On function field Mordell-Lang and Manin-Mumford

TL;DR

This paper reduces the function field Mordell-Lang conjecture to the Manin-Mumford conjecture across all characteristics using model theory, extending quantifier elimination results to all abelian varieties in positive characteristic.

Contribution

It provides a model-theoretic reduction of the Mordell-Lang conjecture to Manin-Mumford, extending quantifier elimination to all abelian varieties in positive characteristic.

Findings

Reduction of Mordell-Lang to Manin-Mumford in all characteristics

Extension of quantifier elimination to all abelian varieties in positive characteristic

Revised proof and reorganization in the latest version

Abstract

We present a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski structures. In this version 2, the quantifier elimination result in positive characteristic is extended from simple abelian varieties to all abelian varieties, completing the main theorem in the positive characteristic case. In version 3, some corrections are made to the proof of quantifier elimination in positive characteristic, and the paper is substantially reorganized.