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Some examples toward a Manin-Mumford conjecture for abelian uniformizable \(T\)-modules. (English. French summary) Zbl 1346.14113

The aim of the paper under review is to study a possible adaptation of the Manin-Mumford Conjecture (first proved by M. Raynaud in [Invent. Math. 74, 207–233 (1983; Zbl 0564.14020)]) to T-modules.
After the introduction the author presents the definition and properties of T-modules, which are affine algebraic varieties over function fields of characteristic \(p\) possessing a module structure that generalize to higher dimension the notion of Drinfeld Module. He then shows that the natural generalizations of the Manin-Mumford Conjecture for T-modules, proposed by L. Denis, fails for arbitrary T-module (even if similar results holds for the finite powers of Drinfeld modules as in [T. Scanlon, J. Number Theory 97, No. 1, 10–25 (2002; Zbl 1055.11037)]). Various counterexamples are constructed, the more general one is given by the product of two different Drinfeld modules, with module structures chosen properly.
Finally the author proposes a new formulation of (various versions of the) Manin-Mumford Conjecture, in the spirit of J. Pila and U. Zannier [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Apple. 19 (2), 149–162 (2008; Zbl 1164.11029)], that reads as follows:
Conjecture: Let \(\mathcal{A} = (\mathbb{G}_a^m,\phi)\) be a uniformizable T-module of dimension \(m > 1\) such that there exists \(i \neq 0\) such that the leading coefficient matrix of \(\phi(T^i)\) is invertible. Let \(X\) be a non-trivial irreducible algebraic sub variety of \(\mathcal{A}\), defined over \(\overline{k}\). If \(X\) does not contain torsion sub varieties of positive dimension, then \(X\) contains at most finitely many torsion points of \(\mathcal{A}\).

MSC:

14K12 Subvarieties of abelian varieties
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11J93 Transcendence theory of Drinfel’d and \(t\)-modules
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References:

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