Curves on K 3 surfaces and modular forms

Modular Curves, Modular Surfaces, and Modular Fourfolds

We begin with some general remarks. Let X be a smooth projective variety of dimension n over a field k. For any positive integer p < n, it is of interest to understand, modulo a natural equivalence, the algebraic cycles Y = ∑ j mjYj lying on X, with each Yj closed and irreducible of codimension p, together with codimension p + 1 algebraic cycles Zj = ∑ i rijZij lying on Yj , for all j. There is...

Elliptic Curves and Modular Forms

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

Modular forms and K3 surfaces

For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over Q associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.

Counting Curves with Modular Forms

We consider the type IIA string compactified on the Calabi-Yau space given by a degree 12 hypersurface in the weighted projective space P(1,1,2,2,6). We express the prepotential of the low-energy effective supergravity theory in terms of a set of functions that transform covariantly under PSL(2,ZZ) modular transformations. These functions are then determined by monodromy properties, by singular...

Elliptic Curves , Arithmetic Geometry , and Modular Forms