Meromorphic Continuation of Eisenstein Series. Gideon Providence

Transcription

1 Meromorphic Continuation of Eisenstein Series by Gideon Providence A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Mathematics University of Toronto c Copyright 217 by Gideon Providence

2 Abstract Meromorphic Continuation of Eisenstein Series Gideon Providence Master of Science Graduate Department of Mathematics University of Toronto 217 The Eisenstein series associated to the cusp a of a discrete subgroup Γ < SL(2; R) is defined by E a (z; s) := Γ a\γ Im(σ 1 a γz) s, where σ a is a certain fractional linear transformation. These sums are originally defined for Re(s) > 1, and generally diverge for Re(s) 1. However they can be extended to all of C \ {a discrete set} if they are reinterpreted in the sense of distributions. In fact, we shall prove that the distribution valued function s E a ( ; s) is meromorphic. To prove this, we begin by investigating modified function spaces on Γ\SL(2; R). These spaces are obtained by truncating Fourier expansions of regular automorphic functions near the cusp a. Versions of the hyperbolic Laplacian on truncated space have discrete spectra, while truncated approximations to E a are expressible as unique solutions to equations in these operators. Since E a differs from the truncated approximations by a holomorphic function, meromorphicity of s E a (z; s) follows. ii

3 Thank you Professor Henry Kim for your time. Acknowledgements iii

4 Contents Acknowledgements iii 1 Introduction 1 2 Automorphy and Eisenstein Series Hyperbolic Surfaces Cusps Automorphic Functions Automorphization The cosets B(Z)\σ 1 a Γσ b The double cosets B(Z)\σ 1 a Γσ b /B(Z) Eisenstein & pseudo-eisenstein series Truncation at a cusp Truncation of function spaces The truncated Laplacian and its extension Extending the usual Laplacian Localized Sobolev spaces Spectrum of Fr The domain of Fr The resolvent operator ( Fr λ) Compact embedding Λ a H 1 (X Γ ) Λ a L 2 (X Γ ) Meromorphic Continuation Another description of E(z, s) The functions E (z, s) Definitions Constant terms The meromorphic continuation formula Bibliography 35 iv

5 Chapter 1 Introduction In [11], Gel fand and Shilov consider the problem of regularizing certain divergent integrals. For instance, they consider the functional x λ + : Cc (R) R defined by ϕ, x λ + := ϕ(x)x λ + dx (1.1) R where x λ for x + = x λ for x >. When Re(λ) > 1 these integrals converge, producing a well-defined functional. Since ϕ has compact support, we can differentiate under the integral sign to see λ ϕ, x λ + is even analytic for Re(λ) > 1: d dλ ϕ, xλ + = d n dx n ϕ, xλ + =. ϕ(x)x λ log(x) dx ϕ(x)x λ log(x) n dx. In order to make sense of x λ + for λ with real part 2 < Re(λ) < 1, the defining equation (1.1) is rewritten using the identity R ϕ(x)x λ dx = 1 (ϕ(x) ϕ())x λ dx + 1 ϕ(x)x λ dx + ϕ() λ + 1. The first integral on the right hand side is defined for provided Re(λ) > 2; since we have subtracted off the first term of the Taylor expansion of ϕ, the resulting integrand is O(x 2+λ ). The last two terms on the right are defined for all λ, and λ 1 respectively. In other words, the distribution-valued function λ x λ + has been extended to Re(λ) > 2, λ 1. Given that ϕ, x λ + is analytic, this is in fact the analytic continuation of λ λ +. So the process of regularizing integrals ϕ, x λ + for 2 < Re(x) 1, λ 1 amounts to the same thing as analytic/meromorphic continuation to this region. In general x λ + can be extended to Re(λ) > n 1, λ 1, 2,..., n by the same principle using 1

6 Chapter 1. Introduction 2 the identities R ϕ(x)x λ dx = 1 ϕ(x) n 1 j=1 ϕ (j) () j! x j x λ dx + 1 ϕ(x)x λ dx + n j=1 ϕ (j 1) () (j 1)!(λ + j). (1.2) In effect, we subtracting the Taylor expansion of the test function ϕ near zero to obtain a convergent integral, then add an appropriate constant to undo the error caused by subtracting. If we adopt the perspective of x λ + as an element of the dual to Cc (R) (ie. we forget the dependence on the specific function ϕ), we can instead view this operation as a truncation of the space Cc (R). Gel fand and Shilov demonstrate the utility of this technique in [11] by regularizing various divergent quantities, like the generalized functions whose integrals are often important in applications. (x ± i ) λ and log(x ± i ) Similar ideas make an appearance in [8] as Lax and Phillips explore spectral properties of the hyperbolic Laplacian on quotients Γ\H of the upper half plane by discrete subgroups Γ < SL(2; R). ves Colin de Verdière makes use of these ideas in [2] to regularize the Eisenstein series associated to Γ\H. The simplest case is Γ = SL(2; Z), where the Eisenstein series is given by E(z; s) := (c,d)=1 y s cz + d 2s. In regularizing E, the Fourier expansion of an automorphic function around the cusp plays the role of the Taylor expansion in the previous example. Instead of truncating functions in Cc (R) near using their Taylor expansion, we truncate functions in Cc (Γ\H) by removing the constant term of the Fourier expansion in a neighbourhood of the cusp. In a more general setting, an arbitrary cusp a takes the place of. It was relatively easy to deduce that the continuation of x λ + given by (1.2) had a discrete set of poles (negative integers) was meromorphic. For the Eisenstein series, the process is more involved and will depend both on spectral properties of the Laplacian as an operator on truncated Cc (Γ\H), and on a characterization of E as an eigenfunction of the Laplacian. To this end, in chapter 2 we review the basics of automorphic functions and the domains Γ\H on which they are defined. In particular, we define Eisenstein series for a broad class of subgroups Γ < SL(2; R). In chapter 3 we describe the truncated version of some important function spaces, including Cc (Γ\H). Friedrichs-Sobolev spaces, which are important for the later spectral theory are introduced here. In chapter 4, we first study properties of the Laplacian as it operates on truncated space, working up to a proof it has discrete spectrum on such space. The resolvent operator λ ( λ) 1 plays an important role since Eisenstein series are (unique) solutions to ( λ s )E = ( s(s 1))E =. The resolvent is shown to be a holomorphic operator in a neighbourhood of any point λ where it is defined. Finally, in chapter 5 we re-express the Eisenstein series in terms of the resolvent ( λ) 1. Since the truncated Laplacian has discrete spectrum, and the resolvent is locally holomorphic, we are able to conclude in that the extended Eisenstein series is meromorphic on C. A formula expressing E in terms of known meromorphic functions is provided by theorem The meromorphicity follows

7 Chapter 1. Introduction 3 from this formula. Coupled with results like this formula provides information on the location of poles of E.

8 Chapter 2 Automorphy and Eisenstein Series In this section we review the elementary theory of automorphic functions and distributions. In particular, we discuss a procedure for automorphic functions, and investigate their Fourier theory. Further details about harmonic analysis on H can be found in [7] and [5]. Fourier theory, especially of Eisenstein series, is well presented in [6]. The theory of the group SL(2; R) and its subgroups, along with quotients of H is excellently explained [9]. Material on distributions was inspired by [11]. 2.1 Hyperbolic Surfaces A standard representation of the hyperbolic plane is the metric space (H, dµ), where H = { z = x + iy y > } and ds 2 = dx2 + dy 2 y 2. The group G = SL(2; R) acts on the extended half-plane Ĥ = H R { } by Möbius transformations: [ a b ] az + b c d z = cz + d. In particular, H is a homogeneous space for the Möbius G-action. Since G factors as the product NAK, where {[ ] N = {[ 1 x 1 ] : x R}, A = y y 1 and since the stabilizer of i H is K, we have the identifications } : y >, K = SO(2; R); H = G/K = NA. Observing that [ ] [ 1 x 1 ] y i = x + iy 2 y 1 gives an explicit correspondence H NA: z = x + iy [ ] [ ] 1 n(x) a(y) 1 a(y) 1 (2.1) 4

9 Chapter 2. Automorphy and Eisenstein Series 5 where n(z) = x and a(z) = y 1/2. Direct computation shows the metric tensor dµ is G-invariant. It gives rise to the invariant volume element dµ = dxdy y 2 used for computing hyperbolic plane integrals. It is often useful to compute such integrals on NA instead of H. From equation (2.1) it follows that dx = dn and dy = 2ada and hence The Laplacian on H is defined by It is G-invariant in the sense that H f(x, y) dxdy y 2 = f(x, y) NA dn 2ada a 4 = 2 f(na i) dnda NA a 3. ( ) := y 2 2 x y 2. (f γ)(z) = ( f) γ(z) f C (H), γ G We will need to investigate quotients X Γ := Γ\H of H by discrete subgroups Γ < G. The action of such a subgroup on H is always properly discontinuous, meaning the spaces X Γ are again manifolds. If F H is a fundamental domain for Γ, and if q : H X Γ is the quotient map, then the restricted map q F gives a coordinate system on X Γ via p = q F (z) z for all z F. Discontinuity also means the orbits Γz, for z H, have no accumulation points in H; however, they may still accumulate on the boundary H = R. Definition If Γ < G is discrete, and if each point in Ĥ is an accumulation point of some orbit Γz, then Γ is said to be a finite volume group. If Γ\H is compact then we say also that Γ is co-compact. 2.2 Cusps Suppose Γ < G is of finite volume. The fundamental domain F can be chosen to be a hyperbolic polygon such that F consists of finitely many geodesic arcs, intersecting at common vertices. If Γ is not co-compact then the closure F in Ĥ must have a vertex a on R. Definition A vertex for F on R is called a cusp. If U a is an open set in Ĉ not containing any other cusp of Γ, then the open set V a := q(u a F ) X Γ is called a cuspidial region. Provided that Γ is of finite volume, we can always choose F so that the cusps are all Γ-inequivalent. If F is such a domain, with cusps a, b,..., the cuspidial regions V a, V b,... are all non-compact. The remaining region X Γ \ (V a V b ) is compact and simply connected. As a result, we shall be mostly interested in the cuspidial regions.

10 Chapter 2. Automorphy and Eisenstein Series 6 In order to study such a function in a cusp neighbourhood V a, we need to first introduce a convenient change of coordinates on H. The stabilizer Γ a the cusp a of F is an infinite cyclic group Γ a = γ a (2.2) generated by a parabolic element γ a G. There exists σ a G such that σ a ( ) = a, and σ 1 a γ a σ a = ( ). (2.3) If we apply the automorphism σa 1 to H, the action of Γ on H is taken that of σa 1 Γσ a on σa 1 (H). In particular, the cusp neighbourhood U a is mapped into a strip of the form S( ) = {x + iy : < x < 1, y > } for some >. The point a itself corresponds to in σa 1 (H). Definition We refer to the coordinates on X Γ defined by p a = q F σ a (ζ) ζ for all ζ σ 1 a (F ) as cusp coordinates for a. Certain cuspidial regions, defined through cusp coordinates, are particularly useful. Definition For > 3 2, set H := {x + iy H y > }. We define the cusp neighbourhoods by V a V a := p a(h ) X Γ. 2.3 Automorphic Functions Definition A function f : H C is said to be automorphic for the finite volume group Γ < G if it satisfies the following transformation rule: f(γz) = f(z) for all γ Γ. Clearly any automorphic function descends to a function on X Γ. functions on X Γ is denoted A(X Γ ). As such, the set of automorphic Definition In the special case that f is smooth and supp(f) is a compact, we say f is an automorphic test function. The space of automorphic test functions is denoted D(X Γ ). In order to topologize D(Γ\H), we first describe it as the direct limit D(Γ\H) = lim n C c (K n ). where K 1 K 2 an exhaustion of Γ\H by compacts (i.e. Γ\H = n K n, and each K n is compact). We then equip D(Γ\H) with the associated locally convex inductive limit topology.

11 Chapter 2. Automorphy and Eisenstein Series 7 Definition Elements of the the dual space D (Γ\H) are called automorphic distributions. We could equivalently characterize the automorphic distributions as those µ D (H) satisfying ϕ, µ = ϕ γ, µ for all automorphic test functions ϕ, and all γ Γ. Parameterized families {µ s } of automorphic distributions, where the parameter s ranges in some domain Ω C, are of particular importance. For any such family, and any automorphic test function ϕ, we can define the ϕ-evaluation function Ω C by M ϕ (s) := ϕ, µ s. Definition If M ϕ is a holomorphic (resp. meromorphic) function for all ϕ D(Γ\H) then we say µ s is a holomorphic (resp. meromorphic) family of distributions. Definition Suppose µ s, ν s are a meromorphic families parameterized by s Ω µ and s Ω ν respectively. If Ω µ Ω ν we say that ν s is a meromorphic continuation of µ s to Ω ν. Suppose we are given an automorphic function f. represented by f a, where f a (ζ) := f(σ a (ζ)) = f(z) In cusp coordinates ζ = σa 1 (z) on H, f is Our observations in section 2.2 show that f a is σa 1 Γσ a -invariant. In particular, f a must be σa 1 Γ a σ a = B(Z)-invariant, meaning f a (x + k + iy) = f a (( 1 k 1 ) ζ) = f a (ζ) = f a (x + iy) (2.4) for all k Z. As an immediate consequence of equation (2.4) we have Proposition Any automorphic function f L 2 (X Γ ) admits a Fourier expansion at the cusp a of the form f a (ζ) = k Z f a k (y)e 2πikx The coefficient functions fk a (y), which are defined on N\H (which we identify with (, )), are given by f a k (y) = 1 f a (x + iy)e 2πikx dx. The th notation: order Fourier coefficient plays an important and recurring role, so it gets some special Definition The constant term projection c a : L 2 (X Γ ) C (N\H) at a is defined by c a [f](y) = 1 f a (x + iy) dx. Making the convention c a [f](x + iy) := c a [f](y), we recover an N-invariant function on X Γ. We can therefore extend c a to distributions µ by setting ϕ, c a [µ] := c a [ϕ], µ ϕ D(X Γ ).

12 Chapter 2. Automorphy and Eisenstein Series 8 It is worthwhile to record how the constant term map interacts with the Laplacian in this context. Proposition The constant term map commutes with the Laplacian in the sense that c a [f] = c a [ f], where f is smooth, and c a [f] is interpreted as an N-invariant function. Proof. Direct computation: ( ) c a [f](x + iy) = y x y 2 f a (t + iy) dt = y 2 1 = y 2 1 = 1 2 f a (t + iy) dt y2 2 f a (t + iy) dt + y2 x2 f a (t + iy) dt = c a [ f](x + iy) 1 2 y 2 f a (t + iy) dt Corollary The distributional versions of the constant term map and Laplacian also commute: c a [µ] = c a [ µ] for all distributions µ. Proof. Immediate from proposition Automorphization The use of cusp coordinates ζ around a cusp a gives rise to the process of automorphization at a. Automorphization allows us to quickly construct functions on H that descend to functions on X Γ. Definition Given a function g satisfying g(x + iy) = g(ζ) = g (( 1 n 1 ) ζ) = g(x + n + iy) for all n Z the autmorphization of g at a, denoted A a [g], is defined in cusp coordinates by A a [g](ζ) = γ B(Z)\σ 1 a Γσ a g(γζ). (2.5) That we need to sum over cosets by B(Z) follows from equations (2.2) and (2.3). Two alternative representations of A a [g] frequently prove useful:

13 Chapter 2. Automorphy and Eisenstein Series 9 1. In standard z-coordinates A a [g] takes the form A a [g](z) = γ Γ a\γ g(σa 1 γσ a σa 1 z) = g(σa 1 γz). γ Γ a\γ 2. In cusp coordinates ζ for some cusp b a A b a[g](ζ) = γ B(Z)\σ 1 a Γσ b f(γζ). (2.6) The fact that the coset sum takes the form B(Z)\σa 1 Γσ b follows from the identity σ 1 a Γ a γσ b = σ 1 Γ a σ a σ 1 γσ b = B(Z)σ 1 γσ b a a a which holds for all γ Γ. In order to compute explicit automorphizations, it is necessary to gain a more explicit description of the set B(Z)\σa 1 Γσ b of cosets. In order to do this, we shall decompose σa 1 Γσ a into double cosets B(Z)\σa 1 Γσ b /B(Z). Computing sums of the form in equation (2.6) requires the case a b, while equation (2.5) corresponds to the special case a = b. We proceed in two steps: first we investigate the left cosets B(Z)\σa 1 Γσ b, then we build on this to describe the double coset situation The cosets B(Z)\σa 1 Γσ b Two matrices ( a b ) ( c d and a1 b 1 ) c 1 d d are in the same coset of B(Z)\σ 1 a Γσ b if and only if c = c 1 and d = d 1, or c = c 1 and d = d 1. Indeed, computing directly for arbitrary γ = ( 1 k 1 ) B(Z), ( 1 k 1 ) ( ) ( a b c d = a+ck b+dk ) c d. In other words, left multiples by γ all have the same bottom row. On the other hand, suppose γ = ( a ) b c d and γ1 = ( ) a 1 b 1 c d are elements of σ 1 a Γσ b with the same bottom row. Since γ1 1 = ( ) d b 1 c a 1, we have γ γ1 1 = ( ) 1 a b 1+a 1b 1 Γ which means that γ γ 1 mod B(Z). Finally, we remark that ( c d ) and ( c d ) are left B(Z)-equivalent since ( ) 1 1 B(Z). So we have shown the cosets B(Z)\σ 1 a Γσ b are parameterized by pairs (c, d) of numbers such that c and σa 1 Γσ b contains ( c d ). It is useful to note that the set Ω := { ( ) σa 1 } Γσ b is non-empty if and only if a and b are equivalent. Indeed, if σ 1 a γ b = σ a (σ 1 a γσ b ) = σ a = a. γσ b σ 1 Γσ b, then Conversely, if a is equivalent to b choose γ Γ such that γb = a and note that a σ 1 a γσ b = σa 1 γ b = σa 1 a =.

14 Chapter 2. Automorphy and Eisenstein Series 1 In other words, γ stabilizes, whence it has the form ( ) The double cosets B(Z)\σa 1 Γσ b /B(Z) For any ( c d ) σ 1 Γσ b we compute directly that a ( c d ) ( 1 n 1 ) = ( c d+cn ) From this we deduce that specifying a double B(Z)-coset requires two choices of parameter. First we must choose c > such that ( c ) σa 1 Γσ b. With c fixed, we must then choose d < c such that ( c d ) σa 1 Γσ b. 2.5 Eisenstein & pseudo-eisenstein series N-invariant functions are evidently B(Z)-invariant, so the space C (N\H) = C (, ) ends up being a prime candidate for the automorphization process. Definition The space PEisen a of pseudo-eisenstein series attached to a is obtained by automorphising C (, ) at a: PEisen a = f f(z) = γ Γ a\γ ϕ ( Im(σa 1 γz) ) for some ϕ C (, ). The automorphization process can be used to construct smooth automorphic -eigenfunctions. It is easy to find a -eigenfunction on H; indeed, the function z (Im(z)) s = y s has eigenvalue λ = s(s 1). It is clearly N-invariant, hence B(Z)-invariant too, making it a viable candidate for automorphization. Definition The Eisenstein series attached to the cusp a is the function E a (z, s) obtained by automorphising z (Im(z)) s = y s : E a (z, s) = γ Γ a\γ In cusp coordinates around b, equation (2.7) looks like E b a(ζ, s) = ( Im(σ 1 a γz) ) s. (2.7) (Im(γζ)) γ B(Z)\σa 1 Γσ b Lemma The Eisenstein series E a (z, s) is convergent for Re(s) > 1. s.

15 Chapter 2. Automorphy and Eisenstein Series 11 Proof. See e.g. [6], theorem Since Im(z) s = y s is a -eigenfunction and is G-invariant, the Eisenstein series E a (z, s) is an automorphic -eigenfunction with eigenvalue λ = s(s 1). Fourier expansions at each cusp. By the techniques in section 2.4, we can write E b a(ζ, s) = γ B(Z)\σ 1 a s (Im(γζ)) Γσ b = δ ab (Im(ζ)) s + c> d (mod c) n Z This means the Eisenstein series have ( ( )) s (2.8) a Im c 1 c(c(ζ + n) + d) where δ ab = 1 if a = b, and δ ab = otherwise. To develop the Fourier expansion, we concentrate on the inner sum over Z. Applying Poisson summation and the change of variables t t x d c (where ζ = x + iy) yields An explicit computation shows Im ( ( )) s a Im c 1 c(c(ζ + n) + d) n Z = ( ( )) s a Im n Z c 1 e 2πint dt c(c(ζ + t) + d) = ( ( )) s e 2πin(x+ d c ) a Im c 1 c 2 e 2πint dt (t + iy) n Z ( ) a c 1 y c 2 (t+iy) = c 2 (t 2 +y 2 ), whence ( ( )) s a Im c 1 ( ) s c 2 e 2πint dt = ys 1 (t + iy) c 2s t 2 + y 2 e 2πint dt This last integral can be evaluated explicitly: ( ) s y Γ(s π 1 t 2 + y 2 e 2πint 2 ) Γ(s) y 1 2s if n = dt = 2π s Γ(s) n s 1 2 y s+ 1 2 K s 1 (2π n y) otherwise 2 Substituting back into equation (2.8), we get E b a(ζ, s) = δ ab y s + c> + c> 1 c 2 1 c 2 where δ ab = 1 if a = b, otherwise δ ab =. d (mod c) d (mod c) n Z n Γ(s 1 π 2 ) y 1 s Γ(s) e 2πin(x+ d c ) 2πs Γ(s) n s 1 2 y 1 2 Ks 1 2 (2π n y)

16 Chapter 3 Truncation at a cusp In this section we modify the space L 2 (X Γ ) and the operator to obtain a new space Λ a L 2 (X Γ ) equipped with a new densely defined operator. Later, by extending the domain of slightly, we will obtain an operator Fr with discrete spectrum. Because the Eisenstein series can be expressed in terms of Fr, the meromorphic continuation will follow from the discrete spectrum property. The main references for this section were [3] and [1]. A detailed study of truncated function spaces on X Γ and associated truncated Laplacians is given, in context of the hyperbolic wave equation, in [8]. An excellent explanation of the truncation process, complete with pictures, can be found in [1]. 3.1 Truncation of function spaces Heuristically, we will construct the Λ a L 2 (X Γ ) spaces by subtracting off constant terms of functions near to the cusp a. While the constant term projection might seem useful for such a purpose, it is easier to construct Λ a L 2 (X Γ ) as an orthogonal complement of pseudo-eisenstein series supported entirely in the cuspidial regions. Definition For > we define the spaces PEisen a PEisen a according to PEisen a := {A a[ϕ] : ϕ C c [, )} Definition The truncated space Λ a L 2 (X Γ ) is the orthogonal complement of PEisen a in L 2 (X Γ ): Λ a L 2 (X Γ ) = (PseudoE a ). To truncate subspaces of L 2 (X Γ ) we simply take intersections; for instance, Λ a C c (X Γ ) = C c (X Γ ) Λ a L 2 (X Γ ) Notice that the truncation Λ a C c (X Γ ) is a subspace of D(X Γ ), the space automorphic test functions. 12

17 Chapter 3. Truncation at a cusp The truncated Laplacian and its extension In this section we first define the truncated Laplacian. We then extend the domain of this new operator using the Friedrichs procedure. The resulting operator will be very useful later on due to its spectral properties. Definition The truncated Laplacian is defined to be the restriction := Λ a C c (XΓ). is a densely defined defined operator Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) in light of the following result: Proposition Λ a Cc (X Γ ) is dense in Λ a L 2 (X Γ ). Proof. See, e.g. [3] page 2. Now that the truncation has been defined, we need some minor machinery for the extension process. Define the map, Fr taking Λ a Cc (X Γ ) Λ a Cc (X Γ ) C according to f, g Fr := f, g 2 f, g 2 = (1 )f, g 2. The pairing, Fr clearly respects scalar multiplication and addition of functions. Since positive operator, we see that f, f Fr = f, f 2 f, f 2. is a Symmetry of yields f, g Fr = f, g 2 f, g 2 = g, f 2 g, f 2 = g, f Fr It follows that, Fr is an inner product on Λ a C c (X Γ ). Definition The inner product, Fr specified above is called the Friedrichs-Sobolev inner product. It determines a norm and hence a metric on Λ a C c respectively. (X Γ ) which we denote by Fr and d Fr Definition The completion of Λ a Cc (X Γ ) in the d Fr -metric is denoted by Λ a H 1 (X Γ ). Spaces like Λ a H 1 (X Γ ) are called Friedrichs-Sobolev spaces. There is a natural identification of Λ a H 1 (X Γ ) with a subset of Λ a L 2 (X Γ ). Indeed, strictly speaking, elements of Λ a H 1 (X Γ ) are equivalence classes of d Fr -Cauchy sequences in Λ a Cc (X Γ ). If {f i } is any such d Fr -Cauchy sequence, then the inequality f i f j 2 f i f j Fr implies {f i } is Cauchy in the L 2 -metric as well. Thus there exists f Λ a L 2 (X Γ ) to which {f i } converges in the usual topology. We make the association {f i } f. (3.1)

18 Chapter 3. Truncation at a cusp 14 Limits in Λ a L 2 (X Γ ) are unique, and Cauchy sequences that are Λ a H 1 (X Γ )-equivalent converge to the same Λ a L 2 (X Γ ) limit since f i g i 2 f i g i Fr. Therefore the identification in equation (3.1) is well-defined. For each g Λ a L 2 (X Γ ) define the linear functional λ g : Λ a H 1 (X Γ ) C by λ g (f) := g, f 2. Note that we use the standard L 2 inner product above. An application of Cauchy-Schwarz shows that λ g (f) f 2 g 2 f Fr g 2 from which we deduce λ g is continuous on H 1 (X Γ ) with operator norm λ g Fr bounded above by g 2. In light of the above, the Riesz representation theorem furnishes a linear operator B : Λ a L 2 (X Γ ) Λ a H 1 (X Γ ) satisfying λ g (f) = Bg, f Fr f Λ a H 1 (X Γ ) (3.2) and Bg Fr = λ g Fr g 2 g Λ a L 2 (X Γ ). (3.3) Equation (3.2) by definition says g, f 2 = (1 )Bg, f 2. The idea is therefore to invert the operator B in order to obtain an extension of (1 ), then subtract off the 1 and multiply by 1 to obtain an extension of. First let us collect some useful properties of B. Proposition The operator B defined above satisfies the following properties: i. B is a bounded operator; ii. B is positive; iii. B is injective; iv. B has dense image; v. B is symmetric. Proof. i. This is an immediate consequence of equation (3.3). ii. Follows from the identity Bg, Bg 2 Bg, Bg Fr = λ g (Bg) = g, Bg 2. iii. Suppose Bf = for some f Λ a L 2 (X Γ ). Then for all g Λ a H 1 (X Γ ) we have = g, Fr = g, Bf Fr = Bf, g Fr = λ f (g) = f, g 2 Density of Λ a H 1 (X Γ ) in Λ a L 2 (X Γ ) implies f. iv. Suppose f Λ a H 1 (X Γ ) is such that Bg, f Fr = for all g Λ a L 2 (X Γ ). Then in particular = Bf, f Fr = λ f (f) = f, f 2. Since, 2 is an inner product, this means f and hence B(Λ a L 2 (X Γ )) is dense in Λ a H 1 (X Γ ).

19 Chapter 3. Truncation at a cusp 15 v. Direct computation: g, Bf 2 = λ g (Bf) = Bg, Bf Fr = Bf, Bg Fr = λ f (Bg) = f, Bg 2 = Bg, f 2. We can exploit proposition to get our inverse to B, and hence the candidate extension of (1 ). Corollary The operator B : Λ a L 2 (X Γ ) Λ a H 1 (X Γ ) defined above admits a positive, potentially unbounded inverse B 1 : dom(b 1 ) Λ a L 2 (X Γ ), where dom(b 1 ) = range(b) is a dense subset of Λ a H 1 (X Γ ). Corollary The operator B : Λ a L 2 (X Γ ) Λ a H 1 (X Γ ) defined above is self-adjoint. In a complex Hilbert space properties (i) - (iv) of proposition are enough to ensure corollary The explicit symmetry of proposition (v), while illustrative, is only necessary if we restrict our attention to R-valued functions. Let us now check that this construction actually worked. Proposition The operator B 1 constructed above is an extension of (1 ) in the sense that Λ a Cc (X Γ ) = dom(1 ) dom(b 1 ) Λ a H 1 (X Γ ). Proof. First we prove Λ a Cc (X Γ ) is contained in dom(b 1 ), then we show the inclusion is strict. Fix any f, g Λ a Cc (X Γ ). By definition we have g, f Fr = (1 )g, f 2 = λ (1 )g(f) = B(1 )g, f Fr. Subtracting one side from the other yields g B(1 )g, f Fr = f, g Λ a C c (X Γ ). (3.4) It follows that g B(1 )g =, and hence g = B(1 )g. In particular, g range(b) = dom(b 1 ), establishing the inclusion Λ a Cc (X Γ ) = dom(1 ) dom(b 1 ). To see that the inclusion is strict, we remark that B injects Λ a L 2 (X Γ ) into Λ a H 1 (X Γ ). Therefore B 1 must surject from its domain onto Λ a L 2 (X Γ ). This is a strictly larger set than the range of (1 ), so the domain of B 1 must be strictly larger than that of (1 ). As we saw above, proposition gives the desired extension of since B 1 extends (1 ) = (B 1 1) extends.

20 Chapter 3. Truncation at a cusp 16 Definition Denote the Friedrichs extension of the truncated Laplacian by Fr := (B 1 1) Fr is negative, since it is negative on a dense subset of its domain. It will also be useful to know that Fr is self-adjoint. In order to prove this it is sufficient to show B 1 is self-adjoint, which is more convenient. Proposition The operator B 1 defined above is self-adjoint. Proof. i. Define U : Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) by U(f, g) := ( g, f). Then graph(b 1 ) = U(graph(B 1 )). Indeed, suppose (h, g) U(graph(B 1 )). Then for all (f, B 1 f) graph(b 1 ), = (h, g), ( B 1 f, f) graph = h, B 1 f 2 + g, f 2 = B 1 h, f 2 + g, f 2 ; in other words, g = B 1 h. So we have shown (h, g) U(graph(B 1 )) = (h, g) = (h, B 1 h) graph(b 1 ). For the reverse inclusion, suppose ( B 1 f, f) U(graph(B 1 )) and (g, B 1 g) graph(b 1 ) are arbitrary. Directly, ( B 1 f, f), (g, B 1 g) graph = B 1 f, g 2 + f, B 1 g 2 = B 1 f, g 2 + B 1 f, g 2 =. ii. Use (i) to prove B 1 is self-adjoint. First we must define another operator: S : Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) S : (f, g) (g, h). The operator S relates the graphs of B and B 1 by graph(b 1 ) = S(graph(B)), and when composed with U satisfies U S = S U. Moreover, we get (S(W )) = S(W ) for any subset W. Thus a computation yields graph(b 1 ) = U ( graph(b 1 ) ) = U (S (graph(b))) = S (U (graph(b))) = S (U (graph(b)) ) = S(graph(B )) = S(graph(B)) = graph(b 1 ) = graph(b 1 ).

21 Chapter 3. Truncation at a cusp Extending the usual Laplacian In section 3.2 we constructed an operator B 1 which extended (1 ); we then defined the extension Fr in terms of B 1. The same procedure can be used to define an extension Fr of the usual Laplacian. We outline the process below. 1. Define H 1 (X Γ ) to be the completion of Cc (X Γ ) with respect to norm derived from f, g H 1 := (1 )f, g Use the linear functional λ g : H 1 (X Γ ) C given by λ g (f) := g, f 2, along with Riesz representation to obtain B : L 2 (X Γ ) H 1 (X Γ ) such that λ g (f) = Bg, f H Prove that B is bounded, positive, injective, symmetric, and has dense image. The proofs of these five properties proceed exactly in section Use the previous point to deduce that B admits an inverse B 1 : dom(b 1 ) L 2 (X Γ ), where dom(b 1 ) is a dense subset of H 1 (X Γ ). 5. Define the Friedrichs extension of the usual Laplacian by Fr := (B 1 1). As before, Fr is negative and self-adjoint. 3.4 Localized Sobolev spaces Fix any compact K (, ) N\H and define the seminorm ρ K on functions f C (H) by ρ K (f) := K [,1] (1 )f f dxdy y 2. A function f C (N\H) can be considered as an N-invariant element of C (H), and by restricting to such functions ρ K can instead be considered as a seminorm C (N\H) R. Definition The local Sobolev space Hloc 1 (N\H) is the completion of the space C (N\H) with respect to the initial topology induced by the family {ρ K K N\H is compact}

22 Chapter 3. Truncation at a cusp 18 of seminorms. We can also understand Hloc 1 (N\H) if we note that f H 1 loc(n\h) f H 1 (K) K (, ) = N\H. Here H 1 (K) is the usual Sobolev space on the compact subset K of the real line. These localized Sobolev spaces are important here mostly because they appear as the codomain of the constant term map: Proposition The constant term map c a (f) is a continuous function mapping c a (f): Λ a H 1 (X Γ ) H 1 loc(n\h). Proof. Indeed, suppose {f n } is a sequence in Λ a H 1 (X Γ ) converging to f and set U n := f n f. Then U n in Λ a L 2 (X Γ ), and since U n has a Fourier expansion Un(ζ) a = u a nk(y)e 2πinx k Z we see that u a n(y) for each n and a.e. y. In particular, c a (f n f)(y) = u a no(y) for a.e. y. It follows that c a (f n f) H1 (K) for all K N\H, so c a is continuous. It is well known that H 1 loc(n\h) = H 1 loc(, ) C (, ), so we have the useful corollary: Corollary If f Λ a H 1 (X Γ ) then c a (f): N\H C is continuous. It is also useful to note a criterion for when an element of PEisen a is in Λ a H 1 (X Γ ). Proposition Let ϕ C (, ) be smooth. Then the pseudo-eisenstein series A a [ϕ](z) = Γ a\γ ϕ ( Im(σ 1 a γz) ) is contained in H 1 (X Γ ) if ϕ 2 + y ϕ 2 dy y y 2 <. Proof. This result follows from the Lie theory of SL(2; R). See e.g. [3].

23 Chapter 4 Spectrum of Fr The main goal of this chapter is to prove the spectrum sp( Fr ) is a discrete subset of C. We will also see that the resolvent (λ Fr ) 1, which only exists for λ sp( Fr ), is a compact operator for all valid λ. The main references for this section were [3] and [4]. 4.1 The domain of Fr Of central importance in describing the domain of Fr is the linear functional T : Λ a H 1 (X Γ ) R defined by T [f] = c a [f]( ). Recall that in section 3.4 we showed that c a [Λ a H 1 (X Γ )] C (, ), so the evaluation c a P [f]( ) in the definition of T is well-defined. Moreover, T is a continuous functional. Indeed, we saw also in section 3.4 that c a : Λ a H 1 (X Γ ) H 1 loc (X Γ) is continuous, and since the evaluation-at- map C (, ) R is continuous, so is the composition T : Λ a H 1 (X Γ ) H 1 loc (X Γ) eval R Proposition The domain of Fr satisfies dom( Fr) { f Λ a L 2 (X Γ ) f Λ a L 2 (X Γ ) + C T } where all derivatives are taken in the distributional sense. Proof. i. First let us suppose f dom( Fr ) and find F Λ a L 2 (X Γ ) such that f = F + α T, α C. (4.1) This will prove dom( Fr ) is contained in the set described in the proposition. We may assume f = (1 Fr) 1 f for some f Λ a H 1 (X Γ ) 19

24 Chapter 4. Spectrum of Fr 2 since dom( Fr ) = dom(1 Fr ). Appealing to proposition 3.2.1, we see that (1 )g, (1 Fr) 1 f 2 = g, f 2 g Λ a C c (X Γ ), f Λ a L 2 (X Γ ). Treating (1 Fr ) 1 f as an element of D (X Γ ) we compute the distributional derivative to obtain g, f 2 = (1 )g, (1 Fr) 1 f 2 = g, (1 )(1 Fr) 1 f 2 (4.2) = g, (1 )f 2 at least for test functions g Λ a C c (X Γ ). Equation (4.2) shows that the distribution ν f := (f f ) f annihilates Λ a C c 1 (X Γ ). It follows that supp(ν f ) is contained in the closed cuspidial neighbourhood V a. Elements of Λ a Cc (X Γ ) are smooth functions with compact support on X Γ whose th Fourier coefficient vanishes above the line y =. However, the n th coefficient for n need not vanish anywhere. The fact that ν f annihilates Λ a Cc (X Γ ) therefore implies that ν f is a distribution whose n th Fourier coefficients all vanish on [, ) for n. ii. If we suppose instead that g PEisen a is perpendicular to PEisen a. The identity then certainly g, f 2 = ; indeed f Λ a L 2 (X Γ ), which = g, f 2 = g, (1 )f 2, derived similarly to equation (4.2), shows ν f also annihilates PEisen a. We conclude supp(ν f ) = { }, and that all Fourier coefficients of ν f at vanish except for the constant term. Restrictions on the order of distributions like c a [ν f ] in the dual space to Hloc 1 (N\X Γ) now imply that c a [ν f ] is some constant multiple of the delta distribution δ. In other words, as distributions ν f = α T for some α C since ϕ, ν f = ϕ, c a [ν f ] = c a [ϕ], α δ = ϕ, α T (4.3) Rearranging the definition of ν f and applying (4.3) yields f = (f f ) α T. Taking F = f f in equation (4.1), we are done. Proposition tells us that, for each f dom( Fr ), the distribution f can be uniquely decomposed as f = F + α T for F Λ a L 2 (X Γ ) and α C. (4.4) This is useful for computing Fr f since it turns out that Fr f = F. In other words, Fr f is the L2 -part of the decomposition 4.4: 1 Note that equations (3.4) and (4.2) are different since the orders of application of the operators (1 ) and (1 Fr ) 1 vary. The computation in equation (4.2) necessarily occurs in the context of D (X Γ ).

25 Chapter 4. Spectrum of Fr 21 Proposition If f dom( Fr ) then Fr f Λ a L 2 (X Γ ), and f decomposes uniquely as: f = Frf + α T for some α C. Proof. Fix any f dom( Fr ), and choose f such that f = (1 Fr ) 1 f In the proof of proposition we saw that substitute f = (1 Fr )f to obtain f = f f α T ; Solving for Frf establishes the desired identity. To equations above to deduce Since f, f Λ a L 2 (X Γ ), so is their difference. f = f (1 Fr)f α T = Frf α T. Frf = f f. see that Fr f Λ a L 2 (X Γ ), combine the two 4.2 The resolvent operator ( Fr λ) 1 Definition The resolvent operator R λ : Λ a L 2 (X Γ ) Λ a L 2 (X Γ ) is defined by R λ := ( Fr λ ) 1 for those values λ C such that the given inverse exists. Proposition For λ (, ], the resolvent R λ is everywhere defined on Λ a L 2 (X Γ ). Proof. i. Fix λ = ξ + iη C \ R. First we show R λ is everywhere defined for such λ. In (ii) we extend to the case λ (, ). For any f dom( Fr ), a direct computation yields ( Fr λ)f 2 2 = ( Fr ξ)f, ( Fr ξ)f 2 iηf, ( Fr ξ)f 2 ( Fr ξ)f, iηf 2 + ξ 2 f, f 2. The middle two terms cancel since Fr self-adjoint and ξ R imply ( Fr ξ)f, iηf 2 = iη f, ( Fr ξ)f 2 = iηf, ( Fr ξ)f 2. We are left with and hence ( Fr λ)f 2 2 = ( Fr ξ)f ξ2 f 2 2 ξ2 f 2 2 (4.5) ( Fr λ)f 2 ξ f 2.

26 Chapter 4. Spectrum of Fr 22 So if ξ and f then ( Fr λ)f. In other words, Fr is injective, hence potentially invertible, whenever λ C \ R. In light of the previous paragraph, to show R λ is everywhere defined for λ C \ R, it is enough to demonstrate the set equality ( Fr λ)dom( Fr) = Λ a L 2 (X Γ ). (4.6) Clearly ( Fr λ)dom( Fr ) Λ a L 2 (X Γ ). Suppose the inclusion were strict. Then there must exist f Λ a L 2 (X Γ ) such that f ( Fr λ)dom( Fr ); ie ( Fr λ)g, f 2 = g dom( Fr). This implies f dom(( Fr λ) ) = dom( Fr λ), and g, ( Fr λ)f 2 = g dom( Fr), whence Frf = λf. In other words, λ C \ R is an Fr -eigenvalue. But Fr is negative, so its eigenvalues are non-positive reals. This is a contradiction, which proves equation (4.6) ii. Now we extend to the case λ (, ). Recall from section 3.2 that Fr is a negative self-adjoint operator, whence Fr is positive and self-adjoint. If λ (, ) then ( Fr λ) is also positive, allowing us to compute ( Fr λ)f 2 2 = Fr f λ 2 f Re(λ) Frf, f 2 λ 2 f 2 2. (4.7) From this we deduce that if λ (, ) then ( Fr λ) is injective. The exact same argument as in part (i) then shows that ( Fr λ)dom( Fr ) = Λ a L 2 (X Γ ). Corollary If λ C \ (, ] then R λ 2 1 Im(λ). Proof. Fix λ C \ R and choose f Λ a L 2 (X Γ ). By proposition we can write f = ( Fr λ)f for some f dom( Fr ). From equation (4.5): f 2 = ( Fr λ)f 2 Im(λ) f 2 = Im(λ) R λ ( Fr λ)f 2 = Im(λ) R λ f 2, whence R λ 2 1/ Im(λ) as claimed. If instead λ (, ) we appeal to equation (4.7) to compute f 2 = ( Fr λ)f 2 λ f 2 = λ R λ ( Fr λ)f 2 = λ R λ f 2 ; this time we deduce R λ 2 1/ λ = 1/ Im(λ).

27 Chapter 4. Spectrum of Fr 23 Proposition Fix λ C \ (, ]. If the resolvent R λ exists, it is holomorphic at λ. Proof. We want to show there exists a linear operator L on Λ a L 2 (X Γ ) such that Consider the identity lim R λ+ɛ R λ ɛ ɛ L =. 2 1 ( Fr (λ + ɛ))r λ = 1 ( Fr λ)r λ ɛr λ = ɛr λ. Applying R λ+ɛ to both sides of this equation yields R λ+ɛ (1 ( Fr (λ + ɛ)))r λ = R λ+ɛ R λ on the left-hand side, and ɛr λ+ɛ R λ on the right-hand side. It follows that R λ+ɛ R λ = ɛr λ+ɛ R λ ; that is, R λ+ɛ R λ ɛ = R λ+ɛ R λ. This suggests Rλ 2 as a candidate for L. Indeed, apply the above equality a few times to produce R λ+ɛ R λ ɛ R 2 λ = R λ+ɛ R λ R 2 λ = (R λ+ɛ R λ )R λ = ɛr λ+ɛ R 2 λ. Finally, compute the limit with an appeal to to corollary for bounds on the operator norm: lim R λ+ɛ R λ ɛ ɛ Rλ 2 = lim ɛrλ+ɛ R 2 λ ɛ 2 lim ɛ 2 ɛ Im(λ + ɛ) Im(λ) 2 =. 4.3 Compact embedding Λ a H 1 (X Γ ) Λ a L 2 (X Γ ) Proposition The inclusion ι: Λ a H 1 (X Γ ) Λ a L 2 (X Γ ) is a compact map. By definition this means that ι(k) is compact in Λ a L 2 (X Γ ) whenever K is compact in Λ a H 1 (X Γ ). In practice it suffices to show ι(b 1 ()) is compact, where B 1 () Λ a H 1 (X Γ ) is the unit ball. If for any ɛ > we can exhibit a finite set of functions {η 1,..., η N } such that ι(b) B ɛ (η 1 ) B ɛ (η 2 ) B ɛ (η N ) (4.8) we get total boundedness of ι(b 1 ()), from which compact closure follows. The condition in equation (4.8) is equivalent to showing f ι(b (1)), f η i 2 < ɛ for at least one η i. This is the approach we take.

28 Chapter 4. Spectrum of Fr 24 Proof of proposition i. Setup. Assume ɛ > is given, and suppose X Γ has cusps a 1,..., a k (so that a {a i }). Fix some > 3/2, and consider cusp neighbourhoods Va 1,..., Va k about these cusps. Let V be an open set such that X Γ \ ( Va 1 Va ) k Vo. With these definitions X Γ Va 1 Va k V o is an open covering, and V is a bounded set. ii. Handle the V part. Let {ϕ i } {ϕ } be a partition of unity on X Γ subordinate to the cover Va 1 Va k V such that supp(ϕ i ) V a i and supp(ϕ ) V. In cusp coordinates around a i we can assume ϕ i (x + iy) = ψ ( ) y where ψ : (, ) R is such that ψ(y) = if y (, 1] ψ(y) 1 if y (1, 2) ψ(y) = 1 if y [2, ) Arbitrary f Λ a H 1 (X Γ ) can be decomposed against our partition as f = ϕ f + k ϕ i f. V is bounded, so Rellich compactness implies the smaller inclusion ι: H 1 (V ) Λ a L 2 (V ) is a compact map. In particular ι(b 1 ()) is totally bounded, and since i=1 ϕ ι(b 1 ()) = ι(ϕ B 1 ()) ι(b 1 ()), it follows all of these sets are totally bounded. We use this to choose {η 1,..., η N } Λ a L 2 (V ) such that ϕ f ϕ ι(b 1 ()), ϕ f η i 2 < ɛ 2 for at least one η i. The η i can be extended to all of X Γ by setting η i on the complement of V. Now, if f ι(b 1 ()) it follows that ϕ f ι(b 1 ()) H 1 (V ) also. Thus, for some η i, we have a chain of inequalities ( ) k f η i 2 = ϕ f + ϕ i f η i i=1 2 k ϕ f η i 2 + ϕ i f ɛ k 2 + ϕ i f 2. i=1 The above argument holds for arbitrary > 3/2. Therefore we are done once we prove that choosing i=1 2

29 Chapter 4. Spectrum of Fr 25 sufficiently large implies k ϕ i f 2 ɛ 2 i=1 f ι(b 1 ()). (4.9) iii Handle the V ai parts. Fix any f ι(b 1 ()) and any > 3/2. For convenience set F i := ϕ i f. One way to prove the bound (4.9) is to show lim F i 2 2 = lim F i 2 dxdy y y V y 2 = a i at each cusp a i. To make computations easier, we change into cusp coordinates ζ = x + iy, apply the Tonelli theorem, and simplify the result: F i 2 2 F i 2 dxdy V y 2 = a i y> y> 1 F ai i (ζ) 2 dxdy. F ai i (ζ) 2 dxdy y 2 The trick is to express this last integral in terms of F. First, liberal application of the Parseval identity for F x, the Fourier transform in x, along with a scaling of the n th Fourier coefficient by (2πn) 2 brings us to 1 2 y> 1 F ai i (ζ) 2 dydx = y> = 1 2 y> = 1 2 y> y> n Z n F x [F ai i (, y)](n) 2 dy (2πn) 2 F x [F ai (, y)](n) 2 dy n Z n F x n Z n 1 [ F a i ] i (, y) x F ai i x (ζ) 2 i dxdy (n) 2 dy Now, notice < y> F ai i y 2 dy = y> 2 F ai i y 2 F ai i dy; similarly < y> 2 F ai i x 2 F ai i dy. Therefore changing the order of integration yields F i = 1 2 [ 1 y> y> 1 2 F ai x 2 F ai i (ζ) F ai F ai i i dxdy y 2 (ζ) + 2 F ai i x 2 F ai i ] dxdy (4.1) F i 2 Fr 2.

30 Chapter 4. Spectrum of Fr 26 Suppose we knew that for some constant C, Since f 2 Fr < 1 by hypothesis, equation (4.1) would say F i 2 Fr = F i, F i 2 F i, F i 2 C f 2 Fr. (4.11) lim F i 2 2 lim C 2 = and we would be done. The immediate fact that F i 2 2 f 2 2 reduces a proof of (4.11) to demonstrating the inequality Proceeding directly, F i, F i C f 2 Fr. F i, F i = I 1 + I 2 where I 1 and I 2 are integrals which have representations in cusp coordinates ζ = x + iy around a i of the form I 1 = I 2 = 2 y> 1 y> 1 ( y ) [ 2 2 f ψ ( y ) ( ψ ψ y x f y 2 ) f ] ( y ) ( f + ψ ψ y ) f 2 2 dxdy f y dxdy. Let s get a bound on I 1 first. Using the fact that y = 2, I y 1 C 1,ψ ( ψ y (2 ) 2 f 2 2 y> C 1,ψ f 2 2 ) ( y ) f 2 ψ dxdy y 2 2 dxdy y> f 2 a f f dxdy y 2 1 ( y ψ is negative definite and ψ vanishes above y> X Γ f f dxdy y 2 = C 1,ψ f Fr. 1 ( y ) 2 ψ f f dxdy y 2 ) 2 f f dxdy Here C 1,ψ is a constant that depends only on ψ, the cutoff function. For I 2 we can use an integration by parts and the fact that ψ (y) = for y > 2 to obtain I 2 = 1 2 = 1 2 ( y ) ( ψ ψ y ( 1 ( y y ψ ) f 2 y ) ( ψ y y 2 dxdy ) ) f 2 dxdy.

31 Chapter 4. Spectrum of Fr 27 Applying absolute values, we obtain a bound: I 2 = = C 2,ψ f 2 2 ( 1 ( y y ψ ( 1 ( y y ψ ( y ) ( y ψ ψ ) ψ ( y ) ( ψ y ) ) f 2 dxdy ) + ψ ( y ) ) f 2 (2 ) 2 dxdy y 2 ) 2 f 2 dxdy y 2 where again C 2,ψ is some constant that depends on ψ. Choosing C := C 1,ψ + C 2,ψ completes the proof of equation (4.11), and hence the proposition. Corollary The spectrum sp( Fr ) is discrete, and the operator ( Fr λ) 1 exists and is compact for λ sp( Fr ). Moreover, ( Fr λ) 1 is meromorphic in λ. Proof. Proposition and corollary imply that ( Fr ) 1 is continuous and self-adjoint respectively, when considered as a map Λ a L 2 (X Γ ) Λ a H 1 (X Γ ). We just proved (proposition 4.3.1) that the inclusion Λ a H 1 (X Γ ) Λ a L 2 (X Γ ) is compact. It follows that the composition ι ( Fr) 1 : Λ a L 2 (X Γ ) Λ a H 1 (X Γ ) Λ a L 2 (X Γ ) is also a compact map. For convenience we abuse notation and denote ι ( Fr ) 1 simply by ( Fr ) 1. The regular theory of compact, self-adjoint operators now says that ( Fr ) 1 has only point spectrum; points of sp(( Fr ) 1 ) can only accumulate at. We claim there is a bijection between sp( Fr ) and sp(( Fr ) 1 ) \ {} taking λ λ 1. Indeed, the point spectra are the same: from ( Fr) 1 λ 1 = ( Fr) 1 (λ Fr)λ 1 = ( Fr) 1 ( Fr λ)λ 1 we can deduce Similarly, from we see that ( Fr) 1 λ 1 injective = Fr λ injective. Fr λ = Fr(λ 1 ( Fr) 1 )λ = Fr(( Fr) 1 λ 1 )λ Fr λ injective = ( Fr) 1 λ 1 injective. Together these imply there is a bijection λ λ 1 at least between the point spectra of Fr and ( Fr ) 1. Moreover, we know that sp( Fr ) simply because ( Fr ) 1 exists and is continuous. If λ 1 sp(( Fr ) 1 ), then ( Fr ) 1 λ 1 is injective with continuous, everywhere defined inverse denoted by (( Fr) 1 λ 1 ) 1 = (λ 1 ( Fr) 1 ). Inverting the identity Fr λ = Fr (( Fr ) 1 λ 1 )λ then allows us to write ( Fr λ) 1 = λ 1 (λ 1 ( Fr) 1 ) 1 ( Fr) 1

32 Chapter 4. Spectrum of Fr 28 from which we deduce that ( Fr λ) 1 is also continuous and everywhere defined. In other words, λ 1 sp(( Fr) 1 ) = λ sp( Fr). Taking the contrapositive of this statement, we conclude sp( Fr ) is in bijection with sp(( Fr ) 1 ) \ {}. The meromorphicity follows immediately from proposition

33 Chapter 5 Meromorphic Continuation Throughout this section we make the notational convention that λ = λ s = s(s 1) where s C, and set Ω := { s C Re(s) > 1 } 2 and λ s sp( Fr ). Since Fr is a positive operator, we know Ω [ 1 2, 1] =. The main references for this section were [2] and [3]. 5.1 Another description of E(z, s) This section presents an alternative characterization of the Eisenstein series E(z, s). This characterization will be valid for s Ω. The key ingredients are a certain pseudo-eisenstein series obtained by truncating y s, and its images under the operators ( λ) and ( Fr λ) 1. More precisely, recall from section 4.3 the cutoff function ψ : (, ) R given by ψ(y) = if y (, 1] ψ(y) 1 if y (1, 2) ψ(y) = 1 if y [2, ). Set ϕ(y) := ψ( y ) and form a pseudo-eisenstein series h(ζ, s) in a-coordinates by automorphising the product y ϕ(y) y s : h(ζ, s) := ϕ(im(γζ)) Im(γζ) s (5.1) γ B(Z)\σa 1 Γσ a Since ϕ is supported in the cuspidial region V a, there can only be one non-zero summand in equation (5.1). Thus the sum defining h(ζ, s) converges everywhere on X Γ. In fact, is can be checked that s h(, s) is a holomorphic distribution on X Γ. 29

34 Chapter 5. Meromorphic Continuation 3 For convenience set H(z, s) := ( λ)h(z, s). Define K : X Γ Ω C according to the formula K(ζ, s) := h(ζ, s) ( Fr λ) 1 H(ζ, s). It is valid to apply ( Fr λ) 1 to H, as a simple computation shows that H L 2 (X Γ ) = dom( Fr λ) 1. We restrict to Re(s) > 1/2 since in this case we know K does not vanish. Indeed, if Re(s) > 1/2: h(, s) L 2 (X Γ ) ( Fr λ) 1 H(, s) dom( Fr λ) L 2 (X Γ ). Thus, when Re(s) > 1/2, K is the difference of an L 2 and non-l 2 function and so cannot vanish. Our claim is that K(z, s) is the same as the usual Eisenstein series E(z, s), at least for s Ω. Proposition For s Ω, the function K(z, s) h(z, s) is the unique element of dom( Fr ) satisfying ( Fr λ)[k(z, s) h(z, s)] = H(z, s). (5.2) Proof. Suppose K h satisfies equation (5.2). Then uniqueness follows from the fact that ( Fr λ) is an injective map. With this in mind we focus on proving (5.2). First, use the definition of K to obtain the identity K(z, s) h(z, s) = h(z, s) ( Fr λ) 1 H(z, s)) h(z, s) = ( Fr λ) 1 H(z, s). This equation implies K h dom( Fr ). Applying ( Fr λ) to both sides therefore yields ( Fr λ)[k(z, s) h(z, s)] = H(z, s). (5.3) On the other hand, if Re(s) > 1 then the series defining E(z, s) converges, and E(z, s) h(z, s) dom( Fr ). Notably, the function E h is smooth, so an application of ( Fr λ) can be replaced by an application of the resolvent ( λ) of the usual Laplacian. We get ( Fr λ)[e(z, s) h(z, s)] = ( λ)[e(z, s) h(z, s)] = ( λ)h(z, s) (5.4) = H(z, s). Combining equations (5.3) and (5.4) with the uniqueness result shows that K(z, s) is merely the usual Eisenstein series E(z, s) for Re(s) > 1. Moreover, K provides a continuation of E (in the s-parameter) to the domain Ω.

35 Chapter 5. Meromorphic Continuation The functions E (z, s) Definitions Let h(z, s) be the same psuedo-eisenstein series as in section 5.1, and again set H(z, s) = ( λ)h(z, s). Inspired by the characterisation of E(z, s) via the identity E = h ( Fr λ) 1 H, let us set E (z, s) := h(z, s) ( Fr λ) 1 H(z, s). This time we use the fact that H(z, s) Λ a L 2 (X Γ ) = dom( Fr λ) to justify the application of ( Fr λ) 1 ) to H. Proposition For s not contained in the discrete set sp( Fr ), E (z, s) h(z, s) is the unique element of dom( Fr ) satisfying ( Fr λ)[e (z, s) h(z, s)] = H(z, s). Proof. Clearly E (z, s) h(z, s) dom( Fr ) when defined. The function h(z, s) is entire in s, and ( Fr λ) = ( Fr s(s 1)) is a meromorphic operator-valued function of s. It follows that s E (z, s) is meromorphic. Thus the expression E (z, s) h(z, s) is valid for s not contained sp( Fr ). For such s, unravelling definitions gives ( Fr λ)[e (z, s) h(z, s)] = ( Fr λ)[h(z, s) ( Fr λ) 1 H(z, s) h(z, s)] = H(z, s). Uniqueness follows from the fact that ( Fr λ) is an injective map Constant terms In order to understand the constant term map c a [E (, s)]: N\H C it is useful to think of N\H (, ) as the union N\H = (, ) { } (, ). Let s investigate what happens on each of these three regions. i. z = x + iy with y (, ): The constant term of ( Fr λ) 1 H(z, s) vanishes above since it s an element of Λ a L 2 (X Γ ). Therefore c a [E (, s)] = c a [h(, s)] + c a [( Fr λ) 1 H(, s)] = c a [h(, s)] = y s. ii. y (, ): Proposition enables the following computation, valid in the sense of distributions: H(z, s) = ( Fr λ)(e (z, s) h(z, s)) = ( λ)(e (z, s) h(z, s)) + C(s) T = ( λ)e (z, s) H(z, s) + C(s) T.