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Modern Analysis of Automorphic Forms By Example [Kietas viršelis]

(University of Minnesota)
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Raktažodžiai:
This is Volume 1 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.

This is Volume 1 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 1 features critical results, which are proven carefully and in detail, including discrete decomposition of cuspforms, meromorphic continuation of Eisenstein series, spectral decomposition of pseudo-Eisenstein series, and automorphic Plancherel theorem. Volume 2 features automorphic Green's functions, metrics and topologies on natural function spaces, unbounded operators, vector-valued integrals, vector-valued holomorphic functions, and asymptotics. 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.

Recenzijos

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

Daugiau informacijos

Volume 1 of a two-volume introduction to the analytical aspects of automorphic forms, featuring proofs of critical results with examples.
Introduction and Historical Notes ix
1 Four Small Examples
1(54)
1.1 Groups G = SL2(R), SL2(C), Sp*1,1, and SL2(H)
2(3)
1.2 Compact Subgroups K ⊂ G, Cartan Decompositions
5(4)
1.3 Iwasawa Decomposition G = PK = NA+K
9(4)
1.4 Some Convenient Euclidean Rings
13(2)
1.5 Discrete Subgroups Γ ⊂ G, Reduction Theory
15(4)
1.6 Invariant Measures, Invariant Laplacians
19(3)
1.7 Discrete Decomposition of L2(Γ\G/K) Cuspforms
22(2)
1.8 Pseudo-Eisenstein Series
24(3)
1.9 Eisenstein Series
27(5)
1.10 Meromorphic Continuation of Eisenstein Series
32(3)
1.11 Truncation and Maaß-Selberg Relations
35(9)
1.12 Decomposition of Pseudo-Eisenstein Series
44(5)
1.13 Plancherel for Pseudo-Eisenstein Series
49(3)
1.14 Automorphic Spectral Expansion and Plancherel Theorem
52(1)
1.15 Exotic Eigenfunctions, Discreteness of Pseudo-Cuspforms
53(2)
2 The Quotient Z+GL2(k)\GL2(A)
55(83)
2.1 Groups Kυ = GL2(oυ) ⊂ Gυ = GL2(kυ)
56(3)
2.2 Discrete Subgroup GL2(k) ⊂ GL2(A), Reduction Theory
59(8)
2.3 Invariant Measures
67(3)
2.4 Hecke Operators, Integral Operators
70(4)
2.5 Decomposition by Central Characters
74(2)
2.6 Discrete Decomposition of Cuspforms
76(3)
2.7 Pseudo-Eisenstein Series
79(6)
2.8 Eisenstein Series
85(11)
2.9 Meromorphic Continuation of Eisenstein Series
96(4)
2.10 Truncation and Maaß-Selberg Relations
100(5)
2.11 Decomposition of Pseudo-Eisenstein Series: Level One
105(9)
2.12 Decomposition of Pseudo-Eisenstein Series: Higher Level
114(5)
2.13 Plancherel for Pseudo-Eisenstein Series: Level One
119(5)
2.14 Spectral Expansion, Plancherel Theorem: Level One
124(1)
2.15 Exotic Eigenfunctions, Discreteness of Pseudo-Cuspforms
125(13)
2.A Appendix: Compactness of J1/kx
127(1)
2.B Appendix: Meromorphic Continuation
128(6)
2.C Appendix: Hecke-Maaß Periods of Eisenstein Series
134(4)
3 SL3(Z), SL4(Z), SL5(Z), ...
138(86)
3.1 Parabolic Subgroups of GLr
139(3)
3.2 Groups Kυ = GLr(φυ) ⊂ Gυ = GLr(kυ)
142(3)
3.3 Discrete Subgroup Gk = GLr(k), Reduction Theory
145(3)
3.4 Invariant Differential Operators and Integral Operators
148(2)
3.5 Hecke Operators and Integral Operators
150(2)
3.6 Decomposition by Central Characters
152(1)
3.7 Discrete Decomposition of Cuspforms
152(3)
3.8 Pseudo-Eisenstein Series
155(3)
3.9 Cuspidal-Data Pseudo-Eisenstein Series
158(2)
3.10 Minimal-Parabolic Eisenstein Series
160(7)
3.11 Cuspidal-Data Eisenstein Series
167(14)
3.12 Continuation of Minimal-Parabolic Eisenstein Series
181(12)
3.13 Continuation of Cuspidal-Data Eisenstein Series
193(2)
3.14 Truncation and Maaß-Selberg Relations
195(9)
3.15 Minimal-Parabolic Decomposition
204(4)
3.16 Cuspidal-Data Decomposition
208(6)
3.17 Plancherel for Pseudo-Eisenstein Series
214(4)
3.18 Automorphic Spectral Expansions
218(6)
3.A Appendix: Bochner's Lemma
220(2)
3.B Appendix: Phragmen-Lindelof Theorem
222(2)
4 Invariant Differential Operators
224(29)
4.1 Derivatives of Group Actions: Lie Algebras
225(4)
4.2 Laplacians and Casimir Operators
229(3)
4.3 Details about Universal Algebras
232(3)
4.4 Descending to G/K
235(1)
4.5 Example Computation: SL2(R) and S
236(3)
4.6 Example Computation: SL2(C)
239(3)
4.7 Example Computation: Sp*1,1
242(2)
4.8 Example Computation: SL2(H)
244(9)
4.A Appendix: Brackets
246(4)
4.B Appendix: Existence and Uniqueness
250(3)
5 Integration on Quotients
253(10)
5.1 Surjectivity of Averaging Maps
254(2)
5.2 Invariant Measures and Integrals on Quotients H\G
256(7)
5.A Appendix: Apocryphal Lemma X ~ G/Gx
258(4)
5.B Appendix: Topology on Quotients H\G or G/H
262(1)
6 Action of G on Function Spaces on G
263(43)
6.1 Action of G on L2(Γ\G)
263(4)
6.2 Action of G on C0c(Γ\G)
267(3)
6.3 Test Functions on Z+Gk\GA
270(4)
6.4 Action of GA on C∞c(Z+Gk\Gk)
274(3)
6.5 Symmetry of Invariant Laplacians
277(2)
6.6 An Instance of Schur's Lemma
279(6)
6.7 Duality of Induced Representations
285(4)
6.8 An Instance of Frobenius Reciprocity
289(1)
6.9 Induction in Stages
290(5)
6.10 Representations of Compact G/Z
295(1)
6.11 Gelfand-Kazhdan Criterion
296(10)
6.A Appendix: Action of Compact Abelian Groups
301(5)
7 Discrete Decomposition of Cuspforms
306(48)
7.1 The Four Simplest Examples
307(13)
7.2 Z+GL2(k)\GL2(A)
320(13)
7.3 Z+GLr(k)\GLr(A)
333(21)
7.A Appendix: Dualities
345(6)
7.B Appendix: Compact Quotients Γ\G
351(3)
8 Moderate Growth Functions, Theory of the Constant Term
354(19)
8.1 The Four Small Examples
354(6)
8.2 GL2(A)
360(2)
8.3 SL3(Z), SU(Z), SL5(Z), ...
362(5)
8.4 Moderate Growth of Convergent Eisenstein Series
367(4)
8.5 Integral Operators on Cuspidal-Data Eisenstein Series
371(2)
8.A Appendix: Joint Continuity of Bilinear Functionals
372(1)
Bibliography 373(8)
Index 381
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.