Computation of p-Adic Heights and Log Convergence

This paper is about computational 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 Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Kedlaya et al., we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E2 on the Hasse domain.