  This is a completely rewritten version of the paper formerly entitled
"Sheaves of Iwasawa modules, moment maps and the $\ell$-adic elliptic
polylogarithm". The proof of the main result is also simplified.
  In the paper we study systematically the $\ell$-adic realization of the
elliptic polylogarithm in the context of sheaves of Iwasawa modules. This leads
to a description of the elliptic polylogarithm in terms of elliptic units. As
an application we prove a precise relation between $\ell$-adic Eisenstein
classes and elliptic Soul\'e elements. This allows to give a new proof of the
formula for the residue of the $\ell$-adic Eisenstein classes at the cusps and
reproves the formula for the cup-product construction in \cite{Huber-Kings99}.
  The paper is the elaboration of lectures given at the Pune-Workshop on the
proof of the Bloch-Kato conjectures for the Riemann zeta function in 2012.
