On the Hasse principle for finite group schemes over global function fields

Abstract

Abstract. Let K be a global function field of characteristic p > 0 and let M be a (commutative) finite and flat K-group scheme. We show that the kernel of the canonical localization mapin flat (fppf) cohomology can be computed solely in terms of Galois cohomology. We then give applications to the case where M is the kernel of multiplication by p m on an abelian variety defined over K.

Brauer groups and Neron class groups

“…where the first map is defined in [8,Remark 3.4] and shown to be an isomorphism in [6, Main Theorem] and the second map is induced by the Tate local duality isomorphisms H 0 (K v , A)…”

Local-global principles for Weil–Châtelet divisibility in positive characteristic

“…All of the results above establishing the local-global principle for divisibility by m in H 0 (k, A) and H 1 (k, A) are obtained by showing that the subgroup X 1 (k, A[m]) of locally trivial classes in H 1 (k, A[m]) vanishes (possibly with A replaced by its dual). Building on [GAT12], we give a necessary and sufficient criterion for the vanishing of X 1 (k, A[m]) in the case that A is an elliptic curve and m = p n is a power of the characteristic of k (see Theorem 7). This allows us to construct the examples in Theorems 1 and 2 demonstrating the failure of the local-global principle for divisibility in H r (k, A).…”

“…). Using results of [GAT12] and a well known argument using Chebotarev's density theorem (e.g., [Ser64, proposition 8]) this may be reduced to a computation in the cohomology of finite groups. In the case that A[m](k s ) is cyclic, the required computation is a classical one used in the proof of the Grunwald-Wang Theorem.…”

“…In this paper we are interested in these questions when m = p n is a power of the characteristic of k. In this case [GAT12,proposition 3•5] shows that for an elliptic curve A/k, the local-global principle for divisibility by p n in H r (k, A) holds except possibly if r ∈ {0, 1}, p = 2 and n 3 (for r 2 see Lemma 5). Their results imply that, for elliptic curves, the only possible counterexamples occur among non-constant elliptic curves with j-invariant in k 8 (see Corollary 10 below).…”

Selmer groups over $\Z_p^d$-extensions

“…Proof. If p is prime to the characteristic of K, the first statement is proved in [Mil86, I.6.26(C)]); otherwise, a proof is given in [GAT12]. The second statement follows from [Mil86, I.3.2 and III.7.8].…”