Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers

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