Twisted Kloosterman sums and cubic exponential sums

Transcription

1 Twisted Kloosterman sums and ubi exponential sums Dissertation zur Erlangung des Doktorgrades der Mathematish-Naturwissenshaftlihen Fakultäten der Georg-August-Universität zu Göttingen vorgelegt von Benoît Louvel aus Rennes Göttingen 2008

2 D7 Referent: S.J. Patterson Korreferent: P. Mihel Tag der mündlihen Prüfung:

3 Contents Introdution 5 1 Asymptoti behaviour of K 2 m, n, Automorphi forms on the upper half-plane Maaß forms and modular forms Poinaré series The Goldfeld-Sarnak method The Kloosterman-Selberg zeta funtion Asymptoti behaviour Theta funtions of half-integral weight A result of Serre and Stark Orthonormalisation of theta funtions Determination of the onstant Choie of the usps and of the level Proof of Theorem Cubi metapleti forms Metapleti forms on SL 2 Z[ω] K-types and metapleti forms Eisenstein series, theta funtions and the spetral deomposition theorem Poinaré series Summation formulae of Kuznetsov type Lebedev transform and the spetral sum formula On the spetrum of the metapleti group Bessel transform and the Kloosterman sum formula Metapleti group and metapleti representations Kubota symbol and metapleti group Metapleti and automorphi representations Shimura orrespondene Definitions and results Classial interpretation

4 Contents Asymptoti behaviour of K m, n, 81.1 The Kubota Patterson theta funtion The Maaß- Selberg relations Properties of the ubi theta funtions Relations between theta funtions Auxiliary results Salar produt of theta funtions Some onlusions Canellation of SaX + bx,, for allmost prime A non-trivial estimate in average Sieve argument on Z[ω] The Selberg sieve Preise estimates An upper bound for SaX + bx, An upper bound for A ± An upper bound for B A lower bound for SaX + bx, Sato-Tate vertial law Absolute lower bound Aknowledgements 15 Lebenslauf 155 Abstrat 157 4

5 Introdution This dissertation is onerned with exponential sums of the form S χ f, = fx χx e, 1 x for in the ring of integers R of a number field k, f a rational funtion, χ a harater modulo and ez = exp2iπ Tr k/q z. They are finite sums, where x runs through the finite set of representative of R/R and where we agree to write x instead of x mod. Historially, a first motivation for the study of these sums arises from the diophantine analysis, where many problems are redued to the evaluation of suh sums. For example, Hilbert s Eleventh Problem asks about the representability of integers in a number field by an integral quadrati form. For the speial ase of diagonal forms, the irle method was introdued by Hardy and Littlewood to study asymptotially the number of solutions of F x = x x x 2 s = m, for s 5. In 1926, Kloosterman [21] studied this problem over Q for s = 4, and was led to introdue the so-alled Kloosterman sums Km, n, = 2iπ mx + nx 1 exp, 2 x,x x x 1 defined for m, n, Z. He sueeded in obtaining the non trivial individual estimate Km, n, p E p /4, whih allowed him to solve the problem for diagonal forms in four variables. Besides the signifiane of individual bound as for Kloosterman sums, it is expeted that bound on average are equally important. Around 1960, working on some additive problems, Linnik introdued a dispersion method in whih Kloosterman sums play a fundamental role, and he onjetured [28] a anellation among these sums. At the same time, in his seminal work on Fourier oeffiients of modular forms [40], Selberg studied the zeta funtion assoiated to Kloosterman sums and formulated the same onjeture, whih asserts that <X Km, n, X 1/2+ε, ε > 0. 4 The major breakthrough in this problem ame from Kuznetsov in 1979 [26], who proved it for any ε > 1/6. These is the kind of problems we shall be interested in, for some speial sums S χ f,. 5

6 Introdution A further motivation omes from the ohomologial interpretation of the Sato-Tate onjeture about ellipti urves without omplex multipliation. For an ellipti urve E over Q, if a p E is defined by a p E = EF p 1 p, where EF p is the number of points on E over F p, then the Hasse upper bound asserts that Aording to 5, we define a family of angles θ p E by a p E 2 p. 5 a p E 2 p = os θ pe, and the question is, if these angles are uniformly distributed for some measure. In the late 1940s, the Weil onjetures about the Hasse-Weil zeta funtion attahed to a variety gave the key to the uniform distribution of the angles θ p E when E has omplex multipliation, sine a result of Deuring asserts that in this ase, the Hasse-Weil zeta funtion is expressed in terms of Heke L-funtion about whih we know enough analyti results. Around 1960, Sato and Tate arrived independently to the onjeture that suh a uniform distribution measure also exists for ellipti urves without multipliation, and is given by µ S T = 2 sin 2 θ dθ on [0, π]. 6 π By a ohomologial approah, Serre see [41], I-25, I-26 proved that one an dedue distribution results from knowledge about the analyti ontinuation of the L-funtion attahed to E. Coming bak to Kloosterman sums, the proof by Weil from the Riemann hypothesis for urves led him to improve the individual bound for Kloosterman sum to Km, n, p E p 1/2, 7 whih is the best possible. Within the ohomologial framework, N. Katz see [18], onj formulated the uniform distribution of the angles of Kloosterman sums aording to the Sato-Tate law, a folklore onjeture whih arose in the 1970s. More preisely, if K1, a, p 2 p then it was onjetured that for any 0 α < β π, {p : X p < 2X, α θ p,1 β} {p : X p < 2X} = os θ p,a 0 θ p,a π, 8 2 π β α sin 2 θ dθ, as X +. 9 Notie that in the angles that we onsider, both the parameter p and the other parameter either E or a an vary. We shall distinguish these ases by speaking from the horizontal ase and from the vertial one, respetively. As well as the numerous numerial verifiations supporting the horizontal onjeture, one other reason to believe in the Sato-Tate onjeture for ellipti urves or for Kloosterman sums ame from Birh [1] and Katz [19] Ex. 1.6, who proved the vertial ase of this onjeture, respetively for ellipti urves and for Kloosterman sums. Finally, the Sato-Tate onjeture for urves was proved in 2006 by Clozel, Harris, Shepherd-Barron & Taylor under the ondition that je is not an integer. We ome now to our objet of interest. Atually in [1], after proving the vertial asymptoti behaviour for ellipti urves, Birh onjetured the same vertial result for the ubi sums 6

7 Introdution SaX + bx, ; this onjeture was latter proved by Livné [29] and [0]. The same dihotomy as the one between ellipti urves with or without omplex multipliation ours for ubi sums, and, as a matter of fat, a uniform distribution over the primes in the ase where b = 0, i.e. for the sums SaX, p, was proved by Heath-Brown and Patterson [15]. This parallel with Heke s work leads naturally to onjeture the horizontal Sato-Tate law for ubi sums with b 0. For both ases, i.e. for Kloosterman sums or for ubi sums, even though the vertial question was solved, the horizontal one remains still open. We shall be here mostly interested in ubi sums, and we shall present some progress towards the anellation and the uniform distribution of suh sums over the Eisenstein integers, i.e. the integers of the Eisenstein ring Z[ω], where ω = exp2iπ/. Another objet of interest will be the twisted Kloosterman sums. These are analogues of the lassial Kloosterman sums when, in the definition 1, we hoose the harater χ to be the Legendre-Jaobi symbol of order j. By this we mean that these sums, that we shall denote by K j, are given by S X 2 +1 j X,, i.e. K j m, n, = x,x xx 1 x j mx + nx e. 10 We shall study the ases where j = 2 or j =, and speak about quadrati and ubi Kloosterman sums, respetively. We remark that the Kloosterman sums defined in 2 orrespond to the ase where j = 1, and therefore, we shall all K 1 m, n, the lassial Kloosterman sum. The reason for studying the ase where j = is that there exists a lose relation between the ubi exponential sum SaX + bx, and the ubi Kloosterman sum K m, n,. The study of the ase where j = 2 follows the same general steps as for the ubi ase, although the tehniques used are easier to deal with. Let us present the method that we follow: The problem of asymptoti behaviour of Kloosterman sums was undertaken by Kuznetsov, along the lines desribed by Selberg in [40]. This is based on the theory of automorphi forms. We shall parallel this, and ombine spetral properties of automorphi forms with properties of the zeta funtion attahed to the sums K j m, n, to obtain a summation formula for the sums K j m, n,. We shall then naturally be onfronted to the minimal eigenvalue problem whih brings into play theta funtions. For j = 2 these theta funtions are the lassial ones, namely the twists by Dirihlet haraters of the funtion θz = n Z exp iπn 2 z. For j =, we shall work with the ubi analogues of θz, whih are metapleti forms onstruted as residues of Eisenstein series; these funtions are desribed in [2]. Our work aims at improving the results of Livné and Patterson Theorem 1.1 of [1] about the asymptoti behaviour of the ubi Kloosterman sums; we shall also give the quadrati analogue result. As for the uniform distribution problems over primes, it is believed that one aquires a better understanding by working first with integers and then by applying a sieve argument. Atually, the uniform distribution of K 2 m, n, is already proved in [17] over the integers, and in [5] over the primes, but suh results are not known for SaX + bx, ; more surprisingly, even the problem of distribution of the signs of the Kloosterman sums Km, n, remains open. Notie that a result of uniform distribution implies the asymptoti distribution in absolute value; for example, the horizontal Sato-Tate onjeture for Kloosterman sums implies by partial summation 7

8 Introdution K1, 1, p 2 p 4 π p X X log X. Indeed, upper bound and lower bound of this kind, i.e. with absolute value, are possible to derive [10], but the small improvements gained ompared with the trivial estimation show that the anellation expeted among Kloosterman sums is more due to the hange of sign than to the smallness of the norm. A way was found by Fouvry and Mihel [11] to prove that this hange of signs ours for Km, n,, at least for being almost prime, i.e. when the number of primes of is bounded. Our goal is to adapt their method to the ubi exponential sums, and one of the main result is to show that the sum of ubi exponential sums an atually be onsidered as a rest term. We prove this fat by using the theory of metapleti forms, but, as in [15], one ould expet a bias toward the SaX +bx, having a main term due to the existene of an exeptional eigenvalue of the Laplaian. This exeptional term omes from the ubi theta funtions, but an be ontrolled in average over the level, leading to some non-trivial estimate. More preisely, it is expeted that the lassial Kloosterman sums satisfy {p X : Km, n, p > 0} = 1 2 πx + O X 1/2, 11 log X the same being true for the Kloosterman sums of negative sign, and one ould onjeture that the ubi exponential sums satisfy { p X : SaX + bx, p > 0 } = 1 X5/6 πx + C 2 log X + O X 1/2. 12 log X This kind of behaviour was already speulated in [1] p We now desribe the ontent of the thesis in more detail. In Chapter 1 we study the asymptoti distribution of K 2 m, n,. The summation formula over Q is presented in its simple form, i.e. without making expliit the ontribution of the whole spetrum; In this way, we obtain quikly a formula for the asymptoti onstant. In Chapter 2 we give all neessary results about automorphi forms as well as a more omplete summation formula over Qω, where all the spetrum of the Laplaian ours. In Chapter, we study the asymptoti onstant of K m, n,, i.e. we look at the basis problem for ubi theta funtions. This should lead us, in a future work, to the determination of the onstant C appearing in 12. In Chapter 4 we develop some sieve argument to obtain an upper bound for SaX + bx, ; a fundamental role is played by the the omplete summation formula of Chapter 2. Nevertheless, as for the Kloosterman sums Km, n,, the remainder term is of the order of X log X log log X, and hides the ontribution of the theta term. We then use the vertial Sato-Tate law for SaX + bx, to obtain a lower bound, and onlude to the hange of signs when is almost prime. 8

9 Introdution Notations Unless otherwise stated, we make the following onvention: - The inverse of x modulo a given integer will be denoted by x. Therefore, we shall write the sum mx + nx mx + nx χx e as χx e, x,x mod xx 1 mod where the star means the restrition to the representatives x oprime with. - For any omplex number z, the omplex onjugate will be written z. α β - For any matrix g =, we define ag = α, bg = β, g = γ and dg = δ. γ δ x - The symbol δ will be the Kroneker delta symbol, i.e. 1 if a = b, δ a,b = 0 if a b, and similarly, 1 if the assertion P is true, δ P = 0 otherwise. - We shall denote the sign of a real number x by sgnx. 9

10

11 1 Asymptoti behaviour of K 2 m, n, The quadrati Kloosterman sums K 2 m, n, appear in the work of Salié see [8], 54 p. 102, where the following relation is proved: m 2iπx K 2 m, m, p = ε p p exp, p p 2 x 2 4m 2 p where ε d = 1 if d 1 mod 4 and ε d = i id d mod 4. This formula an be generalized, see for example 8 p. 48 of [5]. In this hapter, we study the asymptoti behaviour of the sum K 2 m, n,. We shall use the spetral theory of Maaß forms to obtain the asymptoti formula, and the theta funtions to onstrut an expliit basis of the spae of automorphi forms for whih the eigenvalue of the Laplae operator is minimal. Our main goal in this hapter will be to prove the following theorem. Theorem 1.1. Let f be an odd positive integer and let χ be a primitive Dirihlet harater of ondutor f. Let D be an odd positive integer. Let m, n Z. Then for any ε > 0, we have K 2 m, n, ε χ = CD, χ, m, n X /2 + O X 5/4+ε. D 0 D, 2 X with For C = 0, if f is not square-free, or if the square-free parts of m and n are not equal and divisible by f. If these onditions are met, then m and n have to be of the form for some m = tfs 2 m 2 n = tfs 2 n 2 - square-free t oprime with f suh that t D, - some s suh that s 2 D t and p s p t, - some m, n oprime with t. Then, if and if D t = X 0s 2 X 2, with X 0 square-free and X oprime with t { } U = p : p X, ordp gdx, m, n < ord p X, then 11

12 1 Asymptoti behaviour of K 2 m, n, CD, χ, m, n = 2 π 2 t /2 s ϕt D p Df p p + 1 χ tf χ t m n gdx, m, n U ϕu In Setion 1.1 and in Setion 1.2, we shall work in the general ontext of a disrete subgroup G of SL 2 Z, a real weight k and a multipliative system assoiated to k. In Setion 1.1 the theory of automorphi forms will be developed, and in Setion 1.2 we shall present an argument introdued by Goldfeld and Sarnak, whih will enable us to derive the asymptoti formula in a simpler way as the one developed by Kuznetsov in [26]. Namely, suh an asymptoti formula as the one from Theorem 1.1, is lassially dedued by methods from analyti number theory one one has enough analyti properties of the assoiated zeta funtion. The main ontribution of Goldfeld and Sarnak is, preisely, to obtain the polynomial growth of the Kloosterman-Selberg zeta funtion in the ritial vertial strip. In Setion 1., we shall restrit ourselves to the ongruene subgroups Λ and Γ 0 N of SL 2 Z, fix the weight k = 1/2, and study the theta funtions, following the work of Serre and Stark [42]. Finally, we will see in Setion 1.4 how the results of Setion 1. will allow us to ome bak to our arithmetial problem, i.e. to prove the formula on the asymptoti onstant of Theorem

13 1.1 Automorphi forms on the upper half-plane 1.1 Automorphi forms on the upper half-plane In this setion, we shall first reall the definitions and properties of modular forms and Maaß forms, and then study the Poinaré series. Let us define the angle of a omplex number as a real number in [0, 2π[. In partiular, for any α R, we have z α = z α e iαargz, z C, argz [0, 2π[. We shall use the notation g z = z + d 2, where g = a b d SL2 R. Definition 1.1. Let k be a real number. The fator system of weight k is the appliation The following property holds See [5], 2: σ k : SL 2 R SL 2 R C, g, h gh z k/2 g hz k/2 h z k/2. σ k a, b σ k ab, = σ k a, b σ k b,. Definition 1.2. Let k R and let G be a disrete subgroup of SL 2 R. A multipliative system for G of weight k is an appliation ψ : G C satisfying ψg 1 g 2 = ψg 1 ψg 2 σ k g 1, g 2, g 1, g 2 G. Remark 1.1. Let g, h SL 2 Z. If a funtion f defined on H satisfies g z k/2 f gz = ψg fz h z k/2 f hz = ψh fz for two onstants C g and C h, then one dedues that f satisfies s z k/2 f sz = ψs fz, s g, h, where g, h is the group generated by g and h, and that the appliation s ψs is a multipliative system for the group g, h of weight k. Let us define j g z = g z g z Then, onsidering the equality gh z k = g hz k h z k, one shows that j gh z k/2 = j g hz k/2 j h z k/2 σ k g, h It it possible to determine σ k expliitly. Let us onsider g = a b d a b d. Then σk g, h is defined by, h = a b d and gh = z + d 2 k/2 = hz + d 2 k/2 z + d 2 k/2 σk g, h. Defining for any omplex number z the fator 1

14 1.1 Automorphi forms on the upper half-plane one an shows that e ikπ if 0 < argz π, ωz = e 2ikπ if π < argz 2π, 1 if argz = 0, z 2 k/2 = z k ωz. We obtain e ikarg z+d ω z + d = e ikarghz+d ωhz + d e ikarg z+d ω z + d σ k g, h. One sees that arg z +d arghz+d arg z +d is 0 when, > 0 or when < 0 and < 0, and it is 2π otherwise. The ases = 0, = 0 and = 0 have to be onsidered separately. Define xγ by {, if 0, a b xγ = d, if = 0,, for γ = d Then we obtain the following table: signxg signxh signxgh σg, h e ikπ e ikπ e ikπ e ikπ Maaß forms and modular forms The Poinaré upper half plane is H = R R +. If g = a b d GL + 2 R, the ation of g on H is given by gz = a z + b z + d. A point s R { } is a usp of G if it is fixed by some paraboli element of G. Then G ats on H {usps of G}. We say that z 1 and z 2 are G-equivalent if there is some element g of G suh that z 1 = gz 2. As { } is SL 2 Z-equivalent to Q, any usp s of G an be written as s = σ 1 for some σ 1 SL 2 Z. When working with usps, we will assume that we work with equivalent lasses, i.e. if σ 1 and τ 1 are two given usps, whih are G-equivalent, then we take σ = τ. We make the assumption that Id G and that ψ Id = 1. Let us now define the width of a usp σ 1 of G, with σ SL 2 Z, as the smallest positive integer q σ suh that σ 1 T qσ σ G σ, where G σ is the stabiliser of σ 1. Define also κ σ [0, 1[ by 14

15 1.1 Automorphi forms on the upper half-plane ψ σ 1 1 qσ σ = e κ 0 1 σ. With these notations, for any g σ G σ and for any multipliative system ψ, one has ψg σ = ψ g σ. We also have ψg σ = e nκ σ σ k σ, g σ, if g σ = σ 1 1 nqσ σ Definition 1. modular forms. Let G be a disrete subgroup of SL 2 R and ψ a multipliative system for G of weight k. A modular form is a funtion f : H C, holomorphi on H and at the usps of G whih satisfies g z k/2 f gz = ψg fz z H, g G. The spae of modular forms is denoted by Mod G, k, ψ. Here, the ondition that f is holomorphi at the usps of G means that there exists some α 0 suh that for any σ SL 2 R, σ 1 z k/2 f σ 1 z = O Iz α, as z. If α = 0, then f is said to be a usp form. The Fourier expansion of f is given by σ 1 z k/2 f σ 1 z = a f σ, n e n κ σ z q σ n qσ 1 Z It an be shown that the ondition that f is holomorphi to the usps of G is equivalent with a f σ, n = 0 for n κ σ < 0 and that the ondition that f is a usp form is equivalent with a f σ, n = 0 for n κ σ 0, for every usp σ 1. For any real number k, the Laplaian is defined by = y 2 2 x y 2 iky x. Definition 1.4 Maaß forms. A Maaß form is a funtion f : H C on H of polynomial growth at eah usp of G, eigenvalue of the Laplaian, and whih satisfies j g z k/2 f gz = ψg fz z H, g G. The spae of Maaß forms is denoted by Maaß G, ψ, k. Writing z = x + iy, one sees that a Maaß form f Maaß G, k, ψ has a Fourier expansion j σ 1z k/2 f σ 1 z = F σ, ny e n κ σ x, q σ where n qσ 1 Z F σ, ny = 1 q σ qσ 0 j σ 1z k/2 f σ 1 z e n κ σ x dx q σ 15

16 1.1 Automorphi forms on the upper half-plane This is shown in [7] 2. In the ase of a Maaß form f, we know that F n, σ is a multiple of a Bessel funtion. More preisely if the eigenvalue λ satisfies λ = s1 s, where s is alled the spetral parameter, there exists see [7] p.01 oeffiients ρ f σ, n C suh that, ρ f σ, n W k 4π n n κσ F σ, ny = 2 sgn κσ,s 1/2 q σ y if n 0, qσ ρ f σ, 0 y s + ρ f σ, 0 y 1 s if n = For modular forms, a salar produt is defined by k dx dy f 1, f 2 = f 1 z f 2 z y y 2. G\H Similarly, for Maaß forms, a salar produt is defined by f 1, f 2 = f 1 z f 2 z G\H dx dy y 2. The subspae of modular forms whih are square integrable is L 2 Mod G, ψ, k and the subspae of Maaß forms whih are square integrable is L 2 Maaß G, ψ, k. It is the sum over the eigenvalues λ of the subspaes L 2 λ G, ψ, k of forms suh that + λ f = 0. Moreover, if f L2 Maaß G, k, ψ, then ρ f σ, 0 = 0, in the Fourier expansion It is onjetured that eigenvalues λ = s1 s 1 4 do not our, i.e. that the spetral parameters s lie all on the vertial line it. For a given weight k, what one knows is the following lower bound λ k 1 k The bound is derived from the results of Roelke see Satz 5.4 of [7], or [9] Prop Proposition 1.1. Let k = 1 2. Let G be given and let ψ be a multipliative system of weight 1 2 relative to G. Then there is a bijetion L 2 Mod G\H, ψ, 1 = L 2 G\H, ψ, 1, fz y 1 4 fz. If the Fourier expansion of f is given as in and if the Fourier expansion of y 1 4 f is given as in 1.1.7, then ρ f σ, n = a f σ, n 4π n κ σ 1/4, n 0. Proof. Every modular form f of weight k gives a Maaß form gz := fz y k/2 of weight k and of eigenvalue k 2 1 k 2, and this holds a fortiori over the square integrable forms. In the opposite diretion, the ondition for a Maaß form g of minimal eigenvalue to be sent on a modular form f through fz := gz y k/2 is that the onstant term of g should be of the shape ρ0, σ y k/2. Combined with the ondition for g to be square integrable, we see that a Maaß form g of eigenvalue s 1 s with Rs 1/2 gives rise to a modular form fz := gz y k/2 if and only if 1 s = k/2, i.e. k < 1. In partiular, for k = 1 2, there is a bijetion 16 q σ

17 1.1 Automorphi forms on the upper half-plane L 2 Mod G\H, ψ, 1 = L 2 G\H, ψ, 1, fz y 1 4 fz. Suppose that f is given by fσ 1 z = σ 1 z 1/4 a f σ, 0 + a f σ, n enz, then fσ 1 ziσ 1 z 1/4 = j σ 1z 1/4 a f σ, 0y 1/4 + a f σ, ny 1/4 enz On the other side, from the formula we obtain, for any Maaß form g, y 1 4 W 1 4 sgny, 1 4 y = e y/2, g σ 1 z = j σ 1z 1/4 = j σ 1z 1/4 = j σ 1z 1/4 ρ g σ, 0 y 1 4 ρ g σ, 0 y n Z + 0 n Z ρ g σ, 0 y πy 1/4 ρ g σ, n W 1 4 sgnn, 1 4π n y enx 4 ρ g σ, n 4πny 1/4 e 2πny enx 0 n Z ρ g σ, n n 1/4 enz Sine the two expressions in and are equal, we arrive to the relation a f σ, n = ρ f σ, n 4π n 1/4, n Poinaré series As analogues of the non holomorphi Poinaré series known sine Petersson, we present here the Poinaré series as they were introdued by Selberg in [40]. Let σ 1 be a usp of G, σ 1 SL 2 Z. For m qσ 1 Z {0} one defines f m,σ z, s = y s e m κ σ x exp 2π q σ m κ σ q σ y The Poinaré series assoiated to m and to the usp σ 1 is given by P m,σ z, s = g G σ\g ψg σ k σ, g j σg z k/2 f m,σ σgz, s, z H, s C. 17

18 1.1 Automorphi forms on the upper half-plane One verifies that these series are well defined using As a funtion of s, P m,σ z, s is holomorphi in Rs > 1 and as a funtion of z, P m,σ z, s satisfies j g z k/2 f gz = ψg fz z H, g G. Moreover it lies in L 2, but it is not an eigenfuntion of. Atually, it satisfies [ + s1 s ] Pm,σ z, s = 4πm s k 2 Pm,σ z, s As the disrete spetrum of the Laplaian intersets [1/2, 1] in a finite set, R s1 s is holomorphi in Rs 1/2 with at most a finite number of poles in [1/2, 1]. This shows the analyti ontinuation of P m,σ z, s to Rs > 1 2, with a finite number of poles, whih are the spetral parameters of. Inherited from those of P m,σ z, s, Res s=si P m,σ z, s posses the properties of transformation aording to G and to be square integrable. Moreover, Res s=si P m,σ z, s is an eigenfuntion of the Laplaian for the spetral parameter s i. Thus Res s=si P m,σ z, s L 2 λ i G\H, ψ, k, whih means that if {u} forms an orthonormal basis of it, then Res s=si P m,σ z, s = u Res s=si P m,σ z, s, uz uz = u Res s=si P m,σ z, s, uz uz. Proposition 1.2. Let f L 2 λ G\H, ψ, k, with λ = s f 1 s f. Let σ 1 be an essential usp of G. Let the Fourier expansion of f be given by Then fσ 1 z =j σ 1z k/2 ρ f σ, 0 y 1 s f + 0 n Λ σ ρ f σ, n W k 4π n 2 sgn κσ, 1 qσ 2 s f n κ σ y q σ e n κ σ x. q σ P m,σ, s, f = q σ e ikπ ρ f σ, m 4π m κ σ 1 s Γs s f Γs + s f 1 q σ Γ s k 2 sgn m κσ q σ and if B i denotes an orthonormal basis of L 2 λ i G\H, ψ, k, with λ i = s i 1 s i for a spetral parameter s i, then Res s=si P m,σ z, s = q σ e 4π ikπ m κ σ 1 si Γ2 si 1 q σ ρ u σ, m uz. Γ s i k 2 sgn m κσ q σ u B i where ρ u σ, is the Fourier oeffiient of u at σ 1. It is known that twisted Kloosterman sums arise as Fourier oeffiients of Poinaré series. To show this, we need to define a geometri analogue of the Kloosterman sums K 2 m, n,. Definition 1.5. Let σ 1 and τ 1 be two essential usps of G. Let m Λ σ {0} and n Λ τ {0}. Then, for any Z, we define 18

19 1.1 Automorphi forms on the upper half-plane g Gσ\G/Gτ σgτ 1 = K σ,τ m, n, = m κσ ψg σ k σ, g σ k σg, τ 1 q e σ aσgτ 1 e n κτ q τ dσgτ 1. Remark 1.2. The sum K σ,τ m, n, will be the geometrial analogue of the sums K 2 m, n,, one we have hosen a suitable multipliative system ψ and a onvenient group G. We shall use the same notation in Chapter 2 and in Chapter, for the analogue of the sum K m, n,, but the ontext should make lear to whih we refer. Proposition 1.. Let σ 1 and τ 1 be two essential usps of G. Let m Λ σ {0}. The Poinaré series P m,σ z, s possesses at τ 1 a Fourier expansion τ 1 z k/2 P m,σ τ 1 z, s = F n, τy e n κ τ x q τ n qτ 1 Z with F n, τy = δ σ,τ δ m,n e ikπ y s exp 2π m κ σ y + e ikπ y 1 s >0 2s K σ,τ m, n, 1 q τ exp 2π q σ e ikargt+i t s e m κ σ q σ m κ σ q σ t 2 y t y t 2 e n κ τ yt dt. + 1 q τ Proposition 1.4. Let σ 1 and τ 1 be two essential usps of G. Let m, n Λ σ {0}. Let s, t C with Rs, Rt > 1. Then the salar produt of the two Poinaré series P m,σ z, s and P n,τ z, t is given by P m,σ, s, P n,τ, t = δ σ,τ δ m,n q τ 2π m κ σ + n κ τ 1 s t Γ t + s 1 + with >0 K σ,τ m, n, 2s 0 q σ q τ y t s 1 exp 2πy n κ τ Im, j,, y dy, Im, j,, y = e ikargu+i u s e u m m κσ q σ 2 y u 2 exp 2π κ σ q σ y u 2 e n κ τ yu du. + 1 q τ q τ 19

20 1.2 The Goldfeld-Sarnak method 1.2 The Goldfeld-Sarnak method In this setion we still work with a disrete subgroup G of SL 2 R and a multipliative system ψ of weight k. The goal is to obtain a first formula for the asymptoti behaviour of the funtion K σ,τ m, n,. In analyti number theory, one possibility to prove the asymptoti behaviour for an arithmeti funtion t, is to use the analyti properties of its zeta funtion t s. In our ase, the diffiulty omes from the lak of information about the Selberg-Kloosterman zeta funtion Z m,n s. However, the Kuznetsov formula for Kloosterman sums whih led to the formula 4 of Introdution, an also be developed for twisted Kloosterman sums and, as onsequene of the omplete summation formula, one obtains the asymptoti behaviour. All details were given by Proskurin in [6]. Nevertheless, it is possible to derive the desired properties of Z m,n s, the most diffiult being the growth ondition in vertial strip; this was ahieved by Goldfeld and Sarnak in a short and elegant paper see [12]. In the ase of the sums K m, n,, it is still possible to apply suh a method see [1] but sine we shall need all spetral information, we shall have to deal with the omplete formula. In the first part of this setion we shall summarize the ideas of [12]; it onsists in an estimate for the Laplaian, as well as the lassial mahinery build on the Poinaré series. In the seond part, the asymptoti formula is derived in Theorem The Kloosterman-Selberg zeta funtion In the formula of Proposition 1.4, the integral on the right side is given by 0 = = + y t s 1 exp 2πy 0 n κ τ Im, j,, y dy q τ y t s 1 exp 2πy n κ τ e ikargu+i q τ u s e u m m κσ q σ 2 y u 2 exp 2π κ σ q σ y u 2 e y t s 1 exp 2πy n κ τ e ikargu+i q τ u s e u m m κσ q σ 2 y u 2 exp 2π κ σ q σ y u e 0 y t s 1 exp 2πy n κ τ e ikargu+i u s e q τ n κ τ yu du dy q τ n κ τ yu du dy q τ n κ τ yu du dy. q τ Denoting by I 1 the first double integral and by I 2 the seond double integral 1.2.2, independent of. We have shown that 20

21 1.2 The Goldfeld-Sarnak method P m,σ, s, P n,τ, t = δ σ,τ δ m,n q τ 2π m κ σ + n κ τ 1 s t Γ t + s 1 q σ q τ + >0 K σ,τ m, n, 2s I 1 + I 2 Z σ,τ m, n, s The estimation of I 2 is easy to handle. Lemma 1.1. If t = s + 2, then I 2 = n κ τ 4π q τ 2 e ikπ/2 2 2s Γ2s + 1. Γ s + k 2 sgn n κτ q τ Γ s + 2 k 2 sgn n κτ q τ The goal is then to find an upper bound for I 1, whih makes the sum over the s in 1.2. onverge. In 1.2.1, we make appear the dependane in by using the estimate expz 1 1 z 1 whenever z 1. Thus, introduing a onstant α > 0 whih we shall hoose later, we have I 1 2 αt s t s u s 0 { α u s 2 y t s 1 dy + 0 y t s 1 exp 2iπ u im κ 1 2 qyu u s du + 2 u s 1 2 α 1 2πyn exp κ2 q y t s 1 } 2πyn 2 y u exp κ2 dy du q α dy du 2πyn y t s 2 κ2 exp dy du. q The first integral onverges for Rs > 1 2 and the seond integral for Rt s > 1. From this we obtain that 1 I 1 Rs 1 2 α Rt s + 2. We hoose α = 1 and, as in the Lemma 1.1 above, t = s + 2; ombined with the trivial estimate for Kloosterman sums, it shows that >0 K σ,τ m, n, 2s I 1 is holomorphi in Rs > 1 2 and is bounded by Rs We an reformulate this as follows: Z σ,τ 2s 4π e ikπ/2 n κ τ q τ 2 Γ s + sgnn k 2 Γ s + 2 sgnn k 2 2 2s P m,σ, s, P n,τ, s + 2 Γ2s + 1 is holomorphi in Rs > 1 2 and bounded by Rs Therefore the possible poles of Z σ,τ s are loated at s = 2s i, for s i an exeptional spetral parameter. Moreover, for Rs > 1 2, 21

22 1.2 The Goldfeld-Sarnak method P n,τ, s + 2 is bounded, thus one will have all neessary properties of Z σ,τ m, n, one we possess an upper bound for the Poinaré series in the vertial strip 1 2 < Rs < 1. This is proved in [12], using the property Namely, from the upper bound one obtains R λ P m,σ z, s = O 1 distaneλ, spetrum, Rs The above disussion is gathered in the following proposition: Proposition 1.5. Let 0 < k < 1 and let G, ψ, σ, τ, m, n as above. Then The Kloosterman-Selberg zeta funtion Z σ,τ s defined by has the following properties: Z σ,τ s = >0 K σ,τ m, n, s - holomorphy in Rs > 2, - meromorphy in Rs 1, with polynomial growth s Z σ,τ s = O, Rs 1 - poles at s = 2s i, with residue Res s=2si Z σ,τ s =e ikπ/2 4 1 s i π 2s i 1/2 q σ q τ Γ2s i 1 Γs i + sgnn k 2 Γs i sgnm k 2 n κ τ m κ σ q τ u BMod where BMod is an orthonormal basis of L 2 Mod Γ\H, k, ψ. q σ /4 si a u σ, m a u τ, n, Proof. Everything has been already proved above, exept the last statement about the residues, that we prove using Proposition 1.2 and Propositon Asymptoti behaviour The goal of this setion is to derive an asymptoti formula for the funtion K σ,τ m, n, from the analyti properties of its zeta funtion Z m,n s. This argument an be found in [12] or in [1] in the ase of ubi Kloosterman sums K m, n,. We write the exeptional spetral parameters as s 1 > s 2 >... > 1 2. Theorem 1.2. Let 0 < k < 1 and let G, ψ, σ, τ, m, n as above. Then for any ε > 0, 0<<X K σ,τ m, n, β Res s=2s 1 Z σ,τ s 2s 1 β 2s X 1 β + O X max2s 2,5/4+ε β. 22

23 1.2 The Goldfeld-Sarnak method Proof. Let α ]0, 1[ and let ω 1 > α+1 and ω 2 > α+2; onsider the ounterlokwise integral of X Zs α s ss 1 around the retangle with verties ω 1 it, ω 1 +it, ω 2 +it and ω 2 it. From the Phragmén-Lindelhöf Theorem and Proposition 1.5, one has Z σ,τ s = O Is Φs, for a linear funtion Φ satisfying Φ1 + ε = 1 and Φ2 + ε = 0. On the one side, as T goes to infinity, it remains the integral on the vertial lines ω 1 and ω 2. On the other side, by the Cauhy theorem, this is equal to s i R i. Thus, 1 2iπ ω 2 Zs α X s ss 1 ds = X 2si+α R i 2s s i + α 2s i + α X s Zs α 2iπ i ω 1 ss 1 ds, where the sum on the right hand side is taken over the exeptional spetral parameter s 1 > s 2 >... > s i >... > 1 2 and R i = Res s=2si+α Z σ,τ s α. As ω 2 > α + 2, the left hand side onverges, and we an interhange integral and summation. We obtain 1 2iπ ω 2 X s Zs α ss 1 ds = 0< = 0< X K σ,τ m, n, α 1 X/ s 2iπ β ss 1 ds K σ,τ m, n, 1 α X. On the right side, the integral over ω 1 is bounded by X ω 1. Comparing both sides of the equality gives 0<<X K α1,α 2 m, n, 1 α X = s i R i X 2si+α 2s i + α 2s i + α 1 + Xω 1. X We now differentiate this equation. Let 1 << X << X. Substrating X from X + X, we obtain 0<<X K α1,α 2 m, n, 1 α X + X <X+ X K α1,α 2 m, n, 1 α X + X = s i R i X + X 2s i+α X 2s i+α 2s i + α 2s i + α 1 + O X ω 1. The seond sum on the link hand side is bounded by O X α 1/2 X 2. The term orresponding to eah s i in the right hand side is equal to Dividing by X, we obtain 0<<X K α1,α 2 m, n, 1 α = R i 2s i + α 1 X2s i+α 1 X + O X 2s i+α 2 X 2. R 1 2s 1 + α 1 X2s 1+α 1 +OX α 1/2 X+O X 2s 2+α 1 + X ω 1 X 1. 2

24 1.2 The Goldfeld-Sarnak method Choose ω 1 = α ε; then X has to be hosen equal to X /4, and, writing β = 1 α, we obtain 0<<X This finishes the proof of Theorem 1.2. K α1,α 2 m, n, β = R 1 2s 1 β X2s 1 β + O X max2s 2,5/4+ε β. Remark 1.. We know that some spetral gap ours in the exeptional spetrum. For example, Goldfeld and Sarnak see [9], Theorem.6 proved that when G = Γ 0 4N, k = 1 2 and ψ is the multipliative fator assoiated to the theta funtion see next setion then s

25 1. Theta funtions of half-integral weight 1. Theta funtions of half-integral weight In this setion, we shall restrit ourselves to ongruene subgroups Γ 0 N of SL 2 R, to the weight k = 1 2, and to fator systems ψ of the shape κχ, where κ is defined in 1..5 and χ is a primitive Dirihlet harater. Under these onditions, we shall give an expliit orthonormal basis of the modular forms, i.e. of the minimal eigenspae of Maaß forms; this will then allow us to determine expliitly the right hand side of the formula in Theorem 1.2. The main ingredient will be the lassial theta funtion, and we start with some fats about it. It is known that the funtion θz = n Z eiπn2z satisfies always with the hoie of the argument of a omplex number in [0, 2π[ θz + 2 = θz 1..1 i 1 θz = z θ z By Remark 1.1 of Setion 1.1, this implies a modularity property of θ for the group Λ = T 2, S, where T = et S = Proposition 1.6. Let γ Λ. There exists a funtion κ θ on Λ suh that γ z 1/4 θ γz = κ θ γ θz z H, γ Λ. 1.. Beause of κ θ Id = 1, κ θ is determined by its values on the elements γ = a b d Λ, with d > 0; on suh an element, it holds { 2b i if > 0 ε d for even and b 0, d 2 1 if 0 { i if > 0 for even and b = 0, κ θ γ = 1 if 0 e iπ 2a 4 ε for odd and a 0, 2 e iπ 4 for odd and a = 0, where ε x = 1 if x 1 mod 4 and ε x = i if x mod Proof. This property of κ θ on Λ was proved by Kubota in [25]. This result is in fat onsiderably older; atually, it was proved by Hermite see [16] and then by Weber see [44] Our result is different from the result of [25], beause of the hoie of the branh of g z 1/4. This makes from θ a modular form of weight 1 2 for the group Λ. For onveniene, we wish to work in the ontext of ongruene subgroups; for it we define ϑz = θ2z. If we define, for an element γ = a b d, an element γt by γ t = a tb /t d, then 25

26 1. Theta funtions of half-integral weight γ z 1/4 ϑ γz = γ 22z 1/4 θ γ 2 2z γ Γ 0 4 = κ θ γ 2 ϑz beause γ 2 Λ. One sees then that γ κ θ γ 2 =: κγ is a multipliative system for the group Γ 0 4 and for the weight 1 2. The result of the last proposition gives then, for γ Γ 04, { b i if > 0 ε d d 2 1 if 0 κγ = { i if > 0 1 if 0 for even and b 0, for even and b = Thus ϑ is an element of L 2 Mod Γ 04N, 1/2, κχ. By Proposition 1.1 of Setion 1.1.1, we know that y 1/4 ϑ is a non uspidal Maaß form of eigenvalue /16, whih is the smallest possible, by the formula Reall also that for any Dirihlet harater χ modulo 4N, κχ an be made as a multipliative system for Γ 0 4N, by defining χγ as χd, if γ = a b d. This is in partiular true with any Dirihlet harater χ modulo f, with f N. The main result of the last setion, Theorem 1.2, an be applied in this ontext. It gives the following theorem: Theorem 1.. Let χ be a primitive Dirihlet harater modulo f. Let σ 1 and τ 1 be two essential usps of Γ 0 4N, and let m Λ 1 σ {0} and n Λ 1 τ {0}. Let K σ,τ m, n, be the Kloosterman sum assoiated to the multipliative system κχ. Then, if N is an integer so that f N, we have X /2 X K σ,τ m, n, i π q σ q τ sgnm π if sgnm = sgnn 1 if sgnm sgnn a u σ, m a u τ, n, u where u belongs to an orthonormal basis of the spae L 2 Mod Γ 04N, 1/2, κχ and a u, are defined by The rest of this setion is devoted to find an orthonormal basis of L 2 Mod Γ 04N, 1/2, κχ. We introdue theta funtion twisted by a Dirihlet harater, ϑ χ z = n χn e n 2 z We remark that, with the notations of 1.1.5, a ϑχ Id, 0 = 0. This will be needed latter on A result of Serre and Stark One of the main results of Serre and Stark [42] is the fat that, for any Dirihlet harater χ, the spae of modular forms Mod Γ 0 4ondχ 2, χ, 1 2 only ontains one newform, ϑχ. From this, they dedue that there exists a basis of Mod Γ 0 4N, χ, 1 2 formed by theta funtions. Before to state this result, let us introdue some notations. 26

27 1. Theta funtions of half-integral weight In the theory of half-integral modular forms, one has non trivial Heke operators, not for any prime p, but for squares p 2. The Heke operators T p 2 are a defined on Mod Γ 0 4N\H, 1 2, κ, χ. Their ation on a modular form is given by fz = a f n enz n=0 with T p 2fz = a f np 2 b f n = a f np 2 + χp n p p b f n enz, n=0 if p 2N, a f n + χ2 p p a f n/p 2 if p 2N. A useful property of the Heke operators is that χpt p 2 is hermitian; if p 2N, then f T p 2, g = χ 2 p f, g T p 2. The funtion θ χ is an eigenfuntion of any operator T p 2 for p 2N of eigenvalue χp 1 + p 1. For t odd, the Kroneker symbol χ t is the Dirihlet harater n t n of ondutor t or 4t 2 aording to whether t 1 mod 4 or t mod 4. Now let us define an operator V t : Mod Γ 0 4N, χ, 1 Mod Γ 0 4Nt, χχ t, fz ftz. Then, V t and T p 2 ommute if p t. To any Dirihlet harater χ of modulus N, there is an assoiated primitive Dirihlet harater χ of modulus the ondutor of χ, written f χ ; for two Dirihlet haraters χ 1 and χ 2, when we write χ 1 χ 2 we always mean the primitive Dirihlet harater assoiated to the produt of χ 1 and χ 2. Finally, we reall the definition of newform and oldform. Let f Mod Γ 0 4N\H, 1 2, κχ be an eigenform of all but finitely many T p 2. We say that f is an oldform if there exists some prime p N suh that, either f Mod Γ 0 4N/p\H, 1 2, κχ, or f = V p g, for some g Mod Γ 0 4N/p\H, 1 2, κχχ p ; if f is not an oldform, it is said to be a newform. Let New Γ 0 4N\H, 1 2, κχ be the spae spanned by newform. Then Serre and Stark proved [42], Theorem that New Γ 0 4ondχ\H, 1 2, κχ is one dimensional, generated by ϑ χ. This allowed them to prove that any modular form of halfintegral weight is a ombination of theta series. Theorem 1.4. Let N N. Let χ be a primitive Dirihlet harater modulo N. A basis of the spae Mod Γ 0 4N\H, 1 2, κχ is given by the family V du 2 ϑ χχd, where d and u are submitted to the onditions d square-free and fd 2 d N D, u 2 N D fd 2 d d. 27

28 1. Theta funtions of half-integral weight Proof. This is a reformulation of Theorem A of [42], whih states that a basis of the spae Mod Γ 0 4N\H, κχ, 1 2 is given by the family {V t ϑ ψ } ψ,t, where i ψ is a primitive Dirihlet harater, ii ψχ t = χ, as group homomorphisms on Γ 0 4N, iii f ψ 2 t N. Replaing the ondition ii by ψ = χ χ t, we see that iii is equivalent with f χχt 2 t N. Let us deompose t = du 2, with d square-free; then χ t = χ d, and as χ is now fixed, we simplify notations by writing ft for f χχt, the ondutor of χχ t. Corollary 1.1. Let D, f be odd positive integers. Let χ be a primitive Dirihlet harater of ondutor f. The spae Mod Γ 0 4Df, κχ, 1 2 is non-empty only if χ = χf with f 1 mod 4. Any d satisfying D has to be a multiple of f, say d = ft, and χχ tf = χ t. Proof. Let f = p f i i, p i odd. Let d be suh that 4 ondχχ d 2 d 4Df. Then d has to be odd. We then use the produt deomposition with χ i a harater of ondutor p f i i χ = χ i,. Also, we have χ d = ɛ p d where ɛ is the trivial harater if d 1 mod 4 and ɛ is the non trivial harater modulo 4 if d mod 4. Then, χχ d = ɛ p i p, 2 χ i χ p. If some p i does not divide d, then p f i i ondχχ d and therefore p 2f i i divides Df; but f i = ord pi f = ord pi Df. Thus all p i divide d. Moreover, ondχ i χ pi = p f i i, exept if χ i = χ pi, and we obtain the same ontradition as previously, if some χ i χ pi ; thus, χ = χ f. In partiular, f is squarefree and f divides d. As we work with even haraters, we need f 1 mod 4. Let us write d = ft, with t square-free and gdt, f = 1. Then, χ χ d = ɛ p i χ i and the primitive harater assoiated to χχ d is χ χ d = ɛ p d, p f p d, p t p d χ p p d, p t but as the ondutor has to be odd, we need d 1 mod 4 i.e. t f 1 mod 4. One obtains therefore χχ d = χ t, of ondutor t. We remark that the argument used in this proof is no more valid if we onsider the larger spae Mod Γ 0 4Df i, κχ, 1 2, for some i χ p, χ p

29 1. Theta funtions of half-integral weight 1..2 Orthonormalisation of theta funtions Among the set of theta funtions V du 2 ϑ χχd forming a basis of Mod Γ 0 4N\H, 1 2, κχ see Theorem 1.4, some are orthogonal and some are not. The following lemmas desribe preisely the salar produt of two theta funtions. Lemma 1.2. Let d and d satisfy D, and let u and u satisfy D d and D d, respetively. Assume d d. Then V du 2 ϑ χχd, V d u 2 ϑ χχ d = 0. Proof. Sine d and d are square-free, we an hoose p 2N suh that χ d p χ d p. We use the fat that ϑ χχd belongs to New4fd 2, χχ d, and is an eigenfuntion of T p 2 with eigenvalue χχ d p1 + p 1. The operators V du 2 and T p 2 ommute, thus V du 2 ϑ χχd is an eigenfuntion of T p 2, for the same eigenvalue. Then, χp χχ d p 1 + p 1 V du 2ϑ χχd, V d u 2ϑ χχ d = T p 2χpV du 2ϑ χχ d, V d u 2ϑ χχ d = V du 2ϑ χχd, T p 2χpV d u 2ϑ χχ d = χpχχ d p 1 + p 1 V du 2ϑ χχd, V d u 2ϑ χχ d. On the one side, χp χχ d p = χ d p, and on the other side, χpχχ d p = χ d p χ d p, as χ d p is real. One remarks that we used three different Heke operators T p 2, eah one being defined aording to a different harater. Let now d be a fixed integer, satisfying the ondition D. We study the set of funtions V du 2 θ χχd, where u satisfies the ondition D d and ompute the salar produt by the following lemma. Lemma 1. Rankin-Selberg. Let Γ be a subgroup of SL 2 Z. Let Ez, s = γ Γ \Γ Iγzs. Let f, g ModΓ, k, χ, and denote by a f n and a g n their Fourier oeffiients at. Assume that a f 0 a g 0 = 0. Then f, g E, s = Γs + k 1 n>0 an bn 4πn k+s 1. Proof. It is the usual unfolding method, one one remarks that fz gz Iz is Γ-invariant. N Define fd 2 d = X, and define a divisor X d of X as X d = p i fd pe i i, where X = p e i i ; X d will be said to be the divisor of X supported by fd. Lemma 1.4. With the notations as above, let d satisfy Condition D and let u and u satisfy Condition D d, i.e. u 2 X and u 2 X. Denote by s and s, the divisors of u and u, respetively, supported by fd. Then { 0 if s s, V du 2 ϑ χχd z, V du 2 ϑ χχd z = CN, d χχ u d s χχ u gdu,u d s u u if s = s, where we defined the onstant CN, d = π N 2 d p N p + 1 p p fd p 1 p. 29

30 1. Theta funtions of half-integral weight Proof. Lemma 1. applied with k = 1 2 and Γ = Γ 04N gives with V du 2 ϑ χχd, V du 2 ϑ χχd E, s = Γs 1 2 n an a n, 4πn s 1 2 χχ n d if n du 2 Z 2, du an = 2 0 otherwise, and a n = { χχ d n du 2 if n du 2 Z 2 0 otherwise. Let g = gdu, u ; then n has to belong to d u2 u 2 g 2 V du 2 ϑ χχd, V du 2 ϑ χχd E, s = Γs 1 2 m Z 2, and we obtain χχ u m d g 4πdm 2 u 2 u 2 g 2 χχ d um g s 1 2 = Γs 1 u u 4πdu 2 2 χχ u 2 1/2 s χχ d n χχ d n d χχ d g g g 2 n 2 s 1 2 = Γs 1 2 χχ d u Taking the residue at s = 1, we get g u 4πdu 2 u 2 χχ d g g 2 n 1/2 s n>0 n,fd=1 1. n 2 s 1 2 = Res s=1 Ez, s 1 χχ d u = Res s=1 Ez, s 1 χχ d u V du 2 ϑ χχd, V du 2 ϑ χχd g g u χχ d g u χχ d g g 2 u u d Res s=1 g 4 u u d p fd p 1 p. n,fd=1 1 n 2 s 1 2 Let s, s and g d be the divisors of u, u and g, respetively, with support in fd. Then, χχ u d g χχ u d g 0 if and only if g d = s = s. Moreover, χχ u d g χχ u d g = χχ d u /s χχ d u/s. Finally, one shows that Res s=1 Ez, s = π Res s=1,4n=1 = π 4N Res s=1 p 4N ϕ 2s 1 p 2s 1 p 1 2s = 1 2 π N p N p p

31 1. Theta funtions of half-integral weight Let d satisfy D and write X = N fd 2 d ; let X d be the divisor of X supported by fd and write X d = X/X d. Let now s satisfy the ondition D d, with s supported by fd, i.e. s 2 X d. Let finally u and v satisfy u 2, v 2 X d. Then the preeding lemma gives This leads to define These funtions satisfy V ds 2 u 2 ϑ χχ d z, V ds 2 v 2 ϑ χχ d z = CN, d χχ d v χχ d u gdu, v. s u v ϑ 1 d,s,u = CN, d 1/2 s 1 2 u χχd u V ds 2 u 2 ϑ χχ d ϑ 1 d,s,u, ϑ1 d,s,v = gdu, v, for any u, v so that u2, v 2 X d For a ouple d, s as above, we have to orthogonalise the set {ϑ 1 d,s,u }, with u2 X d. This is ahieved by the Möbius formula u µj f = gu fu = gj j j u j u Let us define the funtion ϑ d,s,u = ϕu 1/2 j u µj ϑ 1 d,s, u j Lemma 1.5. Let d satisfy D. Let s satisfy D d and assume that s is supported by fd. Let u, v satisfy D d and assume gdu, fd = gdv, fd = 1. Then { 1 if u = v, ϑ d,s,u, ϑ d,s,v = 0 if u v. Proof. By definition of ϑ d,s,u, the equality to be proven is equivalent to ϕu { 1/2 µj ϑ 1 1 if u = v, d,s, u, ϑ d,s,v = j 0 if u v, j u that we rewrite as By hoosing µj ϑ 1 d,s, u, ϑ d,s,v = j j u fx = ϑ 1 d,s,x, ϑ d,s,v and gx = { ϕu 1/2 if u = v, 0 if u v. in the Möbius formula 1..8, is equivalent to ϑ 1 d,s,u, ϑ d,s,v = { ϕj 1/2 si j = v, 0 si j v. j u { ϕx 1/2 if x = v, 0 if x v,

32 1. Theta funtions of half-integral weight By the definition 1..9 of ϑ d,s,v, and by evaluating the right hand side of 1..11, the equality of Lemma 1.5 is equivalent to ϕv 1/2 j v µj ϑ 1 d,s,u, ϑ1 d,s, v = j { ϕv 1/2 si v u, 0 si v u, what we reformulate as j v We apply the Möbius formula with µj ϑ 1 d,s,u, ϑ1 d,s, v = j { ϕv si v u, 0 si v u. { fx = ϑ 1 d,s,u, ϑ1 d,s,x and gx = ϕx if x u, 0 if x u. Lemma 1.5 is then proved if and only if ϑ 1 d,s,u, ϑ1 d,s,v = j v { ϕj if j u 0, si j u The right hand side is equal to j gdu,v ϕj = gdu, v, therefore is verified and the lemma is proved. Theorem 1.5. Let N N. Let χ be a primitive Dirihlet harater modulo f. Denote by fd the ondutor of the primitive Dirihlet harater assoiated to the produt of the haraters χ and χ d. Define the following onstant: N, d, s, u = 2 d π N p N p p + 1 p fd p p 1 1/2 s 1 2 ϕu 1/ Then an orthonormal basis of Mod 4N, χ, 1 2 is given by the set of funtions ϑd,s,u z defined in 1..9, where the parameters d, s, u satisfy i d square-free, fd 2 d N, ii s 2 N fd 2 d, supps suppfd, iii u 2 N fd 2, gdu, fd = 1. d Their Fourier expansions at infinity are given by ϑ d,s,u z = N, d, s, u a d,s,u Id, m emz m Z 2

33 1. Theta funtions of half-integral weight with Fourier oeffiients { 0 if m / ds 2 Z 2, a d,s,u Id, m = χ t m j u,m µ u j j if m = ds 2 m 2. Proof. As before, write X = N fd 2 d and deompose X = X d X d, with X d supported by d. It remains to study the Fourier expansion of ϑ d,s,u. By 1..6 and 1..9, on a ϑ d,s,u = CN, d 1/2 s 1 2 ϕu 1/2 µj u u j χχ d V j ds 2 u/j 2ϑ χχ d j u = N, d, s, u j u µ u j χχ d j V j ds 2 j 2ϑ χχ d. By the definition of ϑ χχd, we obtain ϑ d,s,u z = d, s, u j u µ u j χχ d j j n χχ d nen 2 ds 2 j 2 z = N, d, s, u n χχ d n j u µ u j χχ d j en 2 ds 2 j 2 z j = N, d, s, u m a m em 2 ds 2 z, where a m = j u,m χχ d m/jµ u j χχ d j. j Sine u 2 X d, then gdu, fd = 1, and therefore, for any j u, we have χχ dm/j = χχ d mχχ d j. Thus a m = χχ d m u µ j. j j u,m Corollary 1.2. Let D, f be odd positive integers, gdd, f = 1. Let χ be a primitive harater of ondutor f. The spae Mod Γ 0 4Df, κχ, 1 2 is non-empty if and only if f 1 mod 4 and χ = J f = χ f, in whih ase an orthonormal basis of it is given by the ϑ tf,s,u z, for t D, t 1 mod 4, s 2 D t, s supported by t, u 2 D t, gdu, t = 1. For suh parameter t, we have χχ tf = J t = χ t. Proof. Reall that from Corollary 1.1, we know that d 0 mod f, for any d satisfying D. Writing d = ft and translating the onditions of Theorem 1.5 we get the result as stated.