### Abstract

Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers O_{K} of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

Original language | English |
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Pages (from-to) | 269-302 |

Number of pages | 34 |

Journal | Nagoya Mathematical Journal |

Volume | 219 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2015 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Nagoya Mathematical Journal*,

*219*(1), 269-302. https://doi.org/10.1215/00277630-2891995

**p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas.** / Bannai, Kenichi; Furusho, Hidekazu; Kobayashi, Shinichi.

Research output: Contribution to journal › Article

*Nagoya Mathematical Journal*, vol. 219, no. 1, pp. 269-302. https://doi.org/10.1215/00277630-2891995

}

TY - JOUR

T1 - p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

AU - Bannai, Kenichi

AU - Furusho, Hidekazu

AU - Kobayashi, Shinichi

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers OK of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

AB - Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers OK of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

UR - http://www.scopus.com/inward/record.url?scp=84945264795&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945264795&partnerID=8YFLogxK

U2 - 10.1215/00277630-2891995

DO - 10.1215/00277630-2891995

M3 - Article

AN - SCOPUS:84945264795

VL - 219

SP - 269

EP - 302

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

SN - 0027-7630

IS - 1

ER -