Cup Products and Pairings for Abelian Varieties
Klaus Loerke

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 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 -part of Grothendieck's pairing in the semistable reduction case, thus proving a conjecture of Bertapelle.
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Taxonomy
Topicssemigroups and automata theory · Cellular Automata and Applications
