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El. knyga: 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway

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  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 10-Feb-2008
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540741190
Kitos knygos pagal šią temą:
1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, NorwayDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 10-Feb-2008
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540741190
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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.

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.

Recenzijos

From the reviews:





"Zagier gives 28 stunning applications, starting with class numbers, ranging through drums of unhearable shape, percolation, nonlinear differential equations, and ending with primes that equal the sum of two cubes. Summing Up: Highly recommended. Upper-division undergraduate through professional collections." (D. V. Feldman, Choice, Vol. 46 (7), March, 2009)

"The authors of The 1-2-3 of Modular Forms succeed in providing a thorough account of the theory of modular forms in various guises and applications, discussing in more detail the topics mentioned here as well as many more. Undoubtedly, due to both its breadth and readability, this book will be a useful source for mathematicians, both novice and expert, wishing to read more about modular forms and a beautiful theory." (Amanda Folsom, Bulletin of the American Mathematical Society, Vol. 46 (3), July, 2009)

If you ever wanted to gain a general understanding of modular forms then you should check out The 1-2-3 of Modular Forms. Each set of lectures includes standard introductory material as well as concrete examples and applications. One could use this as a way to start a graduate student working on modular forms . Whether you are looking for a nice way to begin your study or are already familiar with them, this is a book that you will enjoy. (Suzanne Caulk, The Mathematical Association of America, September, 2010)

Elliptic Modular Forms and Their Applications
1(104)
Don Zagier
Foreword
1(2)
Basic Definitions
3(9)
Modular Groups, Modular Functions and Modular Forms
3(2)
The Fundamental Domain of the Full Modular Group
5(2)
Finiteness of Class Numbers
7(1)
The Finite Dimensionality of Mk(Γ)
8(4)
First Examples: Eisenstein Series and the Discriminant Function
12(12)
Eisenstein Series and the Ring Structure of M*(Γ1)
12(3)
Fourier Expansions of Eisenstein Series
15(3)
Identities Involving Sums of Powers of Divisors
18(1)
The Eisenstein Series of Weight 2
18(2)
The Discriminant Function and Cusp Forms
20(3)
Congruences for T(n)
23(1)
Theta Series
24(13)
Jacobi's Theta Series
25(1)
Sums of Two and Four Squares
26(5)
The Kac-Wakimoto Conjecture
31(1)
Theta Series in Many Variables
31(2)
Invariants of Even Unimodular Lattices
33(3)
Drums Whose Shape One Cannot Hear
36(1)
Hecke Eigenforms and L-series
37(11)
Hecke Theory
37(2)
L-series of Eigenforms
39(2)
Modular Forms and Algebraic Number Theory
41(1)
Binary Quadratic Forms of Discriminant -- 23
42(2)
Modular Forms Associated to Elliptic Curves and Other Varieties
44(2)
Fermat's Last Theorem
46(2)
Modular Forms and Differential Operators
48(18)
Derivatives of Modular Forms
48(1)
Modular Forms Satisfy Non-Linear Differential Equations
49(1)
Moments of Periodic Functions
50(3)
Rankin--Cohen Brackets and Cohen--Kuznetsov Series
53(3)
Further Identities for Sums of Powers of Divisors
56(1)
Exotic Multiplications of Modular Forms
56(2)
Quasimodular Forms
58(2)
Counting Ramified Coverings of the Torns
60(1)
Linear Differential Equations and Modular Forms
61(3)
The Irrationality of ζ(3)
64(2)
An Example Coming from Percolation Theory
66(1)
Singular Moduli and Complex Multiplication
66(39)
Algebraicity of Singular Moduli
67(6)
Strange Approximations to π
73(1)
Computing Class Numbers
74(1)
Explicit Class Field Theory for Imaginary Quadratic Fields
75(1)
Solutions of Diophantine Equations
76(1)
Norms and Traces of Singular Moduli
77(2)
Heights of Heegner Points
79(4)
The Borcherds Product Formula
83(1)
Periods and Taylor Expansions of Modular Forms
83(2)
Two Transcendence Results
85(1)
Hurwitz Numbers
85(4)
Generalized Hurwitz Numbers
89(1)
CM Elliptic Curves and CM Modular Forms
90(2)
Factorization, Primality Testing, and Cryptography
92(3)
Central Values of Hecke L-Series
95(2)
Which Primes are Sums of Two Cubes?
97(2)
References and Further Reading
99(6)
Hilbert Modular Forms and Their Applications
105(76)
Jan Hendrik Bruinier
Introduction
105(1)
Hilbert Modular Surfaces
106(21)
The Hilbert Modular Group
106(3)
The Baily--Borel Compactification
109(2)
Siegel Domains
111(2)
Hilbert Modular Forms
113(5)
Mk(Γ) is Finite Dimensional
118(1)
Eisenstein Series
119(3)
Restriction to the Diagonal
122(1)
The Example Q(√5)
123(2)
The L-function of a Hilbert Modular Form
125(2)
The Orthogonal Group O(2, n)
127(19)
Quadratic Forms
128(1)
The Clifford Algebra
129(4)
The Spin Group
133(2)
Quadratic Spaces in Dimension Four
135(1)
Rational Quadratic Spaces of Type (2, n)
136(1)
The Grassmannian Model
136(1)
The Projective Model
137(1)
The Tube Domain Model
137(1)
Lattices
138(2)
Heegner Divisors
140(1)
Modular Forms for O(2, n)
140(1)
The Siegel Theta Function
141(2)
The Hilbert Modular Group as an Orthogonal Group
143(2)
Hirzebruch--Zagier Divisors
145(1)
Additive and Multiplicative Liftings
146(35)
The Doi--Naganuma Lift
146(4)
Borcherds Products
150(1)
Local Borcherds Products
150(4)
The Borcherds Lift
154(4)
Obstructions
158(2)
Examples
160(2)
Automorphic Green Functions
162(5)
A Second Approach
167(1)
CM Values of Hilbert Modular Functions
168(1)
Singular Moduli
168(3)
CM Extensions
171(1)
CM Cycles
172(1)
CM Values of Borcherds Products
173(2)
Examples
175(1)
References
176(5)
Siegel Modular Forms and Their Applications
181(66)
Gerard van der Geer
Introduction
181(2)
The Siegel Modular Group
183(4)
Modular Forms
187(2)
The Fourier Expansion of a Modular Form
189(3)
The Siegel Operator and Eisenstein Series
192(2)
Singular Forms
194(1)
Theta Series
195(1)
The Fourier--Jacobi Development of a Siegel Modular Form
196(2)
The Ring of Classical Siegel Modular Forms for Genus Two
198(3)
Moduli of Principally Polarized Complex Abelian Varieties
201(3)
Compactifications
204(3)
Intermezzo: Roots and Representations
207(2)
Vector Bundles Defined by Representations
209(1)
Holomorphic Differential Forms
210(2)
Cusp Forms and Geometry
212(1)
The Classical Hecke Algebra
213(2)
The Satake Isomorphism
215(3)
Relations in the Hecke Algebra
218(1)
Satake Parameters
219(1)
L-functions
220(1)
Liftings
221(5)
The Moduli Space of Principally Polarized Abelian Varieties
226(1)
Elliptic Curves over Finite Fields
226(4)
Counting Points on Curves of Genus 2
230(2)
The Ring of Vector-Valued Siegel Modular Forms for Genus 2
232(3)
Harder's Conjecture
235(2)
Evidence for Harder's Conjecture
237(10)
References
241(6)
A Congruence Between a Siegel and an Elliptic Modular Form
247(16)
Gunter Harder
Elliptic and Siegel Modular Forms
247(3)
The Hecke Algebra and a Congruence
250(2)
The Special Values of the L-function
252(1)
Cohomology with Coefficients
253(4)
Why the Denominator?
257(1)
Arithmetic Implications
258(5)
References
259(1)
Appendix
260(3)
Index 263
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