Realisation de Hodge du polylogarithme d'un schema abelien

TL;DR

This paper proves Levin's conjecture that certain currents describe the polylogarithm of an abelian scheme topologically, enabling explicit computation of Eisenstein classes with implications for special values of L-functions.

Contribution

It confirms Levin's conjecture and provides a new method to explicitly describe Eisenstein classes topologically for abelian schemes.

Findings

Currents by Levin describe the polylogarithm topologically.

Method to explicitly compute Eisenstein classes.

Non-vanishing of certain Eisenstein classes in geometric contexts.

Abstract

The main result of this article is the fact that the currents defined by Levin give a description of the polylogarithm of an abelian scheme at the topological level. This result was a conjecture of Levin. This provides a method to explicit the Eisenstein classes of an abelian scheme at the topological level. These classes are of special interest since they have a motivic origin by a theorem of Kings. In a forthcoming article, we use the main result of this paper to prove that the Eisenstein classes of the universal abelian scheme over an Hilbert-Blumenthal variety degenerate at the boundary of the Baily-Borel compactification of the base in a special value of an $L$-function associated to the underlying totally real number field. As a corollary, we get a non vanishing result for some of these Eisenstein classes in this geometric situation.