$p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

Abstract

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\ge 5$ and has complex multiplication by the full ring of integers ${\mathcal{O}}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

v2. The current version is the synthesis of §1-§3 of the first version of this article with the content of arXiv:0807.4008 "The Kronecker limit formulas via the distribution relation." §4,§5 of the first version of this paper will be treated in a subsequent article

References in corpus (5)

• p-adic multiple zeta values I -- p-adic multiple polylogarithms and the p-adic KZ equation
• Towards multiple elliptic polylogarithms
• Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers
• 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 (1)

• 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