#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: John H. Coates' Sixtieth Birthday (2006), 577-614

** Barry Mazur, William Stein, John Tate **
Computation of p-Adic Heights and Log Convergence

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 {\it single
quantity}: the value of the $p$-adic modular form ${\bf E}_2$ 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 ${\bf E}_2$ of an elliptic
curve. We also discuss the $p$-adic convergence rate of canonical expansions
of the $p$-adic modular form ${\bf E}_2$ on the Hasse domain. In particular,
we introduce a new notion of log convergence and prove that $\E_2$ is log
convergent.

2000 Mathematics Subject Classification: 11F33, 11Y40, 11G50

Keywords and Phrases: $p$-adic heights, algorithms, $p$-adic modular forms, Eisenstein series,
sigma-functions

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