Diophantine problems over \(t\)-modules
Let \({\mathbf A}= \mathbb{F}_q [T]\) and let \(C\) be the Carlitz module defined over \({\mathbf A}\). Let \(C^{\otimes n}\) be the \(n\)-th tensor power of \(C\). It is by now well known that \(C\) plays the role of the multiplicative group \(\mathbb{G}_m\) for \({\mathbf A}\) and that its division values generate analogs of classical cyclotomic fields. Thus it makes sense to inquire if one can define a ``Fermat equation'' associated to \(C\); in fact, this is easily done by noting that \(x^n- y^n= y^n (({x \over y})^n -1)\) and that the roots of \(u^n- 1\) \((u:= x/y)\) are the \(n\) division points of \(\mathbb{G}_m\). The fields generated by the division values of \(C\) are actually geometric extensions of \({\mathbf k}\) (i.e., they contain no constant field extensions). Thus one refers to these fields as the geometric cyclotomic extensions, and the above Fermat equation as the geometric Fermat equation. The constant field extensions, which after all are obtained from roots of unity, are called the arithmetic cyclotomic extensions, and the classical Fermat equation, viewed over \({\mathbf A}\), is referred to as the arithmetic Fermat equation. In a striking change from the classical theory, it turns out that the geometric Fermat equations are amenable to a geometric attrack [see \textit{L. Denis}, Trans. Am. Math. Soc. 343, 713-726 (1994; Zbl 0811.11044)]. As one associates to \(C\) a Fermat equation, so too one can attach ``Fermat families'' to \(C^{\otimes n}\) (see \textit{L. Denis}, loc. cit.). In the paper being reviewed the author establishes that, under certain very general hypotheses, one has finitely many relatively prime solutions to these families. This is established through results of the author on heights. Various other results of interest are presented, such as estimates relating the canonical height and the Weil height. Finally, a number of open problems in the diophantine geometry of Drinfeld modules are stated.
scientific article; zbMATH DE number 858017
scientific article