Selmer Groups and Torsion Zero Cycles

on the Selfproduct of a Semistable Elliptic Curve

Andreas Langer

Received: February 25, 1997

Communicated by Ulf Rehmann

Abstract. In this paper we extend the finiteness 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 finiteness 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.

1991 Mathematics Subject Classification: Primary 14H52; Secondary 19E15, 14F30. Key words and phrases: torsion zero cycles, semistable elliptic curve, multiplicative reduction primes, Selmer group of the symmetric square, Hyodo-Kato cohomology.

Introduction.

In this note we extend the main finiteness 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 finiteness 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

ϱ_{p} : Gal(\overline{Q}/Q) ⟶ Aut(E_{p})

is absolutely irreducible (here E_{p} = E_{p}(\overline{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){p} be — for a fixed 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 | N, i.e., E has (bad) multiplicative reduction at p. Assume that the condition H 1) is satisfied. Then CH_{0}(X){p} is a finite group.

Let A = H^{2}(\overline{X}, Q_{p}/ZZ_{p}(2)) be the Q_{p}/ZZ_{p}-realization of the motive H^{2}(X)(2) with its Gal(\overline{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 | N. Then the Selmer group S(Q, A) is finite.

Remarks:

— In [L-S] we showed the finiteness of CH_{0}(X){p} for primes p such that p |̸ 6 and E has good reduction at p. We also proved that CH_{0}(X){p} 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 finite. In order to find a first 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}(\overline{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 finiteness of S(Q, A) for primes p ≥ 5 such that E has good reduction at p and the representation ϱ_{p} is surjective. We were able to remove the latter hypothesis by using a rank-argument of Bloch-Kato and reproved Flach’s finiteness result for primes p such that p |̸ 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 ϱ_{p} can be avoided by applying a certain lemma, due to J. Nekovář, that bounds the order of H^{1}(Gal(Q(E_{p^{n}})/Q), (Sym^{2} H^{1}(\overline{E}, ZZ/p^{n}(1)))(−1)) independently of n.

Selmer Groups and Torsion Zero Cycles … 49

The paper is organized as follows: In the first 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 different. The difference 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 defining 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 étale 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_{g}^{1}(Q_{p}, H^{2}(\overline{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 filtration on the sheaf of p-adic vanishing cycles in terms of modified logarithmic Hodge-Witt sheaves.

This paper was written during a visit at the University of Cambridge. I want to thank J. Coates and J. Nekovář for their invitation and J. Nekovář 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 finiteness 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.

§1

We first fix some notations.

For an Abelian group M let M_{div} be the maximal divisible subgroup of M and M{p} its p-primary torsion subgroup. For a scheme Z over a field k let \overline{Z} = Z ×_{k} \overline{k} where

k is an algebraic closure of k. Denote by G_{k} = Gal(\overline{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_{Zar}^{j}(Z, K_{2}) be its Zariski cohomology. Let E be a semistable elliptic curve over Q with conductor N, ϕ : 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}(\overline{X}, ZZ_{p}(2)) , A = H^{2}(\overline{X}, Q_{p}/ZZ_{p}(2)) , V = H^{2}(\overline{X}, Q_{p}(2)) .

Note that as Abelian groups T ≅ ZZ_{p}^{6}, A ≅ Q_{p}/ZZ_{p}^{6}, 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 field of X. For a prime p let

NH^{3}(X, Q_{p}/ZZ_{p}(2)) := ker(H^{3}(X, Q_{p}/ZZ_{p}(2)) → H^{3}(K, Q_{p}/ZZ_{p}(2)))

and

K_{N}H^{3}(X, Q_{p}/ZZ_{p}(2)) := ker(NH^{3}(X, Q_{p}/ZZ_{p}(2)) → H^{3}(\overline{X}, Q_{p}/ZZ_{p}(2)))

By results of Bloch and Merkurjev-Suslin ([Bl], §5 and [M-S] we have the following exact sequence

(1 − 1) 0 → H^{1}(X, K_{2}) ⊗ Q_{p}/ZZ_{p} → NH^{3}(X, Q_{p}/ZZ_{p}(2)) → CH_{0}(X){p} → 0

Since H^{1}(\overline{X}, K_{2}) ⊗ Q_{p}/ZZ_{p} = 0 we get an exact sequence

(1 − 2)  0 ⟶ H^{1}(X, K_{2}) ⊗ Q_{p}/ZZ_{p} ⟶ K_{N}H^{3}(X, Q_{p}/ZZ_{p}(2))
⟶ ker(CH_{0}(X)\{p\} ⟶ CH_{0}(\overline{X})\{p\}^{G}) ⟶ 0

Since X is identified with its Albanese variety, the map CH_{0}(X)_{tors} ⟶ CH_{0}(\overline{X})_{tors}^{G} is the Albanese map and therefore (CH_{0}(\overline{X})\{p\})^{G} ≅ X(Q)\{p\} is finite. Consider the Hochschild-Serre spectral sequence

E_{2}^{a,b} = H^{a}(Q, H^{b}(\overline{X}, Q_{p}/ZZ_{p}(2)) ⟹ H^{a+b}(X, Q_{p}/ZZ_{p}(2)) .

Then we have

Lemma I: Let the assumptions be as above. Then the composite map

E_{2}^{2,1} ⟶ H^{3}(X, Q_{p}/ZZ_{p}(2)) ⟶ H^{3}(K, Q_{p}/ZZ_{p}(2))

is injective.

This is shown in ([L-S], Lemma (A)) without any assumption on the prime p.

Corollary (1.3) The composite map

φ : K_{N}H^{3}(X, Q_{p}/ZZ_{p}(2)) ⟶ H^{1}(G_{Q}, A)

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 | N and assume that the condition H 1) in the introduction is satisfied. 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 p |̸ 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 [\frac{1}{Np}]. So CH_{0}(X){p} is a subquotient of H^{3}(X , Q_{p}/ZZ_{p}(2)) and one knows that the latter group is co-finitely generated. Therefore CH_{0}(X){p} is co-finitely generated as ZZ_{p}- module. Since the kernel of the canonical map

CH_{0}(X){p} ⟶ CH_{0}(X){p}

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){p} → → CH_{0}(X){p} .

So we also know that CH_{0}(X){p} is co-finitely generated. On the other hand, by (1-2), the finiteness of CH_{0}(\overline{X})\{p\}^{G} and Lemma II we conclude that the maximal divisible subgroup of CH_{0}(X){p} is zero. Therefore CH_{0}(X){p} is a finite group. To complete the proof of Theorem A it remains to show Lemma II.

§2

For each prime ℓ let

H_{e}^{1}(Q_{ℓ}, V) ⊂ H_{f}^{1}(Q_{ℓ}, V) ⊂ H_{g}^{1}(Q_{ℓ}, V) ⊂ H^{1}(Q_{ℓ}, V)

be defined as in ([BK], 3.7)). Let

H_{f}^{1}(Q_{ℓ}, T) ⊂ H_{g}^{1}(Q_{ℓ}, T) ⊂ H^{1}(Q_{ℓ}, T)

be the inverse image of H_{f}^{1}(Q_{ℓ}, V) and H_{g}^{1}(Q_{ℓ}, V). Put

H_{f}^{1}(Q_{ℓ}, A) := H_{f}^{1}(Q_{ℓ}, T) ⊗ Q_{p}/ZZ_{p} ⊂ H^{1}(Q_{ℓ}, A)

and

H_{g}^{1}(Q_{ℓ}, A) := H_{g}^{1}(Q_{ℓ}, T) ⊗ Q_{p}/ZZ_{p} ⊂ H^{1}(Q_{ℓ}, A)

Write ∧_{ℓ} = H^{1}(Q_{ℓ}, T)/H_{f}^{1}(Q_{ℓ}, T). Then we have

∧_{ℓ} ⊗ Q_{p}/ZZ_{p} = H^{1}(Q_{ℓ}, A)_{div}/H_{f}^{1}(Q_{ℓ}, A)

Consider as in ([L-S], §3) the composite map

ψ : H^{1}(X, K_{2}) ⊗ Q_{p}/ZZ_{p} ⟶ K_{N}H^{3}(\overline{X}, Q_{p}/ZZ_{p}(2))_{div} ⟶^{α′} ⊕_{ℓ} ∧_{ℓ} ⊗ Q_{p}/ZZ_{p}

where α′ is the restriction of the map

α : H^{1}(Q, A) ⟶ ⊕_{all ℓ} \frac{H^{1}(Q_{ℓ}, A)}{H_{f}^{1}(Q_{ℓ}, A)}

the kernel of which defines 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 §1. Let p ≥ 3 a prime, such that E has multiplicative reduction at p. Assume that condition (H 1) holds. Then we have

a)  coker ψ = H^{1}(Q_{p}, A)_{div}/H_{g}^{1}(Q_{p}, A)
b)  Im ψ = Im α′

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_{p}, A)_{div}/H_{g}^{1}(Q_{p}, A) and H_{g}^{1}(Q_{p}, A)/H_{f}^{1}(Q_{p}, A). Let

Ω_{p} = H_{g}^{1}(Q_{p}, V)/H_{f}^{1}(Q_{p}, V) and θ_{p} = H^{1}(Q_{p}, V)/H_{g}^{1}(Q_{p}, V)

as in ([L-S], §4). 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 fiber. By local Tate-Duality ([B-K], §3.8), Ω_{p} is the Q_{p}-dual of H_{f}^{1}(Q_{p}, V (−1))/H_{e}^{1}(Q_{p}, V (−1)) and this quotient is — by the computations in [B-K], 3.8 — isomorphic to (B_{crys} ⊗ V (−1))^{G_{Q_{p}}} /1 − f, which is by Kato’s and Tsuji’s proof of the Fontaine-Jannsen-Conjecture ([Ka], §6), ([Tsu]) isomorphic to (D_{2})^{N=0}/1 − f), where

D_{2} = H_{log crys}^{2}((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}φ, where φ is the Frobenius acting on D_{2}. Therefore we have by Poincaré duality for Hyodo-Kato cohomology that Ω_{p} is isomorphic to (coker N : D_{2} → D_{2})^{φ=p}. Since the functor D_{st}(·) = (B_{st} ⊗ ·)^{G_{Q_{p}}} commutes with tensor products and a Tate-elliptic curve has ordinary semistable reduction in the sense of ([Il], Definition 1.4) we have a Hodge-Witt-decomposition ([Il], Proposition 1.5)

D_{2} = ⊕_{i+j=2} H^{i}(X_{p}, W w^{j}) ⊗ Q_{p} .

Here H^{i}(X_{p}, W w^{j}) is the cohomology of the modified Hodge-Witt-sheaves. From the action of the Frobenius φ on D_{2} it is clear that (D_{2})_{φ=p} is contained in H^{1}(X_{p}, W w^{1})_{Q_{p}}. By ([Mo], §6) we know that the monodromy filtration and the weight filtration on D_{2} coincide. Using the formula Nφ = pφN we have that

N(H^{0}(X_{p}, W w^{2})) ⊂ H^{1}(X_{p}, W w^{1})

and the map

N^{2} : H^{0}(X_{p}, W w^{2}) ⟶ H^{2}(X_{p}, W w^{0})

is an isomorphism. Since dim H^{i}(X_{p}, W w^{j})_{Q_{p}} = dim H^{i}(X_{Q_{p}}Ω^{j}) by ([Il], Corollaire 2.6), we see that

dim(coker N : D_{2} → D_{2})^{φ=p} = dim(D_{2})_{φ=p}^{N=0} ≤ 3 .

On the other hand the B_{St}-comparison-isomorphism provides an injection

Pic(X) ⊗ Q_{p} ↪ H^{2}(\overline{X}, Q_{p}(1))^{G_{Q_{p}}} ↪ (D_{2})_{φ=p}^{N=0} .

Since Pic(X) has rank 3 we have Lemma (2.2):

dim Ω_{p} = dim(D_{2})_{φ=p}^{N=0} = 3 .

By the same methods and the proof of ([L-S], Lemma 4.4) we get

Lemma (2.3):

dim θ_{p} = 1 .

From Lemma (2.2) and ([L-S], Lemma 4.1) we get

Lemma (2.4): The image of the composite map

(Pic(X) ⊗ Q^{∗}) ⊗ Q_{p}/ZZ_{p} ⟶ H^{1}(X, K_{2}) ⊗ Q_{p}/ZZ_{p} ⟶^{ψ_{p}} ∧_{p} ⊗ Q_{p}/ZZ_{p}

is

H_{g}^{1}(Q_{p}, A)/H_{f}^{1}(Q_{p}, A) .

Now we will give the proof of Theorem B and we distinguish between two cases.

Case I:

The map α′_{p}, i.e. the p-component of α′ 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 ψ_{ℓ}, i.e. the ℓ-component of ψ, for ℓ ≠ p that follows from Prop. 2.1, and where the condition (H 1) is not needed, we see that coker α has ZZ_{p}-corank 0. Now apply the modified version of ([B-K], Lemma 5.16) that is given in ([L-S], Lemma (3.3)): All the assumptions there are also satisfied for our choice of p:

— V is a de Rham representation of Gal(\overline{Q}_{p}/Q_{p}) by Falting’s proof of the de Rham conjecture.

— For the characteristic polynomial P_{ℓ}(V, t) we have P_{ℓ}(V, 1) ≠ 0. For ℓ ≠ p the proof is the same as in ([L-S], §3). For ℓ = p, we have Crys(V)^{f=1} = (D_{2})_{φ=p^{2}}^{N=0}. By the same methods as in the proof of Lemma (2.2) we have (D_{2})_{φ=p^{2}}^{N=0} = 0. By the same arguments as in the proof of ([L-S], Theorem 3.2) we get the formula corank(ker α) = corank(coker α) = 0. Therefore S(Q, A) = ker α is finite.

Case II:

Im α′_{p} = H_{g}^{1}(Q_{p}, A)/H_{f}^{1}(Q_{p}, A)

By Lemmas (2.3) and (2.4) this is the only remaining case to consider.

Let T′ = Sym^{2} H^{1}(\overline{E}, ZZ_{p}(1)). By Lemma (2.2) and Lemma (2.4) we have H_{g}^{1}(Q_{p}, T′)/H_{f}^{1}(Q_{p}, T′) = 0. Let c(ℓ) for ℓ |̸ 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(ℓ) ∈ H_{f}^{1}(Q_{p}, T′). We get this property with little effort 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(ℓ) in ([Fl], Prop. (1.1)) are also satisfied 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 ϱ_{p} in order to derive the finiteness of S(Q, A(−1)), we use the following Lemma, due to Nekovář, that finishes, 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 coordinates of all p^{n}-torsion points on E and let T′ be as above. Then there exists a c > 0, such that the exponent of H^{1}(Gal(Q(E_{p^{n}})/Q), T′(−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 ϱ_{p}.

Proof: Put G := Im(Gal(\overline{Q}/Q) → Aut_{ZZ_{p}}(T_{p}(E))). Since E is without complex multiplication over \overline{Q}, G is of finite index in Aut_{ZZ_{p}}(T_{p}(E)) = GL_{2}(ZZ_{p}). Put G_{n} := ker(G → GL_{2}(ZZ/p^{n}), T′ := Sym^{2}(T_{p}(E)), G̃ := Im(G → Aut_{ZZ_{p}}(T′)) = G/Z ∩ G,

where Z = center of GL_{2}(ZZ_{p}) = \{( \begin{matrix}λ & 0 \\ 0 & λ \end{matrix} ) , λ ∈ ZZ_{p}^{∗}\}.

Consider the following diagram with horizontal and vertical exact sequences: (note that G/G_{n} ≅ Gal(Q(E_{p^{n}})/Q).

0

↓

H^{1}(G, T′(−1)) ⊗ ZZ/p^{n}

↓

0 → H^{1}(G/G_{n}, T′(−1)/p^{n}) ⟶^{inf} H^{1}(G, T′(−1)/p^{n})⟶^{res}H^{1}(G_{n}, T′(−1)/p^{n})^{G/G_{n}}

↓

H^{2}(G, T′(−1))_{p^{n}}

It is clear that H^{i}(G, T′(−1)) = H_{cont}^{i}(G, T′(−1)) = H_{naive}^{i}(G, T′(−1)) are ZZ_{p}- modules of finite type. Therefore H^{2}(G, T′(−1))_{p^{∞}} is finite. We have an exact sequence

0 → H^{1}(G̃, T′(−1)) ⟶^{inf} H^{1}(G, T′(−1)) ⟶^{res} H^{1}(Z ∩ G, T′(−1))^{G/Z∩G}

=

Hom_{cont}(Z ∩ G, (T′(−1))^{G/Z∩G})

But (T′(−1))^{G/Z∩G} is zero (E has no CM). Thus H^{1}(G̃, T′(−1)) = H^{1}(G, T′(−1)). By result of Lazard there is an injection

\begin{aligned}H^{1}(G̃, T′(−1)) ⊗ Q & ↪ & H^{1}(Lie(G̃), T′(−1) ⊗ Q) \\ & & = H^{1}(sl(2), T′(−1) ⊗ Q) \end{aligned}

and H^{1} vanishes for semisimple Lie-algebras (and every representation). So H^{1}(G, T′(−1)) is finite 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.

§3

The surjectivity of the map

ψ′ = ⊕_{ℓ≠p} ψ_{ℓ} : H^{1}(X, K_{2}) ⊗ Q_{p}/ZZ_{p} ⟶ ⊕_{ℓ≠p} H^{1}(Q_{ℓ}, A)_{div}/H_{f}^{1}(Q_{ℓ}, A)

follows from ([L-S], Lemmas (4.1), (4.3), (4.4) and (4.5)). On the other hand the composite map

Pic(X) ⊗ p^{ZZ}) ⊗ Q_{p}/ZZ_{p} → H^{1}(X, K_{2}) ⊗ Q_{p}/ZZ_{p} →^{ψ_{p}} H_{g}^{1}(Q_{p}, A)/H_{f}^{1}(Q_{p}, A)

is surjective by Lemma (2.2), whereas the image of (Pic(X) ⊗ p^{ZZ}) ⊗ Q_{p}/ZZ_{p} under the map ψ′ is zero. To finish the proof of Proposition (2.1) we therefore have to show that the image of α′_{p}, the p-component of α′ is contained in H_{g}^{1}(Q_{p}, A)/H_{f}^{1}(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_{et}^{3}(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 fiber.

Let H_{et}^{3}(X , τ_{≤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_{et}^{3}(X , τ_{≤2}Rj_{∗}ZZ/p^{n}(2)) .

Proof:

This follows from the proof of ([L-S], Lemma (5.4)).

Lemma (3.2): H^{3}(\overline{X}_{Q_{p}}, Q_{p}(2))^{G_{Q_{p}}} = 0.

Proof:

Using the Künneth formula and the fact that H^{2}(\overline{E}, Q_{p}(1)) ≅ Q_{p} (the Brauer group of a curve over an algebraically closed field is zero), it suffices to show that H^{1}(\overline{E}, Q_{p}(1))^{G_{Q_{p}}} = 0. This follows from ([J], Theorem 5a).

Using Lemma (3.2) and the Hochschild-Serre spectral sequence we get a canonical map

σ : lim_{\overleftarrow{n}} H^{1}(X_{Q_{p}}, K_{2}/p^{n}) ⊗ Q_{p} ⟶ H^{1}(Q_{p}, V) .

When we deal with a variety over a local field, all cohomology groups under consideration are (co-)finitely generated. The map α′_{p} certainly factors through lim_{\overrightarrow{n}} H^{1}(X_{Q_{p}}, K_{2}/p^{n})_{div}. The assertion that lim_{\overrightarrow{n}} H^{1}(X_{Q_{p}}, K_{2}/p^{n})_{div} is contained in

H_{g}^{1}(Q_{p}, A) is therefore equivalent to the assertion that the image of σ is contained in H_{g}^{1}(Q_{p}, V). In view of Lemma (3.1) we see that Proposition (2.1) follows from the following

Lemma (3.3): Under the condition H1) we have: Im σ ⊂ H_{g}^{1}(Q_{p}, V).

To prove Lemma (3.3) it suffices to show that the image of the map

H^{3}(X , τ_{≤2}Rj_{∗}Q_{p}(2)) ⟶ H^{1}(Q_{p}, V)

is contained in H_{g}^{1}(Q_{p}, V).

Let s_{n}^{log}(2) be the log-syntomic complex in D_{et}(X) constructed by Kato ([Ka], §6) and Tsuji [Tsu] together with a canonical map

s_{n}^{log}(2) ⟶ τ_{≤2}i_{∗}i^{∗}Rj_{∗}ZZ/p^{n}(2) .

This gives rise to a composite map

η : H_{et}^{3}(X , s_{Q_{p}}^{log}(2)) ⟶ H^{1}(Q_{p}, V) .

Since (D_{2})_{φ=p^{2}}^{N=0} = (D_{3})_{φ=p^{2}}^{N=0} = 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 η = H_{g}^{1}(Q_{p}, V).

Tsuji has proven that there is a canonical isomorphism between the cohomology H^{2}(i^{∗}s_{n}^{log}(2)) and the sheaf M_{n}^{2} = i^{∗}R^{2}j_{∗}ZZ/p^{n}(2) of p-adic vanishing cycles ([Tsu], Theorem 3.2). His proof relies on a filtration Fil^{·} on M_{n}^{2} that was defined 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_{n}^{2} sits in an extension (change of notation: Y := X_{p}, the closed fiber of X)

0 ⟶ W_{n}w_{Y,log}^{1} ⟶ gr^{0}M_{n}^{2} ⟶ W_{n}w_{Y,log}^{2} ⟶ 0

where W_{n}w_{Y,log}^{i} are the modified logarithmic Hodge-Witt-sheaves ([H] (1.5)). On the other hand Hyodo and Kato ([H-K] Prop. 1.5) constructed an exact sequence of Hodge-Witt-sheaves

0 ⟶ W_{n}w_{Y}^{1} ⟶ W_{n}w̃_{Y}^{2} ⟶ W_{n}w_{Y}^{2} ⟶ 0

and used the connecting homomorphism on the level of cohomology to define the monodromy operator on log-crystalline cohomology. It follows from the work of Tsuji ([Tsu], §2.4) that there is a commutative diagram

0 → W_{n}w_{Y,log}^{1} → gr^{0}M_{n}^{2} → W_{n}w_{Y,log}^{2} → 0
↓ ↓ ↓
0 → W_{n}w_{Y}^{1} → W_{n}w̃_{Y}^{2} → W_{n}w_{Y}^{2} → 0

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-Witt-decomposition of H^{r}(Y, W w^{·}) ([Il], Proposition (1.5)) it is easy to derive a Hodge-Witt-decomposition for H^{r}(Y, W w̃_{Y}^{·})

H^{r}(Y, W w̃_{Y}^{·}) = ⨁_{i+j=r} H^{i}(Y, W w̃_{Y}^{j}) .

From the action of the Frobenius φ on H^{r}(Y, W w̃_{Y}^{·}) we get

H^{3}(Y, W w̃_{Y}^{·})_{φ=p^{2}} = H^{1}(Y, W w̃_{Y}^{2} )^{F =1} .

On the other hand it is shown in the proof of the semistable analogue of the p-adic points conjecture on log-syntomic cohomology [L], (2.6), Prop. (2.9), Prop. (2.13) that there is a surjection

H_{et}^{3}(X , s_{Q_{p}}^{log}(2)) → → (H^{3}(Y, W w̃_{Y}^{·})_{Q_{p}})_{φ=p^{2}} and the above arguments yield a commutative diagram

H^{3}(X , s_{Q_{p}}^{log}(2))
↓  ↘
H_{et}^{3}(X , τ_{≤2}Rj_{∗}Q_{p}(2)) → H^{1}(Y, gr^{0}M_{Q_{p}}^{2}) → (H^{3}(Y, W w̃_{Y}^{·})_{Q_{p}})_{φ=p^{2}}
↓  ↓
H^{1}(Q_{p}, V)  ⟶  H^{1}(Q_{p}, B_{crys} ⊗ V)

It follows from ([L], (2.10)) that the composite

(H^{3}(Y, W w̃_{Y}^{·})_{Q_{p}})_{φ=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_{st}^{1} = H_{g}^{1} (unpublished result of Hyodo, see also Nekovář ([Ne](1.24)) we conclude that the image of the map

H_{et}^{3}(X , τ_{≤2}Rj_{∗}Q_{p}(2)) ⟶ H^{1}(Q_{p}, V)

is H_{g}^{1}(Q_{p}, V) in view of Lemma (3.4). This finishes the proof of Lemma (3.3) and Proposition (2.1).

References

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Andreas Langer

Mathematisches Institut der Universität Münster Einsteinstr. 62 D - 48149 Münster Germany 60

Documenta Mathematica 2 (1997)