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 ≥ 5 and has complex multiplication by the full ring of integers OK 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. K coln−m (0, z, n) := E col m,n (z) to highlight the analogy. Then in analogy with Theorem 1.1 (ii), we have the following.Theorem 1.2 (p-adic second Kronecker limit formula). For any prime p ≥ 5 of good reduction, we have the second limit formulawhere log p θ(z) is a certain p-adic analogue of the function log |θ(z)|−|z| 2 /A defined in Definition 5.1 using the reduced theta function θ(z) and the branch of our p-adic logarithm.