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arXiv · 1805.03850

Asymptotic lower bound of class numbers along a Galois representation

Tatsuya Ohshita
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Abstract

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of K_n, which is the Galois extension field of K fixed by the stabilizer of T/p^n T. By applying this result, we prove an asymptotic inequality which describes an explicit lower bound of the class numbers along a tower K(A[p^\infty])/K for a given abelian variety A with certain conditions in terms of the Mordell-Weil group. We also prove another asymptotic inequality for the cases when A is a Hilbert--Blumenthal or CM abelian variety.

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Tatsuya Ohshita. 2018-05-10. Asymptotic lower bound of class numbers along a Galois representation. https://arxiv.org/abs/1805.03850

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