Relative Trace Formulas (Simons Symposia)

Description

A series of three symposia took place on the topic of trace formulas, each with an accompanying proceedings volume. The present volume is the third and final in this series and focuses on relative trace formulas in relation to special values of L-functions, integral representations, arithmetic cycles, theta correspondence and branching laws. The first volume focused on Arthur’s trace formula, and the second volume focused on methods from algebraic geometry and representation theory. The three proceedings volumes have provided a snapshot of some of the current research, in the hope of stimulating further research on these topics. The collegial format of the symposia allowed a homogeneous set of experts to isolate key difficulties going forward and to collectively assess the feasibility of diverse approaches. Preface Introduction Contents Archimedean Theory and ε-Factors for the Asai Rankin-Selberg Integrals 1 Introduction 1.1 General Notation 2 Preliminaries 2.1 Groups 2.2 Topological Vector Spaces 2.3 Representations 2.4 Uniform Bounds for Matrix Coefficients 2.5 Harish-Chandra Schwartz Spaces of Whittaker Functions 2.6 Uniform Bounds for Families of Whittaker Functions 2.7 Application to the Existence of Good Sectionsfor Whittaker Models 2.8 Holomorphic Continuation of Certain Functions 3 Zeta Integrals and Statement of the Main Theorems 3.1 Notation 3.2 Asai L-Functions and Epsilon Factors 3.3 Definition and Convergence of the Local Zeta Integrals 3.4 Local Functional Equation: The Split Case 3.5 Local Functional Equation: The Inert Case 3.6 Unramified Computation 3.7 Global Zeta Integrals and Their Functional Equation 3.8 A Globalization Result 3.9 Proof of Theorems 3.5.1 and 3.5.2 in the Nearly Tempered Case 3.10 Proof of Theorems 3.5.1 and 3.5.2 in the General Case References The Relative Trace Formula in Analytic Number Theory 1 The Poisson Summation Formula 2 Harmonic Analysis on Upper Half Plane 3 Applications 3.1 Statistics of eigenvalues 3.2 Arithmetic Applications: Kloosterman Sums,Primes and Arithmetic Functions 3.3 Applications to L-Functions I 3.4 Applications to L-Functions II: Subconvexityand Equidistribution 4 Other Groups 4.1 The Group GL(3) 4.2 The Group GSp(4) 4.3 Groups of Unbounded Rank References Dimensions of Automorphic Representations, L-Functions and Liftings 1 Introduction 2 The Dimension Equation 3 Examples of the Dimension Equation 4 Extending the Dimension Equation 5 Examples of the Extended Dimension Equation 6 Integral Kernels and the Dimension Equation 7 Doubling Integrals and Integral Kernels 8 An Example 9 Local Analogues References Relative Character Identities and Theta Correspondence 1 Introduction 2 Spectral Decomposition à la Bernstein 2.1 Direct Integral Decompositions 2.2 Pointwise-Defined and Fine Morphisms 2.3 The Maps ασ(ω) and βσ(ω) 2.4 Harish-Chandra-Schwartz Space of X 2.5 Inner Product 2.6 Pointwise Spectral Decomposition 3 Basic Plancherel Theorems 3.1 Harish-Chandra-Plancherel Theorem 3.2 Whittaker-Plancherel Theorem 3.3 Continuity Properties 4 GL2 and SL2 4.1 Measures on F and F 4.2 The Group GL2 4.3 The Group SL2 4.4 Harish-Chandra-Schwartz Space 5 Theta Correspondence 5.1 Weil Representation 5.2 Smooth Theta Correspondence 5.3 Doubling Zeta Integral 5.4 L2-Theta Correspondence 5.5 The Maps Aσ and Bθ(σ) 6 Periods 6.1 Transfer of Periods 6.2 Unitary Structure on L2(Xa) 6.3 Spectral Decomposition of L2(Xa) 6.4 A Commutative Diagram 7 Relative Characters 7.1 Relative Characters 7.2 Alternative Incarnation 7.3 Jσ as a Relative Character 7.4 Space of Orbital Integrals 8 Transfer of Test Functions 8.1 A Correspondence of Test Functions 8.2 Basic Function and Fundamental Lemmas 8.3 Relation with Adjoint L-Factors 8.4 Orbital Integrals 9 Relative Character Identities 9.1 Proof of Theorem 9.1 9.2 Some Consequences 10 Local L-Factor 10.1 Unramified Setting 10.2 The L-Factor L#X 10.3 Some Constants 10.4 Key Computations 10.5 General Case 11 Transfer in Geometric Terms 12 Factorization of Global Periods 12.1 Tamagawa Measures 12.2 Automorphic Forms 12.3 Global Periods 12.4 The Maps αAut and βAut 12.5 Global Relative Characters 12.6 Quadratic Spaces and Hyperboloids 12.7 Global Weil Representation 12.8 Global Theta Lifting 12.9 The Maps AAut and B()Aut 12.10 Global Transfer of Periods 12.11 Decompositions of Unitary Representations 12.12 Adelic Periods 12.13 Comparison of Automorphic and Adelic Periods 12.14 Global Result 12.15 Avoiding Rallis Inner Product 12.16 Global Relative Character Identity 12.17 End Remarks References Incoherent Definite Spaces and Shimura Varieties 1 Introduction 2 Local and Global Orthogonal Spaces 3 Local and Global Hermitian Spaces 4 Incoherent Definite Orthogonal and Hermitian Data 5 Incoherent Data in Codimension One 6 Orthogonal Lattices 7 Hermitian Lattices 8 The Homogeneous Spaces X, Y, and Z 9 The Fibers of the Map f: Y →Z 10 Shimura Varieties Associated to Incoherent Definite Data 11 The Special Locus Modulo p 12 A Mass Formula References Shimura Varieties for Unitary Groups and the Doubling Method 1 Introduction and Overview 2 Shimura Varieties for Unitary Groups 2.1 Notation and Conventions 2.2 Shimura Data 2.5 Measures and Discriminants 3 Coherent Cohomology 3.1 Automorphic Vector Bundles 3.2 Coherent Cohomology and Period Invariants 3.4.1 Change of Polarization 3.5.1 Period Invariants, n = 1 4 Holomorphic Eisenstein Series 4.1 The Doubled Group and Its Variety 4.1.1 Tube Domains 4.3 Induced Representations and Eisenstein Series 4.4 Automorphic Line Bundles on the Doubled Group 4.5 Automorphic Forms on the Point Boundary Shimura Variety 4.9 Holomorphic Eisenstein Series: Absolutely Convergent Case 4.11 Holomorphic Eisenstein Series: Application of the Siegel-Weil Formula 5 Differential Operators 5.1 Parameters for Differential Operators 5.2 Parameters for Nearly Holomorphic Eisenstein Series 6 The Doubling Integral 6.1 Zeta Integrals 6.3 Review of the Local Theory 6.5 The Archimedean Theory 7 Main Results 8 Unitary and Similitude Periods References Bessel Descents and Branching Problems 1 Introduction 2 Bessel Modules and Bessel Periods 2.1 Discrete Spectrum 2.2 Bessel Modules and Bessel Periods 2.3 A Family of Global Zeta Integrals 3 Bessel Periods and Global Arthur Parameters 3.1 π Has a Generic Global Arthur Parameter 3.2 σ Has a Generic Global Arthur Parameter 3.3 Reformulation in Terms of L-Functions 4 Theory of Twisted Automorphic Descents 5 Branching Problem: Cuspidal Case 6 Reciprocal Branching Problem 6.1 σ Has a Generic Global Arthur Parameter 6.2 Relations with the Global Gan–Gross–Prasad Conjecture References Distinguished Representations of SO(n+1,1) SO(n,1), Periods and Branching Laws 1 Introduction 2 Representations with Nonsingular Integral Infinitesimal Character 2.1 Notation 2.2 Principal Series Representations 2.2.1 The Set A 2.2.2 Classification of the Set A for n Even 2.2.3 Classification of the Set A for n Odd 2.3 The Height of Representations in A 2.4 Signatures of Representations in A 2.4.1 Tempered Representations 2.4.2 Nontempered Representations 2.4.3 Hasse and Standard Sequences 2.4.4 The θ-stable and Enhanced θ-stable Parameters of Irreducible Representations 2.4.5 The Hasse and Standard Sequences in θ-stable Parameters 3 The Restriction of Representations of SO(n+1,1) in Ato the Subgroup SO(n,1) 3.1 Branching Laws for Finite-Dimensional Representations 3.2 Symmetry Breaking Operators 3.3 Branching Laws for Representations in A: First Formulation 3.4 Branching Laws for Representations in A: Second Formulation 3.5 Branching Laws for Representations in A: Third Formulation 4 Gross–Prasad Conjectures for Tempered Representations of (SO(n+1,1), SO(n,1)) 5 Distinguished Representations and Periods 5.1 Periods 5.2 Distinguished Representations 6 Bilinear Forms on (g ,K)-cohomologies Inducedby Symmetry Breaking Operators References Explicit Decomposition of Certain Induced Representations of the General Linear Group 1 Introduction 2 Basics 3 Permutations 4 Calculation 5 Tables 6 Supplement References Mixed Arithmetic Theta Lifting for Unitary Groups 1 Introduction 2 Special Cycles via Incoherent Spaces 3 Mixed Arithmetic Generating Functions: the r=0 Case 4 Mixed Arithmetic Generating Functions: the r>0 Case 5 Product Relation for Arithmetic Generating Functions 6 Mixed Arithmetic Theta Lifting 7 Conjecture on Arithmetic Inner Product Formula for Odd Ranks References Twists of GL(3) L-functions 1 Introduction 2 Petersson Trace Formula and GL(2) δ-Method 3 Outline of the Proof 4 The Off-Diagonal 5 Treating the Old Forms 6 Applying Functional Equations 7 Intertwining Voronoi and Poisson Summations with Reciprocity 8 Applying Cauchy Inequality Followed by Poisson Summation References Modular forms on G2 and their Standard L-Function 1 Introduction 2 Octonions and the Group G2 2.1 The Octonions 2.2 The Lie Algebra g2 2.3 The Heisenberg Parabolic Subgroup 2.4 Cartan Involution 3 Modular forms on G2 3.1 Quaternionic Discrete Series 3.2 Wallach's Result and Fourier Coefficients 3.3 Cubic Rings and Fourier Coefficients 3.4 The Theorems 3.4.1 The Fourier Expansion 3.4.2 The Dirichlet Series 4 The Generalized Whittaker Function 4.1 The Cartan Decomposition 4.1.1 Basis of k 4.1.2 Basis of p 4.2 Iwasawa Decomposition 4.3 Actions of Lie Algebra Elements 4.4 Schmid Equation 4.5 The Formula 5 The Rankin-Selberg Integral 5.1 The Global Integral 5.1.1 The Group GE 5.1.2 The Global Integral 5.2 Embedding in SO(7) 5.2.1 Unfolding 5.3 The Unramified Computation: Overview 5.4 Local Unramifed Integral 5.5 The Fourier Coefficient of the Approximate Basic Function 5.6 The Hecke Operator S 5.7 Hecke Operator Applied to Local Integral 5.8 The Cubic Ring Identity 5.9 The Dirichlet Series 6 Archimedean Zeta Integral 6.1 Setup 6.2 The Computation References