Local heights on Abelian varieties and rigid analytic uniformization

Abstract

We express classical and $p$-adic local height pairings on an abelian variety with split semistable reduction in terms of the corresponding pairings on the abelian part of the Raynaud extension (which has good reduction). Here we use an approach to height pairings via splittings of biextensions which is due to Mazur and Tate. We conclude with a formula comparing Schneider's $p$-adic height pairing to the $p$-adic height pairing in the semistable ordinary reduction case defined by Mazur and Tate.