T-modules and Pila–Wilkie estimates

We present in this work an upper bound estimate of the number of torsion points contained into the transcendent part of an algebraic subvariety of an abelian and uniformizable T-module in function of their degree, taking an analogous shape as in [PW]. We give in particular a version of the Implicit Function Theorem for analytic sets provided by a non-archimedean topology, we present our definition of analytic space and we introduce the notion of dimension on it, showing that it is coherent with the classic notion of dimension on rigid analytic spaces. We then use such results to prove a theorem of density of regular points (which we define in this new context). We finally give some results which allow to apply our main theorem to our strategy to prove Manin–Mumford conjecture in this setting.