On the Manin-Mumford and Mordell-Lang conjectures in positive characteristic

We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.Comment: arXiv admin note: substantial text overlap with arXiv:1103.262

Infinitely pp-divisible points on abelian varieties defined over function fields of characteristic p>0p>0

In this article we consider some questions raised by F. Benoist, E. Bouscaren and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is \mZ then there are no infinitely $p$-divisible points of order a power of $p$

Algebraic dynamics of the lifts of Frobenius

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.Comment: 37 page