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The 1-2-3 of modular forms: lectures at a summer school in Nordfjordeid, Norway
Publisher
Springer
Publication Date
c2008
Language
English
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Table of Contents
From the Book
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
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ISBN
9783540741176
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