Duality for cohomology of curves with coefficients in abelian varieties

TL;DR

This paper establishes a duality theorem for the cohomology of algebraic curves over perfect fields with coefficients in Neron models of abelian varieties, extending local duality results to a global setting in positive characteristic.

Contribution

It generalizes the Cassels-Tate pairing's perfectness to a global function field context using advanced duality theories and cohomological tools.

Findings

Proves a duality for cohomology of curves with abelian variety coefficients.

Extends local duality results to global function fields.

Connects duality with the Cassels-Tate pairing in positive characteristic.

Abstract

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Neron models of abelian varieties. This is a global function field version of the author's previous work on local duality and Grothendieck's duality conjecture. It generalizes the perfectness of the Cassels-Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin-Milne's global finite flat duality, the non-degeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational etale site developed earlier.