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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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