On torsors under abelian varieties

TL;DR

This paper constructs a geometric duality between torsors under abelian varieties and the fundamental group of their Néron models over local fields, extending to positive characteristic fields.

Contribution

It provides a new geometric construction of Shafarevich duality using the relative Picard functor, applicable in mixed and positive characteristic settings.

Findings

Constructs Shafarevich duality via the relative Picard functor.

Extends duality to positive characteristic fields for prime-to-p parts.

Offers a geometric perspective on torsors and duality in abelian varieties.

Abstract

Let A be an abelian variety over a local field K of mixed characteristic and with algebraically closed residue field. We provide a geometric construction (via the relative Picard functor) of the Shafarevich duality between the group of isomorphism classes of torsors under A and the "fundamental group" of the N\'eron model of the dual abelian variety A'. An analogous construction works over fields of positive characteristic p providing a duality on the prime-to-p parts.