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Intersection cohomology and \(L\)-functions of some Shimura varieties. (Cohomologie d’intersection et fonctions \(L\) de certaines variétés de Shimura.) (French) Zbl 0553.12005

This paper presents the results of the authors which generalize the well known results of Eichler-Shimura, Deligne, and Langlands on modular curves to the case of Shimura varieties associated with certain rank-one \(\mathbb Q\)-groups (including the moduli of Hilbert-Blumenthal Abelian varieties). One replaces \(l\)-adic cohomology with the \(l\)-adic intersection cohomology of certain bundles and one uses the truth of the Zucker conjecture (for such groups) to interpret this cohomology as \(L_2\)-cohomology. Given a “good” prime one has two local Hasse-Weil factors: one coming from the intersection cohomology and the other by decomposing \(L_2\)-cohomology. The main result establishes that these two factors are equal and it is proved, á la Langlands, by comparing terms in the Selberg trace formula with terms in the Lefschetz trace formula. The purity theorem in intersection cohomology (established by O. Gabber) is then used to prove a version of the Ramanujan conjecture.

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R39 Langlands-Weil conjectures, nonabelian class field theory
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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