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The cycle structure of a Markoff automorphism over finite fields

Cerbu, Alois; Gunther, Elijah; Magee, Michael; Peilen, Luke

The cycle structure of a Markoff automorphism over finite fields Thumbnail


Alois Cerbu

Elijah Gunther

Luke Peilen


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.


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.

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
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 211
Pages 1-27


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