PUB - Publikationen an der Universität Bielefeld

The hyperbolic 3-manifolds fibered over the circle with fiber a once punctured torus are considered. Surprisingly, in all of about 200 cases, where the class numbers of their trace fields were calculated, they turned out to be equal to one. This is a report on the work of our friend Heinz Helling during the last several years. Unfortunately, for health problems he is unable to write about it himself. All the main ideas and results presented in this paper belong to him, but some technical details may be different. In a series of preprints [1], [3], [4], [5], H. Helling with co-authors studied some class of hyperbolic 3-manifolds fibered over the circle, and calculated the trace fields of these manifolds. Ten years later he came to the idea to investigate arithmetic properties of these fields. This led him to a surprising result: in all the cases, when he managed to calculate the class number (with the help of a computer program), it turned out to be equal to one!