On the Kuznetsov Trace Formula for $\Mathrm {PGL} 2 (\Mathbb {C}) $

Home , Kuznetsov trace formula

arXiv:1606.02477v3 [math.RT] 2 Mar 2018 PSL SO emti ie egtdsmo lotra usaie fr arises sums Kloosterman of sum weighted a side, for geometric trace Petersson the in occurring forms cusp holomorphic hog pcrldcmoiinfruafrteinrprod inner the for formula decomposition spectral a through nte omo i rc oml tevrini [ in tran version (the Bessel formula the trace for his formula of inversion form another an using Then, series. .A xlctKosemnSeta oml h Kuznetso the - Formula Kloosterman-Spectral References Explicit An 5. .KosemnSm,PicreSre n h Kloosterman- the and Poincaré Series Sums, Kloosterman 4. .Seta nlssof Analysis Spectral Theorem of Statement 3. and Notations 2. Introduction 1. p 2 2 nteppr[ paper the In 2010 e od n phrases. and words Key p q{t˘ Z qz nlssi h xsigmto u oBugmnadMotohas and Bruggeman to due method existing the in analysis hpr,w rv h unto rc oml o narbitra an for formula trace Kuznetsov the prove we Shapiro, rvdb h uhr nBse ucin o PGL i for ingredient functions Bessel essential on An author, the by cocompact. proved not but cofinite is that A ahmtc ujc Classification. Subject Mathematics PSL bstract 1 u hr r w om fhsfrua h praht h rtf first the to approach The formula. his of forms two are There . 2 p nteKzesvTaeFruafor Formula Trace Kuznetsov the On R nti oe sn ersnaintertcmto fCo of method theoretic representation a using note, this In . q{ Kuz K nwhich in , L ,Kzesvdsoee i rc oml o PSL for formula trace his discovered Kuznetsov ], 2 h unto rc oml,seta eopsto,Bess decomposition, spectral formula, trace Kuznetsov the p Γ z G q H 2 .Introduction 1. eoe h yeblcuprhl-ln and half-plane upper hyperbolic the denotes 17,11M36. 11F72, C h Qi Zhi ontents 1 2 p DI C q pcrlFrua9 Formula Spectral rc oml 14 Formula Trace v hsapoc vistedi the avoids approach This . smr opeei h es that sense the in complete more is ] ydsrt group discrete ry enlfrua recently formula, kernel a s uaaeas novd.O the On involved). also are mula c of uct hi. fr,Kzesvobtained Kuznetsov sform, mcmuigteFourier the computing om PGL dl n Piatetski- and gdell two 2 shrcl Poincaré(spherical) p Γ lfunctions. el C nPGL in q ffi 2 cult 2 p p C Z ruais ormula q qz H K 2 18 – “ 6 3 1 2 ZHI QI coefficients of Poincaréseries. The spectral side involves the Fourier coefficients of holomorphic and Maaß cusp forms and Eisenstein series along with the Bessel functions associated with their spectral parameters. The second form plays a primary role in the investi- gation of Kuznetsov on sums of Kloosterman sums in the direction of the Linnik-Selberg conjecture.I Along the classical lines, the Kuznetsov trace formula has been studied and generalized by many authors (see, for example, [Bru1, Bru2, Pro, DI, BM2]). Their ideas of generalizing the formula to the non-spherical case are essentially the same as Kuznetsov. It should however be noted that the pair of Poincaréseries is chosen and spectrally decom- posed in the space of a given K-type. In the framework of representation theory, the second form of the Kuznetsov formula for an arbitrary Fuchsian group of the first kind Γ PGL2 R was proved straightfor- Ă p q wardly by Cogdell and Piatetski-Shapiro [CPS]. Their computations use the Whittaker and

Kirillov models of irreducible unitary representations of PGL2 R . They observe that the p q Bessel functions occurring in the Kuznetsov formula should be identiﬁed with the Bessel functions for irreducible unitary representations of PGL2 R given by [CPS, Theorem 4.1], p q in which the Bessel function Jπ associated with a PGL2 R -representation π satisﬁes the p q following kernel formula,

a 1 b ˆ (1.1) W ´ Jπ ab W d b, 1 1 “ ˆ p q 1 ˜˜ ¸˜ ¸¸ żR ˜ ¸ for all Whittaker functions W in the Whittaker model of π. Note that their idea of ap- proaching Bessel functions for GL2 R using local functional equations for GL2 GL1- p q ˆ Rankin-Selberg zeta integrals over R (see [CPS, §8]) also occurs in [Qi]. In the book of Cogdell and Piatetski-Shapiro [CPS], the Kuznetsov trace formula is derived from computing the Whittaker functions (Fourier coeﬃcients) of a (single)

Poincaréseries P f g in two different ways, first unfolding P f g to obtain a weighted p q p q 2 sum of Kloosterman sums, and secondly spectrally expanding P f g in L Γ G and then p q p z q computing the Fourier coefficients of the spectral components in terms of basic representation theory of PGL2 R . The Poincaréseries in [CPS] arise from a very simple type of p q functions that are supported on the Bruhat open cell of PGL2 R and split in the Bruhat p q coordinates, which is surely not of a fixed K-type. In other words, [CPS] suggests that, instead of the Iwasawa coordinates, it would be more pleasant to study the Kuznetsov formula using the Bruhat coordinates. For this, [CPS] works with the full spectral theorem rather than a version, used by all the other authors, that is restricted to a given K-type. In the direction of generalization to other groups, Miatello and Wallach [MW] gave the spherical Kuznetsov trace formula for real semisimple groups of real rank one, which include both SL2 R and SL2 C . It is however much more difficult to extend the formula p q p q

IIt is Selberg who introduced Poincar´eseries and realized the intimate connections between Kloosterman 2 sums and the spectral theory of the Laplacian on PSL2 Z H in [Sel]. p qz ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 3

II to the non-spherical setting for SL2 C . The ﬁrst breakthroughis the work of Bruggeman p q and Motohashi [BM3], where the Kuznetsov trace formula for PSL2 Z i PSL2 C was p r sqz p q found.III Let H3 denote the three dimensional hyperbolic space and let K SU 2 1 . “ p q{t˘ u Featuring a combination of the Jacquet and the Goodman-Wallach operators, the analysis carried out in [BM3] is considerably hard. Nevertheless, similar to [Kuz], the approach of [BM3] is also from considering the inner product of two certain sophisticatedly chosen Poincar´eseries of a given K-type. It is however remarked without proof in [BM3, §15] that their Bessel kernel should be interpreted as the Bessel function of an irreducible unitary representation of PSL2 C . p q In this note, we shall prove the Kuznetsov trace formula for an arbitrary discrete group

Γ in PGL2 C that is coﬁnite but not cocompact. An essential ingredient is a kernel formula p q for PGL2 C (see (4.3)) that is identical to (1.1). This is a consequence of the represen- p q tation theoretic investigations on a general type of special functions, the (fundamental)

Bessel kernels for GLn C , arising in the Vorono¨ısummation formula; see [Qi, §18]. Our p q derivation of the Kuznetsov formula will be in parallel with that in [CPS], at least for the Kloosterman-spectral formula (see §4). As such, this note should be viewed as the sup- plement and generalization of the work of Cogdell and Piatetski-Shapiro over the complex numbers. With this method, we can avoid the very diﬃcult and complicated analysis in [BM3].

Finally, we remark that the only reason for considering PGL2 C is to keep notations p q simple. Without much diﬃculty, we may extend the Kuznetsov formula to SL2 C . In p q another direction, we hope to implement the Cogdell-Piatetski-Shapiro method to PGL2 or

SL2 over an arbitrary number ﬁeld. Acknowledgements . The author would like to thank James W. Cogdell and Stephen D. Miller for valuable comments and helpful discussions. The author wishes to thank the anonymous referee for thorough reading of the manuscript and several suggestions which helped improving the paper.

2. Notations and Statement of Theorem

2.1. We shall adopt the notations in [CPS]. Let G PGL2 C ( PSL2 C ). Let “ p q “ p q 1 u N u : u C G, “ # “ ˜ 1¸ P + Ă a A a : a Cˆ G. “ # “ ˜ 1¸ P + Ă

IIThis spherical formula was generalized to an arbitrary number ﬁeld by Bruggeman and Miatello [BM1]. IIIIn her thesis [LG], Lokvenec-Guleska extended the formula of [BM3] to congruence subgroups of

SL2 C over any imaginary quadratic field. Maga recently generalized the first form of the formula to an ar- p q bitrary number field in [Mag]. 4 ZHI QI

Define A r A : r R . Let B NA denote the Borel subgroup of G. Let 4 ` “ t P P ù “ “ 1 ´ be the long Weyl element of G. Then we have the Bruhat decomposition G ˜1 ¸ “ B N4B. Let Y v w 2 2 K SU 2 1 kv, w : v w 1 1 . “ p q{t˘ u“ “ w v | | ` | | “ {t˘ u # ˜´ ¸ + We have the Iwasawa decomposition G NA K. “ ` Let g denote the Lie algebra of G and U g its universal enveloping algebra. p q 2.2. Let Γ G be a discrete subgroup that is cofinite but not cocompact. Let C Ă Ă H3 denote the set of cusps of Γ. We assume that C. For each cusp a C, we fix B 8 P P ga G such that ga a . Let Pa denote the parabolic subgroup of G stabilizing a. P ¨ “ 8 Let Pa Ua Ma be its Langlands decomposition. Then the conjugation by ga provides an “ „ „ „ isomorphism Pa B, which induces isomorphisms Ua N and Ma A. For each 1 ÝÑ ´1 ÝÑ ÝÑ a, define Γ Γ Pa, Γa Γ Ua. Let gaΓag u : u Λa , with Λa a lattice in a “ X “ X a “ t P u C. Let Λa denote the area of Λa C. According to [EGM, Theorem 2.1.8 (3)], we have 1 | | z Γa ηma Λa, where ma 1, 2, 3, 4, 6 , ηma denotes the group of ma-th roots of unity, – ˙ P t IV u and ηma acts on Λa by multiplication . Let X Cˆ denote the group of multiplicative characters of Cˆ. Every character µ ˆ p q 2s d P X C is of the form µs,d z z z , with s C, d Z and the notation z z z . p q ˆ p q “ | | r s P Pˆ r s “ {| | Hence X C has a structure of a complex manifold, X C Z C. We define Re µs,d p q p q– ˆ “ Re s . Each µ X Cˆ defines a character of A, by µ a µ a , and a character of B p q P p q p q “ p q through B N B A. Through the isomorphisms give above, µ also defines a character Ñ z – ˆ of Ma and Pa. Let Xm C denote the set of characters µs,d such that m d. Note that ˆ p q 1 | characters in Xm C are trivial on Γ . a p q a Let X8 C denote the group of additive characters on Λ8 C. We define the dual 1 p q z lattice Λ8 of Λ8 by Λ1 ω C : Tr ωλ ωλ ωλ Z, for all λ Λ . 8 “ P p q“ ` P P 8 ! ) 1 Then every ψ X C is of the form ψω z e Tr ωz for some ω Λ , with e x P 8p q p q “ p p qq P 8 p q “ e2πix. Furthermore, each ψ X C defines a character of Γ N, by ψ u ψ u . P 8p q 8z p q“ p q 2.3. In the notations of [Qi, §18.2], the irreducible infinite dimensional unitary representations of PGL2 C are p q - (principal series) π` it for t R and d Z, with π` it π` it , d p q P P d p q– ´dp´ q - (complementary series) π t for t 0, 1 . p q P 2 ` In order to unify notations, let us put πd it ` π˘ it and π0 t π t so that we may p q “ d p q p q “ p q denote by πd s either principal series for s it or complementary series for s t and p q “ “ d 0. “ IV When ma 4, respectively 3 or 6, Λa must be the ring of integers in the quadratic number field Q i , “ p q respectively Q ? 3 . p ´ q ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 5

2.4. Let dz be twice the ordinary Lebesgue measure on C, and choose the standard multiplicative Haar measure dˆz dz z 2 on Cˆ Cr 0 . “ {| | “ t u We take the Haar measure dk on K of total mass 1. Writing an element of G in the Iwasawa decomposition as g zrk, with z N, r A and k K, we let dg “ P P ` P “ 2r´3dzdrdk. This Haar measure dg on G induces the hyperbolicmeasure on the hyperbolic space H3 z r j : z C, r R . Moreover, if we use the Bruhat coordinates g “ t ` P P `u “ u14u2 a on the open Bruhat cell N4B, with u1, u2 N and a A, then the measure 2 ˆ P P dg 1 4π du1 du2 d a. “ p { q 2.5. Main Theorem.

heorem 8 ˆ ˆ T 2.1. Let F Cc C be a smooth compactly supported function on C . Let 1 P p q ω1, ω2 Λ r 0 and ψ1 ψω , ψ2 ψω . Then P 8 t u “ 1 “ 2 KlΓ c; ψ1, ψ2 ω ω p q F 1 2 c c c Ω Γ Pÿp q | | ´ ¯

2 Aω2 ϕπ Aω1 ϕπ π Λ8 p q p q F s, d “ | | 2 d 1 Gs,d p q π Π Γ Pÿdp q p | |` q π–πd psq p 1 1 ω2 a a µit,d ZΓ µit,d;0, ω2 ZΓ µit,d;0, ω1 F it, d dt, 4 ma Λa ω ` a d P m Z R 1 p q p q p q ÿ | | ÿa ż ˆ ˙ p where KlΓ c; ψ1, ψ2 are the Kloosterman sums at infinity associated with Γ, ψ1 and ψ2 p q 2 given in §4.1,Aω ϕπ are the Fourier coefficients of a certain L -normalized automorphic p q form in the space of the representation π that occurs discretely in L2 Γ G (see §5.2), a p z q Z µit,d;0, ω are the Kloosterman-Selberg zeta functions arising in the Fourier coefficients Γp q of Eisenstein series (see §5.3),

1, if s it, Gs,d “ “ $Γ 1 2t Γ 1 2t , if s t, d 0, & p ` q{ p ´ q “ “ and ﬁnally F s, d is the Bessel% transform of F given by p q 1 ˆ Fp s, d F z Js,2d 4π ?z J´s,´2d 4π ?z d z, p q“ sin 2πs ˆ p q p p q´ p qq p q żC with p

Js,2d z J 2s d z J 2s d z , p q“ ´ ´ p q ´ ` p q and Jν z the classical Bessel function of the first kind. p q Remark 2.2. When Γ is a congruencegroup for an imaginary quadraticfield, the automorphic forms ϕπ will usually be chosen to be common eigenfunctions of Hecke operators. Furthermore, according to [GJ], there will be no residual spectrum if Γ is congruence. Note that our definition of Kloosterman sums is slightly different from the usual one.

Taking the simplest example of the full modular group Γ PSL2 O , with O the ring of “ p q 6 ZHI QI

1 integers of an imaginary quadratic ﬁeld, then for c O r 0 and ω1, ω2 O r 0 we P t u P t u have

2 ω1a1 ω2a2 KlΓ c ; ψ1, ψ2 e Tr ` . p q“ c a1, a2PO{cO ˆ ˆ ˙˙ a1a2”ÿ1pmod cq

For Γ PSL2 Z i we obtain the summation formula in [BM3]. “ p r sq

3. Spectral Analysis of L2 Γ G p z q The spectral decomposition of L2 Γ G is a consequence of the general theory of p z q Eisenstein series due to Langlands in [Lan]. Here we shall follow the expositions in [CPS]. Let L2 Γ G be the space of all square integrable functions on Γ G with respect to the p z q z measure induced by the Haar measure on G. Let , be the (Petersson) inner product on p¨ ¨q L2 Γ G . G acts on this space by right translation. p z q Let L2 Γ G denote the space of L2-cusp forms. We have a decomposition 0p z q L2 Γ G L2 Γ G L2 Γ G , p z q“ 0p z q‘ ep z q where L2 Γ G is the orthogonal complement of L2 Γ G . e p z q 0p z q 3.1. Spectral Decomposition on L2 Γ G . It is a well known theorem of Gelfand 0p z q and Piatetski-Shapiro that L2 Γ G decomposes into a discrete countable direct sum of 0p z q irreducible unitary representations π, Vπ of G, each isomorphism classes occurring with p q ﬁnite multiplicity (see [GGPS, HC, Lan]). We let Π0 Γ be the set of irreducible con- p q stituents of L2 Γ G . 0p z q

ˆ 3.2. Eisenstein Series. For each µ X C and cusp a C, let Va µ denote the P p q P p q Hilbert space of functions f : G C such that 1 Ñ 2 - f pg µ p δ p f g , p Pa, p q“ p q a p q p q P - f is square integrable on K.

Here δa is the modulus character of Pa acting on Ua by conjugation. For the matrix p ´1 2 “ ga uaga in Pa, we have δa p a . G acts on this space by right translation. We denote p q “ | | this representation by πa µ . There is a non-degenerate G-invariant Hermitian paring on ´1 p q Va µ Va µ given by p qˆ p q

1 1 1 ´1 f, f a f g f g dg, f Va µ , f Va µ . x y “ p q p q P p q P p q żPazG

Note that if Re µ 0 then πa µ is unitary with respect to this inner product. p q“ p q For f Va, we form the Eisenstein series P

Ea g; f, µ f γg . p q“ p q γ Γ Γ Pÿaz ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 7

ˆ Note that Ea g; f, µ 0 if µ Xma C . It converges uniformly and absolutely on p q ” R1 p q compact subsets of G for Re µ . Ea g; f, µ is analytic in µ and admits a meromorphic ą 2 p q continuation to all µ X Cˆ . Furthermore, there exist intertwining operators P p q ´1 M a, b; µ : Va µ Vb µ p q p q Ñ p q such that the Eisenstein series satisfy the functional equation

´1 (3.1) Ea g; f, µ Eb g; M a, b; µ f, µ . p q“ b p p q q ÿ 3.3. The Residual Spectrum. The poles of the Ea g; f, µ in Re µ 0, which are p q ě identical with the poles of M a, b; µ , are all simple and their s-coordinates lie in a ﬁnite 1 p q subset of the segment 0, 2 . The residues of these Eisenstein series at such a pole form a 1 2 2 non-cuspidal irreducible representation occurring discretely in L Γ G . The point µ δa ` ‰ p z q “ always yields the trivial representation, which represents the constant functions, whereas all the other components in the residual spectrum are complementary series. Let Πr Γ 2 p q denote the representations π, Vπ which so occur discretely in L Γ G coming from the p q p z q residues of all Eisenstein series Ea g; f, µ . p q 2 3.4. The Continuous Spectrum. It is known that Le Γ G is the closure of the space V p z q spanned by incomplete Eisenstein series Ea g; f of the form p q Ea g; f f γg , p q“ Γ Γ p q ÿaz 8 8 with f C Ua G . For f C Ua G , Ea g; f is compactly supported on Γ G, then P c p z q P c p z q p q z we may form the inner product

1 Ea ; f , Eb ; f , µ p p¨ q p¨ qq 1 with the Eisenstein series for f Vb µ with Re µ 0. This linear functional will be P p q “ presented by some Φ Ea ; f Vb µ in the sense that πbpµqp p¨ qq P p q 1 1 1 Ea ; f , Eb ; f , µ Φ Ea ; f , f b, all f Vb µ . p p¨ q p¨ qq“x πbpµqp p¨ qq y P p q 2 Therefore Φ gives a G-intertwining projection L Γ G Vb µ . πbpµq ep z q Ñ p q 2 3.5. The Spectral Decomposition of L Γ G . For convenience, let Πd Γ Π0 Γ p z 2q p q“ p qY Πr Γ denote the complete discrete spectrum of L Γ G . p q p z q Theorem 3.1. The spectral decomposition of L2 Γ G is p z q

2 1 1 L Γ G Vπ Va µ dµ, p z q“ ‘ 4πi ma Λa Re µ“0 p q ˜πPΠd pΓq ¸ a ż à ÿ | | where the sections of the continuous spectrum must satisfy the functional equation (3.1) ˆ and the integral over dµ is on the vertical lines Xm C maZ s C : Re s 0 . a p q– ˆt P “ u

VHere we follow the terminologies and notations of [Iwa] rather than [CPS]. 8 ZHI QI

To be concrete, for a function ϕ L2 Γ G , we may write P p z q 1 1 (3.2) ϕ Fπ ϕ Fπapµq ϕ dµ. “ p q` 4πi ma Λa p q π Π Γ a Re µ“0 Pÿdp q ÿ | | ż

For π Πd Γ ,Fπ ϕ is the unique element in Vπ satisfying P p q p q 1 1 1 (3.3) Fπ ϕ , ϕ ϕ, ϕ , all ϕ Vπ. p p q q “ p q P 2 Let Φ be the G-intertwining projection from L Γ G Va µ , which extends onto πapµq e p z q Ñ p q L2 Γ G , then p z q

(3.4) F ϕ Ea g; Φ ϕ , µ . πapµqp q“ p πapµqp q q The spectral decomposition theorem in general is due to Langlands ([Lan]). For

PGL2 C , we may however prove the theorem following [EGM, Chapter 6], [GJ]or[Iwa] p q for PSL2 R . A Fourier series expansion on the circle B SU 2 1 SU 1 1 is p q X p q{t˘ u– p q{t˘ u needed at the end. Let S Γ G be the space of smooth vectors in L2 Γ G , which is endowed with a ﬁner p z q p z q topology than that induced by the L2-norm. This topology may be deﬁned by the semi- norms νX ϕ R X ϕ 2 for all X U g . Recall that g is the Lie algebra of G and U g p q“} p q } P p q8 8 p q is the universal enveloping algebra. Let V and Va µ denote the spaces of smooth vec- π p q tors in Vπ and Va µ , respectively. As consequences of the theorem of Dixmier-Malliavin p q [DM], for ϕ S Γ G , each of its spectral components is smooth and the spectral decom- P p z q position (3.2) converges in S Γ G (see [CPS, Proposition 1.3, 1.4]). p z q

3.6. The Whittaker-Spectral Decomposition. Let ψ X Cˆ r 1 and ϕ P 8p q t u P S Γ G . We deﬁne the ψ-Whittaker function associated with ϕ as p z q

´1 Wϕ, ψ g ϕ ng ψ n dn. p q“ p q p q żΓ8zN Let C N G; ψ denote the space of continuous functions f on G such that f ng p z q p q “ ψ n f g for all n N. Let S N G; ψ denote the space of smooth vectors in C N G; ψ . p q p q P p z q p z q It is clear that if ϕ S Γ G then Wϕ, ψ S N G; ψ . P p z q P p z q Using Sobolev’s lemma ([Eva, §5.6.3 Theorem 6]), we may prove the following ana- logue of [CPS, Lemma 1.1]. For all ϕ S Γ G we have P p z q 4 ´4 Wϕ, ψ a a a R X ϕ 2, | p q| Î p| | ` | | q α } p q } X“X |αÿ|ď4

6 Xα 6 αl where, on choosing a basis Xl l“1 of g, l“1 Xl U g (the order of Xl in the prod- 6 t u “ P p q uct is ﬁxed), α αl, and the implied constant depends only on Γ. Consequently, | | “ l“1 ś the linear functional ϕ Wϕ, ψ 1 is continuous on S Γ G with its natural topology. ř ÞÑ p q p z q ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 9

Theorem 3.2. Let ψ X Cˆ r 1 . Suppose ϕ S Γ G has spectral expansion as P 8p q t u P p z q in (3.2). Then 1 1 (3.5) Wϕ, ψ 1 WFπpϕq,ψ 1 WFπapµq pϕq,ψ 1 dµ, p q“ p q` 4πi ma Λa p q π Π Γ a Re µ“0 Pÿd p q ÿ | | ż ˆ with µ Xm C . P a p q 4. Kloosterman Sums, Poincare´ Series and the Kloosterman-Spectral Formula

ˆ In the following, we fix two nontrivial characters ψ1, ψ2 X C . Let κ C be such P 8p q P that ψ2 z ψ1 κz . p q“ p q 4.1. Kloosterman Sums. We first introduce Ω Γ c Cˆ : N4cN Γ Ø, c A , p q“ P X ‰ P and for c Ω Γ define Γc N 4cN Γ. Γc is both right and left( invariant under Γ . For P p q “ X 8 each γ Γc, we decompose γ according to the Bruhat decomposition, namely P γ n1 γ 4cn2 γ , “ p q p q with n1 γ , n2 γ N. p q p qP ˆ For c Ω Γ and ψ1, ψ2 X C , we define the associated Kloosterman sum as P p q P 8p q KlΓ c; ψ1, ψ2 ψ1 n1 γ ψ2 n2 γ . p q“ Γ Γ Γ p p qq p p qq 8zÿc{ 8 4.2. Poincare´ Series. Given any α, β 0, we let S α, β N G; ψ1 denote the function ą p z q space consisting of functions f S N G; ψ1 satisfying P p z q 1`α 1´β f zrk α, β min r , r | p q| Î for all z N, r A and k K. For f S α, β N G; ψ1 , we( form the Poincaréseries P P ` P P p z q P f g f γg . p q“ Γ Γ p q ÿ8z Using the (spherical) Eisenstein series in [EGM, §3.2] as majorant, it is readily verified that the series for P f g is absolutely convergent,uniformly on compact subsets. Moreover, p q applying similar arguments as in the proof of [EGM, Proposition 3.2.3], we may estimate VI P f g near the cusps of Γ and prove that P f g S Γ G . p q p qP p z q Later, we shall apply the Whittaker-spectral formula (3.5) in Theorem 3.2 to a specific

Poincar´eseries P f that will be constructed in a moment. With this in mind, we need the following results that follow from standard unfolding computations. By decomposing Γ according to the Bruhat decomposition,

1 Γ Γ Γc, “ 8 Y c Ω Γ Pďp q VI In [CPS, §2.3], the notion of rapid decreasing modulo N is introduced in terms of the norm N on N G. } } z However, their arguments in the proofs of Proposition 2.1 and 2.2 are incorrect. Nevertheless, the arguments here work in the PGL2 R context and corrects the errors in [CPS]. p q 10 ZHI QI we ﬁnd on the left hand side of (3.5) a weighted sum of Kloosterman sums.

Lemma 4.1. Let f S α, β N G; ψ1 . Then P p z q

WP ,ψ 1 δψ ,ψ Λ f 1 KlΓ c; ψ1, ψ2 K f, c; ψ1, ψ2 , f 2 p q“ 1 2 | 8| p q` p q p q c Ω Γ Pÿp q where δψ1,ψ2 is the Kronecker symbol for ψ1 and ψ2 lying in the same orbit under the action of ηm , Λ is the area of Λ C, and 8 | 8| 8z ´1 K f, c; ψ1, ψ2 f 4cn ψ n dn p q“ p q 2 p q żN is called the Kloosterman-Jacquet distribution.

For the right hand side of (3.5), we shall need two identities on the inner products with Poincar´eseries. First,

(4.1) P f , ϕ f g Wϕ, ψ g dg, ϕ S Γ G . p q“ p q 1 p q P p z q żNzG Second, for f S α, β N G; ψ1 , as alluded to above, with the arguments in the proof of P p z q [EGM, Proposition 3.2.3], one may show that the Poincaréseries P f g has sufficient 1 p q decay to take its inner product with the Eisenstein series Ea g; f , µ , and p q 1 1 (4.2) P f , Ea ; f , µ f g W 1 g dg, f Va µ . p p¨ qq “ p q Ea p¨; f , µq,ψ1 p q P p q żNzG 4.3. The Specific Choice of Poincare´ Series. Let η S C be a Schwartz function P p q on C and let ν C8 Cˆ be a smooth function of compact support on Cˆ. To η and ν we P c p q shall associate a function fη,ν S N G; ψ1 by defining P p z q

ψ1 u1 η u2 ν a , if g u14u2 a N4NA, fη,ν g p q p q p q “ P p q“ $0, if g B NA. & P “

We have fη,ν ng ψ1 n fη,ν g for n N and it may be shown that fη,ν S α,1 N G; ψ1 p q“ p q% p q P P p z q for all α 0. Note that fη,ν 1 0. ą p q“ Subsequently, for the speciﬁc choice ϕ P f , the left and the right hand side of the “ η,ν identity (3.5) will be referred to as the geometric and the spectral side, respectively.

4.4. The Geometric Side. The Kloosterman-Jacquet distribution K f, c; ψ1, ψ2 in p q Lemma 4.1 for f fη,ν can be computed very easily. “ Since c 1 u 1 cu c fη,ν 4 fη,ν 4 η cu ν c , ˜ ˜ 1¸˜ 1¸¸ “ ˜ ˜ 1 ¸˜ 1¸¸ “ p q p q by deﬁnition, we have 1 u K f , c ψ , ψ η cu ν c ψ´1 u du ν c η u ψ´1 du. η,ν ; 1 2 2 2 2 p q“ C p q p q p q “ c p q C p q c ż | | ż ´ ¯ ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 11

Lemma 4.2. We have 1 1 K fη,ν, c; ψ1, ψ2 ν c η , p q“ c 2 p q c | | ˆ ˙ where η is the Fourier transform with respect to ψ2, p

η z η u ψ´1 uz du. p p q“ p q 2 p q żC 4.5. The Spectral Side. Inp the following, we shall compute the spectral side and show that it can be expressed in terms of Bessel transforms of the weight function. 4.5.1. Bessel functions. Let ψ be a nontrivial additive character of C. For an inﬁnite dimensional unitary representation π of G, let W π, ψ be the Whittaker model of π, that p q is, the image of a nonzero embedding of V8 into the space IndG ψ of C8 functions π N p q W : G C satisfying W ng ψ n W g for all n N, with G acting on IndG ψ Ñ p q “ p q p q P N p q by right translation. Actually, it is known from Shalika’s multiplicity one theorem [Sha] that such a nonzero embedding exists and is unique up to constant. Let K π, ψ be the p q associated Kirillov model of π on L2 Cˆ, dˆz , whose smooth vectors are all of the form p q a W for W W π, ψ . ˜ 1¸ P p q It is shown in [Qi, (18.4)] that there exists a (complex-valued) real analytic function ˆ Jπ, ψ on C which acts as an integral kernel of the action of the Weyl element 4 on the Kirillov model K π, ψ . Namely, for W W π, ψ , p q P p q

a b ˆ (4.3) W 4 Jπ, ψ ab W d b. 1 “ ˆ p q 1 ˜˜ ¸ ¸ żC ˜ ¸

A simple fact is that the Bessel function is conjugation invariant, namely, Jπ, ψ z p q “ J ´1 z Jπ, ψ z (see [Qi, (18.2)]). π, ψ p q“ p q 4.5.2. The constants cψ π, Γ and cψ πa µ , Γ . Let ψ be a nontrivial character of C. p q p p q q2 First, let π, Vπ be a discrete component of L Γ G . The inner product on Vπ is the p q p z q Petersson inner product inherited from L2 Γ G . For smooth vector ϕ V8 S Γ G , p z q P π Ă p z q the map

´1 ϕ Wϕ g ϕ ng ψ n dn ÞÑ p q“ p q p q żΓ8zN deﬁnes a Whittaker model W π, ψ of π, V8 . Let K π, ψ be the associated Kirillov p q p π q p q model. It comes equipped with its canonical L2 inner product on Cˆ. Hence there is a 1 positive constant, which we shall denote c π, Γ cψ π, Γ , such that for all ϕ, ϕ Vπ p q“ p q P a a ˆ 1 (4.4) Wϕ Wϕ1 d a c π, Γ ϕ g ϕ g dg. ˆ 1 1 “ p q p q p q żC ˜ ¸ ˜ ¸ żΓzG

Second, we consider the representation πa µ , Va µ . For f Va µ , the function p p q p qq P p q ´1 W g Ea ng; f, µ ψ n dn Eap¨; f, µqp q“ p q p q żΓ8zN 12 ZHI QI deﬁnes a Whittaker model W πa µ , ψ of πa µ . Recall that there is also a canonical inner p p q q p q product on πa µ , Va µ given by p p q p qq 1 1 f, f a f g f g dg. x y “ p q p q żPazG 1 8 Hence there exists a constant c πa µ , Γ cψ πa µ , Γ such that for all f, f Va µ p p q q “ p p q q P p q we have

a a ˆ 1 1 (4.5) WEa p¨; f, µq WEa p¨; f , µq d a c πa µ , Γ f g f g dg. ˆ 1 1 “ p p q q p q p q żC ˜ ¸ ˜ ¸ żPazG

4.5.3. The constants c π; ψ2 ψ1 and c πa µ ; ψ2 ψ1 . In what follows we shall have p { q p p q { q to compare Whittaker models associated with ψ1 and ψ2.

Let π Πd Γ . Because ψ2 z ψ1 κz , if Wϕ, ψ W π, ψ1 , then the function P p q p q“ p q 1 P p q κ W1 g W g ϕ, ψ2 ϕ, ψ1 p q“ ˜˜ 1¸ ¸

1 1 κ satisﬁes Wϕ, ψ ng ψ2 n Wϕ, ψ g . Since this left multiplication by commutes 2 p q“ p q 2 p q ˜ 1¸ 1 with the action of G, the collection of function W g form another ψ2-Whittaker model. ϕ, ψ2 p q By Shalika’s multiplicity one theorem, there is a constant c π; ψ2 ψ1 such that for all p { q ϕ V8 we have P π κ (4.6) Wϕ, ψ2 g c π; ψ2 ψ1 Wϕ, ψ1 g . p q“ p { q ˜˜ 1¸ ¸ 8 Similarly, there exists a constant c πa µ ; ψ2 ψ1 such that for all f Va µ we have p p q { q P p q κ (4.7) WEap¨; f, µq,ψ2 g c πa µ ; ψ2 ψ1 WEa p¨; f, µq,ψ1 g . p q“ p p q { q ˜˜ 1¸ ¸ 4.5.4. Computing the spectral side.

Lemma 4.3. Let f fη,ν. Suppose that ψ2 z ψ1 κz . “ 2 p q“ p q For π, Vπ occurring discretely in L Γ G , we have p q p z q 1 κ J ˆ (4.8) WFπpP f q,ψ2 1 cΓ π; ψ1, ψ2 ν z η π, ψ1 d z, p q“ p q ˆ p q z z żC ˆ ˙ ˆ ˙ with p c π; ψ2 ψ1 cψ1 π, Γ cΓ π; ψ1, ψ2 p { q p q. p q“ 4π2 2 For πa µ , Va µ occurring in the continuous spectrum of L Γ G , we have p p q p qq p z q 1 κ J ˆ (4.9) WFπa pµqpP f q,ψ2 1 cΓ πa µ ; ψ1, ψ2 ν z η πapµq,ψ1 d z, p q“ p p q q ˆ p q z z żC ˆ ˙ ˆ ˙ with p c πa µ ; ψ2 ψ1 cψ1 πa µ , Γ cΓ πa µ ; ψ1, ψ2 p p q { q p p q q . p p q q“ 4π2 ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 13

Proof. First, in view of (4.6), we have κ WFπ pP f q,ψ2 1 c π; ψ2 ψ1 WFπ pP f q,ψ1 . p q“ p { q ˜ 1¸ κ So it is enough to compute WFπpP f q,ψ1 . Let us now drop ψ1 from notations. By ˜ 1¸ a (3.3), (4.1) and (4.4), we see that WFπpP f q is characterized by ˜ 1¸

a a ˆ 8 WFπ pP f q Wϕ d a c π, Γ f g Wϕ g dg all ϕ Vπ . ˆ 1 1 “ p q p q p q P żC ˜ ¸ ˜ ¸ żNzG Now substitute the deﬁnition of f g fη,ν g on the right and integrate over the open p q “ p q Bruhat cell. We get 1 1 u z f g W g dg ν z η u W 4 dˆzdu. ϕ 2 ϕ p q p q “ 4π ˆ p q p q 1 1 żNzG żC żC ˜ ˜ ¸˜ ¸¸ We now use the identity (4.3) for the Bessel function. 1 u z Wϕ 4 W z u 4 π ϕ ˜ ˜ 1¸˜ 1¸¸ “ 1 p q ˆ ˙

a ˆ Jπ a W z u d a ˆ π ϕ “ C p q 1 ˜ 1¸ ż ˆ ˙

az ˆ ψ1 au Jπ a Wϕ d a “ ˆ p q p q 1 żC ˜ ¸

1 au a a ˆ ψ2 Jπ Wϕ d a. “ ˆ κ z z 1 żC ˆ ˙ ˆ ˙ ˜ ¸ Therefore,

f g Wϕ g dg p q p q “ żNzG 1 au a a ν z η u ψ´1 J W dˆadˆzdu. 2 2 π ϕ 4π ˆ ˆ p q p q κz z 1 żC żC żC ˆ ˙ ˆ ˙ ˜ ¸ Interchanging the order of integrations, this becomes 1 a a a J ˆ ˆ 2 ν z η π d z Wϕ d a. ˆ 4π ˆ p q κz z 1 żC " żC ˆ ˙ ˆ ˙ * ˜ ¸ Note that η is the Fourier transformp of η with respect to ψ2. This yields the identity a c π, Γ a a W ν z η J dˆz, p FπpP f q p 2 q π 1 “ 4π ˆ p q κz z ˜ ¸ żC ˆ ˙ ˆ ˙ and then follows the formula (4.8). p Proceedingexactlyasabove,wemayderive(4.9). Q.E.D. 14 ZHI QI

4.6. The Kloosterman-Spectral Formula. In conclusion, as a consequence of The- orem 3.2 applied to the Poincar´eseries P f with f fη,ν as in §4.3, we have the following “ Kloosterman-spectral formula.

8 ˆ ˆ Theorem 4.4. Let η S C and ν C C . SetAη,ν z ν 1 z η z . Let κ C P p q P c p q p q “ p { q p q P be such that ψ2 z ψ1 κz . Then p q“ p q p KlΓ c; ψ1, ψ2 1 p q A c 2 η,ν c c Ω Γ Pÿp q | | ˆ ˙ ˆ cΓ π; ψ1, ψ2 Aη,ν z Jπ, ψ1 κz d z “ p q ˆ p q p q π Π Γ C Pÿdp q ż 1 1 J ˆ cΓ πa µ ; ψ1, ψ2 Aη,ν z πapµq,ψ1 κz d z dµ, Λ ˆ ` a 4πi ma a Re µ“0 p p q q C p q p q ÿ | | ż "ż * ˆ with µ Xm C . P a p q 5. An Explicit Kloosterman-Spectral Formula - the Kuznetsov Trace Formula

In this section, we shall express the Bessel functions in terms of the classical Bessel functions and relate the constants cΓ π; ψ1, ψ2 to classical quantities. p q

5.1. By [Qi, Proposition 18.5], the Bessel functions Jπ, ψ may be expressed explic- itly in terms of the classical Bessel functions. For π πd s , with either s it purely 1 – p q ˆ “ imaginary or s t on the segment 0, , and ψ ψa, with a C , we have “ 2 “ P 2 2π `2 ˘ Jπ, ψ z a z Js,2d 4πa ?z J s, 2d 4πa ?z , p q“ sin 2πs | | p p q´ ´ ´ p qq p q where Js,2d z J 2s d z J 2s d z . Here ?z is the principal branch of the square root p q“ ´ ´ p q ´ ` p q of z and the expression on the right is independent on the argumentof z modulo 2π.

2 5.2. We ﬁrst consider the discrete component π, Vπ L Γ G with π πd s , p q Ă p z q – p q s it or s t. The restriction of Vπ on K decomposes into the direct sum of the 2l 1 - “ “ p ` q dimensional representations σl of K with l d . Furthermore, each representation σl ě | | decomposes into the direct sum of the one-dimensional weight spaces of weight q with 8 q l. Accordingly, we say that a vector of Vπ is of K-type l, q if it lies in the q-weight | |ď VII 8 p q space of σl. Therefore, V contains a unique vector ϕ of type d , d normalized such π p| | q that ϕπ has Petersson square norm 1. It is known that ϕπ g will have a Fourier expansion s,d p q at inﬁnity in terms of Jacquet’s Whittaker function Wω (see, for example, [JL]),

1 z r s,d ϕπ k aω ϕπ ψω z W r, k , 1 1 “ p q p q ω p q ˜˜ ¸˜ ¸ ¸ ω Λ1 ÿP 8

VII 1 i By convention, the weights q are the eigenvalues for H 2 i as an element in the “ ´ b ˜ i¸ ´ complexified Lie algebra C R su 2 . Note that the definition of K-type coincides with that in [BM3] if q is b p q changed into q. ´ ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 15 with z C, r R , k K. Now, to define W s,d, we first introduce the function P P ` P ω 1 z r r2s`1v2d, if d 0, φs,d kv, w ě ˜˜ 1¸˜ 1¸ ¸ “ $r2s`1v ´2d, if d 0. & ă 1 2 2s`1 d It is readily verified that φs,d is left µs,dδ -variant,% that is, φs,d uag a a φs,d g , 8 p q “ | | r s p q with u N and a A. For Re s 0, W s,d r, k is given by P P ą ω p q s,d ´1 W r, k ψ n φs,d 4nrk dn, ω p q“ ω p q p q żN in which the integral converges absolutely, and W s,d r, k is defined for all s via meromor- ω p q phic continuation, except for the poles that occur in the case ω 0. “ For computing the constant cΓ π; ψ1, ψ2 , we shall be concernedonlywith those nonzero s,d p q1 iφ ω and the values of W r, kv,0 , with v e 2 . With these in mind, we have the following ω p q “ lemma.

1 iφ iφ Lemma 5.1. Let φ R 2πZ and r R . Setv e 2 and a re . Then, for P { P ` “ “ ω Λ1 r 0 , we have P 8 t u d 2s`|d|`1 s,d 2 1 2π 2s`|d| |d|`1 2 d (5.1) W r, kv,0 p´ q p q ω a ω a K 4π ωa . ω p q“ Γ 2s d 1 | | | | r s |d|´2sp | |q p ` | |` q Proof. Onexpressing 4zrkv,0 in the Iwasawa coordinates, changing the variables from z to rz and using the polar coordinates for z, we arrive at 8 2|d|`1 d s,d 1´2s idφ x Iω rx W r, kv, 2r e p q dx, ω 0 2 2s`|d|`1 p q“ 0 1 x ż p ` q with 2π iθ Id x e2πiTrpωxe q´2idθdθ. ωp q“ ż0 We first recall the integral representation of Bessel, 2π m ix cos θìmθ 2πi Jm x e dθ, m Z, p q“ P ż0 then we find that Id x 2π 1 d ω 2d J 4π ω x . ωp q“ p´ q r s 2|d|p | | q Consequently, 8 x2|d|`1J π ω rx s,d d 2d 1´2s idφ 2|d| 4 W r, kv, 4π 1 ω r e p | | qdx. ω 0 2 2s`|d|`1 p q“ p´ q r s 0 1 x ż p ` q Using the formula [GR, 6.565 4] 8 ν`1 µ x Jν ax a 3 p q dx Kν µ a , 2Re µ Re ν 1, a 0, 1 x2 µ`1 “ 2µΓ µ 1 ´ p q ` 2 ą ą´ ą ż0 p ` q p ` q we obtain d 2s`|d| 2d |d|`1 idφ s,d 4π 1 2π ω ω r e W r, kv,0 p´ q p | |q r s K 4π ω r . ω p q“ Γ 2s d 1 |d|´2sp | | q p ` | |` q 16 ZHI QI

Therefore (5.1) is proven for Re s 0 and remains valid by analytic continuation. ą Q.E.D.

It will be convenient to work with renormalized Fourier coeﬃcients

2s 2s d Aω ϕπ 2π ω ω aω ϕπ . p q “ p q | | r s p q Then if ψi z ψω z , i 1, 2, we see that p q“ i p q “ d |d|`1 a 2 Λ8 1 2π |d| d Wϕ ,ψ | |p´ q p q Aω ϕπ a ωia ωia K 4π ωia . π i 1 “ Γ 2s d 1 i p q| || | r s |d|´2sp | |q ˜ ¸ p ` | |` q Let us ﬁrst compute c π; ψ2 ψ1 , which is deﬁned by p { q ω1 ω2 Wϕπ,ψ2 c π; ψ2 ψ1 Wϕπ,ψ1 . ˜ 1¸ “ p { q ˜ 1¸ It follows that

Aω2 ϕπ ω1 c π; ψ2 ψ1 p q| |. p { q“ Aω ϕπ ω2 1 p q| | On the other hand, as ϕπ has Petersson square norm 1, cψ π, Γ is deﬁned by 1 p q

a a ˆ cψ1 π, Γ Wϕπ,ψ1 Wϕπ,ψ1 d a. p q“ ˆ 1 1 żC ˜ ¸ ˜ ¸ We ﬁrst let s it. Integrating in the polar coordinates, we obtain “ 2 2|d|`3 8 Λ8 2π 2 2|d| cψ π, Γ | | p q Aω ϕ ω1 1 p q“ Γ 2it d 1 2 | 1 p q| | | | p ` | |` q| 8 2|d|`1 r K 4π ω1 r K 4π ω1 r dr. |d|´2itp | | q |d|`2itp | | q ż0 As a special case of the Weber-Schafheitlin formula for K-Bessel functions (see [GR, 6.576 4]), we have 8 ρ´1 x Kν ax Kµ ax dx p q p q ż0 2 ρ´3Γ 1 ρ ν µ Γ 1 ρ ν µ Γ 1 ρ ν µ Γ 1 ρ ν µ 2 p ` ` q 2 p ` ´ q 2 p ´ ` q 2 p ´ ´ q , “ a ρΓ ρ ` ˘ ` p˘ q ` ˘ ` ˘ Re ρ Re ν Re µ , a 0. ą | | ` | | ą This yields 2π Λ 2 A ϕ 2 Γ 8 ω1 cψ1 π, | | | p q|2 . p q“ 2 d 1 ω1 p | |` q| | Therefore, we get 2 Λ8 Aω2 ϕπ Aω1 ϕπ cΓ π; ψ1, ψ2 | | p q p q . p q“ 2π 2 d 1 ω1ω2 p | |` q| | Similarly, when s t and d 0, with t 0, 1 , we have “ “ P 2 2 2 2`π Λ˘ Aω ϕ c π, Γ 8 1 , ψ1 | | | 2p q| p q“ Gt,0 ω1 | | ON THE KUZNETSOV TRACE FORMULA FOR PGL2pCq 17 with Γ 1 2t Gt,0 p ` q. “ Γ 1 2t p ´ q Hence

Aω2 ϕπ Aω1 ϕπ cΓ π; ψ1, ψ2 p q p q. p q“ 2πGt,0 ω1ω2 | | Combining these, if we simply put Git,d 1, then “ 2 Λ8 Aω2 ϕπ Aω1 ϕπ cΓ π; ψ1, ψ2 | | p q p q . p q“ 2π 2 d 1 Gs,d ω1ω2 p | |` q | | d 2s 5.3. For the continuous spectrum, we ﬁx a cusp a and let µ z µs,d z z z , 8 p q “ p q“r s | | with s it and ma d. Choose f Va µ to be “ | P p q

f g φs,d gag . p q“ p q

We have f, f a 1 2 d 1 . With f is associated the Eisenstein series Ea g; f, µ . To x y “ {p | |` q p q describe the Fourier coeﬃcients of this Eisenstein series we need some notations. ˆ For each cusp a let Ωa Γ c C : gaΓ N4cN Ø, c A , and for each c a p q “ t P a X ‰ P ´1u P Ωa Γ deﬁne Γ gaΓ N4cN. Then Γ is left invariant under gaΓaga and right invariant p q c “ X c under Γ . For each γ Γa we write 8 P c

γ n1 γ 4cn2 γ , “ p q p q with n1 γ , n2 γ N. p q p qP a For c Ωa Γ we define the Ramanujan sum Kl c;1, ψ by P p q Γp q a KlΓ c;1, ψ ψ n2 γ , p q“ ´1 a p p qq γPgaΓagÿa zΓc {Γ8 and for ω Λ1 we define the Kloosterman-Selberg zeta series P 8 a a KlΓ c;1, ψω ZΓ µ;0, ω p 1q, p q“ 2 c Ω Γ µ c δ8 c Pÿap q p q p q if Re µ is sufficiently large. Recall that δ c c 2 is the modulus character. These series 8p q “ | | Za µ;0, ω have meromorphic continuation to the whole complex plane and have no poles Γp q on the line Re µ 0. “ We may now describe the Fourier expansion of the Eisenstein series Ea g; f, µ . For p q 1 z r g k, we have “ ˜ 1¸˜ 1¸

1 a s,d Ea g; f, µ δa m φs,d g Z µ;0, 0 W r, k p q“ 8 8 p q` Λ Γp q 0 p q | 8| 1 a s,d ψω z Z µ;0, ω W r, k , ` Λ p q Γp q ω p q 8 ω Λ1 r 0 | | P ÿ8 t u where the ﬁrst term exists only if a . “8 18 ZHI QI

1 Let ψ1 ψω and ψ2 ψω , with ω1, ω2 Λ r 0 . From Lemma 5.1, for i 1, 2, “ 1 “ 2 P 8 t u “ we have a 2 1 d 2π 2s`|d|`1 W , µ ,ψ p´ q p q Eap¨; f q i 1 “ Γ 2s d 1 ˜ ¸ p ` | |` q a 2s`|d| |d|`1 2 d Z µ;0, ωi ωi a ω a K 4π ωia . Γp q| | | | r i s |d|´2sp | |q Then we get a 2s´1 d Z µ;0, ω2 ω2 ω2 c π µ ; ψ ψ Γp q| | r s a 2 1 a 2s´1 d p p q { q“ Z µ;0, ω1 ω1 ω1 Γp q| | r s and a 2 2π Z µ;0, ω1 c π µ , Γ | Γp q| . ψ1 a 2 p p q q“ ω1 | | Therefore, a a ZΓ µ;0, ω2 ZΓ µ;0, ω1 ω2 cΓ πa µ ; ψ1, ψ2 p q p qµ . p p q q“ 2π ω1ω2 ω1 | | ˆ ˙ 5.4. In conclusion, with the computations above, the Kuznetsov trace formula in Theorem 2.1 then follows from Theorem 4.4, with the choice of weight function F z p q “ Aη,ν z ω1ω2 z ω1ω2 . It should be remarked that F z of this form actually exhaust all p { q | { | p q functions in C8 Cˆ if one let η and ν vary. c p q

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Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road,Piscataway, NJ 08854-8019, USA E-mail address: [email protected]