Simultaneous diagonalization of vector bundles

TL;DR

This paper investigates the simultaneous splitting of two-dimensional vector bundles over finite fields using Eichler orders, providing a classification based on divisor-valued distances, extending classical results to a new context.

Contribution

It characterizes genera of Eichler orders with only split or finitely many non-split classes, advancing the understanding of vector bundle decompositions over function fields.

Findings

Characterization of genera with only split Eichler orders

Identification of genera with finitely many non-split classes

Use of divisor-valued distances to parametrize genera

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 This article shall not be published in its present form.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].