Hashimoto / Namikawa Automorphic Forms and Geometry of Arithmetic Varieties

1. Auflage 2014
ISBN: 978-1-4832-1807-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 570 Seiten, Web PDF

Reihe: Advanced Studies in Pure Mathematics

Automorphic Forms and Geometry of Arithmetic Varieties deals with the dimension formulas of various automorphic forms and the geometry of arithmetic varieties. The relation between two fundamental methods of obtaining dimension formulas (for cusp forms), the Selberg trace formula and the index theorem (Riemann-Roch's theorem and the Lefschetz fixed point formula), is examined. Comprised of 18 sections, this volume begins by discussing zeta functions associated with cones and their special values, followed by an analysis of cusps on Hilbert modular varieties and values of L-functions. The reader is then introduced to the dimension formula of Siegel modular forms; the graded rings of modular forms in several variables; and Selberg-Ihara's zeta function for p-adic discrete groups. Subsequent chapters focus on zeta functions of finite graphs and representations of p-adic groups; invariants and Hodge cycles; T-complexes and Ogata's zeta zero values; and the structure of the icosahedral modular group. This book will be a useful resource for mathematicians and students of mathematics.

Autoren/Hrsg.

Hashimoto, K.

Namikawa, Y.

Inhaltsverzeichnis

1;Front Cover;1
2;Automorphic Forms and Geometry of Arithmetic Varieties;4
3;Copyright Page;5
4;Table of Contents;16
5;Foreword;8
6;Dedication;10
7;Preface;12
8;PART I;18
8.1;Secton I: Zeta Functions Associated to Cones and their Special Values;18
8.1.1;Introduction;18
8.1.2;1. Self-dual homogeneous cones;19
8.1.3;2. Zeta functions associated to a self-dual homogeneous cone;24
8.1.4;3. Geometric invariants of cusp singularities;33
8.1.5;4. Zeta functions associated to Tsuchihashi singularities;38
8.1.6;References;42
8.2;Secton II: Cusps on Hilbert Modular Varieties and Values of L-Functions;46
8.2.1;1-;46
8.2.2;2.;48
8.2.3;3.;51
8.2.4;4.;53
8.2.5;5.;56
8.2.6;References;57
8.3;Secton III: On Dimension Formula for Siegel Modular Forms;58
8.3.1;0. Introduction;58
8.3.2;1. Dimension formula for Ã2(Í) and ãÍ) with iV> 3;61
8.3.3;2. Dimension formula for Ã2(×) and Ã3(À);72
8.3.4;References;79
8.4;Secton IV: On the Graded Rings of Modular Forms in Several Variables;82
8.4.1;1. A graded ring;83
8.4.2;2. A graded ring and a subring;86
8.4.3;3. Hilbert modular forms;90
8.4.4;4. Siegel modular forms of degree two;93
8.4.5;5. Siegel modular forms of degree three;95
8.4.6;6. Siegel modular forms of degree four;100
8.4.7;References;101
8.5;Secton V: Vector Valued Modular Forms of Degree Two and their Application to Triple L-functions;106
8.5.1;1. Differential operators;106
8.5.2;2. Construction of certain vector valued modular forms;108
8.5.3;3. Triple L-functions;109
8.5.4;References;113
9;PART II;116
9.1;Secton VI: Special Values of L-functions Associated with the Space of Quadratic Forms and the Representation of Sp(2#i9 Fp) in the Space of Siegel Cusp Forms;116
9.1.1;Introduction;116
9.1.2;Chapter I. L-functions of quadratic forms;123
9.1.2.1;1.1. Definition of zeta functions and L-functions;123
9.1.2.2;1.2. Some properties of £*($, rgw))? £2*C*, ^det)> and L?(.y, ø^,ñ);126
9.1.3;Chapter II. Evaluation of special values of L-functions;132
9.1.3.1;2.1. L-functions, and partial zeta functions;132
9.1.3.2;2.2. Integral representations of partial zeta functions I;137
9.1.3.3;2.3. Integral representations of partial zeta functions II;149
9.1.3.4;2.4. Evaluation of special values of Lf(s, øÇ,ñ);164
9.1.3.5;2.5. Evaluation of special values of LftP(s, %det), ?*(s);173
9.1.4;Chapter III. Some applications to the representation of Sp(2n, Fp) in the space of Siegel cusp forms;178
9.1.4.1;3.1. The representatinn ì1ß of Sp(2n, Fp) in the space of cusp forms;178
9.1.4.2;3.2. On the integrals In(IIr(a); k);180
9.1.4.3;3.3. Traces of ìê{á) in the case of degree 4 (n = 2);183
9.1.5;References;185
9.2;Secton VII: Selberg-Ihara's Zeta function for p-adic Discrete Groups;188
9.2.1;0. Introduction;188
9.2.2;1. Groups with axiom;190
9.2.3;2. Tits system and building;193
9.2.4;3. P-adic algebraic groups;196
9.2.5;4 Structure of the discrete subgroups Ã;197
9.2.6;5. j^-conjugacy classes of given degree;201
9.2.7;6. Zeta function Zr(u; p);203
9.2.8;7. Remarks;208
9.2.9;Appendix. Bipartite trees, Hecke algebras, and flowers of groups(by Ki-ichiro Hashimoto);210
9.2.9.1;8. Introduction;210
9.2.9.2;9. Groups with axioms (G, /, I), (G, /, II);212
9.2.9.3;10. Construction of a tree X{qx, q2);214
9.2.9.4;11. Graph of groups over a flower;218
9.2.9.5;12. Tits system and the Hecke algebra;222
9.2.10;References;226
9.3;Secton VIII: Zeta Functions of Finite Graphs and Representations of p-Adic Groups;228
9.3.1;0. Introduction;228
9.3.2;1. Graphs and multigraphs;232
9.3.3;2. Zeta functions of finite multigraphs;237
9.3.4;3. Spectrum of a finite multigraph;243
9.3.5;4. Harmonic functions and the Hodge decomposition;250
9.3.6;5. Representation of C[T19 T2] ; a proof of (3.14;256
9.3.7;6. Representations of /?-adic groups;265
9.3.8;7. Special values of zeta functions;270
9.3.9;8. Miscellaneous results;276
9.3.10;9. Zeta functions of well known families of graphs;283
9.3.11;10. Examples;290
9.3.12;References;296
9.4;Secton IX: Any Irreducible Smooth GL2-Module is Multiplicity Free for any Anisotropie Torus;298
9.4.1;1.;298
9.4.2;2.;298
9.4.3;3.;299
9.4.4;4.;300
9.4.5;5.;300
9.4.6;6.;301
9.4.7;References;302
9.5;Secton X: A Formula for the Dimension of Spaces of Cusp Forms of Weight 1;304
9.5.1;Introduction;304
9.5.2;1. The Selberg eigenspace;304
9.5.3;2. The compact case;306
9.5.4;4. The finite case 2 ( : f 9 - / );313
9.5.5;5. The case of Ã0(ñ);315
9.5.6;References;317
9.6;Secton XI: On Automorphism Groups of Positive Definite Binary Quaternion Hermitian Lattices and New Mass Formula;318
9.6.1;1. New mass formula;321
9.6.2;2. Review on quaternion hermitian lattices and some miscellaneous results;327
9.6.3;3. Classification of conjugacy classes;333
9.6.4;4. Non-existence of D8 and D1Q;338
9.6.5;5. Calculation of local data;340
9.6.6;6. Explicit mass formulae;354
9.6.7;7. Main Theorems;358
9.6.8;References;365
9.7;Secton XII: T-Complexes and Ogata's Zeta Zero Values;368
9.7.1;Introduction;368
9.7.2;1. An equality for a nonsingular complete fan;369
9.7.3;2. J-complexes;371
9.7.4;3. Functors on a graph of cones;374
9.7.5;4. (»-invariant of a J-complex;378
9.7.6;5. Ogata's zeta zero value;379
9.7.7;References;381
9.8;Secton XIII: The Structure of the Icosahedral Modular Group;382
9.8.1;Introduction;382
9.8.2;1. Quotient of P2(C) by SI5.;383
9.8.3;2. Computation of the fundamental group;384
9.8.4;3. The icosahedral modular group;387
9.8.5;References;388
9.9;Secton IVX: Invariants and Hodge Cycles;390
9.9.1;Dedications;390
9.9.2;1. Introduction;390
9.9.3;2. The character-algebra of SX (2, C);395
9.9.4;3. The character-algebra of (a product of) SL2(C);397
9.9.5;4. Chemistry and abelian scheme;398
9.9.6;5. |ËÃ|=2;399
9.9.7;6. Cyclic case, ì=1;402
9.9.8;7. Cyclic case, ì=2ïô ì^2;404
9.9.9;8.;407
9.9.10;References;430
9.10;Secton XV: A Note on Zeta Functions Associated with Certain Prehomogeneous Affine Spaces;432
9.10.1;0. Introduction;432
9.10.2;1. Prehomogeneous affine spaces;433
9.10.3;2. Zeta functions associated with prehomogeneous affine spaces;436
9.10.4;3. Integrals Z(fn(X; X), L, s) and Z*(f*(X; X), £*, s);441
9.10.5;4. The contribution of purely parabolic conjugacy classes to the dimension formula for the space of Jacobi cusp forms;442
9.10.6;References;445
9.11;Secton XVI: On Zeta Functions Associated with the Exceptional Lie Group of Type Ee;446
9.11.1;Introduction;447
9.11.2;Chapter I. Microlocal Analysis of Invariant Hyperfunctions with respect to the Group E6;448
9.11.2.1;1. A realization of the exceptional Lie group of type E6;448
9.11.2.2;2. An invariant holonomic system Wls and its holonomy diagram;451
9.11.2.3;3. Real forms of the prehomogeneous vector spaces and real holonomy diagrams of SK,;452
9.11.2.4;4. Local zeta functions as a solution of the holonomic system SKS;457
9.11.2.5;5. Real principal symbols of the solutions of 2JÎS and their explicit expressions;458
9.11.2.6;6. A relation between real principal symbols on two Lagrangian varieties having an intersection of codimension one;461
9.11.2.7;7. Real principal symbols on the zero section and on the conormal bundle of the origin;463
9.11.2.8;8. The Fourier transforms of local zeta functions;465
9.11.2.9;9. Invariant measures on singular orbits and their Fourier transforms;467
9.11.3;Chapter II. Zeta functions associated with the exceptional group of type E6;471
9.11.3.1;10. Zeta functions and functional equations;472
9.11.3.2;11. Computations of residues;476
9.11.4;References;479
9.12;Secton XVII: On Functional Equations of Zeta Distributions;482
9.12.1;Introduction;482
9.12.2;1. Preliminaries;484
9.12.3;2. Fourier transforms of/?-adic complex powers;488
9.12.4;3. Calculation of JT-matrices;495
9.12.5;4. Zeta functions as distribution;509
9.12.6;References;523
9.13;Secton XVIII: Multi-Tensors of Differential Forms on the Hubert Modular Variety and on Its Subvarieties, II;526
9.13.1;1. Preliminaries;527
9.13.2;2. Résumé of [8];530
9.13.3;3. Vanishing order;530
9.13.4;4. Modular groups;532
9.13.5;5. Irreducibility;533
9.13.6;6. Key proposition;535
9.13.7;7. Proof in case I;537
9.13.8;8. Proof in case II;538
9.13.9;9. Proof of Theorem;538
9.13.10;References;539