Class numbers of central simple algebras over global function fields

TL;DR

This paper introduces an effective method to compute class numbers of hereditary orders in central simple algebras over global function fields, simplifying calculations across different genera.

Contribution

It provides a new computational approach for class numbers and reduces complex genus calculations to simpler principal genus cases.

Findings

Method for evaluating class numbers of hereditary orders

Reduction of non-principal genus class numbers to principal genus

Applicable to arbitrary definite central simple algebras

Abstract

Let $K$ be a global function field together with a place $\infty$, and $A$ the subring of functions regular outside $\infty$. In this paper we present an effective method to evaluate the (locally free) class number of an arbitrary hereditary $A$-order in an arbitrary definite central simple $K$-algebra. We also show that the class number of any non-principal genus for a hereditary order in $D$ can be reduced to that of the principal genus for another hereditary order in $D$.