### 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 |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Annales Scientifiques de l'Ecole Normale Superieure*,

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

**On the de Rham and p-adic realizations of the elliptic polylogarithm for CM elliptic curves.** / Bannai, Kenichi; Kobayashi, Shinichi; Tsuji, Takeshi.

Research output: Contribution to journal › Article

*Annales Scientifiques de l'Ecole Normale Superieure*, vol. 43, no. 2, pp. 185-234. https://doi.org/10.24033/asens.2119

}

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

PY - 2010/1/1

Y1 - 2010/1/1

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.

UR - http://www.scopus.com/inward/record.url?scp=78650910912&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650910912&partnerID=8YFLogxK

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 -