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

Kenichi Bannai, Hidekazu Furusho and Shinichi Kobayashi

Arxiv ID: 0807.4007•Last updated: 9/11/2020

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.

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