Orbital integrals are motivic

Author:
Thomas C. Hales

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1515-1525

MSC (2000):
Primary 22E50

DOI:
https://doi.org/10.1090/S0002-9939-04-07740-8

Published electronically:
December 15, 2004

MathSciNet review:
2111953

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article shows that under general conditions, -adic orbital integrals of definable functions are represented as the trace of a Frobenius operator on a virtual motive. This gives an explicit example of the philosophy of Denef and Loeser, which predicts that all ``naturally occurring'' -adic integrals are motivic.

**1.**Gábor Bodnár and Josef Schicho,*A computer program for the resolution of singularities*, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 231–238. MR**1748621**, https://doi.org/10.1007/978-3-0348-8399-3_7**2.**P. Cartier, Representations of -adic groups: a survey, in Automorphic Forms, Representations, and -functions, Corvallis, Symp. in Pure Math, AMS XXXIII, part 1, 1977.MR**0546593 (81e:22029)****3.**Jan Denef and François Loeser,*Definable sets, motives and 𝑝-adic integrals*, J. Amer. Math. Soc.**14**(2001), no. 2, 429–469. MR**1815218**, https://doi.org/10.1090/S0894-0347-00-00360-X**4.**J. Denef and F. Loeser,*Motivic integration and the Grothendieck group of pseudo-finite fields*, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 13–23. MR**1957016****5.**J. Denef, F. Loeser,*One some rational generating series occurring in arithmetic geometry*, preprint math.NT/0212202.**6.**H. Derksen, Constructive Invariant Theory and the Linearization Problem, Ph.D. thesis, University of Basel, 1997, http://www.math.lsa.umich.edu/ hderksen/preprint.html.**7.**Santiago Encinas and Herwig Hauser,*Strong resolution of singularities in characteristic zero*, Comment. Math. Helv.**77**(2002), no. 4, 821–845. MR**1949115**, https://doi.org/10.1007/PL00012443**8.**Herbert B. Enderton,*A mathematical introduction to logic*, 2nd ed., Harcourt/Academic Press, Burlington, MA, 2001. MR**1801397****9.**M. D. Fried, M. Jarden, Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],**11**. Springer-Verlag, Berlin, 1986.MR**0868860 (89b:12010)****10.**J. Gordon, thesis, University of Michigan, 2002.**11.**Julia Gordon and Thomas C. Hales,*Virtual transfer factors*, Represent. Theory**7**(2003), 81–100. MR**1973368**, https://doi.org/10.1090/S1088-4165-03-00183-3**12.**T. C. Hales. Can -adic integrals be computed? to appear, http://xxx.lanl.gov/abs/ math.RT/0205207.**13.**I. G. Macdonald,*Spherical functions on a group of 𝑝-adic type*, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971. Publications of the Ramanujan Institute, No. 2. MR**0435301****14.**Johan Pas,*Uniform 𝑝-adic cell decomposition and local zeta functions*, J. Reine Angew. Math.**399**(1989), 137–172. MR**1004136**, https://doi.org/10.1515/crll.1989.399.137

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
22E50

Retrieve articles in all journals with MSC (2000): 22E50

Additional Information

**Thomas C. Hales**

Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Email:
hales@pitt.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07740-8

Received by editor(s):
January 7, 2002

Received by editor(s) in revised form:
October 24, 2003

Published electronically:
December 15, 2004

Additional Notes:
This research was supported by NSF grant 245332

Communicated by:
Dan M. Barbasch

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.