Divisibility of Selmer groups and class groups

TL;DR

This paper explores the divisibility properties of class groups in quadratic number fields and develops methods to construct Selmer and Tate-Shafarevich groups for abelian varieties over number fields, linking algebraic number theory and algebraic geometry.

Contribution

It introduces new insights into the divisibility of class groups and provides constructions for Selmer and Tate-Shafarevich groups in the context of abelian varieties.

Findings

New criteria for divisibility of class groups

Construction methods for Selmer groups

Connections between class groups and algebraic geometry

Abstract

In this paper, we study two topics. One is the divisibility problem of class groups of quadratic number fields and its connections to algebraic geometry. The other is the construction of Selmer group and Tate-Shafarevich group for an abelian variety defined over a number field.