Research in Number Theory

, 5:37 | Cite as

Joint distribution of inverses in matrix groups over finite fields

  • Corentin Perret-GentilEmail author


We study the joint distribution of the solutions to the equation \(gh=x\) in \(G(\mathbb {F}_p)\) as \(p\rightarrow \infty \), for any fixed \(x\in G(\mathbb {Z})\), where \(G={\text {GL}}_n\), \({\text {SL}}_n\), \({\text {Sp}}_{2n}\) or \({\text {SO}}_{n}^\pm \). In the special linear case, this answers in particular a question raised by Hu and Li, and improves their error terms. Similar results are derived in certain subgroups, and when the entries of gh lie in fixed intervals. The latter shows for example the existence of \(g\in {\text {GL}}_n(\mathbb {F}_p)\) such that \(g,g^{-1}\) have all entries in \([0, c_np^{1-1/(2n^2+2)+\varepsilon }]\) for some absolute constant \(c_n>0\). The key for these results is to use Deligne’s extension of the Weil conjectures on a sheaf on G, along with the stratification theorem of Fouvry, Katz and Laumon, instead of reducing to bounds on classical Kloosterman sums.

Mathematics Subject Classification

11C20 11K36 11T24 11L05 



  1. 1.
    Ahmadi, O., Shparlinski, I.E.: Distribution of matrices with restricted entries over finite fields. Indag. Math. (N.S.) 18(3), 327–337 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aschbacher, M.: On the maximal subgroups of the finite classical groups. Invent. Math. 76(3), 469–514 (1984)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bourgain, J., Gamburd, A., Sarnak, P.: Affine linear sieve, expanders, and sum-product. Invent. Math. 179(3), 559–644 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Sean Allen Broughton: A note on characters of algebraic groups. Proc. Am. Math. Soc. 89(1), 39–40 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Deligne, P.: Cohomologie étale, séminaire de géométrie algébrique du Bois-Marie SGA 4\(\frac{1}{2}\). Lecture Notes in Mathematics, vol. 569. Springer (1977)Google Scholar
  6. 6.
    Deligne, P.: La conjecture de Weil. II. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 52(1), 137–252 (1980)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Drmota, M., Tichy, R.F.: Sequences, Discrepancies, and Applications. Lecture Notes in Mathematics, vol. 1651. Springer (1997)Google Scholar
  8. 8.
    Ferguson, R., Hoffman, C., Luca, F., Ostafe, A., Shparlinski, I.E.: Some additive combinatorics problems in matrix rings. Rev. Mat. Complut. 23(2), 501–513 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fouvry, E., Katz, N.M.: A general stratification theorem for exponential sums, and applications. J. für die reine Angew. Math. 2001(504), 115–166 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fouvry, É., Kowalski, E., Michel, P.: Algebraic twists of modular forms and Hecke orbits. Geom. Funct. Anal. 25(2), 580–657 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fouvry, É.: Consequences of a result of N. Katz and G. Laumon concerning trigonometric sums. Israel J. Math. 120(1), 81–96 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Su, H., Li, Y.: Gauss sums over some matrix groups. J. Number Theory 132(12), 2967–2976 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Su, H., Li, Y.: On a uniformly distributed phenomenon in matrix groups. J. Number Theory 133(11), 3578–3588 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Katz, N.M.: Sums of Betti numbers in arbitrary characteristic. Finite Fields Appl. 7(1), 29–44 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Katz, N.M., Laumon, G.: Transformation de Fourier et majoration de sommes exponentielles. Inst. Hautes Études Sci. Publ. Math. 62, 361–418 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shparlinski, I.E.: Modular hyperbolas. Jpn. J. Math. 7(2), 235–294 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Steele, J.M.: The Cauchy-Schwarz Master Class. MAA Problem Books Series. Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge (2004). (An introduction to the art of mathematical inequalities)Google Scholar
  18. 18.
    Wilson, R.A.: The Finite Simple Groups, vol. 251 of Graduate Texts in Mathematics. Springer (2009)Google Scholar
  19. 19.
    Xu, J.: Stratification for Multiplicative Character Sums. International Mathematics Research Notices (2018). rny096Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada
  2. 2.ZurichSwitzerland

Personalised recommendations