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

TL;DR

This paper constructs $p$-adic Eisenstein-Kronecker series for CM elliptic curves over imaginary quadratic fields and proves their $p$-adic Kronecker limit formulas, extending classical complex analysis results into the $p$-adic setting.

Contribution

It introduces $p$-adic analogues of Eisenstein-Kronecker series for CM elliptic curves and establishes their Kronecker limit formulas using Coleman functions.

Findings

Construction of $p$-adic Eisenstein-Kronecker series as Coleman functions.

Proof of $p$-adic Kronecker limit formulas.

Extension of classical Kronecker formulas to the $p$-adic context.

Abstract

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ 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.