Three Lectures about Explicit Methods in Number Theory Using Sage

This article is about using the mathematical software Sage to do computations with number fields and modular forms. It was written for the October 2008 Bordeaux meeting on explicit methods in number theory (http://www.math.u-bordeaux.fr/gtem2008/). It assumes no prior knowledge about Sage, but assumes a graduate level background in algebraic number theory.

• Introduction
  • What is Sage?
  • Using Sage
  • The Sage-Pari-Magma Ecosystem

• Number Fields
  • Introduction to Number Fields
  • Number Fields: Galois Groups and Class Groups
  • Orders and Relative Extensions

• A Bird’s Eye View
  • Integer Factorization
  • Elliptic Curves
  • The Matrix of Frobenius on Hyperelliptic Curves
  • Modular Symbols
  • Enumerating Totally Real Number Fields
  • Bernoulli Numbers
  • Polynomial Arithmetic

• Modular Forms
  • Modular Forms and Hecke Operators
  • Modular Symbols
  • Method of Graphs
  • Level One Modular Forms
  • Half Integral Weight Forms
  • Generators for Rings of Modular Forms
  • -series
  • Modular Abelian Varieties