Local-global principles for Weil-Ch\^atelet divisibility in positive characteristic

TL;DR

This paper investigates the conditions under which local-global divisibility principles hold for Weil-Châtelet groups over global fields of positive characteristic, extending previous results and providing counterexamples.

Contribution

It extends the characterization of Weil-Châtelet divisibility to positive characteristic fields and offers examples demonstrating the optimality and limitations of these principles.

Findings

Characterization of local-global divisibility in positive characteristic fields

Examples of elliptic curves with local divisibility but not global divisibility

Counterexamples showing failure of the local-global principle

Abstract

We extend existing results characterizing Weil-Ch\^atelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of Gonz\'alez-Avil\'es 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-Ch\^atelet group can also fail.