The 1-2-3 of modular forms: lectures at a summer school in Nordfjordeid, Norway

Elliptic Modular Forms and Their Applications

Foreword

1. Basic Definitions

1.1. Modular Groups, Modular Functions and Modular Forms

1.2. The Fundamental Domain of the Full Modular Group

Finiteness of Class Numbers

1.3. The Finite Dimensionality of M[subscript k]([Gamma])

2. First Examples: Eisenstein Series and the Discriminant Function

2.1. Eisenstein Series and the Ring Structure of M[subscript *]([Gamma subscript 1])

2.2. Fourier Expansions of Eisenstein Series

Identities Involving Sums of Powers of Divisors

2.3. The Eisenstein Series of Weight 2

2.4. The Discriminant Function and Cusp Forms

Congruences for [tau](n)

3. Theta Series

3.1. Jacobi's Theta Series

Sums of Two and Four Squares

The Kac-Wakimoto Conjecture

3.2. Theta Series in Many Variables

Invariants of Even Unimodular Lattices

Drums Whose Shape One Cannot Hear

4. Hecke Eigenforms and L-series

4.1. Hecke Theory

4.2. L-series of Eigenforms

4.3. Modular Forms and Algebraic Number Theory

Binary Quadratic Forms of Discriminant -23

4.4. Modular Forms Associated to Elliptic Curves and Other Varieties

Fermat's Last Theorem

5. Modular Forms and Differential Operators

5.1. Derivatives of Modular Forms

Modular Forms Satisfy Non-Linear Differential Equations

Moments of Periodic Functions

5.2. Rankin-Cohen Brackets and Cohen-Kuznetsov Series

Further Identities for Sums of Powers of Divisors

Exotic Multiplications of Modular Forms

5.3. Quasimodular Forms

Counting Ramified Coverings of the Torns

5.4. Linear Differential Equations and Modular Forms

The Irrationality of [zeta](3)

An Example Coming from Percolation Theory

6. Singular Moduli and Complex Multiplication

6.1. Algebraicity of Singular Moduli

Strange Approximations to [pi]

Computing Class Numbers

Explicit Class Field Theory for Imaginary Quadratic Fields

Solutions of Diophantine Equations

6.2. Norms and Traces of Singular Moduli

Heights of Heegner Points

The Borcherds Product Formula

6.3. Periods and Taylor Expansions of Modular Forms

Two Transcendence Results

Hurwitz Numbers

Generalized Hurwitz Numbers

6.4. CM Elliptic Curves and CM Modular Forms

Factorization, Primality Testing, and Cryptography

Central Values of Hecke L-Series

Which Primes are Sums of Two Cubes?

References and Further Reading

Hilbert Modular Forms and Their Applications

Introduction

1. Hilbert Modular Surfaces

1.1. The Hilbert Modular Group

1.2. The Baily-Borel Compactification

Siegel Domains

1.3. Hilbert Modular Forms

1.4. M[subscript k]([Gamma]) is Finite Dimensional

1.5. Eisenstein Series

Restriction to the Diagonal

The Example Q([square root]5)

1.6. The L-function of a Hilbert Modular Form

2. The Orthogonal Group O(2, n)

2.1. Quadratic Forms

2.2. The Clifford Algebra

2.3. The Spin Group

Quadratic Spaces in Dimension Four

2.4. Rational Quadratic Spaces of Type (2, n)

The Grassmannian Model

The Projective Model

The Tube Domain Model

Lattices

Heegner Divisors

2.5. Modular Forms for O(2, n)

2.6. The Siegel Theta Function

2.7. The Hilbert Modular Group as an Orthogonal Group

Hirzebruch-Zagier Divisors

3. Additive and Multiplicative Liftings

3.1. The Doi-Naganuma Lift

3.2. Borcherds Products

Local Borcherds Products

The Borcherds Lift

Obstructions

Examples

3.3. Automorphic Green Functions

A Second Approach

3.4. CM Values of Hilbert Modular Functions

Singular Moduli

CM Extensions

CM Cycles

CM Values of Borcherds Products

Examples

References

Siegel Modular Forms and Their Applications

1. Introduction

2. The Siegel Modular Group

3. Modular Forms

4. The Fourier Expansion of a Modular Form

5. The Siegel Operator and Eisenstein Series

6. Singular Forms

7. Theta Series

8. The Fourier-Jacobi Development of a Siegel Modular Form

9. The Ring of Classical Siegel Modular Forms for Genus Two

10. Moduli of Principally Polarized Complex Abelian Varieties

11. Compactifications

12. Intermezzo: Roots and Representations

13. Vector Bundles Defined by Representations

14. Holomorphic Differential Forms

15. Cusp Forms and Geometry

16. The Classical Hecke Algebra

17. The Satake Isomorphism

18. Relations in the Hecke Algebra

19. Satake Parameters

20. L-functions

21. Liftings

22. The Moduli Space of Principally Polarized Abelian Varieties

23. Elliptic Curves over Finite Fields

24. Counting Points on Curves of Genus 2

25. The Ring of Vector-Valued Siegel Modular Forms for Genus 2

26. Harder's Conjecture

27. Evidence for Harder's Conjecture

References

A Congruence Between a Siegel and an Elliptic Modular Form

1. Elliptic and Siegel Modular Forms

2. The Hecke Algebra and a Congruence

3. The Special Values of the L-function

4. Cohomology with Coefficients

5. Why the Denominator?

6. Arithmetic Implications

References

Appendix

Index