TY - JOUR

T1 - On the de Rham and p-adic realizations of the elliptic polylogarithm for CM elliptic curves

AU - Bannai, Kenichi

AU - Kobayashi, Shinichi

AU - Tsuji, Takeshi

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve E defined over an imaginary quadratic field K with complex multiplication by the full ring of integers OK of K. Note that our condition implies that K has class number one. Assume in addition that E has good reduction above a prime p ≥ 5 unramified in OK. In this case, we prove that the specializations of the p-adic elliptic polylogarithm to torsion points of E of order prime to p are related to p-adic Eisenstein-Kronecker numbers. Our result is valid even if E has supersingular reduction at p. This is a p-adic analogue in a special case of the result of Beilinson and Levin, expressing the Hodge realization of the elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. When p is ordinary, then we relate the p-adic Eisenstein-Kronecker numbers to special values of p-adic L-functions associated to certain Hecke characters of K.

AB - In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve E defined over an imaginary quadratic field K with complex multiplication by the full ring of integers OK of K. Note that our condition implies that K has class number one. Assume in addition that E has good reduction above a prime p ≥ 5 unramified in OK. In this case, we prove that the specializations of the p-adic elliptic polylogarithm to torsion points of E of order prime to p are related to p-adic Eisenstein-Kronecker numbers. Our result is valid even if E has supersingular reduction at p. This is a p-adic analogue in a special case of the result of Beilinson and Levin, expressing the Hodge realization of the elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. When p is ordinary, then we relate the p-adic Eisenstein-Kronecker numbers to special values of p-adic L-functions associated to certain Hecke characters of K.

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U2 - 10.24033/asens.2119

DO - 10.24033/asens.2119

M3 - Article

AN - SCOPUS:78650910912

VL - 43

SP - 185

EP - 234

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

SN - 0012-9593

IS - 2

ER -