### Abstract

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.

Original language | English |
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Pages (from-to) | 185-234 |

Number of pages | 50 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 43 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2010 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Annales Scientifiques de l'Ecole Normale Superieure*,

*43*(2), 185-234. https://doi.org/10.24033/asens.2119