Shimura varieties and twisted orbital integrals. (English) Zbl 0533.14009

Langlands’ work on the Shimura varieties attached to quaternion algebras over totally real number fields led him to outline an attack on the problem of expressing the zeta function of a general Shimura variety in terms of automorphic L-functions. He divided his approach into parts, one of them being a combinatorial problem which arises during the comparison of the Selberg trace formula with the number of points on the Shimura variety over a finite field. The author of this paper shows that Langlands’ combinatorial problem can be reduced to a standard problem in local harmonic analysis: that of establishing the spherical function identities needed for base change.


14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G25 Global ground fields in algebraic geometry
11R80 Totally real fields
14K15 Arithmetic ground fields for abelian varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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