Spectral theory of automorphic forms and analytic continuation of Eisenstein series Bart MICHELS Advisor: F. Brumley Dissertation submitted in partial fulfillment of the requirements for the degree of Master de Sciences, Technologies, Sant´e,Mention Math´ematiqueset Applications, Parcours Type Math´ematiquesFondamentales Universit´ePierre et Marie Curie { Sorbonne Universit´e September 2018 2 Acknowledgements Je ne peux que remercier mon encadrant, Farrell Brumley, pour les nombreux conseils et divers faits ainsi que l'acc`es`ades r´ef´erences cruciales parfois non publi´ees.Sans qui ce projet, une exp´eriencede d´ecouverte dans l'analyse fonctionnelle sur les surfaces hyperboliques, n'aurait pas ´et´epossible. J'ai eu le plaisir d'´etudieret de raffiner des d´emonstrationsexistantes sous sa direction. Bart Michels Paris, septembre 2018 3 4 Contents 1 Introduction 8 2 The hyperbolic plane 9 2.1 Isometries and geometry . .9 2.2 Group actions and fundamental domains . 11 2.3 Examples . 14 2.4 Fourier expansions . 14 3 Operators on symmetric spaces 17 3.1 Weakly symmetric spaces . 18 3.2 Point-pair invariants . 20 3.3 Radially symmetric functions . 22 3.4 Isotropic spaces . 23 3.5 Integral operators . 24 3.6 The algebra of invariant differential operators . 25 3.7 Selberg's eigenfunction principle . 26 4 Functions on the quotient ΓnH 29 4.1 Eisenstein series . 29 4.2 Automorphic kernels . 34 4.2.1 Fourier expansions . 36 4.2.2 Truncated kernels . 39 4.3 Maass forms . 41 4.3.1 Cusp forms . 42 4.3.2 Dimensions . 44 5 Analytic continuation of Eisenstein series 47 5.1 Elementary proofs . 47 5.1.1 Proof by Poisson summation . 47 5.1.2 Proof by Fourier expansion . 48 5.2 Proof via Fredholm-theory . 48 5.2.1 A truncated Eisenstein series. 48 5.2.2 A Fredholm equation . 49 5.2.3 Truncated kernels . 50 5.2.4 Uniqueness principle . 52 5.3 Bernstein's continuation principle . 53 5.3.1 Systems of equations . 53 5.3.2 Systems of finite type . 56 5.3.3 Criteria for finiteness . 58 5.3.4 Eisenstein series . 59 5.4 Further analysis . 62 A Functional analysis 63 A.1 Bounded operators . 63 A.2 The adjoint of an operator, C∗-algebras . 64 A.3 Banach-Alaoglu and the Gelfand-transform . 65 A.4 Continuous functional calculus . 65 A.5 Positive operators . 66 A.6 Compact operators . 66 A.7 Diagonalizable operators . 68 A.8 Spectral theory of compact normal operators . 70 A.9 Trace class and Hilbert{Schmidt operators . 71 5 B Functional calculus 73 B.1 Differentiability and holomorphy . 74 B.2 Weak holomorphy . 76 B.3 Three notions of integration . 77 B.4 Power series and meromorphy . 79 B.5 Integration in function spaces . 80 B.6 Holomorphy in function spaces . 81 B.7 Meromorphy in function spaces . 85 C Fredholm integral equations 87 C.1 Regularity . 87 C.2 The Fredholm equation for bounded operators . 88 C.3 Fredholm theory for compact operators . 90 D Riemannian geometry 92 D.1 Riemannian manifolds . 92 D.2 Connections . 93 D.3 Geodesics and parallel transport . 94 D.4 Geodesics on Riemannian manifolds . 95 D.4.1 The exponential map . 95 D.4.2 Geodesics and distance . 96 D.4.3 Completeness . 97 D.5 Integration . 98 D.6 The Laplace{Beltrami operator . 98 D.7 Isometry groups . 99 D.8 Stabilizers . 99 E Symmetries of manifolds 101 E.1 Isotropic manifolds . 101 E.2 Homogeneous spaces . 101 E.3 Symmetric spaces . 102 F Differential operators 103 F.1 Grading . 104 F.1.1 Global grading . 104 F.1.2 Grading at a point . 105 F.1.3 Symbols . 107 F.2 Elliptic regularity . 107 F.3 Invariant differential operators . 108 F.4 Differential operators on Lie groups . 109 G Spectral theory of the Laplacian 112 G.1 Unbounded operators . 112 G.2 The Laplacian as a symmetric operator . 113 G.3 Extensions and essential self-adjointness . 115 G.4 Operators with compact resolvent . 116 G.5 The spectrum of the Laplacian . 117 H Whittaker functions 118 H.1 Asymptotic expansion . 118 H.2 Second solution . 119 6 7 1 Introduction Real analytic Eisenstein series for PSL2(R) in their general form and their meromorphic continuation, were first studied by Selberg, with the aim of determining the decomposition of the spectrum of the Laplace operator ∆ on a finite-volume hyperbolic surface: it measures the default of diagonalizability of ∆. His ultimate objective was to prove a trace formula, with applications to representation theory of PSL2(R). Selberg's work was later generalized to other groups by Langlands, and various proofs were published by Bernstein, Selberg, Colin de Verdi`ereand others. We will limit our attention to Eisenstein series on PSL2(R) whose associated character of the maximal compact subgroup PSO2(R) is trivial. That ∼ is, they are functions on the hyperbolic plane H = PSL2(R)= PSO2(R), which are invariant under the action of a lattice Γ ⊆ PSL2(R). The Eisenstein series becomes a function E(w; s) on H × C, defined for suitable s. Throughout the text, we will assume that Γ has only one cusp located at 1 (for definitions, see the next section). Without this assumption, the notations become heavier and the proofs slightly more technical. This allows us to focus more on the methods used to prove meromorphic continuation, and the fundamental difficulties that arise. One question we need to ask ourselves, is what it means to analytically (or meromorphically) continue the Eisenstein series: it is a function of two variables, so one can for example interpret holomorphy to mean that E(w; s) is holomorphic for fixed w, or the stronger property that it is holomorphic as a function on U with values in the Fr´echet space of smooth functions C1(H). That is: that the.