The dynamical Mordell-Lang problem for etale maps

TL;DR

This paper establishes a dynamical version of the Mordell-Lang conjecture for etale endomorphisms of quasiprojective varieties, using p-adic and algebraic geometry methods, with implications for classical conjectures and dense sequences.

Contribution

It introduces a dynamical approach to the Mordell-Lang conjecture for etale maps, providing new proofs and answering open questions in algebraic geometry.

Findings

Proves a dynamical Mordell-Lang conjecture for etale endomorphisms.

Provides a new proof of the classical Mordell-Lang conjecture for semiabelian varieties.

Answers positively a question about dense sequences of points in schemes.

Abstract

We prove a dynamical version of the Mordell-Lang conjecture for etale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer positively a question of Keeler/Rogalski/Stafford for critically dense sequences of closed points of a Noetherian integral scheme.