The 1-2-3 of Modular Forms

by Bruinier, Jan Hendrik; Van Der Geer, Gerard; Harder, Gunter; Zagier, Don

ISBN13: 9783540741176

ISBN10: 3540741178

Format: Paperback

Pub. Date: 2008-05-04

Publisher(s): Springer Verlag

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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 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.

Author Biography

Jan H. Bruinier is Professor of Mathematics at the University of Darmstadt. His main field of research is number theory. In particular, he is interested in automorphic forms and their applications in geometry and arithmetic Gerard van der Geer is Professor of Algebra at the University of Amsterdam. His main field of research is algebraic geometry, in particular moduli spaces Gunter Harder is Professor em. at the University of Bonn and the Max Planck Institute for Mathematics in Bonn Don Zagier is a scientific member and director of the Max Planck Institute for Mathematics in Bonn and Professor of Number Theory at the College de France, Paris

Elliptic Modular Forms and Their Applications	p. 1
Foreword	p. 1
Basic Definitions	p. 3
Modular Groups, Modular Functions and Modular Forms	p. 3
The Fundamental Domain of the Full Modular Group	p. 5
Finiteness of Class Numbers	p. 7
The Finite Dimensionality of M[subscript k]([Gamma])	p. 8
First Examples: Eisenstein Series and the Discriminant Function	p. 12
Eisenstein Series and the Ring Structure of M[subscript *]([Gamma subscript 1])	p. 12
Fourier Expansions of Eisenstein Series	p. 15
Identities Involving Sums of Powers of Divisors	p. 18
The Eisenstein Series of Weight 2	p. 18
The Discriminant Function and Cusp Forms	p. 20
Congruences for [tau](n)	p. 23
Theta Series	p. 24
Jacobi's Theta Series	p. 25
Sums of Two and Four Squares	p. 26
The Kac-Wakimoto Conjecture	p. 31
Theta Series in Many Variables	p. 31
Invariants of Even Unimodular Lattices	p. 33
Drums Whose Shape One Cannot Hear	p. 36
Hecke Eigenforms and L-series	p. 37
Hecke Theory	p. 37
L-series of Eigenforms	p. 39
Modular Forms and Algebraic Number Theory	p. 41
Binary Quadratic Forms of Discriminant -23	p. 42
Modular Forms Associated to Elliptic Curves and Other Varieties	p. 44
Fermat's Last Theorem	p. 46
Modular Forms and Differential Operators	p. 48
Derivatives of Modular Forms	p. 48
Modular Forms Satisfy Non-Linear Differential Equations	p. 49
Moments of Periodic Functions	p. 50
Rankin-Cohen Brackets and Cohen-Kuznetsov Series	p. 53
Further Identities for Sums of Powers of Divisors	p. 56
Exotic Multiplications of Modular Forms	p. 56
Quasimodular Forms	p. 58
Counting Ramified Coverings of the Torns	p. 60
Linear Differential Equations and Modular Forms	p. 61
The Irrationality of [zeta](3)	p. 64
An Example Coming from Percolation Theory	p. 66
Singular Moduli and Complex Multiplication	p. 66
Algebraicity of Singular Moduli	p. 67
Strange Approximations to [pi]	p. 73
Computing Class Numbers	p. 74
Explicit Class Field Theory for Imaginary Quadratic Fields	p. 75
Solutions of Diophantine Equations	p. 76
Norms and Traces of Singular Moduli	p. 77
Heights of Heegner Points	p. 79
The Borcherds Product Formula	p. 83
Periods and Taylor Expansions of Modular Forms	p. 83
Two Transcendence Results	p. 85
Hurwitz Numbers	p. 85
Generalized Hurwitz Numbers	p. 89
CM Elliptic Curves and CM Modular Forms	p. 90
Factorization, Primality Testing, and Cryptography	p. 92
Central Values of Hecke L-Series	p. 95
Which Primes are Sums of Two Cubes?	p. 97
References and Further Reading	p. 99
Hilbert Modular Forms and Their Applications	p. 105
Introduction	p. 105
Hilbert Modular Surfaces	p. 106
The Hilbert Modular Group	p. 106
The Baily-Borel Compactification	p. 109
Siegel Domains	p. 111
Hilbert Modular Forms	p. 113
M[subscript k]([Gamma]) is Finite Dimensional	p. 118
Eisenstein Series	p. 119
Restriction to the Diagonal	p. 122
The Example Q([square root]5)	p. 123
The L-function of a Hilbert Modular Form	p. 125
The Orthogonal Group O(2, n)	p. 127
Quadratic Forms	p. 128
The Clifford Algebra	p. 129
The Spin Group	p. 133
Quadratic Spaces in Dimension Four	p. 135
Rational Quadratic Spaces of Type (2, n)	p. 136
The Grassmannian Model	p. 136
The Projective Model	p. 137
The Tube Domain Model	p. 137
Lattices	p. 138
Heegner Divisors	p. 140
Modular Forms for O(2, n)	p. 140
The Siegel Theta Function	p. 141
The Hilbert Modular Group as an Orthogonal Group	p. 143
Hirzebruch-Zagier Divisors	p. 145
Additive and Multiplicative Liftings	p. 146
The Doi-Naganuma Lift	p. 146
Borcherds Products	p. 150
Local Borcherds Products	p. 150
The Borcherds Lift	p. 154
Obstructions	p. 158
Examples	p. 160
Automorphic Green Functions	p. 162
A Second Approach	p. 167
CM Values of Hilbert Modular Functions	p. 168
Singular Moduli	p. 168
CM Extensions	p. 171
CM Cycles	p. 172
CM Values of Borcherds Products	p. 173
Examples	p. 175
References	p. 176
Siegel Modular Forms and Their Applications	p. 181
Introduction	p. 181
The Siegel Modular Group	p. 183
Modular Forms	p. 187
The Fourier Expansion of a Modular Form	p. 189
The Siegel Operator and Eisenstein Series	p. 192
Singular Forms	p. 194
Theta Series	p. 195
The Fourier-Jacobi Development of a Siegel Modular Form	p. 196
The Ring of Classical Siegel Modular Forms for Genus Two	p. 198
Moduli of Principally Polarized Complex Abelian Varieties	p. 201
Compactifications	p. 204
Intermezzo: Roots and Representations	p. 207
Vector Bundles Defined by Representations	p. 209
Holomorphic Differential Forms	p. 210
Cusp Forms and Geometry	p. 212
The Classical Hecke Algebra	p. 213
The Satake Isomorphism	p. 215
Relations in the Hecke Algebra	p. 218
Satake Parameters	p. 219
L-functions	p. 220
Liftings	p. 221
The Moduli Space of Principally Polarized Abelian Varieties	p. 226
Elliptic Curves over Finite Fields	p. 226
Counting Points on Curves of Genus 2	p. 230
The Ring of Vector-Valued Siegel Modular Forms for Genus 2	p. 232
Harder's Conjecture	p. 235
Evidence for Harder's Conjecture	p. 237
References	p. 241
A Congruence Between a Siegel and an Elliptic Modular Form	p. 247
Elliptic and Siegel Modular Forms	p. 247
The Hecke Algebra and a Congruence	p. 250
The Special Values of the L-function	p. 252
Cohomology with Coefficients	p. 253
Why the Denominator?	p. 257
Arithmetic Implications	p. 258
References	p. 259
Appendix	p. 260
Index	p. 263
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The 1-2-3 of Modular Forms

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