2017: Geometric theory of modular forms

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