Modular Symbols

Slide 2

Motivation

Birch and Swinnerton-Dyer

The BSD Conjecture

Birch introduced modular symbols

How can we compute in the modular world?

Subgroups of the modular group

Modular Curves

Modular curve for N=3:

Modular curve X (37):

Modular Forms

Examples of modular forms

Computing the space of modular symbols

Explicit presentation of modular symbols

Relations

Example: N=11

Manins Trick

Slide 20

Slide 21

The connection with modular forms

Example

Applications of Modular Symbols

Key References and Software

Author: William Stein

E-mail: [email protected]

Homepage: http://www.math.harvard.edu/~was