On embedding numbers into quaternion orders
type: Item
instance of: scholarly article
title: On embedding numbers into quaternion orders
zbMATH Open document ID: 0733.11047
DOI: 10.1007/BF02566649
author: N. E. Zubov
published in: Commentarii Mathematici Helvetici
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140235
review text: Let \(R\) be a complete discrete valuation ring of characteristic different from 2 with quotient field \(K\). If \(S\) is an \(R\)-order in a quadratic extension \(L\) of \(K\) and \(\Lambda\) is an \(R\)-order in a quaternion algebra \(A\) over \(K\), an embedding \(S\to \Lambda\) is called optimal if its cokernel is a projective \(R\)-module. The primary aim of this paper is to determine the number of such optimal embeddings modulo the action of the unit group \(\Lambda^*\) by conjugation, when \(\Lambda\) is an Eichler order (and \(S\) is arbitrary). The main result is an explicit formula which yields this number in terms of the number of elements in the residue field of \(R\) (which is assumed to be finite), the discriminant of the Eichler order \(\Lambda\) and the conductor of \(S\) in the maximal order of \(L\), generalizing earlier computations due to Eichler and Hijikata. From this result, the author derives a stability property for embedding numbers, showing that for a given Eichler order \(\Lambda\) and a given quadratic extension \(L/K\), the number of optimal embeddings of an order \(S\) into \(\Lambda\) modulo the action of \(\Lambda^*\) does not depend on \(S\), provided its conductor is relatively small and the Gorenstein closure of \(\Lambda\) is a Bass order. In the final section, applications of embedding numbers to representation of integers by ternary quadratic forms are discussed.
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 4212173
zbMATH Keywords: optimal embeddings
zbMATH Keywords: Eichler order
zbMATH Keywords: maximal order
zbMATH Keywords: stability
zbMATH Keywords: representation of integers by ternary quadratic forms
reviewed by: Jean-Pierre Tignol
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4212173