Cup Products and Pairings for Abelian Varieties

TL;DR

This paper explores the relationships between various pairings associated with abelian varieties with semistable reduction, establishing a connection that confirms a conjecture and enhances understanding of their arithmetic properties.

Contribution

It demonstrates that the Bester/Bertapelle pairing can describe the p-part of Grothendieck's pairing, proving Bertapelle's conjecture in the semistable case.

Findings

Bester/Bertapelle pairing describes the p-part of Grothendieck's pairing.

Confirmed Bertapelle's conjecture for semistable reduction.

Established a close connection between different pairings for abelian varieties.

Abstract

Let $A_K$ be an abelian variety with semistable reduction over a strictly henselian field of positive characteristic with perfect residue class field. We show that there is a close connection between the pairings of Grothendieck, Bester/Bertapelle and Shafarevic. In particular, we show that the pairing of Bester/Bertapelle can be used to describe the $p$-part of Grothendieck's pairing in the semistable reduction case, thus proving a conjecture of Bertapelle.