### 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 - 2010 Jun 1 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Duke Mathematical Journal*,

*153*(2), 229-295. https://doi.org/10.1215/00127094-2010-024