Asymptotic lower bound of class numbers along a Galois representation

JOURNAL OF NUMBER THEORY

卷:211

Article

Ohshita, Tatsuya¹ 

[1] Keio Univ, Fac Sci & Technol, Dept Math, Kohoku Ku, 3-14-1 Hiyoshi, Yokohama, Kanagawa 2238522, Japan

关键词: Class number;    Galois representation;    Elliptic curve;    Abelian variety;    Selmer group;    Mordell-Weil group;    Iwasawa theory;   

DOI  :  10.1016/j.jnt.2019.09.024

来源: Elsevier

【 摘 要 】

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(infinity)])/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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free