The Dynamical Mordell-Lang Conjecture for endomorphisms of semiabelian varieties defined over fields of positive characteristic

TL;DR

This paper proves a version of the Dynamical Mordell-Lang Conjecture for semiabelian varieties over fields of positive characteristic, linking it to a Diophantine problem in characteristic zero and confirming cases for algebraic tori.

Contribution

It establishes the conjecture's equivalence with a Diophantine problem in characteristic zero and verifies the conjecture for certain cases of algebraic tori.

Findings

Proves the conjecture for semiabelian varieties in specific cases.

Shows the conjecture's equivalence to a Diophantine problem in characteristic zero.

Validates the conjecture when the subvariety has dimension at most 2 or under certain endomorphism conditions.

Abstract

Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let f be a regular self-map on X defined over K, let V be a subvariety of X defined over K, and let x be a K-point of X. The Dynamical Mordell-Lang Conjecture in characteristic p predicts that the set S consisting of all positive integers n for which f^n(x) lies on V is a union of finitely many arithmetic progressions, along with finitely many p-sets, which are sets consisting of all integers of the form c_1 p^{k_1 n_1} + ... + c_m p^{k_m n_m} (as we vary n_1,..,n_m among all positive integers), for some given positive integer m, some rational numbers c_i and some non-negative integers k_i. We prove that this conjecture is equivalent with some difficult Diophantine problem in characteristic 0. In the case X is an algebraic torus, we can prove the conjecture in two cases: either when the dimension of V is at most 2, or when no iterate of f is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of X.