Counting curves with modular forms

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 and Modular Forms

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

Elliptic Curves , Arithmetic Geometry , and Modular Forms

Counting functions for branched covers of elliptic curves and quasi-modular forms

of the branched covers of an elliptic curve. Here, N (m) g,d is the (weighted) number of isomorphism classes of branched covers, with genus g(> 1), degree d, and ramification index (m,m, . . . ,m), of an elliptic curve. Such a cover is called an m-simple cover. Our aim is to prove that the formal power series F (m) g converges to a function belonging to the graded ring of quasi-modular forms wi...

ELLIPTIC CURVES AND MODULAR FORMS Contents

1. January 21, 2010 2 1.1. Why define a curve to be f rather than V (f) ⊂ P(k)? 3 1.2. Cubic plane curves 3 2. January 26, 2010 4 2.1. A little bit about smoothness 4 2.2. Weierstrass form 5 3. January 28, 2010 6 3.1. An algebro-geometric description of the group law in terms of divisors 6 3.2. Why are the two group laws the same? 7 4. February 2, 2010 7 4.1. Overview 7 4.2. Uniqueness of Weier...