Une borne pour l'action de l'inertie sauvage sur la torsion d'un module de Drinfeld. (A bound for the wild inertia action on the torsion of a Drinfeld module)
The author gives an upper bound for the differents of Galois extensions of global function fields by torsion points of rational Drinfeld modules. Let \(K/\mathbb{F}_q(T)\) be a global function field and \(K\{\tau\}\) the skew polynomial ring with the commutation rule \(\tau c=c^q\tau\) for any \(c\in K\). Consider a rational Drinfeld module \(\phi:A=\mathbb{F}_q[T]\rightarrow K\{\tau\}\), \(a\mapsto\phi_a\), of rank \(r\). Such a module generates the Galois extension \(K[a]=K(\phi[a])\) for any \(a\in A\) where \(\phi[a]=\text{Ker}(\phi_a|K^s)\) is the finite \(A\)-module of \(a\)-torsion points. The author gives an upper bound for the different \(\mathfrak{D}(K[a]/K)\) in terms of the Galois structure of the \(a\)-torsion module \(\phi[a]\). Combining this result with an effective version of the Chebotarev density theorem, the author refines a theorem of \textit{Y. Taguchi} [Ramifications arising from Drinfeld modules, In: The arithmetic of function fields, de Gruyter, Berlin (1992; Zbl 0801.11027)] allowing for the efficient determination of whether two Drinfeld modules are isogenous.
scientific article; zbMATH DE number 1837409
scientific article