# On genera containing non-split Eichler orders over function fields

## TL;DR

This paper investigates the generalization of the Grothendieck-Birkhoff theorem to Eichler orders over function fields, characterizing genera with only split or finitely many non-split classes, and computing related quotient graphs.

## Contribution

It characterizes genera of Eichler orders over finite fields with specific splitting properties and develops methods to compute quotient graphs for certain subgroups of PGL2.

## Key findings

- Characterization of genera with only split Eichler orders
- Identification of genera with finitely many non-split classes over curves
- Method for computing quotient graphs of subgroups of PGL2

## 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 This paper includes material from the unpublished work "Simultaneous diagonalization of vector bundles".