Distribution of torsion points

Description

The Manin-Mumford conjecture (a theorem now) posits that any subvariety of a complex torus (namely (C*)^n), that contains a Zariski dense set of torsion points must equal a finite union of torsion cosets. There are similar results known for products of elliptic curves (and in general, abelian varieties). Such a statement cannot possibly be true over algebraic closures of finite fields, as every point defined over a finite field is torsion! There is recent / ongoing work in progress that partly explains what ought to be true in the setting of setting of torii over finite fields The goal of this project would be to understand what ought to be true for products of elliptic curves defined over finite fields, instead of torii.