Computation of $p$-adic heights and log convergence

Abstract

This paper is about computational and theoretical questions regarding $p$-adic height pairings on elliptic curves over a global field $K$. The main stumbling block to computing them efficiently is in calculating, for each of the completions $K_v$ at the places $v$ of $K$ dividing $p$, a single quantity: the value of the $p$-adic modular form $\mathbf{E}_2$ associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of $\mathbf{E}_2$ of an elliptic curve. We also discuss the $p$-adic convergence rate of canonical expansions of the $p$-adic modular form $\mathbf{E}_2$ on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that $\mathbf{E}_2$ is log convergent.