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

Abstract

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-function 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 Poincare bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for CM abelian varieties. 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 Manin-Vishik and Katz. Our method via theta functions also gives insight for the case when $p$ is supersingular. The method of this paper will be used in subsequent papers to study the precise $p$-divisibility of critical values of Hecke $L$-functions associated to Hecke characters of quadratic imaginary fields for supersingular $p$, as well as explicit calculation in two-variables of the $p$-adic elliptic polylogarithm for CM elliptic curves.

55 pages, 2 figures. Minor misprints and errors were corrected

References in corpus (2)

• On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves
• Algebraic theta functions and Eisenstein-Kronecker numbers

Cited by in corpus (13)

• On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves
• Katz type p-adic L-functions for primes p non-split in the CM field
• Canonical Equivariant Cohomology Classes Generating Zeta Values of Totally Real Fields
• Algebraic theta functions and Eisenstein-Kronecker numbers
• $p$-adic Beilinson conjecture for ordinary Hecke motives associated to imaginary quadratic fields
• Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation
• The de Rham realization of the elliptic polylogarithm in families
• $p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas
• Eisenstein-Kronecker series via the Poincaré bundle
• Integral structures on $p$-adic Fourier theory
• p-adic Eisenstein-Kronecker series and non-critical values of p-adic Hecke L-functions of an Imaginary Quadratic Field when the conductor is divisible by p
• The Kronecker limit formulas via the distribution relation
• Periods of Hilbert Modular forms, Kronecker series and Cohomology