Computation of p-Adic Heights and Log Convergence

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Computation of p-Adic Heights and Log Convergence

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Title: Computation of p-Adic Heights and Log Convergence
Author: Mazur, Barry C.; Stein, William; Tate, John Torrence

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Citation: Mazur, Barry, William Stein, and John Tate. 2006. Computation of p-Adic heights and log convergence. Documenta Mathematica Extra Volume: John H. Coates' Sixtieth Birthday: 577–614.
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Abstract: 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 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 Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer 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. In particular, we introduce a new notion of log convergence
and prove that E2 is log convergent.
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