## Algebra & Number Theory

### Algebraic dynamics of the lifts of Frobenius

Junyi Xie

#### 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.

#### Article information

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

Dates
Revised: 15 June 2018
Accepted: 17 July 2018
First available in Project Euclid: 9 November 2018

https://projecteuclid.org/euclid.ant/1541732439

Digital Object Identifier
doi:10.2140/ant.2018.12.1715

Mathematical Reviews number (MathSciNet)
MR3871508

Zentralblatt MATH identifier
06976300

#### Citation

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. https://projecteuclid.org/euclid.ant/1541732439

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