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

Abstract

The aim of this work is to present a possible adaptation of the Manin-Mumford conjecture to the T −modules, a mathematical object which has been introduced in the 1980's by G. Anderson as the natural analogue of the abelian varieties in the context of modules over rings which are contained in positive characteristic function fields. We propose then a generalisation of such an adapted conjecture to a modified general version of Mordell-Lang conjecture for T −modules which might correct the one proposed for the first time by L. Denis in [Den2] but no longer compatible with the present results.We will remind in our first preliminary section the formulation of the Mordell-Lang and the Manin-Mumford conjectures and the definition of T −modules and sub-T −modules, listing nothing more than the basic definitions and properties which are strictly essential to us in order to state and prove our theorems. All the detailed informations the reader G. Faltings proved the Mordell-Lang conjecture (see [F]) in the following version.

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

“…Moreover, as we showed in [D2,Proposition 2.12], one may still find counterexamples produced by an insufficient "degree of abelianity" of A (see the discussion after [D2, Proposition 2.12]). We had therefore to strenghten the hypotheses on the finiteness of the rank of the T −motive associated to A (see [D2,Theorem 2.13] and [D2,Proposition 2.15]). The final statement we propose is the following one.…”

“…By choosing k ∞ an algebraic closure of k ∞ and calling C := (k ∞ ) ∞ , we let k be the algebraic closure of k in C. D e f i n i t i o n 1.1. A T −module of dimension m and degree d is a pair A = (G m a , Φ), where G a is the algebraic additive group over C and Φ the F q −algebra homomorphism defined from F q [T ] to k m,m {τ } (see [D2,Definition 1.5]) such that (1) Φ(T )(τ ) = a 0 + a 1 τ + ... + a d τ d…”

“…Therefore it is clear that d(Φ(a(T ))) = a(T )1 m + N a(T ) for a nilpotent matrix N a(T ) . We also remind that a T −module A is abelian if its rank (see [D2,Definition 1.12]) is finite. We also denote by k(Φ) the definition field of A: it is the finite field extension of k in k generated by the entries of the coefficients of Φ(T ).…”

“…The main purpose of our work in [D1] was to outline a possible strategy for proving a general adaptation of the Manin-Mumford conjecture to the context of abelian and uniformizable T −modules (we introduce such a special class of T −modules in the next section). We showed in [D2] that a naïve version of this statement is false in general and we exhibited several counterexamples by remarking firstly that a theoretical statement must involve, as analogues of abelian subvarieties, all the possible sub−B−modules of the given A (see [D2,Proposition 2.5]). Moreover, as we showed in [D2,Proposition 2.12], one may still find counterexamples produced by an insufficient "degree of abelianity" of A (see the discussion after [D2, Proposition 2.12]).…”

“…We showed in [D2] that a naïve version of this statement is false in general and we exhibited several counterexamples by remarking firstly that a theoretical statement must involve, as analogues of abelian subvarieties, all the possible sub−B−modules of the given A (see [D2,Proposition 2.5]). Moreover, as we showed in [D2,Proposition 2.12], one may still find counterexamples produced by an insufficient "degree of abelianity" of A (see the discussion after [D2, Proposition 2.12]). We had therefore to strenghten the hypotheses on the finiteness of the rank of the T −motive associated to A (see [D2,Theorem 2.13] and [D2,Proposition 2.15]).…”