On genera containing non-split Eichler orders over function fields

Abstract

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line P1 splits as the sum of one dimensional vector bundles. This can be rephrased, in terms of orders, as stating that all maximal orders over the projective line in a matrix algebra split. In this work we study the extent to which this result can be generalized to Eichler 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 conjugacy classes. The latter characterization is given for arbitrary projective curves over F. The method developed here also allows us to compute quotient graphs for some subgroups of $PGL_2(F[t])$ of arithmetical interest. This paper includes material from the unpublished work "Simultaneous diagonalization of vector bundles".