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E-grāmata: Zeta Functions Of Reductive Groups And Their Zeros

(Kyushu Univ, Japan)
  • Formāts: 556 pages
  • Izdošanas datums: 09-Feb-2018
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813230668
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Zeta Functions Of Reductive Groups And Their ZerosPalielināt
  • Formāts: 556 pages
  • Izdošanas datums: 09-Feb-2018
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813230668
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This book provides a systematic account of several breakthroughs in the modern theory of zeta functions. It contains two different approaches to introduce and study genuine zeta functions for reductive groups (and their maximal parabolic subgroups) defined over number fields. Namely, the geometric one, built up from stability of principal lattices and an arithmetic cohomology theory, and the analytic one, from Langlands' theory of Eisenstein systems and some techniques used in trace formula, respectively. Apparently different, they are unified via a Lafforgue type relation between Arthur's analytic truncations and parabolic reductions of Harder–Narasimhan and Atiyah–Bott. Dominated by the stability condition and/or the Lie structures embedded in, these zeta functions have a standard form of the functional equation, admit much more refined symmetric structures, and most surprisingly, satisfy a weak Riemann hypothesis. In addition, two levels of the distributions for their zeros are exposed, i.e. a classical one giving the Dirac symbol, and a secondary one conjecturally related to GUE.This book is written not only for experts, but for graduate students as well. For example, it offers a summary of basic theories on Eisenstein series and stability of lattices and arithmetic principal torsors. The second part on rank two zeta functions can be used as an introduction course, containing a Siegel type treatment of cusps and fundamental domains, and an elementary approach to the trace formula involved. Being in the junctions of several branches and advanced topics of mathematics, these works are very complicated, the results are fundamental, and the theory exposes a fertile area for further research.
Introduction vii
Non-Abelian Zeta Functions 1(52)
1 Semi-Stable Lattice
3(24)
1.1 Motivation
3(4)
1.2 Projective Modules
7(2)
1.2.1 Invertible Modules
7(1)
1.2.2 General Projective Modules
8(1)
1.3 Stability of Lattices
9(8)
1.3.1 Lattices
9(1)
1.3.2 Covolumes of Lattices
10(1)
1.3.3 Stability of Lattices
11(2)
1.3.4 Canonical Filtration
13(4)
1.4 Volume of Lattice: Special Linear Group
17(3)
1.4.1 Metrics of Lattices
17(1)
1.4.2 Special Metrics of Lattices
18(2)
1.5 Automorphisms of Lattices
20(5)
1.5.1 General Automorphisms
20(1)
1.5.2 Special Automorphisms
21(2)
1.5.3 Unit Automorphisms
23(2)
1.6 Compact Moduli Spaces of Semi-Stable Lattices
25(2)
2 Geometry of Numbers
27(16)
2.1 Global Cohomology
27(3)
2.1.1 Adelic Ring
27(1)
2.1.2 Adelic Cohomology Groups
28(1)
2.1.3 Topological Duality: Local and Global Pairings
29(1)
2.2 Duality and Riemann-Roch Theorem in Arithmetic
30(6)
2.2.1 Nine-Diagram
30(2)
2.2.2 Arithmetic Riemann-Roch Theorem
32(4)
2.3 Arithmetic Vanishing Theorem
36(7)
2.3.1 Elementary Properties of 0-th Cohomology
36(3)
2.3.2 Ampleness and Vanishing Theorem
39(4)
3 Non-Abelian Zeta Functions
43(10)
3.1 Moduli Spaces of Semi-Stable Lattices
43(3)
3.1.1 Stability
43(1)
3.1.2 Effective Vanishing Theorem
44(2)
3.2 Non-Abelian Zeta Function
46(9)
3.2.1 Definition
46(1)
3.2.2 Zeta Properties
46(2)
3.2.3 Relation with Epstein Zeta Function
48(3)
3.2.4 Riemann Hypothesis
51(2)
Rank Two Zeta Functions 53(106)
4 Distances to Cusps and Fundamental Domains
55(48)
4.1 Upper Half Space Model
55(5)
4.1.1 Upper Half Plane
56(1)
4.1.2 Upper Half Space
57(2)
4.1.3 Rank Two OK-Lattices: Upper Half Space Model
59(1)
4.2 Cusps and Ideal Classes
60(8)
4.2.1 Generators of Fractional Ideals
61(1)
4.2.2 Special Transformations
62(1)
4.2.3 Cusps and Ideal Classes for Total Real Fields
63(2)
4.2.4 Cusps and Ideal Classes
65(3)
4.3 Stabilizers of Cusps and Their Fundamental Domains
68(13)
4.3.1 Upper Half Plane
68(2)
4.3.2 Upper Half Space
70(1)
4.3.3 Fundamental Domains for Stabilizer Groups (I)
71(7)
4.3.4 Fundamental Domains for Stabilizer Groups (II)
78(3)
4.4 Distance to Cusp and Fundamental Domain
81(22)
4.4.1 Upper Half Plane
81(2)
4.4.2 Upper Half Space
83(2)
4.4.3 Fundamental Domain (I)
85(8)
4.4.4 Fundamental Domain (II)
93(10)
5 Rank Two Zeta Functions
103(56)
5.1 Distance to Cusp and Stability
103(8)
5.1.1 Parameter Space of Semi-Stable Rank Two Lattices
103(2)
5.1.2 Rank One Sub-Lattices
105(1)
5.1.3 Stability and Distances to Cusps
106(2)
5.1.4 Example: Truncated Fundamental Domain
108(1)
5.1.5 Moduli Space of Rank Two Semi-Stable Lattices
109(2)
5.2 Rank Two Non-Abelian Zeta Function
111(3)
5.2.1 Definition
111(2)
5.2.2 Relations with Epstein Zeta Functions
113(1)
5.3 Epstein Zeta Function and Its Fourier Expansion
114(19)
5.3.1 Automorphic Function in One Variable
114(4)
5.3.2 Upper Half Space
118(7)
5.3.3 Eisenstein Series for Rank Two Lattice
125(8)
5.4 Explicit Formula of Rank Two Zeta Function
133(20)
5.4.1 Classical Rankin-Selberg Method
133(7)
5.4.2 Generalization of Rankin-Selberg Method
140(3)
5.4.3 Rank Two Non-Abelian Zeta Function
143(10)
5.5 Zeros of Rank Two Non-Abelian Zeta Functions
153(8)
5.5.1 Product Formula for Entire Function of Order 1
153(1)
5.5.2 Zeros of Rank Two Non-Abelian Zeta Function of Q
154(2)
5.5.3 A Simple Generalization
156(3)
Eisenstein Periods and Multiple L-Functions 159(48)
6 Multiple L-Functions
161(24)
6.1 Reduction Groups
161(11)
6.1.1 Parabolic Subgroups
161(2)
6.1.2 Roots, Coroots, Weights and Coweights
163(2)
6.1.3 Example with Special Linear Group
165(4)
6.1.4 Logarithmic Map
169(1)
6.1.5 Compatible Haar Measures
170(1)
6.1.6 Classical Reductive Theory
171(1)
6.2 Multiple L-Functions
172(13)
6.2.1 Growth Condition
172(1)
6.2.2 Automorphic Forms
173(3)
6.2.3 Eisenstein Series and Their Constant Terms
176(2)
6.2.4 Stability Truncation on Adelic Space
178(3)
6.2.5 Multiple L-Functions
181(2)
6.2.6 Meromorphic Extension and Functional Equations
183(1)
6.2.7 Holomorphicity and Singularities
183(2)
7 Periods of Reductive Groups
185(22)
7.1 Arthur's Analytic Truncation
185(6)
7.1.1 Positive Cone and Positive Chamber
185(1)
7.1.2 Preliminary Estimations
186(1)
7.1.3 Langlands' Combinatorial Lemma
187(1)
7.1.4 Langlands-Arthur's Partition: Reduction Theory
188(1)
7.1.5 Arthur's Analytic Truncation
189(1)
7.1.6 Basic Properties
190(1)
7.2 Analytic Arthur Periods and Geometric Eisenstein Periods
191(10)
7.2.1 LambdaT1 as Characteristic Function of Compact Set
191(3)
7.2.2 Geometric Eisenstein Period
194(1)
7.2.3 Regularized Integration over Cone
195(2)
7.2.4 Regularized Period of Automorphic Form
197(3)
7.2.5 Regularized Periods
200(1)
7.3 Periods of Reductive Groups
201(8)
7.3.1 Eisenstein Periods for Cusp Forms
201(1)
7.3.2 Gindikin-Karpelevich Formula
202(1)
7.3.3 Periods of Reductive Groups
202(1)
7.3.4 Volumes of Truncated Fundamental Domains
203(4)
Zeta Functions for Reductive Groups 207(72)
8 Zeta Functions for Reductive Groups
209(38)
8.1 Zeta Function for SLn: Genuine but Different
209(8)
8.1.1 Non-Abelian Zeta Function and Eisenstein Period
209(1)
8.1.2 Epstein Zeta Function and Siegel-Eisenstein Series
210(3)
8.1.3 Langlands' Eisenstein and Siegel's Eisenstein
213(1)
8.1.4 Zeta Function for SLn
214(3)
8.2 From SLn to Sp2n: Role of Periods
217(3)
8.2.1 Sp2n-Periods
217(1)
8.2.2 Siegel Eisenstein Series
218(1)
8.2.3 Siegel Eisenstein Series and Langlands Eisenstein Series
219(1)
8.2.4 Siegel-Maabeta-Eisenstein Period and Zeta Function of Sp2n
219(1)
8.3 Zeta Functions for G2
220(22)
8.3.1 Looking for Zeta Functions of G2
220(8)
8.3.2 RH for Zeta Functions of G2 I: Preparations
228(8)
8.3.3 Riemann Hypothesis for G2
236(6)
8.4 Zeta Functions for (G, P)
242(5)
8.4.1 Singular Hyperplanes Located
242(1)
8.4.2 Main Definition
242(3)
8.4.3 Basic Properties
245(2)
9 Zeta Function of (SLn, Pn-1,1)
247(18)
9.1 Special Weyl Elements
247(11)
9.1.1 General Setting
247(2)
9.1.2 Special Weyl Elements
249(3)
9.1.3 Special Permutations
252(1)
9.1.4 Working Site
253(2)
9.1.5 Special Permutations
255(3)
9.2 Explicit Formula for Zeta Function of (SLn, Pn-1,1)
258(7)
9.2.1 Rough Formula for Period of (SLn, Pn_1,1)
258(2)
9.2.2 Applications of Special Permutations
260(3)
9.2.3 Explicit Formula for Zeta Functions of (SLn, Pn-1,1)
263(2)
10 Functional Equation
265(14)
10.1 Preparations
265(4)
10.1.1 Statement
265(2)
10.1.2 Lie Theoretic Relations (I)
267(2)
10.2 Normalization and Micro Functional Equation
269(5)
10.2.1 Overdone Normalization and Micro Functional Equation
269(4)
10.2.2 Normalization
273(1)
10.3 Functional Equation
274(3)
10.3.1 Proof of Functional Equation
274(2)
10.3.2 Constant cG,p
276(1)
10.4 T-Version for SL3
277(2)
Algebraic, Analytic Structures and Rieman Hypothesis 279(88)
11 Conditional Weak RH for finalsigmaQSLn/Pn-1,1(s)
281(18)
11.1 Refined Symmetric Structure
281(3)
11.1.1 Zeta Functions for (SLn, Pn-1,1)/Q
281(2)
11.1.2 Refined Symmetric Structure
283(1)
11.2 Zero-Free Region of Upper Half Function
284(7)
11.2.1 Dominant Term
284(2)
11.2.2 Zero Free Region on Right Half Plane
286(3)
11.2.3 Zero Free Region on Left Half Plane
289(2)
11.3 Refined Hadamard Product for ξQn+1/2(s)
291(4)
11.3.1 Distributions of Zeros for ξQn+1/2(s)
291(2)
11.3.2 Refined Hadamard Product for ξQn+1/2(s)
293(2)
11.4 Conditional Weak Riemann Hypothesis for ζQSLn/Pn-q,1(s)
295(4)
11.4.1 Elementary Lemma
295(3)
11.4.2 Conditional Weak Riemann Hypothesis for ζQSLn/Pn-1,1(s)
298(1)
12 Algebraic and Analytic Structures and Weak Riemann Hypothesis
299(50)
12.1 Criterion for Weak Riemann Hypothesis
299(2)
12.1.1 Discriminant of (G, P)
299(2)
12.1.2 Criterion of Weak Riemann Hypothesis
301(1)
12.2 Refined Symmetries of Zeta Functions
301(3)
12.2.1 Positively Oriented
301(1)
12.2.2 Refined Symmetries
302(2)
12.3 Dominate Term
304(9)
12.3.1 Entire Function Oriented
304(2)
12.3.2 Terms with Maximal Discrepancy
306(3)
12.3.3 Relation between p and Wp
309(3)
12.3.4 Leading Polynomials
312(1)
12.4 Zero Free Regions
313(12)
12.4.1 Lie Theoretic Structures Involved
313(5)
12.4.2 Estimations on Right Half Plane
318(1)
12.4.3 Normalization Factor
319(1)
12.4.4 Zero Free Region on Right Half Plane
320(2)
12.4.5 Zero Free Region on Left Half Plane
322(3)
12.5 Hadamard Product
325(7)
12.5.1 Distributions for Zeros of Ap(s)
325(3)
12.5.2 Hadamard Product
328(4)
12.6 Proof of Theorem
332(1)
12.7 Discriminant and Residue of Period
333(6)
12.7.1 Residue of Period
333(2)
12.7.2 Period of Pseudo Root System
335(2)
12.7.3 Residue and Leading Coefficient
337(2)
12.A Decomposition of σp(1)
339(9)
12.B Geometric Interpretation of cp
348(1)
13 Weak Riemann Hypothesis for Zeta Functions of Exceptional Groups
349(18)
13.1 Weyl Groups
349(7)
13.1.1 Root Systems
349(1)
13.1.2 Weyl Group for D5
350(1)
13.1.3 Weyl Group for E6
351(1)
13.1.4 Weyl Group for E7
352(1)
13.1.5 Weyl Group for E8
353(3)
13.2 Weak Riemann Hypothesis for Zeta Functions of En
356(13)
13.2.1 Conditions for Weak Riemann Hypothesis
356(2)
13.2.2 Special and Very Special Weyl Elements
358(2)
13.2.3 Constants λp,αvw
360(1)
13.2.4 Discriminants ΔQEn,P
360(6)
13.2.5 Weak Riemann Hypothesis for Zeta Functions of Exceptional Groups
366(1)
Geometric Structures and Riemann Hypothesis 367(78)
14 Analytic and Geometric Truncations: Special Linear Groups
369(12)
14.1 Geometric Truncation and Parabolic Reduction
369(6)
14.1.1 Canonical Polygon and Compactness
369(1)
14.1.2 Parabolic Reduction and Geometric Truncation
370(5)
14.2 Analytic and Geometric Truncations
375(6)
14.2.1 Micro Bridge
375(5)
14.2.2 Analytic and Geometric Truncations
380(1)
15 Special Uniformity of Zeta Functions and Weak Riemann Hypothesis
381(6)
15.1 Special Uniformity of Zeta Functions
381(3)
15.1.1 Basic Properties of Truncations
381(2)
15.1.2 Analytic and Geometric Periods
383(1)
15.2 Weak Riemann Hypothesis for Non-Abelian Zeta Functions
384(3)
15.2.1 Parabolic Reduction, Stability and Volumes
384(2)
15.2.2 Existence of Stable OF-Lattices
386(1)
15.2.3 Weak Riemann Hypothesis
386(1)
16 Analytic and Geometric Truncations: Reductive Groups
387(40)
16.1 Lie Theoretic Preparation
387(14)
16.1.1 Maximal Split Sub or Quotient Torus
387(1)
16.1.2 Generalized Root System
388(2)
16.1.3 Partial Order Induced by Positive Weyl Cone
390(4)
16.1.4 Classical Example
394(2)
16.1.5 Relative 'Root System'
396(1)
16.1.6 Relative Positive Chamber and Cone
397(2)
16.1.7 Partial Order in Relative Theory
399(2)
16.2 Arithmetic Principal Torsors and Their Stability
401(14)
16.2.1 Compatible Metrics
401(4)
16.2.2 Principal Lattices
405(2)
16.2.3 Parabolic Reduction
407(3)
16.2.4 Torsors in Arithmetic
410(3)
16.2.5 Canonical Parabolic Subgroup Scheme
413(2)
16.3 Analytic and Geometric Truncations
415(12)
16.3.1 Stability and Parabolic Reduction
416(6)
16.3.2 Equivalence of Analytic and Geometric Truncations
422(1)
16.3.3 Properties of Analytic and Geometric Truncations
422(5)
17 Weak Riemann Hypothesis for Zeta Functions of Reductive Groups
427(6)
17.1 Volumes of Semi-Stable Moduli Spaces
427(4)
17.1.1 Parabolic Reduction, Stability and the Volumes
427(4)
17.2 Riemann Hypothesis for Zeta Functions of Reductive Groups
431(2)
17.2.1 Geometric Interpretation of Discriminant
431(1)
17.2.2 Riemann Hypothesis for Weng Zeta Functions
432(1)
18 Distributions of Zeros for Zeta Functions of Reductive Groups
433(14)
18.1 Distributions of Zeta Zeros
433(4)
18.1.1 Distributions in Classical Style
433(1)
18.1.2 New Secondary Distributions
434(3)
18.2 Proof of Theorems
437(10)
18.2.1 Rank Two Zeta Zeros
437(1)
18.2.2 Proof of Theorems 18.3 and 18.4
438(6)
18.2.3 Proof of Theorems 18.1 and 18.2
444(1)
Five Essays on Arithmetic Cohomology (Joint with K. Sugahara) 445(2)
Appendix A Arithmetic Adelic Complexes 447(12)
A.1 Parshin-Beilinson's Theory
447(5)
A.1.1 Local Fields for Reduced Flags
447(2)
A.1.2 Adelic Cohomology Theory
449(2)
A.1.3 Example
451(1)
A.2 Arithmetic Cohomology Groups
452(7)
A.2.1 Adelic Rings for Arithmetic Surfaces
452(2)
A.2.2 Adelic Spaces at Infinity
454(1)
A.2.3 Arithmetic Adelic Complexes
455(3)
A.2.4 Cohomology Theory for Arithmetic Curves
458(1)
Appendix B Cohomology for Arithmetic Surfaces 459(18)
B.1 Local Residue Pairings
459(2)
B.1.1 Residue Maps for Local Fields
459(2)
B.1.2 Local Residue Maps
461(1)
B.2 Global Residue Pairing
461(2)
B.2.1 Global Residue Pairing
462(1)
B.2.2 Non-Degeneracy
462(1)
B.3 Adelic Subspaces
463(8)
B.3.1 Level Two Subspaces
463(2)
B.3.2 Perpendicular Subspaces
465(6)
B.4 Arithmetic Cohomology Groups for Arithmetic Surfaces
471(6)
B.4.1 Definitions
471(1)
B.4.2 Inductive Long Exact Sequences
472(2)
B.4.3 Duality of Cohomology Groups
474(3)
Appendix C Ind-Pro Topologies in Arithmetic Dimension Two 477(14)
C.1 Ind-Pro Topology on Adelic Spaces
477(9)
C.1.1 Ind-Pro Topology Spaces and Their Duals
477(4)
C.1.2 Adelic Spaces and Their Ind-Pro Topologies
481(1)
C.1.3 Adelic Spaces are Complete
482(3)
C.1.4 Adelic Spaces are Compactly Oriented
485(1)
C.1.5 Dual Adelic Spaces
485(1)
C.2 Adelic Spaces and Their Duals
486(5)
C.2.1 Continuity of Scalar Products
486(2)
C.2.2 Residue Maps are Continuous
488(1)
C.2.3 Adelic Spaces are Self-Dual
488(3)
Appendix D Closeness of Adelic Subspaces and Finiteness of H1ar for Arithmetic Surfaces 491(10)
D.1 Arithmetic Cohomology Groups: A Review
491(2)
D.2 Closeness of Adelic Sub-Quotient Spaces
493(5)
D.3 Topological Duality of Arithmetic Cohomology Groups
498(3)
Appendix E Central Extensions and Reciprocity Laws for Arithmetic Surfaces 501(18)
E.1 Algebraic Aspect
504(4)
E.1.1 Reciprocity Law around Point/Along Vertical Curve
504(1)
E.1.2 K2-Central Extensions
505(2)
E.1.3 Sketch of Proof
507(1)
E.2 Arithmetic Aspect
508(5)
E.2.1 Arithmetic Adelic Complex
508(1)
E.2.2 Numerations in Terms of Arakelov Intersection
509(4)
E.3 Numerations for Arithmetic Adelic Cohomologies
513(2)
E.4 Arithmetic Central Extension
515(1)
E.5 Splitness
516(1)
E.6 Reciprocity Law along Horizontal Curve
517(2)
Bibliography 519(6)
Index 525
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