Algebraic dynamics of the lifts of Frobenius

Xie, Junyi

doi:10.2140/ant.2018.12.1715

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Algebraic dynamics of the lifts of Frobenius

Junyi Xie

Vol. 12 (2018), No. 7, 1715–1748

DOI: 10.2140/ant.2018.12.1715

Abstract

We study the algebraic dynamics of endomorphisms of projective spaces with coefficients in a $p$ -adic field whose reduction in positive characteristic is the Frobenius. In particular, we prove a version of the dynamical Manin–Mumford conjecture and the dynamical Mordell–Lang conjecture for the coherent backward orbits of such endomorphisms. We also give a new proof of a dynamical version of the Tate–Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definition, a dynamical system for a polarized lift of Frobenius on a projective variety can be embedded into a dynamical system for some endomorphism of a projective space.

Keywords

algebraic dynamics, perfectoid space

Mathematical Subject Classification 2010

Primary: 37P55

Secondary: 37P20, 37P35

Milestones

Received: 2 October 2017

Revised: 15 June 2018

Accepted: 17 July 2018

Published: 27 October 2018

Authors

Junyi Xie
Institut de Recherche Mathématique de Rennes CNRS - Université de Rennes 1 Bâtiment 22-23 du campus de Beaulieu Rennes France

Vol. 12, No. 7, 2018