MSRI 2006 Summer Graduate Workshop in Computational Number Theory

Lseries of elliptic curve of rank 2

Each project will involve a combination of:
 Understanding and explaining the theory,
 Improving the (bestknown) relevant algorithms,
 Analyzing the theoretical complexity of relevant
algorithms,
 Implementing algorithms in SAGE,
 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 higherrank group] (Gunnells)
 Databases: Design the ultimate database(s) of modular forms (Stein, Kohel,
Dembele, Gunnells)
 padic modular forms (Wiese, Stein)
 Lfunctions (Kohel, Stein)
 Serre's conjecture mod pq? (if Khare hasn't already (dis)proved it)
 Level raising mod p^{2}? What is the right conjecture?
 Siegel modular forms
