Papers and Preprints

• Appendix to On the pointwise convergence of the number of Abelian varieties over F_p with fixed trace (with Jeff Achter).


• Integrability and singularities of Harish-Chandra characters (with Itay Glazer and Yotam Hendel ).


• Orbital integrals and normalizations of measures (with appendix by Matthew Koster).
  Lecture notes for the winter school at the National University of Singapore (2018/19).


• Counting abelian varieties over finite fields via Frobenius densities (with Jeff Achter , S. Ali Altug , and Luis Garcia , and with an appendix by Wen-Wei Li and Thomas Rud ).
  Algebra and Number Theory, 17 No. 7 (2023), pp. 1239--1280.
  

  Important warning (October 2025): we recently learned that the main definition of Gekeler ratio is incorrect. There is a conceptual error in the paper. There is, however, an easy fix. Erratum and full corrections will appear soon. In the meanwhile, if you are planning to use any part of this paper, please email me for corrections and clarification.

  The correct version for stable orbital integrals appears in the appendix to On the pointwise convergence of the number of Abelian varieties over F_p with fixed trace .


• Uniform analysis on local fields and applications to orbital integrals , (with R. Cluckers and I. Halupczok ).
  Trans. Amer. Math. Soc. Ser. B, Vol. 5, 125166. (2018).


• Elliptic curves, random matrices and orbital integrals (with Jeff Achter and with an appendix by S. Ali Altug ).
  Pacific J. Math. 286 no. 1 (2017), pp 1-24.
  

  Important warning (October 2025): we recently learned that there is a small conceptual error in the paper, in addition to all the corrections discussed in Section 7 of the sequel . None of the main results are affected, and a corrected version will be posted to the ArXiv soon. In the meanwhile, if you are planning to use any part of this paper, please email me for corrections and clarification.

  The preliminary corrected version (new text in blue) is here .


• The canonical measure on a reductive p-adic group is motivic (with David Roe ) (2015).
  Ann. Sci. Ec. Norm. Sup. 50, no. 2, 345-355, 2017.
  This is a companion paper to this one , removing an unnecessary technical difficulty about motivic-ness of Haar measure on general reductive groups.


• Endoscopic transfer of orbital intgerals in large residual characteristic (with T.C. Hales ).
  Amer. J. Math. Volume 138, Number 1, February 2016 pp. 109-148


• Shalika germs for sl(n) and sp(2n) are motivic (with Lance Robson and Sharon Frechette ).
  In the proccedings of WIN-Europe, 2013 .


• Transfer principles for bounds of motivic exponential functions (with R. Cluckers and I. Halupczok ).
  In the proceedings of Simons symposium on Families of Automorphic forms and the Trace Formula (2014) .


• Motivic functions, integrability, and uniform in p bounds for orbital integrals (with R. Cluckers and I. Halupczok ).
  Electronic Math. Research Announcements in Math. Sciences, volume 21, pp. 137-152, 2014. published version .
  This is an announcement (and summary of all the results) of the three papers below, and related work.


• Appendix B to Sato-Tate theorem for families and low-lying zeros of automorphic L-functions by Sug Woo Shin and Nicolas Templier. (with R. Cluckers and I. Halupczok ). Published version in Invent. Math.
  In the appendix, we prove a uniform in p bound for normalized orbital integrals.


• Local integrability results in harmonic analysis on reductive groups in large positive characteristic. (with R. Cluckers and I. Halupczok ).
  Ann. Sci. Ec. Norm. Sup., Volume 47, No. 6 (2014), 1163-1195.


• Integrability of oscillatory functions on local fields: transfer principles. (with R. Cluckers and I. Halupczok ).
  Duke Math J., vol.163 No. 8, pp. 1549-1600, (2014).


• Transfer to characteristic zero -- appendix to "The fundamental lemma of Jacquet-Rallis in positive characteristics" by Zhiwei Yun.
  Duke Math J. 156 No. 2 (2011), 220-227.


• On the computability of some positive-depth characters near the identity (with R. Cluckers , C. Cunningham , and L. Spice ).
  Represent. Theory 15 (2011), 531-567.


• An overview of arithmetic motivic integration. (with Y. Yaffe)
  in "Ottawa Lectures on p-adic Groups", C. Cunnigham and M. Nevins, Eds., Fields Institute Monograph Series, 2009.


• Motivic proof of a character formula for SL(2) (with Clifton Cunningham ).
  Experiment. Math. 18 No.1 (2009), pp.11-44.


• Motivic Haar Measure on Reductive Groups .
  Canadian J. Math. 58 (2006), No. 1, 93--114.


• Motivic nature of character values of depth-zero representations.
  IMRN 2004, no. 34, 1735 - 1760.


• Virtual Transfer Factors , (with T.C. Hales ).
  Represent. Theory 7 (2003), 81--100 (electronic).


• Common hypercyclic vector for the multiples of backward shift , (with E.V. Abakumov).
  J. Funct. Anal. 200 (2003), no. 2, 494--504.


• Composition operators on the space of Dirichlet series with square summable coefficients, (with Haakan Hedenmalm ).
  Michigan Math. Journal, 46 (1999), no. 2, 313--329.

Motivic integration in representation theory

• T.C. Hales, Can p-adic integrals be computed? (this is a good place to start).
• R. Cluckers, T. Hales, and F. Loeser, Transfer Principle for the Fundamental Lemma .
• T.C. Hales, Orbital integrals are motivic.
• C. Cunningham and T.C. Hales, Good orbital integrals .
• J. Diwadkar Nilpotent conjugacy classes in p-adic Lie algebras and definability in Pas' language .

• Motivic Poisson summation by E. Hrushovski and D. Kazhdan.
• The value ring of geometric motivic integration and the Iwahori Hecke algebra of SL_2. by E. Hrushovski and D. Kazhdan.
• Fourier transform of the additive group in algebraically closed valued fields by Yimu Yin.

Motivic Integration links and resources

• 2004 complete motivic Integration bibiography , at Manuel Blickle's old page.
• Other materials at Manuel Blickle's page.
• Wim Veys
• "What is Motivic Measure?" By T.C. Hales
• More resources can be found at an old page for the seminar on motivic integration we had in 2011 at UBC.

Representation Theory links and resources

• David Ben-Zvi's Geometric Langlands page.
• Bill Casselman.
• Paul Garrett.
• Fiona Murnaghan.
• A course on representation theory (D. Barbasch)
• Collected works of Robert Langlands.

Acknowledgment