T-modules and Pila-Wilkie estimates

Abstract: The $T-$modules, introduced by G. Anderson in the '80s, are the natural analogue of abelian varieties in Function Field Arithmetic in positive characteristic. For a special class of them we highlight that a totally similar description of the classic Weierstrass function still holds. In particular, torsion points correspond modulo a finite-rank lattice to rational points of the tangent space. We present in this work an upper bound estimate of the number of rational points of the trascendent part of the analytic set corresponding, into the tangent space, to a nontrivial algebraic subvariety of a $T-$module of this special class. Such an estimate, which takes the same shape of that proved by J. Pila and J. Wilkie, represents our first step of our strategy to prove Manin-Mumford conjecture for such special $T-$modules, based on the ideas developped by U. Zannier and J. Pila.