Imagin\"arquadratische Einbettung von Ordnungen rationaler Quaternionenalgebren, und die nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen

TL;DR

This paper classifies intersections of quaternion orders over imaginary quadratic fields and relates them to finite subgroups of Bianchi groups, providing new insights into their structure and conjugacy classes.

Contribution

It introduces a classification of F-orders as intersections of maximal M-orders and relates discriminants to isomorphism types, extending to non-cyclic subgroups and Eichler orders.

Findings

Discriminant uniquely determines the isomorphism type of maximal M-orders.

Determined which finite subgroups are contained in Bianchi groups.

Calculated conjugacy classes of non-cyclic finite subgroups.

Abstract

Let k be an imaginary quadratic number field, let F be a rational quaternion algebra and M an extension of F as a quaternion k-algebra. We are going to classify the F-orders which arise as an intersection of F with a maximal M-order; and we are going to prove that the discriminant of such an intersection determines uniquely the isomorphism type of the corresponding maximal M-order. Building on this, we are going to relate this intersection to the intersection of a second rational quaternion algebra F' in M with a second maximal M-order. This allows us to determine whether the Bianchi group over the maximal k-order contains 3-dihedral, tetrahedral or 2-dihedral groups which are maximal as a finite subgroup. Additionally, we determine the number of maximal M-orders which respectively admit the same intersection with F. Building on this, we calculate the numbers of conjugacy classes of non-cyclic maximal finite subgroups in the given Bianchi group. In the final two chapters, we investigate non-trivial intersections of non-cyclic finite subgroups of the Bianchi groups and extend our results to Eichler orders and especially to Bianchi congruence subgroups.