×

zbMATH — the first resource for mathematics

Test functions for Shimura varieties: the Drinfeld case. (English) Zbl 1014.20002
Let \((G,X,K)\) be a Shimura datum with reflex field \(E\), and let \(\mathfrak p\) be a prime ideal of \(E\) lying over a prime number \(p\) such that \(E_{\mathfrak p}\) is an unramified extension of \(\mathbb{Q}_p\). Let \(F\) be an unramified extension containing \(E_{\mathfrak p}\), and assume that \(G\) is quasi-split over \(F\). If \(\mu\) is a miniscule cocharacter of the group \(G_{\overline E_{\mathfrak p}}\), then the associated Bernstein function \(z_\mu\) belongs to the center of the Iwahori-Hecke algebra of the \(p\)-adic group \(G(F)\). According to a conjecture of Kottwitz, the Bernstein function \(z_\mu\) plays an important role in the bad reduction of a Shimura variety with Iwahori level structure. More precisely, the function trace of Frobenius on nearby cycles is conjecturally expressible in terms of \(z_\mu\). In this paper the author obtains an explicit formula for \(z_\mu\) in terms of the standard basis for the Iwahori-Hecke algebra. He also uses this formula to prove Kottwitz’s conjecture for a particular Shimura variety attached to \(G=\text{GU}(1,d-1)\) and \(\mu=(1,0^{d-1})\), known as the Drinfeld case.

MSC:
20C08 Hecke algebras and their representations
11G18 Arithmetic aspects of modular and Shimura varieties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J.-F. Boutout and Th. Zink, The \(p\)-adic uniformization of Shimura curves , preprint 95–107, Universität Bielefeld, 1995.
[2] D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles , preprint, http://front.math.ucdavis.edu/math.AG/9912074. · Zbl 1072.14055
[3] U. Görtz, Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models for \(\GL_4\) and \(\GL_5\) , Mathematisches Institut zu Köln, preprint, 1999.
[4] T. Haines, The combinatorics of Bernstein functions , to appear in Trans. Amer. Math. Soc. JSTOR: · Zbl 0962.14018
[5] T. Haines and B. C. Ngô, Nearby cycles for local models of some Shimura varieties , preprint, 1999. · Zbl 1009.11042
[6] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups , Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. · Zbl 0228.20015
[7] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math. 53 (1979), 165–184. · Zbl 0499.20035
[8] R. Kottwitz, Shimura varieties and twisted orbital integrals , Math. Ann. 269 (1984), 287–300. · Zbl 0533.14009
[9] –. –. –. –., Points on some Shimura varieties over finite fields , J. Amer. Math. Soc. 5 (1992), 373–444. JSTOR: · Zbl 0796.14014
[10] G. Lusztig, Some examples of square integrable representations of semisimple \(p\)-adic groups , Trans. Amer. Math. Soc. 277 (1983), 623–653. · Zbl 0526.22015
[11] –. –. –. –., Affine Hecke algebras and their graded version , J. Amer. Math. Soc. 2 (1989), 599–635. JSTOR: · Zbl 0715.22020
[12] M. Rapoport, “On the bad reduction of Shimura varieties” in Automorphic Forms, Shimura Varieties, and \(L\)-functions (Ann Arbor, Mich., 1988), Vol. II , Perspect. Math. 11 , Academic Press, Boston, 1990, 253–321. · Zbl 0716.14010
[13] M. Rapoport and T. Zink, Über die lokale Zetafunktion von Shimuravarietäten, Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik , Invent. Math. 68 (1980), 21–101. · Zbl 0498.14010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.