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

TL;DR

This paper demonstrates that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture for abelian varieties over function fields, providing a proof independent of model theory tools.

Contribution

It establishes a direct implication from Manin-Mumford to Mordell-Lang in positive characteristic without relying on model-theoretic methods.

Findings

Proves the implication in positive characteristic.

Provides a model-theory independent proof of Mordell-Lang.

Clarifies the relationship between the two conjectures in this setting.

Abstract

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.