Grothendieck's pairing on Neron component groups: Galois descent from the semistable case

TL;DR

This paper proves Grothendieck's conjecture on the perfectness of pairings between Neron component groups of abelian varieties and their duals, using Galois descent from the semistable case, extending to tori and 1-motives.

Contribution

It establishes the perfectness of Grothendieck's pairing for a broad class of abelian varieties and related structures via Galois descent from the semistable case.

Findings

Proves Grothendieck's pairing is perfect for general abelian varieties.

Extends the pairing's perfectness to tori and 1-motives.

Utilizes Galois descent from the semistable case to the general case.

Abstract

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This in particular implies Grothendieck's conjecture on the perfectness of his pairing between the Neron component groups of an abelian variety and its dual. The point is that our formulation is well-suited with Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.