Construction of Poincaré-type series by generating kernels
Yasemin Kara, Moni Kumari, Jolanta Marzec, Kathrin Maurischat, Andreea Mocanu, Lejla Smajlović
aa r X i v : . [ m a t h . N T ] F e b CONSTRUCTION OF POINCAR ´E-TYPE SERIES BY GENERATINGKERNELS
YASEMIN KARA, MONI KUMARI, JOLANTA MARZEC, KATHRIN MAURISCHAT, ANDREEAMOCANU AND LEJLA SMAJLOVI ´C
Abstract.
Let Γ ⊂ PSL ( R ) be a Fuchsian group of the first kind having a fundamentaldomain with a finite hyperbolic area, and let e Γ be its cover in SL ( R ). Consider thespace of twice continuously differentiable, square-integrable functions on the hyperbolicupper half-plane, which transform in a suitable way with respect to a multiplier system ofweight k ∈ R under the action of e Γ. The space of such functions admits the action of thehyperbolic Laplacian ∆ k of weight k . Following an approach of [JvPS16] (where k = 0),we use the spectral expansion associated to ∆ k to construct a wave distribution andthen identify the conditions on its test functions under which it represents automorphickernels and further gives rise to Poincar´e-type series. An advantage of this methodis that the resulting series may be naturally meromorphically continued to the wholecomplex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in thediscrete spectrum of ∆ k . Contents
1. Introduction 21.1. Poincar´e-type series 21.2. Our results 31.3. Outline of the paper 41.4. Acknowledgement 42. Preliminaries 52.1. Basic notation 52.2. Weighted Laplacian 52.3. Unitary multiplier system 62.4. Spectral expansion 83. The automorphic kernel 103.1. Selberg Harish-Chandra transform 103.2. Resolvent kernel and pre-trace formula 124. Sup-norm bounds for the eigenfunctions associated to discrete eigenvalues 125. The wave distribution associated to the weighted Laplacian 155.1. The heat and Poisson kernel 155.2. The wave distribution and its integral representation 186. The basic automorphic kernel 256.1. Construction and meromorphic continuation of the basic automorphic kernel 256.2. The geometric automorphic kernel 27References 30
Mathematics Subject Classification.
Key words and phrases.
Wave distribution, automorphic kernel, real weight Laplacian, multipliersystem, sup-norm bound. . Introduction
Let Γ ⊂ PSL ( R ) be a Fuchsian group of the first kind having a fundamental domain F with a finite hyperbolic area. It acts on the complex upper half-plane H and the quotientspace can be identified with the Riemann surface M = Γ \ H .We fix a real weight k , such that there exists a unitary multiplier system χ of weight k on the cover e Γ of Γ in SL ( R ). Throughout the paper we will assume the weight k andthe multiplier system to be arbitrary but fixed.The hyperbolic Laplacian of weight k on M is the operator∆ k = − y ( ∂ x + ∂ y ) + 2 kiy∂ x acting on the space D k of all twice continuously differentiable, square-integrable functionson H which transform in a suitable way with respect to the weight k unitary multipliersystem χ on e Γ (a precise definition is given in Section 2).The operator ∆ k is a special case of a differential operator investigated by Maassin [Maa52] (see also [Roe56]); in some papers (e.g. [Os90]), ∆ k is referred to as theMaass-Laplacian. It is the analogue of the non-Eucledian Laplacian for non-analyticautomorphic forms on Γ \ H of weight k . Namely, the weighted Laplacian preserves thetransformation behavior of functions from D k (cf. formula (7) below); it can be rep-resented as a composition of differential operators of the first order (the lowering andraising Maass operators) mapping weight k forms into forms of weight k ± k ± k − k (1 − k ) is isomorphic to the setof meromorphic differentials on F of weight k with unitary multiplier system, see [Fi87,Section 1.4] and references therein.The spectral theory of ∆ k was carefully developed in [Roe66, Roe67], [El73a, El73b]and [Pa75, Pa76a, Pa76b]. It was proved that the Hilbert space of all square-integrablefunctions on H which transform in a suitable way with respect to the weight k unitarymultiplier system on e Γ is a direct sum of countably many finite dimensional eigenspacesspanned by the eigenfunctions ϕ j associated to the discrete eigenvalues λ j of ∆ k and, inthe case when the surface is non-compact, the Eisenstein series associated to each cuspand corresponding to the continuous spectrum.In the seminal paper [Fay77], Fay computed the basic eigenfunction expansions (in bothrectangular and hyperbolic polar coordinates) of the resolvent kernel for the operator ∆ k acting on the Hilbert space of real weight k automorphic forms. He applied it to theconstruction of an automorphic “prime form” and automorphic functions with prescribedsingularities. Following the ideas of Selberg [Se56], Hejhal developed the trace formulafor ∆ k on the space Γ \ H and derived various applications of it, such as a distribution ofpseudo-primes and the computation of the dimension of the space of classical cusp formsof weight 2 k ∈ N .1.1. Poincar´e-type series.
The resolvent kernel for ∆ k is the special case of Poincar´e-type series, which is the series defined by summing over the group Γ (or over its cover e Γ)the weight k point-pair invariant, multiplied by certain other factors depending on theelement of Γ (or e Γ) in the index of the sum. Loosely speaking, the weight k point-pairinvariant is a function k ( z, w ) of z, w ∈ H which is radial/spherical (meaning it dependsonly upon the hyperbolic distance between z and w ) and which transforms “nicely” withrespect to the weight k pseudo-action of e Γ. In most applications (and so is the case withthe resolvent kernel), the point-pair invariant is taken to depend upon one or two complexparameter(s) with large enough real part (to ensure the convergence of the series). hen the point-pair invariant is well-chosen, Poincar´e-type series become very impor-tant objects of study. This is because their Fourier expansions (in different coordinates)together with a Laurent/Taylor series expansion in an additional complex variable s (usu-ally at s = 1 or s = 0) carry important information. For example, when the multipliersystem is identity, the constant term in the Laurent series expansion of the resolvent ker-nel at s = 1 gives rise to a holomorphic automorphic “prime form”, see [Fay77, Theorem2.3]. Such a form is an important object in the construction of automorphic forms withprescribed singularities. In case when M is compact and the multiplier system is one-dimensional, the constant term in the Laurent series expansion of the resolvent kernel at s = 1 gives rise to a unique Prym differential with multipliers, see [Fay77, p. 163]. Theresolvent kernel asymptotic is used more recently in [FJK19] with k ∈ N to establisheffective sup-norm bounds on average for weight 2 k cusp forms for Γ.In the series of papers [Br85, Br86a, Br86b], Bruggeman constructed families of Eisen-stein and Poincar´e series on the full modular group, which depend on a complex variable s parameterizing the eigenvalue 1 / − s of ∆ k , and also depend continuously (in factreal-analytically) on the weight k (which in this special case can be taken to belong tothe interval (0 , k = 0, all square-integrable modularforms (of a certain type) occur in such families.1.2. Our results.
In this paper, we follow an approach of [JvPS16] and [CJS20] towardsthe construction of Poincar´e-type series associated to the weighted Laplacian. It is similarto Bruggeman’s in the sense that we undertake the “operator” approach, looking atappropriate distributions. However, our starting point is the “wave distribution” (seeDefinition 16), a concept which does not appear in [Br85]–[Br86b].We illustrate this approach in Section 6, by constructing a new Poincar´e-type series K s ( z, w ), for z, w ∈ F and Re( s ) ≫
0, which transforms “nicely” with respect to theweight k multiplier system. We then obtain its meromorphic continuation with respectto the s variable and deduce its representation in terms of the sum over the group of acertain point-pair invariant.More precisely, starting with the spectral expansion theorem ([Fi87, Theorem 1.6.4]),we construct the wave distribution associated to the weighted Laplacian. We prove inProposition 17 that the wave distribution acts on a rather large space of test functionsand that it can be represented as an integral operator with a certain kernel (Theorem18). In Proposition 17, we also derive sufficient conditions on the test function so thatthe wave distribution acting on this test function produces an L -automorphic kernel.To guarantee the absolute convergence of the aforementioned objects we need boundsfor their discrete and, in case when the surface is non-compact, continuous spectra. Itturned out that even though the spectral properties of ∆ k are well studied, both analyti-cally and computationally (see [St08]), the properties of the eigenvalues associated to itsdiscrete spectrum that are different from the minimal eigenvalue | k | (1 − | k | ) did not getmuch attention in the non-compact setting. This is probably because in the non-compactsetting the discrete spectrum still remains very mysterious; for example it is not evenknown in general whether it is finite or infinite.For that reason, in Section 4 we prove the sup-norm bound sup z ∈F | ϕ j ( z ) | ≪ | λ j | forthe eigenfunctions associated to discrete eigenvalues λ j of ∆ k , uniform in j . This resultis of independent interest, because it is proved in a general setting of a possibly non-compact surface, real weight k and vector-valued eigenfunctions ϕ j . In the non-compactsetting, we also derive the sup-norm bound for the growth of a certain weighted integralof the (vector-valued) Eisenstein series along the critical line (Proposition 12.(b)). he definition of the wave distribution enables one to construct automorphic ker-nels through the action of the wave distribution on suitably chosen test functions. InSection 6, a new L -automorphic kernel K s ( z, w ), called the basic automorphic ker-nel, is constructed through the action of the wave distribution on the test function g s ( u ) = Γ( s − / s ) cosh( u ) − ( s − / for Re s ≫
0. The kernel K s ( z, w ) is called the basickernel, because, as will be seen in [KKMMMS20], both the resolvent kernel and, conse-quently, the Eisenstein series, can be expressed in terms of this kernel and its translatesin the s -variable. It is analogous to the basic automorphic kernel constructed in [CJS20]in the setting of smooth, compact, projective K¨ahler varieties.Using the properties of the wave distribution, it is possible to construct Poincar´e-typeseries that are not square-integrable by taking appropriate sums/integrals of the wavedistribution (see e.g. [JvPS16, Section 7] in the special case of the multiplier systemequal to 1). We leave this investigation to the subsequent paper [KKMMMS20].This approach to the construction of Poincar´e-type series has many advantages. Firstly,the construction depends only on the spectral properties of the Laplacian, and we believeit can be applied in more general settings, with the Laplacian replaced by the Casimirelement (see a discussion in Section 2.2 below). Second, the problem of the meromorphiccontinuation of Poincar´e series, which is usually attacked by means of Fourier seriesexpansion and serious analytic considerations related to the coefficients in those series, issimplified. Namely, the meromorphic continuation essentially boils down to establishinga suitable functional relation between the Fourier transform of the test function at s andat s + α , for a suitable translation parameter α (see Lemma 19 below). For this reason,the scaling factor Γ( s − / s ) appears in the test function g s ( u ) above.Moreover, this approach provides additional flexibility in the construction of Poincar´eseries, depending on the desired properties of the series, under the action of ∆ k . Namely,assume that one is interested in the construction of Poincar´e series P s ( z, w ) on M , suchthat ∆ k P s ( z, w ) equals a certain function of P s ( z, w ). Then, representing P s ( z, w ) asthe wave distribution acting on an unknown test function, this construction boils downto solving a second order differential equation satisfied by this test function, with somenatural boundary conditions, such as e.g. decay to zero as Re( s ) → ∞ . This task is noteasy, but it may turn out to be easier than solving the partial differential equation thatis to be satisfied by the point-pair invariant generating the series P s ( z, w ) as a sum overthe group Γ (or its cover e Γ).1.3.
Outline of the paper.
The paper is organized as follows: in Section 2, we intro-duce the basic notation, define the weighted Laplacian, the unitary multiplier system andthe spaces of functions we are interested in and we recall the spectral expansion theo-rem. In Section 3, the construction of the geometric automorphic kernel is presented,following the approach undertaken in [He76] and [He83] and the pre-trace formula for theresolvent kernel is recalled from [Fi87]. Section 4 is devoted to proof of the non-trivialsup-norm bound for the eigenfunctions of the weighted Laplacian, a result necessary forthe construction of the wave distribution associated to ∆ k in Section 5. Properties of thewave distribution are identified in Section 5 and applied to the construction of the basicautomorphic kernel in Section 6.1.4. Acknowledgement.
The authors would like to thank the organizers and sponsorsof WINE3 for providing a stimulating atmosphere for collaborative work. . Preliminaries
Basic notation.
Let Γ ⊂ PSL ( R ) denote a Fuchsian group of the first kind. Itacts by fractional linear transformations on the hyperbolic upper half-plane H := { x + iy | x, y ∈ R ; y > } . We choose once and for all a connected fundamental domain F ⊆ H for Γ. We further assume F (and therefore every fundamental domain) to have finitehyperbolic area. Then M := Γ \ H is a finite volume hyperbolic Riemann surface, whichwe allow to have elliptic fixed points and c Γ cusps. Locally, M is identified with itsuniversal cover H , and each point on M has a unique representative in F . We rely onthis identification of M with F whenever a definition of a function on M uses the choiceof a representative in H . This in particular applies to the kernel functions in this paper.Let e Γ denote the cover of Γ in SL ( R ), i.e. the set of all matrices γ ∈ SL ( R ) such that[ ± γ ] ∈ Γ. Throughout this paper, assume that e Γ contains − I , where I stands for theidentity element of SL ( R ).Let µ hyp denote the hyperbolic metric on M , which is compatible with the complexstructure of M , and has constant negative curvature equal to minus one. The hyperbolicline element ds is given by ds := dx + dy y . Denote the hyperbolic distance from z ∈ H to w ∈ H by d hyp ( z, w ). It satisfies the relationcosh (cid:0) d hyp ( z, w ) (cid:1) = 1 + 2 u ( z, w ) , (1)where u ( z, w ) := | z − w | z )Im( w ) . (2)In the sequel, we will need the displacement function σ ( z, w ), which is defined as σ ( z, w ) := 1 + | z − w | z )Im( w ) = | z − w | z )Im( w ) . (3)2.2. Weighted Laplacian.
For any real k , denote by∆ k = − y ( ∂ x + ∂ y ) + 2 kiy∂ x the hyperbolic Laplacian on M of weight k , which will be applied to twice differentiablefunctions f : H → C . H. Maass [Maa52] introduced in broader generality, for realnumbers α and β , the differential operator∆ α,β = − y (cid:0) ∂ x + ∂ y (cid:1) + ( α − β ) iy∂ x − ( α + β ) y∂ y . Specializing to α + β = 0, we recover the classical Laplace-Beltrami operator on H ofweight α − β (which is, among others, subject of Roelcke’s work [Roe56, Roe66, Roe67]).There is a slight ambiguity in the notation used: the operator ∆ α,β with α − β = α + β = k is also called the weighted Laplacian of weight k in the literature. It is thatone which is used in the theory of mock modular forms (see e.g. [BK18]).Our choice of the weighted Laplacian is the specialization to weight 2 k of the Laplace-Beltrami operator e ∆ = − y (cid:0) ∂ x + ∂ y (cid:1) + y∂ x ∂ θ , which in turn equals the Casimir operator for SL(2 , R ), up to a multiplicative constant.More precisely, the action of SL(2 , R ) on L ( e Γ \ SL(2 , R )) by right translations comesalong with an action of its Lie algebra on C ∞ -vectors, given by differential operators.The Casimir element generates the center of the universal enveloping Lie algebra and, ritten with respect to the coordinates ∂ x , ∂ y , ∂ θ , this operator coincides with the Laplace-Beltrami operator ˜∆ above. By Schur’s lemma, the Casimir acts as a constant on anyirreducible representation of SL(2 , R ). In turn, any eigenfunction of the Casimir, respec-tively the Laplace-Beltrami, together with its SL(2 , R )-translates generates an irreduciblerepresentation. On the other hand, when restricting to eigenfunctions of weight 2 k forthe maximal compact subgroup SO(2) of SL(2 , R ), the Casimir operator specializes toour choice of the weighted Laplacian ∆ k . In turn, the isomorphism of e Γ \ SL(2 , R ) / SO(2)with Γ \ H induces an isomorphism of the SO(2)-eigenfunctions on e Γ \ SL(2 , R ) with auto-morphic forms of weight 2 k on H .2.3. Unitary multiplier system.
For every γ = (cid:18) ∗ ∗ c d (cid:19) ∈ e Γ and every complexnumber z , define j ( γ, z ) := cz + d and J γ,k ( z ) := exp(2 ik arg j ( γ, z )). Definition 1.
A function µ : H → C ∗ satisfying the transformation property µ ( γz, γw ) = µ ( z, w ) J γ,k ( z ) J γ,k ( w ) − for all γ ∈ SL ( R ) and all z, w ∈ H is called a weight k point-pair invariant .Note that, due to the fact that SL ( R ) acts transitively on point-pairs of a fixed hy-perbolic distance, a point-pair invariant of weight zero is just an ordinary point-pairinvariant depending only on the hyperbolic distance of the point-pair ( z, w ). Further, if µ is a weight k point-pair invariant and Φ is a weight zero point-pair invariant, then µ · Φis also a weight k point-pair invariant. Furthermore, if ν is also a weight k point-pairinvariant, then µ/ν is a point-pair invariant of weight zero. Lemma 2.
Decompose the real number k = k + k with k ∈ Z and k ∈ ( − , ] , anddefine z k = z k exp( k log z ) for the principal branch of the complex logarithm log z . Thefunction H k : H → C ∗ given by H k ( z, w ) := (cid:18) − | z − w | ( z − w ) (cid:19) k = (cid:18) z − ww − z (cid:19) k = (cid:18) (1 − ζ ) | − ζ | (cid:19) k , for ζ = z − wz − w , is a weight k point-pair invariant.Proof. The function r : H → C given by r ( z, w ) = 1 − z − wz − w = 2 i Im wz − w transforms under SL ( R ) as(4) r ( γz, γw ) = r ( z, w ) (cid:18) cz + dcw + d (cid:19) . Note that, for all z, w ∈ H , we have z − w ∈ H . In particular, 0 ≤ arg( z − w ) < π , whichimplies that − π < arg( r ) = π − arg( z − w ) ≤ π . We claim that the argument of r transforms asarg( r ( γz, γw )) = arg( r ( z, w )) + arg( cz + d ) − arg( cw + d ) . To see this, notice that both cz + d and cw + d belong either to upper or lower complex half-plane. In particular, either arg( cz + d ) , arg( cw + d ) ∈ (0 , π ) or arg( cz + d ) , arg( cw + d ) ∈ − π, (cid:18) cz + dcw + d (cid:19) = arg( cz + d ) − arg( cw + d ) . Consequently, since in (4) both values of r have arguments in ( − π , π ] and sincearg( r ( γz, γw )) = arg( r ( z, w )) + arg (cid:18) cz + dcw + d (cid:19) + 2 πl for some l ∈ Z , we must have l = 0. Since H k ( z, w ) = H k ( r ( z, w )) = (cid:18) r | r | (cid:19) k · (cid:18) r | r | (cid:19) k , the claim of the lemma follows trivially for k = k ∈ Z , using the definition of J γ,k . For k = k ∈ ( − , ], it follows from the above by noticing that, for exponent 2 k ∈ ( − , e i arg( z ) ) k = e ik arg( z ) . The lemmafollows for arbitrary real k from our choice of the k -th power. (cid:3) For every γ = ( a a a a ) and γ = (cid:0) b b b b (cid:1) ∈ SL ( R ), write γ γ = ( c c c c ). For every z ∈ H , we have a γ z + a = c z + c b z + b and therefore there exists an integer w ( γ , γ ) ∈ {− , , } , which is independent of z ,such that(5) 2 πw ( γ , γ ) = arg( a γ z + a ) + arg( b z + b ) − arg( c z + c ) . The function ω k ( γ , γ ) := exp(4 πikw ( γ , γ )) is called a factor system of weight k .Let ( V, h· , ·i ) be a d -dimensional unitary C -vector space ( d < ∞ ), where the innerproduct h· , ·i is semi-linear in the first argument. Let U ( V ) denote the unitary group,i.e. the automorphisms u of V respecting the scalar product, h u ( v ) , u ( w ) i = h v, w i for all v, w ∈ V . Definition 3. A (unitary) multiplier system of weight k on e Γ is a function χ : e Γ → U ( V )which satisfies the properties:( a ) χ ( − I ) = e − πik id V and( b ) χ ( γ γ ) = ω k ( γ , γ ) χ ( γ ) χ ( γ ).If e Γ contains parabolic elements, then there exists a unitary multiplier system on e Γ forevery weight k ∈ R ; when e Γ does not contain parabolic elements, a unitary multipliersystem on e Γ exists for certain rational values of weight k , depending on the signature ofthe group Γ, see [Fi87, Proposition 1.3.6]. From now on, we fix k ∈ R such that thereexists a unitary multiplier system χ : e Γ → U ( V ) of weight k on e Γ, which we also fix.
Lemma 4.
For every weight k point-pair invariant µ such that the series S Γ ,µ ( z, w ) := X γ ∈ e Γ χ ( γ ) J γ,k ( w ) µ ( z, γw ) is absolutely convergent for all z, w ∈ H , we have the identity S Γ ,µ ( ηz, w ) J η,k ( z ) − = χ ( η ) S Γ ,µ ( z, w ) for all η ∈ e Γ . roof. We have to prove that χ ( η ) X γ ∈ e Γ χ ( γ ) J γ,k ( w ) µ ( z, γw ) = X γ ∈ e Γ χ ( γ ) J γ,k ( w ) J η,k ( z ) − µ ( ηz, γw )for every η in e Γ. Setting γ ′ = η − γ and summing over γ ′ instead of γ by absoluteconvergence of the series, the above follows from the definitions of multiplier system andweight k point-pair invariant, combined with the implication of (5) that, for any w ∈ H and any η, γ ∈ e Γ, ω k ( η, γ ) = J η,k ( γw ) J γ,k ( w ) J ηγ,k ( w ) − . (cid:3) Spectral expansion.
For every γ ∈ e Γ, define the linear operator | [ γ, k ] on the spaceof functions f : H → V by f | [ γ, k ]( z ) := f ( γz ) J γ,k ( z ) − . It is important to notice that ∆ k commutes with | [ γ, k ], in other words∆ k ( f | [ γ, k ]) = (∆ k f ) | [ γ, k ]for every twice continuously differentiable function f : H → V . It follows that, if f issuch a function and it additionally satisfies(6) f | [ γ, k ] = χ ( γ ) f for every γ ∈ e Γ, then(7) (∆ k f ) | [ γ, k ] = χ ( γ )∆ k f. Notice that if f , f : H → V are functions satisfying (6) then h f , f i is a e Γ-invariant,vector-valued function on H . Let F denote an arbitrary fundamental domain of Γ. Let H k denote the space of (equivalence classes of µ hyp -almost everywhere equal) µ hyp -measurablefunctions f : H → V which satisfy the properties:( a ) f | [ γ, k ]( z ) = χ ( γ ) f ( z ) for all γ ∈ e Γ and( b ) k f k := R F h f, f i dµ hyp < ∞ .It follows that H k is a Hilbert space when equipped with the scalar product( f, g ) := Z F h f, g i dµ hyp . For all f , f ∈ H k , the function h f , f i given by the scalar product on V determines analmost everywhere well-defined function on H . In particular, when V = C , h f, g i = ¯ f · g and ( f, g ) = R F f ( z ) g ( z ) dµ hyp ( z ) is the usual L -scalar product. From now on, theequivalence class of a function f : H → V under the equivalence relation µ hyp -almosteverywhere equal will be denoted by f by abuse of notation. Moreover, identify V = C d ,which implies that(8) h x, y i = d X j =1 x j y j for every x = ( x , . . . , x d ) t and y = ( y , . . . , y d ) t in V . Here, X t denotes the transpose ofa matrix X . With these conventions, measurability, differentiability, integrability, etc. ofany function f : H → V are defined component-wise. he norm on V corresponding to the scalar product h· , ·i will be denoted by | · | V . Wewill at times apply the Hermitian inner product to d × d matrices, more precisely to xy t for arbitrary x, y ∈ V . Denote the resulting norm by | · | d × d and note that | xy t | d × d = | x | V | y | V . Let D k denote the set of all twice continuously differentiable functions f ∈ H k suchthat ∆ k f ∈ H k . The operator ∆ k : D k → H k is essentially self-adjoint [Fi87, Theorem1.4.5]. Let ˜∆ k : ˜ D k → H k denote the unique maximal self-adjoint extension of ∆ k with˜ D k as its domain.In case when e Γ contains parabolic elements, let ζ , . . . , ζ c Γ denote a complete system ofrepresentatives of the e Γ-equivalence classes of cusps of e Γ. Choose matrices A , . . . , A c Γ ∈ SL ( R ), such that the stabilizers e Γ ζ j := { γ ∈ e Γ | γζ j = ζ j } are generated by − I and T j := A − j ( ) A j . Let m j denote the multiplicity of the eigenvalue 1 of χ ( T j ). For every j ∈ { , . . . , c Γ } , choose an orthonormal basis { v j , . . . , v jd } of V such that χ ( T j ) v jl = e πiβ jl v jl , with ( β jl = 0 , if 1 ≤ l ≤ m j and β jl ∈ (0 , , if m j < l ≤ d. For every z ∈ H and s ∈ C with Re( s ) >
1, define the parabolic Eisenstein series ofweight k for the cusp ζ j , the multiplier system χ and the eigenvector v jl as the series(9) E jl ( z, s ) := 12 X γ ∈ e Γ ζj \ e Γ ω k ( A j , γ ) − χ ( γ ) − v jl J A j γ,k ( z ) − (Im( A j γz )) s . This series converges uniformly absolutely in ( z, s ) ∈ H × { s ∈ C | Re( s ) > ε } forevery ε >
0, hence it defines a C ∞ -function from H × { s ∈ C | Re( s ) > } to V , whichis holomorphic in s . It was shown in [Roe67] that, for every s ∈ C such that Re( s ) > E jl ( · , s ) is an eigenfunction of ∆ k , with eigenvalue s (1 − s ):(10) ∆ k E jl ( · , s ) = s (1 − s ) E jl ( · , s ) . Furthermore, for every fixed z ∈ H , the series E jl ( z, · ) can be extended to a meromorphicfunction on C , which is denoted in the same way. This function has only simple polesin the half-plane { s ∈ C | Re( s ) > / } , which all lie in the interval (1 / , { s ∈ C | Re( s ) = 1 / } , from which it follows that E jl is continuous on H × { s ∈ C | Re( s ) = 1 / } . The Eisenstein series E jl satisfies (10) in this domain. Recallthe following theorem from [Fi87, pp. 37–38]: Theorem 5 (Spectral expansion) . Every function f ∈ ˜ D k has an expansion of the fol-lowing form: f ( z ) = X n ≥ ( ϕ n , f ) ϕ n ( z ) + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ ( E jl ( · , / it ) , f ) E jl ( z, / it ) dt, where ( ϕ n ) n ≥ is a countable orthonormal system of eigenfunctions of ˜∆ k : ˜ D k → H k .The series P n ≥ ( ϕ n , f ) ϕ n converges uniformly absolutely on compact subsets of H . When Γ is cocompact, the second sum on the right hand side of the above equation is identicallyzero.Remark . In the sequel, whenever we apply the spectral expansion theorem to cocompactΓ, we will assume that the sum over parabolic elements is identically zero and we willnot treat that case separately. et | k | (1 − | k | ) = λ ≤ λ ≤ λ ≤ . . . denote the discrete eigenvalues corresponding tothe orthonormal system ( ϕ n ) n ≥ and write(11) λ n = 1 / t n for every n where t n = p λ n − / t n ∈ (0 , iA ] when λ n < ; here, A is defined as(12) A := max { / , | k | − / } , and note that | k | (1 − | k | ) ≥ − A . Every λ n occurs with finite multiplicity µ n and theseries P n ≥ λ − n converges [Fi87, Theorem 1.6.5].3. The automorphic kernel
In this section, we recall the construction of automorphic forms for Γ with multipliersystem χ , using point-pair kernel functions (i.e. kernel functions depending only uponthe hyperbolic distance between the points).3.1. Selberg Harish-Chandra transform.
Following [He83, pp. 386–387], let Φ be areal-valued function defined on [0 , ∞ ), four times differentiable in this interval and suchthat | Φ ( ℓ ) ( t ) | ≪ ( t + 4) − δ − ℓ , for ℓ = 0 , , , , δ > max { , | k |} . To theweight k point-pair invariant(13) k ( z, w ) := H k ( z, w ) · Φ (cid:18) | z − w | Im( z ) Im( w ) (cid:19) , where z, w ∈ H , we associate the automorphic kernel(14) K Γ ( z, w ) := 12 X γ ∈ e Γ χ ( γ ) J γ,k ( w ) k ( z, γw ) , which takes values in the endomorphism ring End( V ). Note that Φ (cid:16) | z − w | Im( z ) Im( w ) (cid:17) is aweight zero point-pair invariant. Due to the bounds on the derivatives of Φ and toLemma 4, the automorphic kernel K Γ belongs to ˜ D k as a function of z .The Selberg Harish-Chandra transform h Φ of a function Φ satisfying the conditionsstated above can be computed using the following three steps:(i) compute Q ( y ) = Z ∞−∞ Φ( y + v ) (cid:18) √ y + 4 + iv √ y + 4 − iv (cid:19) k dv for y ≥ g ( u ) = Q (2(cosh u − g , i.e. h Φ ( r ) = ∞ Z −∞ g ( u ) e iru du. The Selberg Harish-Chandra transform exists for complex numbers r with suitably boundedimaginary part. Remark . A slightly different, yet equivalent version of the Selberg Harish-Chandratransform of the point-pair invariant is given in [Fay77, Theorem 1.5]. In the cited text,the automorphic kernel constructed from the point-pair invariant is defined as˜ K Γ ( z, w ) = X γ ∈ Γ χ ( γ ) (cid:18) cw + dcw + d (cid:19) k (cid:18) z − γwγw − z (cid:19) k g (cosh( d hyp ( z, γw ))) , nder the assumption that g ( u ) is a continuous function of u >
1, with a majorant g ( u ) ∈ L ∩ L (1 , ∞ ) satisfying the following condition: for any δ > m ( δ ) > z, w ∈ H with d hyp ( z, w ) > δ , g (cosh( d hyp ( z, w ))) ≤ m ( δ ) · Z d hyp ( ζ,w ) <δ g (cosh( d hyp ( ζ , w ))) dµ hyp ( ζ ) . The Selberg Harish-Chandra transform h of the point-pair invariant function g is givenby the formula h ( r ) = 2 π ∞ Z g (cosh( y )) (cid:18) y + 1 (cid:19) r F (cid:18) r − k, r + k ; 1; cosh y − y + 1 (cid:19) d (cosh( y )) , where F ( a, b ; c ; z ) stands for the (Gauss) hypergeometric function.In fact, equation (1) yields that ˜ K Γ ( z, w ) = K Γ ( z, w ), where K Γ ( z, w ) is defined by(14) with the point-pair invariant function Φ in definition (13) given by Φ( x ) = g (1 + x );in particular h = h Φ .For a function h : D → C , where D is a subset of C , and a constant a > h ( r ) is an even function.(S2) h ( r ) is holomorphic in the strip | Im( r ) | < a + ǫ for some ǫ > h ( r ) ≪ (1 + | r | ) − − δ for some fixed δ > | r | → ∞ in the set of definition ofcondition (S2).Choosing a = A as in (12), the conditions (S1)–(S3) are actually the assumptions posedon the test function h in the trace formula [He83, Theorem 6.3]. The following propositionholds: Proposition 8 ([He83, Section 9.7]) . Let A be defined as in (12) and λ j = 1 / t j as in (11) . Suppose that the Selberg Harish-Chandra transform h Φ exists and satisfiesconditions (S1)–(S3) for a = A . Then the automorphic kernel (14) admits a spectralexpansion of the form K Γ ( z, w ) = X λ j ≥| k | (1 −| k | ) h Φ ( t j ) ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ h Φ ( r ) E jl ( z, / ir ) E jl ( w, / ir ) t dr, (15) which converges absolutely and uniformly on compacta. When Γ is cocompact, according to Remark 6, the second sum on the right hand sideof (15) is identically zero.The assumptions on the test function h , which ensure the convergence of the seriesand the integral on the right-hand side of (15), can be relaxed. Namely, we will prove inSection 5 that, if the function h satisfies the conditions (S1),(S2 ′ ) h ( r ) is well-defined and even for r ∈ R ∪ [ − ia, ia ],and (S3) in the set of definition of condition (S2 ′ ) (that is, as | r | → ∞ ), then the seriesand integrals on the right-hand side of (15) are well-defined and converge absolutely anduniformly on compacta. However, the assumptions (S1), (S2 ′ ) and (S3) do not implythat the right-hand side of (15) represents a spectral expansion of some L -automorphickernel for a = A . .2. Resolvent kernel and pre-trace formula.
Let ρ ( ˜∆ k ) denote the resolvent set of˜∆ k , i.e. the set of all complex numbers λ for which the linear operator ( ˜∆ k − λ id ˜ D k ) − : H k → ˜ D k is bounded. According to [Fi87, pp. 25–27], the resolvent kernel associated tothe operator ˜∆ k is the integral kernel of the operator ( ˜∆ k − s (1 − s )) − , defined for all s ∈ C \ { k − n, − k − n | n = 0 , , , ... } with Re( s ) > z, w ∈ H such that z = γw forall γ ∈ Γ as the automorphic kernel(16) G s ( z, w ) := 12 X γ ∈ e Γ χ ( γ ) k s ( σ ( z, γw )) J γ,k ( w ) H k ( z, γw ) , with the point-pair invariant function k s ( σ ) := σ − s Γ( s − k )Γ( s + k )4 π Γ(2 s ) F ( s + k, s − k ; 2 s ; σ ) , where σ := σ ( z, w ) is defined by (3) and F ( α, β ; γ ; z ) denotes the classical Gauss hyper-geometric function.The series on the right-hand side of (16) converges normally in the variables z, w ∈ H such that z = γw and s ∈ C \ { k − n, − k − n | n = 0 , , , ... } with Re( s ) > V .Recall from [Fi87, Formula (2.1.4) on p. 46] the pre-trace formula that follows fromthe computation of the trace of the resolvent kernel G s ( z, w ): Lemma 9.
For all s, t ∈ C \ { k − n, − k − n | n = 0 , , , . . . } with Re( t ) , Re( s ) > and z ∈ H , we have X n ≥ (cid:18) λ n − λ − λ n − µ (cid:19) | ϕ n ( z ) | V + 14 π c Γ X j =1 m j X l =1 ∞ Z −∞ (cid:18) + r − λ − + r − µ (cid:19) (17) × | E jl ( z, + ir ) | V dr = − d π ( ψ ( s + k ) + ψ ( s − k ) − ψ ( t + k ) − ψ ( t − k ))+ 12 X γ ∈ e Γ \{± I } Tr( χ ( γ )) ( k s ( σ ( z, γz )) − k t ( σ ( z, γz ))) J γ,k ( z ) H k ( z, γz ) , where λ := s (1 − s ) , µ := t (1 − t ) and ψ ( x ) := Γ ′ ( x )Γ( x ) is the digamma function.Moreover, by Dini’s theorem, all the sums and integrals in (17) converge uniformly forevery s, t as above and z ∈ H . When Γ is cocompact, the sum over cusps on the left hand side of (17) is identicallyzero.4.
Sup-norm bounds for the eigenfunctions associated to discreteeigenvalues
In this section, we use (17) to derive the sup-norm bounds for the norm | ϕ n ( z ) | V , when z ∈ F . Hence, among others, we need an upper bound for the absolute value of thedifference g k ( s ; z, γz ) := k s ( σ ( z, γz )) − k s +1 ( σ ( z, γz )) analogous to the bound derived in[FJK19, Lemma 6.2]. The proof of [FJK19, Lemma 6.2] could be adopted to our settingwhen the weight k is not a positive integer or a half-integer. However, we give a directproof of a better bound, valid for all real weights k . emma 10. Let k ∈ R and let s > | k | be a real number. Then (18) | g k ( s ; z, γz ) | ≤ s π ( s − k ) σ ( z, γz ) − s . Proof.
From the definition of hypergeometric series in terms of the Pochammer symbol( a ) j := Γ( a + j ) / Γ( a ) and the identity ( a + 1) j = ( a ) j +1 /a , we obtain that k s +1 ( σ ( z, γz )) = σ ( z, γz ) − s Γ( s − k )Γ( s + k )4 π Γ(2 s + 1) ∞ X j =0 ( s + k ) j +1 ( s − k ) j +1 ( j + 1)( j + 1)!(2 s + 1) j +1 σ ( z, γz ) − ( j +1) , so that g k ( s ; z, γz ) = σ ( z, γz ) − s Γ( s − k )Γ( s + k )4 π Γ(2 s ) ∞ X j =0 ( s + k ) j ( s − k ) j j !(2 s + 1) j σ ( z, γz ) − j . Since σ ( z, γz ) ≥
1, application of [GR07, Formula 9.122.1] gives ∞ X j =0 ( s + k ) j ( s − k ) j j !(2 s + 1) j σ ( z, γz ) − j ≤ ∞ X j =0 ( s + k ) j ( s − k ) j j !(2 s + 1) j = Γ(2 s + 1)Γ( s − k + 1)Γ( s + k + 1) , which leads to g k ( s ; z, γz ) ≤ s π ( s − k )( s + k ) σ ( z, γz ) − s . The proof is complete; note that we have omitted the absolute values because all expres-sions are positive, due to the fact that s > | k | is real. (cid:3) Remark . In the case where s = k + ǫ , for some ǫ ∈ (0 ,
1) and some positive integer k , the upper bound from (18) becomes k + ǫ πǫ (2 k + ǫ ) σ ( z, γz ) − s . This is obviously less than πǫ σ ( z, γz ) − s for all positive integers k , hence the bound (18) is better than the oneobtained in [FJK19, Lemma 6.2] using the representation of the resolvent kernel as anintegral transform of the heat kernel.Next, we derive the sup-norm bound for the eigenfunctions of the weighted Laplacianassociated to discrete eigenvalues λ j , j ≥
0, and for the integral of the Eisenstein series,when M in non-compact. Throughout this section, identify the surface M with thefundamental domain F . Let Y > F Yj denote the neighbourhoodof the cusp ζ j , j ∈ { , . . . , c Γ } , characterized by A − j F Yj = { z ∈ H | − / ≤ Re( z ) ≤ / , Im( z ) ≥ Y } , where A j is the scaling matrix associated to the cusp ζ j , for every j ∈ { , . . . , c Γ } . Denoteby F Y the closure of the complement of c Γ S j =1 F Yj with respect to F (note that F = F Y ifΓ is cocompact).We introduce the constant(19) C ( k, M, d ) := d ( | k | + 2)8 π ( | k | + 1) + (cid:18) | k | + 2 | k | + 1 (cid:19) d hyp ( F ) e diam hyp ( F ) , where diam hyp ( F ) denotes the hyperbolic diameter of the fundamental domain F . Theconstant C ( k, M, d ) clearly depends upon the surface and the multiplier system, but noton the eigenvalue. With this notation, the following proposition holds: roposition 12. ( a ) Let ϕ j ( z ) be the eigenfunction of the Laplacian ∆ k associated tothe discrete eigenvalue λ j . Then (20) sup z ∈F | ϕ j ( z ) | V ≤ C ( k, M, d ) | λ j | , where the constant C ( k, M, d ) depends on the surface and the multiplier system, but noton the eigenvalue. When λ j ≥ | k | , one can take C ( k, M, d ) = ( C ( k, M, d )( | k | + 2)) . ( b ) In case when e Γ contains parabolic elements, for any j ∈ { , . . . , c Γ } and l ∈ { , . . . , m j } ,the following bound for the parabolic Eisenstein series (9) of weight k for the cusp ζ j , themultiplier system χ and the eigenvector v jl holds: (21) sup z ∈F ∞ Z −∞ + r + ( | k | + 2) ) − ( | k | + 2) | E jl ( z, + ir ) | V dr ≤ π | k | + 2 C ( k, M, d ) . Proof.
Take s = | k | + 2 and t = | k | + 3 in Lemma 9 (note that s, t / ∈ { k − n, − k − n | n =0 , , , . . . } ). Start with an upper bound for the right-hand side of (17). The sum ofthe values of digamma functions may be evaluated by applying the functional equation ψ ( z + 1) = ψ ( z ) + z − : d π | ψ ( s + k ) + ψ ( s − k ) − ψ ( t + k ) − ψ ( t − k ) | = d ( | k | + 2)8 π ( | k | + 1) . To bound the sum, use inequality (18). Recall that | J γ,k ( z ) H k ( z, γz ) | = 1 for all z and γ and that χ is unitary, so that (cid:12)(cid:12)(cid:12)(cid:12) X γ ∈ e Γ \{± I } Tr( χ ( γ )) ( k s ( σ ( z, γz )) − k t ( σ ( z, γz ))) J γ,k ( z ) H k ( z, γz ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X γ ∈ Γ \{ I } d ( | k | + 2)8 π ( | k | + 1) σ ( z, γz ) − ( | k | +2) . Furthermore, applying [FJK19, Lemma 3.7] with δ = | k | + 2, we deduce that, for any Y > z ∈ F Y , X γ ∈ Γ \{ I } d ( | k | + 2)8 π ( | k | + 1) σ ( z, γz ) − ( | k | +2) ≤ (cid:18) | k | + 2 | k | + 1 (cid:19) dB Y , where B Y = exp (cid:0) diam hyp ( F Y ) (cid:1) vol hyp ( F Y ) − . Note that, for every Y ≥ B Y is boundedby exp (cid:0) diam hyp ( F ) (cid:1) vol hyp ( F ) − . Hence, for all z ∈ F Y and Y ≥
2, the right-hand sideof (17) is bounded from above by the constant C ( k, M, d ) defined in (19).Now, specialize the pre-trace formula (17) to either one summand or one integral onthe left-hand side.( a ) Since there are only finitely many eigenvalues that are less than 3+ | k | , it is sufficientto prove (20) for eigenvalues λ j ≥ | k | . Therefore, assume that λ j ≥ | k | .Our choice of s and t in Lemma 9, together with above computations and the assump-tion on λ j , lead to the inequalitysup z ∈F Y | ϕ j ( z ) | V ≤ C ( k, M, d )( | k | + 2) λ j , which holds for all Y ≥
2. It remains to extend it to z ∈ F . ince all eigenfunctions ϕ j are continuous on F and the area of F is finite (withthe area of the boundary equal to zero, since Γ is of the first kind), one deduces that( z
7→ | ϕ j ( z ) | V ) ∈ L p ( F ) for all p ≥ z ∈F | ϕ j ( z ) | V = lim p →∞ µ ( F ) − /p k| ϕ j ( z ) | V k p = lim p →∞ µ ( F ) − Z F | ϕ j ( z ) | pV dµ hyp ( z ) /p , where k·k p denotes the L p -norm (see e.g. [Ch84, Formula (22) on p. 100] for the analogousstatement related to eigenfunctions of the Laplacian).Let { Y n } n ≥ be an increasing sequence of real numbers bigger than 2, tending to infinity.For every p >
1, the monotone convergence theorem applied to the sequence | ϕ j ( z ) | p · F Yn ( z ), where 1 F Yn ( z ) denotes the characteristic function of the set F Y n , yields that Z F | ϕ j ( z ) | pV dµ hyp ( z ) = lim n →∞ Z F Yn | ϕ j ( z ) | pV dµ hyp ( z ) ≤ C ( k, M, d ) p/ ( | k | + 2) p/ | λ j | p µ ( F ) . Therefore, sup z ∈F | ϕ j ( z ) | V ≤ C ( k, M, d ) / ( | k | + 2) / | λ j | . ( b ) If M has cusps, fix j ∈ { , . . . , c Γ } and l ∈ { , . . . , m j } . The above computationsimply that, for s = | k | + 2 and for all Y ≥ z ∈F Y ∞ Z −∞ + r + s ) − s | E jl ( z, + ir ) | V dr ≤ π | k | + 2 C ( k, M, d ) . Proceeding analogously as above, define the function G ( z ) := Z ∞−∞ + r + s ) − s | E jl ( z, + ir ) | V dr, which is continuous and non-negative on F . Applying the monotone convergence theoremto the sequence G ( z ) p · F Yn ( z ), together with the fact that the sup-norm is the limit of L p − norms, and reasoning as in the proof of part (a), we deduce thatsup z ∈F ∞ Z −∞ + r + s ) − s | E jl ( z, + ir ) | V dr ≤ π | k | + 2 C ( k, M, d ) . This completes the proof. (cid:3) The wave distribution associated to the weighted Laplacian
The heat and Poisson kernel.
In this section, we define the Poisson kernel forthe weighted Laplacian ∆ k via the heat kernel. For any t > ρ ≥
0, define the heatkernel(22) K heat ( t ; ρ ) := √ e − t/ (4 πt ) / Z ∞ ρ re − r / t p cosh( r ) − cosh( ρ ) T k (cid:18) cosh( r/ ρ/ (cid:19) dr, where T k ( x ) = 12 h ( x + √ x − k + ( x − √ x − k i , or any real k . Here the k -th powers are chosen as in Lemma 2. Note that, for k ∈ Z ,the function T k ( x ) coincides with the 2 k -th Chebyshev polynomial. The hyperbolic heatkernel on H is defined by K H ( t ; z, w ) := K heat ( t ; d hyp ( z, w )) ( z, w ∈ H ) . For any t > k ∈ R , the same argument as in [FJK16, p. 136] shows that the heatkernel K H ( t ; ρ ) is strictly monotonic decreasing with respect to ρ >
0. In the spirit of[Fay77, p. 157], the hyperbolic heat kernel on M associated to ∆ k is defined as(23) K hyp ( t ; z, w ) := 12 X γ ∈ e Γ χ ( γ ) (cid:18) cw + dcw + d (cid:19) k (cid:18) z − γwγw − z (cid:19) k K H ( t ; z, γw ) ( z, w ∈ F ) . Lemma 13.
For any k ∈ R and t > , K hyp ( t ; z, w ) converges absolutely and uniformlyon any compact subset of F × F .Proof.
Let U be a compact subset of F × F . Since χ is a unitary multiplier system and (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) cw + dcw + d (cid:1) k (cid:16) z − γwγw − z (cid:17) k (cid:12)(cid:12)(cid:12)(cid:12) = 1 for any γ ∈ e Γ and z, w ∈ F , it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ ( γ ) (cid:18) cw + dcw + d (cid:19) k (cid:18) z − γwγw − z (cid:19) k K H ( t ; z, γw ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) End( V ) = √ d | K H ( t ; z, γw ) | , (24)where | · | End( V ) denotes hermitian norm on End( V ). Therefore, in order to prove theabsolute and uniform convergence of K hyp ( t ; z, w ) for any t > z, w ) ∈ U , we needto prove the convergence of the series X γ ∈ e Γ K H ( t ; z, γw )in C . Introduce the counting function N ( ρ ; z, w ) := { γ ∈ e Γ | d hyp ( z, γw ) < ρ } , which is defined for any ρ > z, w ) ∈ U . Then [PR09] gives a bound (uniformlyfor all ( z, w ) ∈ U ) for the function N ( ρ ; z, w ), namely N ( ρ ; z, w ) = O e Γ ( e ρ ) , where the implied constant depends only on e Γ. By Stieltjes integral representation, wehave(25) X γ ∈ e Γ K H ( t ; z, γw ) = Z ∞ K heat ( t ; ρ ) dN ( ρ ; z, w ) . Using the fact that K heat ( t ; ρ ) is a non-negative, continuous and monotonic decreasingfunction of ρ , write(26) Z ∞ K heat ( t ; ρ ) dN ( ρ ; z, w ) = O (cid:18)Z ∞ K heat ( t ; ρ ) e ρ dρ (cid:19) . Following the idea of the proof of [FJK16, Proposition 3.3], we obtain that(27) K heat ( t ; ρ ) ≤ e − ρ / (8 t ) G k ( t ) , where the function G k ( t ) is given by G k ( t ) := e − t/ (4 πt ) / Z ∞ re − r / (8 t ) sinh( r/ e kr dr. ombining (25), (26) and (27), we obtain that X γ ∈ e Γ K H ( t ; z, γw ) = O e Γ ( G k ( t ) h ( t )) , with h ( t ) := R ∞ e − ρ / (8 t ) e ρ dρ and where the implied constant depends only on e Γ. Hence,the proof is complete. (cid:3)
Notice that using the notations, introduced in Section 2 of the paper, we can rewritethe heat kernel as K hyp ( t ; z, w ) = 12 X γ ∈ e Γ χ ( γ ) J γ,k ( w ) − H k ( z, γw ) − K H ( t ; z, γw ) ( z, w ∈ F ) . Now using the fact that H k is a weight k point-pair invariant, χ is a multiplier systemand the relation J η,k ( γz ) J γ,k ( z ) = ω k ( η, γ ) J ηγ,k ( z ) , ( η, γ ∈ SL ( R ) , z ∈ H ) , one can easily prove the following equations: K hyp ( t ; ηz, w ) = K hyp ( t ; z, w ) J η,k ( z ) − χ ( η ) − ,K hyp ( t ; z, ηw ) = J η,k ( w ) χ ( η ) K hyp ( t ; z, w ) , for t > , z, w ∈ H and η ∈ Γ. The hyperbolic heat kernel K hyp ( t ; z, w ) admits thespectral expansion K hyp ( t ; z, w ) = X λ j ≥ λ e − λ j t ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ e − (1 / r ) t E jl ( z, / ir ) E jl ( w, / ir ) t dr. (28)Let U ⊂ F × F be a compact subset and let t >
0. Using the sup-norm bound (20) forthe eigenfunctions ϕ j ( j ≥
0) and applying the norm arising from the Hermitian innerproduct to a d × d matrix in (8), we obtain that (cid:12)(cid:12)(cid:12)(cid:12) X λ ≤ λ j < / e − λ j t ϕ j ( z ) ϕ j ( w ) t + X λ j ≥ / e − λ j t ϕ j ( z ) ϕ j ( w ) t (cid:12)(cid:12)(cid:12)(cid:12) d × d ≪ X λ j ≥ λ λ − j , where λ = | k | (1 −| k | ) and the implied constant depends only on t and on the set U , apartfrom k and d (this is possible because the number of eigenvalues in the interval [ λ , / P λ j ≥ / λ − j is convergent ([Fi87, Theorem 1.6.5]), and so is theseries on the right-hand side of (28). Similarly, using H¨older’s inequality and Proposition17(a) below, we deduce the absolute and uniform convergence of each integral on theright hand side of (28). Therefore, for any t >
0, the series and the integrals on the righthand side of (28) converge absolutely and uniformly on every compact subset of
F × F .From the integral representation of K H ( t ; d hyp ( z, w )) and the spectral expansion (28),we deduce that K hyp ( t ; z, w ) satisfies the following estimates, stated component-wise:(29) K hyp ( t ; z, w ) = O F ,k ( t − / e − d ( z,w ) / t ) as t → , (30) K hyp ( t ; z, w ) = O F ,k ( e − λ t ) as t → ∞ . or every Z ∈ C with Re( Z ) ≥ −| k | (1 − | k | ), z, w ∈ F and u ∈ C with Re( u ) ≥
0, thetranslated by − Z Poisson kernel P M, − Z ( u ; z, w ) is defined as(31) P M, − Z ( u ; z, w ) := u √ π Z ∞ K hyp ( t ; z, w ) e − Zt e − u / t t − / dt, where the integral is taken component-wise. This kernel is a fundamental solution of theassociated differential operator ∆ k + Z − ∂ u . Furthermore, using the spectral expansionof the heat kernel K hyp ( t ; z, w ) and the identity (see [JLa03]) e − aλ = a √ π Z ∞ e − tλ e − a / t dtt / , λ ≥ , a ∈ C , Re( a ) ≥ , we have the following spectral expansion P M, − Z ( u ; z, w ) = X λ ≤ λ j < / e − u √ λ j + Z ϕ j ( z ) ϕ j ( w ) t + X λ j ≥ / e − u √ λ j + Z ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ e − u | r | E jl ( z, / ir ) E jl ( w, / ir ) t dr. Following the steps of the proof of [JLa03, Theorem 5.2], using the estimates (29)–(30)for each component of the heat kernel K hyp ( t ; z, w ) and the fact that K hyp ( t ; z, w ) − X λ j ≤ / e − λ j t = O ( e − λt ) as t → ∞ , where λ is the first eigenvalue of ∆ k bigger than 1 /
4, one can deduce that, for Re( u ) > u ) >
0, the Poisson kernel P M, − Z ( u ; z, w ) has an analytic continuation for eachentry of the matrix to Z = − /
4. The continuation is given by P M, / ( u ; z, w ) = X λ ≤ λ j < / e − u √ λ j − / ϕ j ( z ) ϕ j ( w ) t + X λ j ≥ / e − ut j ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ e − u | r | E jl ( z, / ir ) E jl ( w, / ir ) t dr, (32)where t j = p λ j − / ≥
0, for λ j ≥ / λ j < / p λ j − / The wave distribution and its integral representation.
Let L ( R ) denote thespace of absolutely integrable functions on R and let C ∞ ( R ) denote its subspace of allinfinitely differentiable functions with compact support. Definition 14.
For any a ≥
0, denote by L ( R , a ) (resp. S ′ ( R , a )) the space of even functions g in L ( R ) (resp. in the Schwartz space on R ) such that g ( u ) exp( | u | a ) isabsolutely dominated by an integrable function on R . Denote the Fourier transform ofevery g ∈ L ( R , a ) by(33) H ( r, g ) = Z ∞−∞ g ( u ) exp( iru ) du, with the domain extended to all r ∈ C for which it is well-defined. otice that, since g is assumed to be even,(34) H ( r, g ) = 2 Z ∞ cos( ur ) g ( u ) du. The following result is a generalization of Lemma 3 in [JvPS16].
Lemma 15.
Let n ≥ be an integer. ( a ) Let g ∈ L ( R , a ) be such that g ( l ) ∈ L ( R ) for ≤ l ≤ n , and lim u →∞ g ( l ) ( u ) = 0 for ≤ l ≤ n − . Then the Fourier transform H ( r, g ) is well-defined for r ∈ { z ∈ C | | Im( z ) | ≤ a } and satisfies the conditions (S1), (S2 ′ ), and (S3) with δ = n − . ( b ) Let η > and let g ∈ S ′ ( R , a + η ) be such that g ( j ) ( u ) exp( | u | ( a + η )) is absolutelybounded by some integrable function on R for ≤ j ≤ n − . Then the function H ( r, g ) satisfies the conditions (S1), (S2) for any < ǫ < η , and (S3) with δ = n − . ( c ) If g ∈ S ′ ( R , a ) , then H ( r, g ) is a Schwartz function in r ∈ R .Proof. ( a ) For all r ∈ { z ∈ C | | Im( z ) | ≤ a } , u ∈ R and g ∈ L ( R , a ), | g ( u ) exp( iru ) | ≤| g ( u ) | e | u | a is dominated by an integrable function on R , and thus H ( r, g ) is well-defined.It is also even with respect to r . Furthermore, for every r ∈ R , using the assumptions onthe decay of g ( l ) for 0 ≤ l ≤ n − g (2 j +1) (0) = 0 (since g is even), weobtain that12 H ( r, g ) = (cid:20) r sin( ur ) g ( u ) (cid:21) ∞ − Z ∞ r sin( ur ) g ′ ( u ) du = (cid:20) r cos( ur ) g ′ ( u ) (cid:21) ∞ − Z ∞ r cos( ur ) g ′′ ( u ) du = . . . = (cid:20) ( − n/ r n − sin( ur ) g ( n − ( u ) (cid:21) ∞ + ( − n/ Z ∞ r n − sin( ur ) g ( n − ( u ) du = (cid:20) ( − n/ r n cos( ur ) g ( n − ( u ) (cid:21) ∞ + ( − n/ Z ∞ r n cos( ur ) g ( n ) ( u ) du, when n is even. For odd n , we obtain a similar series of equations, terminating at ∓ (cid:20) r n sin( ur ) g ( n − ( u ) (cid:21) ∞ ± Z ∞ r n sin( ur ) g ( n ) ( u ) du. Hence, using the definition of H ( r, g ) and the integrability conditions, it follows that(1 + | r | ) n | H ( r, g ) | ≤ c · n X l =0 Z ∞ | g ( l ) ( u ) | du ≪ , for some constant c . This proves that H ( r, g ) satisfies the condition (S3) for r ∈ R with δ = n − b ) By assumption, there is an integrable function G ( u ) dominating g ( u ) exp( | u | ( a + η ))absolutely. In turn, | g ( u ) cos( ur ) | ≤ G ( u ) exp( − ( η − ǫ ) | u | )is uniformily bounded in the strip | Im( r ) | ≤ a + ǫ for 0 < ǫ < η . Hence, the integraldefining H ( r, g ) converges absolutely and uniformly on any compact set contained in sucha strip, and thus defines a holomorphic function on the open strip { r ∈ C | | Im( r ) | < a + η } . In particular, conditions (S1) and (S2) are satisfied. Similarly, for j = 1 , . . . , n −
1, the unctions g ( j ) ( u ) cos( ur ), as well as g ( j ) ( u ) sin( ur ), are bounded absolutely and uniformlyin r by some integrable functions G j ( u ) exp( − ( η − ǫ ) | u | ). Recalling the computationinvolving partial integration from part ( a ), we obtain (S3) as well.( c ) If g ∈ S ′ ( R , a ), then its Fourier transform H ( r, g ) is a Schwartz function in thevariable r ∈ R . (cid:3) We now define the wave distribution.
Definition 16 (Wave distribution) . Let z, w ∈ F . For every g ∈ C ∞ ( R ), the wavedistribution W M,k,χ ( z, w ) applied to g is defined as W M,k,χ ( z, w )( g ) := X λ j ≥| k | (1 −| k | ) H ( t j , g ) ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ H ( r, g ) E jl ( z, / ir ) E jl ( w, / ir ) t dr, (35)where p λ j − / t j ≥ λ j ≥ / t j ∈ (0 , iA ] when λ j < , where A is definedin (12). Proposition 17.
Let z, w ∈ F . ( a ) For every g as in Lemma 15 . ( a ) with a = A and n = 4 , the wave distribution W M,k,χ ( z, w ) is well-defined. ( b ) Let g ∈ S ′ ( R , A ) satisfy the conditions of Lemma 15 . ( b ) with n = 4 . Then W M,k,χ ( z, w )( g ) represents the automorphic kernel K Γ ( z, w ) = K Γ , Φ ( z, w ) for theinverse Selberg Harish-Chandra transform Φ of H ( · , g ) .Proof. ( a ) By Lemma 15, the function H ( r, g ) is well-defined for all r ∈ C with | Im( r ) | ≤ A , which implies that the finite sum X | k | (1 −| k | ) ≤ λ j < H ( t j , g ) ϕ j ( z ) ϕ j ( w ) t converges (recall that lim j →∞ λ j = ∞ ). Furthermore, if λ j ≥ , then t j ∈ R and thus H ( t j , g ) ≪ (1 + | t j | ) − as j → ∞ .Fix z, w ∈ F and observe that H ( t j , g ) ϕ j ( z ) ϕ j ( w ) t ∈ C d × d . By applying the norm | · | d × d and H¨older’s inequality, we obtain that X λ j ≥ | H ( t j , g ) ϕ j ( z ) ϕ j ( w ) t | d × d = X λ j ≥ | H ( t j , g ) || ϕ j ( z ) | V | ϕ j ( w ) | V ≪ (cid:18) X λ j ≥ | H ( t j , g ) || ϕ j ( z ) | V (cid:19) / (cid:18) X λ j ≥ | H ( t j , g ) || ϕ j ( w ) | V (cid:19) / . Note that, due to the estimate on | H ( t j , g ) | , each of the factors on the right-hand sidecan be compared with the sum occurring in the pre-trace formula (17). Use (17) with s = | k | + 2 and t = | k | + 3, as in the proof of Proposition 12, to obtain the following ound: X λ j ≥ | H ( t j , g ) || ϕ j ( z ) | V ≪ X λ j ≥ | t j | ) | ϕ j ( z ) | V ≪ X λ j ≥ t j + t j ( + 2 s ) + + s + s ( s − | ϕ j ( z ) | V ≤ C ( k, M, d )2( | k | + 2) . Convergence of the integral (uniform on compact subsets of
F × F ) can be proved ina similar way, completing the proof.( b ) Because the properties of g imposed by assumption imply those claimed in part ( a ),the wave distribution is well-defined by the spectral expansion with coefficients H ( r, g ).The claim follows from Proposition 8, once we establish that H ( r, g ) belongs to the imageof the of the Selberg Harish-Chandra transform. Hence, we have to invert steps (i)–(iii)of page 10. The inverse of H ( r, g ) under Fourier transformation (iii) is trivially g . Since g is an even C ∞ -function, the inverse Q : R + → C of g under (ii) exists and it is given by Q ( y ) = g (cid:18) (cid:18) p y + 4 + 12 √ y (cid:19)(cid:19) . It belongs to C ∞ ( R + ). Since g ( u ) exp(( A + η ) u ) → u → ∞ , we obtain that Q ( y ) ≪ exp (cid:18) − ( A + η )2 log (cid:18) p y + 4 + 12 √ y (cid:19)(cid:19) , i.e. Q ( y ) ≪ ( y + 4) − ( A + η ) . Similarly, for its first derivative Q ′ ( y ) = g ′ (cid:18) (cid:18) p y + 4 + 12 √ y (cid:19)(cid:19) · p y ( y + 4)we find that Q ′ ( y ) ≪ ( y + 4) − ( A + η ) − , and for its second one Q ′′ ( y ) = g ′′ (cid:18) (cid:18) p y + 4 + 12 √ y (cid:19)(cid:19) · p y ( y + 4) − g ′ (cid:18) (cid:18) p y + 4 + 12 √ y (cid:19)(cid:19) · y + 2( y ( y + 4)) we obtain the estimate Q ′′ ( y ) ≪ ( y + 4) − ( A + η ) − . By [He76, pp. 455–457], the inverse of Q under (i) is given by(36) Φ( x ) = − π ∞ Z −∞ Q ′ ( x + t ) √ x + 4 + t − t √ x + 4 + t + t ! k dt , where Φ ∈ C ( R + ) satisfies(37) | Φ( x ) | ≪ ( x + 4) − α and | Φ ′ ( x ) | ≪ ( x + 4) − α − , or some α > max { , | k |} . We have to show that the integral in (36) is C and satisfiesthe two conditions (37).Let β = A + η + 1. The bound on Q ′ together with its differentiability allows us toconclude that the first condition of (37) holds, once we prove that(38) − π ∞ Z ( x +4+ t ) − β √ x + 4 + t − t √ x + 4 + t + t ! k + √ x + 4 + t − t √ x + 4 + t + t ! − k dt ≪ ( x +4) − ( β − / . Let x = x + 4 and introduce the change of variables y = √ x + t − t √ x + t + t in the integral on theleft-hand side of (38). Using t = √ x (1 − y )2 √ y , x + t = x (1 + y ) y , dt = − √ x (1 + y )4 y dy , the integral becomes − π β − x − ( β − / Z (1 + y ) − β +1 ( y β + k − / + y β − k − / ) dy , which is finite if and only if β − | k | − / > −
1. This inequality in turn holds, due to ourchoice of A and η >
0. This proves (38) and the first part of (37).Next, we prove that Φ( x ) is C and that the second bound of (37) holds. In order toprove that Φ( x ) is C , it is sufficient to show that the integrand in (36) is differentiablein x and that the derivative of the integrand is bounded by some integrable function. Ifso, then Φ ′ ( x ) = − π ∞ Z −∞ ddx Q ′ ( x + t ) √ x + 4 + t − t √ x + 4 + t + t ! k dt . Differentiability of the integrand with respect to x is obvious, so it remains to prove that(39) − π ∞ Z −∞ ddx Q ′ ( x + t ) √ x + 4 + t − t √ x + 4 + t + t ! k dt ≪ ( x + 4) − ( β +1 / . Analogously to the above, starting with the bound for Q ′′ , we immediately deduce that − π ∞ Z −∞ Q ′′ ( x + t ) √ x + 4 + t − t √ x + 4 + t + t ! k dt ≪ ( x + 4) − ( β +1 / . Therefore, to prove (39) and complete the proof of part ( b ), it suffices to show that − π ∞ Z Q ′ ( x + t ) ddx √ x + 4 + t − t √ x + 4 + t + t ! k + √ x + 4 + t − t √ x + 4 + t + t ! − k dt ≪ ( x +4) − ( β +1 / . This can be done analogously to the proof of the first bound in (37), i.e. take x = x + 4and change variables to y = √ x + t − t √ x + t + t . Using the bound for Q ′ , it follows that the above ntegral is bounded by kπ x − β − / β − Z (1 + y ) − β (1 − y ) (cid:0) y β + k − / − y β − k − / (cid:1) ≪ x − β − / , because β − | k | − / > − (cid:3) Theorem 18.
Let z, w ∈ F be such that z = w . Then there exists a continuous d × d matrix-valued function W ( u ; z, w ) on R + such that the following hold:(a) W ( u ; z, w ) = P λ j < / e u √ / − λ j ( p / − λ j ) − ϕ j ( z ) ϕ j ( w ) t + O ( u ) as u → ∞ ;(b) W ( j ) ( u ; z, w ) = O ( u − j ) as u → j = 0 , , , ;(c) For any g ∈ S ′ ( R , A ) such that g ( j ) ( u ) exp( Au ) has a limit as u → ∞ and isbounded by some integrable function on R for j = 0 , , , , , we have (40) W M,k,χ ( z, w )( g ) = Z ∞ W ( u ; z, w ) g (4) ( u ) du. Proof.
For every ζ ∈ C \ { } and t = 0 in the strip { t ∈ C : | Im( t ) | ≤ A } , define w ( ζ , t ) := e − tζ − P l =0 h l, (sin t )( − ζ ) l t and set w ( ζ ,
0) := lim t → w ( ζ , t ). In the above definition, h l, ( x ) is a polynomial of degreeat most 3 such that, for all t ∈ R such that t → h l, (sin t ) = t l l ! + O ( t ) . For the explicit construction of h l, ( x ), see the proof of [CJS20, Theorem 2]. It is easy tosee that w ( ζ ,
0) is well-defined and equal to ζ . If t ∈ R + , then, using the above estimateof h l, for 0 ≤ l ≤
3, we obtain that w ( ζ , t ) = O ζ (1) as t → . Furthermore, if Re( ζ ) ≥ t ∈ R + , then e − tζ and sin( t ) are bounded functions as t → ∞ . Therefore,(41) w ( ζ , t ) = O ζ ( t − ) as t → + ∞ . Note that the above estimates also hold for w ( j ) ( ζ , t ), the j -th derivative of w ( ζ , t ) withrespect to ζ , for j = 1 , , ,
4. For z, w ∈ F with z = w and ζ ∈ C with Re( ζ ) ≥
0, definethe following matrix-valued function: f W ( ζ ; z, w ) = X λ j ≥ λ w ( ζ , t j ) ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ w ( ζ , | r | ) E j,l ( z, / ir ) E j,l ( w, / ir ) t dr, (42)where t j = p λ j − / j ≥ λ j < /
4, then t j ∈ (0 , iA ], otherwise t j ≥
0. Also, in case when Γ iscocompact, the second sum on the right hand side of (42) is identically zero.Following the same reasoning as in the proof of Proposition 17, part ( a ) (i.e., usingthe H¨older inequality, the estimate (41), comparing with the pre-trace formula (17) and sing the bounds obtained in Proposition 12), it is clear that for Re( ζ ) ≥ X λ j ≥ λ w ( ζ , t j ) ϕ j ( z ) ϕ j ( w ) t converges absolutely and uniformly on F × F . The same holds for the integral Z ∞−∞ w ( ζ , | r | ) E j,l ( z, / ir ) E j,l ( w, / ir ) t dr, for any pair ( j, l ) with 1 ≤ j ≤ c Γ and 1 ≤ l ≤ m j , when Γ is non-compact.Therefore, for every arbitrary fixed ζ ∈ C with Re( ζ ) ≥ f W ( ζ ; z, w ) is a well-definedmatrix-valued function which converges absolutely and uniformly on F × F . Moreover,any of the first 4 derivatives with respect to ζ of f W ( ζ ; z, w ) converges uniformly andabsolutely, provided Re( ζ ) >
0. Therefore, term by term differentiation is valid. Bydifferentiating component-wise four times and using the fact that d dζ w ( ζ , t ) = e − ζt , weobtain that d dζ f W ( ζ ; z, w ) = X λ ≤ λ j < / e − ζ √ λ j − / ϕ j ( z ) ϕ j ( w ) t + X λ j ≥ / e − ζt j ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ e − ζ | r | E jl ( z, / ir ) E jl ( w, / ir ) t dr = P M, / ( ζ ; z, w ) , where P M, / is defined in (31). For Re( ζ ) >
0, define(43) P ( k ) ( ζ ; z, w ) = P M, / ( ζ ; z, w ) , if k = 0 and Z ζ P ( k − ( ξ ; z, w ) dξ, if k ≥ . In the above definition, the integral is taken component-wise over a ray contained in theupper half-plane Re( ζ ) >
0. With this definition, we have(44) P (4) ( ζ ; z, w ) = f W ( ζ ; z, w ) + q ( ζ ; z, w ) , where q ( ζ ; z, w ) is a d × d matrix-valued function consisting of degree 3 polynomials in ζ , with coefficients depending on z and w at each component. For z = w and ζ →
0, thefunction P (0) ( ζ ; z, w ) has a limit; therefore,(45) P ( k ) ( ζ ; z, w ) = O ( ζ k ) as ζ → . For every u ∈ R + , define(46) W ( u ; z, w ) = 12 i h(cid:16)f W ( iu ; z, w ) + q ( iu ; z, w ) (cid:17) + (cid:16)f W ( − iu ; z, w ) + q ( − iu ; z, w ) (cid:17)i . We claim that the function W ( u ; z, w ) satisfies all the required conditions given in thestatement. Using the spectral expansion (32) of the Poisson kernel P M, / ( ζ ; z, w ) andintegrating it four times, we obtain the property ( a ). Assertion ( b ) follows using thebound (45) in (44). For a given g as in the statement, we can derive assertion ( c ) usingintegration by parts four times on the right-hand side of (40). (cid:3) . The basic automorphic kernel
In this section, we study two automorphic kernels, namely the basic and the geomet-ric automorphic kernels. After defining the basic automorphic kernel K s ( z, w ) for any z, w ∈ F in an appropriate complex half s -plane in terms of the wave distribution appliedto a test function, we prove that it has a meromorphic continuation to the whole com-plex s -plane. Then we introduce the geometric automorphic kernel ˜ K s ( z, w ) and showthat K s ( z, w ) = ˜ K s ( z, w ) for Re( s ) > max { , | k |} , thus also obtaining the meromorphiccontinuation of ˜ K s ( z, w ) to the whole complex s -plane.6.1. Construction and meromorphic continuation of the basic automorphickernel.
For z, w ∈ F and s ∈ C with Re( s ) > max { , | k |} , we define the basic automor-phic kernel K s ( z, w ) by(47) K s ( z, w ) := Γ( s − )Γ( s ) W M,k,χ ( z, w ) (cid:16) cosh( u ) − ( s − ) (cid:17) . Here, W M,k,χ ( z, w ) is the wave distribution and it is applied to the test function(48) g s ( u ) = Γ( s − )Γ( s ) cosh( u ) − ( s − ) . Notice that g s ( u ) satisfies the conditions in Lemma 15.( a ) with a = A and n = 4 where A is defined in (12). Thus, K s ( z, w ) is well-defined by Proposition 17.( a ). Lemma 19.
For all s ∈ C with Re( s ) > max { , | k |} , n ∈ N , and r ∈ R ∪ [ − Ai, Ai ] , theFourier transform H ( r, · ) (see (33) ) of g s given by (48) satisfies the functional equation (49) H ( r, g s ) = 2 − n ( s ) n (cid:0) s − − ir (cid:1) n (cid:0) s − + ir (cid:1) n H ( r, g s +2 n ) , where ( . ) n denotes the Pochhammer symbol.Proof. Let n ∈ N and s ∈ C with Re( s ) > max { , | k |} . Definitions (34) and (48) implythat H ( r, g s ) = 2Γ( s − )Γ( s ) Z ∞ cos( ur ) cosh( u ) − ( s − / du. When r ∈ R \ { } or r ∈ [ − Ai, Ai ] \ { } , or r = 0, we have (see [GR07, Formulas 3.985.1,3.512.1 and 3.512.2, respectively])(50) Z ∞ cos( ur ) cosh( u ) − v du = 2 ν − Γ( ν ) Γ (cid:18) ν − ir (cid:19) Γ (cid:18) ν + ir (cid:19) , where Re( ν ) > A . Hence, for r ∈ R ∪ [ − Ai, Ai ], using (50) with ν = s − / ν = s − / n and the Pochhammer symbol ( s ) n = Γ( s + n )Γ( s ) , we obtain the identity (49). (cid:3) The functional equation (49) enables us to deduce the meromorphic continuation ofthe kernel K s ( z, w ) to the whole complex s -plane. Theorem 20.
For any z, w ∈ F , the basic automorphic kernel K s ( z, w ) admits a mero-morphic continuation to the whole complex s -plane. The possible poles of the function Γ( s )Γ( s − / − K s ( z, w ) are located at the points s = 1 / ± it j − n , where n ∈ N and λ j = 1 / t j is a discrete eigenvalue of ∆ k . When M is non-compact, possible poles of K s ( z, w ) are also located at the points s = 1 − ρ − n , where n ∈ N and ρ ∈ (1 / , is apole of the parabolic Eisenstein series E jl ( z, s ) , and the points s = ρ − n , where n ∈ N and ρ is a pole of E jl ( z, s ) with Re( ρ ) < / . roof. The proof we present here follows closely the proof of [JvPS16, Theorem 10].We assume M is non-compact; in case of cocompact Γ, the sums over cusps below areidentically zero and there are no poles stemming from poles of the Eisenstein series.Let B := max { , | k |} . First we prove that K s ( z, w ) has a meromorphic continuationto the half-plane Re( s ) > B − n for any n ∈ N . For s ∈ C with Re( s ) > B we use thewave representation (35) of K s ( z, w ) to obtain K s ( z, w ) = X λ j ≥| k | (1 −| k | ) H ( t j , g s ) ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ H ( r, g s ) E jl ( z, / ir ) E jl ( w, / ir ) t dr, (51)where λ j = 1 / t j .Letting h n ( r, s ) := (cid:0) s − − ir (cid:1) n (cid:0) s − + ir (cid:1) n and using formula (49) in (51), we get2 n Γ( s )Γ( s + 2 n ) K s ( z, w ) = X λ j ≥| k | (1 −| k | ) H ( t j , g s +2 n ) h n ( t j , s ) ϕ j ( z ) ϕ j ( w ) t + 14 π c Γ X j =1 m j X l =1 Z ∞−∞ H ( r, g s +2 n ) h n ( r, s ) E jl ( z, / ir ) E jl ( w, / ir ) t dr. (52)It can be easily seen that ( a ) n = Q n − j =0 ( a + j ). This implies that h n ( r, s ) ∼ r n as r → ∞ . Hence, the series in (52) arising from the discrete spectrum is locally absolutelyand uniformly convergent as a function of s for Re( s ) > B − n away from the poles of h n ( r, s ) − , i.e. away from the zeros of h n ( r, s ). Using ( a ) n = Q n − j =0 ( a + j ), we calculatethe zeros of h n ( r, s ) for Re( s ) > B − n , which occur at the points s = 1 / ± it j − m for m = 0 , . . . , n − r / ir so that the integral is nowover the vertical line whose real part is 1 / s ∈ C theintegral, denote it by I / ,jl ( s ), is holomorphic for s ∈ C with Re( s ) > B − n satisfyingRe( s ) = 1 / − m , where m = 0 , . . . , n −
1. In order to get the meromorphic continuationof this function across the lines Re( s ) = 1 / − m , we will use the same method appliedin the proofs of [JKvP10, Theorem 2] and [JvPS16, Theorem 10] or in [vP10].As a first step, let m = 0 and choose ǫ > E jl ( z, s )has no poles in the strip 1 / − ǫ < Re( s ) < / ǫ . For s ∈ C with 1 / < Re( s ) < / ǫ , we apply the residue theorem to the function I / ,jl ( s ) to obtain the meromorphiccontinuation I (1)1 / ,jl ( s ) of it in the strip 1 / − ǫ < Re( s ) < / ǫ . Then, assuming1 / − ǫ < Re( s ) < / I (2)1 / ,jl ( s ) of the integral I (1)1 / ,jl ( s ) to the strip − / < Re( s ) < /
2. Finally,adding the formulas coming from the applications of the residue theorem we obtain themeromorphic continuation of the integral I / ,jl ( s ) to the strip − / < Re( s ) ≤ / I / ,jl ( s ) to the strip − / − m < Re( s ) ≤ / − m for m = 1 , . . . , n − s = 1 − ρ − m , where ρ is a pole of the Eisensteinseries E jl ( z, s ) belonging to the line segment (1 / , s = ρ − m , where ρ is apole of the Eisenstein series E jl ( z, s ) such that Re( s ) < /
2, and m = 0 , . . . , n − his completes the proof of the meromorphic continuation of K s ( z, w ) to the whole s -plane, as n ∈ N was chosen arbitrarily. (cid:3) The geometric automorphic kernel.
For Re( s ) sufficiently large and for any twopoints z, w ∈ F , define the geometric automorphic kernel by˜ K s ( z, w ) := Γ( s − k )Γ( s + k ) √ π Γ( s ) X γ ∈ e Γ χ ( γ ) cosh( d hyp ( z, γw )) − s × F ( − k, k ; s ; (1 + cosh( d hyp ( z, γw ))) − ) J γ,k ( w ) H k ( z, γw ) , (53)where F ( − k, k ; s ; (1 + cosh( d hyp ( z, γw ))) − ) stands for the (Gauss) hypergeometric func-tion.Note that it is possible to extend the above definition to z, w ∈ H . Then, for any fixed w ∈ H , the function ˜ K s ( z, w ) can be viewed as a map from H to End( V ) (which can beidentified with C d × d ). The following proposition shows that (for sufficiently large Re( s ))for any fixed w ∈ H , the columns of the d × d matrix ˜ K s ( z, w ) belong to the space H k (see Section 2.4), when viewed as maps from H to V . Proposition 21. ( a ) The series in formula (53) converges normally with respect tothe operator norm in the ring
End( V ) of endomorphisms of V in the variables ( z, w ; s ) , with s in the half-plane Re( s ) > and z, w ∈ F . It defines a holomorphicfunction of s in the half-plane Re( s ) > . The convergence is uniform when z, w ∈ F are restricted to any compact subset of F . ( b ) The kernel ˜ K s ( z, w ) is a meromorphic function of s in the half-plane Re( s ) > ,possessing simple poles in this half-plane only when | k | > . When | k | > , thesimple poles are located at s = | k | − n , for integers n ∈ [0 , | k | − . ( c ) For each w ∈ H and s ∈ C with Re( s ) > max { , | k |} each column of the matrix ˜ K s ( · , w ) defines a function in H k .Proof. ( a ) For any γ ∈ e Γ, the hypergeometric function F ( − k, k ; s ; (1+cosh( d hyp ( z, γw ))) − )is well-defined for all s ∈ C , due to the fact that0 < (1 + cosh( d hyp ( z, w ))) − ≤ for any two points z, w ∈ F (equality being attained when z = w ). Moreover, for all s with Re( s ) >
1, the function F ( − k, k ; s ; (1 + cosh( d hyp ( z, γw ))) − ) is holomorphic, sinceit is the sum of uniformly convergent holomorphic functions. For all s ∈ C such thatRe( s ) >
1, it is uniformly bounded by F (cid:0) − k, k ; 1 , (cid:1) .Since χ is a unitary multiplier system and | J γ,k ( w ) | = | H k ( z, w ) | = 1, when d = 1, theseries appearing in the definition of the kernel ˜ K s ( z, w ) is dominated (uniformly in s , forRe( s ) >
1) by the series(54) X γ ∈ e Γ cosh( d hyp ( z, γw )) − Re( s ) , which converges in the half-plane Re( s ) > z, w ∈ F are restricted to any compact subset of F .When d >
1, in order to prove the normal convergence of the series in (53), it issufficient to notice that χ can be identified with a unitary d × d matrix, with matrix norminduced from the Hilbert space norm obviously equal to √ d . The normal convergencefollows again from the convergence of the series (54) for Re( s ) >
1, which is uniformwhen z, w ∈ F are restricted to any compact subset of F . This proves part ( a ) of theProposition. b ) From part ( a ) it follows that the sum over e Γ on the right-hand side of (53) is aholomorphic function in s , for Re( s ) >
1. Therefore the poles of ˜ K s ( z, w ) in the half-plane Re( s ) > s − k )Γ( s + k ) / Γ( s ) . Thisfactor can have poles in the half-plane Re( s ) > | k | >
1, and they are locatedat s = | k | − n , for integers n ∈ [0 , | k | − L (Γ \ H , m, W ) = H k in Hejhal’s notation. Therefore, it suffices to show that the function(55) Φ s (4 u ) := r π · Γ( s − k )Γ( s + k )Γ( s ) (1 + 2 u ) − s F ( − k, k ; s ; u ) ) , where u = u ( z, w ) is given by (2), satisfies [He83, Assumption 6.1 on p. 387].It is clear from the definition of F ( − k, k ; s ; u ) ) that it is four times differentiablein u ≥ F (cid:0) − k, k ; 1 , (cid:1) . The function F s ( u ) = (1 + 2 u ) − s is also four times differentiable as a function of u for any fixed s ∈ C with Re( s ) > max { , | k |} and satisfies the bound (cid:12)(cid:12) F ( j ) s ( u ) (cid:12)(cid:12) ≪ (1 + u ) − j − Re( s ) , for j = 0 , , , ,
4, where the implied constant is independent of u . It follows that Φ s ( t )is four times differentiable as a function of the real parameter t ≥
0. Furthermore, forevery s ∈ C such that Re( s ) > max { , | k |} , the estimate (cid:12)(cid:12) Φ ( j ) s ( t ) (cid:12)(cid:12) ≪ (4 + t ) − j − Re( s ) holds for j = 0 , , , , t ≥
0. Thus, the proof is complete. (cid:3)
According to Proposition 21, for any fixed complex number s with Re( s ) > max { , | k |} ,the kernel ˜ K s ( z, w ) can be viewed as a map from F × F to End( V ). In the followingproposition we prove that for all such s , automorphic kernels ˜ K s ( z, w ) and K s ( z, w ) areequal on F × F . Proposition 22.
For all s ∈ C with Re( s ) > max { , | k |} and for all z, w ∈ F , ˜ K s ( z, w ) = K s ( z, w ) . Proof.
In view of Proposition 21.( c ), it suffices to show that, for Re( s ) > max { , | k |} ,the functions ˜ K s ( z, w ) and K s ( z, w ) have the same coefficients in the spectral expansion,which amounts to showing that the function Φ s (4 u ) defined in (55) is the inverse SelbergHarish-Chandra transform of the Fourier transform h s of the function(56) g s ( u ) = Γ( s − )Γ( s ) (cosh u ) − ( s −
12 ) = Q s ( e u + e − u − . Based on [He83, Formula (6.6) on p. 386], this is equivalent to showing that(57) Φ s (4 u ) = Γ( s + ) π Γ( s ) 2 s − ∞ Z −∞ (4 u + t + 2) − ( s + 12 ) √ u + 4 + t − t √ u + 4 + t + t ! k dt. Substitute y = 4 u and denote by I ( y ) the integral on the right-hand side of (57). Itfollows immediately that I ( y ) = ∞ Z ( α + t − − ( s + 12 ) √ α + t − t √ α + t + t ! k + √ α + t − t √ α + t + t ! − k dt, here α = y + 4. Introducing a new variable x = √ α + t − t √ α + t + t , we obtain that I ( α −
4) = α − s s −
12 1 Z ( x + 2 x (1 − α ) + 1) − ( s + 12 ) (( x s + k + x s − k − ) + ( x s − k + x s + k − )) dx. Substituting x = 1 /x in the two integrals containing exponents x s − k − and x s + k − yieldsthe following simplified expression: I ( y ) = α − s s − ∞ Z x s + k + x s − k ( x + 2 x yy +4 + 1) s + 12 dx. Next, write I ( y ) = I (4 u ) = I + (4 u ) + I − (4 u ), where I ± (4 u ) = 12 ( u + 1) − s ∞ Z x s ± k ( x + 2 x uu +1 + 1) s + 12 dx. The integral on the right-hand side of the above equation appears in Formula 8.714.2 of[GR07] for the integral representation of the Legendre function, with cos( ϕ ) = u/ ( u +1) ∈ (0 , µ = s and ν = ± k . For Re( s ) > max { , | k |} , the conditions Re( µ ± ν ) > I ± (4 u ) = 12 ( u + 1) − s ∞ Z x s ± k ( x + 2 x uu +1 + 1) s + 12 dx = 2 s − Γ( s + 1)Γ( s ± k + 1)Γ( s ∓ k )Γ(2 s + 1)(1 + 2 u ) s/ P − s ± k (cid:18) uu + 1 (cid:19) , where P µν stands for the Legendre function. Inserting the above expression for I ± (4 u )into (57) after applying the doubling formula Γ(2 s + 1) = 2 s π − Γ( s + )Γ( s + 1) for thegamma function, we obtain thatΦ s (4 u ) = 2 − √ π Γ( s + k )Γ( s − k )Γ( s ) (cid:18) ( s + k ) P − sk (cid:18) uu + 1 (cid:19) + ( s − k ) P − s − k (cid:18) uu + 1 (cid:19)(cid:19) (1+2 u ) − s . In order to prove (55), it is left to show that( s + k ) P − sk (cid:18) uu + 1 (cid:19) + ( s − k ) P − s − k (cid:18) uu + 1 (cid:19) = 2Γ( s ) (1 + 2 u ) − s F (cid:16) − k, k ; s ; u ) (cid:17) . Apply [GR07, Formula 8.704], with x = u/ ( u + 1) ∈ (0 , µ = − s and ν = ± k , in orderto express the Legendre function in terms of the hypergeometric function: P − s ± k (cid:18) uu + 1 (cid:19) = (1 + 2 u ) − s s Γ( s ) F (cid:16) ∓ k, ± k + 1; s + 1; u ) (cid:17) . Therefore, proof of (55) reduces to proving that1 s (cid:16) ( s + k ) F (cid:16) − k, k + 1; s + 1; u ) (cid:17) + ( s − k ) F (cid:16) k, − k + 1; s + 1; u ) (cid:17)(cid:17) = 2 F (cid:16) − k, k ; s ; u ) (cid:17) . The above identity follows immediately from the definition of the hypergeometric functionand the property ( a + 1) j = ( a ) j a + ja of the Pochammer symbol ( a ) j = Γ( a + j ) / Γ( a ), forall non-negative integers j , applied with a = k and a = − k . (cid:3) emark . When k = 0 and χ is the identity, the hypergeometric series F (cid:16) − k, k ; s ; u ) (cid:17) is identically equal to one, hence the series ˜ K s ( z, w ) coincides with the automorphic kernel K s ( z, w ) defined in [JvPS16, Formula (19)], up to the constant √ π .By the uniqueness of meromorphic continuation, combining the above proposition withTheorem 20, we arrive at the following corollary: Corollary 24.
For any z, w ∈ F , the geometric kernel ˜ K s ( z, w ) admits a meromorphiccontinuation to the whole complex s -plane. The possible poles of the function Γ( s )Γ( s − / − ˜ K s ( z, w ) are located at the points s = 1 / ± it j − n , where n ∈ N and λ j = 1 / t j is a discrete eigenvalue of ∆ k . In case when M is non-compact, possible poles of ˜ K s ( z, w ) are also located at the points s = 1 − ρ − n , where n ∈ N and ρ ∈ (1 / , is a pole ofthe parabolic Eisenstein series E jl ( z, s ) , and at the points s = ρ − n , where n ∈ N and ρ is a pole of E jl ( z, s ) with Re( ρ ) < / .Remark . Corollary 24 illustrates the strength of the approach to constructing Poincar´eseries using generating kernels. Namely, in order to deduce the meromorphic continuationof the automorphic kernel ˜ K s ( z, w ) from its geometric definition (53), one would have toconsider some type of Fourier expansion (e.g. an expansion in rectangular or sphericalcoordinates at a certain point) and investigate certain properties of the coefficients in theexpansion (e.g. uniform boundedness, analyticity, etc.). This is a heavy task, which weovercome by considering the wave distribution acting on the function g s defined in (56). References [BK18] Bringmann, K. and Kudla, S. A classification of harmonic Maass forms.
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E-mail address : [email protected] Moni Kumari , School of Mathematics, Tata Institute of Fundamental Research, India
E-mail address : [email protected] Jolanta Marzec , Department of Mathematics, Technical University of Darmstadt, Schlossgarten-straße 7, 64289 Darmstadt, Germany
E-mail address : [email protected] Kathrin Maurischat , Lehrstuhl A Mathematik, RWTH Aachen University, Templergraben 55,52062 Aachen, Germany
E-mail address : [email protected] Andreea Mocanu
E-mail address : [email protected] Lejla Smajlovi´c , Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71000Sarajevo, Bosnia and Herzegovina
E-mail address : [email protected]; [email protected]@efsa.unsa.ba; [email protected]