Divisibility questions in commutative algebraic groups

TL;DR

This paper investigates conditions under which divisibility by a prime in commutative algebraic groups over number fields aligns locally and globally, impacting the structure of Tate-Shafarevich groups and Weil-Châtelet groups.

Contribution

It provides new sufficient conditions for local-global divisibility by p and triviality of Sha(k, A[p]) in commutative algebraic groups, especially abelian varieties.

Findings

Conditions imply divisibility of Sha(k, A) by p in H^1(k, A).

Local-global divisibility by p holds in all H^r(k, A).

Results connect divisibility properties with Tate-Shafarevich groups.

Abstract

Let $k$ be a number field, let ${\mathcal{A}}$ be a commutative algebraic group defined over $k$ and let $p$ be a prime number. Let ${\mathcal{A}}[p]$ denote the $p$-torsion subgroup of ${\mathcal{A}}$. We give some sufficient conditions for the local-global divisibility by $p$ in ${\mathcal{A}}$ and the triviality of $Sha (k,{\mathcal{A}}[p])$. When ${\mathcal{A}}$ is an abelian variety principally polarized, those conditions imply that the elements of the Tate-Shafarevich group $Sha(k,{\mathcal{A}})$ are divisible by $p$ in the Weil-Ch\^atelet group $H^1(k,{\mathcal{A}})$ and the local-global principle for divisibility by $p$ holds in $H^r(k,{\mathcal{A}})$, for all $r\geq 0$.