Computation of -adic heights and log convergence

  • Barry Mazur

  • William Stein

  • John Tate


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