairings onT=pk=Apk are non-degenerate for allk.
This condition is satised in the case we are considering by [Nek92, proposition 3.1]. It is mostly made at this point so that we do not have to consider bothT and its Kummer dual. We have the following well known results:
Proposition 2.2. The pairing above induces local Tate pairings, for each place v of K:
H1(Kv;T)H1(Kv;A)!H1(Kv;Qp=Zp(1))=Qp=Zp; H1(Kv;Apk)H1(Kv;Apk)!H1(Kv;Z=pk(1))=Z=pk;
which are both perfect and will be denoted by h; iv (for the torsion coecients case see [Mil86, Chap. I, Cor. 2.3]). The following properties hold:
1. [BK90, Proposition 3.8] The pairing h ; iv makes Hf1(Kv;T) and Hf1(Kv;A) exact annihilators of each other (this is true even in the casepjv).
2. Ifx andy belong toH1(K;Apk) then
X
v
hxv;yviv= 0;
where the sum is over all places v ofK but is in fact a nite sum.
We remark that it is possible to neglect the innite places in all the discussions if we assume thatp6= 2 or ifK is totally imaginary. Both conditions will in fact hold.
Definition 2.3. LetF be a local eld. We deneHf1(F;Apk) to be the preimage in H1(F;Apk) ofHf1(F;A). We deneHf1(F;Apk) to be the annihilator ofHf1(F;Apk) in H1(F;Apk) under local Tate duality. We will call the classes in Hf1(F;Apk) the dual nite classes. We dene the singular part of the cohomology as
Hsin1 (F;Apk) =H1(F;Apk)=Hf1(F;Apk)
(this denition is due to Mazur). Ifx2H1(F;Apk) we denote by xsin its projection on the singular part. WhenK is a number eld we let
Sel(K;Apk) := KerH1(K;Apk) !Y
v H1(Kv;Apk)=Hf1(Kv;Apk):
Lemma 2.4. The group Hf1(F;Apk) is the image of Hf1(F;T) under the canonical mapH1(F;T)!H1(F;Apk). There is a perfect pairing, induced by h; iv:
h ; iv :Hf1(F;Apk)Hsin1 (F;Apk)!Z=pk
Proof. This is a formal consequence of the preceding denition and proposition 2.2.
For a Gal( F=F)-moduleB and F K F we denote BGal (F=K ) byB(K). If B0 is a subset of B we denote byF(B) the xed eld of the subgroup of Gal( F=F) xingB0.
3 Method of proof
The Kolyvagin method, as applied toMf by Nekovar, works as follows: Letf have q-expansionf =Panqn. LetE be the eld generated overQ by theai. It is known thatEis a totally real nite extension ofQ. LetOE be the ring of integers ofE. As explained in [Nek92, Proposition 3.1], the invariant latticeTp(Mf) can be taken to be a free rank 2 module overOEZp=QOEp, where the product is over all primes
pof E dividing p. To prove the result about X it is sucient to choose one such primepand consider only the direct summand of Tp(Mf) corresponding top. This summand will be denotedTf;p. For the rest of this section we xT =Tf;p and let as usualV =TQp andA=V=T.
As the Tate-Safarevic group is (obvious with the above denition)p-torsion, we wish to show that its part killed bypk is killed by the xed powerp2Ip for eachk. We look at the short exact sequence
0!Apk !A p!k A !0 and the induced sequence on cohomology
0!A(K)=pk !H1(K;Apk)!H1(K;A)pk !0
The conditions we will impose on the prime p imply, as we will see in part 2 of proposition 6.3, thatA(K) = 0, and henceH1(K;A)pk =H1(K;Apk). It follows that the preimage inH1(K;Apk) of Selp(T=K) is Sel(K;Apk). SinceP(1)2Hf1(K;V) it will be enough to show that Sel(K;Apk)=(OEp=pk)P(1) is killed byp2Ip.
Choose once and for all a complex conjugation 2Gal(Q=Q). LetS(k) be the set of primes`satisfying:
`-NDp;
`is inert inK;
pk dividesa` and`+ 1;
`+ 1a` are not divisible bypk+1.
Remark 3.1. The rst 3 conditions are equivalent to Frob(`) and being conjugates in Gal(K(Apk)=Q). The last condition can be arranged for innitely many `'s (see proposition 6.10).
Letn be a product of distinct primes` 2S(k). Nekovar associates with n a coho- mology classyn 2H1(Kn;T), where Kn is the ring class eld ofK of conductor n. The classesyn are dened as the images of certain CM cycles under the Abel-Jacobi map ofMf. When n=m`the relation
corKn;Km(yn) =a`ym
holds, as well as some local congruence condition which we will not discuss here.
Let Gn := Gal(Kn=K1). Then Gn = Q`jnG`. For each prime ` 2 S(k) we associate the elementD`2Z[G`] which is given by
D`=X`
i=1ii; G`=hi;
and let Dn = Q`jnD` 2 Z[Gn]. One now notices, following Kolyvagin, that Dn(redpkyn)2H1(Kn;Apk) isGn-invariant. By [Nek92, Proposition 6.3]
pMApk(Kn) = 0; (3.1)
with some constantMindependent ofnandk. An application of the ination restric- tion sequence shows that there is a canonically dened class zn 2 H1(K1;Apk 2M) such that
resK1;Knzn=Dn(redpk 2Myn):
Indeed, one has the commutative diagram with exact ination restriction rows:
H1(K1;Apk) resK1;Kn! H1(Kn;Apk)Gn ! H2(Gn;Apk(Kn))
redpk M
?
?
y
redpk M
?
?
y
redpk M
?
?
y
H1(K1;Apk M) resK1;Kn! H1(Kn;Apk M)Gn ! H2(Gn;Apk M(Kn)) and the rightmost vertical map is 0 by (3.1) because the reduction map kills pM torsion. It follows that
redpk M yn2Im resK1;Kn :H1(K1;Apk M)!H1(Kn;Apk M):
We get the canonical classzn by further reduction as in [Nek92,x7]. Finally, dene P(n) := corK1;Kzn:
Note the important dierence between Nekovar's denition of the same classes and ours: in Nekovar's denition resK1;Knzn=pMDn(redpk Myn). To simplify the nota- tion, we may notice that the denition is entirely independent of the value ofM. To dene classes in the cohomology ofApr we need to start withnwhose prime divisors satisfy certain congruences depending onrandMand we may freely assume that we have chosen then correctly whatever the congruences are. It will be convenient to make the change of variablek=k 2Mhere. Note thatP(1) can be considered mod pk for any kand its denition is independent ofM.
Proposition 3.2. The classes P(n) enjoy certain fundamental properties:
1. P(n) belongs to the ( 1)par(n)"L-eigenspace of the complex conjugation acting onH1(K;Apk), wherepar(n) is the parity of the number of prime factors inn and"L is the negative of the sign of the functional equation of L(f;s).
2. For a place v of K such that v-Nn,P(n)2Hf1(Kv;Apk).
3. Ifn=m`andis the unique prime ofKabove`, then there is an isomorphism between Hf1(K;Apk) and Hsin1 (K;Apk) which takes P(m) to P(n);sin. In particular, ifP(m)6= 0, thenP(n);sin6= 0.
Proof. This is [Nek92, Proposition 10.2] with a couple of modications. First of all we remark that there is a miss-print in [Nek92] and the eigenvalue of on P(n) is indeed ( 1)par(n)"Las can be seen from the proof. To get the second statement when v- pwe note that if such av is a prime of good reduction one has Hf1(Kv;Apk) = Hf1(Kv;Apk) =Hur1 (Kv;Apk) (see lemma 4.4) and that the auxiliary power ofpthat appear in [Nek92] is not needed here because of the change in the denition ofP(n) alluded to above. The casevjpfollows from [Nek92, Lemma 11.1]. Here, two remarks are in place: First of all, Nekovar uses the comparison theorem of Faltings for open varieties [Fal89]. As is well known, this result is not universally accepted. However, in the last 2 years Nekovar himself [Nek96] and Nizio l [Niz97, Theorem 3.2] have supplied alternative proofs that the image of the Abel-Jacobi map lies insideHf in the case of good reduction. The second remark is that this is all we need because our assumptionp-2N imply thatvjpis a place of good reduction.
One of the main points of this work is to analyze the dual nite conditions at primes of bad reduction and to show that by further reduction (i.e. by possibly increasingM) one may assume that the classes P(n) are dual nite at these primes (see corollary 5.2).
4 Finite and dual finite conditions at `
LetF be a nite extension ofQ` (`6=p) and letT be a freeZp-module of nite rank with a continuous action ofG= Gal( F=F). Again letV =TQp andA=V=T. Let I= Gal( F=Fur) be the inertial group. We assume the following condition is satised (as is in the case at hand, see [Nek92, proposition 3.1]):
Condition 4.1. There is a Galois invariant, non-degenerate bilinear pairingVV !
Qp(1) andVI( 1) has no nontrivial xed vector with respect to any power of Frobenius (true ifVI has no part of weight 2).
Proposition 4.2. Under the above condition there exists a constantM such that for any nite unramied extensionL=F we have
1. pMH1(Lur;T)Gal(Lur=L)= 0;
2. Hf1(L;V) =H1(L;V);
3. V(L) = 0.
Proof. The second statement immediately follows from the rst. For the rst state- ment we begin by noticing thatI is independent ofL. By making a nite ramied extension we may assume that the action ofI factors through thep-primary part of its tame quotient. It then follows thatH1(I;T)=TI( 1) as Gal(Lur=L)-modules. The condition now implies thatTI( 1) is a direct sum of a torsion group and a Zp-free module on which Frobenius has no invariants. Finally, the third statement follows since by duality one gets that 1 is not an eigenvalue of any power of Frobenius on VI.
Remark 4.3. If T is the Tate module of an elliptic curve with split semi-stable re- duction, then the constant M is essentially the p-adic valuation of the number of components of the special ber ofE.
It follows from part 2 of proposition 4.2 that for any nite unramied extension L=F we haveHf1(L;T) =H1(L;T), and therefore by lemma 2.4 we get
Hf1(L;Apk) = ImH1(L;T) red!H1(L;Apk):
Lemma 4.4. If theG-moduleT is unramied, then for any Las above
Hf1(L;Apk) =Hf1(L;Apk) =Hur1 (L;Apk) := KerH1(L;Apk)!H1(Lur;Apk): Proof. It is enough to show the second equality as the condition of being unramied is self dual. It is clear that any class inHf1(L;Apk) is unramied. Conversely, a class in Hur1 (L;Apk) is inated from H1(Lur=L;Apk). Since Gal(Lur=L)= ^Z, H1 is just coinvariants. It follows that the reduction mapH1(Lur=L;T)!H1(Lur=L;Apk) is surjective.
5 The local condition under restriction
Keeping the assumption of the previous section, suppose now that L=F is a nite unramied extension with Galois group . The short exact sequence 0 ! T p!k T redpk!Apk !0 gives rise to the following commutative diagram with exact rows:
0 ! H1(F;T)=pk redpk! H1(F;Apk) ! H2(F;T)pk ! 0
resF;L
?
?
y
resF;L
?
?
y
resF;L
?
?
y
0 ! H1(L;T)=pk redpk! H1(L;Apk) ! H2(L;T)pk
(5.1) Given x 2 H1(F;Apk) such that resF;Lx is in Hf1(L;Apk), we would like to know how far isxfrom being inHf1(F;Apk). In view of (5.1) the obstruction is given by
KerH2(F;T)pk resF;L!H2(L;T)pk: (5.2) Proposition 5.1. The kernel (5.2) is annihilated by a constantpM independent of kandL.
Proof. Since is nite, there is a Hochschild-Serre spectral sequence E2i;j=Hi(;Hj(L;T)))Hi+j(F;T):
Note that the cohomology here is the continuous cohomology. The Hochschild-Serre spectral sequence does not exist in general for continuous cohomology. A proof that it does exits in our case is found in the appendix. Fori+j= 2 the spectral sequence converges to a ltrationF0F1F20 on H2(F;T) with
F1= KerH2(F;T) resF;L!H2(L;T);
F1=F2=E11;1=E31;1= KerH1(;H1(L;T))!H3(;T(L))
=H1(;H1(L;T));
F2=E12;0E22;0=H2(;T(L)) = 0; sinceT(L) = 0 by part 3 of proposition 4.2. Therefore,
KerH2(F;T)pk resF;L!H2(L;T)pk
=H1(;H1(L;T))pk:
Applying the ination restriction sequence to Gal(Lur=L)/Gal(L=L) andT we nd 0 !H1(Lur=L;T(Lur)) !H1(L;T)!H1(Lur;T)Gal(Lur=L) !0:
The right exactness is a consequence of the fact that Gal(Lur=L) = ^Z has co- homological dimension 1. Applying the Hochschild-Serre spectral sequence to Gal(Lur=L)/Gal(Lur=F) and T(Lur) we nd that H1(;H1(Lur=L;T(Lur))) in- jects into H2(Lur=F;T(Lur)) and is therefore 0 since Gal(Lur=F) = ^Z. Therefore, H1(;H1(L;T)),!H1(;H1(Lur;T)Gal(Lur=L)) and the result follows from propo- sition 4.2
Corollary 5.2. Let pM be the constant given by proposition 5.1. Then, if x H1(F;Apk+M) and resF;Lx2Hf1(L;Apk+M), then redpkx2Hf1(F;Apk).
Proof. The commuting diagram with exact rows
0 ! T pk+M! T redpk+M! Apk+M ! 0
pM
?
?
y
=
?
?
y
redpk
?
?
y
0 ! T pk! T redpk! Apk ! 0 gives rise to
H1(F;T) redpk+M! H1(F;Apk+M) ! H2(F;T)pk+M
=
?
?
y
redpk
?
?
y pM
?
?
y
H1(F;T) redpk! H1(F;Apk) ! H2(F;T)pk
The corollary now follows by a diagram chase on this last diagram as well as on (5.1) withkreplaced byk+M.
6 Proof of theorem 1.2
In this section we give the proof of the main theorem using a variant of the Kolyvagin argument following mostly [Gro91]. By proposition 3.2 and corollary 5.2 we may assume that the classP(n) is dual nite at all primes which do not dividen. Recall that this involves xing some large integerM, constructing the classes modulopk+M and then reducing them modpk.
We will concentrate on the case wheref has no CM. The CM case can be handled similarly (see the remark in [Nek92] page 121). Recall thatE is the eld generated by the Fourier coecients of the formf. We rst exclude primespwhich are ramied in E. Ifpis not excluded, letpbe a prime ofEabovepand recall that we are considering T =Tf;p which is a rank 2 freeOEp-module with an action of Gal(Q=Q). Let again f;p be the p-adic representation associated with f. Consider the p component of f;p which is a representation of Gal(Q=Q) on a 2-dimensionalEp vector spaceVf;p. According to a result of Ribet [Rib85, theorem 3.1] if p is outside a nite set of primes then there is a subeldE0 ofEp such that in an appropriate basis the image of Gal(Q=Q) in Aut(Vf;p)= GL2(Ep) contains
fg2GL2(OE0); detg2((Zp)2r 1)g
(in fact, the result of Ribet is stronger and treats the image of Galois in all the completions ofE abovepsimultaneously), and therefore contains in particular
fg2GL2(Zp); detg2((Zp)2r 1)g: (6.1) We exclude all other primes and the prime 2. This concludes our exclusions which we may sum up in:
Definition 6.1. The set (f) of excluded primes for theorem 1.2 is the set contain- ing the primes dividing 2N, primes that ramify inE =Q(ai) and primes where the image of Gal(Q=Q) in Aut(Vf;p) does not contains (6.1) (in some basis).
We consider non excluded primes from now onward.
Lemma 6.2. Let ~Gp be the image of Gal(Q=Q) in Aut(T) = GL2(OEp) (p not ex- cluded). Then, ~Gp contains a subgroup conjugate toGL2(Zp).
Proof. By (2.1), T Ep is just the r-th Tate twist of Vf;p. From that and Ribet's theorem it follows easily that after xing an appropriate basis for T every matrix A 2 GL2(OEp) has a scalar multiple in ~Gp. Since SL2(OEp) is the commutator subgroup of GL2(OEp), it follows that SL2(OEp)G~p. The lemma follows because for almost all`, Frob(`) has determinant` 1 and because ~Gp is closed.
Let F = OEp=pk. Let Gpk = Gal(Q(Apk)=Q) be the image of Gal(Q=Q) in Aut(Apk)= GL2(F). Then,Gpk contains a groupG0pk conjugate to SL2(Z=pk).
Proposition 6.3. LetL=K(Apk).
1. When k= 1, Ap is an irreducible F[Gal(L=K)]-module.
2. Hi(Gal(L=K);Apk) = 0 for all i0.
3. There is a natural pairing [ ; ] : H1(K;Apk)Gal(Q=L)! Apk inducing an isomorphism ofF-modules H1(K;Apk)= HomGal (L=K)(Gal(Q=L);Apk).
4. The F-module Apk is the direct sum of its 1 eigenspaces with respect to the generator ofGal(K=Q), each free of rank 1.
Proof. Since SL2(Fp) has no nontrivialZ=2 quotients whenp > 2 and Gal(L=K) is of index at most 2 in Gp, it follows that Gal(L=K) contains G0p and therefore that Ap is an irreducible F[Gal(L=K)]-module. It also follows that Gal(L=K), consid- ered as embedded in Aut(Apk), contains the central Subgroup of order 2 generated by 1. Since p 6= 2, Hi(1;Apk) = 0 for all i 0 and the second assertion fol- lows from the Hochschild-Serre spectral sequenceHi(Gal(L=K)=1;Hj(1;Apk))) Hi+j(Gal(L=K);Apk). An ination restriction sequence now implies that
H1(K;Apk)=H1(L;Apk)Gal(L=K)= HomGal (L=K)(Gal(Q=L);Apk)
hence the third assertion. Finally, part 4 follows because the determinant of on T is 1.
Let S be a nitely generated F-submodule of H1(K;Apk). We consider the elements of S as elements of HomGal(L=K)(Gal(Q=L);Apk) and let LS be the eld xed by the common kernel of these elements. The following lemma is immediate:
Lemma 6.4. The pairing[; ] induces a pairing
[; ]S :SGal(LS=L)!Apk;
which in turn induces an injection
Gal(LS=L),!HomF(S;Apk) as Gal(L=K)-modules. (6.2) This injection has the property that
x2S and[x;Gal(LS=L)]S= 0 =)x= 0: In addition, this pairing induces an injection
S ,!HomGal (L=K)(Gal(LS=L);Apk) asF-modules
Remark 6.5. Unlike the situation for elliptic curves [Gro91, proposition 9.3] we can not in general expect the injection (6.2) to be an isomorphism. For instance, ifGpk is contained in GL2(Z=pk), then there might exist a homomorphism: Gal(Q=L)! Apk whose image is contained in (Z=pk)2. If we take S to be theF-span of, then Gal(LS=L) = (Z=pk)2 and is not in general an F-module whereas HomF(S;Apk) is.
The failure of (6.2) to be an isomorphism forces some changes in the nal arguments.
Our chosen complex conjugation acts on all the groups above. We will denote byG the1-eigenspace of acting on an abelian groupG.
Lemma 6.6. LetCHomF(S;Apk) be a Gal(L=K)-submodule with the property that x2S and[x;C]S = 0 implyx= 0. Let 06=s2S and leta2HomF(S;Apk)+. Let
C0=a+C+; C0 0=fc2C0; [s;c]S 6= 0g:
Then,C0 andC00 have the same property asC with respect to eigenvectors of inS, that is, ifx2S and[x;C0]S = 0 or [x;C0 0]S= 0, then x= 0.
Proof. Suppose rst that [x;C+]S= 0. ThenF[x;C]S is anF[Gal(L=K)]-submodule ofApk which is contained in the proper submoduleApk. Consideringp-torsion and using part 1 of proposition 6.3 one nds that F [x;C]S is trivial. It follows in particular that [s;C+]S is non trivial and since p3 it contains at least 3 elements.
From that it follows that for anyc2C+one may always nd c1; c22C+ such that c= (a+c1) (a+c2) and [s;a+ci]S 6= 0 fori= 1;2. The lemma follows easily.
Lemma 6.7. Let`be a prime inS(M+k). Then,`is inert inK. Letbe the unique prime ofK above `. Then, for any choice of Frob() in a decomposition group of , Frob() acts trivially on Apk and therefore splits completely inL.
Proof. Both assertions follow from remark 3.1. In Gal(K=Q), Frob(`) = hence`is inert inK. It now follows that Frob() is conjugate to2and is therefore the identity onApk.
Let` and be as in the previous lemma, let0 be a prime of LS above and let Frob(0)2Gal(LS=L) be the associated Frobenius substitution. It is easy to see that the formula
0(x) := [x;Frob(0)]S
denes an element of HomF(S;Apk) which depends only on ` up to conjugation on Apk by some element of Gal(L=K). Using lemma 6.7 one has:
Lemma 6.8. There is aGal(K=Q)-equivariant isomorphism
Hf1(K;Apk)=H1(Kur=K;Apk)=Apk; (6.3) where the last step is evaluation at the Frobenius. If x 2 H1(K;Apk) and x 2 Hf1(K;Apk) then, up to conjugation as before, the image of x under this isomor- phism is0(x).
Lemma 6.9. Letbe as above.
1. The pairing h; i dened in lemma 2.4 induces nondegenerate pairings:
h; i :Hf1(K;Apk)Hsin1 (K;Apk) !Z=pk:
2. Both Hf1(K;Apk) and Hsin1 (K;Apk) are direct sums of their 1 eigenspaces with respect to. All eigenspaces are free of rank1 over F.
Proof. The rst assertion follows since h ; i is Gal(L=K) equivariant. The second assertion follows forHf1(K;Apk) by lemma 6.8 and part 4 of proposition 6.3 and the same now follows forHsin1 (K;Apk) by the rst assertion.
Proposition 6.10. Let x; y 2 S and suppose that y 6= 0. Then there exists some
`2S(M+k) such that y6= 0. If for almost all `2S(M+k) with y 6= 0 we have x= 0, then x= 0.
Proof. Let LM = K(ApM+k+1). Let C be the image of Gal(Q=LM) in Gal(LS=L).
We rst claim that when considered in HomF(S;Apk), C satises the assumption of lemma 6.6. To show that, we rst notice that the same argument used to prove that Hi(Gal(L=K);Apk) = 0 for all i 0 in proposition 6.3 shows that Hi(Gal(LM=K);Apk) = 0 for all suchi. An ination restriction sequence now shows that
HomGal(L=K)(Gal(LM=L);Apk) =H1(Gal(LM=L);Apk)Gal (L=K)= 0:
This implies that ifx2S satises [x;C]S = 0, then in fact [x;Gal(LS=L)]S = 0 and the claim follows from lemma 6.4.
By lemma 6.2 the image of Gal(Q=K) in Aut(ApM+k+1) = GL2(OEp=pM+k+1) contains an element of the formaI such thata21+pM+k(Z=p). One checks that this element denes0 2Gal(LM=LM 1) with the property that if Frob(`) contains 0, then`2S(M+k).
Now let L0 =LM \LS. ThenC= Gal(LS=L0). Consider 2C+. SinceC has odd order we can nd2C such that =. Let02Gal(LM LS=K) be the element whose restriction to Gal(LM=K) is 0 and whose restriction to Gal(LS=L0) is. By Cebotarev's density theorem, we may nd innitely many primes ` whose Frobenius conjugacy class in Gal(LM LS=Q) contains 0. Every such ` is in S(M +k). In addition, after projecting to Gal(LS=L0) we nd Frob() = ()2 = =. Thus, we are able to generate a full coset of C in Gal(LS=L) with these Frob(). By lemma 6.8 we are also able to generate all elements of this coset for which [y;]S = 0 with fFrob(); y 6= 0g. The proposition therefore follows from lemma 6.6.