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The cycle structure of a Markoff automorphism over finite fields.

Cerbu, Alois and Gunther, Elijah and Magee, Michael and Peilen, Luke (2020) 'The cycle structure of a Markoff automorphism over finite fields.', Journal of number theory., 211 . pp. 1-27.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1016/j.jnt.2019.09.022
Publisher 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/
Date accepted:09 September 2019
Date deposited:29 October 2019
Date of first online publication:28 October 2019
Date first made open access:30 November 2019

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