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arXiv · 1009.3546

A Grunwald-Wang type theorem for abelian varieties

Brendan Creutz
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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.

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Brendan Creutz. 2012-09-26. A Grunwald-Wang type theorem for abelian varieties. https://doi.org/10.4064/aa154-4-2

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