Some examples toward a Manin-Mumford conjecture for T-modules

TL;DR

This paper explores an adaptation of the Manin-Mumford conjecture for T-modules, aiming to generalize and correct previous conjectures in the context of positive characteristic function fields.

Contribution

It proposes a new formulation of the Manin-Mumford conjecture for T-modules and a revised version of the Mordell-Lang conjecture compatible with current results.

Findings

Proposed an adapted Manin-Mumford conjecture for T-modules

Suggested a corrected general Mordell-Lang conjecture for T-modules

Provided a framework aligning with recent mathematical results

Abstract

The aim of this work is to present a possible adaptation of the Manin-Mumford conjecture to the $T-$modules, a mathematical object which has been introduced in the 1980's by G. Anderson as the natural analogue of the abelian varieties in the context of modules over rings which are contained in positive characteristic function fields. We propose then a generalisation of such an adapted conjecture to a modified general version of Mordell-Lang conjecture for $T-$modules which might correct the one proposed for the first time by L. Denis but no longer compatible with the present results.