This book grew out of three series of lectures given at the summer school on Modular Forms and their Applications at the Sophus Lie Conference Center in Nordfjordeid in June 2004.

The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture.

Each part treats a number of beautiful applications, and together they form a comprehensive survey for the novice and a useful reference for a broad group of mathematicians.

• Elliptic Modular Forms and Their Applications Don Zagier
  • Basic Definitions
  • First Examples : Eisenstein Series and the Discriminant Function
  • Theta Series
  • Hecke Eigenforms and L-series
  • Modular Forms and Differential Operators
  • Singular Moduli and Complex Multiplication
• Hilbert Modular Forms and Their Applications - Jan Hendrik Bruinier
  • Hilbert Modular Surfaces
  • The Orthogonal Group O(2, n)
  • Additive and Multiplicative Liftings
• Siegel Modular Forms and Their Applications - Gerard van der Geer
  • Introduction
  • The Siegel Modular Group
  • Modular Forms
  • The Fourier Expansion of a Modular Form
  • The Siegel Operator and Eisenstein Series
  • Singular Forms
  • Theta Series
  • The Fourier-Jacobi Development of a Siegel Modular Form
  • The Ring of Classical Siegel Modular Forms for Genus Two
  • Moduli of Principally Polarized Complex Abelian Varieties
  • Compactifications
  • Intermezzo : Roots and Representations
  • Vector Bundles Defined by Representations
  • Holomorphic Differential Forms
  • Cusp Forms and Geometry
  • The Classical Hecke Algebra
  • The Satake Isomorphism
  • Relations in the Hecke Algebra
  • Satake Parameters
  • L-functions
  • Liftings
  • The Moduli Space of Principally Polarized Abelian Varieties
  • Elliptic Curves over Finite Fields
  • Counting Points on Curves of Genus 2
  • The Ring of Vector-Valued Siegel Modular Forms for Genus 2
  • Harder's Conjecture
  • Evidence for Harder's Conjecture
• ACongruence Between a Siegel and an Elliptic Modular Form - Gunter Harder