Garrett's introduction presents a diagram of the tortuous sequence of implications leading to the Arithmetic Structure Theorem, and one realizes with some astonishment that his derivation of this result uses little more than elementary complex analysis, measure theory, and elementary algebraic number theory. One also realizes that these simple ingredients have been combined to present exemplary applications of most of the standard techniques of the analytic theory of automorphic forms. Better still, starting from scratch, Garrett has succeeded in developing his subject matter to the point of presenting results which, if not exactly the cutting edge of the field, certainly come close. Such results include not only the Arithmetic Structure Theorem but also simplified versions of a series of theorems of Shimura, dating from the '70s, which relate special values of L-functions to periods of integrals. Apart from their intrinsic interest, Shimura's theorems are the starting point for the kind of arithmetic applications discussed above, and Garrett's exposition makes Shimura's difficult theorems seem completely natural.
Anyone interested in the arithmetic of number fields will eventually have to learn something about Hilbert modular forms. As an introduction to the analytic and arithmetic aspects of the subject, Garrett's book may be the best place to start.
Author(s): Paul B. Garrett
Series: Wadsworth & Brooks/Cole Mathematics Series
Publisher: Chapman and Hall/CRC
Year: 1989
Language: English
Commentary: Improvements with respect to 18DD8A4DBE2463BF58EA14A2A1B4249C: pagenated and added OCR by adobe ACROBAT + bookmarks + review
Pages: 304
Introduction
Contents
1. Classical Theory of Hilbert Modular Forms
1.1 The Hilbert Modular Group
1.2 Hilbert Modular Forms
1.3 Class Numbers and Cusps
1.4 Koecher's Principle
1.5 Holomorphic Eisenstein Series of Level One
1.6 Siegel Sets: An Approximate Fundamental Domain
1.7 Finite Dimensionality of Spaces of Cuspforms, Estimates on Cuspforms
1.8 Holomorphic Eisenstein Series and Cuspforms
1.9 Dirichlet Series Associated to Cuspforms
1.10 Some Integration Theory
1.11 A Volume Computation
1.12 The Petersson Inner Product
1.13 Poincaré Series
1.14 A Reproducing Kernel for Cuspforms
1.15 Hecke Operators in a Special Case
2. Automorphic Forms on GL(2, ?)
2.1 Structure of GL(2, ?)
2.2 A Volume Computation
2.3 The Spherical Hecke Algebra
2.4 Invariant Differential Operators
2.5 Adelic Fourier Expansions, Cuspforms, and Hecke Operators
2.6 General Definition of Adelic Automorphic Forms
2.7 Formalism of L-Functions Associated to Cuspforms
3. Comparison of Classical and Adelic Viewpoints
3.1 Comparison of Function Spaces via Strong Approximation
3.2 Hecke Operators
3.3 Holomorphic Automorphic Forms
3.4 Holomorpbic Hecke Eigenfunctions
3.5 Fourier Expansions of Holomorphic Automorphic Forms
3.6 Further Remarks
4. Eisenstein Series
4.1 Definitions and Integral Representations
4.2 Analytic Continuation and Functional Equation
4.3 Moderate Growth of Eisenstein Series
4.4 Fourier Expansion
4.5 Hecke-Theoretic Aspects
4.6 Application of Differential Operators
4.7 Eisenstein Series of Low Weight
4.8 Fourier Expansion of Certain Holomorphic Eisenstein Series
4.9 Eisenstein Series and Cuspforms
4.10 Rankin's Integral Representations of L-Functions
5. Theta Series
5.1 A Simple Theta Series, Reciprocity Laws
5.2 Hecke's Identity
5.3 Pluriharmonic Theta Series
5.4 Some Theta Series of Level One
5.5 Action of GL^+(2, F) on Theta Series
5.6 Cuspforms Obtained as Pluriharmonic Theta Series
5.7 L-Functions with Grossencharacters
6. Arithmetic of Hilbert Modular Forms
6.1 The Arithmetic Structure Theorem
6.2 Special Values of L-Functions of Totally Real Number Fields
6.3 Special Values of Grossencharacter L-Functions
6.4 Special Values of Product L-Functions
6.5 Special Values of Standard L-Functions Attached to Cuspforms
7. Proof of the Arithmetic Structure Theorem
7.1 Siegel's Holomorphic Eisenstein Series
7.2 Cell Decomposition
7.3 Convergence
7.4 Two Archimedean Integrals
7.5 The Small-Cell Contribution
7.6 The Middle-Cell Contribution
7.7 The Big-Cell Contribution
7.8 Summary of Fourier Coefficient Computations
7.9 The Main Formula
7.10 Proof of the Arithmetic Structure Theorem
Appendix A.1 Integration on Homogeneous Spaces
Appendix A.2 Harmonic Analysis on the Adeles
Appendix A.3 Strong Approximation for SL(n)
Appendix A.4 Invariant Differential Operators
Appendix A.5 Dirichlet L-Functions over ℚ
Bibliography
[GGP]
[Sh3]
Index
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[Review] Michael Harris, Holomorphic Hilbert modular forms, by Paul B. Garrett., Bull. Amer. Math. Soc. 25 (1991), 184-195
