# 2017: Geometric theory of modular forms

## Overview

During 2017 we will run a reading group on the geometric theory of modular forms. The ultimate goal will be to read Hida’s book *Geometric Theory of Modular Forms*. Before tackling this, we will review the material from Chapters 2, 3, and 6-9 of Diamond and Shurman’s *A First Course in Modular Forms*. This reading group is likely to run for several semesters.

### Location and Time:

Martin O-010, Mondays 11:30 – 13:00. The first meeting will be January 16, 2017.

### Participants:

- Jim Brown (faculty)
- Soumendra Ganguly (grad)
- Hugh Geller (grad)
- Kevin James (faculty)
- Huixi Li (grad)

### Lectures:

- Line bundles on Riemann surfaces with a view towards modular forms (Jim Brown)
- The non-compact modular curve associated to a congruence subgroup as a Riemann surface (Jim Brown)
- Cusps (Patrick Dynes)
- Automorphic forms and meromorphic differentials (Jim Brown)
- Integration, homology, the Jacobian, and modularity (Hugh Geller)
- Maps between Jacobians (Jim Brown)
- Modular Jacobians and Hecke operators (Jim Brown)
- Algebraic eigenvalues (Jim Brown)
- Eigenforms, abelian varieties, and modularity (Jim Brown)
- Elliptic curves as algebraic curves (Jim Brown)
- Algebraic curves and their function fields (Jim Brown)
- Divisors on curves
- The Weil pairing algebraically
- Function fields over the complex numbers
- Function fields over the rational numbers
- Modular curves as algebra curves and modularity
- Isogenies algebraically
- Hecke operators algebraically
- Elliptic curves in arbitrary characteristic
- Algebraic curves in arbitrary characteristic
- Elliptic curves over the rational nubmers and their reductions
- Elliptic curves over the algebraic closure of the rational numbers and their reductions
- Reduction of algebraic curves and maps
- Modular curves in characteristic p: Igusa’s theorem
- The Eichler-Shimura relation
- Fourier coefficients, L-functions, and modularity