Asymptotic lower bound of class numbers along a Galois representation

Abstract

Let T be a free Zp-module of finite rank equipped with a continuous Zp-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 Kn, which is the Galois extension field of K fixed by the stabilizer of T/pnT. 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∞])/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.