Some Siegel threefolds with a Calabi-Yau model

We describe some examples of projective Calabi-Yau manifolds which arise as desingularizations of Siegel threefolds. There is a certain explicit product of six theta constants which defines a cusp form of weight three for a certain subgroup of index two of the Hecke group Gamma(2,0)[2]. This form defines an invariant differential form for this group and for any subgroup of it. We study the question whether the Satake compactification for such a subgroup admits a projective desingularization on which this differential form is holomorphic and without zeros. Then this desingularization is a Calabi-Yau manifold. We shall prove: For any group between Gamma(2)[2] and Gamma(2,0)[2] there exists a subgroup of index two which produces a (projective) Calabi-Yau manifold. The proof rests on a detailed study of this cusp form and on Igusa's explicit desingularization of the Siegel threefolds with respect to the principal congruence subgroup of level q > 2 (we need q = 4). For a particular case we produce the equations for the corresponding Siegel threefold.