Algebraic dynamics of the lifts of Frobenius

TL;DR

This paper explores algebraic dynamics of lifts of Frobenius endomorphisms over p-adic fields, proving key conjectures like Manin-Mumford and Mordell-Lang in this setting using perfectoid space theory.

Contribution

It establishes dynamical versions of Manin-Mumford and Mordell-Lang conjectures for lifts of Frobenius, and provides a new proof of the dynamical Tate-Voloch conjecture, employing perfectoid spaces.

Findings

Proved dynamical Manin-Mumford conjecture for lifts of Frobenius.

Established dynamical Mordell-Lang conjecture in this context.

Provided a new proof of the dynamical Tate-Voloch conjecture.

Abstract

We study the algbraic dynamics for endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteritic 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 for 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 embedding into a dynamical system for some endomorphism of a projective space.