On Neron class groups of abelian varieties

TL;DR

This paper investigates the Neron S-class group of an abelian variety over a global field, exploring its analogy with class groups and the Birch and Swinnerton-Dyer conjecture.

Contribution

It provides a detailed study of the Neron S-class group, highlighting its role as an analog of the ideal class group in the context of abelian varieties over global fields.

Findings

Establishes the analogy between C_{A,F,S'} and class groups.

Analyzes properties of the Neron S-class group.

Connects the group to conjectures in number theory.

Abstract

Let F be a global field and let S denote a nonempty finite set of primes of F containing the set S' of archimedean primes of F. In this paper we study the Neron S-class group C_{A,F,S} of an abelian variety A defined over F. In the well-known analogy that exists between the Birch and Swinnerton-Dyer conjecture for A over F and Dirichlet's analytic class number formula for the field F (in the number field case), the finite group C_{A,F,S'} (not the Tate-Shafarevich group of A) is a natural analog of the ideal class group of F.