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

Abstract

Abstract. We extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

On the local-global principle for divisibility in the cohomology of elliptic curves

“…Due to its connection with a question of Cassels [Cas62, Problem 1.3], the analogous question where E(k) = H 0 (k, E) is replaced by the Galois cohomology group H 1 (k, E) has also recieved much attention [C ¸S15, Cre13,Cre16]. Function field analogues of these questions were studied in [CV17]. In all cases the positive results in the literature concerning local-global divisibility in the groups E(k) and H 1 (k, E) have relied on the same technique, which considers a more general local-global principle for the N-torsion subgroup of E (see Definition 1.1 below).…”

Local-global questions for divisibility in commutative algebraic groups

“…In the case of global fields of positive characteristic, the problem was treated by Creutz and Voloch in the mentioned [38]. In particular they showed some counterexamples to Problem 1.2 in elliptic curves and also some counterexamples to Cassels' question.…”

“…Then the local-global divisibility by q holds in H r (k, A). Theorem 5.2 has an extension to the case when k has positive characteristic, that was implemented by Creutz and Voloch in [38]. The triviality of X(k, A[ p l ]), for every l 1, implies an affirmative answer in A ∨ over k to Cassels' question for p and to Problem 1.3 for every power of p. When A is a principally polarized abelian variety, then A A ∨ .…”

Selmer groups are intersection of two direct summands of the adelic cohomology

“…Proof This follows from [, Lemma 5], but for the sake of completeness we give a short proof. By an elementary cohomological argument, if , then the map is injective.…”