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E-grāmata: Modern Analysis of Automorphic Forms By Example: Volume 2

(University of Minnesota)
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Modern Analysis of Automorphic Forms By Example: Volume 2Palielināt
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This is Volume 2 of a two-volume book that provides a self-contained introduction to the theory and application of automorphic forms, using examples to illustrate several critical analytical concepts surrounding and supporting the theory of automorphic forms. The two-volume book treats three instances, starting with some small unimodular examples, followed by adelic GL2, and finally GLn. Volume 2 features critical results, which are proven carefully and in detail, including automorphic Green's functions, metrics and topologies on natural function spaces, unbounded operators, vector-valued integrals, vector-valued holomorphic functions, and asymptotics. Volume 1 features discrete decomposition of cuspforms, meromorphic continuation of Eisenstein series, spectral decomposition of pseudo-Eisenstein series, and automorphic Plancherel theorem. With numerous proofs and extensive examples, this classroom-tested introductory text is meant for a second-year or advanced graduate course in automorphic forms, and also as a resource for researchers working in automorphic forms, analytic number theory, and related fields.

This is Volume 2 of a two-volume book that provides a self-contained introduction to the analytical aspects of automorphic forms by proving several critical results carefully and in detail. With extensive examples, it will be useful for graduate students and researchers in automorphic forms, number theory, and other related fields.

Recenzijas

Review of Multi-volume Set: 'Any researcher working in the analytic theory of automorphic forms on higher rank groups will want to own this book. It is a treasure trove of examples and proofs that are well known to experts but very difficult to find in the open literature.' Dorian Goldfeld, Columbia University Review of Multi-volume Set: 'Written by a leading expert in the field, this volume provides a valuable account of the analytic theory of automorphic forms. The author chooses his examples to provide a middle road between the general theory and the most classical cases that do not exhibit all of the subject's more general phenomena. What makes this book special is this compromise and the subsequent aim, 'to discuss analytical issues at a technical level truly sufficient to convert appealing heuristics to persuasive, genuine proofs'.' John Friedlander, University of Toronto Review of Multi-volume Set: 'It is marvelous to see how Garrett goes about presenting such deep and broad material in what is certainly a superbly holistic manner. It's really a wonderful example of what I think is the right pedagogy for this part of number theory. The examples he uses are lynchpins for an increasingly elaborate development of the subject, and the reader has a number of accessible places to hang his hat as the story unfolds.' Michael Berg, MAA Reviews Review of Multi-volume Set: 'Garrett's volumes fill an important void in the literature of automorphic forms' Matthew P. Young, Bulletin of the AMS

Papildus informācija

Volume 2 of a two-volume introduction to the analytical aspects of automorphic forms, featuring proofs of critical results with examples.
Introduction and Historical Notes ix
9 Unbounded Operators on Hilbert Spaces
1(64)
9.1 Unbounded Symmetric Operators on Hilbert Spaces
2(5)
9.2 Friedrichs's Self-Adjoint Extensions of Semibounded Operators
7(3)
9.3 Example: Incommensurable Self-adjoint Extensions
10(2)
9.4 Unbounded Self-adjoint Operators with Compact Resolvents
12(2)
9.5 Example: Δ on L2(T) and Sobolev Spaces
14(10)
9.6 Example: Exotic Eigenfunctions on T
24(2)
9.7 Example: Usual Sobolev Spaces on R
26(2)
9.8 Example: Discrete Spectrum of -Δ + x2 on L2(R)
28(7)
9.9 Essential Self-adjointness
35(3)
9.10 Example: Essentially Self-adjoint Operator
38(2)
9.A Appendix: Compact Operators
40(8)
9.B Appendix: Open Mapping and Closed Graph Theorems
48(2)
9.C Appendix: Irreducibles of Compact Groups
50(4)
9.D Appendix: Spectral Theorem, Schur's Lemma, Multiplicities
54(8)
9.E Appendix: Tietze-Urysohn-Brouwer Extension Theorem
62(3)
10 Discrete Decomposition of Pseudo-Cuspforms
65(26)
10.1 Compact Resolvents in Simplest Examples
66(1)
10.2 Compact Resolvents for SL3(Z), SL4(Z), SL5(Z),...
67(2)
10.3 Density of Domains of Operators
69(2)
10.4 Tail Estimates: Simplest Example
71(3)
10.5 Tail Estimates: Three Additional Small Examples
74(3)
10.6 Tail Estimate: SL3(Z), SL4(Z), SL5(Z)
77(5)
10.7 Compact B1a → L2a in Four Simple Examples
82(6)
10.8 Compact B1a → L2a for SL3(Z), SL4(Z), SL5(Z),...
88(2)
10.9 Compact Resolvents and Discrete Spectrum
90(1)
11 Meromorphic Continuation of Eisenstein Series
91(51)
11.1 Up to the Critical Line: Four Simple Examples
92(2)
11.2 Recharacterization of Friedrichs Extensions
94(3)
11.3 Distributional Characterization of Pseudo-Laplacians
97(4)
11.4 Key Density Lemma: Simple Cases
101(3)
11.5 Beyond the Critical Line: Four Simple Examples
104(8)
11.6 Exotic Eigenfunctions: Four Simple Examples
112(2)
11.7 Up to the Critical Line: SLr(Z)
114(4)
11.8 Distributional Characterization of Pseudo-Laplacians
118(4)
11.9 Density Lemma for Pr,r ⊂ SL2r
122(4)
11.10 Beyond the Critical Line: Pr,r ⊂ SL2r
126(9)
11.11 Exotic Eigenfunctions: Pr,r ⊂ SL2r
135(2)
11.12 Non-Self-Associate Cases
137(3)
11.A Appendix: Distributions Supported on Submanifolds
140(2)
12 Global Automorphic Sobolev Spaces, Green's Functions
142(45)
12.1 A Simple Pretrace Formula
143(7)
12.2 Pretrace Formula for Compact Periods
150(3)
12.3 Global Automorphic Sobolev Spaces Hl
153(8)
12.4 Spectral Characterization of Sobolev Spaces Hs
161(5)
12.5 Continuation of Solutions of Differential Equations
166(5)
12.6 Example: Automorphic Green's Functions
171(2)
12.7 Whittaker Models and a Subquotient Theorem
173(5)
12.8 Meromorphic Continuation of Intertwining Operators
178(2)
12.9 Intertwining Operators among Principal Series
180(5)
12.A Appendix: A Usual Trick with Γ(s)
185(2)
13 Examples: Topologies on Natural Function Spaces
187(61)
13.1 Banach Spaces Ck[ a, b]
188(2)
13.2 Non-Banach Limit C∞[ a, b] of Banach Spaces Ck[ a, b]
190(6)
13.3 Sufficient Notion of Topological Vectorspace
196(4)
13.4 Unique Vectorspace Topology on Cn
200(2)
13.5 Non-Banach Limits Ck(R), C∞(R) of Banach Spaces Ck[ a, b]
202(3)
13.6 Banach Completion Ck0(R) of Ckc(R)
205(1)
13.7 Rapid-Decay Functions, Schwartz Functions
206(5)
13.8 Non-Frechet Colimit C∞ of Cn, Quasi-Completeness
211(4)
13.9 Non-Frechet Colimit C∞c(R) of Frechet Spaces
215(2)
13.10 LF-Spaces of Moderate-Growth Functions
217(1)
13.11 Seminorms and Locally Convex Topologies
218(6)
13.12 Quasi-Completeness Theorem
224(5)
13.13 Strong Operator Topology
229(1)
13.14 Generalized Functions (Distributions) on R
229(6)
13.15 Tempered Distributions and Fourier Transforms on R
235(3)
13.16 Test Functions and Paley-Wiener Spaces
238(3)
13.17 Schwartz Functions and Fourier Transforms on Qp
241(7)
14 Vector-Valued Integrals
248(23)
14.1 Characterization and Basic Results
249(3)
14.2 Differentiation of Parametrized Integrals
252(1)
14.3 Fourier Transforms
253(2)
14.4 Uniqueness of Invariant Distributions
255(2)
14.5 Smoothing of Distributions
257(4)
14.6 Density of Smooth Vectors
261(1)
14.7 Quasi-Completeness and Convex Hulls of Compacts
262(2)
14.8 Existence Proof
264(1)
14.A Appendix: Hahn-Banach Theorems
265(6)
15 Differentiable Vector-Valued Functions
271(22)
15.1 Weak-to-Strong Differentiability
271(1)
15.2 Holomorphic Vector-Valued Functions
272(3)
15.3 Holomorphic Hol(Omega;, V)-Valued Functions
275(2)
15.4 Banach-Alaoglu: Compactness of Polars
277(1)
15.5 Variant Banach-Steinhaus/Uniform Boundedness
278(1)
15.6 Weak Boundedness Implies (strong) Boundedness
279(1)
15.7 Proof That Weak C1 Implies Strong C0
280(1)
15.8 Proof That Weak Holomorphy Implies Continuity
281(1)
15.A Appendix: Vector-Valued Power Series
282(2)
15.B Appendix: Two Forms of the Baire Category Theorem
284(1)
15.C Appendix: Hartogs's Theorem on Joint Analyticity
285(8)
16 Asymptotic Expansions
293(40)
16.1 Heuristic for Stirling's Asymptotic
294(1)
16.2 Watson's Lemma
295(1)
16.3 Watson's Lemma Illustrated on the Beta Function
296(1)
16.4 Simple Form of Laplace's Method
297(3)
16.5 Laplace's Method Illustrated on Bessel Functions
300(3)
16.6 Regular Singular Points Heuristic: Freezing Coefficients
303(2)
16.7 Regular Singular Points
305(2)
16.8 Regular Singular Points at Infinity
307(1)
16.9 Example Revisited
308(1)
16.10 Irregular Singular Points
309(5)
16.11 Example: Translation-Equivariant Eigenfunctions on
314(2)
16.12 Beginning of Construction of Solutions
316(2)
16.13 Boundedness of K(x, t)
318(2)
16.14 End of Construction of Solutions
320(2)
16.15 Asymptotics of Solutions
322(5)
16.A Appendix: Manipulation of Asymptotic Expansions
327(2)
16.B Appendix: Ordinary Points
329(4)
Bibliography 333(8)
Index 341
Paul Garrett is Professor of Mathematics at the University of Minnesota. His research focuses on analytical issues in the theory of automorphic forms. He has published numerous journal articles as well as five books.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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