Uniform analysis on local fields and applications to orbital integrals

Cluckers, Raf; Gordon, Julia; Halupczok, Immanuel

doi:10.1090/btran/25

Uniform analysis on local fields and applications to orbital integrals
HTML articles powered by AMS MathViewer

by Raf Cluckers, Julia Gordon and Immanuel Halupczok;

Trans. Amer. Math. Soc. Ser. B 5 (2018), 125-166

DOI: https://doi.org/10.1090/btran/25

Published electronically: October 2, 2018

HTML | PDF

Abstract:

We study upper bounds, approximations, and limits for functions of motivic exponential class uniformly in non-Archimedean local fields whose characteristic is $0$ or sufficiently large. Our results together form a flexible framework for doing analysis over local fields in a field-independent way. As corollaries, we obtain many new transfer principles, for example, for local constancy, continuity, and existence of various kinds of limits. Moreover, we show that the Fourier transform of an $L^2$-function of motivic exponential class is again of motivic exponential class. As an application in representation theory, we prove uniform bounds for the Fourier transforms of orbital integrals on connected reductive $p$-adic groups.

References

Jeffrey D. Adler and Alan Roche, An intertwining result for $p$-adic groups, Canad. J. Math. 52 (2000), no. 3, 449–467. MR 1758228, DOI 10.4153/CJM-2000-021-8
James Ax and Simon Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630. MR 184930, DOI 10.2307/2373065
W. Casselman, J. Cely, and T. Hales, The spherical Hecke algebra, partition functions, and motivic integration, arXiv:1611.05773, to appear in Trans. Amer. Math. Soc., DOI: https://doi.org/10.1090/tran/7465.
Raf Cluckers, Presburger sets and $p$-minimal fields, J. Symbolic Logic 68 (2003), no. 1, 153–162. MR 1959315, DOI 10.2178/jsl/1045861509
Raf Cluckers, Georges Comte, Daniel J. Miller, Jean-Philippe Rolin, and Tamara Servi, Integration of oscillatory and subanalytic functions, Duke Math. J. 167 (2018), no. 7, 1239–1309. MR 3799699, DOI 10.1215/00127094-2017-0056
Raf Cluckers, Clifton Cunningham, Julia Gordon, and Loren Spice, On the computability of some positive-depth supercuspidal characters near the identity, Represent. Theory 15 (2011), 531–567. MR 2833466, DOI 10.1090/S1088-4165-2011-00403-9
Raf Cluckers, Julia Gordon, and Immanuel Halupczok, Integrability of oscillatory functions on local fields: transfer principles, Duke Math. J. 163 (2014), no. 8, 1549–1600. MR 3210968, DOI 10.1215/00127094-2713482
Raf Cluckers, Julia Gordon, and Immanuel Halupczok, Local integrability results in harmonic analysis on reductive groups in large positive characteristic, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 6, 1163–1195 (English, with English and French summaries). MR 3297157, DOI 10.24033/asens.2236
Raf Cluckers, Julia Gordon, and Immanuel Halupczok, Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups, Electron. Res. Announc. Math. Sci. 21 (2014), 137–152. MR 3356593, DOI 10.3934/era.2014.21.137
Raf Cluckers, Julia Gordon, and Immanuel Halupczok, Transfer principles for bounds of motivic exponential functions, Families of automorphic forms and the trace formula, Simons Symp., Springer, [Cham], 2016, pp. 111–127. MR 3675165
Raf Cluckers, Thomas Hales, and François Loeser, Transfer principle for the fundamental lemma, On the stabilization of the trace formula, Stab. Trace Formula Shimura Var. Arith. Appl., vol. 1, Int. Press, Somerville, MA, 2011, pp. 309–347. MR 2856374
Raf Cluckers and Immanuel Halupczok, Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero, J. Éc. polytech. Math. 5 (2018), 45–78 (English, with English and French summaries). MR 3732692, DOI 10.5802/jep.63
R. Cluckers and L. Lipshitz, Fields with analytic structure, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1147–1223. MR 2800487, DOI 10.4171/JEMS/278
Raf Cluckers and Leonard Lipshitz, Strictly convergent analytic structures, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 1, 107–149. MR 3584560, DOI 10.4171/JEMS/662
Raf Cluckers and François Loeser, Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math. (2) 171 (2010), no. 2, 1011–1065. MR 2630060, DOI 10.4007/annals.2010.171.1011
Ju. L. Eršov, On the elementary theory of maximal normed fields, Dokl. Akad. Nauk SSSR 165 (1965), 21–23 (Russian). MR 190140
I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Generalized Functions, vol. 6, Academic Press, Inc., Boston, MA, 1990. Translated from the Russian by K. A. Hirsch; Reprint of the 1969 edition. MR 1071179
Julia Gordon and Thomas Hales, Endoscopic transfer of orbital integrals in large residual characteristic, Amer. J. Math. 138 (2016), no. 1, 109–148. MR 3462882, DOI 10.1353/ajm.2016.0005
Julia Gordon and David Roe, The canonical measure on a reductive $p$-adic group is motivic, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 2, 345–355 (English, with English and French summaries). MR 3621433, DOI 10.24033/asens.2322
Thomas C. Hales, Can $p$-adic integrals be computed?, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 313–329. MR 2058612
Thomas C. Hales, Orbital integrals are motivic, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1515–1525. MR 2111953, DOI 10.1090/S0002-9939-04-07740-8
Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR 1702257, DOI 10.1090/ulect/016
Rebecca A. Herb, Bounds for Fourier transforms of regular orbital integrals on $p$-adic Lie algebras, Represent. Theory 5 (2001), 504–523. MR 1870601, DOI 10.1090/S1088-4165-01-00125-X
Robert E. Kottwitz, Harmonic analysis on reductive $p$-adic groups and Lie algebras, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 393–522. MR 2192014
Kien Huu Nguyen, Uniform rationality of the Poincaré series of definable, analytic equivalence relations on local fields, arXiv:1610.07952, to appear in Trans. Amer. Math. Soc., DOI: https://doi.org/10.1090/tran/7357.
Sug Woo Shin and Nicolas Templier, Sato-Tate theorem for families and low-lying zeros of automorphic $L$-functions, Invent. Math. 203 (2016), no. 1, 1–177. Appendix A by Robert Kottwitz, and Appendix B by Raf Cluckers, Julia Gordon and Immanuel Halupczok. MR 3437869, DOI 10.1007/s00222-015-0583-y
Zhiwei Yun, The fundamental lemma of Jacquet and Rallis, Duke Math. J. 156 (2011), no. 2, 167–227. With an appendix by Julia Gordon. MR 2769216, DOI 10.1215/00127094-2010-210