T-modules and Pila–Wilkie estimates

Abstract

We finally give some results in order to find a construction analogous to the one that has been built up by J. Pila and U. Zannier in their work. In the third section we prove our main result. In the fourth and last section we make some little consideration about our work.

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

“…We also remark (see [5], Remark 3) that if A is an abelian, uniformizable T −module of rank d, then Lie(A) can be written as a direct sum of a k ∞ −vector space of dimension d, into which the associated lattice Λ = Ker(e A ) is cocompact (this is called the torsion part of Lie(A)), and a k ∞ −vector space of infinite dimension (the free part of Lie(A)). In other words, by calling {ω 1 , ..., ω d } a generating set of periods of:…”

“…We refer to [5], section 1, for a complete discussion. Roughly speaking, we essentially work on the decomposition briefly described after Remark 1 of the tangent space Lie(A) of an abelian uniformizable T −module A in its torsion part and in its free part.…”

A few remarks on a Manin-Mumford conjecture in function field arithmetic and generalized Pila-Wilkie estimates

“…Our results in [D1] still apply by considering A as a T j(A) −module or, equivalently, for the F q [T j(A) ]−torsion points of A. By the way, even if this was not explicitly written and [D1, Proposition 5] is false for non diagonal differentials dΦ(a(T )), it is easy to see that the estimate we provided in [D1,Theorem 10] still applies to all the a(T )−torsion points of an algebraic subvariety X of A exactly in the way we intended, without any real change. We first recall here the statement of such a theorem, which is the main result in [D1].…”

“…Then the two-dimensional C−vector space Lie(A) is homeomorphic via the exponential map to a k−entire subset of C 2 of dimension 1 (see [D1,Definition 16]).…”

“…Since the exponential map is invertible and the logarithm is k−analytic as well, such a subset e(Lie(A)) is a proper k−entire subset of C 2 . Thus its dimension is the same at each point in which e(Lie(A)) is a k−analytic space (see [D1,Definition 13]). By Tate theory, it is easy to see that this dimension is 1.…”