Alois Cerbu
The cycle structure of a Markoff automorphism over finite fields
Cerbu, Alois; Gunther, Elijah; Magee, Michael; Peilen, Luke
Abstract
We begin an investigation of the action of pseudo-Anosov elements of Out(F2) on the Marko-type varieties X : x2 + y2 + z2 = xyz + 2 + over nite elds Fp with p prime. We rst make a precise conjecture about the permutation group generated by Out(F2) on X????2(Fp) that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting `generically'. We prove that this conjecture is sharp. We show that for a xed pseudo-Anosov g 2 Out(F2), there is always an orbit of g of length C log p + O(1) on X(Fp) where C > 0 is given in terms of the eigenvalues of g viewed as an element of GL2(Z). This improves on a result of Silverman from [26] that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic (p ! 1) behavior of the longest orbit of a xed pseudo-Anosov g acting on X????2(Fp) is dictated by a dichotomy that we describe both in combinatorial terms and in algebraic terms related to Gauss's ambiguous binary quadratic forms, following Sarnak [23]. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture in Conjecture 1.10.
Citation
Cerbu, A., Gunther, E., Magee, M., & Peilen, L. (2020). The cycle structure of a Markoff automorphism over finite fields. Journal of Number Theory, 211, 1-27. https://doi.org/10.1016/j.jnt.2019.09.022
Journal Article Type | Article |
---|---|
Acceptance Date | Sep 9, 2019 |
Online Publication Date | Oct 28, 2019 |
Publication Date | Jun 30, 2020 |
Deposit Date | Oct 29, 2019 |
Publicly Available Date | Oct 28, 2020 |
Journal | Journal of Number Theory |
Print ISSN | 0022-314X |
Electronic ISSN | 1096-1658 |
Publisher | Elsevier |
Peer Reviewed | Peer Reviewed |
Volume | 211 |
Pages | 1-27 |
DOI | https://doi.org/10.1016/j.jnt.2019.09.022 |
Public URL | https://durham-repository.worktribe.com/output/1280623 |
Files
Accepted Journal Article
(529 Kb)
PDF
Publisher Licence URL
http://creativecommons.org/licenses/by-nc-nd/4.0/
Copyright Statement
© 2019 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
You might also like
Automorphism-invariant positive definite functions on free groups
(2018)
Presentation / Conference Contribution
Quantum Unique Ergodicity for Cayley graphs of quasirandom groups
(2023)
Journal Article
Random Unitary Representations of Surface Groups II: The large n limit
(2023)
Journal Article
Matrix group integrals, surfaces, and mapping class groups II: O(n) and Sp(n)
(2022)
Journal Article
Core surfaces
(2022)
Journal Article
Downloadable Citations
About Durham Research Online (DRO)
Administrator e-mail: dro.admin@durham.ac.uk
This application uses the following open-source libraries:
SheetJS Community Edition
Apache License Version 2.0 (http://www.apache.org/licenses/)
PDF.js
Apache License Version 2.0 (http://www.apache.org/licenses/)
Font Awesome
SIL OFL 1.1 (http://scripts.sil.org/OFL)
MIT License (http://opensource.org/licenses/mit-license.html)
CC BY 3.0 ( http://creativecommons.org/licenses/by/3.0/)
Powered by Worktribe © 2024
Advanced Search