Simple Fourier Trace Formulas of Cubic Level and Applications

Home , Kuznetsov trace formula

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GL P 2 fcubic of (2) L -values. k n level and Theorem 1.1. Let n1 and n2 be two positive integers with (n1n2,N) = 1. We have 2 λπ(n1)λπ(n2) (2k 1)N ϕ(N) 2 =δ(n1,n2) − 2 3 Lfin(1, π, sym ) 2π π∈AX(2k,N ) k ( 1) (2k 1) A (c) 4π√n1n2 + − − N J S(n ,n ; N 2c) π c 2k−1 N 2c 1 2 Xc≥1 and 3 λ (n )λ (n ) (2k 1)N 3/2 S(N n ,n ; c) 4π√n n ǫ(π) π 1 π 2 = 1 2 J 1 2 , 2 − 2k−1 3/2 Lfin(1, π, sym ) π c N c π∈A(2k,N 3) c 1 X (c,NX≥)=1 where δ(n1,n2) is the diagonal symbol of Kronecker, ϕ(N) is Euler’s totient function, S(n1,n2; c) is the classical Kloosterman sum, and AN (c)= p|N Ap(c) with Q 1, if p ∤ c, A (c)= − p p 1, if p c. ( − | The first result in Theorem 1.1 is Petersson’s formula over newforms, and the second one is the formula twisted by the root number. Petersson’s formula over newforms was first established by Iwaniec- Luo-Sarnak [IwLuSa2000] when the level is square-free, and was generalized by Rouymi [Rou2011] to the case of prime’s power level. Recently, Nelson [Nel2017, Theorem 4] has showed the existence of the local test function which gives a standard projection to the space of local newvectors for a given level, and then express Petersson’s formula over newforms in terms of averages over all forms of some levels. Different from Nelson’s method, we establish Petersson’s formulas for given cuspidal parameters at the ramified places firstly, and then deduce Theorem 1.1 by summing over cuspidal parameters. We refer to Proposition 3.2 for Petersson’s formula for the given cuspidal parameters. One of the important applications of Petersson’s formula for newforms is to investigate non-vanishing of automorphic L-functions and there are many advances in the past two decades (for example, see [Du1995], [KoMi1999], [IwSa2000], [Kh2010], [Rou2012], [Luo2015] and [BaFr2018]). For holomorphic cusp newforms of square-free level, Iwaniec and Sarnak [IwSa2000] proved that for any ǫ > 0, any square-free integer N with ϕ(N) N as N , ∼ →∞ 1 1 ǫ 1. ≥ 4 − π (2k,N) π∈F(2k,N) L ∈F(1X/2,π)=0 X fin 6 Here new(2k, N) is the Hecke basis of newforms of weight 2k, level N and of trivial nebentypus. In fact,F the constant 1/4 is a natural barrier. Iwaniec and Sarnak [IwSa2000] had proved that any constant bigger than 1/4 with some lower bound on Lfin(1/2,f) would imply that there are no Landau- Siegel zeros for Dirichlet L-functions of real primitive characters. In this paper, we combine Theorem 1.1 and the method in [BaFr2016, BaFr2018] to give the following result. Theorem 1.2. Let the notation be as in Theorem 1.1. For π (2k, N 3), we have that as N , ∈A →∞ 1 1 1 2 ǫ 2 . Lfin(1, π, sym ) ≥ 4 − Lfin(1, π, sym ) π (2k,N3) π∈A(2k,N 3) L ∈A(1X/2,π)=0 X fin 6 2 Remark 1. If we apply Proposition 3.1, the Petersson formula for the given cuspidal parameters, we can prove that for some M with 1 M N and (M,N) = 1, similar result also holds for the subsets (2k, N 3, M) (see (2.3) for the notation).≤ ≤ A Next, we turn to the case of cuspidal automorphic representations associated to Hecke-Maass cusp newforms of cubic level and derive some Kuznetsov’s formulas.

Theorem 1.3. Let h(z) be an even function such that h(z) is holomorphic in the region (z) < A in which it satisfies |ℑ | h(z) (1 + z )−B ≪ | | for some positive constant A and sufficiently large B. We have ∞ λπ(n1)λπ(n2) 2 1 h(tπ) 2 = δ(n1,n2)N ϕ(N) 2 h(t) tanh(πt)tdt 3 Lfin(1, π, sym ) 2π −∞ π∈AX(0,N ) Z +∞ AN (c) 2 h(t)t 4π√n1n2 +i S(n1,n2; N c) J2it 2 dt, c −∞ cosh(πt) N c Xc≥1 Z where it is the set of Langlands parameters of π, and the other notation is as in Theorem 1.1. {± π} Note that each π (0,N 3) is identified to a Hecke-Maass cusp newform of level N 3 and trivial ∈ A 1 2 nebentypus with Laplacian eigenvalue 4 + tπ. Theorem 1.3 is the Kuznetsov’s formula over newforms of level N 3, which is decued by summing over the Kunzetsov’s formula for given cuspidal parameters in Proposition 3.2. As an application of Theorem 1.3, we have the following weighted Weyl’s law. Theorem 1.4. We have 2 1 2 T T 2 = N ϕ(N) 2 + ON . 3 Lfin(1, π, sym ) 2π log T π ∈ A(0,N ) 0

The weighted Weyl’s law follows from a truncated Kuznetsov’s formula for the given cuspidal parameters in Proposition 5.1. By Theorem 1.4, for given N, we have that the density of Hecke-Maass newforms in the space of Maass cusp forms of level N 3 with trivial nebentypus is ϕ2(N) . N 2 This paper is arranged as follows. In Section 2, we introduce our notation and recall simple supercuspidal representations over non-Archimedean places. After that, we specify our choices of test functions over each local place and recall the local Whittaker newforms. Then we compute the geometric sides and spectral sides of the relative trace formula respectively, and establish the Fourier formulas for the given cuspidal parameters in Propositions 3.1 and 3.2. In Section 4, we apply Theorem 1.1 to obtain the asymptotic formulas for the first and second moments, and then prove our non-vanishing results in Theorem 1.2. The weighted Wely law in Theorem 1.4 follows from the truncated Kuznetsov’s trace formula in Proposition 5.1, which is discussed in Section 5. For the self-containess of this paper, we give a detailed computation for the global matrix coefficient of cuspidal automorphic representations in Appendix A. 3 2. Notation and Preliminaries Throughout this article we use the following notation. Let Q be the field of rational numbers. For a place v of Q, let Q be the local complete field of Q with respect to the valuation . If v = p is a v | · |v non-Archimedean place, we denote by Zp the ring of integers of Qp, pZp the maximal ideal of Zp, vp the discrete valuation and kQp the residue field. Let A = AQ be the Adele ring of Q. Let G = GL2 be the general linear algebraic group defined over Q, Z be the center of G and G = G/Z. Denote Gv = GL2(Qv). If v = p is a non-Archimedean place, we denote Kp = GL2(Zp) and a b K (n)= K : c, d 1 pnZ . (2.1) p c d ∈ p − ∈ p

2.1. Haar measures. We fix a non-trivial additive character ψ = v ψv of A/Q with e2πix, if v = , Q ψv(x)= ∞ (2.2) ( e−2πirp(x), if v = p< , ∞ where r (x) Q is the p-principle part of x so that x r (x)+ Z . Let dx be the additive Haar p ∈ ∈ p p v measure on Qv which is self-dual with respect to ψv and let dx d×x = L (1, 1 ) v v v Qv x | v|v × be the multiplicative Haar measure on Qv . Then over Qp one has × × Vol(Zp, dxp) = Vol(Zp , d xp) = 1. × × × Let dx = v dxv and d x = v d xv be measures on A and A , respectively. Over Gv, by the Iwasawa decomposition, for gv Gv, Q Q ∈ z 1 x y g = v v v κ v z 1 1 v v where z ,y Q×, x Q and κ K with K = SO(2) and K = GL (Z ). The measure dg on v v ∈ v v ∈ v v ∈ v ∞ p 2 p v Gv is defined by d×y dg = d×z dx v dκ . v v v y v | v|v In such case, we have the measure on Kv normalized by Vol(Kv, dκv) = 1. Moreover, we give Gv = G /Z the quotient measure by Z Q×. v v v ≃ v Let dg = v dgv be the measure on G(A). Similarly we define the quotient measure on G(A) by Z(A) A×. ≃ Q 2.2. Cuspidal automorphic representations. For k > 1 an integer and N a square-free number, let (2k, N 3) be the set of cuspidal automorphic representations of G(A) which are holomorphic of weightA 2k and level N 3. Each π = π (2k, N 3) satisfies the following conditions. ⊗v v ∈A For v = , π = π is a discrete series representation of G of weight 2k. • ∞ ∞ 2k ∞ For v = p< with p ∤ N, πp is an unramified representation of Gp. • ∞ 4 3 Kp(3) For v = p with p N, π has conductor p , i.e. dim πp = 1 where K (n) is defined in • | p p (2.1). By the result in [KnLi2015], πp = πmp,ζp is a simple supercuspidal representation of Gp, characterized by (mp,ζp) with 1 mp p 1 and ζp 1 . Here ζp is the local root number and the classification depends on≤ the choice≤ − of the local∈ {± addi} tive character −1 ψ˜p(x) := ψp(p x). Let (0,N 3) be the set of cuspidal automorphic representations of G(A) which are level N 3, un- ramifiedA at v = and v = p with p ∤ N. For π = π (0,N 3), π is an irreducible unramified ∞ ⊗v v ∈A ∞ unitary infinite dimensional representation of G∞, which can be realized as the normalized induced representation

G(R) π(ǫπ, itπ) = IndB(R)χǫπ,itπ , where B is the standard parabolic subgroup of G and χǫπ,itπ is a character of B(R) given by

a b a itπ a b χ = sgn(a)ǫπ sgn(d)ǫπ , B(R). ǫπ,itπ 0 d d 0 d ∈ Here ǫ 0, 1 and it is the set of Langlands parameters of π such that π ∈ { } {± π} either tπ R, in which case π∞ = π(ǫπ, itπ) is a principal series, • or t iR∈with 0 < t < 1 , in which case π = π(ǫ , it ) is a complementary series. • π ∈ | π| 2 ∞ π π For given 1 M N with (M,N) = 1, let ≤ ≤ m = (mp)p|N be a tuple of cuspidal parameters with m M mod p for each p N. Let ( ,N 3, M) be the set of p ≡ | A ∗ cuspidal automorphic representations in ( ,N 3) whose local components are characterized by m, A ∗ ( ,N 3, M)= π ( ,N 3) : π = π , ζ 1 , p N . (2.3) A ∗ { ∈A ∗ p mp,ζp p ∈ {± } ∀ | } One has ( ,N 3)= ( ,N 3, M). (2.4) A ∗ A ∗ 1 M N (M,N≤[≤)=1

2.3. The choice of the test function. Let L1(G(A),Z(A)) be the L1-space of Z(A) invariant functions, which are absolutely integrable over G(A). For f L1(G(A),Z(A)), let R(f) act on ∈ φ L2(G(Q) G(A)) by ∈ \

R(f)φ(x)= Kf (x,y)φ(y)dy, ZG(Q)\G(A) where −1 Kf (x,y)= f(x γy) γ∈XG(Q) is the automorphic kernel function. We choose f = v fv so that R(f) give simple trace formulas on (2k, N 3, M) and (0,N 3, M), respectively. Such f will be chosen in the following subsections. A A 5 Q 2.3.1. Non-Archimedean places. For v = p with p ∤ N, we choose fp = 1ZpKp , which is the characteristic function of ZpKp.

For all v = p with p N, πp = πmp,ζp with mp M mod p and ζp 1 . We choose the local test functions as | ≡ ∈ {± }

f = d π (g)u , u , p,M πmp,ζp h mp,ζp πmp,ζp πmp,ζp i ζp∈{±X1} f˜ = ζ d π (g)u , u , p,M p πmp,ζp h mp,ζp πmp,ζp πmp,ζp i ζp∈{±X1}

where dπmp ,ζp is the formal degree of πmp,ζp , and uπmp ,ζp is a unit new vector in πmp,ζp . We have the following explicit formulas (see [KnLi2015, Theorem 7.1]).

Proposition 2.1. For p N, f and f˜ vanish outside the sets | p,M p,M Z× p−1Z Z p−2Z× Z p p ,Z p p p p2Z Z× p pZ× Z p p p p respectively, and, for z Z , ∈ p −1 × −1 ˜ mpc −1 a p b Zp p Zp ψ bℓ + ℓ , if g = 2 2 , a p c d ∈ p Zp 1+ pZp fp,M (zg) = (p + 1)  ℓ∈k× XQp   0, otherwise, −2 −2 ×  ˜ c mpb −1 c p d Zp p Zp  ψ ℓ + ℓ , if g = × , ˜ a d pa b ∈ pZp Zp fp,M (zg) = (p + 1)  ℓ∈k× XQp   0, otherwise.

−1  Here ψ˜(x)= ψp(p x) with ψp in (2.2).

2.3.2. The Archimedean place - weight 2k case. Assume π∞ = π2k is a discrete series of weight 2k. We choose the test function f∞ = f2k as

f (g)= d π (g)u , u 2k 2kh 2k 2k 2ki where u2k is a unit lowest vector and d2k is the formal degree of π2k. Such test function has been explicitly calculated in [KnLi2006b, Propositions 14.5] (or see [RaRo2005]). We list the result in the following proposition.

Proposition 2.2. The test function f2k is 2k 1 det(g)k(2i)2k − , if det(g) > 0, f (g)= 4π ( b + c + (a + d)i)2k 2k  −  0, if g GL(2, R)−, ∈ a b  where g = c d . Moreover,f2k is integrable over G∞ if k > 1. 6 2.3.3. The Archimedean place - weight 0 case. In this case, π∞ = π(ǫπ, itπ) and we choose the test function f = f C∞(GL (R)+,Z K ), the space of smooth functions on GL (R)+, compactly ∞ 0 ∈ c 2 ∞ ∞ 2 supported modulo Z∞ and bi-Z∞K∞-invariant. Let φ0 be a non-zero vector of weight 0 in π(ǫπ, itπ). We have the following result (see [KnLi2013, Proposition 3.6]). Proposition 2.3. For f C∞(GL (R)+,Z K ), the action π (f ) on φ is a scalar given by 0 ∈ c 2 ∞ ∞ ǫπ,itπ 0 0 ∞ ∞ 1 x y1/2 dy (f )(it ) := y−1/2 f dx yitπ , S 0 π 0 1 y−1/2 y Z0 Z−∞ where (f ) is called the spherical transform of f . Moreover, the spherical transform deﬁnes a map S 0 0 S : C∞(GL (R)+,Z K ) PW ∞(C)even, f (f ) S c 2 ∞ ∞ → 0 7→ S 0 which is an isomorphism to the Paley-Wiener space of even functions. 2.3.4. The global test functions. For the local test functions as above, we let

fM = fv fv,M , f˜M = fv f˜v,M . vY∤N vY|N vY∤N vY|N

Then R(fM ) and R(f˜M ) are of the trace class and φ (x)φ (y) π π , if k> 1, φπ, φπ  π∈A(2k,N 3,M) h i K (x,y) =  X (2.5) fM   φπ(x)φπ(y)  (f0)(itπ) , if k = 0, S φπ, φπ π∈A(0,N 3,M) h i  X   φπ(x)φπ(y) ǫfin(π) , if k > 1, φπ, φπ  π∈A(2k,N 3,M) h i K ˜ (x,y) =  X (2.6) fM   φπ(x)φπ(y)  (f0)(itπ)ǫfin(π) , if k = 0. 3 S φπ, φπ  π∈A(0X,N ,M) h i   2 Here ǫ (π)= ζp, and φπ= φ∞ φp L (G(Q) G(A)), where φp is a local new vector if fin p|N × p<∞ ∈ π \ p N, φ is unramified if p ∤ N, and φ is a lowest weight vector in π . | p Q ∞Q ∞ For φπ as above, by the strong approximate theorem (see [Bu1998, Theorem 3.3.1]), φπ is identified to a holomorphic Hecke cusp newform of weight 2k and level N 3 with trivial nebentypus if π (2k, N 3, M), and to a Hecke-Maass cusp newform of level N 3 with the eigenvalue 1 + t2 of the∈ A 4 π Laplace operator and with trivial nebentypus if π (0,N 3, M). Moreover, one has ∈A α φ (g)= W g , π φ 1 αX∈Q× where 1 x W (g)= φ g ψ(x)dx φ π 1 ZA/Q 7 is the Jacquet-Whittaker function of φπ associated to ψ. We choose φπ such that W = W W , φ ∞ × p p<∞ Y where Wp are local Whittaker newforms and W∞ is a lowest weight Whittaker function.

2.4. Local Whittaker newforms. The local Whittaker newform for πv of Gv is a function in the Whittaker model whose Mellin transform equals the local L-function. We choose Wp and W∞ in the following subsections.

2.4.1. Non-Archimedean places. Let (πp, ψp) be the Whittaker model of πp. Let cp be the smallest non-negative integer such that W dim (π , ψ )Kp(cp) = 1, W p p where Kp(n) is deﬁned in (2.1). A Whittaker newform W is the function in (π , ψ )Kp(cp) so that the zeta integral p W p p 1 a s− 2 × Z (s,W ,g,ψ )= W g a p d a p p p p 1 | | ZQp× satisﬁes

Zp(s,Wp,I2, ψp)= Lp(s,πp).

The values of Wp on diagonals are given in the following proposition (see [Popa, Corollary 1]). Proposition 2.4. Let α , α be a set of complex numbers such that { p,1 p,2} 2 L(s,π )= (1 α p−s)−1. p − p,i Yi=1 a For a Q×, W depends only on a and ∈ p p 1 | |p pn 0, if n< 0, Wp = − n n 1 p 2 λ (p ), otherwise, ( πp where λ (pn)= αl1 αl2 and 00 = 1 in case one or both of α , α is 0. πp l1+l2=n p,1 p,2 1 2

2.4.2. The ArchimedeanP place - weight 2k case. Let (π2k, ψ∞) be the Whittaker model of π2k and let W cos θ sin θ = W (π , ψ ) : R(κ )W = ei2kθW, κ = SO(2) W2k ∈W 2k ∞ θ θ sin θ cos θ ∈ − be the subspace of weight 2k. We choose W2k 2k as in [Zh2004, (4.7)], whose values on diagonals are given by ∈W a 2ake−2πa, if a> 0, W2k = (2.7) 1 0, if a< 0. ( It is also a Whittaker newform in the sense that 2k 1 2k + 1 Z (s,W ,I , ψ )=Γ s + − Γ s + ∞ 2k 2 ∞ R 2 R 2 8 − s where ΓR(s)= π 2 Γ(s/2). 2.4.3. The Archimedean place - weight 0 case. Let (π , ψ ) be the Whittaker model of π = W ∞ ∞ ∞ π(ǫπ, itπ). We choose the Whittaker function Wǫπ,0 of weight 0 as in [Zh2004], whose values on diagonals are given by a W = 2sgn(a)ǫπ a 1/2K (2π a ), (2.8) ǫπ,0 1 | | itπ | | where ∞ 1 − y (t+t 1) u dt K (y)= e 2 − t , (y) > 0 u 2 t ℜ Z0 is the K-Bessel function. In this sense,

ΓR(s + itπ)ΓR(s itπ), if ǫπ = 0, Z (s,W ,I , ψ )= − ∞ ǫπ,0 2 ∞ 0, if ǫ = 1. π With respect to the choice of W = W W as above, we have the following proposition. ∞ × p p Proposition 2.5. One has Q 41−2kπ−(2k+1)Γ(2k), if π (2k, N 3), 2 p ∈A φπ, φπ = 2Lﬁn(1, π, sym ) 1 h i p + 1  , if π (0,N 3). p|N  Y  cosh(πtπ) ∈A A proof for π (2k, N) in the classical language can be found in [IwLuSa2000, Lemma 2.5]. To be self-contained,∈ we A give a proof of Proposition 2.5 in the representation language in Appendix A.

3. The Fourier Trace formulas for given cuspidal parameters For 1 M N with (M,N) = 1, let (2k,N,M) and (0,N,M) be the sets of automorphic ≤ ≤ A A cuspidal representations for P GL2(A) deﬁned in (2.3). Let M˜ be a natural number with 1 M˜ N and satisfy the congruence conditions ≤ ≤ N 3 M˜ M mod p, for each p N. (3.1) ≡− p | In this section, following Knightly-Li [KnLi2006a, KnLi2013], we prove the following results.

Proposition 3.1. For (n1n2,N) = 1, we have 2 λπ(n1)λπ(n2) (2k 1)N 2 =δ(n1,n2) − 2 3 Lfin(1, π, sym ) 2π π∈A(2Xk,N ,M) k ( 1) (2k 1) A (c) 4π√n1n2 + − − N,M J S(n ,n ; N 2c) π c 2k−1 N 2c 1 2 Xc≥1 and 3/2 3 λ (n )λ (n ) (2k 1)N S(N n ,n ; c) 4π√n1n2 ǫ(π) π 1 π 2 = − 1 2 J , L (1, π, sym2) π c 2k−1 3/2 3 fin c 1 N c π∈A(2k,N ,M) ≥ X Mc˜ 2 nXn mod N ≡ 1 2 9 where M e , if p ∤ c A (c)= p p,M  1, if p c.  | Proposition 3.2. Let h(z) be an even function such that h(z) is holomorphic in the region (z) < A in which it satisfies |ℑ | h(z) (1 + z )−B ≪ | | for some positive constant A and sufficiently large positive constant B. One has ∞ λπ(n1)λπ(n2) 2 1 h(tπ) 2 = δ(n1,n2)N 2 h(t) tanh(πt)tdt 3 Lfin(1, π, sym ) 2π −∞ π∈A(0X,N ,M) Z +∞ AN,M (c) 2 h(t)t 4π√n1n2 +i S(n1,n2; N c) J2it 2 dt, c −∞ cosh(πt) N c Xc≥1 Z where AN,M (c) is defined in Proposition 3.1. By (2.4), we can deduce Theorem 1.1 (resp. Theorem 1.3) from Proposition 3.1 (resp. Proposition 3.2) by summing over the cuspidal parameters 1 M N with (M,N) = 1. ≤ ≤ 3.1. The relative trace formula. Let n1 and n2 be two natural numbers with (n1n2,N) = 1. For r r any r> 0, definen ˜1 andn ˜2 to be two Adele elements given by pvp(ni), when v = p, n˜r = for i = 1, 2. i,v r, when v = , ( ∞ For f = fM and f = f˜M , we consider the following integral 1 x n˜r 1 y n˜r J(ñr, n˜r,f) = K 1 , 2 ψ(x)ψ(y)dxdy. 1 2 f 1 1 1 1 ZA/Q ZA/Q r r The spectral decompositions in (2.5) and (2.6) lead to the spectral side of J(ñ1, n˜2,f). For the geometric sides, by the Bruhat decomposition, we have two different types of sets of orbits,

a × a × δa = , a Q and δaw = − , a Q , 1 ∈ 1 ∈ 1 where w = is the non-trivial Weyl’s element. It gives the geometric decomposition as 1 − r r r r r r J(ñ1, n˜2,f)= Jδa (ñ1, n˜2,f)+ Jδaw(ñ1, n˜2,f), aX∈Q× aX∈Q× where (ñr)−1 a ay x n˜r J (ñr, n˜r,f)= f 1 2 ψ(x)ψ(y)dxdy (3.2) δa 1 2 M 1 1− 1 ZA ZA/Q and (ñr)−1 x a xy n˜r J (ñr, n˜r,f)= f 1 2 ψ(x)ψ(y)dxdy (3.3) δaw 1 2 M 1 −1 − −y 1 ZA ZA are orbital integrals of the first type and of the second type, respectively. 10 3.2. The spectral side. By the spectral decompositions (2.5) and (2.6), we have the spectral side of r r ˜ J(ñ1, n˜2,f) for f = fM and f = fM in the following proposition. Proposition 3.3. We have

r r −1 1 J(ñ1, n˜2,fM )= (1 + p ) √n1n2 pY|N 4k−1 2k+1 2 π λπ(n1)λπ(n2) 4π 2 , if f∞ = f2k and r = 1, Γ(2k)e Lfin(1, π, sym )  π∈A(2k,N 3,M)  X ×   λπ(n1)λπ(n2)  2r (f0)(itπ) 2 Kitπ (2πr)Kitπ (2πr) cosh(πtπ), if f∞ = f0. S Lfin(1, π, sym ) π∈A(0,N 3,M)  X  r r ˜  −1 1 J(ñ1, n˜2, fM )=  (1 + p ) √n1n2 pY|N 4k−1 2k+1 2 π λπ(n1)λπ(n2) 4π ǫfin(π) 2 , if f∞ = f2k and r = 1, Γ(2k)e Lfin(1, π, sym )  π∈A(2k,N 3,M)  X ×  λ (n )λ (n )  π 1 π 2  2r (f0)(itπ)ǫfin(π) 2 Kitπ (2πr)Kitπ (2πr) cosh(πtπ), if f∞ = f0. S Lfin(1, π, sym ) π∈A(0,N 3,M)  X  Proof. Consider the case f = f f . We have  M 2k × p p,M Q r r n˜1 n˜2 Wφ Wφ r r 1 1 J(ñ1, n˜2,fM )= , 3 φπ, φπ π∈A(2Xk,N ,M) h i where W = W W . By Proposition 2.4 and (2.7), φ 2k × p<∞ p r k −2πr Q n˜i 2r e Wφ = λπ(ni). 1 √n i Note that λπ(n) is real. By Proposition 2.5 we obtain 4k−1 2k+1 −1 2k 2 π p|N 1+ p r λ (n )λ (n ) J(ñr, n˜r,f )= π 1 π 2 . 1 2 M Γ(2k) n n e4πr L (1, π, sym2) Q√ 1 2 3 fin π∈A(2Xk,N ,M) Consider the case f = f f , where f C∞(GL(2, R)+,Z K ). By (2.5) we have M 0 × p<∞ p,M 0 ∈ c ∞ ∞ Q r r n˜1 n˜2 Wφ Wφ r r 1 1 J(ñ1, n˜2,fM )= (f0)(itπ) , 3 S φπ, φπ π∈A(0X,N ,M) h i where W = W W . By (2.8) and Proposition 2.4, φ ǫπ,0 × p<∞ p r 1/2 Q n˜i 2r Kitπ (2πr) Wφ = λπ(ni). 1 √n i 11 This together with Proposition 2.5 gives

r r −1 1 λπ(n1)λπ(n2) J(˜n1, n˜2,fM ) =2r (1 + p ) (f0)(itπ) 2 √n1n2 3 S Lﬁn(1, π, sym ) pY|N π∈A(0X,N ,M) K (2πr)K (2πr) cosh(πt ). × itπ itπ π

˜ r r 3.3. Orbital integrals of the ﬁrst type. For f = fM and f = fM , the orbital integral Jδa (˜n1, n˜2,f) in (3.2) is

n˜r −1 a x n˜r J (ñr, n˜r,f)= f 1 2 ψ(x)dx ψ((a 1)y)dy. δa 1 2 1 1 1 − ZA ! ZA/Q It vanishes except for the case a = 1, in which we have r r JI2 (ñ1, n˜2,f)= JI2,v(ñ1,v, n˜2,v,fv), v Y where −1 n˜r 1 x n˜r J (ñr , n˜r ,f )= f 1,v 2,v ψ (x)dx. I2,v 1,v 2,v v v 1 1 1 v ZQv ! We evaluate the local orbital integrals as follows.

For the case v = and f∞ = f2k, we assume r = 1. By [KnLi2006a, Proposition 3.4] we • obtain ∞ (4π)2k−1 J (1, 1,f )= e−4π. I2,∞ 2k (2k 2)! − For the case v = and f = f , • ∞ ∞ 0 1 r−1x J (r, r, f )= f e2πixdx. I2,∞ 0 0 1 ZR For v = p with p N, by Proposition 2.1, • |

JI2,p(1, 1,fp,M ) = (p + 1)(p 1) (p + 1) ψp(x)dx − − 1 Zp− Zp−Zp = p(p + 1), ˜ JI2,p(1, 1, fp,M ) = 0. For v = p with p ∤ N, f = 1 and • p ZpKp pvp(n2)−vp(n1) p−vp(n1)x J (pvp(n1),pvp(n2), 1 )= 1 ψ (x)dx I2,p ZpKp ZpKp 1 p ZQp

= δ(vp(n1), vp(n2)) dx Z|x|p≤|n1|p = δ(v (n ), v (n )) n . p 1 p 2 | 1|p Therefore we have the following result. 12 Proposition 3.4. We have

r r 1 Jδa (˜n1, n˜2,fM ) = δ(n1,n2) N (p + 1) √n1n2 aX∈Q× pY|N (4π)2k−1 e−4π, if f = f and r = 1, (2k 2)! ∞ 2k  − ×  1 r−1x  f e2πixdx, if f = f ,  0 1 ∞ 0 ZR and   r r ˜ Jδa (˜n1, n˜2, fM ) = 0. aX∈Q×

3.4. Orbital integrals of the second type. For Jδaw in (3.3), it splits into product of local orbital integrals Jδaw = v Jδaw,v, where r −1 r r r Q n˜1,v x a xy n˜2,v Jδ w,v(˜n , n˜ ,fv)= fv − − − ψv(x)ψv(y)dxdy. a 1,v 2,v 1 1 y 1 ZQv ZQv !

We compute Jδaw,v as follows.

3.4.1. Archimedean places. For f∞ = f2k, by [KnLi2006a, Proposition 3.6] we have e−4π(4πi)2k √aJ (4π√a), if a> 0, J (1, 1,f )= 2(2k 2)! 2k−1 (3.4) δaw,∞ 2k  −  0, if a< 0, where Jn(z) is the J-Bessel function. For f = f C∞(GL (R)+,Z K ), we have ∞ 0 ∈ c 2 ∞ ∞ x r−1(a + xy) J (r, r; f )= f e2πi(y−x)dxdy (3.5) δaw,∞ 0 0 −r− y ZR ZR which is non-vanishing if and only if a> 0.

3.4.2. Non-Archimedean places p N. For v = p with p N, it is enough to evaluate Jδaw,p(1, 1,fp,M ) ˜ | | and Jδaw,p(1, 1, fp,M ) since (n1n2,N) = 1. We summarize the computation in the following lemma. Lemma 3.5. For p N, | ψ mp p(p + 1)S (1,p4a; p2), if v (a)= 4, p − p p p − 2t t Jδaw,p(1, 1,fp,M ) = p(p+ 1)Sp(1,p a; p ), if vp(a)= 2t, t 3, − ≥ 0, otherwise,  2 3 ˜ p (p + 1), if vp(a)= 3 and p a mp mod p, Jδaw,p(1, 1, fp,M ) = − ≡− (0, otherwise. Here θ x + θ x S (θ ,θ ; pt)= ψ 1 2 . (3.6) p 1 2 p − pt x,x¯∈Z/ptZ xx¯≡1X mod pt 13 Proof. By Proposition 2.1, f −x −a−xy vanishes except that there exists z Q× such that p,M 1 y ∈ p × −1 zx za zxy Zp p Zp − − − 2 × . z zy ∈ p Zp Z p It gives that v (z) 2, v (x)= v (y)= v (z), v (a + xy) 1 t. p ≥ p p − p p ≥− − Without lost of generality, we assume z = pt with t 2 and thus ≥ v (x)= v (y)= t, v (a + xy) 1 t p p − p ≥− − which implies v (a)= 2t. By writting p − a = p−2ta + p−ta , a (Z /ptZ )×, a Z , 0 1 0 ∈ p p 1 ∈ p x = p−tx + x , x (Z /ptZ )×, x Z , 0 1 0 ∈ p p 1 ∈ p y = p−ty + y , y (Z /ptZ )×, y Z , 0 1 0 ∈ p p 1 ∈ p we have p−3 f −x −a−xy = (p + 1) ψ z(a + xy)ℓ m ℓ−1 p,M 1 y p − − x p ℓ∈k× XQp

(a0 + x0y0) 1 −1 = (p + 1) ψp t ℓ 3−t mpℓ . − p − p x0 ℓ∈k× XQp

We evaluate Jδaw,p(1, 1,fp,M ) in the following cases. 1. Case: t = 2 and v (a + xy)= 3. One has v (a + x y ) = 1, i.e. p − p 0 0 0 a + x y pu mod p2 0 0 0 ≡ for some u (Z /pZ )×. Let x Z be the inverse element of x in (Z /p2Z )×, i.e., ∈ p p 0 ∈ p 0 p p xx 1 mod p2. Then 0 ≡ y x¯ (pu a ) mod p2. 0 ≡ 0 − 0 In this case, uℓ m ℓ−1x f −x −a−xy = (p + 1) ψ ψ p 0 p,M 1 y p − p p − p ℓ∈k× XQp and the contribution of this part is −1 ∗ x0 ∗ x0(pu a0) uℓ mpℓ x0 = (p + 1) ψp 2 ψp 2− ψp ψp 2 − p p − p − p x0 mod p u mod p ℓ∈k× X X XQp −1 ∗ x0 + a0x0 ∗ x0mpℓ ∗ u(x0 ℓ) = (p + 1) ψp 2 ψp ψp − 2 − p − p p x0 Xmod p ℓ modX p u Xmod p mp ∗ x0 + a0x0 = (p + 1) 1+ pψp ψp 2 . − p 2 − p x0 modX p 14 2. Case: t = 2 and v (a + xy) 2. One has p ≥− a + x y 0 mod p2 0 0 0 ≡ and the test function is x m f −x −a−xy = (p + 1) ψ 0 p ℓ−1 = (p + 1). p,M 1 y p − p − ℓ∈k× XQp The contribution of this part is

∗ x0 x0a0 = (p + 1) ψp 2 ψp − 2 − 2 − p p x0 modX p 4 ∗ x0 + p ax0 = (p + 1) ψp 2 . − 2 − p x0 modX p 3. Case: t 3 and v (a + xy)= t 1. One has v (a + x y )= t 1, i.e. ≥ p − − p 0 0 0 − a + x y pt−1u mod pt, for some u (Z /pZ )×. 0 0 0 ≡ ∈ p p t × Denote x0 to be the inverse element of x0 in (Zp/p Zp) . Then y x (pt−1u a ) mod pt. 0 ≡ 0 − 0 The test function is uℓ f −x −a−xy = (p + 1) ψ = (p + 1) p,M 1 y p − p − ℓ∈k× XQp and the contribution in this case is ∗ ∗ = (p + 1) ψ ( p−tx ) ψ (p−t pt−1x u x a ) − p − 0 p 0 − 0 0 x mod pt u mod p 0 X X ∗ x0 + x0a0 ∗ ux0 = (p + 1) ψp t ψp − t − p p x0 Xmod p u Xmod p 2t ∗ x0 + p ax0 = (p + 1) ψp t . t − p x0 Xmod p 4. Case: t 3 and v (a + xy) t. One has ≥ p ≥− a + x y 0 mod pt. 0 0 0 ≡ t × Denote x0 to be the inverse element of x0 in (Zp/p Zp) . Then y x a mod pt. 0 ≡− 0 0 The test function is f −x −a−xy = (p + 1)(p 1) p,M 1 y − and the contribution in this case is 2t ∗ x0 + p ax0 = (p + 1)(p 1) ψp t . − t − p x0 Xmod p 15 The result on Jδaw,p(1, 1,fp,M ) follows immediately. ˜ ˜ −x −a−xy Consider Jδaw,p(1, 1, fp,M ). By Proposition 2.1, fp,M 1 y vanishes except that there exists × z Qp such that ∈ −2 × zx za zxy Zp p Zp − − − × . z zy ∈ pZ Zp p It gives that v (z) = 1, v (x) 1, v (y) 1, v (a + xy)= 3 p p ≥− p ≥− p − and thus v (a)= 3. Without lost of generality, we assume z = p. By writing p − × a = p−3a + p−1a , a Z /p2Z , a Z , 0 1 0 ∈ p p 1 ∈ p x = p−1x + x , x Z /pZ , x Z , 0 1 0 ∈ p p 1 ∈ p y = p−1y + y , y Z /pZ , y Z , 0 1 0 ∈ p p 1 ∈ p we have m y f˜ −x −a−xy = (p + 1) ψ xℓ p ℓ−1 p,M 1 y p − − p3(a + xy) ℓ∈k× XQp

x0ℓ mpy0 −1 = (p + 1) ψp ℓ . − p − pa0 ℓ∈k× XQp Thus x y x ℓ m y J (1, 1, f˜ ) = (p + 1) ψ 0 ψ 0 ψ 0 p 0 ℓ−1a δaw,p p,M p − p p p p − p − p 0 x0 mod p y0 mod p ℓ∈k× X X XQp y x (ℓ + 1) = (p + 1) ψ 0 (1 m ℓ−1a ) ψ 0 p p − p 0 p − p ℓ∈k× y0 mod p x0 mod p XQp X X y = (p + 1)p ψ 0 (1 + m a ) p p p 0 y0 Xmod p = p2(p + 1)δ(a m mod p). 0 ≡− p We ﬁnish the proof of this lemma.

r vp(ni) 3.4.3. Non-Archimedean places p ∤ N. In this case, fp = 1ZpKp ,n ˜i,p = p for i = 1, 2 and

vp(n2)−vp(n1) −vp(n1) vp(n1) vp(n2) p x p (a + xy) Jδ w,p(p ,p ,fp)= fp − − ψp(x)ψp(y)dxdy. a pvp(n2) y ZQp ZQp We have the following result.

vp(n1) vp(n2) Lemma 3.6. For v = p with p ∤ N, Jδaw,p(p ,p ,fp) vanishes except that

vp(a)= vp(n1)+ vp(n2) 2t, for some t 0 −16 ≥ in which case

vp(n1) vp(n2) 1 vp(n1) 2t−vp(n1) t Jδaw,p(p ,p ,fp)= Sp(p , ap ; p ). pvp(n1)pvp(n2) t Here Sp(θ1,θ2; p ) is deﬁned in (3.6). Proof. Since f = 1 , J (˜n , n˜ ,f ) is non-zero if and only if there exists z Q× such that p ZpKp δaw,p 1,p 2,p p ∈ p pvp(n2)−vp(n1)x p−vp(n1)(a + xy) g = z − − Kp. (3.7) pvp(n2) y ∈ Let r1 = vp(n1) and r2 = vp(n2). Condition (3.7) is equivalent to the following two conditions: 1. det g = p2vp(z)+r2−r1 ( xy + a + xy) Z×, i.e. − ∈ p v (a) + 2v (z)+ r r = 0, p p 2 − 1 which implies that v (a)+ r r v (a) r + r mod 2, v (z)= p 2 − 1 . (3.8) p ≡ 1 2 p − 2 2. zy,zpr2 , zpr2−r1 x, zp−r1 (a + xy) Z , i.e. − − ∈ p v (z)+ v (y) 0, v (z)+ r 0, p p ≥ p 2 ≥ v (z)+ r r + v (x) 0, v (z) r + v (a + xy) 0. p 2 − 1 p ≥ p − 1 p ≥ These together with (3.8) give that

vp(a)+ r2 r1 vp(y) − , vp(a) r1 + r2, ≥ 2 ≤ (3.9) v (a) r + r v (a)+ r + r v (x) p − 2 1 , v (a + xy) p 2 1 . p ≥ 2 p ≥ 2 By (3.8) and (3.9), a Q× satisﬁes ∈ v (a) r + r , v (a) r + r mod 2. p ≤ 1 2 p ≡ 1 2 So we can assume that v (a)= r + r 2t for t 0 and thus x and y satisfy p 1 2 − ≥ v (x) r t, v (y) r t, p ≥ 1 − p ≥ 2 − v (a + xy) r + r t. (3.10) p ≥ 1 2 − Assume t = 0, i.e. vp(a)= r1 + r2. The condition (3.10) is automatically satisﬁed. In this case,

r1 r2 1 Jδ w,v(p ,p ,fp)= ψv(x)ψv(y)dxdy = . a pr1 pr2 Zvp(x)≥r1 Zvp(y)≥r2 Assume v (a)= r + r 2t for some t> 0. One has p 1 2 − v (a)= r + r 2t, v (xy) r + r 2t, v (a + xy) r + r t. p 1 2 − p ≥ 1 2 − p ≥ 1 2 − The condition v (a + xy) r + r t is satisﬁed if and only if p ≥ 1 2 − vp(x)= r1 t and vp(y)= r2 t. − 17 − So we can assume that a = pr1+r2−2ta + pr1+r2−ta , a (Z /ptZ )×, a Z , 0 1 0 ∈ p p 1 ∈ p x = pr1−tx + pr1 x , x (Z /ptZ )×, x Z , 0 1 0 ∈ p p 1 ∈ p y = pr2−ty + pr2 y , y (Z /ptZ )×, y Z . 0 1 0 ∈ p p 1 ∈ p It gives that v (a + x y ) t, i.e. a + x y 0 mod pt and then p 0 0 0 ≥ 0 0 0 ≡ y x a mod pt, 0 ≡− 0 0 t × where x0 is the inverse element of x0 in (Zp/p Zp) . Therefore,

∗ r1 r2 r1−t r1 r1 Jδaw,v(p ,p ,fp) = ψp(p x0 + p x1)d(p x1) t x1∈Zp x0 Xmod p Z ∗ ψ (pr2−ty + pr2 y )d(pr2 y ) × p 0 1 1 t y1∈Zp y0 modX p Z y x a mod pt 0 ≡− 0 0 r1 r2 1 ∗ p x0 p a0x0 = ψp − − r1 r2 t p p t p x0 Xmod p r1 2t−r1 1 ∗ p x0 + ap x0 = ψp . r1 r2 t p p t − p x0 Xmod p We ﬁnish the proof of this lemma.

r r 3.4.4. The global result. Consider Jδa,w(ñ1, n˜2,fM ) firstly. By formulas (3.4), (3.5) and Lemmas 3.5, 3.6, it is non-vanishing if and only if a Q× satisfies a> 0 and ∈ v (a)= v (n )+ v (n ) v (N 4) 2t p p 1 p 2 − p − p for some t 0 at any place p. By the local-global principle, it is equivalent to that p ≥ n n 1 a = 1 2 for some c N. N 4 c2 ∈ Moreover, we have the following result. Proposition 3.7. One has 1 J (ñr, n˜r,f )= N (p + 1) A (c)S(n ,n ; N 2c) δaw 1 2 M n n N,M 1 2 1 2 c≥1 aX∈Q× pY|N X e−4π(4πi)2k √n n 4π√n n 1 2 J 1 2 , if f = f and r = 1, 2(2k 2)! N 2c 2k−1 N 2c ∞ 2k  − ×  x 1 ( n1n2 + xy)  f r N 4c2 e2πi(y−x)dxdy, if f = f ,  0 −r− y ∞ 0 ZR ZR  where AN,M (c) is defined in Proposition 3.1. 18 n1n2 Proof. Recall that a = N 4c2 . For each p< , the local Kloosterman sums in Lemmas 3.5 and 3.6 can be written as ∞

2 vp(n1) vp(n2) a vp(N c) Sp p ,p 2 ; p pvp(n1)+vp(n2)−2vp(N c) 2 2vp(N c) pvp(n1)x + pvp(n2) n1 n2 p x ∗ 0 pvp(n1) pvp(n2) (N 2c)2 0 = ψp 2  vp(N c)  2 − p x mod pvp(N c) 0 X   2 2  pvp(N c) pvp(N c)  ∗ N 2c n1x0 + N 2c n2x0 = ψp 2 − pvp(N c)  vp(N2c) x0 modXp   2 2 2 N c N c vp(N c) = Sp 2 n1, 2 n2; p . pvp(N c) pvp(N c) !

By the property of the Kloosterman sum and the choice of ψ = v ψv in (2.2), we have

2 2 2 Q N c N c vp(N c) 2 Sp 2 n1, 2 n2; p = S n1,n2; N c . vp(N c) vp(N c) p<∞ p p ! Y The proposition follows immediately.

r r ˜ Next, we consider Jδa,w(˜n1, n˜2, fM ). It is non-vanishing if and only if a> 0 and satisﬁes for v = p with (p, N) = 1, • v (a)= v (n )+ v (n ) 2t for some t 0; p p 1 p 2 − p p ≥ for v = p with p N, v (a)= 3 and p3a M mod p, which is equivalent to • | p − ≡− N 3 N 3a M mod p ≡− p for each p N. | Thus a is of the form a = n1n2 1 for some c N with (c, N)=1 and n n Mc˜ 2 mod N, where M˜ N 3 c2 ∈ 1 2 ≡ is determined by (3.1). This gives the following result. Proposition 3.8. One has

1 3 J (˜nr, n˜r, f˜ )= N 2 (p + 1) S(N n ,n ; c) δaw 1 2 M n n 1 2 1 2 c 1 a∈Q× p|N ≥ X Y Mc˜ 2 nXn mod N ≡ 1 2 e−4π(4πi)2k √n n 4π√n n 1 2 J 1 2 , if f = f and r = 1, 2(2k 2)! 3 2k−1 3 ∞ 2k  − N 2 c N 2 c ×  x 1 ( n1n2 + xy)  f r N 3c2 e2πi(y−x)dxdy, if f = f ,  0 −r− y ∞ 0 ZR ZR  where 1 M˜ N is determined by (3.1). ≤ ≤ 19 n1n2 Proof. Recall that a = N 3c2 . For each p< with p ∤ N, the local Kloosterman sums in Lemma 3.6 can be written as ∞

vp(n1) vp(n2) a vp(c) Sp p ,p ; p pvp(n1)+vp(n2)−2vp(c) 2vp(c) pvp(n1)x + pvp(n2) n1 n2 p x ∗ 0 pvp(n1) pvp(n2) N 3c2 0 = ψp − pvp(c)  vp(c) x0 modXp  pvp(c) pvp(c) n  ∗ 2 c n1x0 + c N 3 x0 = ψp − pvp(c) vp(c) ! x0 modXp

c c n2 vp(c) = Sp n1, ; p . pvp(c) pvp(c) N 3

By the property of the Kloosterman sum and the choice of ψ = v ψv in (2.2), we have Q c c n2 vp(c) 3 3 Sp n1, ; p = S n1,n2N ; c = S N n1,n2; c . vp(c) vp(c) N 3 p<∞ p p Y

3.5. Proofs of the Fourier trace formulas. For f = fM with f∞ = f2k, on taking r = 1, by Propositions 3.3, 3.4 and 3.7, we have

4k−1 2k+1 −1 2 π p|N 1+ p λπ(n1)λπ(n2) e4πΓ(2k)√n n L (1, π, sym2) Q 1 2 3 ﬁn π∈A(2Xk,N ,M) 1 e−4π(4π)2k−1 = δ(n1,n2) N (p + 1) √n1n2 (2k 2)! − pY|N −4π 2k e (4πi) 1 AN,M (c) 4π√n1n2 2 +N (p + 1) 2 J2k−1 2 S n1,n2; N c . 2(2k 2)! √n1n2 N c N c p|N − c≥1 Y X

Similarly, for f = f˜M with f∞ = f2k, we have

4k−1 2k+1 −1 2 π p|N 1+ p λ (n )λ (n ) ǫ (π) π 1 π 2 e4πΓ(2k) n n ﬁn L (1, π, sym2) Q √ 1 2 3 ﬁn π∈A(2Xk,N ,M) −4π 2k 2 e (4πi) 1 4π√n1n2 3 = N (p + 1) J2k−1 3 S N n1,n2; c . 2(2k 2)! n n 2 √ 1 2 c 1 N c p|N − ≥ Y Mc˜ 2 nXn mod N ≡ 1 2 Thus Proposition 3.1 follows immediately. We also have the following pre-Kuznetsov trace formula for f = f with f = f C∞(GL(2, R)+,Z K ). M ∞ 0 ∈ c ∞ ∞

20 Proposition 3.9. Let n1 and n2 be two natural numbers with (n1n2,N) = 1, and let r > 0 be a real number. For f C∞(GL(2, R)+,Z K ), we have 0 ∈ c ∞ ∞ λπ(n1)λπ(n2) 2r (f0)(itπ) 2 Kitπ (2πr)Kitπ (2πr) cosh(πtπ) 3 S Lﬁn(1, π, sym ) π∈A(0X,N ,M) 1 xr−1 = δ(n ,n )N 2 f e2πixdx 1 2 0 1 ZR N 2 −1 n1n2 2 x r ( N 4c2 + xy) 2πi(y−x) + AN,M (c)S(n1,n2,N c) f0 − − e dxdy. √n1n2 R R r y Xc≥1 Z Z × Let h(t)= (f0)(it). To establish the simple Kuznetsov trace formula, we integrate in r over R+ on both sides ofS the pre-Kuznetsov trace formula and deal with the integral transforms. This work has been done by Knightly-Li in [KnLi2013]. To be self-contained, we brieﬂy review their work as follows.

3.5.1. The spectral side. Consider the spectral side of the pre-Kuznetsov trace formula in Proposition × 3.9. Integrating in r on R+, by (A.3) we have ∞ 1 ∞ 1 2 K (2πr)K (2πr)rd×r = K (r)K (r)rd×r = . itπ itπ π itπ −itπ 4 cosh(πt ) Z0 Z0 π It gives

1 λπ(m1)λπ(m2) Spectral side = h(tπ) 2 . (3.11) 4 3 Lﬁn(1, π, sym ) π∈A(0X,N ,M) ∞ + 3.5.2. The diagonal term on the geometric side. Note that f0 Cc (GL(2, R) ,Z∞K∞). By Propo- a b ∈ sition 3.4 in [KnLi2013], for g = GL(2, R)+, there exists a function V C∞([0, )) such c d ∈ ∈ c ∞ that a2 + b2 + c2 + d2 a b V (u)= V 2 = f . ad bc − 0 c d − By [KnLi2013, Proposition 3.7], we have 1 ∞ u V (u)= P− 1 +it(1 + )h(t) tanh(πt)tdt 4π 2 2 Z−∞ and 1 ∞ V (0) = h(t) tanh(πt)tdt, 4π Z−∞ where Ps(z) is the Legendre polynomial (see [Iw2002, (1.43)]). It gives (see [KnLi2013, Page 73]) ∞ ∞ 1 xr−1 ∞ ∞ x2 1 f e2πixdx d×r = V e2πixdxd×r = V (0), 0 1 r2 2 Z0 Z−∞ Z0 Z−∞ and thus ∞ 2 1 Diagonal term = δ(n1,n2)N 2 h(t) tanh(πt)tdt. (3.12) 8π −∞ 21 Z 3.5.3. The non-diagonal term on the geometric side. For the non-diagonal term on the geometric side, one needs to deal with the integral

∞ −1 n1n2 x r ( N 4c2 + xy) 2πi(y−x) × := f0 − − e dxdy d r I r y Z0 ZR ZR which has been discussed in [KnLi2013, Proposition 7.1]. We list the result in the following lemma. Lemma 3.10. One has i √n n ∞ √n n h(t) = 1 2 J 4π 1 2 dt. I 4 N 2c 2it N 2c cosh(πt) Z−∞ By the above lemma, the non-diagonal term on the geometric side is +∞ i AN,M (c) 2 h(t)t 4π√n1n2 Non-Diag = S(n1,n2,N c) J2it 2 dt. (3.13) 4 c −∞ cosh(πt) N c Xc≥1 Z By (3.11), (3.12) and (3.13), we can establish the formula in Prposition 3.2 for those h(iz) PW∞(C)even. The work of Knightly-Li in [KnLi2013, Section 8] indicated that the formula is also∈ valid for the function h(t) which satisﬁes the conditions in Proposition 3.2. This proves Proposition 3.2.

4. Non-vanishing of modular L-values Combining the first and the second moments together with the mollification method is the classical and effective approach to study the non-vanishing problems. Traditionally, the approximate functional equation plays an important role in calculating the first and second moments. Recently, Bykovskii- Frolenkov [ByFr2017] and Balkanova-Frolenkov [BaFr2018] derived a new method and got better understanding of error terms in the asymptotic formulas for the first and second moments. In this section, we combine the simple Petersson trace formula in Theorem 1.1 and the method in [ByFr2017, BaFr2018] to establish the asymptotic formulas for the first and the second moments of modular L-values, and prove Theorem 1.2. To be more precisely, for a natural number m with (m, N) = 1, we let 1 λπ(m)Lfin( 2 + u, π) M1(m, u) = 2 , 3 Lfin(1, π, sym ) π∈AX(2k,N ) 1 2 λπ(m)Lfin( 2 + u, π) M2(m, u) = 2 3 Lfin(1, π, sym ) π∈AX(2k,N ) To serve our purpose, we establish the following asymptotic formulas (2k 1)N 2ϕ(N) M (m, 0) = − + O 3Ω(N)m1/2+ǫ , (4.1) 1 2π2√m k (2k 1)Nϕ2(N) N 3 M (m, 0) = − τ(m) log + 2g (N) + O (m1/2(mN)ǫ). (4.2) 2 2π2√m m k k Here Ω(N) denotes the number of prime factors of N and log p Γ′(k) g (N)= + + γ log(2π). (4.3) k p 1 Γ(k) 0 − Xp|N − 22 These formulas together with the mollification method lead to the non-vanishing result in Theorem 1.2. The proofs of (4.1) and (4.2) follow from [BaFr2016, BaFr2018] with suitable modifications. We give a sketch of the proof of (4.2) in the following section.

4.1. Asymptotic formula for the second moment. Consider M2(m, u). For (u) > 3/4, by Ramanujan-Deligne’s bound and the Hecke relation, we have ℜ

1 1/2+u N N 1 λπ (dn1) λπ(n2) M2(m, u)= 1/2+u d 1/2+u 2 . m (n1n2) 3 Lﬁn(1, π, sym ) Xd|m nX1≥1 nX2≥1 π∈AX(2k,N ) Applying Theorem 1.1 we obtain M (m, u)= (m, u)+ (m, u), 2 D2 N D2 where 2 (2k 1)N ϕ(N) 1 (N) 2(m, u)= − τ(m)ζ (1 + 2u), D 2π2 m1/2+u 1 ( 1)k(2k 1) N N 1 (m, u)= d1/2+u 2 1/2+u − − 1/2+u N D m π (n1n2) nX1≥1 nX2≥1 Xd|m A (c) 4π√dn n N J 1 2 S(dn ,n ; N 2c). × c 2k−1 N 2c 1 2 Xc≥1 Clearly, we have the following proposition.

Proposition 4.1. 2(m, u) has meromorphic continuation for u C except for a simple pole at u = 0. Moreover, D ∈ 1 (m, u)= c + c + h (u), D2 u −1 0,1 1 where h1(u) is an analytic function in u with h1(u) = 0, c−1 and c0,1 are constants given by (2k 1)Nϕ2(N) τ(m) c−1 = − , (4.4) 4π2 m1/2 (2k 1)Nϕ2(N) τ(m) 1 log p c0,1 = − γ0 log m + . 2π2 m1/2  − 2 p 1  Xp|N −  (m, u) is much more complicated and we follow the approach in [BaFr2016]. N D2   Proposition 4.2. 2(m, u) has meromorphic continuation for u C except for the case u = 0 which is a simple pole.N D Moreover, ∈ 1 (m, u)= c + c + h (u)+ 1(m, u) N D2 − −1 u 0,2 2 N D2 where h2(u) is an analytic function in u with h2(0) = 0, c−1 is in (4.4), c0,2 is given by

(2k 1)Nϕ2(N) 4π2 1 p log p Γ′(k) c = − τ(m) γ log log m + + 2 0,2 2π2√m  0 − N 2 − 2 p 1 Γ(k)   Xp|N −  23   and 1(m, u) satisﬁes N D2 1(m, 0) (kmN)ǫk2m1/2. N D2 ≪ Proof. To prove Proposition (4.2), we apply [Roy2001, Lemma A.12] and Mobius inversion to remove the coprime condition (n1n2,N) = 1 and get k 1 ( 1) (2k 1) µ(ℓ) 1/2+u 2(m, u)= − − d 1/2+u 1 +u N D m π ℓ 2 Xℓ|N Xd|m 1 A (c) 4π√dℓn n N J 1 2 S(dℓn ,n ; N 2c). 1/2+u 2k−1 2 1 2 × (n1n2) c N c nX1≥1 nX2≥1 Xc≥1 Next, we recall Mellin-Barnes representation of J-Bessel function (see [GaHoSe2009, (17)] for instance). For any 1 2k<α< 0, − α+i∞ 2k−1+s −s 4π√mn 1 Γ 2 2π√mn J2k−1 = ds. (4.5) N 2c 4πi Γ 2k+1−s N 2c Zα−i∞ 2 Then, for max 1 2k, 1 2 (u) <α< 0, by (4.5), we have { − − ℜ } k 1 ( 1) (2k 1) µ(ℓ) 1 +u 2(m, u)= − − d 2 1/2+u 1 +u N D m π ℓ 2 Xℓ|N Xd|m α+i∞ 2k−1+s 2s 1 Γ 2 N 2k+1−s s N,dℓ(u, s)ds, × 4πi Γ (2π)s(ℓd) 2 T Zα−i∞ 2 where 1 AN (c) 2 N,dℓ(u, s)= S(dℓn1,n2; N c). 1+s +u 1−s T (n1n2) 2 c nX1≥1 nX2≥1 Xc≥1 Following the proofs of [BaFr2016, Lemma 6.5-6.7], we get that (u, s) has analytic continuation TN,dℓ for 3 < (u) < k 1, and, for 1 2k < (s) < 1 2 (u), 4 ℜ − − ℜ − − ℜ Γ2 1−s u s (u, s) = 2N 2−2s−4u 2 − sin π + u P − (u, s)+ P + (u, s) , TN,dℓ (2π)1−s−2u 2 N,dℓ N,dℓ n o where 2 τ a N r 1 1 1 s ℓ AN (c) ± u+ −2 ∓ PN,dℓ(u, s)= 1 s 1 s µ(d/r)r 1 s . − −u u+ − −−u 2u ℓ 2 d 2 1 r N 2 2 c r|d a≥ ± a r c|a X XN2/ℓ ℓ ∓ X Note that AN (c) is a ﬁnite sum except for the case a = 0. We can express (m, u) as c|a c2u N D2 P (m, u)= 0(m, u)+ 1(m, u), N D2 N D2 N D2 where

k 2 2 u 0 ( 1) (2k 1)N 4π µ(ℓ) 2u AN (c) (m, u)= − − µ(d/r)r τ (r) I1(u; k), N D2 2π2m1/2 mN 4 ℓ    c2u  Xℓ|N Xd|m Xr|d Xc≥1 24     and k 2 2 u 1 ( 1) (2k 1)N 4π µ(ℓ) 1 +u (m, u) = − − µ(d/r)r 2 N D2 2π2m1/2 mN 4 ℓ Xℓ|N Xd|m Xr|d

N 2 τ a ℓ + r N 2  AN (c)I2 u, k, a + 1  1 −u ℓr ×  1 r N 2 2  a − a + r c|a  ≥XN2/ℓ ℓ X a=0 6   N 2  τ a r 2 ℓ − N + 1 AN (c)I3 u, k, a 1  . 2 2 −u ℓr − 1+r N c|a a≥ 2 a r  XN /ℓ ℓ − X Here  α+i∞ 2k−1+s 1 Γ 2 2 1 s s I1(u, k) = 2k+1−s Γ − u sin π + u ds, 2πi α−i∞ Γ 2 − 2 Z 2 α+i∞ 2k−1+s 1 Γ 2 2 1 s s s/2 I2(u; k, z+) = 2k+1−s Γ − u sin π + u z+ ds, 2πi α−i∞ Γ 2 − 2 Z 2 α+i∞ Γ 2k−1+s s 1 2 2 1 s 2 I3(u, k, z−) = Γ − u z ds 2πi Γ 2k+1−s 2 − − Zα−i∞ 2 with α and (u) satisfying 1 2k<α< 1 2 (u). By [BaFr2016, Lemma 3.6], we have ℜ − − − ℜ Γ2(k u) I (u, k) =2 cos(π(k u))Γ(2u) − , 1 − Γ2(k + u) cos π(k u) Γ2(k u) 1 I2(u, k, z+) =2 1− − 2F1 k u, k u, 2k, , k− 2 Γ(2k) − − z+ z+ 2 Γ2(k u) 1 I3(u, k, z−)= − 2F1 k u, k u, 2k, − , k− 1 2 Γ(2k) − − z− z− where 2F1(α,β,γ,z) is the Gauss Hypbergeometric function. Note that A (c) 1 N = ζ(2u) 1 . c2u p2u−1 − Xc≥1 pY|N 0 By the above argument, 2(u, s) has meromorphic continuation for u C except for a simple pole at u = 0 and N D ∈ 1 0(m, u)= c + c + h (u), N D2 −u −1 0,2 2 where c−1, c0,2 and h2(u) are in Proposition 4.2. For 1(m, u), it has analytic continuation for u C and N D2 ∈ 2 1 (2k 1)N µ(ℓ) (m, 0) = − V˜ 2 (m) N D2 2π2 ℓ N /ℓ Xℓ|N 25 where 2( 1)k V˜ 2 (m)= − µ(d/r)(V˜ 2 (r)+ V˜ 2 (r)+ V˜ 2 (r)) (4.6) N /ℓ m1/2 N /ℓ,1 N /ℓ,2 N /ℓ,3 Xd|m Xr|d with N 2 τ a r 2 k ℓ − Γ (k) r V˜ 2 (r)=r A (c) F k,k, 2k; , N /ℓ,1 k N 2 1 N−2 N 2 Γ(2k) a r a≥ 1+r a r c|a ℓ ! XN2/ℓ ℓ − X − N 2 τ a + r 2 ˜ k ℓ Γ (k) r VN 2/ℓ,2(r) =( r) k AN (c) 2F1 k,k, 2k, 2 , 2 Γ(2k) N − 1 r N a + r ! − ≤a≤−1 a + r c||a| ℓ N2/ℓX ℓ X N 2 τ a + r 2 k ℓ Γ (k) r V˜ 2 (r) =( r) A (c) F k,k, 2k, . N /ℓ,3 k N 2 1 N 2 − N 2 Γ(2k) a + r a≥1 a + r c|a ℓ ! X ℓ X ˜ For VN 2/ℓ(m) in (4.6), we have

V˜ 2 (m) ϕ(c)V 2 (m) N /ℓ ≪ cN /ℓ Xc|N where VcN 2/ℓ(m) is defined in [BaFr2016, (6.15)]. By [BaFr2016, Theorem 8.4], we have 1/2 1+ǫ ǫ m V˜ 2 (m) k (mN) . N /ℓ ≪ N 2/ℓ The estimation of 1(m, 0) in Proposition 4.2 follows immediately. N D2 4.2. The mollification method. For π (2k, N 3), we choose the standard mollifier ∈A N x λ (m) X(π)= m π , √m mX≤M ∆ where M = N with ∆ the length of the mollifier, and xm are real coefficients defined on square-free numbers m with (m, N) = 1 and satisfies x (τ(m) log N)2. m ≪ Let ωπ be the weight given by (2k 1)N 2ϕ(N) ω−1 = − L (1, π, sym2). (4.7) π 2π2 fin We consider the first and second mollified moments, ˜ h M1 = ωπLfin (1/2,π) X(π), 3 π∈AX(2k,N ) ˜ h 2 2 M2 = ωπLfin (1/2,π) X (π). 3 π∈AX(2k,N ) 26 By (4.1) and (4.2) we have N x M˜ h = m + O N −3+∆+ǫ , 1 m k m≤M X ϕ(N) N N N x x ˜ h dm1 dm2 −3+2∆+ǫ M2 = τ(m1m2)S(m1m2)+ Ok N , N dm1m2 d≤M m ≤ M m ≤ M X X1 d X2 d where N 3 S(m1m2) = log + 2gk(N) m1m2 ∆ 3 with gk(N) in (4.3). Here M = N with 0 < ∆ 2 ǫ. We recall the general principle on the mollification≤ − method in [Ko1998, Rou2012]. By

τ(m1m2)= µ(ℓ)τ(m1/ℓ)τ(m2/ℓ), ℓ|(mX1,m2) we express the mollified second moment as ϕ(N) M˜ h = (Π 2Π 2Π )+ O N −3+2∆+ǫ , 2 N − 0,1 − 1,0 k where N S N N τ(m1)τ(m2) Π = (3 log N + 2gk(N)) (n) xm1nxm2n, m1m2 n≤M m ≤ M m ≤ M X X1 n X2 n N S N N τ(m1)τ(m2) Π0,1 = (n) log(m1)xm1nxm2n, m1m2 n≤M m ≤ M m ≤ M X X1 n X2 n N S1 N N τ(m1)τ(m2) Π1,0 = (n) xm1nxm2n m1m2 n≤M m ≤ M m ≤ M X X1 n X2 n with 1 µ(ℓ) 1 µ(ℓ) S(n)= , S1(n)= log ℓ. n ℓ n ℓ Xℓ|n Xℓ|n To diagonalize the quadratic forms above, we take N τ(m) N τ(m) log m y = x , y1 = x , n m nm n m nm m≤ M m≤ M Xn Xn and thus N µ µ(m) x = ∗ y , n m mn m≤ M Xn where µ µ(m)= ab=m µ(a)µ(b). ∗ 27 P ˜ h To optimize the value of Π with respect to the linear form in M1 , we take n µ(n), if (n, N) = 1,n M, y = ϕ(n) ≤ (4.8) n   0, otherwize. Then, following [Rou2012, Proposition 7] with careful computations to remove the dependence of N  in the implied constant, we have the following. Proposition 4.3. Let y be as in (4.8). For M = N ∆ with 0 < ∆ 3 ǫ we have n ≤ 2 − ϕ(N) M˜ h = ∆ log N + O (log log N)6 , 1 N k,∆ ϕ(N) 2 M˜ h = (3∆ + 2∆2) (log N)2 + O log N(log log N)6 . 2 N k,∆ By Cauchy’s inequality, ˜ h 2 (M1 ) ∆ ωπ ǫ ≥ M˜ h ≥ 3+2∆ − π (2k,N3) 2 L ∈A(1X/2,π)=0 fin 6 as N tends to infinity. Recall ω in (4.7). On taking ∆ = 3 ǫ, we prove Theorem 1.2. π 2 − 5. The truncated Kunzetsov trace formula In this section, following Li [Li2011], we establish a truncated Kuznetsov trace formula in the following proposition. Proposition 5.1. Let the notation be as in Proposition 3.2. We have 2 T λπ(m1)λπ(m2) N θ+ǫ 2 =δ(m1,m2) 2 tanh(πt)tdt + ON (m1m2) 3 Lfin(1, π, sym ) 2π −T π ∈ A(0,N , M) Z 0

ǫ 1 +ǫ −2 + O (NT ) (m1m2) 4 min T, √m1m2N N 2T + O (m m )θ+ǫ . 1 2 log(N 2T ) where θ = 7/64 by Kim-Sarnak’s bound. θ+ǫ The error term ON ((m1m2) ) in Proposition 5.1 comes from exceptional eigenvalues, which vanishes if the Selberg smallest eigenvalue conjecture is true. The weighted Weyl’s law in Theorem 1.4 follows immediately by letting m1 = m2 = 1 and applying the fact T tanh(πt)tdt = T 2 + O(1). Z−T To prove Proposition 5.1, following [DuGu1975, Li2011], we fix h to be an even function such that its Fourier transform h satisfies supp(h) ( 1, 1) and h(0) = 1. b ⊂ − Such test function satisfies the condition in Proposition 3.2 and we use the bound b b h(z) (1 + z 4)−1. ≪ 28 | | 5.1. The weighted local estimation. Assume that µ> 0 and L 1. We let ≫ h (z) := h(L(µ + z)) + h(L(µ z)). µ,L − Applying Proposition 3.2 one has

λπ(m1)λπ(m2) hµ,L(tπ) 2 = δ(m1,m2) (µ,L)+ (m1,m2; µ,L), 3 Lﬁn(1, π, sym ) D N D π∈A(0X,N ,M) where N 2 ∞ (µ,L)= h (t) tanh(πt)tdt, (5.1) D 2π2 µ,L Z−∞ +∞ AN,M (c) 2 hµ,L(t)t 4π√m1m2 (m1,m2; µ,L)= i S(m1,m2; N c) J2it 2 dt. (5.2) N D c −∞ cosh(πt) N c Xc≥1 Z For the diagonal term, we have N 2 ∞ µ 1 (µ,L)= h(L(µ t)) tanh(πt)tdt + N 2. D π2 − ≪ L L2 Z−∞ For (m ,m ; µ,L), by the asymptotic expansion of J (2x), we have (see [Li2011, (2.4)]) N D 1 2 2it xe 2i ∞ h (t)t µ1/2 xe 2iµ log J (x) µ,L dt = (2i)1/2 h − 4µ + lower order terms. π 2it cosh πt πL ℑ 4πµ πL Z−∞ ( !) Note that supp(h) ( 1, 1). Thus b ⊂ − µ1/2 A (c) b (m ,m ; µ,L) | N,M | S(m,n; N 2c) N D 1 2 ≪ L c | | 1≤c< π√mn eπL+1 NX2µ which is negligible if π√mn µ eπL+1. (5.3) ≥ N 2 Moreover, by Weil’s bound of the Kloosterman sum, 1/2 1/2 µ 1+2ǫ (m,n,c) (m1,m2; µ,L) N N D ≪ L c1/2−ǫ 1≤c< π√mn eπL+1 NX2µ

1 +ǫ 1 πL +ǫL (mn) 4 e 2 . (5.4) ≪ µǫL i i Therefore, as in [DuGu1975, Lemma 2.3], we can assume h(z) 0 for z R [ 2 , 2 ] and h(x) > 0 for x ( 1, 1), and establish the weighted local estimation in the≥ following∈ lemma.∪ − ∈ − Lemma 5.2. For µ 1 one has ≥ L 1 µ 1 πL 2 2 +ǫL 2 N + ǫ e . 3 Lﬁn(1, π, sym ) ≪ L µ L π ∈ A(0,N , M) tπX> 0 1 |µ − tπ| < L 29 5.2. The proof of Proposition 5.1. Applying the fact that h(0) = 1, we have

λπ(m1)λπ(m2) λπ(mb1)λπ(m2) 2 = 2 L h(L(µ tπ))dµ 3 Lﬁn(1, π, sym ) 3 Lﬁn(1, π, sym ) R − π ∈ A(0,N , M) π ∈ A(0,N , M) Z 0 T

λπ(m1)λπ(m2) 2 = L 2 h(L(µ tπ))dµ. E 3 Lﬁn(1, π, sym ) R−[−T,T ] − π ∈ A(0,N , M) Z 0

Note that 0 involves the exceptional eigenvalues. By the density theorem (See [Iw2002, Theorem 11.7]) we haveE

θ+ǫ ǫ θ+ǫ 0 (m1m2) N 1 N (m1m2) . (5.6) E ≪ 3 ≪ π ∈ A(0,N , M) tπX∈ iR

For and , by Lemma 5.2 and [Li2011, Lemmas 2.3 and 2.4], we have the following result. E1 E2 Lemma 5.3. For L 1 log(N 2T ) one has ≤ 2π T + (m m )θ+ǫN 2 . E1 E2 ≪ 1 2 L For in (5.5), by Proposition 3.2 we have M T T = δ(m ,m )L (µ,L)dµ + L (m ,m ; µ,L)dµ, M 1 2 D N D 1 2 Z0 Z0 where (µ,L) and (m1,m2; µ,L) are given in (5.1) and (5.2), respectively. D N D 30 Note that T L 1 T ∞ L (µ,L)dµ = N 2 h (t) tanh(πt)tdtdµ D 2 2π2 µ,L Z0 Z−T Z−∞ N 2 T N 2T = tanh(πt)tdt + O . 2π2 L Z−T By (5.3) and (5.4), T 1−ǫ 1 +ǫ πL +ǫL √mn πL L (m ,m ; µ,L)dµ (mn) 4 e 2 min T, e . N D 1 2 ≪ N 2 Z0 1 2 Therefore, on taking L = 2πK log(N T ) with K > 1 we have N 2 T N 2T = tanh(πt)tdt + O K M 2π2 log(N 2T ) Z−T 1 +ǫ 1 +ǫ 1+ 1 +ǫ −2+ 1 +ǫ 1 +ǫ +O (mn) 4 min N 2K T 4K , √mnN K T 2K . (5.7)

n 1 o Thus Proposition 5.1 follows from (5.6), (5.7) and Lemma 5.3 with K = ǫ and ǫ L = log(N 2T ) 2π for suﬃciently small ǫ.

Appendix A. The Rankin-Selberg Theory In this part, we recall the Rankin-Selberg theory in [Ja1972] to prove Proposition 2.5. A.1. Eisenstein Series. We recall Eisenstein series in [Co2004]. For Φ in the Bruhat-Schwartz space (A2), let S T Φ((x1,x2)) := Φ((y1,y2))ψ (y1,y2) (x1,x2) dy1dy2 2 · ZA be the Fourier transformb of Φ and let η(g, Φ; s) := det g s Φ((0, 1)tg) t 2sd×t. | | | | ZA× 1 G(A) s− 2 It gives a smooth section of the normalized induced representation IndB(A)δB , where B is the standard parabolic subgroup of GL2. For (s) > 1, we have the Eisenstein series ℜ E(g, Φ,s)= η(δg, Φ; s). δ∈B(XQ)\G(Q) It has a meromorphic continuation to all s C with simple poles at s = 0 and s = 1 and satisﬁes the functional equation ∈ E(g, Φ,s)= E(T g−1, Φ, 1 s). − The residues of E(g, Φ,s) at the simple poles are b 1 1 Res E(g, Φ,s)= Φ((0, 0)) and Res E(g, Φ,s)= Φ((0, 0)). s=1 2 s=0 −2 31 b 2 A.2. The Rankin-Selberg Integral. Let φπ Lπ(G(Q) G(A)). For (s) > 1, we consider the Rankin-Selberg integral ∈ \ ℜ

I(s, φπ, Φ) = φπ(g)φπ(g)E(g, Φ,s)dg. ZG(Q)Z(A)\G(A) It has meromorphic continuation to all s C and satisﬁes the functional equation ∈ I(s, φ , Φ) = I(1 s, φ , Φ), π − π where φ (g)= φ (T g−1). Moreover, s = 1 is a simple pole of I(s, φ , Φ) with the residue π π e b π Φ((0, 0)) e Res I(s, φ , Φ) = φ , φ . (A.1) s=1 π 2 h π πi For (s) > 1 we have b ℜ I(s, φπ, Φ) = Iv(s,Wv, Φv), v Y where Iv(s,Wv, Φv) are local Rankin-Selberg integrals given by

A.3. The choice of the test function. We choose Φ = v Φv as in Jacquet [Ja1972] as follows. 2 2 For v = , let Φ (x,y)= e−π(x +y ). It is invariant under the right action by K and • ∞ ∞ Q ∞ −π(x2+y2) Φ∞((0, 0)) = e dxdy = 1. ZR×R For v = p with p ∤ N,b we choose Φp to be the characteristic function of Zp Zp and thus • ×

Φv((0, 0)) = Φ(x,y)dxdy = 1. ZQp×Qp For v = p with p N,b let Φ be the characteristic function of p3Z Z×. In this case, for • | p p × p k K ,Φ ((0, z)k ) = 0 unless z Z× and k Z×K (3). Thus we have p ∈ p p p ∈ p p ∈ p p 1 p−1 Φ ((0, 0)) = − . p p3 Therefore b p 1 Φ((0, 0)) = Φ ((0, 0)) = − . (A.2) v p4 v Y pY|N b b 2 A.4. The local Rankin-Selberg integrals. Recall that we choose φπ Lπ(G(Q) G(A)) such that W = W W , where W and W are given in Section 2.4. ∈ \ φ ∞ × p p p ∞ 32 Q A.4.1. Non-Archimedean places p N. By the choice of Φ we have | p × 1, if kp Zp Kp(3), ξp(s, R(kp)Φp)= ∈ ( 0, otherwise. Thus a a I (s,W , Φ ) =Vol(Z×K (3)) W W a s−1d×a p p p p p p 1 p 1 | |p ZQp× 1 = . p2(p + 1)

3 Note that πp is a simple supercuspidal representation of conductor p . By the classiﬁcation of supercuspidal representations (see [BuHe2006]) one has

2 Lp(s,πp, sym ) = 1. It gives that 1 p−s I (s,W , Φ )= − L (s, 1 )L (s,π , sym2). p p p p2(p + 1) p Qp p p

A.4.2. Non-Archimedean places p ∤ N. In this case,

= L (2s, 1 ) p−ms αl1 αl2 α l3 α l4 p Qp  p,1 p,2  p,1 p,2  mX≥0 l1+Xl2=m l3+Xl4=m 2    = Lp(s, 1Qp )Lp(s,πp, sym ).

A.4.3. The Archimedean place. For k K we have ∞ ∈ ∞ ∞ 2 dz ξ (s, k .Φ ) = 2 z2se−πz = π−sΓ(s). ∞ ∞ ∞ z Z0 Hence ∞ a a da I (s,W , Φ )= π−sΓ(s) W W as−1 ∞ 2k ∞ 2k 1 2k 1 a Z0 ∞ da = π−sΓ(s) 4a2k+s−1e−4πa a Z0 = 42−2k−sπ−(2s+2k−1)Γ(s)Γ(s + 2k 1), 33 − and

−s a a s−1 da I∞(s,Wǫπ,0, Φ∞)= π Γ(s) Wǫπ,0 Wǫπ,0 a ∞ 1 1 | | a ∞ ZR× | | ∞ da = 23−sπ−2sΓ(s) K (a)K (a)as itπ itπ a Z0 s s s 2 = π−2sΓ it Γ + it Γ . 2 − π 2 π 2 In the last step, we have used the formula in [Go2006, page 212], ∞ dy Γ s−µ−ν Γ s−µ+ν Γ s+µ−ν Γ s+µ+ν K (y)K (y)ys = 2s−3 2 2 2 2 . (A.3) µ ν y Γ(s) Z0 By the above argument, the global Rankin-Selberg integral is 1 p−s I(s, φ , Φ) = ζ(s)L (s,π, sym2) − 4−(s+2k−2)π−(2s+2k−1)Γ(s)Γ(s + 2k 1) π fin p2(p + 1) − pY|N n o if π (2k, N 3), and ∈A 1 p−s s s s 2 I(s, φ , Φ) = ζ(s)L (s,π, sym2) − π−2sΓ it Γ + it Γ π fin p2(p + 1) 2 − π 2 π 2 pY|N if π (0,N 3). Thus we have ∈A Ress=1I(s, φπ, Φ) 4−(2k−1)π−(2k+1)Γ(2k), if π (2k, N 3), 2 p 1 ∈A = Lfin(1, π, sym ) − 1 (A.4) p3(p + 1)  , if π (0,N 3). p|N  Y  cosh(πtπ) ∈A Proposition 2.5 follows immediately from formulas (A.1), (A.2) and (A.4). References [BaFr2016] O. Balkanova and D. 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35 School of Mathematics and Statistics, Shandong Univeristy, Weihai, Weihai 264209, China E-mail address: [email protected]

Shenzhen Key Laboratory of Advanced Machine Learning and Applications, College of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong 518060, China E-mail address: [email protected]

Department of Mathematics, National University of Singapore, Singapore 119076 E-mail address: [email protected]