Computation of pp-adic heights and log convergence

  • Barry Mazur

  • William Stein

  • John Tate


This paper is about computational and theoretical questions regarding pp-adic height pairings on elliptic curves over a global field KK. The main stumbling block to computing them efficiently is in calculating, for each of the completions KvK_v at the places vv of KK dividing pp, a single quantity: the value of the pp-adic modular form E2\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 E2\mathbf{E}_2 of an elliptic curve. We also discuss the pp-adic convergence rate of canonical expansions of the pp-adic modular form E2\mathbf{E}_2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E2\mathbf{E}_2 is log convergent.