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.

The proof of the main Theorem 10 contains a serious gap, which invalidates it

Cited by in corpus (3)

• Rational points of rigid-analytic sets: a Pila-Wilkie type theorem
• Some examples toward a Manin-Mumford conjecture for T-modules
• A few remarks on a Manin-Mumford conjecture in function field arithmetic and generalized Pila-Wilkie estimates