Hecke Operators on Modular Symbols

Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Linear Algebra With an Eye Toward Computing Modular Forms

William Stein

Department of Mathematics University of Washington

August 7, 2006 / MSRI Workshop

Outline

1 Hecke Operators on Modular Symbols

2 Hecke Operators on Manin Symbols

3 Using Hecke Operators to Compute Modular Forms

Reduced Row Echelon Form Many Linear Algebra Problems Can be Expressed in terms of Echelon Form Examples Matrix Multiplication

Strassen-Winograd

Rational Reconstruction Multimodular Algorithm (Good for Sparse Matrices)

Echelon Forms via Matrix Multiplication Matrix Inverse The Problem Characteristic and Minimal Polynomials Wiedemann’s Algorithm p-adic Nullspace