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Consider an elliptic curve defined over an imaginary quadratic field $K$ withgood reduction at the primes above $p\geq 5$ and has complex multiplication bythe 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 curveas Coleman functions on the elliptic curve. We then prove $p$-adic analogues ofthe first and second Kronecker limit formulas by using the distributionrelation of the Kronecker theta function.

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