The geometry and arithmetic of banana nano-manifold

The Hodge numbers of a Calabi-Yau threefold X are determined by the two numbers h^{1,1}(X) and h^{1,2}(X) which can be interpreted respectively as the dimensions of the space of Kähler forms and complex deformations respectively. We construct four new examples X_N, where N \in {5,6,8,9}, of rigid Calabi-Yau threefolds (h^{2,1}=0) with Picard number 4 (h^{1,1}=4). These manifolds are of “banana type” and have among the smallest known values for Calabi-Yau Hodge numbers. We (partially) compute the Donaldson-Thomas partition functions of these manifolds and in particular show that the associated genus g Gromov-Witten potential is given by a weight 2g-2 Siegel paramodular form of index N. These manifolds are also modular in the arithmetic sense: there is a weight 4 modular form whose Fourier coefficients are obtained by counting points over finite fields on X_N. We compute this form and observe that it is the unique cusp form of weight 4 and index N. This is joint work with Stephen Pietromonaco.