Modular Calabi-Yau Threefolds
The Taniyama-Shimura conjecture (used in the proof of Fermat’s last theorem) relates the number of points on an elliptic curve over a finite field to Fourier coefficients of a modular form of weight two. Calabi-Yau manifolds are generalizations of elliptic curves to higher dimensions and they play a prominent role in the contemporary development of string theory. Their arithmetic properties in dimensions two (i.e. properties of K3 surfaces) have been studied recently. This book is devoted to the case of Calabi-Yau manifolds in dimension three. The dimension of the middle étale cohomology of a Calabi-Yau three-fold M gives key information on its arithmetic properties. If it equals two, the three-fold is called rigid. There is a precise conjecture on a connection of a rigid three-fold with modular forms, whereas the situation is much more complicated for non-rigid ones. The book contains hundreds of examples of rigid and non-rigid cases. The first chapter reviews basic facts on Calabi-Yau manifolds and their arithmetic. The next four chapters contain a discussion of a considerable number of explicit examples (including many different quintics, double octics and various complete intersections). In the last chapter, the author describes the correspondences between Calabi-Yau three-folds having the same modular form in their L-series. The book is based on the author’s PhD thesis.