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Research in Number Theory

, 5:37 | Cite as

Joint distribution of inverses in matrix groups over finite fields

  • Corentin Perret-GentilEmail author
Research
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Abstract

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 

Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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