Asymptotic lower bound of class numbers along a Galois representation

TL;DR

This paper establishes explicit asymptotic lower bounds for class numbers in number field towers using Galois representations and Selmer groups, with applications to abelian varieties including CM and Hilbert--Blumenthal types.

Contribution

It introduces a method to derive lower bounds of class numbers along Galois extensions via Selmer groups and applies it to various classes of abelian varieties.

Findings

Derived explicit lower bounds for class numbers in Galois towers.

Applied bounds to abelian varieties with specific endomorphism structures.

Provided asymptotic inequalities for class numbers in different abelian variety cases.

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.