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Sup-norm problem of certain eigenfunctions on arithmetic hyperbolic manifolds Jana, Subhajit 2015

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Sup-norm problem of certain eigenfunctions on arithmetichyperbolic manifoldsbySubhajit JanaB.Math. (Hons.), Indian Statistical Institute, 2013a thesis submitted in partial fulfillmentof the requirements for the degree ofMASTER OF SCIENCEinthe faculty of graduate and postdoctoral studies(Mathematics)The University Of British Columbia(Vancouver)April 2015c© Subhajit Jana, 2015AbstractWe prove a power saving over the local bound for the L∞ norm of uniformly non-tempered Hecke-Maass forms on arithmetic hyperbolic manifolds of dimension 4and 5. We use accidental isomorphism and use the Hecke theory of the correspond-ing groups to show that if the automorphic form is non-tempered at positive densityof finite places then the Hecke eigenvalues are large; amplifying the saving comingfrom the non temperedness we get a power saving.iiPrefaceThis thesis is an unpublished and original work of the author. The results in chapter3 and 4 were obtained by the author of the thesis. The results in chapter 5 wereobtained in collaboration with my supervisor Professor Lior Silberman.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Clifford Algebras, Vahlen Matrices and Real Hyperbolic Spaces . 72.2 Arithmetic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . 133 Hecke Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 SO(4,1) Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 SO(5,1) Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Amplified Pre-trace Formula . . . . . . . . . . . . . . . . . . . . . . 404.1 Spherical Transform and Automorphic Kernel . . . . . . . . . . . 404.2 Archimedean Amplification and Bounding k . . . . . . . . . . . . 435 Diophantine Analysis and Bounds . . . . . . . . . . . . . . . . . . . 486 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 54ivBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56vAcknowledgementsFirst of all, I am extremely thankful and obliged to my supervisor Professor LiorSilberman, without whom this project would not have started. I am really grate-ful for his patience, generous amount of time and several discussion sessions whichwent in diverse field of mathematics. Not only regarding my thesis, but general dis-cussion regarding various type of problems in mathematics motivated me towardsfields like dynamical system, mathematical physics, quantum mechanics and so onwhich were, to me, apparently disconnected from number theory.Secondly, I want to thank Professor Valentin Blomer, discussion with whomat IHE´S enhanced my visualization towards the problem I was doing and also forhis enough kindness for explaining all the questions of mine through e-mails. I amgrateful to Professor Aloys Krieg who was extremely helpful for my understandingof his work on quaternionic Hecke algebras. I also want to thank John Voight,Professor Ameya Pitale and Julia Gordon for several helpful discussion.I want to specially thank Nishant Chandgotia and Vasu Tewary who werefriends and brothers to me at UBC. Without their advice and help in mathemat-ical and non-mathematical aspects this thesis would not have completed. Alongwith them, I want to thank Gourab Ray, Myrtw Mavraki, Shen Ning Tung, QiangZhang, Navid Ghadermarzy and many more who helped me in several aspect, evenoutside academic, of my life at Vancouver.I am grateful to my parents whose bliss and moral support were necessary tocomplete this thesis. Even they were physically thousands of miles away fromme, they never let me feel alone here in Vancouver. I want to heartily thank SadafAkhtar for her love, friendship and tolerance which kept me alive outside the aca-demic world.viI want to dedicate this thesis to my mother who first blossomed my interest inmathematics.viiChapter 1IntroductionThe eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold arevery important to study and create links between various fields in mathematics suchas spectral geometry, harmonic analysis, quantum mechanics, thermodynamics, orglobal analysis. The limiting behaviour of the eigenfunctions and the distributionof their mass play an important role in both physics and mathematics.Let M be a compact Riemannian manifold of dimension n and ψ be a functionon M which satisfies ∆ψ + λψ = 0 and ||ψ||L2(M) = 1, where ∆ is the Laplace-Beltrami operator on M. By ||ψ|| and ||ψ||p we will denote ||ψ||L∞(M) and ||ψ||Lp(M)for 1 ≤ p < ∞ unless mentioned otherwise. It is known that (see e.g. [15],[52])one can have a general bound||ψ||  λν(M) with ν(M) =dim(M) − 14. (1.0.1)The above bound is sharp for round sphere M = S n or on a surface of revolutionthat is diffeomorphic to S 2, but is far from the true bound on flat tori. It is usuallybelieved that if the geodesic flow on the unit cotangent bundle of M is chaotic thenthe bound (1.0.1) can be improved. One result of this kind is due to Be´rard [3],who proves that if M has negative sectional curvature (so geodesic flow is ergodic)then one has,||ψ|| λν(M)√log λ.1Similar results on the assumption on the geodesic flow of the manifold (ergodicity,positive entropy etc.) can be found in [56] or [55]. In case of negatively curvedmanifold understanding the behaviour of ||ψλ||p as λ→ ∞ is an important questionregarding quantum chaos (see [45]).The case of bounding sup norm of eigenfunctions on congruence quotients ofmanifolds is an interesting problem because due to automorphy and symmetriesarising from underlying Hecke algebra one expects a power saving in (1.1). Thefirst breakthrough was done by Iwaniec and Sarnak [27] who proved that for com-pact (see [5] for non-compact) arithmetic hyperbolic surface, such as quotient ofH2 by the group of units in an order in a quaternion division algebra over Q, for aHecke-Maass eigenform ψ with Laplace eigenvalue λ the supnorm ||ψ||  λ524 + .They also provided a lower bound of sup-norm, namely ||ψλ|| √log log λ forinfinitely many eigenfunctions. In fact, it has been conjectured by Hejhal-Rackner[23] (using a random wave model and numerical computation on large deviationsupport) that the bound ||ψ||  λ holds (modified in non-compact case) on com-pact hyperbolic surface. This is compactible with the results of Iwaniec-Sarnak.Not only the eigenvalue aspect, but the volume of the underlying manifold (so-called level aspect) is also an important aspect to bound the eigenforms as thecomplexity of the chaotic system formed by the quantum eigenstates does increasein level direction, for instance, see [25], [26], [5].There are several generalizations of Iwaniec-Sarnak type result in higher di-mensional and higher rank arithmetic locally symmetric spaces, i.e. a power sav-ing in the bound of L∞ norm from the so called ’convexity‘ bound. In a letter toMorawetz Sarnak [44] proved that if X = G/K is a locally symmetric space of di-mension n and rank r and Ω ⊆ X compact then an L2 normalized joint eigenfuctionof the ring of invariant differential operators ψ should satisfy||ψ|Ω||  λn−r4 .Note that this is the ’convexity‘ bound and the natural replacement of the boundin equation (1.0.1). Blomer, Harcos and Milic´evic´ [4] recently proved a strongsupnorm bound for cuspidal eigenfunctions on arithmetic hyperbolic 3-manifolds.There are also results where higher rank eigenfunctions are proved to have saving2fro the standard bound in their supnorm; for instance SL3(Z)\SL3(R) by Holowin-sky, Ricotta and Royer [24], Sp4(Z)\Sp4(R) by Blomer and Pohl [11] and SLn(Z)\PGLn(R)by Blomer and Maga [6] and [7]. Marshall [37] in his wonderful work has provedthe saving in sup norm for eigenfunctions on symmetric spaces arising from a largeclass of semisimple Lie groups. Bounds for Lp norm of the eigenfunctions can befound in [36]. There are also results on bounding eigenfunctions on round spheresand arithmetic ellipsoids, namely, [8] and [9].There also have been a lot of works regarding lower bounds of the eigenfunc-tions. The naive expectation of ||ψ||  λ which is suggested by the random wavemodel, does not hold in higher dimension and higher rank case. It has been shownin [39] that on any arithmetic hyperbolic 3-manifold of Maclachlan-Reid type thereexists an infinite orthonormal family of cusp forms ψ with lower bound||ψ||  λ1/4− .Large values of GLn Maass forms are also established in the recent work of [13].In this thesis, we focus our attention to a particular class of eigenfunctionson arithmetic hyperbolic manifolds of dimension 4 and 5, i.e. non-compact con-gruence quotients of H4 and H5 respectively. In general, one can view the ndimensional hyperbolic space as a locally symmetric space arising from SOn,1(R),precisely,Hn = SO+n,1(R)/SOn(R). But due to sporadic isogenies of special orthog-onal groups one may view SO5,1(R) as SL2(H) and SO4,1(R) as Sp∗1,1(H) (precisedescription in chapter 3), where H is the division algebra Hamilton quaternionsover real. Let D be the unique quaternion algebra over Q ramified at {2,∞} so thatH = D ⊗Q R is the division algebra of Hamilton’s quaternions. Let G be the alge-braic group defined over Q such that G(Q)  GL2(D) (correspondingly Sp∗1,1(D)).The real points G(R) is a real Lie group which is isomorphic to GL2(H) (corre-spondingly Sp∗1,1(H)) and acts on H5 (correspondingly H4) by isometries withrespect to the usual hyperbolic metric. Let O be the Hurwitz maximal order in D.Let Γ = SL2(O) (correspondingly Sp∗1,1(O)) be the subgroup which acts discretelyon the corresponding hyperbolic space (our results in fact apply with D being anyquaternion algebra ramified at infinity and O any maximal order in D, and Γ anycongruence subgroup in SL2(O), but we make a specific choice for ease of presen-3tation). We consider the Hecke-Maass forms on M = Γ\Hn. These are the eigen-functions of the Laplace-Beltrami operator. They are naturally also eigenfunctionsof the Hecke operators arising from Γ which commute among themselves and alsowith Laplace operator; so one may choose a basis of the Hilbert space which is thecuspidal part of L2(Γ\Hn) consisting joint eigenfunctions of Laplace and Heckeoperators and they will be denoted as Hecke-Maass cusp forms. We parametrizethe eigenvalue as follows: let −∆φ = λφ be a Maass form on Hn. As λ ≥ 0 (theLaplace-Beltrami operator is non-positive in L2(M)),t =√λ −(n − 1)24∈ R ∪[−n − 12,n − 12]i.Throughout the thesis we denoteT := max(1, |t|)  1 +√λ.Using the accidental isomorphism of the groups SO4,1(R) and SO5,1(R) withSp∗1,1(H) and SL2(H) respectively we can explicitly work out the Hecke algebra forcorresponding cases. The Hecke algebras in respective cases are basically Heckealgebras of GSp4 and GL4 at odd places. As the Hecke eigenfunctions preservefurther symmetries we expect them not to grow very large. The Hecke-algebraeigenvalues at a prime p of a Hecke eigenfunction as described by a spectral pa-rameter, a vector of complex numbers. We say an eigenfunction is η-non-temperedat p if it has a spectral parameter with real part at least η (for the definition of thespectral parameters see definition 9). We say that a sequence of eigenfunctions isuniformly-nontempered if there is a set P of primes of natural density δ > 0 and aconstant η > 0 such that for every eigenfunction in the sequence and for any p ∈ P,it is η-non-tempered at p. In this setting we will describe our main theorem whichis an improvement of (1.0.1) in the eigenvalue aspect for non-tempered eigenforms.Theorem 1.(i) Suppose we have a sequence of uniformly nontempered L2-normalized Hecke-Maass cuspidal eigenform on Sp∗1,1(O)\H4. Let φ be an element in the se-4quence with Laplace eigenvalue λ. Then for Ω ∈ Sp∗1,1(O)\H4 compact,||φ|Ω||  λ34− ,for some  > 0.(ii) Suppose we have a sequence of uniformly nontempered L2-normalized Hecke-Maass cuspidal eigenform on SL2(O)\H5. Let φ be an element in the se-quence with Laplace eigenvalue λ. Then for Ω ∈ SL2(O)\H5,||φ|Ω||  λ1− ,for some  > 0.Remark. (i) Note that in the theorem the exponents 34 and 1 in the exponentsare from 1.0.1 for respective dimensions.(ii) While the Generalized Ramanujan Conjecture (GRC) predicts that the genericcuspidal representations of quasi-split groups (e.g. GLn) should be temperedat all places [47], there actually exist uniformly non-tempered cusp forms incase of SO(4, 1) and SO(5, 1). It is also believed that a representation is non-tempered i.e. does not satisfy the GRC if it comes from a functorial lift froma smaller group. The most evident counter-exaples are the CAP (Cuspidalrepresentation Associated to Parabolic) representation defined by Howe andPiatetski-Shapiro [22] and the Kurokawa lift for Sp4 [32].(iii) Pitale [41] has constructed an example of CAP representation through a liftfrom S˜L2 (metaplectic) to Spin(1, 4) which gives a counter-example to theGRC for Spin(1, 4). Also recently in [40] an example is constructed of CAPrepresentation through a lift from SL2 to Spin(1, 5) which gives a counter-example to the GRC for Spin(1, 5). So our result automatically proves apower saving in the sup norm of the sequence of Hecke-Maass forms theyhave constructed (in their cases η = 12 ) in both cases.Our proof starts with a pre-trace formula. We define an automorphic kernel on5M × M byK(P,Q) =∑γ∈Γk(u(γP,Q))where u is a point pair invariant and and Γ is the corresponding discrete subgroupand k is a rapid decay smooth function on positive real numbers. Then we choosean orthonormal basis of cuspidal eigenfunction which contains our favourite φ andwrite K in this basis. After we apply suitable Hecke operators on both sides (mak-ing sure that each term in spectral side is positive) the problem reduces to a dio-phantine counting problem where we count, at least at identity, how many of γ inHecke double coset lie in the maximal compact. Then we show that, if the eigen-function is nontempered then there exists a Hecke operator whose eigenvalue forthat eigenfunction is big enough than the Hecke operator returns to its maximalcompact and that gives us a power saving.In the first chapter we describe basic formulation of Clifford algebra and Vahlenmatrices and also describe the Spin(1, n) groups. In the second chapter we describenotion of automorphic forms. In the third chapter we devote ourselves on describ-ing Hecke theory for both Spin(1, 4) and Spin(1, 5) cases. In the fourth chapter wedescribe pre-trace formula and develop the Archimedean amplification and alsoprove the ’trivial bound’. In fifth chapter we give the prove of our main theorem.6Chapter 2Basic Notations2.1 Clifford Algebras, Vahlen Matrices and RealHyperbolic SpacesIn this section we will briefly describe a model of k + 1-dimensional real hy-perbolic space Hk+1 following an approach of Vahlen [57]; for details reader maylook at [18]. For integer k > 0 we construct Ck := Ck(R) to be the Clifford alge-bra over R associated with the negative definite unit form Ik = (−δi j)1≤i, j≤k. Leti1, . . . , ik be the standard basis of Rk. We embed canonically Rk ↪→ Ck we note thatthe elements in satisfy:i2n = −1, inim = −imin (m, n = 1, . . . , k,m , n). (2.1.1)The Clifford algebra is an associative algebra generated by i1, . . . , ik. Hence the 2kelements in1 . . . inp where 1 ≤ n1 ≤ · · · ≤ np, 0 ≤ p ≤ k form a basis of Ck over thereal. One may note C0 = R,C1 = C and C3 = H =Hamilton’s quaternion. Thereare three involutions defined on Ck by means of,∀a = a0 +k∑p=1∑1≤n1≤···≤np,0≤p≤kan1...np in1 . . . inp ∈ Ck7a 7→ a′ := a = a0 +k∑p=1(−1)p∑1≤n1≤···≤npan1...np in1 . . . inp ,a 7→ a¯ := a = a0 +k∑p=1(−1)p(p+1)2∑1≤n1≤···≤npan1...np in1 . . . inp ,a 7→ a∗ := a0 +k∑p=1(−1)p(p−1)2∑1≤n1≤···≤npan1...np in1 . . . inp .(2.1.2)These maps satisfyx¯′ = x∗, ∀x ∈ Ck,x¯ = −x, x′ = −x, x∗ = x, ∀x ∈ Ri1 ⊕ · · · ⊕ Rik,¯(xy)y¯x¯, (xy)′ = y′x′, (xy)∗ = y∗x∗, ∀x, y ∈ Ck.(2.1.3)Vahlen’s model of k + 1−dimensional hyperbolic space is constructed as follows.Hk+1 is embedded in the k + 1−dimensional subspace of Ck:Vk := R.1 ⊕ Ri1 ⊕ · · · ⊕ Rik.For all v ∈ Vk we denote,real part of v = <(v0 + v1i1 + . . . vkik) = v0,trace of v = tr(v) = v + v¯ = 2<(v)norm of v = ||v||2 = v.v¯.We equip Vk with the quadratic formqk(x) := ||x||2 (x ∈ Vk)and SO(qk,R) be the special orthogonal group associated to qk. The set underlying8the model of (k + 1)−dimensional hyperbolic space is the upper half-spaceHk+1 := {x0 + x1i1 + · · · + xkik : x0 . . . , xk ∈ R, xk > 0}.We define for P := x0 + x1i1 + · · · + xkik ∈ Hk+1x(P) = x0 + x1i1 + · · · + xk−ik−1y(P) = xkr(P) = ||P||2 = ||x(P)||2 + y(P)2.(2.1.4)We endowHk+1 with the Riemannian metric whose line element isds2 =dx20 + · · · + dx2kx2kand obtain a model of the (k + 1)−dimensional hyperbolic space.Definition 1. For P,Q ∈ Hk+1 we define a point pair invariant byu(P,Q) :=||z(P) − z(Q)||2 + (y(P) − y(Q))22y(P)y(Q), (2.1.5)and note thatcosh(d(P,Q)) = 1 + 2u(P,Q), (2.1.6)where d(P,Q) is the distance between P and Q coming from Riemannian metric.Now we will describe the orientation preserving isometry group Iso+(Hk+2) bymeans of a group of certain (2 × 2) matrices over Ck. See [57] for details.Definition 2. An element 0 , v ∈ Ck is called a transformer if there exists a linearautomorphism φv : Vk → Vk such thatvx = φv(x)v′ ∀x ∈ Vk.Following the language of Maass we denote Tk to be the set of all transformersof 0 , v ∈ Ck.9Definition 3. For an integer k ≥ 0 we define Vahlen group S Vk byS Vk :=α βγ δ ∈ Mat2(Ck) : α, β, γ, δ ∈ Tk ∪ {0}, αβ∗, δγ∗ ∈ Vk, αδ∗ − βγ∗ = 1.One may check that S V0 = SL2(R), S V1 = SL2(C) and S V3 = SL2(H).Proposition 1.(1) The group S Vk is generated by1−1 ,1 x1 x ∈ Vk.(2) Suppose σ =α βγ δ ∈ S Vk and P ∈ Hk+2, then γP + δ ∈ Tk andσP := (αP + β)(γP + δ)−1 ∈ Hk+2.The above formula defines an action of S Vk on Hk+2 by orientation pre-serving isomerism. The corresponding group homomorphism of S Vk intoIso+(Hk+2) induces an isomorphism of groupsSO◦k+2,1(R)  Iso+(Hk+2)  S Vk/{±I}. (2.1.7)The action of S Vk on Hk+2 is transitive on pairs of points with fixed hyper-bolic distance.Proof. See page 381 in [18]. As we can thinkHn as the Minkowski space (Rn+1, q)where q(x) = x20− x21− · · · − x2n is a quadratic form of signature (1, n), it is clear thatIso+(Hk+2)  SO◦(Rk+1, q)  SO◦n,1.The map P 7→ σP is so-called Mo¨bius transformation, the coordinates ofσP can be recovered from the Iwasawa decomposition of S V2 or equivalentlySO◦n,1(R).10For k ≥ 0, let Uk ⊂ S Vk denote the stabilizer of ik+1 ∈ Hk+2. Then we have,Proposition 2. Uk is a maximal compact subgroup of S Vk and(1)Uk =α −β′β α′ ∈ S Vk : ||α||2 + ||β||2 = 1, (2.1.8)(2) SO(K + 2)  Uk/ ± I andf : S Vk/Uk → Hk+2, f (vUk) = vik+1is an S Vk-equivariant isometry.Proof. The description of Uk is clear from the definition of transformers. From2.1.7 we may thinkUk/ ± I = StabS Vk/±Iik+1 = StabSO◦k+2,1(R)ek+2 = SO(k + 2) × SO(1)  SO(k + 2),where ek+2 = (0, . . . , 0, 1). The rest follows easily. Note that the isomorphism 2.1.7 and above proposition recover the accidentalisomorphisms betweenSOn,1(R)  PSL2(F)where F = R,C and H for n = 2, 3, and 5 respectively. In a similar fashionU0 = SO(2),U1 = S U(2),U3 = Sp(2).From the Iwasawa decomposition of SOn,1(R) = NAK where N  Rn, A  R× andK = SO(n) using above isomorphism one can deduce the Iwasawa decompositionofS Vk/ ± I = NkAkUk (2.1.9)where,Nk =n(x) :=1 x1 |x ∈ Vkand Ak =a(y) :=√y√y−1 |y ∈ R+.11Thus for any P ∈ Hk+2  S Vk/ ± IUk we have that P : (x, y) = (x(P), y(P)) 7→n(x(P))a(y(P)) from 2.1.4. By doing a re-Iwasawa decomposition of σP for σ ∈S Vk and recalling the definition of point pair invariant from 2.1.7 one can easilydeduce the following.Proposition 3. For σ =α βγ δ ∈ S Vk and P = (x, y) ∈ Hk+2 we have,(1)x(σP) =(αx + β)(γx + δ) + αγ¯y2||γx + δ||2 + ||γ||2andy(σP) =y||γx + δ||2 + ||γ||2.(2)2u(ik+1, σik+1) + 1 = ||α||2 + ||β||2 + ||γ||2 + ||δ||2. (2.1.10)2.2 Arithmetic SubgroupsLet Ck(Q) be the Clifford algebra over Q. Then Ck(Q) ⊗Q R = Ck(R) is aClifford algebra over R.Definition 4. A Z-order O ⊂ Ck(Q) is called compatible if it is stable under theinvolutions¯and ′ defined in 2.1.2. For a compatible order O let us also define,Vk+1(O) := O ∩ Vk+1,S Vk+1(O) := Mat2(O) ∩ S Vk(Ck(Q)).It is clear that for any compatible Z-order O then Vk+1(O) is a lattice in Vk+1and as S Vk(O) is the stabilizer of the lattice O2 of S Vk action on Ck(Q)2, S Vk(O)is a discrete arithmetic subgroup of S Vk. The group S Vk(O) can be thought as ahigher dimensional Fuschian group which acts onHk+2 discontinuously. Let P bethe standard minimal parabolic subgroup of S Vk. Then for any discrete subgroupΓ < S Vk(Ck(Q)) we define the set Γ\S Vk(Ck(Q))/P ∩ S Vk(Ck(Q)) to be the set of12Γ cusps. It is well known that set of Γ cusps is finite (see proposition 15.6 in [10]).In particular, for Γ = S Vk(O) we know that Γ is cofinite i.e.Vol(S Vk(O)\Hk+2) =∫Γ\Hk+2dµ < ∞,where dµ is the usual (k + 2)-dimensional hyperbolic volume measure dxdyyk+2, for dxusual (k + 1)-dimensional Lebesgue measure (see corollary 6.4 [17]).2.3 Automorphic FormsNow let us fix some notations for this section. Let q0 be the rational quadratic formdefined by q0((x1, x2, . . . , xn+1)) = −x20 + x21 + · · · + x2n. For any rational quadraticform q we have that its isometry groupG = SO(q), which is a connected and almostsimple linear algebraic group over Q. Let A be the ring of adeles of Q. From thestrong approximation theorem of SO(n, 1) (see theorem 104:4 [38]) we getG(A)  G(Q)G+(R)K f , where K f =∏p<∞G(Zp). (2.3.1)For q = q0 which is a quadratic form of signature (n, 1), we let G = SO◦n,1(R) =G+(R) be a non-compact real Lie group with trivial center. K  SO(n) be a maxi-mal compact subgroup in G. S = G/K  Hn be the n dimensional real hyperbolicspace. G admits an Iwasawa decomposition ofG = NAKwhereN  Rn−1 and A  R+.Thus any point P ∈ S = G/K  N ×A can be uniquely described as (x(P), y(P)) :=(x1, . . . xn−1, y) ∈ Rn−1×  R+.There is a unique G invariant (up to scaling) Riemannian metric onHn whoseline element isds2 =dx21 + · · · + dx2n−1 + dy2y213and obtain a model of the n-dimensional real hyperbolic space.Proposition 4. For P,Q ∈ Hn the point pair invariant defined in 2.1.5 gives thatu(P,Q) :=||z(P) − z(Q)||2 + (y(P) − y(Q))22y(P)y(Q). (2.3.2)From the Riemannian metric on S as defined above, in the same coordinatesthe Laplace-Beltrami operator on C∞(S ) is given by∆n = −xnnn∑i=1∂∂xix2−nn∂∂xifor (x, y) := (x1, . . . xn),and this is a positive operator. Γ is a discrete subgroup of G acting on S properlydiscontinuously with finite covolume.Definition 5. A complex-valued fuction φ ∈ C∞(S ) is called automorphic formwith respect to Γ if φ satisfies following conditions:• ∆nφ = λφ. We define λ =(n−1)24 + t2 for t ∈ R ∪ i(−n−12 ,n−12).• φ(γP) = φ(P) for all P ∈ S and γ ∈ Γ.• φ is of moderate growth.In sense of Maass we call the above functions as Maass forms which has aFourier expansion (see [35])φ(P) = u(y) + yn−12∑m∈L\{0}amKit(2pi|m|y)e(〈m, x〉), (2.3.3)where L is the dual lattice of Γ ∩ N  Zn−1 in Rn−1 with respect to standard innerproduct of the same and hence, L  Zn−1. Also e(z) = exp(2piiz), u ∈ C∞(R+) andKα is the modified Bessel function defined as,Kα(y) =∫ ∞0exp(−y cosh(u)) cosh(αu)du.We let u(y) = 0 and therefore φ ∈ L2(Γ\S ) for rapid decay of K-Bessel function;we normalize ||φ||L2 = 1. We call such function as Maass cusp forms. In fact by atheorem of Harish-Chandra gives that (see [21] chapter 1 lemma 12),14Proposition 5. If φ ∈ C∞(Γ\S ) is a cusp form then φ rapidly decays in every Siegeldomain.The hyperbolic volume element onHn is denoted asµ(P) =dx1 . . . dxn−1dyyn.Definition 6. On the space of automorphic form we define Petersson inner productas〈 f , g〉 =∫Γ\Hnf (P)g(P)dµ(P). (2.3.4)We would like adelize the automorphic form due to ease of explaining Heckeaction. For a given congruence subgroup Γ there is an algebra of Hecke operators,which commute with the laplacian, acting on Maass forms. We will assume thatour Maass form is an eigenform of full Hecke algebra (detailed description is insection 3). Given a Hecke-Maass eigen-cusp form φ, write g = gQg∞k f where g ∈G(A), gQ ∈ G(Q) and k f ∈ K f and define Φ := Φφ : G(A) → C as Φ(g) = φ(g∞).Φ satisfies the following:• Φ(zγgk f k) = Φ(g) for (z, γ, g, k f , k) ∈ A× × G(Q) × G(A) × K f × SO(n),• Φ has moderate growth.15Chapter 3Hecke TheoryIn this section we will describe the action of the Hecke algebra on automorphicforms on the congruence hyperbolic 4 and 5 manifolds only. We confine our dis-cussion only to dimension 4 and 5 because we want to use the help of accidentalisomorphism in those cases. We will give both classical and adelic viewpoint to-wards the Hecke theory. For standard details reader may look at Krieg’s work [29]and [30]. For the purpose of the paper we will only discuss the Hecke theory forodd primes.Let D be Hamilton’s quaternions over Q, that is the Q-algebra spanned by{1, i, j, k} subject toi2 = j2 = k2 = −1 and i j = −k.Let O be the Hurwitz order, defined byO = Zi + Z j + Zk + Z1 + i + j + k2.For a prime p ≤ ∞ we define Dp = D ⊗Q Qp, i.e. D∞ = D ⊗ R = H the usualHamiltom quaternion, and for an odd prime p we have Dp  Mat2(Qp) and Op =O ⊗Z Zp  Mat2(Zp).163.1 SO(4,1) CaseLet G be the Q algebraic group such that G(Q) = Sp∗1,1(D), whereSp∗1,1(D) =g ∈ GL2(D) : g∗11 g =11.and g∗ is g¯t entrywise quaternion conjugation. The first part of the following propo-sition will allow us to realize an automorphic form of SO4,1(R) as an automorphicform on G(R)Proposition 6. (1) There is a 2 to 1 homomorphism between G(R) = Sp∗1,1(H)and SO4,1(R).(2) For an odd prime pG(Qp)  GSp4(Qp) and G(Zp)  GSp4(Zp).Proof.(1) Let gσ = S g∗S −1, where S =11. Then for g ∈ G(R) we have gσg = I2.Let G(R) acts on Mat2(H) by g.x = gσxg. Let t be the reduced trace on Mat2(H)defined byta bc d =12(a + a¯ + d + d¯).(x, y) = t(xy) is a bilinear form on Mat2(H) ×Mat2(H). Note that(g.x, g.y) = t(gσxggσyg) = t(xy) = (x, y)The 5 dimensional R vector spaceV = {x ∈ Mat2(H)|xσ = x and (x, S ) = 0} =a b−b a¯17is stable under this action and has an orthogonal basis11 ,i−i ,j− j ,k−k ,1−1 ;one can check it has the desired signature (4, 1).(2) Noting that for odd prime the division algebra and the Hurwitz order split overQp and Zp respectively i.e.D ⊗ Qp = Mat2(Qp) and O ⊗ Zp = Mat2(Zp)the second part is immediate. Let us fix Γ = Sp∗1,1(O). For an odd rational prime p let us fix pi, a primitivequaternion integer (i.e. pi < nO for any rational intger n) with |pi|2 = p. Let usdefineM(n) = {γ ∈ Mat2(O) : γ∗S γ = nS };so Γ = M(1). Also defineMp = ∪∞m=0M(pm).Following Shimura [51] we define the classical p-Hecke algebra Hp over Γ as thealgebra generated by the double cosets{ΓMΓ : M ∈ Mp}.From theorem 7 [30] we have the following generating elements ofHpProposition 7. Hp is the polynomial ring over Z generated by the elementsT (p) := Γ1p Γ, S (p) := Γpipip Γ, and I(p) := pΓ,which are algebraically independent.To know how big the order of support of a Hecke operators is, we need to de-compose the double cosets into single cosets. Next lemma is describes the single18coset decomposition. For this purpose we will map the classical p-Hecke algebrato the canonical convolution p-adic Hecke algebra Hp = H(GSp4(Qp),GSp4(Zp))which is the convolution algebra of GSp4(Zp)-biinvariant compactly supportedfunction of GSp4(Qp).One may note that,G(Z[p−1]) ∩G(Zp) = G(Z)andG(Qp)  G(Q)GSp4(Zp)  G(Z[p−1])G(Qp),where G(Z) = Γ ∪1−1 Γ. This shows thatΓ\G(Z[p−1])/Γ  G(Z)\G(Z[p−1])/G(Z)  G(Zp)\G(Qp)/G(Zp)and hence p-adic Hecke algebra is isomorphic to p-part of classical Hecke algebra.Lemma 1. With the notation above,Hp  Hp,where one can mapT (p) 7→ Char(Kp11ppKp), S (p)a 7→ Char(Kp1ppp2Kp), and I(p) 7→ Char(pKp).(3.1.1)where Kp = G(Zp) = GSp4(Zp) with Vol(Kp) = 1 under the usual Haar measureof Gp.Let φ be a Hecke-Maass cuspidal eigenform of Γ\H4. Recall that there is anautomorphic form Φ which we can produce from φ as described in the last para-graph of section 2. We consider the representation piφ := piΦ of G(A) on righttranslation of Φ. Let pi be a irreducible component of piΦ which is a cuspidal (as φ19and hence Φ is cuspidal) automorphic representation of G(A). pi has trivial centralcharacter as φ is invariant by the central action. Write pi = ⊗′ppip where pip is rep-resentation of Gp := G(Qp). We note that, for an odd prime p, pip is an irreducibleunramified representation of Gp since Kp is the maximal compact subgroup of Gp.From [14] we know that there exists an unramified character χ of the Borel sub-group of Gp, unique up to the Weyl group orbit, such that pip is isomorphic to theunique spherical constituent piχ of the normalized induced representation IndGpB (χ).We will now decompose the double coset into single cosets to find the eigenvaluesas a polynomial in the components of χ (see [53]).In the following lemma we will describe the single coset decomposition for ourcase. Note that a single coset decomposition of a Hecke double coset is computa-tionally hard problem in general. However for small groups one may try to invertthe Satake map to get such decomposition, for the general inversion of Satake see[48]. For symplectic and general linear groups Hecke decomposition were done byseveral people, e.g. for the following decompositions one may look at [1], [43], or[49]Lemma 2.(1)Kp11ppKp =⋃p ∗ ∗p ∗ ∗11Kp⋃11ppKp⋃b∈Z/pZ1 ∗ ∗−b p ∗ ∗p b1Kp⋃p ∗ ∗1 ∗ ∗1pKp,where in the first term the upper right corner has p3 choices and third andfourth term the upper right corner have p choices each.20(2)Kp1ppp2Kp =⋃b∈Z/pZ1−b pp2 bppKp⋃p1pp2Kp⋃p2 ∗ ∗p ∗ ∗1pKp⋃b∈Z/pZp ∗ ∗−bp p2 ∗ ∗p b1Kp⋃p ∗ ∗p ∗ ∗ppKp,where in the third and fourth term the upper right corners have p3 choicesand the fifth term the upper right corner has p2 − 1 choices.The following explicit Iwasawa decomposition and Satake Isomorphism for Gpmay be found in the paper of Asgari-Schmidt [2]. Gp has an Iwasawa decomposi-tion of the form Gp = BKp and B = NA whereN =A XD |A−1 = Dt and Xt − Xis the unipotent radical andA =a =a1a2a−11 a0a−12 a0: ai ∈ Q×p.Let χ0, χ1, χ2 be unramified characters on Q×p . We define a character on A byχ(a) = χ0(a0)χ1(a1)χ2(a2). (3.1.2)21We extend χ from A to B = NA by setting χ to be trivial on N. Now let us defineI(χ) := IndGpB (χ) = { f ∈ Cc(Gp) : f (nag) = δ1/2(a)χ(a) f (g) for (n, a, g) ∈ N×A×Gp}where δ, the modular function from the usual Haar measure on Gp defined byδ(a) = |a−30 a21a42|p. (3.1.3)We will choose χ in such a way so that pip becomes ismorphic to the unique spher-ical constituent piχ of I(χ).Let Fp be the unramified vector in the space piχ with Fp(e) = 1. ThenFp(nak) = δ1/2(a)χ(a)for n ∈ N, a ∈ A and k ∈ Kp. Any φ ∈ Hp acts on Fp by convolution as following;for h ∈ Gp(φ ∗ Fp)(h) :=∫Gpφ(hg)Fp(g)dg.Now let φ = Char(KpapKp) ∈ Hp with KpapKp =∐i HiKp and also let Hi = niaiwhere (niai) ∈ N × A. Then(φ ∗ Fp)(e) =∫Gpφ(g)Fp(g)dg=∫KpapKpFp(g)dg=∑i∫HiKpFp(g)dg=∑iFp(niai)=∑iδ1/2(ai)χ(ai).(3.1.4)Along with above and lemma 2. we arrive at he following conclusion.Proposition 8. Suppose that piχ has trivial central character. Then,22(1) Char(Kp11ppKp)Fp = p3/2χ0(p)[χ1(p)χ2(p)+1+χ1(p)+χ2(p)]Fp,(2) χ20(p)χ1(p)χ2(p) = 1,(3) Char(Kp1ppp2Kp)Fp = [p2χ20(p){χ1(p)+χ2(p)+χ1(p)χ2(p)(χ1(p)+χ2(p))} + (p2 − 1)]Fp.Proof. As computed in 3.1.4 combining with first decomposition in lemma 2 weget that,Char(Kp11ppKp)Fp = p3δ1/2(pp11χ(pp11+ δ1/2(11ppχ(11pp) + pδ1/2(1pp1)χ(1pp1)+ δ1/2(p11p)χ(p11p)23From 3.1.2 and 3.1.3 we get that,Char(Kp11ppKp)Fp = [p3 p−3/2χ0(p)χ1(p)χ2(p) + p3/2χ0(p)+ p2 p−1/2χ0(p)χ2(p) + pp1/2χ0(p)χ1(p)]Fp= p3/2χ0(p)[χ1(p)χ2(p) + 1 + χ1(p) + χ2(p)]Fp.This proves (1). Similary the second decomposition in lemma 2 givesChar(Kp1ppp2Kp)Fp = p2χ20(p)[{χ1(p) + χ2(p) + χ1(p)χ2(p)(χ1(p) + χ2(p))}+ (p2 − 1)χ20(p)χ1(p)χ2(p)]FpNow using φ = Char(Kp pI4Kp) in 3.1.4 and assuming piχ has trivial central char-acter one getsFp = Char(Kp pI4Kp)Fp = χ20(p)χ1(p)χ2(p)Fp,which gives (2) and (3). Now we will be proving our main lemma to get the amplification. On a sep-arate note the component characters arising from the Borel subgroup are calledthe Satake parameters of the corresponding Hecke-Maass form, in particular, theeigenvalues of the same are polynomials in Satake parameters as shown above. Ourgoal is to show that if a representation is non-tempered then there exist at least oneHecke operator whose eigenvalue is much large than its support. This is usuallycalled an amplification; the idea of amplication is inspired by [16]. For generalamplification scheme one may look at [54].Definition 7 (Temperedness). We say a Hecke Maass form is tempered at a place2 < p < ∞ if all its Satake parameters at p are unitary i.e. |χi(p)| = 1 for all i.24So in our setting where G(Qp) = GSp4(Qp) temperdness is equivalent to thatfact that χi’s are unitary (see [46]). our definition is equivalent to the usual notionof temperedness, that if pi = ⊗vpiv is the global cuspidal representation of a Q groupG then pi is tempered at the place p < ∞ if all the matrix coefficients of pip lie inL2+(G(Qp) for all  > 0.Definition 8. We say a Hecke-Maass form φ is η non-tempered at p if at least onecorresponding Satake parameter at p has absolute value pη i.e. there is a characterχi such that |χi(p)| ≥ pη.Now we will be proving our main amplification lemma. Let φ is the Hecke-Maass form in the question withT (p)φ = λ(p)φ and S (p)φ = µ(p)φ.Lemma 3. Let φ be a Hecke-Maass of Γ\H4 form such that φ is η non-temperedat positive density of odd primes p for some η > 0. Then, either|λ(p)|  p3/2+η/2 or |µ(p)|  p2+ηas p→ ∞.Proof. Combining lemma 1 and proposition 8 we get that,χ0(p)2χ1(p)χ2(p) = 1,p−3/2λ(p) = χ0(p)[χ1(p)χ2(p) + 1 + χ1(p) + χ2(p)]= χ0(p) + χ0(p)−1 + χ0(p)χ1(p) + χ0(p)χ2(p)= χ0(p)2χ1(p)χ2(p)[χ0(p)−1 + χ0(p) + (χ0(p)χ1(p))−1 + (χ0(p)χ2(p))−1]andp−2µ(p) + 1 + p−2 = χ20(p)[χ1(p) + χ2(p) + χ1(p)χ2(p)(χ1(p) + χ2(p)) + 2]= χ20(p)χ1(p) + χ20(p)χ2(p) + χ1(p) + χ2(p) + χ0(p)χ0(p)−1 + χ20(p)χ1(p)χ2(p).25Thus the polynomial equationx4 − p−3/2λ(p)x3 + (p−2µ(p) + 1 + p−2)x2 − p−3/2λ(p)x + 1 = 0 (3.1.5)has rootsχ0(p), χ0(p)−1, χ0(p)χ1(p) and χ0(p)χ2(p)Claim: The equation 3.1.5 has a root ν(p) such that |ν(p)| ≥ pη/2.Note that if either |χ0(p)| = p±η or |χi(p)| = pη for i = 1, 2 we are done. Asthe η non-temperedness forces at least one of the character to have magnitude p±η, using symmetry of χ1 and χ2 the only remaining possible case is|χ0(p)| = ps, |χ1(p)| = p−η, |χ2(p)| = pη−2s.If s <(− η2 ,η2)then we have nothing to do, as either |χ0(p)| ≥ pη/2 or |χ0(p)−1| ≥pη/2. Now let s ∈(− η2 ,η2). Then|χ0(p)χ2(p)| = pη−a ≥ pη/2.So the claim is proved.Now using the root ν(p) in the claim the equation 3.1.5 givesp−3/2λ(p)(ν(p)3 + ν(p)) = ν(p)4 + (p−2µ(p) + 1 + p−2)ν(p)2 + 1=⇒ p−3/2|λ(p)|(|ν(p)|3 + |ν(p)|) ≥ |ν(p)|4 − 1 − |p−2µ(p) + 1 + p−2||ν(p)|2=⇒ p−3/2|λ(p)||ν(p)|3 + |p−2µ(p) + 1 + p−2||ν(p)|2  |ν(p)|4so eitherp−3/2|λ(p)||ν(p)|3  |ν(p)|4=⇒ |λ(p)|  p3/2+η/2or|p−2µ(p) + 1 + p−2||ν(p)|2  |ν(p)|4=⇒ |p−2µ(p)|  pη=⇒ |µ(p)|  p2+η.26This proves the lemma. 3.2 SO(5,1) CaseLet G be an algebraic group over Q such that G(Q) = GL2(D). The first part ofthe following proposition would let us realize a Maass form of hyperbolic 5 spaceas a form on GL2(H).Proposition 9.(1) There is an isomorphism between G(R)/R = PGL2(H) = PSL2(H) andSO5,1(R).(2) For an odd prime p we haveG(Qp)  GL4(Qp) and G(Zp)  GL4(Zp).Proof. (1) It is enough to prove that SO5,1(R)  SL2(H)/±I. The usual embed-ding of H to Mat2(C) bya + bi + c j + dk 7→a + bi c + di−c + di a − bican be characterized by as a subset of Mat2(C)H = {x ∈ Mat2(CC) : x¯ = wxw−1}, where w =−11 .Thus we may identifySL2(H) = {g ∈ SL4(C) : g¯ = WgW−1}, where W =−11−11.27Let e1, e2, e3, e4 be the standard basis of C4 and we give∧2 C4 a C-valuedSL4(C) invariant symmetric form〈x ∧ y, z ∧ w〉.e1 ∧ e2 ∧ e3 ∧ e4 = x ∧ y ∧ z ∧ w.A 6-dimensional R-subspace of∧2 C4 stable under S U(4) will be identied asthe xed vectors of an C-conjugate-linear isomorphism C4 → C4 commutingwith SL2(H), on which 〈, 〉 takes real values.Let’s define a conjugate linear map byJ :2∧C4 →2∧C4x ∧ y 7→ Wx¯ ∧Wy¯.Note that J commutes with the action of SL2(H) as,g.J(x ∧ y) = gWx¯ ∧ gWy¯= WW−1g¯Wx ∧WW−1g¯Wx= Wgx ∧Wgy= J(g.x ∧ y).Since,We1 = e2,We2 = −e1,We3 = e4,We4 = −e3we can readily haveJ2(ei ∧ e j) = ei ∧ e j for i , j.Since J is conjugate linear we have that J2 = 1 on∧2 C4. The +1 eigenspacehas an orthogonal basis consistinge1 ∧ e2 + e3 ∧ e4, e1 ∧ e2 − e3 ∧ e4, e1 ∧ e3 + e2 ∧ e4,ie1 ∧ e3 − ie2 ∧ e4, e1 ∧ e4 − e2 ∧ e3, ie1 ∧ e4 + ie2 ∧ e328Computing 〈, 〉 one can check that it has desired signature (5, 1).(2) Noting that for odd prime the divison algebra and the Hurwitz order splitover Qp and Zp respectively i.e.D ⊗ Qp = Mat2(Qp) and O ⊗ Zp = Mat2(Zp)the second part is immediate.Let Γ = SL2(O). For an odd rational prime p let us fix pi, a primitive quaternioninteger (i.e. pi < nO for any rational integer n) with |pi|2 = p. As we have previouslydiscussed that there is a natural embedding of Mat2(H) to Mat4(C) and hence wedefine determinant of a matrix in Mat2(H) by determinant of its image in Mat4(C).Let us defineM(n) = {γ ∈ Mat2(O) : det(γ) = n};so Γ = M(1). Also defineMp = ∪∞m=0M(pm).Following Shimura [51] we define the classical p-Hecke algebra Hp over Γ as thealgebra generated by the double cosets{ΓMΓ : M ∈ Mp}.From theorem 3b of [30] we have the following generating elements ofHpProposition 10. Hp is the polynomial ring over Z generated by the elementsT (p) := Γpi1 Γ, S (p) := Γp1 Γ,T∗(p) = Γppipi Γ and I(p) := pΓ,which are algebraically independent.To know how big the order of support of a Hecke operators is, we need to de-compose the double cosets into single cosets. Next lemma is describes the single29coset decomposition. For this purpose we will map the classical p-Hecke algebrato the canonical convolution p-adic Hecke algebra Hp = H(GL4(Qp),GL4(Zp))which is the convolution algebra of GL4(Zp)-biinvariant compactly supported func-tion of GL4(Qp).One may note that,G(Z[p−1]) ∩G(Zp) = G(Z)andG(Qp)  G(Q)GL4(Zp)  G(Z[p−1])G(Qp),where G(Z) = Γ ∪1−1 Γ. This shows thatΓ\G(Z[p−1])/Γ  G(Z)\G(Z[p−1])/G(Z)  G(Zp)\G(Qp)/G(Zp)and hence p-adic Hecke algebra is isomorphic to p-part of classical Hecke algebra.Lemma 4. With the notation above,Hp  Hp,where one can mapT (p) 7→ Char(Kpp111Kp), S (p)a 7→ Char(Kppp11Kp),T ∗(p) 7→ Char(Kpppp1Kp), and I(p) 7→ Char(pKp).(3.2.1)where Kp = G(Zp) = GL4(Zp) with Vol(Kp) = 1 under the usual Haar measure ofGp.30Let φ be a Hecke-Maass cuspidal eigenform of Γ\H5. Recall that there isan automorphic form Φ which we can produce from φ as described in the lastparagraph of section 2. We consider the representation pi := piΦ of G(A) on righttranslation of Φ. Note that pi would be irreducible due to strong multiplicity oneof G(A) (see [12]) and cuspidal as φ is a cusp form. pi has trivial central characteras φ is invariant under central action. Write pi = ⊗′ppip where pip is representationof Gp := G(Qp). We note that, for an odd prime p, pip is an irreducible admissiblerepresentation of Gp since Kp is the maximal compact subgroup of Gp. From[14] we know that there exists an unramified character χ of the Borel subgroup ofGp, unique up to the Weyl group orbit, such that pip is isomorphic to the uniquespherical constituent piχ of the normalized induced representation IndGpB (χ). Wewill now decompose the double coset into single cosets to find the eigenvalues asa polynomial in the components of χ (see [53]).Lemma 5.(i)Kpppp1Kp =⊔x14,x24,x34∈Zp/pZpppp11 p−1x141 p−1x241 p−1x341Kpunionsq⊔x12∈Zp/pZpp1pp1 p−1x12111Kpunionsq⊔x13,x23∈Zp/pZppp1p1 p−1x131 p−1x2311Kp unionsq1pppKp.31(ii)Kpp111Kp =⊔x12,x13,x14∈Zp/pZpp1111 p−1x12 p−1x13 p−1x14111Kpunionsq⊔x34∈Zp/pZp11p1111 p−1x341Kpunionsq⊔x23,x24∈Zp/pZp1p1111 p−1x23 p−1x2411Kpunionsq111pKp.32(iii)Kppp11Kp =⊔x13,x14,x23,x24∈Zp/pZppp111 p−1x13 p−1x141 p−1x23 p−1x2411Kpunionsq⊔x24,x34∈Zp/pZp1pp111 p−1x241 p−1x341Kpunionsq⊔x23∈Zp/pZp1p1p11 p−1x2311Kpunionsq⊔x12,x14,x34∈Zp/pZpp1p11 p−1x12 p−1x1411 p−1x341Kpunionsq⊔x12,x13∈Zp/pZpp11p1 p−1x12 p−1x13111Kpunionsq11ppKp.Proof. Fix g =IkpIn−k, for 0 ≤ k ≤ n. We would like to compute the group33P = gKpg−1 ∩ Kp. We see that, if p ∈ P is a typical element then,p =Ak×k Bk×n−kpCn−k×k Dn−k×n−kmod p≡A¯k×k B¯k×n−kD¯n−k×n−k = p¯,where A, B,C,D are Zp matrices of corresponding dimensions. Consider the Grass-mannian G(k, n) = Gr(k,Fnp). Now GLn(Fp) acts on G(k, n) (by left multiplication)transitively. This action induces an action of GLn(Zp), simply by left multipli-cation and reducing mod p. We note that, StabGLn(〈e1, e2, . . . , ek〉) = P¯, and soGLn(Fp)/P¯  G(k, n). Also we know G(k, n) ↪→ P(∧kFnp) (Plucker embedding) iscomplete, hence P¯ is parabolic.Now we will try to list out the elements of G(k) by so-called Plucker matricesthrough the Plucker embedding. Hence finding a set of representatives for KgK/Ksuffices finding set of representatives for K/gKg−1 ∩ K suffices finding ’good set’of representatives for G(k, n). We are providing proof for n = 4 and k = 2; proof ingeneral case would follow similarly.Theory of Plucker matrices tells us that any t ∈ G(2, 4) are 2 × 4 matrices withrank = 2. We may embed the 2×4 matrices in GL4(Fp) by completing empty rowsin obvious manner. Therefore they would look like (in certain base) as follows:1 0 ∗ ∗0 1 ∗ ∗ ,1 ∗ 0 ∗0 0 1 ∗ ,1 ∗ ∗ 00 0 0 10 1 0 ∗0 0 1 ∗ ,0 1 ∗ 00 0 0 1 ,0 0 1 00 0 0 1(3.2.2)We use a known fact as follows:Let H be a subgroup of G. Let a ∈ G. Suppose that [a−1Ha ∩ H : H] < ∞ athen,HaH/H  H/a−1Ha ∩ H.That is obvious as HaH has a transitive right action of H. The stabilizer of HaHfor this action is H ∩ a−1Ha. In fact the representatives of HaH/H can be ob-tained from representatives of H/a−1Ha ∩ H. Now suppose that H = Kp anda = diag(p, p, 1, 1), then using the isomorphism one may deduce the decomposi-34tion. For example, the first element in (3.2.2) would be mapped to the first coset inthe decomposition (3). From now on by Gp and Kp we will mean GL4(Qp) and GL4(Zp) respectively.There is an Iwasawa decomposition of Gp = BKp where B = NA;N =1 ∗ ∗ ∗1 ∗ ∗1 ∗1|∗ ∈ Qpis the unipotent radical and A is the maximal torus i.e.A =a =a1a2a3a4|ai ∈ Q×p.Given four unramified characters χ1, χ2, χ3, χ4 of Q×p we define a character χ of Aby definingχ(a) = χ1(a1)χ2(a2)χ3(a3)χ4(a4). (3.2.3)We extend χ from A to B by setting χ to be N invariant. The unramified principalseries representation corresponding to χ is I(χ) which isI(χ) := IndGpB (χ) = { f ∈ Cc(Gp) : f (nag) = δ1/2(a)χ(a) f (g) for (n, a, g) ∈ N×A×Gp}where δ, the modular function from the usual Haar measure on Gp defined byδ(a) = |a31a2a−13 a−34 |p. (3.2.4)We will choose χ in such a way so that pip becomes isomorphic to the uniquespherical constituent piχ of I(χ).Let Fp be the unramified vector in the space piχ with Fp(e) = 1. ThenFp(nak) = δ1/2(a)χ(a)35for n ∈ N, a ∈ A and k ∈ Kp. Any φ ∈ Hp acts on Fp by convolution as following;for h ∈ Gp(φ ∗ Fp)(h) :=∫Gpφ(hg)Fp(g)dg.Now let φ = Char(KpapKp) ∈ Hp with KpapKp =∐i HiKp and also let Hi = niaiwhere (niai) ∈ N × A. Then(φ ∗ Fp)(e) =∫Gpφ(g)Fp(g)dg=∫KpapKpFp(g)dg=∑i∫HiKpFp(g)dg=∑iFp(niai)=∑iδ1/2(ai)χ(ai).(3.2.5)Along with above and lemma 5 we arrive at he following conclusion.Proposition 11. Suppose that piχ has trivial central character. Then,(i) χ1(p)χ2(p)χ3(p)χ4(p) = 1(ii) Char(Kpp111Kp)Fp = p3/2(χ1(p) + χ2(p) + χ3(p) + χ4(p))Fp(iii) Char(Kpppp1Kp)Fp = p3/2(χ1(p)−1+χ2(p)−1+χ3(p)−1+χ4(p)−1)Fp36(iv)(Kppp11Kp)Fp = p2(χ1(p)χ2(p) + χ2(p)χ3(p) + χ3(p)χ4(p) + χ4(p)χ1(p)+ χ1(p)χ3(p) + χ2(p)χ4(p))Fp.Proof. As piχ has trivial central character (1) is immediate from 3.2.5. We aregiving a proof of (3); (2) and (4) would be similar.Note that the first decomposition in lemma 5 can also be wriiten as⊔x14,x24,x34∈Zp/pZp1 x141 x241 x341ppp1Kpunionsq⊔x12∈Zp/pZp1 x12111p1ppKpunionsq⊔x13,x23∈Zp/pZp1 x131 x2311pp1pKp unionsq1pppKp.From 3.2.5 we get thatChar(Kpppp1Kp)Fp = [p3 p−3/2χ1(p)χ2(p)χ3(p) + pp1/2χ1(p)χ3(p)χ4(p)p2 p−1/2χ1(p)χ2(p)χ4(p) + p3/2χ2(p)χ3(p)χ4(p)]Fp= p3/2χ1(p)χ2(p)χ3(p)χ4(p)(χ1(p)−1 + χ2(p)−1 + χ3(p)−1 + χ4(p)−1)Fp= (χ1(p)−1 + χ2(p)−1 + χ3(p)−1 + χ4(p)−1)Fp.37As in the previous section here also we will use non-temperedness to have anamplification. We will show that if a representation is non-tempered then thereexist at least one Hecke operator whose eigenvalue is much large than its support.We have similar definition of nontemperedness here as well.Definition 9. We say a Hecke-Maass form φ is η non-tempered at p if at least onecorresponding Satake parameter at p has absolute value pη i.e. there is a characterχi such that |χi(p)| ≥ pη.Now we will be proving our main amplification lemma. Let φ is the Hecke-Maass form in the question withT (p)φ = λ(p)φ, and S (p)φ = µ(p)φ.Note that T ∗(p) is the adjoint Hecke operator of T (p) with respect to the Peterssoninner product defined in 2.3.4. So that gives us T ∗(p)φ = λ(p)φ.Lemma 6. Let φ be a Hecke-Maass of Γ\H5 form such that φ is η non-temperedat positive density of odd primes p for some η > 0. Then, either|λ(p)|  p3/2+η/2 or |µ(p)|  p2+ηas p→ ∞.Proof. Note that from lemma 4 and proposition 11 the polynomial equationx4 − p−3/2λ(p)x3 + p−2µ(p) − p−3/2λ(p)x + 1 = 0 (3.2.6)has roots χi(p) for i = 1, 2, 3, 4. As at least one of χi is η away from unitary axisand∏i χi(p) = 1, there is an i, say i = 1, such that |χ1(p)| = pη. Now letting38x = χ1(p) in equation (3.2.6) we get that,|p−3/2λ(p)x3 + p−3/2λ(p)x| = |x4 + p−2µ(p)x2 + 1|=⇒ p−3/2|λ(p)(|x|3 + |x|) ≥ |x|4 − p−2|µ(p)||x|2 − 1=⇒ |p−3/2λ(p)|p3η + p2η|p−2µ(p)  p4ηeither=⇒ |λ(p)|  p3/2+ηor=⇒ |µ(p)|  p2+2η.This proves the lemma. 39Chapter 4Amplified Pre-trace FormulaIn this chapter we will construct an identity which will allow us to reduce ourproblem of estimating a Maass form inHn to a diophantine analysis problem. Thepre-trace formula is the beginning of general Arthur-Selberg’s Trace formula, see[50]. For that, first we need an explicit Selberg/Harish-Chandra spherical transformpair (h, k). For dimension 2 i.e. on upper half the following proposition can befound in [20]. where as the idea of amplification is inspired by [16]. For generalamplification scheme one may look at [54].4.1 Spherical Transform and Automorphic KernelProposition 12. Let (h, k) be the Selberg/Harish-Chandra spherical transform pairinHn and u is the point pair invariant defined in 2.3.2. Thenh(t) =(n − 2)(2pi)n−122Γ(n+12)∫ +∞−∞eiαt∫v(u − v)(n−32 )k(u)dudα, (4.1.1)where v = (eα−1)22eα .Proof. Let hˆ be the spherical transform of h inHn. Recall from 2.1.6 that the point-pair invariant u and distance d(P) := d(P, e) of P = (x, y) from origin e = (0, 1) isrelated by,u =||x||2 + (y − 1)22y= sinh2(d2).40Then by definition of spherical transform and letting hˆ(d) = k(u) we have,h(t) =∫Hnhˆ(d(P))y(P)n−12 +itdµ(P)=∫ ∞0yn−12 +it∫Rn−1hˆ(d)dn−1xdyyn=∫ ∞0y−n+12 +it∫ ∞0k(r2 + (y − 1)22y)drdy∫||x||=rdn−1x= Vol(S n−2)∫ ∞0y−n+12 +it∫ ∞0rn−2k(r2 + (y − 1)22y)drdy.Now letting v = (y−1)22y and y 7→ eα, as u = v + r22y we get,h(t) = Vol(S n−2)∫ ∞0y−n+12 +it∫ ∞v(2y(u − v))n−32 k(u)ydudy= 2n−32 Vol(S n−2)yit∫ ∞v(u − v)n−32 k(u)dudyy=(n − 2)(2pi)n−122Γ(n+12)∫ +∞−∞eiαt∫ ∞v(u − v)n−32 k(u)dudα.Note that we can describe 4.1.1 in a three step relation as following,h(t) =∫ +∞−∞g(α)eiαtdαv =12sinh2(α2)⇔ α = log(1 + v +√v2 + 2v)g(α) =(n − 2)(2pi)n−122Γ(n+12)∫ ∞v(u − v)(n−32 )k(u)du.(4.1.2)For the pre trace formula we need a basis of the Hilbert space L2(Γ\S ) which withPetersson inner product defined in 2.3.4 decomposes into (see [34], [33]),L2(Γ\S ) = L2res(Γ\S ) ⊕ L2cusp(Γ\S ) ⊕ L2cont(Γ\S ),where L2cusp is the closure of the space generated by the cusp forms, L2res is the41residual Eisenstein series; those two spaces combine to L2disc correspond to discretespectrum, L2cont is the continuous spectrum.Let there be m cusps in Γ\S . The Eisenstein series Ek(P, s) correspondingto k′th cusp is holomorphic at <(s) = 1/2. Let {φ}∞i=0 be an orthonormal basis(containing φ the mass form we want to estimate) consisting of joint eigenfunctionof Laplacian and full Hecke algebra. Then from the spectral theorem we can saythe following (see [34]).Proposition 13. Let f ∈ C∞(Γ\S ).Then f has the following spectral expansionf (P) =∞∑i=0〈 f , φi〉φi(P) +m∑k=114pi∫ +∞−∞〈 f , Ek(·, 1/2 + it)〉Ek(P, 1/2 + it)dt, (4.1.3)which converges in C∞-topology.Let us choose f to be our automorphic kernel in P variable, namely for somesmooth k ∈ S(R≥0),K(P,Q) =∑γ∈Γk(u(γP,Q)).After doing a similar computation of respective inner products as in 7.4 [28] wegetK(P,Q) =∞∑i=0φi(P)φi(Q)h(t j)++m∑k=114pi∫ +∞−∞Ek(P, 1/2+ it)Ek(Q, 1/2 + it)h(t)dt,(4.1.4)where h is the Selberg/Harish-Chandra tranform of k as in 4.1.2. The above equa-tion is called pre-trace formula.424.2 Archimedean Amplification and Bounding kWe will use a specific choice for the spherical transform pair (h, k) which will beconstructed so as to emphasize the contribution of the term φ in the basis. Sothat the function h(t) becomes concentrated on the the spectral parameter T of φ.Inspired by [27], we considerg(x) =A cos(T x)cosh(Ax), (4.2.1)For some A > 2 large constant, h and k will be constructed as in 4.1.2.Lemma 7. For above choice of g,(1) the function h(t) is even, holomorphic, and rapidly decaying in the strip|=(t)| < A. It satisfies the boundh(t) >0, t ∈ R ∪ (−A, A)i1/8, t ∈ R ∪ (−A/2, A/2)i and ||t| − T | < A/2.. (4.2.2)(2) The Selberg/Harish-Chandra transform k(u) of h(t), as in 4.1.2, satisfies theboundk(u) A min(T [n2 ](1 + u)Au[n2 ],T n−1)(4.2.3)Proof. The function g(x) defined in 4.2.1 is even and satisfies the boundg(x) A eA|x|,hence it Fourier transformation h(t) in 4.1.2 is even and holomorphic in the strip43|=(t)| < A by Morera’s theorem. We first calculate the Fourier transform of sechx∫ ∞−∞dx sech(pix) eikx = 2∫ ∞−∞dxeikxepix + e−pix= 2∫ 0−∞dxeikxepix + e−pix+ 2∫ ∞0dxeikxepix + e−pix= 2∞∑m=0(−1)m[∫ ∞0dx e−[(2m+1)pi+ik]x +∫ ∞0dx e−[(2m+1)pi−ik]x]= 2∞∑m=0(−1)m[1(2m + 1)pi − ik+1(2m + 1)pi + ik]= 4pi∞∑m=0(−1)m(2m + 1)(2m + 1)2pi2 + k2=12pi∞∑m=−∞(−1)m(2m + 1)(m + 12)2+(k2pi)2By the residue theorem, the sum is equal to the negative sum of the residues atthe non-integer poles ofpi csc (piz)12pi2z + 1(z + 12)2+(k2pi)2which are at z± = −12 ± ik2pi . The sum is therefore−12csc (piz+) −12csc (piz−) = −<1sin pi(− 12 + ik2pi) = sech(k2)By this reasoning, the Fourier transform of sechx is pi sech(pik2). Thus we calculateh(t) =14sech(pi(t + T )2A)+14sech(pi(t − T )2A)=cosh(pit2A)cosh(piT2A)cosh(pitA)+ cosh(piTA) ,(4.2.4)which shows that h(t) is rapidly decaying in the strip |=(t)| < A.44Using this we will prove 4.2.2. Assume first that t ∈ R. From (4.8) it is clearthat h(t) > 0, because both terms are positive. Moreover, if |t| ∈ (T − A2 ,T +A2 )then one of these terms exceeds 14 sech(pi4)> 18 , so that h(t) >18 . Assume nowthat t ∈ (−A, A)i. From (4.8) it is clear that h(t) > 0, because cosh(pit2A)> 0 andcosh(pitA)> −1. Moreover, if t ∈ (−A2 ,A2 )i then cosh(pit2A)> cos(pi4)and cosh(pitA)≤1, so thath(t) >cos(pi4)cosh(piT2A)1 + cosh(piTA) =cos(pi4)2 cosh(piT2A) .If, in addition, |t| ∈ (T − A2 ,T +A2 ), then T = T − |t| + |t| < A/2 + A/2 = A, whenceh(t) >cos(pi4)2 cosh(pi2) >18.We will now prove 4.2.3. Recalling 4.1.2 et us define(n − 2)(2pi)n−122Γ(n+12) q(v) = g(α).Hence we get,q(v) =∫ ∞0xn−32 k(v + x)dx.For ν = 0, 1 according to n ≡ 1, 0 (mod 2),∫ ∞uq(n+1−ν2 )(v)√(v − u)νdv =∫ ∞u1√(v − u)ν∫ ∞0xn−32 k(n+1−ν2 )(x + v)dxdvby integration by parts and using rapid decay of k,=(−1)n+1−ν22νΓ(n − 12) ∫ ∞0∫ ∞0k′(u + x + v)√(vx)νdxdv.Thus integrating left side using polar cordinate we get,k(u) =(−1)n+1−ν22νpiΓ(n−12)∫ ∞uq(n+1−ν2 )(v)√(v − u)νdv. (4.2.5)45Now note that from 4.1.2 and 4.2.1 we have precise bounds of following,α = log(1 + u +√u2 + 2u) ≤√u2 + 2u,cosh(Bx)  (1 + u)B,g(m)(α) T m(1 + u)C1for some constant C1 > 0,dmαdum(1 + u)m−1(√u2 + 2u)2m−1.Thus we get the derivative bound,|q(m)(u)|  |g(m)(α)(dαdu)m| + |g′(α)dmαdum|1(1 + u)C2T m(√u2 + 2u)m+T | sin(Tα)|(√u2 + 2u)2m−11(1 + u)C2T m(√u2 + 2u)m+ Tmin((Tα)m−1, (Tα)2m−1)(√u2 + 2u)2m−11(1 + u)Amin(T mum,T 2m),For some large enough A.Now for odd n from 4.2.5 we get that,k(u)  |q(n−12 )(u)|  minTn−12(1 + u)Aun−12,T n−1 .And for even n we divide the integral in 4.2.5 in two parts and estimate as follow-ing,k(u) ∫ η0|q(n2 )(u + v)|√vdv +|q(n2−1)(u + η)|√η+∫ ∞η|q(n2−1)(u + v)|√v3dv min(T n/2(1 + u)Aun/2,T n1 + u)√η +T (n−2)/2(1 + u + v)A(u + v)n/2√η min(T n/2(1 + u)Aun/2,T n1 + u)√η +T (n−2)/2(1 + u)Au(n−2)/2√η.46Now choosing√η = T−1√u we get that,k(u)  min(Tn2(1 + u)Aun2,T n−1),which proves our claim. 47Chapter 5Diophantine Analysis and BoundsIn this chapter we fix the dimension n of the arithmetic hyperbolic space to be 4and 5. By Γ we would mean Sp∗1,1(O) or SL2(O) in case ofH4 andH5 respectively,where O is Hurwitz integer quaternions. Recall the equation 4.1.4 of pre-trace for-mula. Let {φ j} with Laplace eigenvalue λ j be orthonormal basis of L2disc(Γ\Hn)which contains φ the Maass cusp form we want to bound and Ek be the Eisensteinseries in the continuous spectrum, and K(P,Q) =∑γ∈Γ k(u(γP,Q)) be the automor-phic kernel. Then,∞∑i=0φi(P)φi(Q)h(t j)+m∑k=114pi∫ +∞−∞Ek(P, 1/2+it)Ek(Q, 1/2 + it)h(t)dt =∑γ∈Γk(u(γP,Q)).Let T be a Hecke operator on Γ\Hn which can be thought as a Γ double coset ΓaΓ,such that φ j’s and Ek’s are Hecke eigenfunction withTφ j(P) =∑γ∈Γ\ΓaΓφ(γP) = λ jφ j(P) and T Ek(P, ·) =∑γ∈Γ\ΓaΓEk(γP, ·) = Λ jEk(P, ·).Let T ∗ be the adjoint Hecke operator of T with respect to the Petersson inner prod-uct. We now shall apply T ◦ T ∗ on the P variable of the pre-trace formula. And48then letting P = Q we get.∞∑i=0h(t j)|λ j|2|φi(P)|2 +m∑k=114pi∫ +∞−∞h(t)|Λk|2|Ek(P, 1/2 + it)|2 =∑γ∈ΓaΓk(u(γP, P)).(5.0.1)Recalling lower bound of h(t) from 4.2.2 and noting the positivity of each term inLHS of (5.1) we conclude that,|λ|2|φ(P)|2 ∑γ∈ΓaΓk(u(γP, P))=∑γ∈ΓaΓu(γP,P)<δk(u(γP, P)) +∑γ∈ΓaΓu(γP,P)≥δk(u(γP, P))(5.0.2)for some δ > 0 to be fixed latter. We fix a compact Ω where we bound the L∞norm, in the fundamental domain and let P ∈ Ω. Let us call e = (0, 1) ∈ Γ\Hnand as the ambient group acts isometrically on this space there exist a g such thatg.e = P. Hence,δ > u(γP, P) = u(g−1γg.e, e)which implies g−1γg ∈ Bδ(K) where K is the maximal compact in respective cases.Note that saying P ∈ Ω compact in fundamental domain is equivalent to sayingg ∈ Ω′ for some compact Ω′ ⊂ G.Lemma 8. For e = (0, 1)u(ge, e) =12||g||2HS −√det(g), (5.0.3)where for a matrix A ∈ Matn(H) Hilbert-Schmidt norm of A is defined by||A||2HS =∑i, j|Ai j|2.Proof. We give a proof in SL2(H) case; Sp∗1,1(H) case would be similar. Let g =a bc d be in SL2(H). From Iwasawa decomposition of the corresponding group it49can be seen that,g(0, 1) =bd¯ + ac¯|c|2 + |d|2,√det(g)|c|2 + |d|2 .Therefore from 2.3.2 we calculate,u(ge, e) =|bd¯ + ac¯|2 + (|c|2 + |d|2 − det(g))22(|c|2 + |d|2)=|b|2|d|2 + |a|2|c|2 + 2<(ac¯db¯ + (|c|2 + |d|2)2 − 2(|c|2 + |d|2)√det(g) + det(g)2(|c|2 + |d|2)=|b|2|d|2 + |a|2|c|2 + |c|2|d|2 + |a|2|b|2 + (|c|2 + |d|2)22(|c|2 + |d|2)−√det(g)=(|a|2 + |b|2 + |c|2 + |d|2)(|c|2 + |d|2)2(|c|2 + |d|2)−√det(g)=12||g||2HS −√det(g)Definition 10. Let T be a Hecke operator defined by the double coset ΓaΓ. wedefine support of T bySupp(T ) = {γ ∈ Γ\ΓaΓ}.If T ′ is another Hecke operator then clearlySupp(TT ′) ⊆ {tt′|t ∈ Supp(T ), t′ ∈ Supp(T ′)}and hence |Supp(TT ′)| ≤ |Supp(T )||Supp(T ′)|. In particular we have the followinglemma.Lemma 9.|Supp(T ◦ T ∗)| ≤ |Supp(T )|2. (5.0.4)Combining lemma 1 and lemma 2 one can compute the size of the support ofthe chosen generators in proposition 6 of the corresponding Hecke algebra; namelyfor odd prime p,Supp(T (p))  p3 and Supp(S (p))  p4. (5.0.5)50Also Combining lemma 4 and lemma 5 one can compute the size of the supportof the chosen generators in proposition 10 of the corresponding Hecke algebra;namely for odd prime p,; namely for odd prime p,Supp(T (p)) = Supp(T ∗(p))  p3 = and Supp(S (p))  p4. (5.0.6)Proof of theorem 1: As the argument for the saving in the both dimensions aresame we will give the proof in dimension 5 only.Let the Hecke Maass form φ is η non-tempered at ν > 0 density of primes. Sofor L sufficiently large in the interval [L, 2L] there will be  νLlog L primes p whereφ is non-tempered by the prime number theorem. We assume L to be sufficientlylarge so that νLlog L > 1. Recalling lemma lemma 6 it is clear that at each place thereis a Hecke operator H = Hp with Hφ = λφ such that|λ|2  Supp(H)pη.In (5.0.1) we use T = H ◦ H∗ and so λ = |ν|2. Let us assume H = T (p) defined inproposition 10, if T (p) does not have required property we will chooseΓpi21 Γ = pT2(p) + S (p)which has the required property. As proof in both cases are same we will prove forT (p) only.First we will bound the second summand in (5.2). Let us defineM(t) = M(P, t) := #{γ ∈ Supp(T ) : u(γP, P) < t}.51and for g ∈ Ω′ we get from lemma 8 that,||g−1γg||2HS − det(γ)1/2 < t=⇒ ||γ||2HS Ω′ pk + t=⇒ |γi j| Ω√pk + t=⇒ #γ ∈ M(t) Ω L8k + L8for some absolute positive integer k. Now from 4.2.3 integrating by parts we getthat,∑γ∈Supp(T )u(γP,P)≥δk(u(γP, P)) ∫ ∞δT [n2 ](1 + u)Au[n2 ]dM(u) T [n2 ](L8kδ−[n2 ]+1 + δ−[n2 ]+9).(5.0.7)Now we will bound the first summand.Lemma 10. Let T be a p-Hecke operator on Γ\H5 defined by Γpi1 Γ for someprimitive quaternion pi with norm p for an odd prime p. Then for g ∈ G and Kmaximal compact|{γ ∈ Γpi1 Γ : g−1γg ∈ Bδ(K)}| Ω′ L3/2+ , (5.0.8)for sufficiently small δ.Proof. Now if Q = (gg∗)−1, theng−1γg ∈ Bδ(K)↔ γ∗Qγ = det(γ)1/2Q + O(det(γ)1/2δ).So the first column of Γ satisfies a quadratic form of 8 variables with O(1) coef-ficients, therefore number of solutions will be  L3/2+ (see e.g. [6] Lemma 8a).Fixing the first column the other column can be obtained in L ways. 52Again by 4.2.3, (5.4) and (5.7) we get,∑γ∈Supp(T )u(γP,P)<δk(u(γP, P))  T n−1|{γ ∈ ΓaΓ : g−1γg ∈ Bδ(K)}|Ω T n−1Supp(T ) T n−1|λ|2L−η.Choosing δ = min(T−1/20, 12)and L = T 1/20k and from (5.2) we get|φ(P)|2 Ω T n−1− .53Chapter 6Concluding Remarks(i) It seems natural to expect the techniques of this thesis would generalize toother groups, and in particular to other hyperbolic space. To do this wouldrequire developing an explicit Hecke theory, perhaps generalizing Krieg’swork by starting from a Z-order in the Clifford algebra over Q. The diophan-tine analysis would also be more difficult as the maximal compact subgroupSO(n) would be larger.(ii) The main Theorem is stated for the supremum of the eigenfunctions on afixed compact set Ω. A similar bound probably does not hold in the cusp dueto the behaviour of the K-Bessel function in its transition region (see [5]). Inthe case of GL3 this is discussed in [13]. However, a typical no Siegel zeroresult of the Rankin-Selberg L-function should give supnorm bound nearcusps.(iii) We have already discussed the examples of CAP representaiton in H4 andH5 such that they are tempered in infinite place and nontempered in everyfinite place. 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