Orbital integrals are motivic

Hales, Thomas

doi:10.1090/S0002-9939-04-07740-8

Orbital integrals are motivic
HTML articles powered by AMS MathViewer

by Thomas C. Hales PDF

Proc. Amer. Math. Soc. 133 (2005), 1515-1525 Request permission

Abstract:

This article shows that under general conditions, $p$-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” $p$-adic integrals are motivic.

References

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, DOI 10.1007/978-3-0348-8399-3_{7}
P. Cartier, Representations of $p$-adic groups: a survey, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
Jan Denef and François Loeser, Definable sets, motives and $p$-adic integrals, J. Amer. Math. Soc. 14 (2001), no. 2, 429–469. MR 1815218, DOI 10.1090/S0894-0347-00-00360-X
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
J. Denef, F. Loeser, One some rational generating series occurring in arithmetic geometry, preprint math.NT/0212202.
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.
Santiago Encinas and Herwig Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821–845. MR 1949115, DOI 10.1007/PL00012443
Herbert B. Enderton, A mathematical introduction to logic, 2nd ed., Harcourt/Academic Press, Burlington, MA, 2001. MR 1801397
Michael D. Fried and Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 11, Springer-Verlag, Berlin, 1986. MR 868860, DOI 10.1007/978-3-662-07216-5
J. Gordon, thesis, University of Michigan, 2002.
Julia Gordon and Thomas C. Hales, Virtual transfer factors, Represent. Theory 7 (2003), 81–100. MR 1973368, DOI 10.1090/S1088-4165-03-00183-3
T. C. Hales. Can $p$-adic integrals be computed? to appear, http://xxx.lanl.gov/abs/math.RT/0205207.
I. G. Macdonald, Spherical functions on a group of $p$-adic type, Publications of the Ramanujan Institute, No. 2, University of Madras, Centre for Advanced Study in Mathematics, Ramanujan Institute, Madras, 1971. MR 0435301
Johan Pas, Uniform $p$-adic cell decomposition and local zeta functions, J. Reine Angew. Math. 399 (1989), 137–172. MR 1004136, DOI 10.1515/crll.1989.399.137