Computation of p-Adic Heights and Log Convergence

by Barry Mazur, William Stein, and John Tate


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 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 E_2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E_2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E_2 is log convergent.

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