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

Kenichi Bannai, Shinichi Kobayashi

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10 Citations (Scopus)

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 languageEnglish
Pages (from-to)229-295
Number of pages67
JournalDuke Mathematical Journal
Volume153
Issue number2
DOIs
Publication statusPublished - 2010 Jun 1

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ASJC Scopus subject areas

  • Mathematics(all)

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