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

Kottwitz, Robert E.

Math. Ann. 269, 287-300 (1984).

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.

MSC:

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

Keywords:

twisted orbital integrals; zeta-function of Shimura variety; Langlands combinatorial problem; automorphic L-functions

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