Representation Theory And Automorphic Forms
by Kobayashi, Toshiyuki; Schmid, Wilfried; Yang, Jae-hyunEdition: 1st
Format: Hardcover
Pub. Date: 2007-11-01
Publisher(s): Birkhauser
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Summary
This volume uses a unified approach to representation theory and automorphic forms. The invited papers, written by leading mathematicians, track recent progress in the ever expanding fields of representation theory and automorphic forms, and their association with number theory and differential geometry. Representation theory relates to number theory through Langlands' conjecture, which illuminates the deep properties of primes in number fields. The Langlands program is further analyzed in this work through automorphic functions and automorphic distributions. The relation between representation theory and differential geometry is explored via the Dirac cohomology of Index theory. Also discussed are the subjects of modular forms and harmonic analysis. The volume also branches off from representation theory into self-dual representations, and includes work from the non-standard geometric view of viable action on complex manifolds towards multiplicity-free representation theory. Both graduate students and researchers will find inspiration in this volume.
Table of Contents
| Preface | p. vii |
| Irreducibility and Cuspidality | p. 1 |
| Preliminaries | p. 5 |
| The first step in the proof | p. 15 |
| The second step in the proof | p. 16 |
| Galois representations attached to regular, selfdual cusp forms on GL(4) | p. 18 |
| Two useful lemmas on cusp forms on GL(4) | p. 20 |
| Finale | p. 21 |
| References | p. 25 |
| On Liftings of Holomorphic Modular Forms | p. 29 |
| Basic facts | p. 29 |
| Fourier coefficients of the Eisenstein series | p. 30 |
| Kohnen plus space | p. 32 |
| Lifting of cusp forms | p. 33 |
| Outline of the proof | p. 34 |
| Relation to the Saito-Kurokawa lifts | p. 35 |
| Hermitian modular forms and hermitian Eisensetein series | p. 37 |
| The case m = 2n + 1 | p. 39 |
| The case m = 2n | p. 40 |
| L-functions | p. 40 |
| The case m = 2 | p. 41 |
| References | p. 42 |
| Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs | p. 45 |
| Introduction and statement of main results | p. 45 |
| Main machinery from complex geometry | p. 56 |
| Proof of Theorem A | p. 61 |
| Proof of Theorem C | p. 68 |
| Uniformly bounded multiplicities - Proof of Theorems B and D | p. 70 |
| Counterexamples | p. 77 |
| Finite-dimensional cases - Proof of Theorems E and F | p. 83 |
| Generalization of the Hua-Kostant-Schmid formula | p. 89 |
| Appendix: Associated bundles on Hermitian symmetric spaces | p. 103 |
| References | p. 105 |
| The Rankin-Selberg Method for Automorphic Distributions | p. 111 |
| Introduction | p. 111 |
| Standard L-functions for SL(2) | p. 115 |
| Pairings of automorphic distributions | p. 121 |
| The Rankin-Selberg L-function for GL(2) | p. 128 |
| Exterior Square on GL(4) | p. 137 |
| References | p. 149 |
| Langlands Functoriality Conjecture and Number Theory | p. 151 |
| Introduction | p. 151 |
| Modular forms, Galois representations and Artin L-functions | p. 152 |
| Lattice point problems and the Selberg conjecture | p. 156 |
| Ramanujan conjecture for Maass forms | p. 158 |
| Sato-Tate conjecture | p. 159 |
| Functoriality for symmetric powers | p. 161 |
| Functoriality for classical groups | p. 163 |
| Ramanujan conjecture for classical groups | p. 164 |
| The method | p. 166 |
| References | p. 169 |
| Discriminant of Certain K3 Surfaces | p. 175 |
| Introduction - Discriminant of elliptic curves | p. 175 |
| K3 surfaces with involution and their moduli spaces | p. 178 |
| Automorphic forms on the moduli space | p. 180 |
| Equivariant analytic torsion and 2-elementary K3 surfaces | p. 182 |
| The Borcherds products | p. 184 |
| Borcherds products for odd unimodular lattices | p. 186 |
| K3 surfaces of Matsumoto-Sasaki-Yoshida | p. 188 |
| Discriminant of quartic surfaces | p. 200 |
| References | p. 209 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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