THE FIFTH MOMENT OF MODULAR L-FUNCTIONS EREN MEHMET KIRAL, MATTHEW P. YOUNG Abstract. We prove a sharp bound on the fifth moment of modular L-functions of fixed small weight, and large prime level. 7 1 0 2 b 1. Introduction e F Letq beaprimeandκ 4,6,8,10,14 . LetH (q)bethesetofweightκHeckeeigenforms κ ∈ { } 6 on Γ (q). For any f H (q) (note that any such f is automatically a newform), let λ (n) 0 κ f denote its nth Hecke e∈igenvalue. Our main result is the following theorem: ] T Theorem 1.1. We have N . (1.1) L(1/2,f)5 q1+θ+ε, h ≪ t a f∈XHκ(q) m as q among primes. Here θ is the best-known progress towards the Ramanujan- [ → ∞ Petersson conjecture. 2 v The currently best-known value θ = 7/64 is given by the work of Kim and Sarnak [22]. 7 The central value L(1/2,f) is nonnegative by [23, Corollary 2] and [35], so upon dropping 0 5 all but one term, we deduce: 7 0 Corollary 1.2. For any ε > 0, we have . 01 (1.2) L(1/2,f) ε q1+5θ+ε. ≪ 7 1 Previously, Duke, Friendlander, and Iwaniec [10] bounded the amplified fourth moment in : this family, and Kowalski, Michel, and VanderKam [24] asymptotically evaluated a mollified v i fourth moment. Recently, Balkanova and Frolenkov [2] improved the error term in these X fourth moment problems, and thereby deduced the so-far best-known subconvexity result r a 27−30θ of L(1/2,f) q112−128θ. Corollary 1.2 improves this further. Our method of proof takes a ≪ different course from these works, and we never solve a shifted convolution problem. This paper has some common features with the cubic moment studied by Conrey and Iwaniec [7]. Their work also bounds a moment that is one larger than what one may consider the “barrier” to subconvexity. That is, for the family of L-functions they consider, an upper boundonthesecondmomentthatisLindelo¨f-on-averageleadsbackpreciselytotheconvexity bound on an individual L-value. Similarly, the fourth moment is the “barrier” in the family of Theorem 1.1. In a sense, going one full moment beyond the barrier is a way of amplifying with the L-function itself. As far as the authors are aware, prior to Theorem 1.1, the only ThismaterialisbaseduponworkofM.Y.supportedbytheNationalScienceFoundationunderagreement No. DMS-1401008. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The first author was partially supported by an AMS-Simons travel grant. 1 2 EREN MEHMET KIRAL,MATTHEW P. YOUNG known instances of a sharp upper bound on a moment that is one larger than the barrier moment are the cubic moment and its generalizations [7] [17] [31] [36] [32]. In Section 2, we give a simplified sketch of the argument. The main overall difficulty in the problem is that we require a significant amount of cancellation in multivariable sums with divisor functions and Kloosterman sums. The main thrust of the argument is to apply summation formulas that shorten the lengths of summation, eventually obtaining a sum of Kloosterman sums. To this, we apply the Bruggeman-Kuznetsov formula, which leads to a fourth moment of Hecke-Maass L-functions twisted by λ (q), with an additional average j over the level. This is another incarnation of a Kuznetsov/Motohashi-type formula where a moment probleminonefamilyofL-functionsisrelatedtoanothermoment ina“dual”family (see [26, Section 1.1.3]). Along the way, we encounter many “fake” main terms, which turn out to be surprisingly difficult to estimate. A straightforward bound on these would only lead to O(q5/4+ε) in Theorem 1.1, which would be trivial. We expect that all the “fake” main terms calculated in this paper should essentially cancel, but doing so is a daunting prospect. Instead, we show that with an appropriate choice of weight functions in the approximate functional equations, then all the fake main terms are bounded consistently with Theorem 1.1. The amplified/mollified fourth moment (cf. [10] [24]) also required a difficult analysis of the main terms, which arose from solving the shifted convolution problem. As such, it is not clear how to compare the main term calculations here with [10] [24]. The article [3, Section 1.2] has a more through discussion of the main term analysis with the shifted convolution approach. One of the practical difficulties in applying the Bruggeman-Kuznetsov formula in appli- cations is that one needs to recognize the particular shape of sum of Kloosterman sums one encounters (with coprimality and congruence conditions, etc.) as one associated to a group Γ, pair of cusps a,b, and nebentypus χ. To this end, in Section 4 we have identi- fied all the Kloosterman sums belonging to the congruence subgroup Γ (N) and at general 0 Atkin-Lehner cusps (i.e., those cusps equivalent to under an Atkin-Lehner involution) ∞ with general Dirichlet characters. It turns out that there is a “correct” choice of scaling matrix to use when computing the Fourier coefficients and Kloosterman sums, a choice that ensures the multiplicativity of Fourier coefficients at the Atkin-Lehner cusps. Another practical difficulty is estimating oscillatory integral transforms with weight func- tions depending on multiple variables, with numerous parameters. It is desirable to perform stationary phase analysis on a single variable at a time, yet still keep track of the behavior of the remaining variables in a succinct way. In Section 5 we have codified some properties of a family of weight functions that allows us to do this efficiently. The key stationary phase result, which is a modest generalization of work in [4], is stated as Lemma 5.5 below (this result also appears in [32] which was prepared in tandem with this document). In the spectral analysis of a sum of Kloosterman sums, it is necessary to treat the Maass forms, holomorphic forms, and continuous spectrum. In our situation, the Maass forms and holomorphic forms are rather similar, and lead to a twisted fourth moment of GL cuspidal 2 L-functions. The continuous spectrum is similar in many respects, but a key difference is that the “dual” family of L-functions is essentially a sum of a product of eight Dirichlet L-functions at shifted arguments. One naturally wishes to treat the continuous spectrum on the same footing as the discrete spectrum, which requires shifting some contour integrals past the polesof theDirichlet L-functions(which occur onlywhen the character isprincipal). There is potentially a large loss in savings from these poles on the 1-line compared to the THE FIFTH MOMENT OF MODULAR L-FUNCTIONS 3 contribution from the 1/2-line. Luckily, it turns out that there is some extra savings in the residues of the Dirichlet series (essentially, from considering only the principal characters) that balances against this loss. This savings ultimately arises from a careful calculation of the Fourier expansion of the Eisenstein series on Γ (N) with arbitrary N, attached to an 0 arbitrary cusp, expanded around any Atkin-Lehner cusp. An astute reader may note that κ = 2 is not covered by Theorem 1.1. In fact, there is only a single instance where our proof requires that κ > 2, namely in the study of the continuous part of the spectrum at (11.26). Perhaps with further analysis one might incorporate the weight 2 case, by a more careful analysis of the residues of the Dirichlet L-functions. The restrictionthatq isaprimeandthatκ 14,κ = 12,meansthatthecuspformsf H (Γ (q)) κ 0 ≤ 6 ∈ are automatically newforms. It is reasonable to expect that using a more general Petersson formula for newforms (e.g., see [32]) could relax these assumptions, but the arithmetical complexity would be increased. 2. High-level sketch Here we include an outline of the major steps used in the proof, intended for an expert audience. By approximate functional equations and the Petersson formula, we arrive at τ(m)τ (n) 4π√mn 3 (2.1) := S(m,n;c)J , κ 1 S c√mn − c mX≪qn≪Xq3/2c≡0X(modq) (cid:16) (cid:17) and we wish to show qθ+ε. The hardest case to consider is m q, n q3/2, and S ≪ ≍ ≍ c √mn q5/4, in which case J (x) 1. In practice, one needs to treat the two ranges κ 1 ≍ ≍ − ≈ of the Bessel function (i.e., x 1 and x 1) differently. In this sketch, we focus on the ≪ ≫ transition region of the J-Bessel function where x 1. ≍ The Weil bound applied to the Kloosterman sum shows q7/8+ε, far from qθ+ε. S ≪ The immediate problem with (2.1) is that Voronoi summation applied to m or n leads to a dual sum that is longer than before. The conventional wisdom is that this is a bad move. However, there do not seem to be any other moves available, so it may be necessary to take a loss in the first step. We may attempt to minimize the loss here by opening τ(m) = 1, supposing m m by symmetry, and applying Poisson summation in m1m2=m 1 ≤ 2 m modulo c. This leads to 2 P τ (n) m nk 3 1 e . S ≈ √m knc c 1 m1X≪q1/2c≡c0X(mq5o/d4q)k≪Xq3/4nX≍q3/2 (cid:16) (cid:17) ≍ Note that the trivial bound now gives q, so we lost a factor q1/8 from going the “wrong S ≪ way” in Poisson. However, now we may gain from the structure of the arithmetical part by applying the well-known reciprocity formula m nk m nc m n 1 1 1 e = e e . c − k ck (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) This effectively switches the roles of c and k, at the expense of introducing the potentially- oscillatory factor e (m n) into the weight function. However, when all variables are near ck 1 their maximial sizes, then this factor is not oscillatory, so we shall ignore it in this sketch. Side remark. If one applies Voronoi to the sum over m (which is more in line with the previous works on the amplified/mollified fourth moment), then one encounters a shifted 4 EREN MEHMET KIRAL,MATTHEW P. YOUNG divisor sum of the form τ (m)τ (n). Such sums have been considered by various m n=h 2 3 authors, with the most advan−ced results being the recent work of B. Topac¸og˘ulları [34]. P One way to proceed next would be to convert the additive character into Dirichlet char- acters (modulo k), which has a nice benefit of separating the variables, a key step in [7]. This would lead to a fifth moment of Dirichlet L-functions twisted by Gauss sums, with an averaging over the modulus. One may check that Lindelo¨f applied to these L-functions only gives q1/4+ε which in a sense gets back to the convexity bound. S ≪ Now it is beneficial to apply Voronoi summation in n modulo k (one may view this as opening τ (n) = 1, and applying Poisson in each n ). This leads to 3 n1n2n3=n i τ (p) P 3 (2.2) S(p,cm ,k). 1 S ≈ k√m pc 1 m1X≪q1/2c≡c0X(mq5o/d4q)k≪Xq3/4p≪Xq3/4 ≍ The trivial bound now is q5/8, consistent with saving q1/8 in each of the n variables, just i as we lost q1/8 by Poisson in the initial m variable. One could also apply Poisson in m to 2 1 save another q1/8, but then the arithmetical sum becomes a hyper-Kloosterman sum, which increases the difficulty of the problem (N. Pitt has studied this problem [33], but it seems very hard to obtain enough cancellation using this approach). Here we have a Kloosterman sum to which we may apply the Bruggeman-Kuznetsov formula of level m . Using this, we 1 obtain ν (p c)ν (p p ) j 1 j 2 3 (2.3) . S ≈ √p p p c m1X≪q1/2letvXje≪lmqǫ1c≡c0X(mq5o/d4q)p1,p2X,p3≪q1/4 1 2 3 ≍ We can essentially write this as λ (q) (2.4) ν (1)2 j L(1/2,u )4. j j S ≈ √q m1X≪q1/2letvXje≪lmqǫ1 Here the scaling on the spectral data is that ν (1)2 T2mε. Thus we have converted tj≪T j ≪ 1 to a twisted fourth moment of Maass form L-functions, and one can see how qθ+ε emerges P by bounding λ (q) qθ+ε, and using a Lindelo¨f-on-average bound for the spectral fourth j | | ≪ moment (which in turn is “easy”, following from the spectral large sieve inequality). The above discussion implicitly assumed that the p are nonzero. The zero frequencies i (wheresomeorallp = 0)turnouttobethe“fake”maintermsalludedtointheintroduction. i To handle these, we compute the weight function explicitly, and evaluate the sums over k,m , and c as zeta quotients. We later bound the integral by moving lines of integration, 1 and apparent poles of the integrand are cancelled by a choice of the weight function in the approximate functional equation. To elaborate on this point, consider an overly-simplified model with a sum of the form S = 1 V( n ), where V(x) = 1 G(s)x sds, and n 1 √n √q 2πi (1) s − ≥ G(s) is analytic satisfying G(0) = 1, with rapid decay in the imaginary direction. The P R trivial bound applied to S gives S = O(q1/4), using that V(x) (1+x) 100. Alternatively, − ≪ we have S = 1 ζ(1/2+s)qs/2G(s)ds, which by shifting contours to the line Re(s) = ε > 0 2πi (1) s gives S = G(1/2)q1/4 +O(qε). If G(1/2) = 0 (which one is free to assume in the context of R the approximate functional equation), then in fact one has an improved bound of S = O(qε). This is the principal idea behind the estimation of the fake main terms. The main difficulty THE FIFTH MOMENT OF MODULAR L-FUNCTIONS 5 in practice is that one has a much more complicated sum with multiple variables and weight functions that arise as integral transforms, and it requires significant work to recognize instances of this basic idea. One should also observe that the above method of estimating S is highly reliant on the specific form of the weight function V; if it were multiplied by a compactly-supported bump function (say one part of a dyadic partition of unity), then one could not deduce S = O(qε) anymore. Since we shall apply dyadic partitions of unity in the forthcoming treatment, for the purposes of estimating these fake main terms, it is crucial to re-assemble the partitions. The role of the m -variable within the proof has some curious features. In the sketch 1 above up through (2.4), the m variable was hardly used. Precisely, we never applied a 1 summation formula nor obtained any cancellation from this variable. Nor did we use any reciprocity involving m to lower a modulus. However, non-trivial estimations involving m 1 1 do appear in other parts of the proof. In the evaluation of one type of fake main term in Section 13.7, we evaluate the m -sum similarly to the discussion in the previous paragraph; 1 the lack of pole at s = 1/2 amounts to square-root cancellation in this variable. The other location is in estimating the continuous spectrum analog of (2.3) which so far was neglected in this sketch. One may show that the continuous spectrum analog of (2.4) is O(qε) using 1/2+ε that the number of cusps on Γ (m ) is at most O(m ). However, on average over m , 0 1 1 1 the number of cusps is O(mε), which leads to a bound that saves an additional factor q1/4. 1 In a sense, this discussion indicates that the continuous spectrum is smaller in measure in the level aspect than the discrete spectrum, on average over the level. 3. Preliminaries 3.1. Petersson Trace Formula. We normalize the Petersson inner product by dxdy f,g = f(z)g(z)yκ . h i y2 ZZΓ0(q) H \ For f a Hecke-normalized (holomorphic or Maass) cusp form, one has the estimate q1 ε f,f q1+ε − κ,ε κ,ε ≪ h i ≪ for any ε > 0, by [18] and [15]. The Petersson trace formula reads S(m,n;c) 4π√mn w λ (n)λ (m) = δ +2πi κ J , f f f n=m − κ 1 c − c f∈XHκ(q) c≡0X(modq) (cid:16) (cid:17) where wf = (4πΓ)κ(κ−−11f),f = q−1+o(1) are the Petersson weights. Define h i = (q) = w L(1,f)5. M M f 2 f∈XHκ(q) Our main result, Theorem 1.1, is equivalent to (3.1) qθ+ε. κ,ε M ≪ 6 EREN MEHMET KIRAL,MATTHEW P. YOUNG 3.2. The approximate functional equations. Let κbe a positive even integer, q a prime, and f a Hecke cusp form of weight κ and level q. Put s+ κ 1 s+ κ+1 − γ(s,κ) = π sΓ 2 Γ 2 . − 2 2 (cid:18) (cid:19) (cid:18) (cid:19) Let G (i = 1,2) be an even entire function decaying rapidly in vertical strips such that i G (0) = 1. Define i 1 G (u)γ(1 +u,κ) 1 G (u)γ(1 +u,κ)2 V (x) = 1 2 x udu, V (x) = 2 2 x udu. 1 2πi u γ(1,κ) − 2 2πi u γ(1,κ)2 − Z(1) 2 Z(1) 2 If x qε then by shifting the contour of integration arbitrarily far to the right, we obtain ≫ that V (x) (xq) A. Here and throughout, we view κ as fixed, and q as becoming large. i κ,A − ≪ For later use, it will be important to assume G (1/2) = 0. i Proposition 3.1. With notation as above, we have ∞ λ (m)τ (m) m L(1,f)2 = 2 f 2 V , 2 √m q m=1 (cid:18) (cid:19) X where τ (m) is the (two-fold) divisor function, and 2 ∞ 1 V(x) = V (e2x)/e = V (u)ζ (1+2u)x udu, 2 2 q − 2πi (e,q)=1 Z(1) X where ζ (s) = (1 q s)ζ(s) is the Riemann zeta funcetion with the qth Euler factor missing. q − − Proof. By the Hecke relation, we have ∞ λ (m m /e2) ∞ 1 ∞ τ (m)λ (m) (3.2) L2(s,f) = f 1 2 = 2 f . (m m )s e2s ms 1 2 m1X,m2=1e|(mX1,m2) (eX,q)=1 mX=1 (e,q)=1 Then from the functional equation L2(s,f)γ(s,κ)2qs =: Λ(s,f)2 = Λ(1 s,f)2 we get the − (cid:3) formula, as in [21, Theorem 5.3]. Proposition 3.2. Let ε be the sign of the functional equation for L(s,f). Then f ∞ µ(a) ∞ λ (na) (3.3) L(1,f)3 = (1+ε )3 f τ (n,F ), 2 f a3/2 √n 3 a,√q a=1 n=1 (aX,q)=1 X where n n n 1 2 3 (3.4) τ (n,F ) = F , , , 3 a,√q a √q √q √q n1nX2n3=n (cid:18) (cid:19) and 1 (3.5) F (x ,x ,x ) = V (ax e e )V (ax e e )V (ax e e ) a 1 2 3 1 1 1 2 1 2 1 3 1 3 2 3 e e e e1,e2,e3 1 2 3 (e1e2Xe3,q)=1 3 γ(1 +u ,κ)G(u ) du du du = 2 i i ζ (1+u +u )ζ (1+u +u )ζ (1+u +u ) 1 2 3. ZZZ i=1 (axi)uiγ(21,κ)ui q 1 2 q 1 3 q 2 3 (2πi)3 (1)(1)(1)Y THE FIFTH MOMENT OF MODULAR L-FUNCTIONS 7 Remark. One may easily check that ∂j1+j2+j3 3 (3.6) xj1xj2xj3 F (x ,x ,x ) (ax ) ε(1+ax ) A. 1 2 3 ∂xj1∂xj2∂xj3 a 1 2 3 ≪j1,j2,j3,A,ε i − i − 1 2 3 i=1 Y In the terminology introduced later in Section 5, the property (3.6) means that F satisfies a the same derivative bounds as an X-inert function with X qε, in the region x q 1/2, i − ≪ ≫ for all i. Similar derivative bounds hold for V(x). Proof. UsingtheapproximatefunctionalequationforeachL(1/2,f),andtheHeckerelations, we obtain L(1,f)3 = (1+εf)3 λf n1ne222n3 V n1e1 V n2e1 V n3 2 e √(cid:16)n n n (cid:17) 1 √q 1 √q 1 √q e1,e2 1 n1,n2,n3 1 2 3 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (e1eX2,q)=1 e2(nX1n2,n3) | (1+ε )3 λ (n n n ) n e f n e f n f f f f 1 2 3 1 1 1 2 1 2 3 1 2 = V V V . 1 1 1 e e √n n n √q √q √q (e1eeX12,,eq2)=1 1 2 f1Xf2=e2(nn11X,,nf22),n=31 1 2 3 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Using M¨obius inversion to detect the coprimality condition with µ(a), re-ordering a(n1,f2) the summations, and renaming the summation variables gives the mo|re symmetric form P ∞ µ(a) 1 L(1,f)3 = (1+ǫ )3 2 f a3/2 e e e a=1 e1,e2,e3 1 2 3 (aX,q)=1 (e1eX2e3,q)=1 λ (an n n ) an e e an e e an e e f 1 2 3 1 1 2 2 1 3 3 2 3 V V V . 1 1 1 × √n n n √q √q √q n1X,n2,n3 1 2 3 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:3) This is seen to be equivalent to (3.3). Now apply Propositions 3.1 and 3.2 to . There is a nice simplification in Proposition M 3.2, whereby we may replace 1+ǫ by 2, because if ǫ = 1, then L(1/2,f)2 = 0 anyway. f f − Applying the Petersson trace formula then yields 1 = +2πi κ , − 16M D S where is the diagonal term, and D (3.7) ∞ µ(a) τ2(m)τ3(n,Fa,√q)S(m,na;c) 4π√mna m = J V . S a3/2 c√mn κ−1 c q (a,q)=1 c 0(modq) n,m (cid:18) (cid:19) (cid:18) (cid:19) X ≡ X X It is easy to bound the diagonal term. Lemma 3.3. We have qε. ε D ≪ This follows easily from the fact that the functions V (y) and V (y) decay rapidly as 1 2 y , and using the bound J (x) x for κ 2. κ 1 →Pro∞ving Theorem 1.1 is reduce−d to ≪showing ≥ qθ+ε. We will return to in Section 6. S ≪ S Meanwhile, Sections 4 and 5, which are self-contained, develop some material necessary for our manipulations of . S 8 EREN MEHMET KIRAL,MATTHEW P. YOUNG 4. Kloosterman Sums and Bruggeman-Kuznetsov Summation Formula 4.1. Motivation. We seek a formula for the sum S(Nm,n;c)f(c), (c,N)=1 X c 0(modq) ≡ where f is a smooth function on the positive reals with sufficient decay. In this section we show that this sum of Kloosterman sums has a spectral decomposition since it can be realized asthe Kloosterman sum associated to the pair of cusps ,1/q for the groupΓ (Nq). 0 ∞ This precise form of Kloosterman sum seems to not appear in the standard sources in the literature, such as [20] or [8]. Moreover, we show that the scaling matrices implicit in the definition of the Kloosterman sum as well as in the Fourier expansion at certain cusps (which we call Atkin-Lehner cusps) may be chosen to be Atkin-Lehner operators. We record the Bruggeman-Kuznetsov formula in Section 4.5. We will specify the choice of a basis of automorphic forms in this expansion in Section 4.7. 4.2. Cusps, scaling matrices, and Kloosterman sums. We mostly follow the notation of [20]. Let N be a positive integer and Γ = Γ (N). An element a P1(Q) is called a cusp. 0 ∈ Two cusps a and a are equivalent under Γ if there is a γ Γ satisfying a = γa. Let a be a ′ ′ ∈ cusp and Γ = γ Γ : γa = a be the stabilizer of the cusp a in Γ. A matrix σ SL (R), a a 2 { ∈ } ∈ satisfying (4.1) σ = a, and σ 1Γ σ = (1 n) : n Z a∞ a− a a {± 0 1 ∈ } is called a scaling matrix for the cusp a. Remark. For two equivalent cusps a and a = γa, with γ Γ, the stabilizers of the cusps ′ ∈ are conjugate subgroups in Γ, namely Γa′ = γΓaγ−1. Furthermore, σa′ = γσa is a scaling matrix for the cusp a. ′ Remark. The matrix σ is not uniquely defined by the above properties: for any x R, a ∈ σ (1 x) a 0 1 also satisfies (4.1). The choice of the scaling matrix σ is important in what follows. a Definition 4.1. Let f be a Maass form for the group Γ. The Fourier coefficients of f at a cusp a, relative to a choice of σ , and denoted ρ (σ ,n), are defined by a f a (4.2) f(σ z) = ρ (σ ,n)W (nz), a f a 21+itj n=0 X6 where W (x+iy) = 2 y K (2π y )e(x). s | | s−12 | | Remark. If σ is replaced with σ (1 α)p, then the Fourier coefficients relative to the new a a 0 1 scaling matrix are given by ρ (σ (1 α),n) = e(nα)ρ (σ ,n). f a 0 1 f a If the Fourier coefficients of f at a cusp a are multiplicative with respect to σ , then for a another scaling matrix as above, the Fourier coefficients will typically not be multiplicative. We found the reference [13] useful for its discussions in this context. THE FIFTH MOMENT OF MODULAR L-FUNCTIONS 9 Fourier coefficients at equivalent cusps, however, behave more predictably. If a = γa and ′ we choose the scaling matrix as σa′ = γσa, then due to Γ-invariance of f, we have ρf(σa′,n) = ρf(σa,n). When the scaling matrix σ is understood, we may write ρ (σ ,n) = ρ (n). a f a a,f We now define Kloosterman sums with respect to a pair of cusps. Even though we will not need characters in our work, we state the definition with nebentypus, since we expect this will be useful in other contexts. Let χ be a Dirichlet character modulo N. We extend χ to Γ via χ : Γ S1 −→ γ = ( a b) χ(d). Nc d 7→ If χ is an even character, it can be seen as a multiplier system on Γ with weight 0. Definition 4.2. The cusp a is called singular for the character χ if χ is trivial on Γ . For a a and b singular cusps for χ, we define the Kloosterman sum associated to a,b and χ with modulus c as am+dn (4.3) S (m,n;c;χ) = χ(σ γσ 1)e . ab a b− c γ=(ac db)∈ΓX∞\σa−1Γσb/Γ∞ (cid:18) (cid:19) Remarks. This definition oftheKloostermansum slightly differsfromthatof[20, (2.23)], in that the roles of m and n are reversed. Also note that one requires a and b to be singular cusps with respect to χ for the sum to be well-defined, so in particular χ is even. From Definition 4.2 one may directly deduce (4.4) S (m,n;c;χ) = S (n,m;c;χ). ab ba This corrects a formula in [20, p.48] which omitted the complex conjugation. In many important cases the Kloosterman sums is real, e.g. see Theorem 4.6 below. Definition 4.3. Let the set of allowed moduli be defined as (4.5) = γ > 0 : ( ) σ 1Γσ . Cab γ∗ ∗∗ ∈ a− b Notice that if γ / ab the Klooste(cid:8)rman sum of modulus γ(cid:9)is an empty sum. ∈ C The definition (4.3) is a natural one, as it occurs in the Fourier expansion of the Poincar´e series which is defined as (4.6) Pa(z,s;χ) = χ(γ)Im(σ 1γz)se(nσ 1γz). n a− a− γ∈XΓa\Γ For n = 0, Pa(z,s;χ) L2(Γ H,χ) (to be clear, it transforms by f(γz) = χ(γ)f(z)). 6 n ∈ \ Remark. The Kloosterman sum associated to the pair of cusps a,bdepends on the choice of pair of scaling matrices σ and σ (so it might be better to denote it as S (m,n;c)). If a b σa,σb one changes the choice of the scaling matrix, the Kloosterman sum also changes by (4.7) S (m,n;c) = e( αm+βn)S (m,n;c). σa(10 α1),σb(cid:16)10 β1(cid:17) − σa,σb This corrects a formula of [20, p.48] which has α in place of our α. However, changing − the cusp a to an equivalent one does not alter the Kloosterman sum, if one also changes the scaling matrices accordingly. Indeed, if a = γ a and b = γ b for γ ,γ Γ, then ′ 1 ′ 2 1 2 ∈ S (m,n;c;χ) = S (m,n;c;χ). σa,σb γ1σa,γ2σa 10 EREN MEHMET KIRAL,MATTHEW P. YOUNG If one applies the Bruggeman-Kuznetsov formula (see Section 4.5) to sums of Kloosterman sums associated to the choice of scaling matrices σ ,σ , then the Fourier coefficients at cusps a b of automorphic forms on the spectral side must also be computed using the same scaling matrices. 4.3. Atkin-Lehner cusps and scaling matrices. Assume that N = rs with (r,s) = 1. We call a cusp of the form a = 1/r (with (r,s) = 1) an Atkin-Lehner cusp. The stabilizer of such a cusp is given as 1 Nt st 1 N s (4.8) Γ = − : t Z = − . 1/r ± rNt 1+Nt ∈ ± rN 1+N (cid:26) (cid:18)− (cid:19) (cid:27) (cid:28) (cid:18)− (cid:19)(cid:29) In particular we see that any even Dirichlet character χ (mod N) is singular at such a cusp. Definition 4.4. Let N = rs with (r,s) = 1 as above, and put xs y (4.9) W = , zN ws (cid:18) (cid:19) with det(W) = s. On automorphic forms of weight κ, the Atkin-Lehner operator is defined by (f )(z) = det(W)κj(W,z) κf(Wz). W 2 − | The Atkin-Lehner operator is independent of the choices of x,y,z,w. Since the matrix W normalizes Γ, it preserves automorphic forms on Γ. Furthermore, an Atkin-Lehner operator commutes with all the Hecke operators and hence a newform will be an eigenfunction of all the Atkin-Lehner operators. The cusps of the form a = 1/r with (r,s) = 1 are precisely those that are equivalent to under an Atkin-Lehner operator, which justifies naming them ∞ Atkin-Lehner cusps. The matrix can be normalized to have determinant 1, without changing the operator, via 1 x√s y/√s x y √s 0 W = = . √s zr√s w√s zr ws 0 1/√s (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) Here the determinant condition is xws rzy = 1. We have the freedom to choose x = z = 1, − and then if s is any integer that satisfies ss 1 (mod r), put w = s and y = (ss 1)/r. ≡ − Therefore the matrix 1 (ss 1)/r √s 0 (4.10) σ = τ ν with τ = − , ν = , 1/r r s r r ss s 0 1/√s (cid:18) (cid:19) (cid:18) (cid:19) is an acceptable choice for an Atkin-Lehner operator. Note τ Γ (r) (in particular, it has r 0 ∈ integer entries and determinant 1). From the theory of Atkin-Lehner operators, a newform f will satisfy (4.11) f(σ z) = f(z). 1/r ± One may check directly that σ also satisfies the conditions in (4.1), i.e. it is a scaling 1/r matrix for the cusp 1/r. If f is a Maass form satisfying (4.11) then its Fourier coefficients at the cusp a with respect to the choice (4.10) for its scaling matrix has the Fourier coefficients ρ (σ ,n) = ρ (n). f 1/r ,f ± ∞ Therefore, with this choice of the scaling matrix the Fourier coefficients satisfy the multi- plicative Hecke relations. This will be made more explicit in Section 4.7.