Cup Products and Pairings for Abelian Varieties

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.