In this seminar I will discuss what I know (and don’t know) about the arithmetic of Calabi-Yau 3-folds. The main goal is to explore whether there are questions of common interest in this context to physicists, number theorists and geometers. The main quantities of interest in the arithmetic context are the numbers of points of the manifold considered as a variety over a finite field. We are interested in the computation of these numbers and their dependence on the moduli of the variety. The surprise for a physicist is that the numbers of points over a finite field are also given by expression that involve the periods of a manifold. The number of points are encoded in the local zeta function, about which much is known in virtue of the Weil conjectures. I will discuss a number of interesting topics related to the zeta function and the appearance of modularity for one parameter families of Calabi-Yau manifolds.

A topic I will stress is that for these families there are values of the parameter for which the manifold becomes singular and for these values the zeta function degenerates and exhibits modular behaviour. Furthermore, it is well known that there are certain classical enumerative problems to do with Calabi Yau manifolds of which the simplest is counting the number of holomorphically embedded lines, that are solved by manipulations involving the periods of the mirror manifold. I will discuss briefly on the relation between zeta functions of a Calabi-Yau and its mirror (this is work in progress with Philip Candelas). Finally, I will report (on joint work with Philip Candelas, Mohamed Elmi and Duco van Straten) on an example for which the quartic numerator of the zeta function factorises into two quadrics at special values of the parameter which satisfy an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. We note that these factorisations are due to a splitting of the Hodge structure and that these special values of the parameter are rank two attractor points in the sense of type IIB supergravity. Modular groups and modular forms arise in relation to these attractor points.

To our knowledge, the rank two attractor points that were found by the application of these number theoretic techniques, provide the first explicit examples of such points for Calabi-Yau manifolds of full SU(3) holonomy.