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 with 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 an 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.