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1 June 2010 Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers
Kenichi Bannai, Shinichi Kobayashi
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Duke Math. J. 153(2): 229-295 (1 June 2010). DOI: 10.1215/00127094-2010-024

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

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Kenichi Bannai. Shinichi Kobayashi. "Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers." Duke Math. J. 153 (2) 229 - 295, 1 June 2010. https://doi.org/10.1215/00127094-2010-024

Information

Published: 1 June 2010
First available in Project Euclid: 26 May 2010

zbMATH: 1205.11076
MathSciNet: MR2667134
Digital Object Identifier: 10.1215/00127094-2010-024

Subjects:
Primary: 11G40, 14K25
Secondary: 11E95, 14K05, 14K22

Rights: Copyright © 2010 Duke University Press

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Vol.153 • No. 2 • 1 June 2010
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