Periods of Calabi-Yau Manifolds in Physics and Number Theory

A Calabi-Yau manifold has a naturally defined holomorphic three-form and the integrals of this over a basis of homology cycles are the periods of the manifold. These periods depend on the complex structure parameters.  It transpires that there are two communities that think they own the periods. String theorists compute quantities to do with the effective four dimensional theory and these are usually computed in terms of the periods.  Number theorists also regard the periods as their own, since they encode important arithmetic information about the manifold. I will show how the number of F_p - rational points is calculated in terms of the periods and comment about the form of the zeta-function and its relation to mirror symmetry.