Eisenstein classes, elliptic Soul\'e elements and the $\ell$-adic elliptic polylogarithm

TL;DR

This paper explores the $\, ext{ell}\,$-adic realization of the elliptic polylogarithm using sheaves of Iwasawa modules, establishing new relations between Eisenstein classes and elliptic Soulé elements, and simplifying key proofs.

Contribution

It provides a systematic description of the $\, ext{ell}\,$-adic elliptic polylogarithm in terms of elliptic units and relates Eisenstein classes to elliptic Soulé elements, with simplified proofs.

Findings

Relation between $\, ext{ell}\,$-adic Eisenstein classes and elliptic Soulé elements established.

New proof of the residue formula for $\, ext{ell}\,$-adic Eisenstein classes at cusps.

Reproves the cup-product formula from prior work.

Abstract

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.