Realizations of the Elliptic Polylogarithm for CM elliptic curves

In these notes, we give an overview of our paper [BKT] which gives an explicit description of the de Rham and p-adic realizations of the elliptic polylogarithm, for a general elliptic curve defined over a subfield of C in the de Rham case and for a CM elliptic curve defined over its field of complex multiplication and with good reduction at the primes above p ≥ 5 in the p-adic case. As explained in the appendix of [BKT], our method also gives a simple proof of the description of the real Hodge realization of the elliptic polylogarithm for a general elliptic curve defined over C. In these notes, we introduce the real Hodge and p-adic cases in a parallel fashion to highlight the analogy.