Simultaneous diagonalization of vector bundles

Abstract

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line splits as the sum of one dimensional vector bundles. In this work we study simultaneous splittings of two dimensional vector bundles over a finite field using the theory of Eichler orders. In the sheaf-theoretical context, an Eichler order in a matrix algebra is the intersection of the sheaves of endomorphisms of two vector bundles, so characterizing split Eichler orders solves the problem of simultaneous splitting. We caracterize both the genera of Eichler orders containing only split Eichler orders and the genera containing only a finite number of non-split classes, in terms of a divisor-valued distance parametrizing the genera. This article shall not be published in its present form. These results will be included in the article "On genera containing non-split Eichler orders over function fields" [arXiv:1905.08244].