Oct 28:  Higher weight modular symbols:
         Definition, basic properties, manin symbols.
                
Nov 2, 4: No class

Nov 9:   Computing presentations for spaces of 
         higher weight modular symbols, and the action of Hecke 
         operators on them.

Nov 11:  Heilbronn matrices and Merel's theorem that
         they can be used to compute Hecke actions on Manin symbols.
 
Nov 16:  No Class (I'm at Banff)

Nov 18:  Duality between cusp forms and modular symbols.
         Examples.

Nov 23:  Computing a basis of q-expansions of eigenforms.
         Efficient representation of a q-expansion.

Nov 25:  No class -- Thanksgiving

Nov 30:  Numerical algorithms: period mapping, period lattice,
         real volume, special values of L-functions.

Dec 2:   Rationality results about special values of L-functions at
         integers points.  Computing the Birch and Swinnerton-Dyer 
         ratio L(A,1)/Omega, and higher weight motivic analogues.  
         Computing BSD invariants of A.

Dec 7:   How to enumerate all elliptic curves of given conductor.
             - Cremona's algorithm
             - Finding S-integral points on auxiliary elliptic curve
             - Sketch of generalizations to dimension > 1.

Dec 9:   Overview lecture on Serre's conjectures about modularity
         of mod-p Galois representations.   Definition of the Serre
         invariants of a representation.

Dec 14:  Deciding whether two modular forms are congruent (the Sturm
         bound), and determining the Serre level of a modular Galois
         representation.  I will mention results of Ribet, Diamond,
         Taylor, and an algorithm for finding the minimal level.
         I will also prove Sturm's theorem.    Application: Computing
         Z-module generators for the Hecke algebra.

Dec 16:  ???

Topics

1. Introduction to Modular Forms: basic definitions, motivation, and applications
2. Computing modular forms of level 1
3. Structure theorems for modular forms (Hecke operators, newforms, Atkin-Lehner theory), and how they are relevant to computation
4. Dimension formulas and how to compute them
5. Modular symbols: what they are and how to compute with them
6. Periods and special values of L-functions, and how to compute them using modular symbols
7. Congruences between modular forms, and how to compute them
8. Serre's Conjecture: Computing the Serre level and weight of a modular mod-p representation
9. The Mestre Method of Graphs

Grade