The dynamical Mordell-Lang conjecture in positive characteristic

TL;DR

This paper proves a structure theorem for the set of times when iterates of a point under a self-map on an algebraic torus intersect a curve in positive characteristic, revealing a complex pattern involving arithmetic and p-arithmetic sequences.

Contribution

It establishes the first structure theorem for the dynamical Mordell-Lang conjecture in positive characteristic, including p-arithmetic sequences in the description.

Findings

The set of intersection times is a union of finitely many arithmetic progressions, a finite set, and finitely many p-arithmetic sequences.

The result is sharp; the set can be infinite without containing an arithmetic progression.

This is the first proven structure theorem for such sets in positive characteristic.

Abstract

Let K be an algebraically closed field of prime characteristic p, let N be a positive integer, let f be a self-map on the algebraic torus T=G_m^N defined over K, let V be a curve in T defined over K, and let x be a K-point of T. We show that the set S consisting of all positive integers n for which f^n(x) is contained in V is a union of finitely many arithmetic progressions, along with a finite set and with finitely many p-arithmetic sequences, which are sets of the form {b + ap^{kn}: n is a positive integer} where a and b are given rational numbers and k is a positive integer. We also prove that our result is sharp in the sense that S may be infinite without containing an arithmetic progression. Our result addresses a positive characteristic version of the dynamical Mordell-Lang conjecture and it is the first known instance when a structure theorem is proven for the set S which includes p-arithmetic sequences.