On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields
aa r X i v : . [ m a t h . N T ] N ov On the Fourier transform of regularized Bessel functions on complexnumbers and Beyond Endoscopy over number fields
Zhi Qi A bstract . In this article, we prove certain Weber-Schafheitlin type integral formulae forBessel functions over complex numbers. A special case is a formula for the Fourier trans-form of regularized Bessel functions on complex numbers. This is applied to extend thework of A. Venkatesh on Beyond Endoscopy for Sym on GL from totally real to arbitrarynumber fields.
1. Introduction
A special case of the discontinuous integrals of Weber and Schafheitlin on the Fouriertransform of Bessel functions on R ` “ p , is as follows (see [ Wat , 13.42 (2), (3)]),(1.1) ż J ν p π x q e p˘ xy q x dx “ $’’’’&’’’’% ν ν ` a ´ y ¯ iy ˘ ν , if 0 ď y ď , ν e ˘ π i ν ν ` a y ´ ` y ˘ ν , if y ą , valid for Re ν ą
0, where J ν p z q is the Bessel function of the first kind of order ν and asusual e p z q “ e π iz ; it is assumed here that arg `a ´ y ¯ iy ˘ P ¯ “ , π ‰ . Subsequently,we shall call a formula of this kind special Weber-Schafheitlin integral formula .In Venkatesh’s work on Beyond Endoscopy [
Ven2 ], the second formula in (1.1) arisesin his local computation over R , particularly, in his analysis for the B -transform. Theresults of Venkatesh were proven for totally real number fields, but he pointed out thatthe extension to complex places would only require verifying a similar formula for Besselfunctions over C . This amounts to a “local fundamental lemma” over C . Unfortunately, itseems to resist a proof in every direct way—Venkatesh was not able to prove it at that time.The purpose of this paper is to establish an integral formula for Bessel functions overcomplex numbers which is analogous to the special Weber-Schafheitlin formula as in (1.1)(after regularization). Our approach is a rather indirect method that combines asymptotic Mathematics Subject Classification.
Key words and phrases.
Fourier transform, Bessel functions, hypergeometric function, Beyond Endoscopy. The first line on the right hand side of (1.1) should read p { ν q exp p˘ i ν arcsin p y { qq by bookkeeping [ Wat ,13.42 (2), (3)], but we feel that the formulation here in (1.1) is more suggestive. analysis and di ff erential equations. As an application, the validity of Venkatesh’s work onBeyond Endoscopy is extended from totally real to arbitrary number fields. Bessel kernels and the special Weber-Schafheitlin formula.
First of all, some remarkson the special Weber-Schafheitlin formula (1.1) are in order, as the motivation of our in-vestigation of its complex analogue.The Bessel function of concern has pure imaginary order ν “ ˘ it ( t real). Indeed,the Bessel kernel in the B -transform is B it p π x q (see [ Ven2 , § B ν p x q “ p πν { q p J ´ ν p x q ´ J ν p x qq , x P R ` , (1.2)and the B -transform is defined by φ p x q “ ż B it p π x q h p t q sinh p π t q tdt . (1.3)The formula (1.1) however is only valid for Re ν ą
0. The reason is that J ν p π x q „p π x q ν { Γ p ν ` q as x Ñ Wat , 3.1 (8)]), say for Re ν ě
0, and hence the integralin (1.1) is convergent near zero only when Re ν ą
0. It should be noted that the integralis always absolutely convergent in the vicinity of infinity since J ν p π x q “ O p { ? x q as x Ñ 8 (see [
Wat , 7.21 (1)]). Nevertheless, this convergence issue may be easily addressedby shifting t to t ´ i σ or t ` i σ ( σ ą
0) for J it p π x q or J ´ it p π x q respectively.The Bessel kernel in the K -transform is p { π q cos p π t q K it p π x q (see [ Ven2 , § K ν p x q “ π p πν q p I ´ ν p x q ´ I ν p x qq , x P R ` , (1.4)where I ν p z q and K ν p z q are the modified Bessel functions . The K -transform is defined by φ p x q “ π ż K it p π x q h p t q sinh p π t q tdt . (1.5)The integral in (1.1) would diverge if the J ν p π x q were replaced by I ν p π x q sinceit is of exponential growth ( I ν p x q „ exp p x q{ ? π x , [ Wat , 7.23 (2), (3)]) as x Ñ 8 .Although I ν p x q and I ´ ν p x q conspire in (1.4) so that K ν p x q decays exponentially ( K ν p x q „ exp p´ x q{ a x { π , [ Wat , 7.23 (1)]) as x Ñ 8 , the integral in (1.1) would still diverge if the J ν p π x q were replaced by K ν p π x q . This is because I ν p π x q „ p π x q ν { Γ p ν ` q as x Ñ Wat , 3.7 (2)]) and it would require both Re ν ą p´ ν q ą not exist for the Bessel kernel of the K -transform. Moreover,in the case ν “ it , the order-shifting trick as above applied to I it p π x q and I ´ it p π x q would not work: after the order-shifting their exponential growth could not be canceledcompletely and the resulting di ff erence I ´ it ` σ p π x q ´ I it ` σ p π x q ( σ ą
0) still growsexponentially (compare their asymptotic expansions as in [
Wat , 7.23 (2), (3)]). As such, Note that the sin p πν q is mistakenly written as sin p πν { q in [ Ven2 ]. ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 3 the analysis for the K -transform in [ Ven2 , § ff erent from that for the B -transform in [ Ven2 , § K it p x q .Similar obstacles described above for K it p π x q would arise when analyzing the Fouriertransform of Bessel functions on C . Thus before we initiate our study over C it is necessaryto overcome these obstacles over R with new ideas. Regularizing the special Weber-Schafheitlin formula.
Next, we introduce the regular-ized Bessel kernels and their special Weber-Schafheitlin formulae and briefly explain howthey are applied to unify the analysis for the B -transform and that for the K -transform. Themerit of unified analyses for the B - and the K -transform and also for their Bessel kernelsis that their complex extensions are usually admissible.We first define P ν p x q “ p x { q ν Γ p ν ` q . (1.6)Note that P ν p x q is simply the leading term in the series expansion of J ν p x q or I ν p x q at x “ Wat , 3.1 (8), 3.7 (2)]). By [ GR , 3.764], we have ż x ν ´ e p˘ xy q dx “ Γ p ν q e ˘ π i ν p π y q ´ ν , ă Re ν ă . (1.7)In the following, let Re ν “ ν ‰
0. The results may be extendedto ν “ ν with | Re ν | ă R ν p x q “ p πν { q p P ´ ν p x q ´ P ν p x qq , (1.8)and the regularized Bessel kernels D ν p x q “ B ν p x q ´ R ν p x q , M ν p x q “ p { π q cos p πν { q K ν p x q ´ R ν p x q . (1.9)Consider the following two integrals ż D ν p π x q e p˘ xy q x dx , ż M ν p π x q e p˘ xy q x dx . (1.10)We have J ν p x q´ P ν p x q , I ν p x q´ P ν p x q “ O p x q as x Ñ
0. Thus these integrals are absolutelyconvergent at zero and become so after integration by parts in the vicinity of infinity (theregion of convergence may actually be extended to | Re ν | ă ν in a compact set).For the first integral in (1.10), we apply (1.1) and (1.7) to the Fourier transform of p J ´ ν ` σ p π x q ´ P ´ ν ` σ p π x qq { x and p J ν ` σ p π x q ´ P ν ` σ p π x qq { x and then let σ Ñ σ may be easily verified. In this way, we obtain for y ą ż D ν p π x q e p˘ xy q x dx “ ´ ν sin p πν { q e ¯ π i ν ˜ ` y ` a y ´ ˘ ν ν ´ y ν ¸ ` e ˘ π i ν ˜ ν ` y ` a y ´ ˘ ν ´ y ν ¸ + . (1.11)The formula in the case 0 ď y ď ZHI QI
For the second integral in (1.10), as indicated before, there is no special Weber-Schafheitlin formula for either I ν p π x q or K ν p π x q . We propose an alternative approach bymodifying the integrals by the factor x ρ with 0 ă Re ρ ă
1. By [ GR , 6.699 3, 4] and thetransformation formula for hypergeometric functions with respect to z Ñ { z (see [ MOS , § ρ ą ż K ν p π x q e p˘ xy q x ρ ´ dx “ π p π q ρ sin p πν q " Γ p ρ ´ ν q e ˘ π i p ρ ´ ν q Γ p ´ ν q y ρ ´ ν F ˆ ρ ´ ν , ` ρ ´ ν ´ ν ; ´ y ˙ ´ Γ p ρ ` ν q e ˘ π i p ρ ` ν q Γ p ` ν q y ρ ` ν F ˆ ρ ` ν , ` ρ ` ν ` ν ; ´ y ˙ * . (1.12)A formula of this kind will be called general Weber-Schafheitlin integral formula . Wenow apply (1.7) and (1.12) to the Fourier transform of p { π q cos p πν { q K ν p π x q x ρ ´ ´ ` P ρ ´ ν p π x q ´ P ρ ` ν p π x q ˘ { sin p πν { q and then let ρ Ñ
0. The resulting hypergeometricfunctions may be evaluated by the formula (see [
MOS , § ` ` a ´ z ˘ ´ a “ ´ a F ` a , a ` {
2; 2 a ` z ˘ . (1.13)Finally we obtain ż M ν p π x q e p˘ xy q x dx “ ´ ν sin p πν { q e ¯ π i ν ˜ ` y ` a y ` ˘ ν ν ´ y ν ¸ ` e ˘ π i ν ˜ ν ` y ` a y ` ˘ ν ´ y ν ¸ + , (1.14)which is very similar to (1.11). This somewhat indirect approach to (1.14) would alsolead us to (1.11); we only record here the general Weber-Schafheitlin integral formula for J ν p π x q as below in the case y ą GR , 6.699 1, 2]), ż J ν p π x q e p˘ xy q x ρ ´ dx “ Γ p ρ ` ν q e ˘ π i p ρ ` ν q p π q ρ Γ p ` ν q y ρ ` ν F ˆ ρ ` ν , ` ρ ` ν ` ν ; 4 y ˙ , (1.15)valid for 0 ă Re ρ ă .An observation is that the paired functions in (1.11) and (1.14), say ` y ` a y ´ ˘ ν { ν and y ν , are asymptotically equivalent as y Ñ 8 . In other words, the Fourier transform of B ν p π x q{ x or 4 cos p πν { q K ν p π x q{ π x and that of R ν p π x q{ x , in the formal sense, have thesame asymptotic at infinity. Note that this is also true if they are modified by the factor x ρ .A similar observation, Lemma 4.2, is crucial to our analysis over complex numbers. See § ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 5
For the analysis for the B -transform in [ Ven2 , § ż cos p π kx q φ p x q dxx “ ż cos p π kx q ż D it p π x q h p t q sinh p π t q tdt dxx ` i ż cos p π kx q ż P it p π x q h p t q tdt dxx , (1.16)instead of ż cos p π kx q φ p x q dxx “ i ż cos p π kx q ż J it p π x q h p t q tdt dxx . (1.17)Second, we apply (1.11) directly to the first integral on the right of (1.16), and shift theorder of P it p π x q in (1.16) instead of that of J it p π x q in (1.17). Note that the B -transformturns into the inverse Mellin transform if J it p π x q is substituted by P it p π x q . It is clearthat the analysis for the K -transform may also be done in this way. Since these will beelaborated in the complex setting in §
6, we do not discuss here any further details.The ideas outlined above will be executed in §§ § We now introduce the definition of Bessel functions over com-plex numbers (see [
Qi1 , § BM5 , (6.21), (7.21)]). Let µ be a complex number and d be an integer. We define(1.18) J µ, d p z q “ J µ ` d p z q J µ ´ d p s z q . The function J µ, d p z q is well defined and even on C r t u in the sense that the expression onthe right of (1.18) is independent on the choice of the argument of z modulo π . Next, wedefine(1.19) J µ, d p z q “ p πµ q p J ´ µ, ´ d p π z q ´ J µ, d p π z qq . It is understood that in the non-generic case when µ is an integer the right hand side shouldbe replaced by its limit. It is clear that J µ, d p z q is also an even function on C r t u .According to [ Qi1 , § p C q . We shall not restrict ourselves to the Bessel functions of trivial SU -type ( d “
0) arising in the Kuznetsov-Bruggeman-Miatello formula (see [
Ven2 , § BM4 ]) . For Bessel functions for GL p C q with non-trivial central characters, our resultsand method would still be valid but the formulae would be more involved. For these werefer the reader to Appendix A. General Weber-Schafheitlin formula.
First, we have the general Weber-Schafheitlinintegral formula for J µ, d p z q as follows.T heorem Suppose that | Re µ | ă Re ρ ă . Define (1.20) C ρµ, d “ ´ sin p π p ρ ` µ qqp π q ρ sin p πµ q Γ p ρ ` µ ` d q Γ p ρ ` µ ´ d q Γ p ` µ ` d q Γ p ` µ ´ d q , For succinctness, we shall suppress d from our notation if d “
0, so in particular J µ p z q “ J µ, p z q . ZHI QI andF p q ρ,ν p z q “ F ˆ ρ ` ν ` ρ ` ν ` ν ; z ˙ , F p q ρ,ν p z q “ F ˆ ρ ´ ν ` ρ ´ ν ´ ν ; z ˙ . (1.21) We have the identity ż π ż J µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ “ C ρµ, d F p q ρ,µ ` d ` { y e i θ ˘ F p q ρ,µ ´ d ` e i θ { y ˘ y ρ ` µ e id θ ` C ρ ´ µ, ´ d F p q ρ,µ ` d ` { y e i θ ˘ F p q ρ,µ ´ d ` e i θ { y ˘ y ρ ´ µ e ´ id θ (1.22) for y P r , and θ P r , π q ; for d “ , the right hand side of the identity is to bereplaced by its limit if µ “ . Moreover, the identity (1.22) is valid under the weakercondition | Re µ | ă Re ρ ă if we further assume that y ą . R emark Since F p q ρ,ν p { z q and F p q ρ,ν p { z q are defined via analytic continuationwithin the unit circle | z | ă , the formula (1.22) is not quite illuminating for y ă itis not so clear a priori that the right hand side of (1.22) is a well-defined function on thecomplex plane. An alternative expression of (1.22) obtained from Proposition wouldbe more transparent in terms of the Gauss hypergeometric series for y ă . By the Gaussian formula for F p a , b ; c ; 1 q (see [ MOS , § F p a , b ; c ; 1 q “ Γ p c q Γ p c ´ a ´ b q Γ p c ´ a q Γ p c ´ b q , Re p a ` b ´ c q ă , c ‰ , ´ , ´ , ..., together with the duplication and the reflection formula for the gamma function, it isstraightforward to derive the following corollary.C orollary For | Re µ | ă Re ρ ă , we have ż π ż J µ, d ` xe i φ ˘ e p´ x cos φ q x ρ ´ dxd φ “ ` { π ˘ p π q ´ ρ cos p πρ q Γ p { ´ ρ q ¨ sin p π p ρ ` µ qq sin p π p ρ ´ µ qq Γ p ρ ` µ ` d q Γ p ρ ` µ ´ d q Γ p ρ ´ µ ` d q Γ p ρ ´ µ ´ d q . We shall prove Theorem 1.1 by exploiting a soft method that combines asymptoticanalysis of oscillatory integrals and a uniqueness result for di ff erential equations. Precisely,it will be shown that the two sides of (1.22) satisfy the same asymptotic as y Ñ 8 and alsothe same (hypergeometric) di ff erential equations and hence are forced to be equal.This method of proof grew out of the author’s previous work [ Qi3 ] on a similar-looking integral which, in the notation of this paper, may be written as follows ż π ż J µ, d ` x e i φ ˘ e p´ xy cos p φ ` θ qq dxd φ. (1.23)This Fourier-transform integral is however entirely di ff erent in nature. Roughly speaking,it has an explicit formula in terms of the Bessel function of halved order J µ, d ` { ye i θ ˘ (see [ Qi3 ] for the formula (in slightly di ff erent notation)). That formula was used to extendthe Waldspurger formula of Baruch and Mao from totally real to arbitrary number fields;see [ BM1, BM2, BM3 ] and [
CQ1, CQ2 ]. The x e i φ above in (1.23) is squared into ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 7 the xe i φ in (1.22) (and also in (1.28)), since it is the symmetric square lift Sym underconsideration.Any direct method seems impossible as there is no formula available in the literature todeal with the radial integration—the most outstanding obstacle in the extension of integralformulae from real to complex numbers. It is extremely lucky that the formula for (1.23)can be proven by known formulae in the spherical case ( d “ Qi2 ]. For the integralunder consideration, however, even the spherical case is inaccessible by direct method.
Special Weber-Schafheitlin formula after regularization.
Define(1.24) P µ, d p z q “ p z { q µ ` d p s z { q µ ´ d Γ p µ ` d ` q Γ p µ ´ d ` q , and(1.25) R µ, d p z q “ p πµ q p P ´ µ, ´ d p π z q ´ P µ, d p π z qq . Again, the right hand side is replaced by its limit when µ is an integer. We now state theregularized special Weber-Schafheitlin integral formula for J µ, d p z q as follows.T heorem Suppose that | Re µ | ă . Define the regularized Bessel function M µ, d p z q “ J µ, d p z q ´ R µ, d p z q . (1.26) Let Y p z q “ ˇˇ z ` ? z ´ ˇˇ , E p z q “ z ` ? z ´ ˇˇ z ` ? z ´ ˇˇ . (1.27) We have ż π ż M µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq dxd φ x “ d ´ µ ¨ Y ` ye i θ ˘ µ E ` ye i θ ˘ d ´ y µ e id θ ¯ ` ˜ Y ` ye i θ ˘ µ E ` ye i θ ˘ d ´ y µ e id θ ¸+ (1.28) for y P p , and θ P r , π q ; for d “ , the right hand side of the identity is to be replacedby its limit if µ “ . In the workof Venkatesh [
Ven2 ], Langlands’ proposal of Beyond Endoscopy [
Lan ] is executed forthe symmetric square lift Sym on GL , giving the classification of dihedral forms—formswhose symmetric square has a pole. It is proven that dihedral forms correspond to Gr¨ossen-characters of quadratic field extensions. This result is originally due to Labesse and Lang-lands [ LL ] by endoscopic methods.The fundamental tool used by Venkatesh at the beginning is the Kuznetsov relativetrace formula of Bruggeman and Miatello. Poisson summation is then applied at the stageafter Kuznetsov-Bruggeman-Miatello. Afterwards, the local exponential sums are evalu-ated and units of the quadratic field extension enter into the analysis. The Archimedeantheory of Venkatesh is contained in his Proposition 7, in which arises naturally the Fouriertransform of Bessel functions due to Kuznetsov-Bruggeman-Miatello and Poisson. ZHI QI
Venkatesh works over a number field. As such, the method is extremely notationallycomplicated in his §
4. He however takes great care in guiding the reader by illustrating themain ideas over Q in his §
3. More details may be found in his thesis [
Ven1 ].The main results of Venkatesh (in his §
4) are stated in the setting of a totally realnumber field. The only serious obstacle in the general case (involving complex places)is the validity of a certain integral transform as in his Proposition 7. Except for this, hismethod clearly does not rely on the totally real assumption in any important way.In the present paper, Proposition 7 of Venkatesh is extended to complex places so thathis main results are generalized to arbitrary number fields.For other works on Beyond Endoscopy, see for example [
Sar, Her1, Her2, Her3,FLN, Sak1, Sak2, Whi, Alt1, Alt2, Alt3 ]. Notation.
Let F be an (arbitrary) number field. Let stand for a place of F and F denote the completion of F at . Let S be the set of Archimedean places. Write |8 asthe abbreviation for P S . Define F to be F b R “ ś |8 F . For x P F let Norm p x q denote the norm of x . We fix the Haar measure on F corresponding to the product of dx at real places and | dx ^ d s x | at complex places. Let ψ : F Ñ C be the additive character ψ p x q “ e ` Tr F { R p x q ˘ .Let a be the vector space R | S | . Accordingly, we denote a typical element by t “p t q |8 . Let a C be its complexification. Let dt be the usual Lebesgue measure on a . Fol-lowing [ BM4 ], we also equip a with the positive measure d µ p t q “ ś |8 t sinh p π t q dt . Let Pl : a Ñ C be defined byPl p t q “ ź real p π t q ź complex p π t q{ t . (1.29) Note that d µ p t q{ Pl p t q is the Plancherel measure on the set of spherical tempered repre-sentations of PGL p F q . Define the logarithm function log F : F ˆ8 Ñ a bylog F p x q “ p log | x |q |8 . (1.30) Bessel kernel and Bessel transform.
Let M ą N ą
6. We set H p M , N q to be thespace of functions h : a Ñ C that are of the following form, h p t q “ ź |8 h p t q , where each h : R Ñ C extends to an even holomorphic function on the strip s “ t ` i σ : | Im s | ď M ( such that, on the horizontal line Im s “ σ ( | σ | ď M ), we have uniformly h p t ` i σ q Î e ´ π | t | p| t | ` q ´ N . We define the Bessel kernel B : F ˆ8 ˆ a C Ñ C as follows. For x P F ˆ8 and ν P a C “ C | S | , B p x , ν q “ ź |8 B p x , ν q , In § Ven2 ], the 1 { p π t q here insteadof cosh p π t q . ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 9 where B p x , ν q “ p πν q ` J ´ ν p π ? x q ´ J ν p π ? x q ˘ , B p´ x , ν q “ p πν q ` I ´ ν p π ? x q ´ I ν p π ? x q ˘ “ p πν q π K ν p π ? x q , if is real and x ą
0, and B p x , ν q “ p πν q ` J ´ ν p π ? x q J ´ ν p π ? s x q ´ J ν p π ? x q J ν p π ? s x q ˘ , if is complex; by definition, we have B p x , ν q “ J ν p? x q . R emark For complex , our definition of B p x , ν q is slightly di ff erent from thatof [ Ven2 ] , in which he uses I ν p x q instead of J ν p x q . But we have the formulae I ν p x q “ e ´ π i ν J ν p ix q and I ν p x q “ e π i ν J ν p´ ix q ( see [ Wat , 3.7 (2)] and it is understood that i “ e π i and ´ i “ e ´ π i ) . So the di ff erence is only up to the sign of x . This sign di ff erence howeverwould cause a little inconsistency between the Kuznetsov formula for SL p C q in [ MW ]( or [ BM4 ]) and that in [ BM5 ] . From various sources in the literature, it is suggested that thelatter ( and hence the formula of B p x , ν q here in terms of J ν p x q ) should be the correctone. Let h p t q be a test function on a belonging to H p M , N q and define its Bessel integraltransform ϕ : F ˆ8 Ñ C by ϕ p x q “ ż a h p t q B p x , it q d µ p t q . (1.31)This Bessel transform arises in the Kuznetsov-Bruggeman-Miatello trace formula as itsArchimedean component. Proposition 7 of Venkatesh.
The following theorem was proven for totally real F byVenkatesh [ Ven2 , Proposition 7], as the main ingredient in the Archimedean theory ofhis work. He mentioned that at the time he had not managed to accomplish the generalcase. Given this theorem, the main results of Venkatesh for Sym may be now extended toarbitrary number field F .T heorem Let F be a number field. Suppose thath p t q P H p M , N q and ϕ p x q is the Bessel transform of h p t q defined by (1.31) . Let Pl : a Ñ C be defined as in (1.29) . Define z h ¨ Pl : a Ñ C to be the Fourier transform of h p t q Pl p t q , sofor v P a , z h ¨ Pl p v q “ ż a e i x t , v y h p t q Pl p t q dt , (1.32) where x t , v y “ ř |8 t v . Defining log F : F ˆ8 Ñ a as in (1.30) , we have z h ¨ Pl p log F | κ |q “ ż F ϕ ˆ ` κ ` κ ´ x ˙ ψ p x q dx Norm p x q ;(1.33) the convergence is guaranteed as long as M , N are su ffi ciently large. Since the situation in the presence of multiple Archimedean places is a “product” ofsituations involving just one Archimedean place, it su ffi ces to check it in the case that F “ R or F “ C . The case F “ R has already been settled by Venkatesh (see also § F “ C will be proven in § J µ p z q in Theorem 1.4.C orollary The main results of Venkatesh on Beyond Endoscopy for
Sym on GL , in particular his Proposition and Theorem , are valid over an arbitrary numberfield F. A cknowledgements . I thank all the participants of my analytic number theory sem-inar, especially, Dongwen Liu and Zhicheng Wang, at Zhejiang University in the autumnof 2018—Venkatesh’s paper was the first one we studied in the seminar. I thank RomanHolowinsky and Akshay Venkatesh for their comments. I also thank the referees for carefulreadings and helpful comments.
2. Preliminaries2.1. Classical Bessel functions.
Basic properties of J ν p z q , H p q ν p z q and H p q ν p z q . Let ν be a complex number.Let J ν p z q , H p , q ν p z q denote the Bessel function of the first kind and the Hankel functions oforder ν . They all satisfy the Bessel di ff erential equation(2.1) z d wdz p z q ` z dwdz p z q ` ` z ´ ν ˘ w p z q “ . The function J ν p z q is defined by the series (see [ Wat , 3.1 (8)])(2.2) J ν p z q “ ÿ n “ p´q n p z { q ν ` n n ! Γ p ν ` n ` q . When ν ‰ ´ , ´ , ´ , ... , it follows from [ Wat , 3.13 (1)] that(2.3) | J ν p z q| “ p z { q ν Γ p ν ` q ` ` O ν ` | z | ˘˘ , | z | Î , and, together with the Bessel di ff erential equation and the recurrence formula [ Wat , 3.2(4)], zJ ν p z q “ ν J ν p z q ´ zJ ν ` p z q , that(2.4) z r p d { dz q r J ν p z q Î r ,ν | z ν | , | z | Î , with the implied constants uniformly bounded when ν lies in a given compact subset of C r t´ , ´ , ´ , ... u .We have the following connection formulae (see [ Wat , 3.61 (1, 2)]) J ν p z q “ H p q ν p z q ` H p q ν p z q , J ´ ν p z q “ e π i ν H p q ν p z q ` e ´ π i ν H p q ν p z q . (2.5)According to [ Wat , 7.2 (1, 2)] and [
Olv , 7.13.1], we have Hankel’s expansion of H p q ν p z q and H p q ν p z q as follows, H p q ν p z q “ ˆ π z ˙ e i p z ´ πν ´ π q ˜ N ´ ÿ n “ p´q n ¨ p ν, n qp iz q n ` E p q N p z q ¸ , (2.6) ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 11 H p q ν p z q “ ˆ π z ˙ e ´ i p z ´ πν ´ π q ˜ N ´ ÿ n “ p ν, n qp iz q n ` E p q N p z q ¸ , (2.7)with p ν, n q “ Γ p ν ` n ` { q { n ! Γ p ν ´ n ` { q , of which (2.6) is valid when z is suchthat ´ π ` δ ď arg z ď π ´ δ , and (2.7) when ´ π ` δ ď arg z ď π ´ δ , δ being anypositive acute angle, and z r p d { dz q r E p , q N p z q Î δ, r , N ,ν {| z | N (2.8)for | z | Ï z in the range indicated as above. In view of the error bounds in [ Olv ,7.13.1], the dependence of the implied constant on ν is uniform in any given compact set.2.1.2. Some integral formulae.
When the order ν “ m is an integer, we have theintegral representation of Bessel for J m p z q as follows (see [ Wat , 2.2 (1)]), i m J m p z q “ p´ i q m J ´ m p z q “ π ż π e iz cos φ ´ im φ d φ. (2.9)We have the following formula due to Weber, Sonine and Schafheitlin (see [ Wat , 13.24(1)] and [ GR , 6.561 14]), ż J ν p x q dxx ν ´ µ ` “ Γ p µ { q ν ´ µ ` Γ p ν ´ µ { ` q , (2.10)in which 0 ă Re µ ă Re ν ` (this is the domain of convergence in [ GR , 6.561 14], whileit is literally 0 ă Re µ ă Re ν ` in [ Wat , 13.24], for Watson only considers the domainof absolute convergence). J µ, d p z q . Replacing d { dz by B{B z , we de-note by ∇ ν the di ff erential operator that occurs in (2.1), namely,(2.11) ∇ ν “ z B B z ` z BB z ` z ´ ν . Its conjugation will be denoted by ∇ ν ,(2.12) ∇ ν “ s z B B s z ` s z BB s z ` s z ´ ν . From the definition of J µ, d p z q as in (1.18, 1.19), we infer that ∇ µ ` d p J µ, d p z { π qq “ , ∇ µ ´ d p J µ, d p z { π qq “ . (2.13)Recall the definition of R µ, d p z q given by (1.24, 1.25). Suppose at the moment that | z | ď
2, say. It follows from (2.3) that if µ is not an integer, then J µ, d p z q ´ R µ, d p z q Î µ, d ˇˇ | z | ´ µ ˇˇ ` ˇˇ | z | ` µ ˇˇ . (2.14)Some calculations by the formulae of pB J ν p z q{B ν q| ν “˘ m ( m “ , , , ... ) in [ Wat , § µ is an integer, J µ, d p z q ´ R µ, d p z q Î µ, d | z | ´ | µ | log p {| z |q . (2.15)It will be convenient to unify (2.14) and (2.15) in a slightly weaker form as follows, J µ, d p z q ´ R µ, d p z q Î µ, d , λ | z | ´ λ , (2.16)with λ “ | Re µ | if µ is not an integer and λ ą | Re µ | if otherwise. Further, we have z r s z s pB{B z q r pB{B s z q s J µ, d p z q Î r , s ,µ, d , λ | z | ´ λ . (2.17) For example, in the generic case, this is a direct consequence of the bounds in (2.4).In view of the connection formulae in (2.5), we have another expression of J µ, d p z q interms of Hankel functions,(2.18) J µ, d p z q “ i ´ e π i µ H p q µ, d p π z q ´ e ´ π i µ H p q µ, d p π z q ¯ , with the definition(2.19) H p , q µ, d p z q “ H p , q µ ` d p z q H p , q µ ´ d p s z q . The reader should be warned that the product in (2.19) is not well-defined as function on
C r t u . By (2.6)-(2.8), we may write J µ, d p z q “ e p z q W p z q ` e p´ z q W p´ z q ` E N p z q , (2.20)where W p z q and E N p z q are real analytic functions on C r t u satisfying z r s z s pB{B z q r pB{B s z q s W p z q Î r , s , N ,µ, d {| z | , (2.21) pB{B z q r pB{B s z q s E N p z q Î r , s , N ,µ, d {| z | N ` , (2.22)for | z | ě Our reference for the hypergeometric functionis Chapter II of [
MOS ].The hypergeometric function F p a , b ; c ; z q is defined by the Gauss series F p a , b ; c ; z q “ Γ p c q Γ p a q Γ p b q ÿ n “ Γ p a ` n q Γ p b ` n q Γ p c ` n q n ! z n (2.23)within its circle of convergence | z | ă
1, and by analytic continuation elsewhere. The seriesis absolutely convergent on the unit circle | z | “ p a ` b ´ c q ă
0. The function F p a , b ; c ; z q is a single-valued analytic function of z in the complex plane with a branchcut along the positive real axis from 1 to . Moreover, F p a , b ; c ; z q is analytic in a , b and c , except for c “ , ´ , ´ , ... .The hypergeometric di ff erential equation satisfied by F p a , b ; c ; z q is as follows, z p ´ z q d wdz p z q ` p c ´ p a ` b ` q z q dwdz p z q ´ abw p z q “ . (2.24)It has three regular singular points z “ , , . In the generic case when none of c , a ´ b and c ´ a ´ b is an integer, two linearly independent solutions of (2.24) in the vicinity of z “ 8 are given by G p a , b ; c ; z q “ z ´ a F p a , a ´ c ` a ´ b `
1; 1 { z q , G p a , b ; c ; z q “ z ´ b F p b , b ´ c ` b ´ a `
1; 1 { z q . (2.25)Finally, we record here the transformation formula with respect to z Ñ { z (see [ MOS , § F p a , b ; c ; z q “ Γ p c q Γ p b ´ a q Γ p b q Γ p c ´ a q p´ z q ´ a F p a , a ´ c ` a ´ b `
1; 1 { z q` Γ p c q Γ p a ´ b q Γ p a q Γ p c ´ b q p´ z q ´ b F p b , b ´ c ` b ´ a `
1; 1 { z q , | arg p´ z q| ă π, a ´ b ‰ ˘ m , m “ , , , .... (2.26) ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 13
The following lemma will be very useful inour later analysis.L emma
Let a , b and θ be real numbers such that | b | ă | a | . Let ă A ă B ď 8 , γ ă , and M be a positive integer. Suppose that f p z q is a smooth function on C which issupported on the annulus | z | P r A , B s and satisfiesx r pB{B x q r pB{B φ q s f p xe i φ q Î r , s x γ ´ for all nonnegative integers r , s such that r ` s ď M ( equivalently,z r s z s pB{B z q r pB{B s z q s f p z q Î r , s | z | γ ´ for all r ` s ď M ) . DefineI p a , b , θ q “ ż π ż f ` xe i φ ˘ e p x p a cos p φ ` θ q ` b cos φ qq xdxd φ. Then the integral I p a , b , θ q is convergent after integration by parts ( it is already absolutelyconvergent if γ ă and I p a , b , θ q Î M , γ A M ´ γ p| a | ´ | b |q M , with the convergence and the implied constant uniform for γ in a compact set. Moreover,the integral I p a , b , θ q gives rise to a continuous function in p a , θ q ( the continuity extends toall real values of a if γ ă . P roof . Put p p x , φ ; a , b , θ q “ x p a cos p φ ` θ q ` b cos φ q . Define the di ff erential operatorD “ pB{B x q ` pB{B x q{ x ` pB{B φ q { x so thatD ` e p p p x , φ ; a , b , θ qq ˘ “ ´ π p a ` b ` ab cos θ q ¨ e p p p x , φ ; a , b , θ qq . Note that D is self-adjoint. In view of the conditions on f p z q , it follows from an applicationof the partial integration with respect to D that I p a , b , θ q “ ´ π p a ` b ` ab cos θ q ż π ż D f ` xe i φ ˘ e p p p x , φ ; a , b , θ qq xdxd φ. Since x D f p xe i φ q Î x γ ´ , the integral above is absolutely convergent as γ ă . Repeatingthe partial integration above M times and then bounding the resulting integral trivially,along with a ` b ` ab cos θ ě p| a | ´ | b |q and x D M f p xe i φ q Î M x γ ´ M ´ , we obtainthe estimates for I p a , b , θ q in the lemma. Finally, the continuity of I p a , b , θ q is obvious.Q.E.D.
3. More on the hypergeometric function
We shall be concerned with the hypergeometric di ff erential equation for a “ ρ ` ν , b “ ρ ´ ν , c “ . (3.1)Define the corresponding hypergeometric di ff erential operator ∇ ρ,ν “ z p ´ z q B B z ` p ´ p ρ ` q z q BB z ´ ` ρ ´ ν ˘ , (3.2) and its conjugate ∇ ρ,ν “ s z p ´ s z q B B s z ` p ´ p ρ ` q s z q BB s z ´ ` ρ ´ ν ˘ . (3.3)For the choice of a , b , c in (3.1) such that neither ν nor ρ ´ is an integer, we denote G p q ρ,ν p z q “ G p a , b ; c ; z q , G p q ρ,ν p z q “ G p a , b ; c ; z q . (3.4)Let F p q ρ,ν p z q and F p q ρ,ν p z q be defined as in (1.21). Note that their hypergeometric seriesexpansions are absolutely convergent for all | z | ď ρ ă . Further, we have G p q ρ,ν p z q “ z ´ p ρ ` ν q F p q ρ,ν p { z q , G p q ρ,ν p z q “ z ´ p ρ ´ ν q F p q ρ,ν p { z q . (3.5)L emma Let d be an integer. Let ρ , µ be such that ρ ´ is not an integer and that | Re µ | ă and µ ‰ . Let f p z q be a continuous function on the complex plane which is asolution of the following two di ff erential equations, (3.6) ∇ ρ,µ ` d w “ , ∇ ρ,µ ´ d w “ , with the di ff erential operators ∇ ρ,µ ` d and ∇ ρ,µ ´ d defined as in (3.2) and (3.3) . Supposefurther that f p z q admits the following asymptotic,f p z q „ c z ´ p ρ ` µ ` d q s z ´ p ρ ` µ ´ d q ` c z ´ p ρ ´ µ ´ d q s z ´ p ρ ´ µ ` d q , | z | Ñ 8 . (3.7) Then f p z q “ c G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q ` c G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q (3.8) for all z in the complex plane. P roof . By the theory of di ff erential equations, it follows from (3.6) that f p z q may beuniquely written as a linear combination of G p k q ρ,µ ` d p z q G p l q ρ,µ ´ d p s z q , with k , l “ ,
2, namely, f p z q “ ÿ ÿ k , l “ , c kl G p k q ρ,µ ` d p z q G p l q ρ,µ ´ d p s z q , for all z away from the branch cut from 0 to 1.In view of (2.25) and (3.4), it is clear that G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q and G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q are both single-valued real analytic functions when | z | ą
1, and so are the quotient of G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q and p z {| z |q ´ µ and that of G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q and p z {| z |q µ . Since µ is not an integer, the functions p z {| z |q ´ µ and p z {| z |q µ are not single-valued. Choosingarg z “ , π , it follows from simple considerations that any nontrivial linear combina-tion of G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q and G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q is not independent on arg z modulo 2 π .So we must have c “ c “ c “ c and c “ c . For this, we deduce from (2.25)and (3.4) that G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q “ | z | ´ ρ ´ µ p s z {| z |q d ` O `ˇˇ | z | ´ ρ ´ µ ´ ˇˇ˘ , G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q “ | z | ´ ρ ` µ p z {| z |q d ` O `ˇˇ | z | ´ ρ ` µ ´ ˇˇ˘ , as | z | Ñ 8 . Since | Re µ | ă , the two error terms have order strictly lower than the leastorder of the two main terms. This forces c “ c and c “ c in order for f p z q to havethe prescribed asymptotic in (3.7). ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 15
It is now proven that the identity (3.8) is valid except on the branch cut from 0 to 1.However f p z q is continuous on the whole complex plane, the identity may be extended tothe branch cut. Q.E.D.We remark that the continuity of f p z q is a very strong condition (it is not even clearnow whether exists such a continuous solution f p z q of the di ff erential equations). It doesnot only force c “ c “ c { c has to be unique as neither G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q nor G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q is single-valued in | z | ă
1. Indeed, the following proposition demonstrates that the ratio must be equal to4 ´ µ C ρµ, d { µ C ρ ´ µ, ´ d . This proposition however is not required in our proof of Theorem 1.1.P roposition Let C ρµ, d be defined by (1.20) . Let G p q ρ,ν p z q and G p q ρ,ν p z q be given by (1.21) and (3.5) . Define (3.9) E p q ρ,ν p z q “ F ˆ ρ ` ν , ρ ´ ν z ˙ , E p q ρ,ν p z q “ F ˆ ρ ` ν ` , ρ ´ ν `
12 ; 32 ; z ˙ . We have ´ µ C ρµ, d G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q ` µ C ρ ´ µ, ´ d G p q ρ,µ ` d p z q G p q ρ,µ ´ d p s z q“ B ρ,µ, d A p q ρ,µ, d E p q ρ,µ ` d p z q E p q ρ,µ ´ d p s z q ` B ρ,µ, d ` A p q ρ,µ, d ¨ | z | E p q ρ,µ ` d p z q E p q ρ,µ ´ d p s z q , (3.10) for arg z P p , π q , with B ρ,µ, d “ p´q d cos p πµ q ´ cos p πρ q π ρ ` , (3.11) A p q ρ,µ, d “ Γ ˆ ρ ` µ ` d ˙ Γ ˆ ρ ` µ ´ d ˙ Γ ˆ ρ ´ µ ´ d ˙ Γ ˆ ρ ´ µ ` d ˙ , (3.12) and similarlyA p q ρ,µ, d “ Γ ˆ ` ρ ` µ ` d ˙ Γ ˆ ` ρ ` µ ´ d ˙ Γ ˆ ` ρ ´ µ ´ d ˙ Γ ˆ ` ρ ´ µ ` d ˙ . (3.13) Moreover, for Re ρ ă , the combinations on both sides of (3.10) give rise to a continuousfunction of z on the whole complex plane, real-analytic except at z “ . P roof . By the transformation law (2.26), we have G p q ρ,ν p z q “ ? π Γ p ` ν q e ´ π i p ρ ` ν q E p q ρ,ν p z q Γ pp ` ρ ` ν q{ q Γ pp ´ ρ ` ν q{ q ` ? π Γ p ` ν q e ´ π i p ρ ` ν ` q ? zE p q ρ,ν p z q Γ pp ρ ` ν q{ q Γ pp ´ ρ ` ν q{ q , G p q ρ,ν p z q “ ? π Γ p ´ ν q e ´ π i p ρ ´ ν q E p q ρ,ν p z q Γ pp ` ρ ´ ν q{ q Γ pp ´ ρ ´ ν q{ q ` ? π Γ p ´ ν q e ´ π i p ρ ´ ν ` q ? zE p q ρ,ν p z q Γ pp ρ ´ ν q{ q Γ pp ´ ρ ´ ν q{ q , for arg z P p , π q . For G p q ρ,ν p s z q and G p q ρ,ν p s z q , the formulae are similar but in a subtle way—the factors like e ´ π i p ρ ` ν q should be replaced by e π i p ρ ` ν q . This is because, in view of theconditions in (2.26), we need to let | arg p´ { s z q| ă π and hence ´ “ e ´ i π .Applying these to the left hand side of (3.10). By the duplication formula for thegamma function, we find that, up to sign, the coe ffi cients of E p q ρ,µ ` d p z q ¨ ? s zE p q ρ,µ ´ d p s z q and Note that, according to [
MOS , § E p q ρ,ν p z q and ? zE p q ρ,ν p z q form a system of linearly independent solutionsof the hypergeometric equation with a , b , c as in (3.1) in the vicinity of the singular point z “ ? zE p q ρ,µ ` d p z q ¨ E p q ρ,µ ´ d p s z q are both equal to i d ´ π ρ sin p πµ q ˆ sin p π p ρ ` µ qq Γ pp ρ ` µ ` d q{ q Γ pp ` ρ ` µ ´ d q{ q Γ pp ´ ρ ` µ ` d q{ q Γ pp ´ ρ ` µ ´ d q{ q´ sin p π p ρ ´ µ qq Γ pp ρ ´ µ ´ d q{ q Γ pp ` ρ ´ µ ` d q{ q Γ pp ´ ρ ´ µ ´ d q{ q Γ pp ´ ρ ´ µ ` d q{ q ˙ , which in turn is equal to zero by Euler’s reflection formula for the gamma function. Weobtain (3.10) by calculating the coe ffi cients of E p q ρ,ν p z q E p q ρ,ν p s z q and | z | E p q ρ,ν p z q E p q ρ,ν p s z q in asimilar way.It was noted before that the left hand side of (3.10) gives rise to a single-valued con-tinuous function on | z | ě
1, real-analytic in the interior | z | ą
1, provided that Re ρ ă . Itis also clear that the same is true for the right hand side of (3.10) on | z | ď
1. Thus (3.10)is valid and defines a continuous function (real-analytic except at z “
1) on the wholecomplex plane if Re ρ ă . Q.E.D.
4. Proof of Theorem 1.1
Recall that J µ, d p z q and R µ, d p z q were defined as in (1.18, 1.19) and (1.24,1.25) respec-tively. Let(4.1) F ρµ, d ` ye i θ ˘ “ ż π ż J µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ, and(4.2) P ρµ, d ` ye i θ ˘ “ ż π ż R µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ. Since J µ, d p z q is an even function, the integral F ρµ, d ` ye i θ ˘ is also even in the sense that it isindependent on θ modulo π .First of all, the convergence of F ρµ, d ` ye i θ ˘ and P ρµ, d ` ye i θ ˘ may be easily examined withthe help of Lemma 2.1, along with (2.17) and (2.20)-(2.22) for the former.L emma Let the integrals F ρµ, d ` ye i θ ˘ and P ρµ, d ` ye i θ ˘ be defined as above. (1). F ρµ, d ` ye i θ ˘ is absolutely convergent if | Re µ | ă Re ρ ă , and, for y ą , conver-gent if | Re µ | ă Re ρ ă , uniformly for Re µ and Re ρ in compact sets. (2). P ρµ, d ` ye i θ ˘ is convergent for y ą if | Re µ | ă Re ρ ă ´ | Re µ | , uniformly for Re µ and Re ρ in compact sets.Moreover, both F ρµ, d ` ye i θ ˘ and P ρµ, d ` ye i θ ˘ are continuous functions when y is in the indi-cated ranges above. P roof . We partition the integrals F ρµ, d ` ye i θ ˘ and P ρµ, d ` ye i θ ˘ according to a partition ofunity p ´ w p x qq ` w p x q ” p ,
8q “ p , s Y r , , say.By (2.17), it is obvious that the integral ż π ż p ´ w p x qq J µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ is convergent if | Re µ | ă Re ρ ; the same statement is true if J µ, d is substituted by R µ, d . ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 17
Next, we consider the following integral ż π ż w p x q J µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ. Since J µ, d p z q “ O p {| z |q for | z | ě
1, this integral is absolutely convergent providedthat Re ρ ă . According to (2.20), we divide it into three similar integrals which con-tain W ` xe i φ ˘ , W ` ´ xe i φ ˘ and E ` xe i φ ˘ respectively. For the first two integrals, we applyLemma 2.1 with a “ ´ y , b “ ˘ f p z q “ w p| z |q W p˘ z q| z | ρ ´ and γ “ Re ρ ´ (by(2.21)), and it follows that these two integrals are convergent if Re ρ ă γ ă ). By(2.22), the third integral is absolutely convergent when Re ρ ă a “ ´ y , b “ f p z q “ w p| z |q R µ, d p z q| z | ρ ´ and γ “ Re ρ ` | Re µ | , we infer that the integral ż π ż w p x q R µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ is convergent provided that Re ρ ă ´ | Re µ | . Q.E.D. F ρµ, d ` ye i θ ˘ . In this section, we assume that λ “ | Re µ | ă , β “ Re ρ and λ ă β ă ´ λ so that F ρµ, d ` ye i θ ˘ and P ρµ, d ` ye i θ ˘ are both convergent.Moreover, let µ ‰ F ρµ, d p ye i θ q ,(4.3) F ρµ, d ` ye i θ ˘ „ C ρµ, d y ´ µ ´ ρ e ´ id θ ` C ρ ´ µ, ´ d y µ ´ ρ e id θ , y Ñ 8 , in which C ρµ, d is defined by (1.20) in Theorem 1.1. It is clear that (4.3) follows from thefollowing two lemmas.L emma We have F ρµ, d ` ye i θ ˘ “ P ρµ, d ` ye i θ ˘ ` o ` y ´ λ ´ β ˘ , y Ñ 8 . (4.4)L emma We have P ρµ, d ` ye i θ ˘ “ C ρµ, d y ´ µ ´ ρ e ´ id θ ` C ρ ´ µ, ´ d y µ ´ ρ e id θ . (4.5)P roof of L emma y ą ffi ciently large. All the implied constants in ourcomputation will only depend on β , λ and d .We split F ρµ, d ` ye i θ ˘ ´ P ρµ, d ` ye i θ ˘ as the sum F ρµ, d ` ye i θ ˘ ´ P ρµ, d ` ye i θ ˘ “ D ρµ, d ` ye i θ ˘ ` ´ E ρµ, d ` ye i θ ˘ ` ` E ρµ, d ` ye i θ ˘ ´ ˆ E ρµ, d ` ye i θ ˘ “ ż π ż y ´ u p x q ` J µ, d ` xe i φ ˘ ´ R µ, d ` xe i φ ˘˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ ` ż π ż y ´ v p x q J µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ ` ż π ż w p x q J µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ ´ ż π ż y ´ p v p x q ` w p x qq R µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ, where u p x q ` v p x q ` w p x q ” p , such that u p x q , v p x q and w p x q are smooth functions supported on ` , y ´ ‰ , “ y ´ , ‰ and r , , respectively, andthat x r u p r q p x q , x r v p r q p x q , x r w p r q p x q Î r D ρµ, d ` ye i θ ˘ Î ż π ż y ´ x β ´ λ ` dxd φ Î y λ ´ β ´ “ y ´ λ ´ β y λ ` β ´ . Since β ă ´ λ and λ ă , we have 3 λ ` β ´ ă λ ´ ă D ρµ, d ` ye i θ ˘ “ o ` y ´ λ ´ β ˘ as desired.Second, applying Lemma 2.1 with a “ ´ y , b “ A “ y ´ , B “ M “ f p z q “ v p| z |q J µ, d p z q| z | ρ ´ and γ “ β ´ λ (see (2.17)), we deduce that´ E ρµ, d ` ye i θ ˘ Î y p ´ β ` λ q´ “ y λ ´ β ´ . As above, we also have ´ E ρµ, d ` ye i θ ˘ “ o ` y ´ λ ´ β ˘ . Likewise, ˆ E ρµ, d ` ye i θ ˘ “ o ` y ´ λ ´ β ˘ .Third, according to (2.20), we divide ` E ρµ, d ` ye i θ ˘ into three similar integrals which con-tain W ` xe i φ ˘ , W ` ´ xe i φ ˘ and E ` xe i φ ˘ respectively. For the first two integrals, we applyLemma 2.1 with a “ ´ y , b “ ˘ A “ B “ 8 , M “ f p z q “ w p| z |q W p˘ z q| z | ρ ´ and γ “ β ´ (by (2.21)), and it follows that these two integrals are both O p y ´ q . Forthe third integral containing E ` xe i φ ˘ , we apply Lemma 2.1 with a “ ´ y , b “ A “ B “ 8 , M “ f p z q “ w p| z |q E p z q| z | ρ ´ and γ “ β (by (2.22)), and it follows that thethird integral is also O p y ´ q . We conclude that` E ρµ, d ` ye i θ ˘ “ O p y ´ q “ o ` y ´ λ ´ β ˘ as β ă ´ λ .Finally, combining the foregoing results, we obtain the asymptotic formula (4.4) inLemma 4.2. Q.E.D.The formula (4.5) in Lemma 4.3 is an immediate consequence of the following lemma,applied with m “ ˘ d and ν “ ρ ˘ µ .L emma Let m be an integer. Let ´ | m | ă Re ν ă . We have ż ż π e p´ xy cos p φ ` θ qq x ν ´ e im φ d φ dx “ $’’&’’% sin p πν q Γ p ν ` m { q Γ p ν ´ m { qp π q ν y ν e im θ , i cos p πν q Γ p ν ` m { q Γ p ν ´ m { qp π q ν y ν e im θ , according as m is even or odd ; the integral on the left is convergent as iterated integral for ´ | m | ă Re ν ă and as double integral only for ă Re ν ă . P roof . The inner integral over φ may be evaluated by (2.9) so that ż ż π e p´ xy cos p φ ` θ qq e im φ d φ x ν ´ dx “ π p´ i q | m | e ´ im θ ż J | m | p π xy q x ν ´ dx . Even more, it may be proven that that ´ E ρµ, d ` ye i θ ˘ , ` E ρµ, d ` ye i θ ˘ and ˆ E ρµ, d ` ye i θ ˘ are all arbitrarily small, namely, O A p y ´ A q for arbitrary A ą ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 19
In view of (2.10), the integral in the right hand side is equal to Γ p ν ` | m |{ q p π y q ν Γ p ´ ν ` | m |{ q “ $’’’&’’’% sin p πν q Γ p ν ` | m |{ q Γ p ν ´ | m |{ q π p π y q ν i | m | , if m is even,cos p πν q Γ p ν ` | m |{ q Γ p ν ´ | m |{ q π p π y q ν i | m |` , if m is odd,as desired. Q.E.D. ff erential equations for F ρµ, d . Since F ρµ, d ` ye i θ ˘ is even, it would be convenientto set 4 u “ y e i θ and consider F ρµ, d p ? u q . Assume that | Re µ | ă Re ρ ă so that theintegral F ρµ, d p ? u q is always convergent and gives rise to a continuous function. We arenow going to verify ∇ ρ,µ ` d ` F ρµ, d p ? u q ˘ “ , ∇ ρ,µ ´ d ` F ρµ, d p ? u q ˘ “ , (4.6)with the hypergeometric di ff erential operators ∇ ρ,µ ` d and ∇ ρ,µ ´ d defined as in (3.2) and(3.3) respectively. By symmetry, we only need to verify the former, which, if we set ν “ µ ` d for simplify, may be explicitly written as4 u p ´ u q B F ρµ, d p ? u qB u ` p ´ p ρ ` q u q B F ρµ, d p ? u qB u ´ ` ρ ´ ν ˘ F ρµ, d p ? u q “ . For s , r “ ,
1, 2, with s ` r “ , ,
2, we introduce I ρ s , r ,µ, d p u q “ ? u s ij Cr t u z ρ ` s ` r ´ s z ρ ´ pB{B z q r J µ, d p z q e ` ´ ` z ? u ` s z ? s u ˘˘ idz ^ d s z . The integral I ρ s , r ,µ, d p u q should be regarded as distribution on C . Precisely, for any Schwartzfunction f p u q P S p C q (the Schwartz space on C is denoted by S p C q as usual), let @ u t I ρ s , r ,µ, d , f D “ ij Cr t u z ρ ` s ` r ´ s z ρ ´ pB{B z q r J µ, d p z q f p z q idz ^ d s z , with f p z q P S p C q given by f p z q “ ij Cr t u u ` t ´ s s u f p u q e p´ p zu ` s z s u qq idu ^ d s u , in which t “ , , t “ ´ s “ ,
1. Note here that f p z { q is the Fouriertransform of u ` t ´ s s u f p u q and hence is also Schwartz. In the theory of distributions, weare free to di ff erentiate under the integral and integrate by parts in the formal manner aswhat we shall do in the following.To start with, note that F ρµ, d p ? u q “ I ρ , ,µ, d p u q . For brevity, we put I s , r “ I ρ s , r ,µ, d and I s “ I s , .First, for s “ ,
1, we have B I s B u “ ´ s u I s ´ π i I s ` and B I B u “ ´ π i B I B u “ π iu I ´ π I . Second, for s , r “ ,
1, with s ` r “ ,
1, by partial integration,4 π i ? u ij Cr t u z ρ ` s ` r s z ρ ´ pB{B z q r J µ, d p z q e ` ´ ` z ? u ` s z ? s u ˘˘ idz ^ d s z “ p ρ ` s ` r q ij Cr t u z ρ ` s ` r ´ s z ρ ´ pB{B z q r J µ, d p z q e ` ´ ` z ? u ` s z ? s u ˘˘ idz ^ d s z ` ij Cr t u z ρ ` s ` r s z ρ ´ pB{B z q r ` J µ, d p z q e ` ´ ` z ? u ` s z ? s u ˘˘ idz ^ d s z . It follows that 4 π iu I s ` , r “ p ρ ` s ` r q I s , r ` I s , r ` , and hence ´ π u I “ π i p ρ ` q u I ` π iu I , “ p ρ ` ρ q I ` p ρ ` q I , ` I , . Third, since ∇ µ ` d ` J µ, d ` z { π ˘˘ “ I , ` I , ` π u I , ´ ν I , “ p ν “ µ ` d q . Finally, combining these, we have4 u p ´ u q B I B u ` p ´ p ρ ` q u q B I B u ´ ` ρ ´ ν ˘ I “ u p ´ u q ˆ π iu I ´ π I ˙ ´ π i p ´ p ρ ` q u q I ´ ` ρ ´ ν ˘ I “ ´ π u I ` π u I ` π i p ρ ` q u I ´ ` ρ ´ ν ˘ I “ ´ π u I ´ pp ρ ` ρ q I ` p ρ ` q I , ` I , q ` p ρ ` qp ρ I ` I , q ´ ` ρ ´ ν ˘ I “ ν I ´ π u I ´ I , ´ I , “ , as desired. Combining (4.3) and (4.6), we deduce from Lemma 3.1 that, underthe conditions | Re µ | ă Re ρ ă and µ ‰ F ρµ, d p ? u q “ ´ µ ´ ρ C ρµ, d ¨ G p q ρ,µ ` d p u q G p q ρ,µ ´ d p s u q ` µ ´ ρ C ρ ´ µ, ´ d ¨ G p q ρ,µ ` d p u q G p q ρ,µ ´ d p s u q , for all u in the complex plane (the choice of the square root ? u is not essential since F ρµ, d is even). Let 2 ? u “ ye i θ . In view of the formulae for G p q ρ,ν and G p q ρ,ν in (3.5) andthe definition of F ρµ, d in (4.1), it is clear that (4.7) is equivalent to (1.22) in Theorem 1.1.Finally, thanks to the principal of analytic continuation, it follows from Lemma 4.1 (1)that the condition µ ‰ | Re µ | ă Re ρ ă may beimproved into | Re µ | ă Re ρ ă y ą
2. Note that in the definition of C ρµ, d the zero ofsin p πµ q is annihilated by the pole of Γ p ` µ ` d q Γ p ` µ ´ d q at µ “ d ‰
5. Proof of Theorem 1.4
Consider F ρµ, d ` ye i θ ˘ ´ P ρµ, d ` ye i θ ˘ “ ż π ż M µ, d ` xe i φ ˘ e p´ xy cos p φ ` θ qq x ρ ´ dxd φ, in which M µ, d p z q “ J µ, d p z q ´ R µ, d p z q is the regularized Bessel function defined in (1.26).It is examined in Lemma 4.1 that for the convergence of this integral at infinity we needRe ρ ă ´ | Re µ | and also y ą
0. On the other hand, in view of (2.16), the convergenceat zero is secured if | Re µ | ´ ă Re ρ . ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 21 L emma The integral F ρµ, d ` ye i θ ˘ ´ P ρµ, d ` ye i θ ˘ is convergent for y ą if | Re µ |´ ă Re ρ ă ´ | Re µ | , uniformly for Re µ and Re ρ in compact sets. Let C ρµ, d and F p , q ρ,ν p z q be defined in (1.20) and (1.21). By Theorem 1.1 and Lemma4.3, we have F ρµ, d ` ye i θ ˘ ´ P ρµ, d ` ye i θ ˘ “ C ρµ, d y µ ` ρ e id θ ´ F p q ρ,µ ` d ` { y e i θ ˘ F p q ρ,µ ´ d ` e i θ { y ˘ ´ ¯ ` C ρ ´ µ, ´ d y µ ´ ρ e id θ ´ F p q ρ,µ ` d ` { y e i θ ˘ F p q ρ,µ ´ d ` e i θ { y ˘ ´ ¯ . (5.1)provided that | Re µ | ă Re ρ ă ´ | Re µ | . We claim that (5.1) remains valid in theextended range | Re µ | ´ ă Re ρ ă ´ | Re µ | . For d ‰
0, the validity of (5.1) extendsto | Re µ | ´ ă Re ρ ă ´ | Re µ | , since C ρµ, d is analytic in this range. Now consider thecase d “
0. For simplicity, let us assume µ ‰
0. Then C ρµ, has a simple pole at ρ “ ´ µ .However, we have F p q ρ,µ p z q ´ ” ρ “ ´ µ and similarly F p q ρ,µ p z q ´ ” ρ “ µ . Sothe extension of (5.1) to the range | Re µ | ´ ă Re ρ ă ´ | Re µ | is still permissible.Assume that | Re µ | ă . Let ρ Ñ
0. Since C µ, d “ { ` d ´ µ ˘ , and F p q ,ν p z q “ F ˆ ν ` ν ` ν ; z ˙ “ ˆ ` ? ´ z ˙ ν , F p q ,ν p z q “ F ˆ ´ ν ´ ν ´ ν ; z ˙ “ ˆ ` ? ´ z ˙ ν , we find that the limiting form of (5.1) as ρ Ñ
6. Proof of Theorem 1.6 over complex numbers
To start with, we translate the formula (1.33) into our language for F “ C . Accord-ing to the notation of Venkatesh, set φ p z q “ ϕ p z q and ? κ “ ` k ´ ? k ´ ˘ { | k | ą ij Cr t u φ p z q e p Tr p kz qq idz ^ d s z | z | “ ż h p t q sinh p π t q ˇˇˇˇˇ k ` ? k ´ ˇˇˇˇˇ ´ it dtt , (6.1)where φ p z q “ ż h p t q J it p z q sinh p π t q tdt . (6.2)Recall here that h p s q is a holomorphic even function in the strip | Im s | ď M , satisfyingdecay estimates h p t ` i σ q Î e ´ π | t | p| t | ` q ´ N . (6.3)Define ω p z q to be ω p z q “ ż h p t q R it p z q sinh p π t q tdt “ i ż h p t q| π z | it tdt Γ p ` it q , (6.4) where R it p z q “ R it , p z q is defined as in (1.24, 1.25). Note that ω p z q is simply the (horizon-tal) Mellin inverse transform of the function i p π q it h p t q t { Γ p ` it q .We quote from Lemma 4.1 in [ Qi4 ] the following uniform estimate for J it p z q , t J it p z q Î p| t | ` q min , {| z | ( . (6.5)Further, it is clear from [ Wat , 3.13 (1)] (note that | Γ p ` it q| “ π t { sinh p π t q ) that t p J it p z q ´ R it p z qq Î | z | , | z | ď . (6.6)We first write the integral on the left of (6.1) as follows, ij Cr t u p φ p z q ´ ω p z qq e p Tr p kz qq idz ^ d s z | z | ` ij Cr t u ω p z q e p Tr p kz qq idz ^ d s z | z | (6.7)Applying (6.2) and (6.4), the first integral in (6.7) is equal to12 ij Cr t u ˆż h p t q ` J it p z q ´ R it p z q ˘ sinh p π t q tdt ˙ e p Tr p kz qq idz ^ d s z | z | . Making crucial use of (6.3), (6.5) and (6.6), we verify that the double integral is absolutelyconvergent except for the contribution from R it p z q in the vicinity of z “ 8 . Nevertheless,we may switch the order of integration. This is because the integral becomes absolutelyconvergent after integration by part in a neighborhood of z “ 8 (see Lemma 2.1); we stillneed (6.3) to secure this. We then obtain12 ż h p t q ˆ ij Cr t u ` J it p z q ´ R it p z q ˘ e p Tr p kz qq idz ^ d s z | z | ˙ sinh p π t q tdt . The inner integral is evaluated in Theorem 1.4. Thus ij Cr t u p φ p z q ´ ω p z qq e p Tr p kz qq idz ^ d s z | z | “ ż h p t q ´ˇˇ` k ` ? k ´ ˘ { ˇˇ ´ it ´ | k | ´ it ¯ sinh p π t q dtt . (6.8)Now consider the second integral in (6.7). Applying (6.4), we have ij Cr t u ω p z q e p Tr p kz qq idz ^ d s z | z | “ i ij Cr t u ˆż h p t q| π z | it tdt Γ p ` it q ˙ e p Tr p kz qq idz ^ d s z | z | . Let σ ą σ ă . In view of (6.3), we may shift the line of integration inthe inner integral to Im t “ ´ σ , then the integral on the right turns into i ij Cr t u ˆż ´ i σ `8´ i σ ´8 h p t q| π z | it tdt Γ p ` it q ˙ e p Tr p kz qq idz ^ d s z | z | . Again, we may switch the order of integration, although this integral is not absolutelyconvergent in the vicinity of z “ 8 , obtaining i ż ´ i σ `8´ i σ ´8 h p t q ˆ ij Cr t u | π z | it e p Tr p kz qq idz ^ d s z | z | ˙ tdt Γ p ` it q . The inner integral is evaluated in Lemma 4.4 (it is convergent actually as a double integral,since 0 ă Re p it q “ σ ă ). Thus we have ij Cr t u ω p z q e p Tr p kz qq idz ^ d s z | z | “ ż ´ i σ `8´ i σ ´8 h p t q| k | ´ it sinh p π t q dtt . ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 23
Utilizing again (6.3), we are now free to move the line of integration back to Im t “ ij Cr t u ω p z q e p Tr p kz qq idz ^ d s z | z | “ ż h p t q| k | ´ it sinh p π t q dtt . (6.9)In conclusion, the formula (6.1) is proven by summing (6.8) and (6.9). Appendix A.
In this appendix, we record some extensions of the general Weber-Schafheitlin for-mula in Theorem 1.1. These results may be proven by modifying our arguments in §§ µ be a complex number and d be an integer or half-integer. Define J µ, d p z q “ J µ ` d p z q J µ ´ d p s z q . and J µ, d p z q “ $’’&’’% p πµ q p J µ, d p π z q ´ J ´ µ, ´ d p π z qq , if 2 d is even , i cos p πµ q p J µ, d p π z q ` J ´ µ, ´ d p π z qq , if 2 d is odd . We need to take the limit in the non-generic case when µ ´ d is an integer. Note that J µ, d p z q is an even or odd function according as 2 d is even or odd.Let ρ be a complex number and m be an integer. We consider the following integrals F ρ, m µ, d ` ye i θ ˘ “ ż π ż J µ, d ` xe i φ ˘ cos p π xy cos p φ ` θ qq x ρ ´ e im φ dxd φ, G ρ, m µ, d ` ye i θ ˘ “ ż π ż J µ, d ` xe i φ ˘ sin p π xy cos p φ ` θ qq x ρ ´ e im φ dxd φ. By simple parity considerations, it is clear that the former or the latter integral is zeroaccording as m ` d is odd or even (as the integrand is an odd function).We have the following generalization of Theorem 1.1.T heorem A.1.
Let notation be as above. DefineF λ ν p z q “ F ˆ λ ` ν ` λ ` ν ` ν ; z ˙ . SetC ρ, m µ, d “ ´ Γ p ρ ` µ ` m { ` d q Γ p ρ ` µ ´ m { ´ d q Γ p ` µ ` d q Γ p ` µ ´ d q ¨ $’’&’’% sin p π p ρ ` µ qqp π q ρ sin p πµ q , if d is even , i sin p π p ρ ` µ qqp π q ρ cos p πµ q , if d is odd , D ρ, m µ, d “ ´ Γ p ρ ` µ ` m { ` d q Γ p ρ ` µ ´ m { ´ d q Γ p ` µ ` d q Γ p ` µ ´ d q ¨ $’’&’’% cos p π p ρ ` µ qqp π q ρ sin p πµ q , if d is even , i cos p π p ρ ` µ qqp π q ρ cos p πµ q , if d is odd . For all complex z, we have F ρ, m µ, d p z q “ C ρ, m µ, d F ρ ` m µ ` d ` { z ˘ F ρ ´ m µ ´ d ` { s z ˘ | z | ρ ` µ p z {| z |q m ` d ` C ρ, m ´ µ, ´ d F ρ ` m ´ µ ´ d ` { z ˘ F ρ ´ m ´ µ ` d ` { s z ˘ | z | ρ ´ µ p z {| z |q m ´ d , if m ` d is even, and G ρ, m µ, d p z q “ D ρ, m µ, d F ρ ` m µ ` d ` { z ˘ F ρ ´ m µ ´ d ` { s z ˘ | z | ρ ` µ p z {| z |q m ` d ` D ρ, m ´ µ, ´ d F ρ ` m ´ µ ´ d ` { z ˘ F ρ ´ m ´ µ ` d ` { s z ˘ | z | ρ ´ µ p z {| z |q m ´ d , if m ` d is odd, of which the former is valid when | Re µ | ă Re ρ ă and the latter when | Re µ | ´ ă Re ρ ă ( we need to take the limit in the identities when either µ “ d “ or µ “ | d | “ ) . Moreover, for | z | ą , the former identity is valid when | Re µ | ă Re ρ ă and the latter when | Re µ | ´ ă Re ρ ă . C orollary A.2.
Set ∆ ρµ, d “ cos p πρ q sin p π p ρ ` µ qq sin p π p ρ ´ µ qq ¨ , if d is even , i tan p πµ q , if d is odd , Λ ρµ, d “ cos p πρ q cos p π p ρ ` µ qq cos p π p ρ ´ µ qq ¨ cot p πµ q , if d is even , i , if d is odd , and Γ λ ν “ Γ p { ´ λ q Γ p λ ` ν q Γ p λ ´ ν q . We have F ρ, m µ, d p q “ ` { π ˘ p π q ´ ρ ∆ ρµ, d Γ ρ ` m µ ` d Γ ρ ´ m µ ´ d for | Re µ | ă Re ρ ă and m ` d even, and G ρ, m µ, d p q “ ` { π ˘ p π q ´ ρ Λ ρµ, d Γ ρ ` m µ ` d Γ ρ ´ m µ ´ d for | Re µ | ´ ă Re ρ ă and m ` d odd.In particular, for | Re µ | ă Re ρ ă and half-integer d, we have ż π ż J µ, d ` xe i φ ˘ e p´ x cos φ q x ρ ´ dxd φ “ ` { π ˘ p π q ´ ρ cos p πρ q Γ p { ´ ρ q ¨ cos p π p ρ ` µ qq cos p π p ρ ´ µ qq Γ p ρ ` µ ` d q Γ p ρ ` µ ´ d q Γ p ρ ´ µ ` d q Γ p ρ ´ µ ´ d q , which is an analogue of the formula in Corollary . References [Alt1] S. A. Altu˘g. Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of specialrepresentations.
Compos. Math. , 151(10):1791–1820, 2015.[Alt2] S. A. Altu˘g. Beyond endoscopy via the trace formula, II: Asymptotic expansions of Fourier transformsand bounds towards the Ramanujan conjecture.
Amer. J. Math. , 139(4):863–913, 2017.[Alt3] S. A. Altu˘g. Beyond endoscopy via the trace formula, III: The standard representation.
J. Inst. Math.Jussieu , 2018. doi:10.1017 / S1474748018000427.[BM1] E. M. Baruch and Z. Mao. Bessel identities in the Waldspurger correspondence over a p -adic field. Amer.J. Math. , 125(2):225–288, 2003.[BM2] E. M. Baruch and Z. Mao. Bessel identities in the Waldspurger correspondence over the real numbers.
Israel J. Math. , 145:1–81, 2005.
ESSEL FUNCTIONS AND BEYOND ENDOSCOPY 25 [BM3] E. M. Baruch and Z. Mao. Central value of automorphic L -functions. Geom. Funct. Anal. , 17(2):333–384, 2007.[BM4] R. W. Bruggeman and R. J. Miatello. Sum formula for SL over a number field and Selberg type estimatefor exceptional eigenvalues. Geom. Funct. Anal. , 8(4):627–655, 1998.[BM5] R. W. Bruggeman and Y. Motohashi. Sum formula for Kloosterman sums and fourth moment of theDedekind zeta-function over the Gaussian number field.
Funct. Approx. Comment. Math. , 31:23–92,2003.[CQ1] J. Chai and Z. Qi. Bessel identities in the Waldspurger correspondence over the complex numbers. arXiv:1802.01229, to appear in Israel J. Math. , 2018.[CQ2] J. Chai and Z. Qi. On the Waldspurger formula and the metaplectic Ramanujan conjecture over numberfields.
J. Funct. Anal. , 2019. doi:10.1016 / j.jfa.2019.05.013.[FLN] E. Frenkel, R. Langlands, and B. C. Ngˆo. Formule des traces et fonctorialit´e: le d´ebut d’un programme. Ann. Sci. Math. Qu´ebec , 34(2):199–243, 2010.[GR] I. S. Gradshteyn and I. M. Ryzhik.
Table of Integrals, Series, and Products . Elsevier / Academic Press,Amsterdam, Seventh edition, 2007.[Her1] P. E. Herman. Beyond endoscopy for the Rankin-Selberg L -function. J. Number Theory , 131(9):1691–1722, 2011.[Her2] P. E. Herman. The functional equation and beyond endoscopy.
Pacific J. Math. , 260(2):497–513, 2012.[Her3] P. E. Herman. Quadratic base change and the analytic continuation of the Asai L -function: a new traceformula approach. Amer. J. Math. , 138(6):1669–1729, 2016.[Lan] R. P. Langlands. Beyond endoscopy. In
Contributions to automorphic forms, geometry, and numbertheory , pages 611–697. Johns Hopkins Univ. Press, Baltimore, MD, 2004.[LL] J.-P. Labesse and R. P. Langlands. L -indistinguishability for SL p q . Canad. J. Math. , 31(4):726–785,1979.[MOS] W. Magnus, F. Oberhettinger, and R. P. Soni.
Formulas and Theorems for the Special Functions of Math-ematical Physics . Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band52. Springer-Verlag New York, Inc., New York, 1966.[MW] R. Miatello and N. R. Wallach. Kuznetsov formulas for real rank one groups.
J. Funct. Anal. , 93(1):171–206, 1990.[Olv] F. W. J. Olver.
Asymptotics and Special Functions . Academic Press, New York-London, 1974. ComputerScience and Applied Mathematics.[Qi1] Z. Qi. Theory of fundamental Bessel functions of high rank. arXiv:1612.03553, to appear in Mem. Amer.Math. Soc. , 2016.[Qi2] Z. Qi. On the Fourier transform of Bessel functions over complex numbers—I: the spherical case.
Monatsh. Math. , 186(3):471–479, 2018.[Qi3] Z. Qi. On the Fourier transform of Bessel functions over complex numbers—II: the general case.
Trans.Amer. Math. Soc. , 372:2829–2854, 2019.[Qi4] Z. Qi. Subconvexity for twisted L -functions on GL over the Gaussian number field. Trans. Amer. Math.Soc. , 2019. doi:10.1090 / tran / Automorphic repre-sentations and L-functions , volume 22 of
Tata Inst. Fundam. Res. Stud. Math. , pages 521–590. Tata Inst.Fund. Res., Mumbai, 2013.[Sak2] Y. Sakellaridis. Beyond endoscopy for the relative trace formula II: Global theory.
J. Inst. Math. Jussieu ,18(2):347–447, 2019.[Sar] P. Sarnak. Comments on Langlands’ lecture “Endoscopy and Beyond”. 2001.http: // publications.ias.edu / sites / default / files / SarnakLectureNotes-1.pdf.[Ven1] A. Venkatesh.
Limiting Forms of the Trace Formula . Ph.D. Thesis. Princeton University, 2002.[Ven2] A. Venkatesh. “Beyond Endoscopy” and special forms on GL p q . J. Reine Angew. Math. , 577:23–80,2004.[Wat] G. N. Watson.
A Treatise on the Theory of Bessel Functions . Cambridge University Press, Cambridge,England; The Macmillan Company, New York, 1944. [Whi] P.-J. White. The base change L -function for modular forms and Beyond Endoscopy. J. Number Theory ,140:13–37, 2014.S chool of M athematical S ciences , Z hejiang U niversity , H angzhou , 310027, C hina E-mail address ::