  In this paper, we give an explicit description of the de Rham and p-adic
polylogarithms for elliptic curves using the Kronecker theta function. We prove
in particular that when the elliptic curve has complex multiplication and good
reduction at p, then the specializations to torsion points of the p-adic
elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers,
proving a p-adic analogue of the result of Beilinson and Levin expressing the
complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series.
Our result is valid even if the elliptic curve has supersingular reduction at
p.
