The Arithmetic of Calabi-Yau Threefolds

Louisiana State University, Baton Rouge LA

Investigators

Abstract

The aim of this project is to extend the state of knowledge of the arithmetic of Calabi-Yau varieties, especially Calabi-Yau threefolds, with a strong emphasis on algorithms and computations. A large database of Calabi-Yau threefolds will be constructed, which will include arithmetic data, such as modular forms associated with rigid and other modular Calabi-Yau threefolds. Examples will be constructed using various methods, especially Batyrev's toric geometry construction. This database will be used to test the validity of conjectures in arithmetic geometry, such as the Fontaine-Mazur conjecture (the conjectured modularity of 2-dimensional geometric Galois representations), and the Bloch-Kato conjecture (about periods of L-series), for a large number of examples. The project will also study the variation of Hodge structure of families of Calabi-Yaus, and the relationship between the arithmetic and the Picard-Fuchs equations. Calabi-Yau threefolds were originally studied by mathematical physicists, because of their role in mirror symmetry and superstring theory. In these theories, the structure of the universe is partly described in terms of a Calabi-Yau threefold. Calabi-Yau varieties may also be considered a generalisation of elliptic curves, which have a very rich arithmetic structure. An important arithmetic question is that of modularity. This refers to the conjectured relationship between counting points on varieties, and the coefficients of modular forms. Counting points essentially means counting the number of solutions to a polynomial equation. Modularity predicts that for certain varieties, such as elliptic curves, or rigid Calabi-Yau threefolds, there will be a modular form corresponding to the variety, such that the number of points on the variety, i.e., the number of solutions of the polynomial defining the variety, is given in terms of the coefficients of a Taylor series of the corresponding modular form. Modularity, a deep relationship between two quite different kinds of objects, has important applications and implications in arithmetic. For example, the proof of the modularity of elliptic curves (Wiles, Taylor, et al) was the key to the proof of Fermat's last theorem. Modularity for other varieties remains an open problem.