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

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.