| Résume | (Joint work with Shinichi Kobayashi andTakeshi Tsuji) In this talk, we explicitly describe the de Rham realization of the elliptic polylogarithm for a single ellipitic curve, using rational functions derived from the theta function associated to the Poincare bundle. Using this description, we calculate the $p$-adic (rigid syntomic) realization of the elliptic polylogarithm, when the elliptic curve has complex multiplication and good reduction at the prime $p$. When $p$ is an ordinary prime, we relate the specialization of the elliptic polylogarithm to the special values of the two-variable $p$-adic $L$-functions defined by Manin-Vishik and Katz, giving a $p$-adic Beilinson conjecture type result extending previous calculations of Coleman-de Shalit and the speaker concerning the one-variable case. The case when $p$ is supersingular will also be discussed. |
