On the p-adic realization of elliptic polylogarithms for CM-elliptic curves

Let E be a CM-elliptic curve over Q with good ordinary reduction at a prime p greater than or equal to 5. The purpose of this paper is to construct the p-adic elliptic polylogarithm of E, following the method of A. Beilinson and A. Levin. Our main result is that the specializations of this object at torsion points give the special values of the one-variable p-adic L-function of the Grossencharakter associated to E.