A Grunwald-Wang type theorem for abelian varieties

TL;DR

This paper proves a Lang-Tate conjecture showing weak approximation in the Weil-Châtelet group of an abelian variety over a number field, with mild failures in the n-torsion subgroup, extending ideas similar to the Grunwald-Wang theorem.

Contribution

It establishes a new analog of the Grunwald-Wang theorem for abelian varieties, demonstrating weak approximation properties and their limitations in the Weil-Châtelet group.

Findings

Weak approximation holds in the Weil-Châtelet group of A/k.

Failure of weak approximation occurs in the n-torsion subgroup but is limited.

The methods extend to arbitrary finite Galois modules.

Abstract

Let A be an abelian variety over a number field k. We show that weak approximation holds in the Weil-Ch\^atelet group of A/k but that it may fail when one restricts to the n-torsion subgroup. This failure is however relatively mild; we show that weak approximation holds outside a finite set of primes which is generically empty. This proves a conjecture of Lang and Tate that can be seen as an analog of the Grunwald-Wang theorem in class field theory. The methods apply, for the most part, to arbitrary finite Galois modules and so may be of interest in their own right.