On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves

Details On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves

2007-11-12

arxiv.org/abs/0711.1701v2

Mathematics

Number Theory

61 pages, v2. Sections concerning the Hodge realization was moved to the appendix

Scientific paper

In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular reduction at p.

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