MSRI 2006 Summer Graduate Workshop
in Computational Number Theory

Computing With Modular Forms
L-series of elliptic curve of rank 2

Each project will involve a combination of:
  1. Understanding and explaining the theory,
  2. Improving the (best-known) relevant algorithms,
  3. Analyzing the theoretical complexity of relevant algorithms,
  4. Implementing algorithms in SAGE,
  5. Improving / auditing existing implementations.

General Projects

  • Arithmetic in quaternion algebras (Kohel)
  • Quaternion algebras over totally real fields (Dembele, Kohel)
  • Mestre method of graphs (Kohel, Stein)
  • Modular symbols mod p (Stein, Wiese)
  • Linear algebra algorithms relevant to computing modular forms (Stein)
  • Modular symbols for [specific higher-rank group] (Gunnells)
  • Databases: Design the ultimate database(s) of modular forms (Stein, Kohel, Dembele, Gunnells)
  • p-adic modular forms (Wiese, Stein)
  • L-functions (Kohel, Stein)
  • Serre's conjecture mod pq? (if Khare hasn't already (dis-)proved it)
  • Level raising mod p2? What is the right conjecture?
  • Siegel modular forms