Duke Mathematical Journal
- Duke Math. J.
- Volume 153, Number 2 (2010), 229-295.
Algebraic theta functions and the -adic interpolation of Eisenstein-Kronecker numbers
We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke -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 -adic properties of reduced theta functions for abelian varieties with complex multiplication (CM). As a corollary, when the prime is ordinary, we give a new construction of the two-variable -adic measure interpolating special values of Hecke -functions of imaginary quadratic fields, originally constructed by Višik-Manin and Katz. Our method via theta functions also gives insight for the case when is supersingular. The method of this article will be used in subsequent articles to study in two variables the -divisibility of critical values of Hecke -functions associated to imaginary quadratic fields for inert , as well as explicit calculation in two variables of the -adic elliptic polylogarithms for CM elliptic curves
Duke Math. J., Volume 153, Number 2 (2010), 229-295.
First available in Project Euclid: 26 May 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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