Algebra & Number Theory

Algebraic dynamics of the lifts of Frobenius

Junyi Xie

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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.

Article information

Algebra Number Theory, Volume 12, Number 7 (2018), 1715-1748.

Received: 2 October 2017
Revised: 15 June 2018
Accepted: 17 July 2018
First available in Project Euclid: 9 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37P55: Arithmetic dynamics on general algebraic varieties
Secondary: 37P20: Non-Archimedean local ground fields 37P35: Arithmetic properties of periodic points

algebraic dynamics perfectoid space


Xie, Junyi. Algebraic dynamics of the lifts of Frobenius. Algebra Number Theory 12 (2018), no. 7, 1715--1748. doi:10.2140/ant.2018.12.1715.

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