A Grunwald-Wang type theorem for abelian varieties

Abstract

Let A be an abelian variety over a number field k. We show that weak approximation holds in the Weil-Châtelet 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.

arXiv commentVersion 3: minor edits to incorporate suggestions of the referee

Cited by in corpus (1)

• A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers