The 1-2-3 of modular forms : lectures at a summer school in Nordfjordeid, Norway / Jan Hendrik Bruinier ... [and others].

Format:

Book

Contributor:

Bruinier, Jan H. (Jan Hendrik), 1971-

Series:

Universitext

Language:

English

Subjects (All):

Forms, Modular--Congresses.

Forms, Modular.

Hilbert modular surfaces--Congresses.

Hilbert modular surfaces.

Genre:

Conference papers and proceedings.

Physical Description:

x, 266 pages : illustrations ; 24 cm.

Other Title:

One two three of modular forms

Place of Publication:

Berlin : Springer, 2008.

Summary:

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, Norway in June 2004.

The first series treats the classical one-variable theory of elliptic modular forms; the second 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 section treats a number of beautiful applications, and together they form a comprehensive survey for beginners and a useful reference work for a broad range of mathematicians.

Contents:

Elliptic Modular Forms and Their Applications / Don Zagier 1

1.1 Modular Groups, Modular Functions and Modular Forms 3

1.2 The Fundamental Domain of the Full Modular Group 5

Finiteness of Class Numbers 7

1.3 The Finite Dimensionality of M[subscript k]([Gamma]) 8

2 First Examples: Eisenstein Series and the Discriminant Function 12

2.1 Eisenstein Series and the Ring Structure of M[subscript *]([Gamma subscript 1]) 12

2.2 Fourier Expansions of Eisenstein Series 15

Identities Involving Sums of Powers of Divisors 18

2.3 The Eisenstein Series of Weight 2 18

2.4 The Discriminant Function and Cusp Forms 20

Congruences for [tau](n) 23

3 Theta Series 24

3.1 Jacobi's Theta Series 25

Sums of Two and Four Squares 26

The Kac-Wakimoto Conjecture 31

3.2 Theta Series in Many Variables 31

Invariants of Even Unimodular Lattices 33

Drums Whose Shape One Cannot Hear 36

4 Hecke Eigenforms and L-series 37

4.1 Hecke Theory 37

4.2 L-series of Eigenforms 39

4.3 Modular Forms and Algebraic Number Theory 41

Binary Quadratic Forms of Discriminant -23 42

4.4 Modular Forms Associated to Elliptic Curves and Other Varieties 44

Fermat's Last Theorem 46

5 Modular Forms and Differential Operators 48

5.1 Derivatives of Modular Forms 48

Modular Forms Satisfy Non-Linear Differential Equations 49

Moments of Periodic Functions 50

5.2 Rankin-Cohen Brackets and Cohen-Kuznetsov Series 53

Further Identities for Sums of Powers of Divisors 56

Exotic Multiplications of Modular Forms 56

5.3 Quasimodular Forms 58

Counting Ramified Coverings of the Torns 60

5.4 Linear Differential Equations and Modular Forms 61

The Irrationality of [zeta](3) 64

An Example Coming from Percolation Theory 66

6 Singular Moduli and Complex Multiplication 66

6.1 Algebraicity of Singular Moduli 67

Strange Approximations to [pi] 73

Computing Class Numbers 74

Explicit Class Field Theory for Imaginary Quadratic Fields 75

Solutions of Diophantine Equations 76

6.2 Norms and Traces of Singular Moduli 77

Heights of Heegner Points 79

The Borcherds Product Formula 83

6.3 Periods and Taylor Expansions of Modular Forms 83

Two Transcendence Results 85

Hurwitz Numbers 85

Generalized Hurwitz Numbers 89

6.4 CM Elliptic Curves and CM Modular Forms 90

Factorization, Primality Testing, and Cryptography 92

Central Values of Hecke L-Series 95

Which Primes are Sums of Two Cubes? 97

Hilbert Modular Forms and Their Applications / Jan Hendrik Bruinier 105

1 Hilbert Modular Surfaces 106

1.1 The Hilbert Modular Group 106

1.2 The Baily-Borel Compactification 109

Siegel Domains 111

1.3 Hilbert Modular Forms 113

1.4 M[subscript k]([Gamma]) is Finite Dimensional 118

1.5 Eisenstein Series 119

Restriction to the Diagonal 122

The Example Q([square root]5) 123

1.6 The L-function of a Hilbert Modular Form 125

2 The Orthogonal Group O(2, n) 127

2.1 Quadratic Forms 128

2.2 The Clifford Algebra 129

2.3 The Spin Group 133

Quadratic Spaces in Dimension Four 135

2.4 Rational Quadratic Spaces of Type (2, n) 136

The Grassmannian Model 136

The Projective Model 137

The Tube Domain Model 137

Lattices 138

Heegner Divisors 140

2.5 Modular Forms for O(2, n) 140

2.6 The Siegel Theta Function 141

2.7 The Hilbert Modular Group as an Orthogonal Group 143

Hirzebruch-Zagier Divisors 145

3 Additive and Multiplicative Liftings 146

3.1 The Doi-Naganuma Lift 146

3.2 Borcherds Products 150

Local Borcherds Products 150

The Borcherds Lift 154

Obstructions 158

3.3 Automorphic Green Functions 162

A Second Approach 167

3.4 CM Values of Hilbert Modular Functions 168

Singular Moduli 168

CM Extensions 171

CM Cycles 172

CM Values of Borcherds Products 173

Siegel Modular Forms and Their Applications / Gerard van der Geer 181

2 The Siegel Modular Group 183

3 Modular Forms 187

4 The Fourier Expansion of a Modular Form 189

5 The Siegel Operator and Eisenstein Series 192

6 Singular Forms 194

7 Theta Series 195

8 The Fourier-Jacobi Development of a Siegel Modular Form 196

9 The Ring of Classical Siegel Modular Forms for Genus Two 198

10 Moduli of Principally Polarized Complex Abelian Varieties 201

11 Compactifications 204

12 Intermezzo: Roots and Representations 207

13 Vector Bundles Defined by Representations 209

14 Holomorphic Differential Forms 210

15 Cusp Forms and Geometry 212

16 The Classical Hecke Algebra 213

17 The Satake Isomorphism 215

18 Relations in the Hecke Algebra 218

19 Satake Parameters 219

20 L-functions 220

21 Liftings 221

22 The Moduli Space of Principally Polarized Abelian Varieties 226

23 Elliptic Curves over Finite Fields 226

24 Counting Points on Curves of Genus 2 230

25 The Ring of Vector-Valued Siegel Modular Forms for Genus 2 232

26 Harder's Conjecture 235

27 Evidence for Harder's Conjecture 237

A Congruence Between a Siegel and an Elliptic Modular Form / Gunter Harder 247

1 Elliptic and Siegel Modular Forms 247

2 The Hecke Algebra and a Congruence 250

3 The Special Values of the L-function 252

4 Cohomology with Coefficients 253

5 Why the Denominator? 257

6 Arithmetic Implications 258.

Notes:

Includes bibliographical references and index.

ISBN:

9783540741176

3540741178

OCLC:

173239471