Hilbert symbols, class groups and quaternion algebras

Abstract

© Université Bordeaux 1, 2000. tous droits réservés.Let B be a quaternion algebra over a number field k. To a pair of Hilbert symbols {a, b} and {c, d} for B we associate an invariant ρ = ρR([D(a, b)], [D(c, d)]) in a quotient of the narrow ideal class group of k. This invariant arises from the study of finite subgroups of maximal arithmetic Kleinian groups. It measures the distance between orders D(a, b) and D(c, d) in B associated to {a, b} and {c,d}. If a = c, we compute ρR([D(a, b)], [D(c, d)]) by means of arithmetic in the field k((Formula Presented)). The problem of extending this algorithm to the general case leads to studying a finite graph associated to different Hilbert symbols for B. An example arising from the determination of the smallest arithmetic hyperbolic 3-manifold is discussed.