We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.

Durham Research Online
You are in:

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.


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.
Download PDF
Publisher Web site:
Publisher statement:© 2019 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
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

Save or Share this output

Look up in GoogleScholar