Automorphic Representations of Low Rank Groups.
Title:
Automorphic Representations of Low Rank Groups.
Author:
Flicker, Yuval Z.
ISBN:
9789812773623
Personal Author:
Physical Description:
1 online resource (499 pages)
Contents:
CONTENTS -- PREFACE -- PART 1. ON THE SYMMETRIC SQUARE LIFTING -- INTRODUCTION -- I. FUNCTORIALITY AND NORMS -- I.1 Hecke algebra -- I.2 Norms -- I.3 Local lifting -- I.4 Orthogonality -- II. ORBITAL INTEGRALS -- II.1 Fundamental lemma -- II.2 Differential forms -- II.3 Matching orbital integrals -- II.4 Germ expansion -- III. TWISTED TRACE FORMULA -- III.1 Geometric side -- III.2 Analytic side -- III.3 Trace formulae -- IV. TOTAL GLOBAL COMPARISON -- IV.1 The comparison -- IV.2 Appendix: Mathematica program -- V. APPLICATIONS OF A TRACE FORMULA -- V.1 Approximation -- V.2 Main theorems -- V.3 Characters and genericity -- VI. COMPUTATION OF A TWISTED CHARACTER -- VI.1 Proof of theorem, anisotropic case -- VI.2 Proof of theorem, isotropic case -- PART 2. AUTOMORPHIC REPRESENTATIONS OF THE UNITARY GROUP U(3,E/F) -- INTRODUCTION -- 1. Functorial overview -- 2. Statement of results -- I. LOCAL THEORY -- I.1 Conjugacy classes -- I.2 Orbital integrals -- I.3 Fundamental lemma -- I.4 Admissible representations -- I.5 Representations of U(2,1 -- C/R) -- I.6 Fundamental lemma again -- II. TRACE FORMULA -- II.1 Stable trace formula -- II.2 Twisted trace formula -- II.3 Restricted comparison -- II.4 Trace identity -- II.5 The σ-endo-lifting e' -- II.6 The quasi-endo-lifting e -- II.7 Unitary symmetric square -- III. LIFTINGS AND PACKETS -- III.l Local identity -- III.2 Separation -- III.3 Specific lifts -- III.4 Whittaker models and twisted characters -- III.5 Global lifting -- III.6 Concluding remarks -- PART 3. ZETA FUNCTIONS OF SHIMURA VARIETIES OF U(3) -- INTRODUCTION -- 1. Statement of results -- 2. The Zeta function -- I. PRELIMINARIES -- I.1 The Shimura variety -- I.2 Decomposition of cohomology -- I.3 Galois representations -- II. AUTOMORPHIC REPRESENTATIONS -- II.1 Stabilization and the test function.
II.2 Functorial overview of basechange for U(3) -- II.3 Local results on basechange for U(3) -- II.4 Global results on basechange for U(3) -- II.5 Spectral side of the stable trace formula -- II.6 Proper endoscopic group -- III. LOCAL TERMS -- III.1 The reflex field -- III.2 The representation of the dual group -- III.3 Local terms at p -- III.4 The eigenvalues at p -- III.5 Terms at p for the endoscopic group -- IV. REAL REPRESENTATIONS -- IV.1 Representations of the real GL(2) -- IV.2 Representations of U(2,1) -- IV.3 Finite-dimensional representations -- V. GALOIS REPRESENTATIONS -- V.1 Stable case -- V.2 Unstable case -- V.3 Nontempered case -- REFERENCES -- INDEX.
Abstract:
The area of automorphic representations is a natural continuation of studies in number theory and modular forms. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called "liftings". This book concentrates on two initial examples: the symmetric square lifting from SL(2) to PGL(3), reflecting the 3-dimensional representation of PGL(2) in SL(3); and basechange from the unitary group U(3, E/F) to GL(3, E), [E : F] = 2. The book develops the technique of comparison of twisted and stabilized trace formulae and considers the "Fundamental Lemma" on orbital integrals of spherical functions. Comparison of trace formulae is simplified using "regular" functions and the "lifting" is stated and proved by means of character relations. This permits an intrinsic definition of partition of the automorphic representations of SL(2) into packets, and a definition of packets for U(3), a proof of multiplicity one theorem and rigidity theorem for SL(2) and for U(3), a determination of the self-contragredient representations of PGL(3) and those on GL(3, E) fixed by transpose-inverse-bar. In particular, the multiplicity one theorem is new and recent. There are applications to construction of Galois representations by explicit decomposition of the cohomology of Shimura varieties of U(3) using Deligne's (proven) conjecture on the fixed point formula. Sample Chapter(s). Chapter 1: Functoriality and Norms (963 KB). Contents: On the Symmetric Square Lifting: Functoriality and Norms; Orbital Integrals; Twisted Trace Formula; Total Global Comparison; Applications of a Trace Formula; Computation of a Twisted Character; Automorphic Representations of the Unitary Group U(3, E/F): Local Theory; Trace
Formula; Liftings and Packets; Zeta Functions of Shimura Varieties of U(3): Automorphic Representations; Local Terms; Real Representations; Galois Representations. Readership: Graduate students and researchers in number theory, algebra and representation theory.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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