Duke Mathematical Journal

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

Kenichi Bannai and Shinichi Kobayashi

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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

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Duke Math. J., Volume 153, Number 2 (2010), 229-295.

First available in Project Euclid: 26 May 2010

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Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 14K25: Theta functions [See also 14H42]
Secondary: 11E95: $p$-adic theory 14K22: Complex multiplication [See also 11G15] 14K05: Algebraic theory


Bannai, Kenichi; Kobayashi, Shinichi. Algebraic theta functions and the $p$ -adic interpolation of Eisenstein-Kronecker numbers. Duke Math. J. 153 (2010), no. 2, 229--295. doi:10.1215/00127094-2010-024. https://projecteuclid.org/euclid.dmj/1274902081

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