On the classification of Hilbert modular threefolds

Letk be a totally real number field with ring of integersO _k . The Hilbert modular variety overk is a desingularization of the (natural) compactification of PSL₂(O _k )∖H ^k . The purpose of this paper is to present specific numerical bounds on the size of the discriminantd _k of a cubic fieldk with Hilbert modular variety of particular classifications. specifically, it is shown that ifd _k>2.12×10⁷, then the Hilbert modular variety overk is not rational and further, ifd _k>2.77×10⁸, then Hilbert modular variety overk is of general type. This material is based on work supported by the National Science Foundation under Grant No. DMS-9008689