On genera containing non-split Eichler orders over function fields

Arenas-Carmona, Luis; Bravo, Claudio

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 647-677.

Le théorème de Grothendieck–Birkhoff établit que tout faisceaux vectoriel de dimension finie sur la droite projective ${\mathbb{P}}^1$ se scinde en somme de faisceaux vectoriels unidimensionnels (fibrés en droites). Il peut être reformulé en termes d’ordres comme l’énoncé que tous les ${\mathbb{P}}^1$-ordres maximaux se scindent. Ceci est utile, car les ordres scindés jouent un rôle important dans le calcul des graphes quotients. Dans ce travail, on étudie dans quelle mesure ce résultat se généralise aux ${\mathbb{P}}^1$-ordres d’Eichler, lorsque le corps de base $𝔽$ est fini. Pour être précis, on caractérise, d’une part, les genres des ordres d’Eichler contenant uniquement des ordres scindés et, d’autre part, les genres ne contenant qu’un nombre fini de classes d’isomorphie non scindées. La méthode développée ici nous permet également de calculer les graphes quotients pour certains sous-groupes de ${\mathrm{PGL}}_2\left(𝔽\left[t\right]\right)$ d’intérêt arithmétique.

Grothendieck–Birkhoff Theorem states that every finite dimensional vector bundle over the projective line ${\mathbb{P}}^1$ splits as the sum of one dimensional vector bundles (line bundles). This can be rephrased, in terms of orders, as stating that all maximal ${\mathbb{P}}^1$-orders in a matrix algebra split. This is useful, since split orders play an important role when computing quotient graphs. In this work we study the extent to which this result can be generalized to Eichler ${\mathbb{P}}^1$-orders when the base field $𝔽$ is finite. To be precise, we characterize both the genera of Eichler orders containing only split orders and the genera containing only a finite number of non-split isomorphism classes. The method developed here also allows us to compute quotient graphs for some subgroups of ${\mathrm{PGL}}_2\left(𝔽\left[t\right]\right)$ of arithmetical interest.

Classification : 11R58, 14H60, 14G15, 20E08
Mots clés : Global function fields, eichler orders, quotient graphs, vector bundles

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