Selmer Groups and Torsion Zero Cycles on the Selfproduct of a Semistable Elliptic Curve

In this paper we extend the niteness result on the p-primary torsion subgroup in the Chow group of zero cycles on the selfproduct of a semistable elliptic curve obtained in joint work with S. Saito to primes p dividing the conductor. On the way we show the niteness of the Selmer group associated to the symmetric square of the elliptic curve for those primes. The proof uses p-adic techniques, in particular the Fontaine-Jannsen conjecture proven by Kato and Tsuji.


Introduction.
In this note we extend the main niteness result on p-primary torsion zero-cycles on the selfproduct of a semistable elliptic curve in L-S] to primes p 3 where E has (bad) multiplicative reduction, at least under a certain standard assumption.In the course of the proof we will also derive the niteness of the Selmer group of the symmetric square Sym 2 H 1 (E)(1) for these primes.However, this latter result has already been proven, under the additional condition that the Galois representation % p : Gal(Q=Q) !Aut(E p ) is absolutely irreducible (here E p = E p (Q) is the subgroup of p-torsion elements of E), in a much more general context by Wiles in his main paper ( W] Theorem 3.1) for Selmer groups associated to deformation theories.
To state the Theorems, let E be a semistable elliptic curve over Q with conductor N and let X = E Q E be its self-product.Consider the Chow group CH 0 (X) of zero-cycles on X modulo rational equivalence and let CH 0 (X)fpg be | for a xed prime p | its p-primary torsion subgroup.For a prime p dividing N consider the following hypothesis: H 1) The Gersten-Conjecture holds for the Quillen-(Milnor)-sheaf K 2 on a regular model X of X over ZZ p .

Then we have
Theorem A: Let E be a semistable elliptic curve and p 3 a prime such that p j N, i.e., E has (bad) multiplicative reduction at p. Assume that the condition H 1) is satis ed.Then CH 0 (X)fpg is a nite group.
Let A = H 2 (X; Q p =ZZ p (2)) be the Q p =ZZ p -realization of the motive H 2 (X)(2) with its Gal(Q=Q)-action.Then we have Theorem B: Let E be a semistable elliptic curve over Q and p 3 a prime such that p j N. Then the Selmer group S(Q; A) is nite. Remarks: | In L-S] we showed the niteness of CH 0 (X)fpg for primes p such that p6 j 6 and E has good reduction at p. We also proved that CH 0 (X)fpg is zero for almost all p.
Therefore Theorem A extends this result to bad primes and provides a further step towards a proof that the full torsion subgroup CH 0 (X) tors is nite.In order to nd a rst example where this is true it remains to consider the 2-and 3-primary torsion in CH 0 (X).| The Selmer group S(Q; A) coincides with S(Q; Sym 2 H 1 (E; Q p =ZZ p (1))) that was studied by Fl], because S(Q; Q p =ZZ p (1)) is zero.In Fl] Flach proved the niteness of S(Q; A) for primes p 5 such that E has good reduction at p and the representation % p is surjective.We were able to remove the latter hypothesis by using a rank-argument of Bloch-Kato and reproved Flach's niteness result for primes p such that p 6 j 6N (compare L-S]).In the proof of Theorem B we combine the criterium of Bloch-Kato with Kolyvagin's argument that was used in Flach's paper.Flach's additional condition on the surjectivity of % p can be avoided by applying a certain lemma, due to J. Nekov a r, that bounds the order The paper is organized as follows: In the rst paragraph we reduce the proof of Theorem A to two Lemmas I and II.Lemma I was already proven in ( L-S], Lemma A).Lemma II is similar to ( L-S], Lemma B), but the statement is di erent.The di erence is caused by the particular semistable situation.In the second paragraph we derive Lemma II and Theorem B from a key proposition that bounds the possible corank (at most 1 !) of the cokernel of the map de ning the Selmer group.Finally this proposition is proven in the last paragraph.The methods of the proof are similar to those developed in L-S].At the point where the crystalline conjecture was used in the good reduction case, we now use the Fontaine-Jannsen conjecture (proven by Kato/Tsuji for p 3) that relates the log-crystalline cohomology to the p-adic etale cohomology.The role of the syntomic cohomology in the context of Schneider's p-adic points conjecture is now replaced by a semistable analog relating log-syntomic cohomology to H 1 g (Q p ; H 2 (X; Q p (2))) (compare L]).When we apply this argument we will also need the computation, due to Hyodo and used by Tsuji, on a ltration on the sheaf of p-adic vanishing cycles in terms of modi ed logarithmic Hodge-Witt sheaves.
This paper was written during a visit at the University of Cambridge.I want to thank J. Coates and J. Nekov a r for their invitation and J. Nekov a r for many discussions and the permission to include his proof of Lemma (2.5) in this paper.Finally I thank S. Saito for encouraging me to look at the remaining semistable reduction case of our main niteness result in L-S] and I consider this work as having been done very much in the spirit of our joint paper and a continuation of it. x1 We rst x some notations.
For an Abelian group M let M div be the maximal divisible subgroup of M and Mfpg its p-primary torsion subgroup.For a scheme Z over a eld k let Z = Z k k where k is an algebraic closure of k.Denote by G k = Gal(k=k) the absolute Galois group of k.We will consider the Zariski sheaf K 2 associated to the presheaf U ! K 2 (U) of Quillen (-Milnor) K-groups on Z and let H j Zar (Z; K 2 ) be its Zariski cohomology.Let E be a semistable elliptic curve over Q with conductor N, : X 0 (N) !E a modular parametrization of E, X = E Q E. Let T; A; V be the following G = G Q -modules: T = H 2 (X; ZZ p (2)) ; A = H 2 (X; Q p =ZZ p (2)) ; V = H 2 (X; Q p (2)) : Note that as Abelian groups T = ZZ 6 p , A = Q p =ZZ 6 p , because the integral cohomology of an Abelian variety is torsion-free and the second Betti number of X b 2 is 6.Let K be the function eld of X.For a prime p let NH 3 (X; )) By results of Bloch and Merkurjev-Suslin ( Bl], x5 and M-S] we have the following exact sequence Since X is identi ed with its Albanese variety, the map CH 0 (X) tors !CH 0 (X) G tors is the Albanese map and therefore (CH 0 (X)fpg) G = X(Q)fpg is nite.
Consider the Hochschild-Serre spectral sequence Then we have Lemma I: Let the assumptions be as above.Then the composite map is injective.This is shown in ( L-S], Lemma (A)) without any assumption on the prime p.
Corollary (1. 3) The composite map that is obtained by the Hochschild-Serre spectral sequence is injective.
The Corollary will play an important role in the proof of Lemma II: Under the above assumptions let p 3 be a prime such that p j N and assume that the condition H 1) in the introduction is satis ed.Then we have H 1 (X; K 2 ) Q p =ZZ p = K N H 3 (X; Q p =ZZ p (2)) div : Remark: Lemma II was proven for primes p6 j 6N in ( L-S, Lemma (B)) because in this case K N H 3 (X; Q p =ZZ p (2)) div coincides with H 1 (Q; A) div .This is not stated there explicitly but follows from the proof of Lemma (B) in L-S].Now we deduce Theorem A from Lemma II.
The exact sequence (1-1) also holds for a smooth proper model X of X over ZZ ) and one knows that the latter group is co-nitely generated.Therefore CH 0 (X )fpg is co-nitely generated as ZZ pmodule.Since the kernel of the canonical map CH 0 (X )fpg !CH 0 (X)fpg is a torsion group by the main result in Mi], the localization sequence in the Zariski K-cohomology over X yields a surjection CH 0 (X )fpg !!CH 0 (X)fpg : So we also know that CH 0 (X)fpg is co-nitely generated.
On the other hand, by (1-2), the niteness of CH 0 (X)fpg G and Lemma II we conclude that the maximal divisible subgroup of CH 0 (X)fpg is zero.Therefore CH 0 (X)fpg is a nite group.
To complete the proof of Theorem A it remains to show Lemma II. x2 For each prime `let where 0 is the restriction of the map the kernel of which de nes the Selmer group S(Q; A).
In analogy to ( L-S], Lemma 3.1) we will prove the following Proposition (2.1): Let the notations be as in x1.Let p 3 a prime, such that E has multiplicative reduction at p. Assume that condition (H 1) holds.Then we have coker = H 1 (Q We will give the proof of Proposition 2.1 in the next section.
In the following we will compute the coranks of H 1 (Q ).It is well known that X Q p = E E Q p has a regular proper model X over ZZ p with semistable reduction.Let X p be its closed ber.By local Tate-Duality ) and this quotient is | by the computations in B-K], 3.8 | isomorphic to (B crys V ( 1)) GQ p =1 f, which is by Kato's and Tsuji's proof of the Fontaine-Jannsen-Conjecture ( Ka], x6), ( Tsu]) isomorphic to (D 2 ) N=0 =1 f), where D 2 = H 2 log crys ((X p ; M 1 )=W (IF p ); W(L); O crys ) Q p denotes the log-crystalline cohomology introduced by Hyodo-Kato H-K], N = 0 denotes the kernel under the action of the monodromy operator N, and f acts as p 1 ', where ' is the Frobenius acting on D 2 .Therefore we have by Poincar e duality for Hyodo-Kato cohomology that p is isomorphic to (coker N : D 2 !D 2 ) '=p .Since the functor D st ( ) = (B st ) GQ p commutes with tensor products and a Tate-elliptic curve has ordinary semistable reduction in the sense of ( Il], De nition 1.4) we have a Hodge-Witt-decomposition ( Il], Proposition 1.5) D 2 = i+j=2 H i (X p ; Ww j ) Q p : Here H i (X p ; Ww j ) is the cohomology of the modi ed Hodge-Witt-sheaves.From the action of the Frobenius ' on D 2 it is clear that (D 2 ) '=p is contained in H 1 (X p ; Ww 1 ) Q p .By ( Mo], x6) we know that the monodromy ltration and the weight ltration on D 2 coincide.Using the formula N' = p'N we have that N(H 0 (X p ; Ww 2 )) H 1 (X p ; Ww 1 ) and the map N 2 : H 0 (X p ; Ww 2 ) ! H 2 (X p ; Ww 0 ) is an isomorphism.Since dim H i (X p ; Ww j ) Q p = dim H i (X Q p j ) by ( Il], Corollaire 2.6), we see that dim(coker N : D 2 !D 2 ) '=p = dim(D 2 ) N=0 '=p 3 : On the other hand the B St -comparison-isomorphism provides an injection Pic(X) Q p , !H 2 (X; Q p (1)) GQ p , ! (D 2 ) N=0 '=p : Since Pic(X) has rank 3 we have Documenta Mathematica 2 (1997) 47{59 Lemma (2.2): dim p = dim(D 2 ) N=0 '=p = 3 : By the same methods and the proof of ( L-S], Lemma 4.4) we get Lemma (2.3): dim p = 1 : From Lemma (2.2) and ( L-S], Lemma 4.1) we get Lemma (2.4):The image of the composite map Now we will give the proof of Theorem B and we distinguish between two cases.
Case I: The map 0 p , i.e. the p-component of 0 is surjective.
This case is actually obstructed by the Gersten-conjecture as we will see in the proof of Proposition (2.1).Since we do not assume (H 1) in Theorem B we also consider this case.Using the surjectivity-property of `, i.e. the `-component of , for `6 = p that follows from Prop.2.1, and the condition (H 1) is not needed, we see that coker has ZZ p -corank 0. Now apply the modi ed version of ( B-K], Lemma 5.16) that is given in ( L-S], Lemma (3.3)):All the assumptions there are also satis ed for our choice of p: | V is a de Rham representation of Gal(Q p =Q p ) by Falting's proof of the de Rham conjecture.
By the same methods as in the proof of Lemma (2.2) we have (D 2 ) N=0 '=p 2 = 0.By the same arguments as in the proof of ( L-S], Theorem 3.2) we get the formula corank(ker ) = corank(coker ) = 0. Therefore S(Q; A) = ker is nite.
Case II: By Lemmas (2.3) and (2.4) this is the only remaining case to consider.
Let T 0 = Sym 2 H 1 (E; ZZ p (1)).By Lemma (2.2) and Lemma (2.4) we have H 1 g (Q p ; T 0 )=H 1 f (Q p ; T 0 ) = 0. Let c(`) for `6 j N be the elements in H 1 (X; K 2 ) that were constructed by Mildenhall and Flach.In the notation of ( Fl], Prop.(1.1)) we therefore have res r=p c(`) 2 H 1 f (Q p ; T 0 ).We get this property with little e ort whereas in Fl] this was one of the harder parts in the whole paper.It is now easy to check that all the other required properties on the elements c(`) in ( Fl], Prop.(1.1)) are also satis ed for our choice of p. Thus we apply Kolyvagin's argument in ( Fl], Prop. (1.1)).At the point where Flach needs the surjectivity of the Galois representation % p in order to derive the niteness of S(Q; A( 1)), we use the following Lemma, due to Nekov a r, that nishes, after applying Poitou-Tate Duality, the proof of Theorem B.
Lemma (2.5): Let Q(E p n )=Q be the Galois extension obtained by adjoining the co- ordinates of all p n -torsion points on E and let T 0 be as above.Then there exists a c > 0, such that the exponent of H 1 (Gal(Q(E p n )=Q); T 0 ( 1)=p n ) divides p c for all n 0.
Remark: Flach uses the vanishing of this cohomology group that follows from his additional assumption on the surjectivity of % p .
By result of Lazard there is an injection and H 1 vanishes for semisimple Lie-algebras (and every representation).So H 1 (G; T 0 ( 1)) is nite and Lemma 2.5 follows.
Finally it is easy to see that Corollary (1.3), Proposition (2.1) b) and Theorem B imply Lemma II and as a consequence also Theorem A. It remains to show Proposition (2.1).This will be accomplished in the next paragraph. x3 The surjectivity of the map follows from ( L-S], Lemmas (4.1), (4.3), (4.4) and (4.5)).On the other hand the composite map 2), whereas the image of (Pic(X) p ZZ ) Q p =ZZ p under the map 0 is zero.To nish the proof of Proposition (2.1) we therefore have to show that the image of 0 p , the p-component of 0 is contained in H 1 g (Q p ; A)=H 1 f (Q p ; A).
By the theory of Bloch-Ogus and the work of Merkurjev-Suslin M-S] we have an isomorphism H 1 (X; K 2 =p n ) = NH 3 et (X; ZZ=p n (2)) : Let X be a proper regular semistable model of X Q p over ZZ p , i : X p ! X and j : X Q p , !X the inclusions of the closed and generic ber.Let H 3 et (X ; 2 Rj ZZ=p n (2)) be the cohomology of the truncated complex of p-adic vanishing cycles.Then we have Lemma (3.1): Assume that the Gersten-Conjecture holds for the Zariski sheaf K 2 on the regular scheme X.Then we have the inclusion H 1 (X Q p ; K 2 =p n ) H 3 et (X ; 2 Rj ZZ=p n (2)) : Proof: This follows from the proof of ( L-S], Lemma (5.4)).
Using Lemma (3.2) and the Hochschild-Serre spectral sequence we get a canonical map : lim When we deal with a variety over a local eld, all cohomology groups under consideration are (co-) nitely generated.The map 0 p certainly factors through lim Since (D 2 ) N=0 '=p 2 = (D 3 ) N=0 '=p 2 = 0 (D i denotes the i-th log-crystalline cohomology of X p ) we may apply the main result in L] on a semistable analogue of Schneider's p-adic points conjecture to get Lemma (3.4) Im = H 1 g (Q p ; V ).
Tsuji has proven that there is a canonical isomorphism between the cohomology H 2 (i s log n (2)) and the sheaf M 2 n = i R 2 j ZZ=p n (2) of p-adic vanishing cycles ( Tsu], Theorem 3.2).His proof relies on a ltration Fil on M 2 n that was de ned by Hyodo ( H], (1.4)) and is induced by a symbol map on Milnor K-Theory.Hyodo has shown ( H], Theorem (1.6)) that the highest graded quotient gr 0 M 2 n sits in an extension (change of notation: Y := X p , the closed ber of X) is therefore equivalent to the assertion that the image of is contained in H 1 g (Q p ; V ).In view of Lemma (3.1) we see that Proposition (2.1) follows from the ! W n w2 Y !W n w 2 Y !0 and used the connecting homomorphism on the level of cohomology to de ne the monodromy operator on log-crystalline cohomology.It follows from the work of Tsuji( Tsu], x2.4) that there is a commutative diagram such that the upper exact sequence is obtained by taking the kernel of 1 F acting on the lower exact sequence, where F is the Frobenius.From the Hodge-Wittdecomposition of H r (Y; Ww )( Il], Proposition (1.5)) it is easy to derive a Hodge-Witt-decomposition for H r (Y; W w Y )From the action of the Frobenius ' on H r (Y; W w Y ) we getH 3 (Y; W w Y ) '=p 2 = H 1 (Y; W w2 Y ) F =1 :On the other hand it is shown in the proof of the semistable analogue of the p-adic It follows from ( L], (2.10)) that the composite(H 3 (Y; W w Y ) Qp ) '=p 2 !H 1 (Q p ; B crys V ) ! H 1 (Q p ; B st V )is the zero map.Using the fact that H 1 st = H 1 g (unpublished result of Hyodo, see also Nekov a r( Ne](1.24))weconclude that the image of the mapH 3 et (X ; 2 Rj Q p (2)) !H 1 (Q p ; V ) is H 1 g (Q p ; V ) in view ofLemma (3.4).This nishes the proof of Lemma (3.3) and Proposition (2.1). Y