Calabi-Yau manifolds play an important role in physics owing to the part they play in passing from a 10-dimensional string theory vacuum to the observed world of four dimensions. The process of calculating the four - dimensional quantities usually involves a calculation of certain period integrals. These period integrals encode remarkable geometric and arithmetic information about the manifolds. In the arithmetic context, the principle quantity of interest is the zeta function for the manifold. This also is determined by the periods. This speaker will comment on the mysterious modular behaviour of the zeta-function for the case that the Calabi-Yau manifold is singular.