Local heights on Abelian varieties and rigid analytic uniformization (Q1276545)

Summary: 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) [\textit{M. Raynaud}, Actes Congr. Int. Math. 1970, Tome 1, 473-477 (1971; Zbl 0223.14021)]. Here we use an approach to height pairings via splittings of biextensions which is due to \textit{B. Mazur} and \textit{J. Tate} [Arithmetic and geometry, Prog. Math. 35, 195-237 (1983; Zbl 0574.14036)]. We conclude with a formula comparing \textit{P. Schneider}'s \(p\)-adic height pairing [Invent. Math. 69, 401-409 (1982; Zbl 0509.14048)] to the \(p\)-adic height pairing in the semistable ordinary reduction case defined by Mazur and Tate [see \textit{B. Mazur, J. Tate} and \textit{J. Teitelbaum}, Invent. Math. 84, 1-48 (1986; Zbl 0699.14028)].