Brauer groups and Neron class groups

TL;DR

This paper generalizes a duality theorem for Neron class groups of abelian varieties over global fields by removing finiteness assumptions and relates these groups to Brauer groups via an exact sequence.

Contribution

It extends the duality theorem for Neron class groups without assuming finiteness of the Tate-Shafarevich group and establishes a new exact sequence involving Brauer groups.

Findings

Generalized duality theorem for Neron class groups

Derived an exact sequence linking Jacobian Neron groups to Brauer groups

Removed finiteness hypothesis on Tate-Shafarevich group

Abstract

Let K be a global field, let S be a finite set of primes of K containing the archimedean primes and let A be an abelian variety over K. We generalize the duality theorem established in our paper "On Neron class groups of abelian varieties" by removing the hypothesis in [op.cit.] that the Tate-Shafarevich group of A is finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve X over K to a certain finite subquotient of the Brauer group of X. The sequence alluded to above may be regarded as a global analog of an exact sequence of S.Biswas.