Class groups and Selmer groups

TL;DR

This paper explicitly computes local and global Selmer groups and class groups, providing sharp bounds and a framework applicable to various abelian varieties, isogenies, and number fields.

Contribution

It introduces a method to compute and relate Selmer groups and class groups locally and globally, enhancing understanding of their intersections in number theory.

Findings

Explicit local computations of Selmer and class groups

Sharp upper bounds for global group intersections

Applicability to elliptic and hyperelliptic Jacobians

Abstract

It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is to their intersection. In this paper we compute the two groups and their intersection explicitly in the local case and put together the local information to get sharp upper bounds in the global case. The techniques in this paper can be used for arbitrary abelian varieties, isogenies and number fields assuming a frequently occurring condition. Several examples are worked out for the Jacobians of elliptic and hyperelliptic curves.