The class number of hereditary orders in non-Eichler algebras over global function fields
type: Item
instance of: scholarly article
title: The class number of hereditary orders in non-Eichler algebras over global function fields
zbMATH Open document ID: 0627.16003
DOI: 10.1007/BF01460041
author: Marleen Denert
author: Jan Van Geel
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164469
review text: Let A be a central simple algebra over a global function field K, R a ring of ``integers'' in K. A is supposed not to satisfy the Eichler condition with respect to R (i.e. A is a non-Eichler division algebra over R). For hereditary R-orders \(\Theta\) in A the class numbers of the isomorphism classes of locally free left \(\Theta\)-ideals are studied. In the case where \(K=F_ q(t)\), \(R=F_ q[t]\) and A a non-Eichler \((F_ q[t])\)-algebra of prime index over K, an explicit class number formula is obtained. The main ingredients are Eichler's formula for the stably free measure and the study of the different embeddings of rings of integers in splitting fields of A. In the final section it is indicated how the formulae can be extended to the general case.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16Kxx
zbMATH DE Number: 4019252
zbMATH Keywords: central simple algebra
zbMATH Keywords: global function field
zbMATH Keywords: Eichler condition
zbMATH Keywords: non- Eichler division algebra
zbMATH Keywords: hereditary R-orders
zbMATH Keywords: locally free left \(\theta \) -ideals
zbMATH Keywords: class number formula
MaRDI profile type: Publication
cites work: Orders, in non-Eichler ($R$)-algebras over global function fields, having the cancellation property.
cites work: Q3699803
cites work: Orders in quaternion algebras over global function fields having the cancellation property
cites work: Über die Idealklassenzahl total definierter Quaternionenalgebren
cites work: Zur Zahlentheorie der Quaternionen-Algebren.
cites work: Locally free modules over arithmetic orders.
cites work: Class Groups and Picard Groups of Orders
cites work: Type numbers of Eichler orders.
cites work: Q4061092
cites work: Q3934492
cites work: Q3919832
cites work: Q3845400
OpenAlex ID: W2018937833
description: scientific article; zbMATH DE number 4019252