Algebra & Number Theory
- Algebra Number Theory
- Volume 12, Number 7 (2018), 1715-1748.
Algebraic dynamics of the lifts of Frobenius
We study the algebraic dynamics of endomorphisms of projective spaces with coefficients in a -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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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