Modular forms, elliptic curves and all that jazz
Manasvi
Given the vastness of the topics under consideration, it is imperative to understand what the final purpose of the talk is. We shall start with looking at the familiar elliptic functions and integrals, and see how they lead to elliptic curves. The connection between elliptic curves, modular invariance and modular groups is then studied. The next section will pick up from here, and introduce the theory of modular forms briefly. This will involve the definition of modular forms, and a summary of some of their properties. This is connected with the Eisenstein series, and the discriminant cusp form, and we conclude with an outline of theta functions.
If time permits (or more likely in a second talk) we will connect these up with physics, in particular CFT. Connections would be made between these two areas (modular forms & elliptic curves vs CFT) by examining scenarios such as the free massless scalar field, Weyl and Maxwell field, etc. The importance of modular forms in the Bianchi classification, self-dual Yang-Mills, 3D quantum gravity and statistical physics (the Yang-Baxter equations) will be highlighted.
References
TBA at the talk
- For a simple and concise introduction to elliptic curves TWF-13