MATH 583: Computing With Modular Forms, Spring 2006, UW.
Lecture Plan
[x] (Fri Apr 7) Level 1 modular forms 3: Structure theorem, Victor Miller Basis
[x] SAGE 1: Software for Algebra and Geoemtry Experiementation -- intro
[x] Level 1 modular forms 4: Hecke Operators; Maeda, Edixhoven
[x] Special Topic: Fast computation of Bernoulli numbers
[x] Modular symbols 1: basic definitions
[x] Modular symbols 2: modular symbols
[x] Modular symbols 3: Hecke operators
[x] Modular symbols 4: computing modular forms using modular symbols
[x] Dirichlet characters: intro and algorithm
[x] (April 28) C. Doran -- elliptic modular surfaces
[x] Dirichlet characters: more
[x] Movie -- "The Proof"
[x] Eisenstein series: computing an explicit basis
[x] Dimension formulas: statement and how to compute
[x] Linear algebra 1: computing echelon forms
[x] Linear algebra 2: decomposition algorithms
[x] Higher Weight Modular Symbols 1: basic definitions; how to compute:
Sections 8.1 -- 8.2
[x] Higher Weight Modular Symbols 2: Hecke operators on them
Sections 8.3
[x] Higher Weight Modular Symbols 3: the integration pairing
Sections 8.5
[x] Newforms 1: Atkin-Lehner-Li theory
Sections 9.1 -- 9.3
[x] Newforms 2: Computing (and storing!) systems of eigenvalues
Section 9.4
[] Special Values of L-functions using modular symbols
Sections 10.1-10.4
[] (may 29 -- memorial day holiday)
[] Enumeration of all elliptic curves of given conductor (Cremona's program):
Sections 10.6-10.7
[] Sturm's bound: Congruences between modular forms; gen. Hecke algebras
Chapter 11