### Abstract

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke L-functions of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced ("normalized" or "canonical" in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and p-adic properties of reduced theta functions for abelian varieties with complex multiplication (CM). As a corollary, when the prime p is ordinary, we give a new construction of the two-variable p-adic measure interpolating special values of Hecke L-functions of imaginary quadratic fields, originally constructed by Višik-Manin and Katz. Our method via theta functions also gives insight for the case when p is supersingular. The method of this article will be used in subsequent articles to study in two variables the p-divisibility of critical values of Hecke L-functions associated to imaginary quadratic fields for inert p, as well as explicit calculation in two variables of the p-adic elliptic polylogarithms for CM elliptic curves.

Original language | English |
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Pages (from-to) | 229-295 |

Number of pages | 67 |

Journal | Duke Mathematical Journal |

Volume | 153 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**Algebraic theta functions and the p-adic interpolation of eisenstein-kronecker numbers.** / Bannai, Kenichi; Kobayashi, Shinichi.

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 153, no. 2, pp. 229-295. https://doi.org/10.1215/00127094-2010-024

}

TY - JOUR

T1 - Algebraic theta functions and the p-adic interpolation of eisenstein-kronecker numbers

AU - Bannai, Kenichi

AU - Kobayashi, Shinichi

PY - 2010/6/1

Y1 - 2010/6/1

N2 - We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke L-functions of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced ("normalized" or "canonical" in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and p-adic properties of reduced theta functions for abelian varieties with complex multiplication (CM). As a corollary, when the prime p is ordinary, we give a new construction of the two-variable p-adic measure interpolating special values of Hecke L-functions of imaginary quadratic fields, originally constructed by Višik-Manin and Katz. Our method via theta functions also gives insight for the case when p is supersingular. The method of this article will be used in subsequent articles to study in two variables the p-divisibility of critical values of Hecke L-functions associated to imaginary quadratic fields for inert p, as well as explicit calculation in two variables of the p-adic elliptic polylogarithms for CM elliptic curves.

AB - We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke L-functions of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced ("normalized" or "canonical" in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and p-adic properties of reduced theta functions for abelian varieties with complex multiplication (CM). As a corollary, when the prime p is ordinary, we give a new construction of the two-variable p-adic measure interpolating special values of Hecke L-functions of imaginary quadratic fields, originally constructed by Višik-Manin and Katz. Our method via theta functions also gives insight for the case when p is supersingular. The method of this article will be used in subsequent articles to study in two variables the p-divisibility of critical values of Hecke L-functions associated to imaginary quadratic fields for inert p, as well as explicit calculation in two variables of the p-adic elliptic polylogarithms for CM elliptic curves.

UR - http://www.scopus.com/inward/record.url?scp=77957800225&partnerID=8YFLogxK

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U2 - 10.1215/00127094-2010-024

DO - 10.1215/00127094-2010-024

M3 - Article

AN - SCOPUS:77957800225

VL - 153

SP - 229

EP - 295

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 2

ER -