A Grunwald-Wang type theorem for abelian varieties

Details A Grunwald-Wang type theorem for abelian varieties A Grunwald-Wang type theorem for abelian varieties

2010-09-18

arxiv.org/abs/1009.3546v2

Mathematics

Number Theory

Improved several results. Theorem 1.3 is new. Added references

Scientific paper

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 can be seen as an analog of the Grunwald-Wang theorem in class field theory which asserts that similar results hold for abelian extensions of number fields. 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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