  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 .
