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We give an explicit description of the syntomic elliptic polylogarithm on theuniversal elliptic curve over the ordinary locus of the modular curve in termsof certain $p$-adic analytic moment functions associated to Katz' two-variable$p$-adic Eisenstein measure. The present work generalizes previous results ofBannai-Kobayashi-Tsuji and Bannai-Kings on the syntomic Eisenstein classes.

The syntomic realization of the elliptic polylogarithm via the Poincar\'e bundle