Abstract
Let k be a totally real cubic number field with ring of integers Ok. The Hilbert modular threefold of k is a desingularization of the (natural) compactification of PSL2(Ok)\\H3. The goal of this paper is to prove that all rational Hilbert modular threefolds arise from fields with discriminant less than 75125. Specifically, it is shown that if k is a cubic field of discriminant at least 75125, then the arithmetic genus of the Hilbert modular variety of k is negative and hence the variety is not rational. Smaller bounds on the size of the discriminant are obtained for some special classes of cubic fields.
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