Jorgenson / Lang The Heat Kernel and Theta Inversion on SL2(C)

1. Auflage 2009
ISBN: 978-0-387-38032-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 319 Seiten

Reihe: Springer Monographs in Mathematics

The purpose of this text is to provide a complete, self-contained development of the trace formula and theta inversion formula for SL(2,Z[i])\SL(2,C). Unlike other treatments of the theory, the approach taken here is to begin with the heat kernel on SL(2,C) associated to the invariant Laplacian, which is derived using spherical inversion. The heat kernel on the quotient space SL(2,Z[i])\SL(2,C) is gotten through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the Gauss transform.

Autoren/Hrsg.

Jorgenson, Jay

Lang, Serge

Inhaltsverzeichnis

1;Preface;5
2;Contents;6
3;Introduction;10
3.1;Spherical Inversion;11
3.2;Fourier and Eigenfunction Expansions;11
3.3;Gaussians and the Trace Formula;12
3.4;The General Path to Theta Inversion;12
3.5;Zetas;14
3.6;Ladders;15
3.7;Connection with Analytic Number Theory;15
3.8;Connections with Geometry;16
3.9;Towers of Ladders;18
4;Part I: Gaussians, Spherical Inversion, and the Heat Kernel;19
4.1;Chapter 2;20
4.1.1;Spherical Inversion on SL2(C);20
4.1.1.1;1.1 The Iwasawa Decomposition, Polar Decomposition,and Characters;22
4.1.1.1.1;1.1.1 Characters;23
4.1.1.1.2;1.1.2 K-bi-invariant Functions;24
4.1.1.2;1.2 Haar Measures;26
4.1.1.3;1.3 The Harish Transform and the Orbital Integral;30
4.1.1.4;1.4 The Mellin and Spherical Transforms;32
4.1.1.5;1.5 Computation of the Orbital Integral;35
4.1.1.6;1.6 Gaussians on G and Their Spherical Transform;39
4.1.1.6.1;1.6.1 The Polar Height;42
4.1.1.7;1.7 The Polar Haar Measure and Inversion;44
4.1.1.8;1.8 Point-Pair Invariants, the Polar Height,and the Polar Distance;48
4.2;Chapter 3;51
4.2.1;The Heat Gaussian and Heat Kernel;51
4.2.1.1;2.1 Dirac Families of Gaussians;51
4.2.1.1.1;2.1.1 Scaling;52
4.2.1.1.2;2.1.2 Decay Property;54
4.2.1.2;2.2 Convolution, Semigroup, and Approximations Properties;55
4.2.1.2.1;2.2.1 Approximation Properties;57
4.2.1.3;2.3 Complexifying t and the Null Space of Heat Convolution;60
4.2.1.4;2.4 The Casimir Operator;61
4.2.1.4.1;2.4.1 Scaling;68
4.2.1.5;2.5 The Heat Equation;69
4.2.1.5.1;2.5.1 Scaling;71
4.3;Chapter 4;73
4.3.1;QED, LEG, Transpose, and Casimir;73
4.3.1.1;3.1 Growth and Decay, QEDinfinity and LEGinfinity;73
4.3.1.2;3.2 Casimir, Transpose, and Harmonicity;76
4.3.1.3;3.3 DUTIS;82
4.3.1.4;3.4 Heat and Casimir Eigenfunctions;84
4.3.1.5;Onward;87
5;Part II: Enter cap gamma: The General Trace Formula;88
5.1;Chapter 5;89
5.1.1;Convergence and Divergence of the Selberg Trace;89
5.1.1.1;4.1 The Hermitian Norm;90
5.1.1.2;4.2 Divergence for Standard Cuspidal Elements;93
5.1.1.2.1;4.2.1 Cuspidal and Parabolic Subgroups;93
5.1.1.3;4.3 Convergence for the Other Elements of Cap Gamma;96
5.1.1.4;What Next?;99
5.2;Chapter 6;100
5.2.1;The Cuspidal and Noncuspidal Traces;100
5.2.1.1;5.1 Some Group Theory;101
5.2.1.1.1;5.1.1 Conjugacy Classes;104
5.2.1.2;5.2 The Double Trace and its Decomposition;105
5.2.1.3;5.3 Explicit Determination of the Noncuspidal Terms;109
5.2.1.3.1;5.3.1 The Volume Computation;110
5.2.1.3.2;5.3.2 The Orbital Integral;111
5.2.1.4;5.4 Cuspidal Conjugacy Classes;113
6;Part III: The Heat Kernal on cap gamma\G/K;118
6.1;Chapter 7;119
6.1.1;The Fundamental Domain;119
6.1.1.1;6.1 SL2(C) and the Upper Half-Space H3;120
6.1.1.2;6.2 Fundamental Domain and cap gamma infinity;123
6.1.1.3;6.3 Finiteness Properties;126
6.1.1.4;6.4 Uniformities in Lemma 6.2.3;132
6.1.1.5;6.5 Integration on cap gamma\G/K;133
6.1.1.6;6.6 Other Fundamental Domains;135
6.2;Chapter 8;137
6.2.1;Cap Gamma-Periodization of the Heat Kernel;137
6.2.1.1;7.1 The Basic Estimate;137
6.2.1.1.1;7.1.1 Convolution;138
6.2.1.2;7.2 Heat Convolution and Eigenfunctions on Cap Gamma\G/K;142
6.2.1.3;7.3 Casimir on Cap Gamma\G/K;147
6.2.1.4;7.4 Measure-Theoretic Estimate for Convolution on Cap Gamma\G;149
6.2.1.5;7.5 Asymptotic Behavior of KCap Gammat for t.Infinity;151
6.3;Chapter 9;153
6.3.1;Heat Kernel Convolution on L2cusp (cap gamma\G/K);153
6.3.1.1;8.1 General Criteria for Compactness;154
6.3.1.2;8.2 Estimates for the (cap gamma cap gamma infinity)-Periodization;157
6.3.1.3;8.3 Fourier Series for the cap gamma"U , cap gamma infinity-Periodizations of Gaussians;159
6.3.1.3.1;8.3.1 Preliminaries: The cap gamma"U and cap gamma infinity-Periodizations;159
6.3.1.4;8.4 The Convolution Cuspidal Estimate;162
6.3.1.5;8.5 Application to the Heat Kernel;163
7;Part IV: Fourier - Eisenstein Eigenfunction Expansions;166
7.1;Chapter 10;167
7.1.1;The Tube Domain for cap gamma infinity;167
7.1.1.1;9.1 Differential-Geometric Aspects;167
7.1.1.2;9.2 The Tube of FR and its Boundary Relation with deltaFR;169
7.1.1.3;9.3 The F-Normalizer of cap gamma;171
7.1.1.4;9.4 Totally Geodesic Surface in H3;172
7.1.1.4.1;9.4.1 The Half-Plane H2j;173
7.1.1.5;9.5 Some Boundary Behavior of F in H3 Under cap gamma;175
7.1.1.5.1;9.5.1 The Faces Bi of F and their Boundaries;175
7.1.1.5.2;9.5.2 H-triangle;176
7.1.1.5.3;9.5.3 Isometries of F;178
7.1.1.6;9.6 The Group ˜ and a Basic Boundary Inclusion;180
7.1.1.7;9.7 The Set I, its Boundary Behavior, and the Tube T;181
7.1.1.8;9.8 Tilings;182
7.1.1.8.1;9.8.1 Coset Representatives;184
7.1.1.9;9.9 Truncations;185
7.2;Chapter 11;190
7.2.1;The Cap GammaU\U-Fourier Expansion of Eisenstein Series;190
7.2.1.1;10.1 Our Goal: The Eigenfunction Expansion;190
7.2.1.2;10.2 Epstein and Eisenstein Series;192
7.2.1.3;10.3 The K-Bessel Function;196
7.2.1.3.1;10.3.1 Gamma Function Identities;198
7.2.1.3.2;10.3.2 Differential and Difference Relations;200
7.2.1.4;10.4 Functional Equation of the Dedekind Zeta Function;201
7.2.1.5;10.5 The Bessel–Fourier Cap GammaU\U-Expansion of Eisenstein Series;205
7.2.1.5.1;10.5.1 The Constant Term;210
7.2.1.6;10.6 Estimates in Vertical Strips;212
7.2.1.7;10.7 The Volume–Residue Formula;215
7.2.1.8;10.8 The Integral over F and Orthogonalities;217
7.3;Chapter 12;222
7.3.1;Adjointness Formulaand the cap gamma\G-Eigenfunction Expansion;222
7.3.1.1;11.1 Haar Measure and the Mellin Transform;223
7.3.1.1.1;11.1.1 Appendix on Fourier Inversion;225
7.3.1.2;11.2 Adjointness Formula and the Constant Term;228
7.3.1.2.1;11.2.1 Adjointness Formula;229
7.3.1.3;11.3 The Eisenstein Coefficient E * f and the Expansion for f member of C infinity c (cap gamma\G/K);231
7.3.1.4;11.4 The Heat Kernel Eigenfunction Expansion;236
8;Part V: The Eisenstein - Cuspidal Affair;240
8.1;Chapter 13;241
8.1.1;The Eisenstein Y-Asymptotics;241
8.1.1.1;12.1 The Improper Integral of Eigenfunction Expansion over Cap Gamma\G;241
8.1.1.1.1;12.1.1 L2-Cuspidal Trace;242
8.1.1.2;12.2 Green’s Theorem on FLessthen EqualY;245
8.1.1.3;12.3 Application to Eisenstein Functions;249
8.1.1.4;12.4 The Constant-Term Integral Asymptotics;253
8.1.1.4.1;12.4.1 Appendix;255
8.1.2;12.5 The Nonconstant-Term Error Estimate;256
8.2;Chapter 14;258
8.2.1;The Cuspidal Trace Y-Asymptotics;258
8.2.1.1;13.1 The Nonregular Cuspidal Integral over F less than equal to Y;259
8.2.1.2;13.2 Asymptotic Expansion of the Nonregular Cuspidal Trace;264
8.2.1.3;13.3 The Regular Cuspidal Integral over F lass than equal to Y;269
8.2.1.4;13.4 Nonspecial Regular Cuspidal Asymptotics;272
8.2.1.5;13.5 Action of the Special Subset;274
8.2.1.6;13.6 Special Regular Cuspidal Asymptotics;277
8.3;Chapter 15;284
8.3.1;Analytic Evaluations;284
8.3.1.1;14.1 Partial Sums Asymptotics for ZetaQ and the Euler Constant;284
8.3.1.2;14.2 Estimates Using Lattice-Point Counting;287
8.3.1.3;14.3 Partial-Sums Asymptotics for ZetaQ(i) and the Euler Constant;289
8.3.1.4;14.4 The Hurwitz Constant;293
8.3.1.4.1;14.4.1 The Complex Case, with Z[i];294
8.3.1.4.2;14.4.2 Average of the Hurwitz Constant;295
8.3.1.5;14.5 f 80 f phi(r)rh(r)dr when phi = gt;298
8.3.1.6;14.6 Evaluation of C PrimeYo and C1;300
8.3.1.7;14.7 The Theta Inversion Formula;305
9;References;307
10;Index;312