Towards the full Mordell-Lang conjecture for Drinfeld modules

TL;DR

This paper proves a finiteness result for intersections of generic affine subvarieties with finite rank submodules in the context of Drinfeld modules, advancing the understanding of the Mordell-Lang conjecture in positive characteristic.

Contribution

It establishes a finiteness theorem for intersections involving generic subvarieties and finite rank submodules of Drinfeld modules, extending previous conjectures.

Findings

Finite intersection of generic subvarieties with finite rank submodules

Advancement towards the full Mordell-Lang conjecture for Drinfeld modules

New techniques in positive characteristic algebraic geometry

Abstract

Let $\phi$ be a Drinfeld module of generic characteristic, and let $X$ be a sufficiently generic affine subvariety of $\mathbb{G}_a^g$. We show that the intersection of $X$ with a finite rank $\phi$-submodule of $\mathbb{G}_a^g$ is finite.