U (z,s, N and v, (z,s, N fulfill (i) - (iii) in a > 0 unless g = 0 , N = 0 and s = 1/2 . With the exception of the case just mentioned they are also linearly independent in view of (4.15), (4.16), (4.21), (4.22) and (4.24) - (4.27).

Author: Salahoddin ShokranianPublish On: 2006-11-14

The purpose of this chapter is to explain the geometric expansion of Arthur's trace formula. This, as we shall see, is based on the local study of weighted orbital integrals, predicted by Langlands. Weighted orbital integrals were ...

Author: Salahoddin Shokranian

Publisher: Springer

ISBN: 9783540466598

Category: Mathematics

Page: 99

View: 125

This book based on lectures given by James Arthur discusses the trace formula of Selberg and Arthur. The emphasis is laid on Arthur's trace formula for GL(r), with several examples in order to illustrate the basic concepts. The book will be useful and stimulating reading for graduate students in automorphic forms, analytic number theory, and non-commutative harmonic analysis, as well as researchers in these fields. Contents: I. Number Theory and Automorphic Representations.1.1. Some problems in classical number theory, 1.2. Modular forms and automorphic representations; II. Selberg's Trace Formula 2.1. Historical Remarks, 2.2. Orbital integrals and Selberg's trace formula, 2.3.Three examples, 2.4. A necessary condition, 2.5. Generalizations and applications; III. Kernel Functions and the Convergence Theorem, 3.1. Preliminaries on GL(r), 3.2. Combinatorics and reduction theory, 3.3. The convergence theorem; IV. The Ad lic Theory, 4.1. Basic facts; V. The Geometric Theory, 5.1. The JTO(f) and JT(f) distributions, 5.2. A geometric I-function, 5.3. The weight functions; VI. The Geometric Expansionof the Trace Formula, 6.1. Weighted orbital integrals, 6.2. The unipotent distribution; VII. The Spectral Theory, 7.1. A review of the Eisenstein series, 7.2. Cusp forms, truncation, the trace formula; VIII.The Invariant Trace Formula and its Applications, 8.1. The invariant trace formula for GL(r), 8.2. Applications and remarks

Press & Univ. of Tokyo Press, 1975 [16] A. Good: Local Analysis of Selberg's Trace Formula; Lecture Notes in Math. 1040, Springer-Verlag, 1983 [17] H. Grauert, R. Remmert: Coherent Analytic Sheaves, Grundl. math. Wiss.

Author: Roelof W. Bruggeman

Publisher: Springer Science & Business Media

ISBN: 9783034603362

Category: Mathematics

Page: 318

View: 985

Automorphic forms on the upper half plane have been studied for a long time. Most attention has gone to the holomorphic automorphic forms, with numerous applications to number theory. Maass, [34], started a systematic study of real analytic automorphic forms. He extended Hecke’s relation between automorphic forms and Dirichlet series to real analytic automorphic forms. The names Selberg and Roelcke are connected to the spectral theory of real analytic automorphic forms, see, e. g. , [50], [51]. This culminates in the trace formula of Selberg, see, e. g. , Hejhal, [21]. Automorphicformsarefunctionsontheupperhalfplanewithaspecialtra- formation behavior under a discontinuous group of non-euclidean motions in the upper half plane. One may ask how automorphic forms change if one perturbs this group of motions. This question is discussed by, e. g. , Hejhal, [22], and Phillips and Sarnak, [46]. Hejhal also discusses the e?ect of variation of the multiplier s- tem (a function on the discontinuous group that occurs in the description of the transformation behavior of automorphic forms). In [5]–[7] I considered variation of automorphic forms for the full modular group under perturbation of the m- tiplier system. A method based on ideas of Colin de Verdi` ere, [11], [12], gave the meromorphic continuation of Eisenstein and Poincar ́ e series as functions of the eigenvalue and the multiplier system jointly. The present study arose from a plan to extend these results to much more general groups (discrete co?nite subgroups of SL (R)).

[Go] A. Good, Local Analysis of Selberg's Trace Formula. ... [Ha-Ra] G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. ... [He1] D. A. Hejhal, The Selberg Trace Formula for PSL(2,R).

Author: Henryk Iwaniec

Publisher: American Mathematical Society, Revista Matemática Iberoamericana (RMI), Madrid, Spain

ISBN: 9781470466220

Category: Mathematics

Page: 220

View: 750

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style. The first edition of this book was an underground classic, both as a textbook and as a respected source for results, ideas, and references. Iwaniec treats the spectral theory of automorphic forms as the study of the space of $L^2$ functions on the upper half plane modulo a discrete subgroup. Key topics include Eisenstein series, estimates of Fourier coefficients, Kloosterman sums, the Selberg trace formula and the theory of small eigenvalues. Henryk Iwaniec was awarded the 2002 Cole Prize for his fundamental contributions to number theory.

A. Ghosh , On the Riemann zeta function - mean value theorems and the distribution of S ( t ) | , J. Numb . Thy . ... A. Good , Local analysis of Selberg's trace formula , LNM 1040 ( Springer Verlag , Berlin , 1983 ) .

Author: N.E. Hurt

Publisher: Springer Science & Business Media

ISBN: 0792344596

Category: Mathematics

Page: 362

View: 628

4. 2 Variance of Quantum Matrix Elements. 125 4. 3 Berry's Trick and the Hyperbolic Case 126 4. 4 Nonhyperbolic Case . . . . . . . 128 4. 5 Random Matrix Theory . . . . . 128 4. 6 Baker's Map and Other Systems 129 4. 7 Appendix: Baker's Map . . . . . 129 5 Error Terms 133 5. 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 133 5. 2 The Riemann Zeta Function in Periodic Orbit Theory 135 5. 3 Form Factor for Primes . . . . . . . . . . . . . . . . . 137 5. 4 Error Terms in Periodic Orbit Theory: Co-compact Case. 138 5. 5 Binary Quadratic Forms as a Model . . . . . . . . . . . . 139 6 Co-Finite Model for Quantum Chaology 141 6. 1 Introduction. . . . . . . . 141 6. 2 Co-finite Models . . . . . 141 6. 3 Geodesic Triangle Spaces 144 6. 4 L-Functions. . . . . . . . 145 6. 5 Zelditch's Prime Geodesic Theorem. 146 6. 6 Zelditch's Pseudo Differential Operators 147 6. 7 Weyl's Law Generalized 148 6. 8 Equidistribution Theory . . . . . . . . . 150 7 Landau Levels and L-Functions 153 7. 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 153 7. 2 Landau Model: Mechanics on the Plane and Sphere. 153 7. 3 Landau Model: Mechanics on the Half-Plane 155 7. 4 Selberg's Spectral Theorem . . . . . . . . . . . 157 7. 5 Pseudo Billiards . . . . . . . . . . . . . . . . . 158 7. 6 Landau Levels on a Compact Riemann Surface 159 7. 7 Automorphic Forms . . . . . 160 7. 8 Maass-Selberg Trace Formula 162 7. 9 Degeneracy by Selberg. . . . 163 7. 10 Hecke Operators . . . . . . . 163 7. 11 Selberg Trace Formula for Hecke Operators 167 7. 12 Eigenvalue Statistics on X . . . . 169 7. 13 Mesoscopic Devices. . . . . . . . 170 7. 14 Hall Conductance on Leaky Tori 170 7.

[13] Th.Estermann, On Kloosterman's sum, Mathematika 8 (1961) 83-86. [14] E.Freitag, Hilbert Modular Forms, Springer Verlag, 1980. [15] A.Good, Local analysis of Selberg's trace formula, Lect. Notes in Math. 1040, Springer-Verlag 1984.

Author: Roelof W. Bruggeman

Publisher: American Mathematical Soc.

ISBN: 9780821842027

Category: Mathematics

Page: 81

View: 204

The authors prove a general form of the sum formula $\mathrm{SL}_2$ over a totally real number field. This formula relates sums of Kloosterman sums to products of Fourier coefficients of automorphic representations. The authors give two versions: the spectral sum formula (in short: sum formula) and the Kloosterman sum formula. They have the independent test function in the spectral term, in the sum of Kloosterman sums, respectively.

Author: Clay Mathematics Institute. Summer SchoolPublish On: 2005

The transfer of 7 to a ' is achieved by the trace formula . ... Now , Jacquet and Shalika JS show that for tv generic the local Rankin - Selberg L - function of Ty with its contragredient hus ( 11 ) L ( s , Ty X Tv ) det ( I – a ( Tv ) ...

Author: Clay Mathematics Institute. Summer School

Publisher: American Mathematical Soc.

ISBN: 082183844X

Category: Mathematics

Page: 708

View: 472

Langlands program proposes fundamental relations that tie arithmetic information from number theory and algebraic geometry with analytic information from harmonic analysis and group representations. This title intends to provide an entry point into this exciting and challenging field.

1983 Vol. 1040: A Good, Local Analysis of Selberg's Trace Formula. Ill, 128 pages. 1983. Vol. 1041: Lie Group Representations II. Proceedings 1982-1983. Edited by R. Herb, S. Kudla, R. Lipsman and J. Rosenberg. lx, 340 pages. 1984. Vol.