Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0796.14014
Kottwitz, Robert E.
Points on some Shimura varieties over finite fields. (English)
[J] J. Am. Math. Soc. 5, No.2, 373-444 (1992). ISSN 0894-0347; ISSN 1088-6834/e

The author proves the Langlands conjecture on zeta functions of Shimura varieties of PEL-type in cases $A$ and $C$. More precisely, he computes the number of fixed points of twisted Hecke correspondences $\Phi\sp j\sb \wp \circ f$ inside a single isogeny class in terms of orbital and twisted orbital integrals.\par An important rôle is played by a triple $(\gamma\sb 0, \gamma, \delta)$ associated to points $(A, \lambda, i, \eta)$ on the Shimura variety in characteristic $p$. It is shown that the Kottwitz invariant $\alpha (\gamma\sb 0, \gamma, \delta)$ of such a triple vanishes; conversely, if $\alpha (\gamma\sb 0, \gamma, \delta) = 1$ and two other obvious conditions are satisfied then $(\gamma\sb 0, \gamma, \delta)$ comes from some $(A, \lambda,i)$. The proof of the vanishing of $\alpha$ involves a consideration of the filtered Dieudonné module associated to a crystalline representation of the Galois group of a local field.\par Apart from a lot of useful lemma's, the article also includes Honda-Tate theory for abelian varieties with endomorphisms.
[A.J.de Jong (Utrecht)]

MSC 2000:: *14G35 Modular and Shimura varieties
14G15 Finite ground fields
14G10 Zeta-functions and related questions
11S40 Zeta functions and L-functions of local number fields
14K05 Algebraic theory of abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties

Keywords: Langlands conjecture on zeta functions of Shimura varieties; fixed points of twisted Hecke correspondences; Honda-Tate theory for abelian varieties with endomorphisms

Cited in: Zbl 1213.11128 Zbl 1141.22005 Zbl 1087.14022 Zbl 1051.14052 Zbl 1036.11025 Zbl 0983.14024 Zbl 0906.14021 Zbl 0796.14015

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]