Les classes d'Eisenstein des varietes de Hilbert-Blumenthal

TL;DR

This paper studies Eisenstein classes in Hilbert-Blumenthal varieties, providing explicit coordinate expressions, analyzing their degeneration at cusps, and linking these degenerations to special values of L-functions, with applications to classical theorems.

Contribution

It offers an explicit topological description of Eisenstein classes and connects their degeneration to special L-values, providing new geometric proofs of classical results.

Findings

Eisenstein classes degenerate at special L-values

Explicit coordinate expressions for Eisenstein classes

Non-vanishing results for certain Eisenstein classes

Abstract

This article deals with the Eisenstein classes of Hilbert-Blumenthal families of abelian varieties. We first give a coordinate expression of these one at the topological level, using currents defined by Levin. Then we study the degeneration of these Eisenstein classes at a cusp of the Baily-Borel compactification of the Hilbert-Blumenthal variety. We show, using the explicit description of the Eisenstein classes obtained previously, that these classes degenerate in special values of an $L$-function associated to the underlying totally real number field. We deduce then both a geometric proof the Klingen-Siegel Theorem and a non vanishing result for some of these Eisenstein classes .