Les classes d'Eisenstein des varietes de Hilbert-Blumenthal

Details Les classes d'Eisenstein des varietes de Hilbert-Blumenthal Les classes d'Eisenstein des varietes de Hilbert-Blumenthal

2007-06-17

arxiv.org/abs/0706.2455v2

Mathematics

Number Theory

Constantes rationnelles de la Proposition 4.3 et du Theoreme 5.2 corrigees

Scientific paper

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 .

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