STATISTICS FOR LOW-LYING ZEROS OF SYMMETRIC POWER L-FUNCTIONS IN THE LEVEL ASPECT

Transcription

1 STATISTICS FO LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS IN THE LEVEL ASPECT GUILLAUME ICOTTA AND EMMANUEL OYE ABSTACT We stdy one-level and two-level densities for low lying zeros of symmetric power L-fnctions in the level aspect It allows s to completely determine the symmetry types of some families of symmetric power L-fnctions with prescribed sign of fnctional eation We also compte the moments of one-level density and exhibit mock-gassian behavior discovered by Hghes & dnick CONTENTS Introdction and statement of the reslts Description of the families of L-fnctions stdied Symmetry type of these families 3 Signed asymptotic expectation of the one-level density 4 Sketch of the proof 5 3 Signed asymptotic expectation of the two-level density 6 3 Asymptotic moments of the one-level density 8 4 Organisation of the paper 9 Atomorphic and probabilistic backgrond 0 Atomorphic backgrond 0 Overview of holomorphic csp forms 0 Chebyshev polynomials and Hecke eigenvales 3 Overview of L-fnctions associated to primitive csp forms 3 4 Overview of symmetric power L-fnctions 3 Probabilistic backgrond 4 3 Main technical ingredients of this work 6 3 Large sieve inealities for Kloosterman sms 6 3 iemann s explicit formla for symmetric power L-fnctions 8 33 Contribtion of the old forms 0 4 Linear statistics for low-lying zeros 4 Density reslts for families of L-fnctions 4 Asymptotic expectation of the one-level density 3 43 Signed asymptotic expectation of the one-level density 6 5 Qadratic statistics for low-lying zeros 7 5 Asymptotic expectation of the two-level density and asymptotic variance 7 5 Signed asymptotic expectation of the two-level density and signed asymptotic variance 30 6 First asymptotic moments of the one-level density 3 Date: Version of March 6, 007

2 G ICOTTA AND E OYE 6 One some sefl combinatorial identity 33 6 Proof of the first bllet of proposition Proof of the third bllet of proposition Proof of the second bllet of proposition 6 38 Appendix A Analytic and arithmetic toolbox 4 A On smooth dyadic partitions of nity 4 A On Bessel fnctions 43 A3 Basic facts on Kloosterman sms 43 eferences 43 Acknowledgements This work began while both athors shared the hospitality of Centre de echerches Mathématies Montréal dring the theme year "Analysis in Nmber Theory" first semester of 006 We wold like to thank C David, H Darmon and A Granville for their invitation This work was essentially completed in september 006 in CIM Lminy at the occasion of J-M Deshoillers sixtieth birthday We wold like to wish him the best INTODUCTION AND STATEMENT OF THE ESULTS Description of the families of L-fnctions stdied The prpose of this paper is to compte varios statistics associated to low-lying zeros of several families of symmetric power L-fnctions in the level aspect First of all, we give a short description of these families To any primitive holomorphic csp form f of prime level and even weight κ see for the atomorphic backgrond say f Hκ, one can associate its r -th symmetric power L-fnction denoted by LSym r f, s for any integer r It is given by an explicit absoltely convergent Eler prodct of degree r + on e s > see 4 The completed L-fnction is defined by ΛSym r f, s := r s/ L Sym r f, slsym r f, s where L Sym r f, s is a prodct of r + explicit Γ -factors see 4 and r is the arithmetic condctor We will need some control on the analytic behavior of this fnction Unfortnately, sch information is not crrently known in all generality We sm p or main assmption in the following statement Hypothesis Nicer, f The fnction Λ Sym r f, s is a completed L-fnction in the sense that it satisfies the following nice analytic properties: it can be extended to an holomorphic fnction of order on C, it satisfies a fnctional eation of the shape ΛSym r f, s = ε Sym r f ΛSym r f, s where the sign ε Sym r f = ± of the fnctional eation is given by ε Sym r f := { + if r is even, ε f εκ,r otherwise In this paper, the weight κ is a fixed even integer and the level goes to infinity among the prime nmbers

3 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 3 with εκ,r := i r + κ + r + = i κ if r mod 8, if r 3 mod 8, i κ if r 5 mod 8, + if r 7 mod 8 and ε f = ± is defined in and only depends on f and emark Hypothesis Nicer, f is known for r = E Hecke [0 ], r = thanks to the work of S Gelbart and H Jacet [8] and r = 3,4 from the works of H Kim and F Shahidi [0 ] We aim at stdying the low-lying zeros for the family of L-fnctions given by F r := { LSym r f, s, f Hκ } prime for any integer r Note that when r is even, the sign of the fnctional eation of any LSym r f, s is constant of vale + bt when r is odd, this is definitely not the case As a conseence, it is very natral to nderstand the low-lying zeros for the sbfamilies given by Fr ε := { LSym r f, s, f Hκ,ε Sym r f = ε } prime for any odd integer r and for ε = ± Symmetry type of these families One of the prpose of this work is to determine the symmetry type of the families F r and Fr ε for ε = ± and for any integer r see 4 for the backgrond on symmetry types The following theorem is a ick smmary of the symmetry types obtained Theorem A Let r be any integer and ε = ± We assme that hypothesis Nicer, f holds for any prime nmber and any primitive holomorphic csp form of level and even weight κ The symmetry grop GF r of F r is given by { Sp if r is even, GF r = O otherwise If r is odd then the symmetry grop GF ε r of F ε r is given by { SOeven if ε = +, GFr ε = SOodd otherwise emark It follows in particlar from the vale of ε Sym r f given in that, if r is even, then Sym r f has not the same symmetry type than f and, if r is odd, then f and Sym r f have the same symmetry type if and only if or or r mod 8 and κ 0 mod 4 r 5 mod 8 and κ mod 4 r 7 mod 8

4 4 G ICOTTA AND E OYE emark 3 Note that we do not assme any Generalised iemann Hypothesis for the symmetric power L-fnctions In order to prove theorem A, we compte either the signed asymptotic expectation of the one-level density or the signed asymptotic expectation of the two-level density The reslts are given in the next two sections in which ε = ±, ν will always be a positive real nmber, Φ,Φ and Φ will always stand for even Schwartz fnctions whose Forier transforms, and are compactly spported in [ ν,+ν] and f will always be a primitive holomorphic csp form of prime level and even weight κ for which hypothesis Nicer, f holds We refer to for the probabilistic backgrond Signed asymptotic expectation of the one-level density The one-level density relatively to Φ of Sym r f is defined by log r D, [Φ;r ]f := Φ e ρ iπ + i Im ρ ρ, ΛSym r f,ρ=0 where the sm is over the non-trivial zeros ρ of LSym r f, s with mltiplicities The asymptotic expectation of the one-level density is by definition lim ω f D, [Φ;r ]f prime + f H κ where ω f is the harmonic weight defined in 5 and similarly the signed asymptotic expectation of the one-level density is by definition lim ω f D, [Φ;r ]f when r is odd prime + f H κ εsym r f =ε Theorem B Let r be any integer and ε = ± We assme that hypothesis Nicer, f holds for any prime nmber and any primitive holomorphic csp form of level and even weight κ and also that θ is admissible see hypothesis H θ page 6 Let ν,max r,κ,θ := κ θ r If ν < ν,max r,κ,θ then the asymptotic expectation of the one-level density is Let + r 0 + Φ0 ν ε,max ν r,κ,θ := inf 3,max r,κ,θ, r r + If r is odd and ν < ν ε,max r,κ,θ then the signed asymptotic expectation of the one-level density is + r 0 + Φ0

5 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 5 emark 4 The first part of Theorem B reveals that the symmetry type of F r is Sp if r is even, GF r = O if r =, SOeven or O or SOodd if r 3 is odd We cannot decide between the three orthogonal grops when r 3 is odd since in this case ν,max r,κ,θ < bt the comptation of the two-level densities will enable s to decide Note also that we go beyond the spport [,] when r = as Iwaniec, Lo & Sarnak [8] Theorem bt withot doing any sbtle arithmetic analysis of Kloosterman sms Also, A Gülogl in [9, Theorem ] established some density reslt for the same family of L-fnctions bt when the weight κ goes to infinity and the level is fixed It trns ot that we recover the same constraint on ν when r is even bt we get a better reslt when r is odd This can be explained by the fact that the analytic condctor of any LSym r f, s with f in Hκ which is of size { κ r r if r is even κ r + otherwise is slightly larger in his case than in ors when r is odd emark 5 The second part of Theorem B reveals that if r is odd and ε = ± then the symmetry type of F ε r is GFr ε = SOeven or O or SOodd Here ν is always strictly smaller than one and we are not able to recover the reslt of [8, Theorem ] withot doing some arithmetic on Kloosterman sms Sketch of the proof We give here a sketch of the proof of the first part of Theorem B namely we briefly explain how to determine the asymptotic expectation of the one-level density assming that hypothesis Nicer, f holds for any prime nmber and any primitive holomorphic csp form of level and even weight κ and also that θ is admissible The first step consists in transforming the sm over the zeros of ΛSym r f, s which occrs in D, [Φ;r ]f into a sm over primes This is done via some iemann s explicit formla for symmetric power L-fnctions stated in Proposition 38 which leads to + r D, [Φ;r ]f = 0 + Φ0 + P r [Φ;r ]f + m P [Φ;r,m]f + o where P [Φ;r ]f := log r p P p m=0 λ f p r log p log p p log r The terms P [Φ;r,m]f are also sms over primes which look like P [Φ;r ]f bt can be forgotten in first approximation since they can be thoght as sms over sares of primes which are easier to deal with The second step consists in averaging over all the f in Hκ While doing this, the asymptotic expectation of the one-level density + r 0 + Φ0

6 6 G ICOTTA AND E OYE natrally appears and we need to show that log r p P p f H κ ω f λ f p r log p p log p log r is a remainder term provided that the spport ν of Φ is small enogh We apply some sitable trace formla given in Proposition in order to express the previos average of Hecke eigenvales We cannot directly apply Peterson s trace formla since there may be some old forms of level especially when the weight κ is large Nevertheless, these old forms are atomatically of level since is prime and their contribtion remains negligible So, we have to bond 4πi κ log r p P p S, p r ;c 4π p r J κ c c c c log p p log p log r where S, p r ;c is a Kloosterman sm and which can be written as 4πi κ log S,m;c r a m g m;c c m c c where { a m := [, r ν ] m logm if m = p r r m /r for some prime p, 0 otherwise and 4π m logm g m;c := J κ c r log r We apply the large sieve ineality for Kloosterman sms given in proposition 34 It entails that if ν /r then sch antity is bonded by κ ε θ r ν +ε κ + θ r ν κ θ +ε This is an admissible error term if ν < ν,max r,κ,θ We focs on the fact that we did any arithmetic analysis of Kloosterman sms to get this reslt Of corse, the power of spectral theory of atomorphic forms is hidden in the large sieve inealities for Kloosterman sms 3 Signed asymptotic expectation of the two-level density The two-level density of Sym r f relatively to Φ and Φ is defined by D, [Φ,Φ ;r ]f := Φ ρ j Φ f,r ρ j f,r j,j E f,r j ±j For more precision on the nmbering of the zeros, we refer to 3 The asymptotic expectation of the two-level density is by definition lim ω f D, [Φ,Φ ;r ]f prime + f H κ and similarly the signed asymptotic expectation of the two-level density is by definition lim ω f D, [Φ,Φ ;r ]f prime + f H κ εsym r f =ε

7 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 7 when r is odd and ε = ± Theorem C Let r be any integer and ε = ± We assme that hypothesis Nicer, f holds for any prime nmber and any primitive holomorphic csp form of level and even weight κ If ν < /r then the asymptotic expectation of the two-level density is + + ] r r [ 0 + Φ 0 ][ 0 + Φ 0 + d Φ Φ 0 + r + N+ r Φ 0Φ 0 If r is odd and ν < /r r + then the signed asymptotic expectation of the two-level density is [ 0 + ][ Φ ] Φ 0 + d Φ Φ 0 Φ 0Φ 0 + { } εφ 0Φ 0 emark 6 We have jst seen that the comptation of the one-level density already reveals that the symmetry type of F r is Sp when r is even The asymptotic expectation of the two-level density also coincides with the one of Sp see [9, Theorem AD] or [6, Theorem 33] When r 3 is odd, the first part of Theorem C together with a reslt of Katz & Sarnak see [9, Theorem AD] or [6, Theorem 3] imply that the symmetry type of F r is O emark 7 The second part of Theorem C and a reslt of Katz & Sarnak see [9, Theorem AD] or [6, Theorem 3] imply that the symmetry type of Fr ε is as in Theorem A for any odd integer r and ε = ± In order to prove Theorem C, we need to determine the asymptotic variance of the one-level density which is defined by lim ω f D, [Φ;r ]f ω g D, [Φ;r ]g prime + f H κ g H κ and the signed asymptotic variance of the one-level density which is similarly defined by lim prime + ω f D,[Φ;r ]f f H κ εsym r f =ε when r is odd and ε = ± ω g D, [Φ;r ]g g H κ εsym r g=ε Theorem D Let r be any integer and ε = ± We assme that hypothesis Nicer, f holds for any prime nmber and any primitive holomorphic csp form of level and even weight κ If ν < /r then the asymptotic variance of the one-level density is d

8 8 G ICOTTA AND E OYE If r is odd and ν < /r r + then the signed asymptotic variance of the one-level density is d 3 Asymptotic moments of the one-level density Last bt not least, we compte the asymptotic m-th moment of the one-level density which is defined by lim ω f D, [Φ;r ]f m ω g D, [Φ;r ]g prime + f H κ for any integer m g H κ Theorem E Let r be any integer and ε = ± We assme that hypothesis Nicer, f holds for any prime nmber and any primitive holomorphic csp form of level and even weight κ If mν < 4/r r + then the asymptotic m-th moment of the one-level density is 0 if m is odd, m! d m/ m otherwise! emark 8 This reslt is another evidence for mock-gassian behavior see [3 5] for instance emark 9 We compte the first asymptotic moments of the one-level density These comptations allow to compte the asymptotic expectation of the first level-densities [3, ] We will se the specific case of the asymptotic expectation of the two-level density and the asymptotic variance in 5 Let s sketch the proof of Theorem E by explaining the origin of the main term We have to evalate m l l α E h P l α [Φ;r ]m l P [Φ;r ]α 3 0 l m 0 α l where P [Φ;r ] has been defined in, P [Φ;r ]f = log r and satisfies r r j j = = O p P p log λ f p j log p log p p log r The main term comes from the contribtion l = 0 in the sm 3 Using a combinatorial lemma, we rewrite this main contribtion as m m s ϖ σ log m r E h λ f p r i s= = σ Pm,s i,,i s where Pm, s is the set of srjective fnctions σ: {,,α} {,, s}

9 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 9 sch that for any j {,, s}, either σj = or there exists k < j sch that σj = σk + and for any j {,, s} ϖ σ := #σ {j } j pi i stands for the increasing seence of prime nmbers different from ϖ σ Linearising each λ f p r in terms of λ i f p j with j i rns over integers in [0,r ϖ σ ] and sing a trace formla to prove that the only σ Pm, s leading to a principal contribtion satisfy ϖ σ = for any j {,, s}, we have to estimate j m m s log p i log pi log m r p i log r 4 s= σ Pm,s i,,i s = j {,,s},ϖ σ = j This sm vanishes if m is odd since s j = ϖ σ = m j and it remains to prove the formla for m even In this case, and since we already compted the moment for m =, we dedce from 4 that the main contribtion is { } E h P [Φ;r ] # σ Pm,m/: ϖ σ = j j and we conclde by compting { } # σ Pm,m/: ϖ σ m! = j = j m/ m! Proving that the other terms lead to error terms is done by implementing similar ideas, bt reires especially for the doble prodcts namely terms implying both P and P mch more combinatorial technicalities 4 Organisation of the paper Section contains the atomorphic and probabilistic backgrond which is needed to be able to read this paper In particlar, we give here the accrate definition of symmetric power L-fnctions and the properties of Chebyshev polynomials sefl in section 6 In section 3, we describe the main technical ingredients of this work namely large sieve inealities for Kloosterman sms and iemann s explicit formla for symmetric power L-fnctions In section 4, some standard facts abot symmetry grops are given and the comptation of the signed asymptotic expectation of the one-level density is done The comptations of the signed asymptotic expectation, covariance and variance of the two-level density are done in section 5 whereas the comptation of the asymptotic moments of the one-level density is provided in section 6 Some well-known facts abot Kloosterman sms are recalled in appendix A Notation We write P for the set of prime nmbers and the main parameter in this paper is a prime nmber, whose name is the level, which goes to infinity among P Ths, if f and g are some C-valed fnctions of the real variable then the notations f A g or f = O A g mean that f is smaller than a "constant" which only depends on A times g at least for a large enogh prime nmber and similarly, f = o means that f 0 as goes to infinity among the prime nmbers We will denote by ε an absolte positive constant whose

10 0 G ICOTTA AND E OYE definition may vary from one line to the next one The characteristic fnction of a set S will be denoted S AUTOMOPHIC AND POBABILISTIC BACKGOUND Atomorphic backgrond Overview of holomorphic csp forms In this section, we recall general facts abot holomorphic csp forms A reference is [6] Generalities We write Γ 0 for the congrence sbgrop of level which acts on the pper-half plane H A holomorphic fnction f : H C which satisfies az + b a b c d Γ 0, z H, f cz + d = cz + d κ f z and vanishes at the csps of Γ 0 is a holomorphic csp form of level, even weight κ We denote by S κ this space of holomorphic csp forms which is eipped with the Peterson inner prodct f, f := Γ 0 \H y κ dx dy f zf z y The Forier expansion at the csp of any sch holomorphic csp form f is given by z H, f z = n ψ f nn κ / enz where ez := expiπz for any complex nmber z The Hecke operators act on S κ by T l f z := az + b f l d ad=l 0 b<d a,= for any z H If f is an eigenvector of T l, we write λ f l the corresponding eigenvale We can prove that T l is hermitian if l is any integer coprime with and that T l T l = T l l /d d l,l d,= for any integers l,l By Atkin & Lehner theory [], we get a splitting of S κ into Sκ o, Sκ n where S o κ := Vect C { f z, f Sκ } S κ, S n κ := S o κ, where "o" stands for "old" and "n" for "new" Note that Sκ o = {0} if κ < or κ = 4 These two spaces are T l -invariant for any integer l coprime with A primitive csp form f Sκ n is an eigenfnction of any operator T l for any integer l coprime with which is new and arithmetically normalised namely ψ f = Sch an element f is atomatically an eigenfnction of the other Hecke operators and satisfies ψ f l = λ f l for any integer l Moreover, if p is a prime nmber, define α f p, β f p as the complex roots of the adratic eation X λ f px + ε p = 0

11 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS where ε denotes the trivial Dirichlet character of modls Then it follows from the work of Eichler, Shimra, Igsa and Deligne that for any prime nmber p and so α f p, β f p l, λ f l τl 3 The set of primitive csp forms is denoted by Hκ It is an orthogonal basis of Sκ n Let f be a holomorphic csp form with Hecke eigenvales λ f l l,= The composition property entails that for any integer l and for any integer l coprime with the following mltiplicative relations hold: ψ f l λ f l = / ψ f l l d, 4 ψ f l l = d l,l d,= d l,l d,= µdψ f l /dλ f l /d and these relations hold for any integers l,l if f is primitive The adjointness relation is λ f l = λ f l, ψ f l = ψ f l for any integer l coprime with and this remains tre for any integer l if f is primitive Trace formlas We need two definitions The harmonic weight associated to any f in S κ is defined by ω f := Γκ 4π κ f, f 5 For any natral integer m and n, the -symbol is given by m,n := δ m,n + πi κ c c Sm,n;c c J κ 4π mn c where Sm,n;c is a Kloosterman sm defined in appendix A3 and J κ is a Bessel fnction of first kind defined in appendix AThe following proposition is Peterson s trace formla Proposition If H κ is any orthogonal basis of S κ then ω f ψ f mψ f n = m,n 6 f H κ for any integers m and n H Iwaniec, W Lo & P Sarnak proved in [8] a sefl variation of Peterson s trace formla which is an average over only primitive csp forms This is more convenient when there are some old forms which is the case for instance when the weight κ is large Let ν be the arithmetic fnction defined by νn := n + /p p n for any integer n

12 G ICOTTA AND E OYE Proposition H Iwaniec, W Lo & P Sarnak 00 If n, and m then f H κ ω f λ f mλ f n = m,n νn, l l ml,n 7 emark 3 The first term in 7 is exactly the term which appears in 6 whereas the second term in 7 will be sally very small as an old form comes from a form of level! Ths, everything works in practice as if there were no old forms in S κ Chebyshev polynomials and Hecke eigenvales Let p a prime nmber and f Hκ The mltiplicativity relation 4 leads to It follows that λ f p r t r = λ f pt + t r 0 λ f p r = X r λf p 8 where the polynomials X r are defined by their generating series They are also defined by X r xt r = xt + t r 0 sinr + θ X r cosθ = sinθ These polynomials are known as the Chebyshev polynomials of second kind Each X r has degree r, is even if r is even and odd otherwise The family {X r } r 0 is a basis for Q[X ], orthonormal with respect to the inner prodct P,Q ST := π PxQx In particlar, for any integer ϖ 0 we have x 4 dx X ϖ r r ϖ = xϖ,r, j X j 9 j =0 with xϖ,r, j := X ϖ r, X j ST = π π 0 sin ϖ r + θsinj + θ sin ϖ dθ θ The following relations are sefl in this paper if j = 0 and ϖ is even, xϖ,r, j = 0 if j is odd and r is even, 0 if j = 0, ϖ = and r 0

13 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 3 3 Overview of L-fnctions associated to primitive csp forms Let f in Hκ We define Lf, s := λ f n n n s = α f p p s β f p p s p P which is an absoltely convergent and non-vanishing Dirichlet series and Eler prodct on e s > and also L f, s := Γ s + κ /Γ s + κ + / where Γ s := π s/ Γs/ as sal The fnction Λf, s := s/ L f, slf, s is a completed L-fnction in the sense that it satisfies the following nice analytic properties: the fnction Λf, s can be extended to an holomorphic fnction of order on C, the fnction Λf, s satisfies a fnctional eation of the shape where Λf, s = i κ ε f Λf, s ε f = λ f = ± 4 Overview of symmetric power L-fnctions Let f in Hκ For any natral integer r, the symmetric r -th power associated to f is given by the following Eler prodct of degree r + LSym r f, s := L p Sym r f, s where L p Sym r f, s := i=0 p P r α f p i β f p r i p s for any prime nmber p Let s remark that the local factors of this Eler prodct may be written as r L p Sym r f, s = α f p i r p s for any prime nmber p and i=0 L Sym r f, s = λ f r s = λ f r s as α f p + β f p = λ f p and α f pβ f p = ε p for any prime nmber p according to On e s >, this Eler prodct is absoltely convergent and non-vanishing We also defines [4, 36 and 37] a local factor at which is given by a prodct of r + Gamma factors namely L Sym r f, s := Γ s + a + κ /Γ s + + a + κ / if r is odd and 0 a r / L Sym r f, s := Γ s + µ κ,r a r / Γ s + aκ Γ s + + aκ

14 4 G ICOTTA AND E OYE if r is even where µ κ,r := { if r κ / is odd, 0 otherwise All the local data appearing in these local factors are encapslated in the following completed L-fnction ΛSym r f, s := r s/ L Sym r f, slsym r f, s Here, r is called the arithmetic condctor of ΛSym r f, s and somehow measres the size of this fnction We will need some control on the analytic behavior of this fnction Unfortnately, sch information is not crrently known in all generality Or main assmption is given in hypothesis Nicer, f page Indeed, mch more is expected to hold as it is discssed in details in [4] namely the following assmption is strongly believed to be tre and lies in the spirit of Langlands program Hypothesis Sym r f There exists an atomorphic cspidal self-dal representation, denoted by Sym r π f = p P { } Symr π f,p, of GL r + AQ whose local factors L Sym r π f,p, s agree with the local factors L p Sym r f, s for any p in P { } Note that the local factors and the arithmetic condctor in the definition of Λ Sym r f, s and also the sign of its fnctional eation which all appear withot any explanations so far come from the explicit comptations which have been done via the local Langlands correspondence by J Cogdell and P Michel in [4] Obviosly, hypothesis Nicer, f is a weak conseence of hypothesis Sym r f For instance, the cspidality condition in hypothesis Sym r f entails the fact that Λ Sym r f, s is of order which is crcial for s to state a sitable explicit formla As we will not exploit the power of atomorphic theory in this paper, hypothesis Nicer, f is enogh for or prpose In addition, it may happen that hypothesis Nicer, f is known whereas hypothesis Sym r f is not Let s overview what has been done so far For any f in H κ, hypothesis Symr f is known for r = E Hecke, r = thanks to the work of S Gelbart and H Jacet [8] and r = 3,4 from the works of H Kim and F Shahidi [0 ] Probabilistic backgrond The set Hκ can be seen as a probability space if the measrable sets are all its sbsets, the harmonic probability measre is defined by µ h A := h := ω f for any sbset A of H κ f A Indeed, there is a slight abse here as we only know that lim P + µ h H κ = see remark 3 which means that µ h is an asymptotic probability measre If X is a measrable complex-valed fnction on Hκ then it is very natral to compte its expectation defined by h X := X f, E h f H κ f A

15 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 5 its variance defined by V h X := E h X E h X and its m-th moments given by M h,m X := E h X E h m X for any integer m If X := X is a seence of sch measrable complexvaled fnctions then we may legitimely wonder if the associated complex se- P ences E h X, V h X, M h,m X P P P converge as goes to infinity among the primes If yes, the following general notations will be sed for their limits E h X, Vh X, Mh,m X for any natral integer m In addition, these potential limits are called asymptotic expectation, asymptotic variance and asymptotic m-th moments of X for any natral integer m For the end of this section, we assme that r is odd We may remark that the sign of the fnctional eations of any LSym r f, s when goes to infinity among the prime nmbers and f ranges over Hκ is not constant as it depends on ε f Let Hκ ε := { f Hκ,εSymr f = ε } where ε = ± If f H κ +, then Symr f is said to be even whereas it is said to be odd if f Hκ It is well-known that lim µ h { f H P k : ε f = ε } = + Since εsym r f is ε f p to a sign depending only on κ and r by hypothesis Nicer, f, it follows that lim µ h H ε κ = P + For X as previos, we can compte its signed expectation defined by h X := X f, E h,ε its signed variance defined by V h,ε X := E h,ε f H ε κ X E h,ε X and its signed m-th moments given by M h,ε,m X := E h,ε X E h,ε m X for any natral integer m In case of existence, we write E h,ε X, V h,ε X and M h,ε,mx for the limits which are called signed asymptotic expectation, signed

16 6 G ICOTTA AND E OYE asymptotic variance and signed asymptotic moments The signed expectation and the expectation are linked throgh the formla E h,ε X = h f H κ + ε εsym r f X f = E h X ε εκ,r h f H κ 3 MAIN TECHNICAL INGEDIENTS OF THIS WOK λ f X f 3 3 Large sieve inealities for Kloosterman sms One of the main ingredients in this work is some large sieve inealities for Kloosterman sms which have been established by J-M Deshoillers & H Iwaniec in [5] and then refined by V Blomer, G Harcos & P Michel in [] The proof of these large sieve inealities relies on the spectral theory of atomorphic forms on GL AQ In particlar, the athors have to nderstand the size of the Forier coefficients of these atomorphic csp forms We have already seen that the size of the Forier coefficients of holomorphic csp forms is well nderstood 3 bt we only have partial reslts on the size of the Forier coefficients of Maass csp forms which do not come from holomorphic forms We introdce the following hypothesis which measres the approximation towards the amanjan-peterson-selberg conjectre Hypothesis H θ If π := p P { } π p is any atomorphic cspidal form on GL A Q with local Hecke parameters α π p, α π p at any prime nmber p and µ π, µ π at infinity then j {,}, α j π p p θ for any prime nmber p for which π p is nramified and j {,}, e µ j π θ provided π is nramified Denition 3 We say that θ is admissible if H θ is satisfied emark 3 The smallest admissible vale of θ is crrently θ 0 = 7 64 thanks to the works of H Kim, F Shahidi and P Sarnak [0, ] The amanjan-peterson- Selberg conjectre asserts that 0 is admissible Denition 33 Let T : 3 + and M, N,C \ {0} 3, we say that a smooth fnction h : 3 3 satisfies the property PT ; M, N,C if there exists a real nmber K > 0 sch that [ ] [ ] [ ] M N C i, j,k N 3, x, x, x 3,M,N,C, x i x j i+j +k i+j +k h M N xk 3 x i x j x, x, x 3 K T M, N,C + xk C 3 With this definition in mind, we are able to write the following proposition which is special case of a large sieve ineality adapted from the one of Deshoillers & Iwaniec [5, Theorem 9] by Blomer, Harcos & Michel [, Theorem 4]

17 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 7 Proposition 34 Let be some positive integer Let M, N,C and g be a smooth fnction satisfying property P; M, N,C Consider two seences of complex nmbers a m m [M/,M] and b n n [N/,N] If θ is admissible and M N C then c c m n for any ε > 0 Sm,±n;c a m b n g m,n;c c ε M NC ε C M N θ + M / + N / a b We shall se a test fnction For any ν > 0 let s define S ν as the space of even Schwartz fnction Φ whose Forier transform ξ := F [x Φx]ξ := Φxe xξ dx is compactly spported in [ ν, +ν] Thanks to the Forier inversion formla: Φx = ξexξdx = F [ξ ξ] x, sch a fnction Φ can be extended to an entire even fnction which satisfies expν Im s s C, Φs n + s n for any integer n 0The version of the large sieve ineality we shall se several times in this paper is then the following Corollary 35 Let be some prime nmber, k,k > 0 be some integers, α,α,ν be some positive real nmbers and Φ S ν Let h be some smooth fnction satisfying property PT ; M, N,C for any M k α ν, N k α ν and C Let a p p P and b p p P be some complex nmbers seences If θ is p α ν p α ν admissible and ν / k α + k α then c c p P p p P p ε a p b p Sp k, pk M να k N να k C / c ;c h p k, pk log ;c p log p log α log α M N C θ + + T M, N,C a b M N where indicates that the sm is on powers of The constant implied by the symbol depends at most on ε, k, k, α, α and ν Proof Define â m m N, bn n N and g m,n;c by â m := a m /k P k m [, να k ] m b n := b n /k P k n [, να k ] n logm g m,n;c := hm,n,c log α k logn log α k

18 8 G ICOTTA AND E OYE Using a smooth partition of nity, as detailed in A, we need to evalate T M, N,C Sm,n;c g M,N,C m,n;c â m bn 3 c m n c T M, N,C c M να k N να k C / Since ν / α k + α k, the first smmation is restricted to M N C hence, sing proposition 34, the antity in 3 is a b ε M N C θ T M, N,C + + M N M να k N να k C / 3 iemann s explicit formla for symmetric power L-fnctions In this section, we give an analog of iemann-von Mangoldt s explicit formla for symmetric power L-fnctions Before that, let s recall some preliminary facts on zeros of symmetric power L-fnctions which can be fond in section 53 of [7] Let r and f Hκ for which hypothesis Nicer, f holds All the zeros of ΛSym r f, s are in the critical strip {s C: 0 < e s < } The mltiset of the zeros of ΛSym r f, s conted with mltiplicities is given by { } ρ j f,r = βj f,r + iγj f,r : j E f,r where and E f,r := { Z if Sym r f is odd Z \ {0} β j = e ρj f,r f,r, γ j = Im ρj f,r f,r if Sym r f is even for any j E f,r We enmerate the zeros sch that the seence j γ j is increasing f,r we have j 0 if and only if γ j f,r 0 3 we have ρ j = ρ j f,r f,r Note that if ρ j f,r is a zero of ΛSymr f, s then ρ j f,r, ρj and ρj are also f,r f,r some zeros of ΛSym r f, s In addition, remember that if Sym r f is odd then the fnctional eation of LSym r f, s evalated at the critical point s = / provides a trivial zero denoted by ρ 0 It can be shown [7, Theorem 58] that the nmber f,r of zeros ΛSym r f, s satisfying γ j T π log f,r T is r T r + πe r + +O logt as T goes to infinity We state now the Generalised iemann Hypothesis which is the main conjectre abot the horizontal distribtion of the zeros of ΛSym r f, s in the critical strip

19 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 9 Hypothesis GHr For any prime nmber and any f in Hκ, all the zeros of ΛSym r f, s lie on the critical line {s C: e s = /} namely β j = / for any r,f j E f,r emark 36 We do not se this hypothesis in or proofs Under hypothesis GHr, it can be shown that the nmber of zeros of the fnction ΛSym r f, s satisfying γ j is given by f,r π log r + o as goes to infinity Ths, the spacing between two consective zeros with imaginary part in [0,] is roghly of size π log r We aim at stdying the local distribtion of the zeros of ΛSym r f, s in a neighborhood of the real axis of size /log r since in sch a neighborhood, we expect to catch only few zeros bt withot being able to say that we catch only one Hence, we normalise the zeros by defining ρ j f,r log r := iπ β j f,r + iγj f,r Note that ρ j = ρ j f,r f,r Denition 37 Let f Hκ for which hypothesis Nicer, f holds and let Φ S ν The one-level density relatively to Φ of Sym r f is D, [Φ;r ]f := j E f,r Φ ρ j f,r To stdy D, [Φ;r ]f for any Φ S ν, we transform this sm over zeros into a sm over primes in the next proposition In other words, we establish an explicit formla for symmetric power L-fnctions Since the proof is classical, we refer to [8, 4] or [9, ] which present a method that has jst to be adapted to or setting Proposition 38 Let r and f Hκ for which hypothesis Nicer, f holds and let Φ S ν We have D, [Φ;r ]f = E[Φ;r ] + P r [Φ;r ]f + m=0 m P [Φ;r,m]f +O log r We refer to Miller [5] and Omar [7] for works related to the first zero

20 0 G ICOTTA AND E OYE where + r E[Φ;r ] := 0 + Φ0, P [Φ;r ]f := log r p P p P [Φ;r,m]f := log r for any integer m {0,,r } p P p λ f p r log p p λ f p r m log p p log p log r, log p log r 33 Contribtion of the old forms In this short section, we prove the following sefl lemmas Lemma 39 Let p and p be some prime nmbers and a, a, a be some nonnegative integers Then l a p, pa a l l a/ the implied constant depending only on a and a Proof Using proposition and the fact that H κ = Hκ, we write l p a, pa a = h By Deligne s bond 3 we have f Hκ h f H κ λ f l p a λ f pa a λ f l p a λ f pa λ f a 3 λ f l p a λ f pa τl p a τpa a + a + τl 33 By the mltiplicativity relation 4 and the vale of the sign of the fnctional eation, we have λ f a 34 a/ We obtain the reslt by reporting 34 and 33 in 3 and by sing and τl l l = + / / Lemma 30 Let m,n be some coprime integers Then, mn /4 mn log if mn > m,n δm,n mn κ / mn/4 if mn κ / Proof This is a direct conseence of the Weil-Estermann bond A5 and lemma A

21 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS Corollary 3 For any prime nmber, we have where f H κ h f H κ κ δ κ := 5 λ f δ κ if κ 0 or κ = 4 otherwise Proof of corollary 3 Let K = {κ N: κ 4, κ } By proposition, we have h λ f =, δκ K l, 35 ν l l The term δκ K comes from proposition with the fact that there is no csp forms of weight κ K and level Lemma 30 gives and lemma 39 gives, κ/ 36 l, l l 37 Since ν >, the reslt follows from reporting 36 and 37 in 35 emark 3 In a very similar fashion, one can prove that where µ h H κ = E h = +O γ κ κ if κ 0 or κ = 4 γ κ := otherwise Corollary 3, 38 and 3 imply E h,ε = +O where β κ κ if κ 0 or κ = 4 β κ := otherwise A direct conseence of lemma 39 is the following one Lemma 33 Let α,α,β,β,γ,γ, w be some nonnegative real nmbers Let Φ and Φ be in S ν Then, p P p p P p log p log p p α p α with δ given in table log p log log p β log l p β l γ, pγ w l δν w/+ε

22 G ICOTTA AND E OYE α α ]0,] [,+ [ ]0,] β α + β α β α [,+ [ β α 0 TABLE Vales of δ 4 LINEA STATISTICS FO LOW-LYING ZEOS 4 Density reslts for families of L-fnctions We briefly recall some wellknown featres that can be fond in [8] Let F be a family of L-fnctions indexed by the arithmetic condctor namely F = Q F Q where the arithmetic condctor of any L-fnction in F Q is of order Q in the logarithmic scale It is expected that there is a symmetry grop GF of matrices of large rank endowed with a probability measre which can be associated to F sch that the low-lying zeros of the L-fnctions in F namely the non-trivial zeros of height less than /logq are distribted like the eigenvales of the matrices in GF In other words, there shold exist a symmetry grop GF sch that for any ν > 0 and any Φ S ν, lim Q + F Q π F Q 0 β π γ π Lπ,β π +iγ π=0 logq Φ β π iπ + iγ π = ΦxW GF xdx where W GF is the one-level density of the eigenvales of GF In this case, F is said to be of symmetry type GF and we said that we proved a density reslt for F For instance, the following densities are determined in [9]: W SOevenx = + sinπx, πx W Ox = + δ 0x, W SOoddx = sinπx πx W Spx = sinπx πx + δ 0 x, where δ 0 is the Dirac distribtion at 0 According to Plancherel s formla, ΦxW GF xdx = xŵ GF xdx

23 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 3 and we can check that where Ŵ SOevenx = δ 0 x + ηx, Ŵ Ox = δ 0 x +, Ŵ SOoddx = δ 0 x ηx +, Ŵ Spx = δ 0 x ηx if x <, ηx := if x = ±, 0 otherwise As a conseence, if we can only prove a density reslt for ν, the three orthogonal densities are indistingishable althogh they are distingishable from Sp Ths, the challenge is to pass the natral barrier ν = 4 Asymptotic expectation of the one-level density The aim of this part is to prove a density reslt for the family F r := { LSym r f, s, f Hκ } P for any r which consists in proving the existence and compting the asymptotic expectation E h D [Φ;r ] of D [Φ;r ] := D, [Φ;r ] for any r and P for Φ in S ν with ν > 0 as large as possible in order to be able to distingish between the three orthogonal densities if r is small enogh ecall that E[Φ;r ] has been defined in proposition 38 Theorem 4 Let r and Φ S ν We assme that hypothesis Nicer, f holds for any prime nmber and any f Hκ and also that θ is admissible Let ν,max r,κ,θ := κ θ r If ν < ν,max r,κ,θ then E h D [Φ;r ] = E[Φ;r ] emark 4 We remark that ν,max r,κ,θ 0 = 6 3κ 7 r 8 57r, ν,max r,κ,0 = κ r 3 r and ths ν,max,κ,θ 0 > whereas ν,max r,κ,θ 0 for any r emark 43 Note that E[Φ;r ] = x δ 0 x + r + dx

24 4 G ICOTTA AND E OYE Ths, this theorem reveals that the symmetry type of F r is Sp if r is even, GF r = O if r =, SOeven or O or SOodd if r 3 is odd Some additional comments are given in remark 4 page 5 Proof of theorem 4 The proof is detailed and will be a model for the next density reslts According to proposition 38 and 38, we have E h D, [Φ;r ] = E[Φ;r ] + E h P [Φ;r ] + r m=0 m E h P [Φ;r,m] +O log r 4 The first term in 4 is the main term given in the theorem We now estimate the second term of 4 via the trace formla given in proposition E h P [Φ;r ] = P,new [Φ;r ] + P,old [Φ;r ] 4 where P,new [Φ;r ] = log r p P p P,old [Φ;r ] = log r l l p r, log p log p p log r, p P p Let s estimate the new part which can be written as P,new [Φ;r ] = πi κ log r c p P c Thanks to A3, the fnction p r l, log p log p p log r log p p δ p [, r ν ] p Sp r,;c c hm;c := J κ 4π m c 4π p r J κ satisfies hypothesis PT ; M,,C with / κ κ M M T M,,C = + C C M νr C / c log p log r Hence, if ν /r then corollary 35 leads to κ θ P,new [Φ;r ] ε ε M M + 43 C ε ε M νr M κ θ κ θ + M κ θ κ θ

25 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 5 thanks to A Smming over M via A leads to P,new [Φ;r ] ε κ θ r ν +ε + κ θ r ν κ θ +ε 44 which is an admissible error term if ν < ν,max r,κ,θ According to lemma 33 with α = + we have P,old [Φ;r ] ε r ν +ε 45 which is an admissible error term if ν < /r eporting 44 and 45 in 4 we obtain P [Φ;r ] 46 E h for some δ > 0 depending on ν and r as soon as ν < ν,max r,κ,θ We now estimate the third term of 4 If 0 m r then the trace formla given in proposition implies that P [Φ;r,m] = P,new [Φ;r,m] + P,old [Φ;r,m] where E h P,new [Φ;r,m] = log r p P p P,old [Φ;r,m] = log r l l δ p r m, log p p p P p log p log r p r m l, log p p Let s estimate the new part which can be written as P,new [Φ;r,m] = πi κ log r c p P c, log p log r log p S p r m δ p [ p, r ν ],;c p c 4π p J κ p c r m The fnction 4π m hm,c := J κ c m/4r m satisfies hypothesis PT ; M,,C with / κ κ M M T M,,C = + C C M /4r m Hence, if ν /r then corollary 35 leads to P,new [Φ;r,m] ε ε νr r m M C / M /4r 4m log p log r / κ θ M M + C This is smaller than the bond given in 43 and hence is an admissible error term if ν < ν,max r,κ,θ According to lemma 33, we have P,old [Φ;r ] ε +ε

26 6 G ICOTTA AND E OYE We obtain E h P [Φ;r,m] δ 47 for some δ > 0 depending on ν and r as soon as ν < ν,max r,κ,θ Finally, reporting 47 and 46 in 4, we get E h D, [Φ;r ] = E[Φ;r ] +O 48 log 43 Signed asymptotic expectation of the one-level density In this part, we prove some density reslts for sbfamilies of F r on which the sign of the fnctional eation remains constant The two sbfamilies are defined by Fr ε := { LSym r f, s, f Hκ ε } P Indeed, we compte the asymptotic expectation E h,ε D [Φ;r ] Theorem 44 Let r be an odd integer, ε = ± and Φ S ν We assme that hypothesis Nicer, f holds for any prime nmber and any f Hκ and also that θ is admissible Let ν ε,max ν r,κ,θ := inf 3,max r,κ,θ, r r + If ν < ν ε,max r,κ,θ then E h,ε D [Φ;r ] = E[Φ;r ] Some comments are given in remark 5 page 5 Proof of theorem 44 By 3, we have E h,ε D, [Φ;r ] = E h D, [Φ;r ] ε εk,r E h λ D, [Φ;r ] The first term is the main term of the theorem thanks to theorem 4 According to proposition 38 and corollary 3, the second term withot the epsilon factors is given by E h λ P [Φ;r ] + r m=0 m E h λ P [Φ;r,m] +O log r 49 Let s focs on the first term in 49 knowing that the same discssion holds for the second term with even better reslts on ν We have E h λ P [Φ;r ] = P,new [Φ;r ] + P,old [Φ;r ] where P,new [Φ;r ] = log r p P p P,old [Φ;r ] = νlog r l l p r, log p log p p log r, p P p p r l, log p log p p log r

27 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 7 Lemma 33 implies P 4/,old [Φ;r ] νr which is an admissible error term if ν < 4/r The new part is given by P,new [Φ;r ] = πi κ log log p S p r,;c 4π p r r J κ c p c c c and can be written as where πi κ log r c c m p P p Sm,;c 4π m â m J κ c c log m/ log r { 0 if m or m p r for some p in P, log p log r â m := [, +νr ] log p p if m = p r for some p in P Ths, if ν /r then we obtain P,new [Φ;r,m] ε ε M +νr C / M +r ν κ θ M M + C as in the proof of corollary 35 Smming over C via A gives P,new [Φ;r,m] ε ε M κ θ κ θ + M κ θ κ θ Smming over M via A leads to P,new [Φ;r,m] κ ε θ r ν κ θ+ε κ + θ r ν κ θ +ε which is an admissible error term if ν < r κ θ 5 QUADATIC STATISTICS FO LOW-LYING ZEOS 5 Asymptotic expectation of the two-level density and asymptotic variance Denition 5 Let f Hκ and Φ, Φ in S ν The two-level density relatively to Φ and Φ of Sym r f is D, [Φ,Φ ;r ]f := Φ ρ j Φ f,r ρ j f,r j,j E f,r j ±j emark 5 In this definition, it is important to note that the condition j j does not imply that ρ j f,r ρj It only implies this if the zeros are simple ecall f,r however that some L-fnctions of elliptic crves hence of modlar forms have mltiple zeros at the critical point [3, 4] The following lemma is an immediate conseence of definition 5

28 8 G ICOTTA AND E OYE Lemma 53 Let f H κ and Φ, Φ in S ν Then, D, [Φ,Φ ;r ]f = D, [Φ ;r ]f D, [Φ ;r ]f D, [Φ Φ ;r ]f + H κ f Φ 0Φ 0 We first evalate the prodct of one-level statistics on average Lemma 54 Let r Let Φ and Φ in S ν We assme that hypothesis Nicer, f holds for any prime nmber and any f Hκ and also that θ is admissible If ν < /r then E h D [Φ ;r ]D [Φ ;r ] = E[Φ ;r ]E[Φ ;r ] + d emark 55 Since theorem 4 implies that E h D [Φ ;r ]D [Φ ;r ] E[Φ ;r ]E[Φ ;r ] = E h D [Φ ;r ]D [Φ ;r ] E h D [Φ ;r ]E h D [Φ ;r ], lemma 54 reveals that the term C h D [Φ ;r ],D [Φ ;r ] := d measres the dependence between D [Φ ;r ] and D [Φ ;r ] This term is the asymptotic covariance of D [Φ ;r ] and D [Φ ;r ] In particlar, taking Φ = Φ, we obtain the asymptotic variance Theorem 56 Let Φ S ν If ν < /r then the asymptotic variance of the random variable D, [Φ;r ] is V h D [Φ;r ] = d Proof of lemma 54 From proposition 38, we obtain E h D, [Φ ;r ]D, [Φ ;r ] = E[Φ ;r ]E[Φ ;r ] + C h + r + i,j {,} i j r m =0 m =0 r m=0 m +m E h m E h P [Φ i ;r ]P [Φ j ;r,m] P [Φ ;r,m ]P [Φ ;r,m ] +O log r with C h := Eh P [Φ ;r ]P [Φ ;r ] The error term is evalated by se of theorem 4 and eations, 46 and 47 We first compte C h Using proposition, we compte Ch = E n 4E o with E n := 4 log r p P p p P p log p log p log p log p p log p r log r p r, pr

29 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 9 and E o := log r p P p p P p log p log p log p log p p log p r log r l By definition of the -symbol, we write E n = E n p + and E n e := c c p P p p P p 8πi κ log r E n e with Ep n := 4 log p log p log r p log r p P p log p log p log p log p p log p r log r Spr, pr ;c c J κ l pr 4π c, pr l p r pr We remove the condition p from Ep n at an admissible cost and obtain, after integration by parts, Ep n = d +O log 5 r Using corollary 35, we get E n e log r 5 as soon as ν /r Finally, sing lemma 33, we see that E o is an admissible error term for ν < /r so that eations 5 and 5 lead to C h = d +O log 53 r Let {i, j } = {,} We prove next that each E h P [Φ i ;r ]P [Φ j ;r,m] is an error term when ν < /r Using proposition and lemma 33 we have E h P [Φ i ;r ]P [Φ j ;r,m] = 8πi κ log p log p log p log r i p p log r j log p log r / Sp r r m, p ;c c c c J κ p P p P p p 4π p r c pr m +O log r We se corollary 35 to conclde that E h P [Φ i ;r ]P [Φ j ;r,m] log 54

30 30 G ICOTTA AND E OYE when ν < /r Finally, E h term in the same way P [Φ ;r,m ]P [Φ ;r,m ] is shown to be an error Using lemmas 53 and 54, theorem 4, hypothesis Nicer, f and remark 3, we prove the following theorem Theorem 57 Let r Let Φ and Φ in S ν We assme that hypothesis Nicer, f holds for any prime nmber and any f Hκ and also that θ is admissible If ν < ν,max r,κ,θ then E h D [Φ,Φ ;r ] = [ 0 + r + Φ 0 ][ d Φ Φ 0 + Some comments are given in remark 6 page 7 r + r + N+ r ] Φ 0 Φ 0Φ 0 5 Signed asymptotic expectation of the two-level density and signed asymptotic variance In this part, r is odd Lemma 58 Let Φ and Φ in S ν If ν < /r then E h,ε D [Φ ;r ]D [Φ ;r ] = E[Φ ;r ]E[Φ ;r ] + d emark 59 By theorem 44 and lemma 58 we have E h,ε D [Φ ;r ]D [Φ ;r ] E[Φ ;r ]E[Φ ;r ] = E h,ε D [Φ ;r ]D [Φ ;r ] E h,ε D [Φ ;r ]E h,ε D [Φ ;r ] Ths, C h,ε D [Φ ;r ],D [Φ ;r ] := d is the signed asymptotic covariance of D [Φ ;r ] and D [Φ ;r ] In particlar, taking Φ = Φ, we obtain the signed asymptotic variance Theorem 50 Let Φ S ν If ν < /r then the signed asymptotic variance of D [Φ;r ] is V h,ε D [Φ;r ] = d Proof of lemma 58 From proposition 38 and 39, we obtain with E h,ε D, [Φ ;r ]D, [Φ ;r ] = E[Φ ;r ]E[Φ ;r ] + C h,ε + r r + m =0 m =0 i,j {,} i j r m=0 m +m E h,ε C h,ε m E h,ε P [Φ i ;r ]P [Φ j ;r,m] P [Φ ;r,m ]P [Φ ;r,m ] +O log r := E h,ε P [Φ ;r ]P [Φ ;r ] 55

31 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 3 Assme that ν < /r Then eations 3, 53 and proposition lead to C h,ε = d ε εκ,r G n 4G o 56 with G n := and G o := 4 log r p P p log r p P p p P p p P p log p log p log p log p p log p r log r p r, p r log p log p log p log p p log p r log l p r, pr r l l Lemma 30 implies that if ν < /r then G n νr [r κ +]/ κ / 57 hence G n is an error term as soon as ν /r Lemma 33 implies f H κ G o 3/+νr +ε 58 which is an error term eporting eations 57 and 58 in 56 we obtain C h,ε = d 59 for ν /r r + Next, we prove that each E h,ε P [Φ i ;r ]P [Φ j ;r,m] is an error term as soon as ν /r From eations 3 and 54, we obtain E h,ε P [Φ i ;r ]P [Φ j ;r,m] = ε εκ,r h λ f P [Φ i ;r ]P [Φ j ;r,m] +O 50 log We se proposition and lemmas 33 and 30 to have h f H κ λ f P [Φ i ;r ]P [Φ j ;r,m] It follows from 5 and 50 that E h,ε νr r m+/4 /4 log P [Φ i ;r ]P [Φ j ;r,m] + νr /+ε log 5 = 0 5 for ν /r r + In the same way, we have, for ν in the previos range, P [Φ ;r,m ]P [Φ ;r,m ] = 0 53 E h,ε eporting 59, 5 and 53 in 55, we have the annonced reslt Using lemmas 53, 58, theorem 44, hypothesis Nicer, f and 39, we prove the following theorem

32 3 G ICOTTA AND E OYE Theorem 5 Let f Hκ and Φ, Φ in S ν If ν < /r r + then [ E h,ε D [Φ,Φ ;r ] = 0 + ][ Φ ] Φ 0 + d Φ Φ 0 Φ 0Φ 0 + { } εφ 0Φ 0 emark 5 emark 43 together with theorem 5 and a reslt of Katz & Sarnak see [9, Theorem AD] or [6, Theorem 3] imply that the symmetry type of Fr ε is as in table Some additional comments are given in remark page 3 r ε even odd SOodd Sp SOeven TABLE Symmetry type of F ε r 6 FIST ASYMPTOTIC MOMENTS OF THE ONE-LEVEL DENSITY In this section, we compte the asymptotic m-th moment of the one level density namely M h,m D, [Φ;r ] := lim M h,m D, [Φ;r ] P + where M h,m D, [Φ;r ] = E h m D, [Φ;r ] E h D,[Φ;r ] for m small enogh regarding to the size of the spport of Φ The end of this section is devoted to the proof of theorem E Note that we can assme that m 3 since the work has already been done for m = and m = Thanks to eation 48 and proposition 38, we have M h,m D, [Φ;r ] = m m l=0 l E h P [Φ;r ]m l P [Φ;r ] +O log l where = 0 l m 0 α l P [Φ;r ]f := r log r j = log r m l l α E h l α j =0 p P p r r j j = p P p P [Φ;r ]m l P [Φ;r ]α λ f p r j log p log p p log r λ f p j log p log p p log r 6

33 LOW-LYING ZEOS OF SYMMETIC POWE L-FUNCTIONS 33 and is a positive fnction satisfying log Ths, an asymptotic formla for M h,m D, [Φ;r ] directly follows from the next proposition Proposition 6 Let r be any integer We assme that hypothesis Nicer, f holds for any prime nmber and any primitive holomorphic csp form of level and even weight κ Let α 0 and l 0 be any integers If α and αν < 4/r then P [Φ;r ]α = O E h log If α l m and α + m lν < 4/r r + then E h P [Φ;r ]m l P [Φ;r ]α = O log If α and αν < 4/r r + then O E h P [Φ;r ]α log = d α! α/ α! +O log if α is odd, otherwise 6 One some sefl combinatorial identity In order to se the mltiplicative properties of Hecke eigenvales in the proof of proposition 6, we want to reorder some sms over many primes to sms over primes We follow the work of Hghes & dnick [4, 7] see also [3] and the work of Soshnikov [8] to achieve this Let Pα, s be the set of srjective fnctions σ: {,,α} {,, s} sch that for any j {,,α}, either σj = or there exists k < j sch that σj = σk + This can be viewed as the nmber of partitions of a set of α elements into s nonempty sbsets By definition, the cardinality of Pα, s is the Stirling nmber of second kind [9, 4] For any j {,, s}, let ϖ σ := #σ {j } j Note that ϖ σ j for any j s and s j = ϖ σ j = α 6 The following lemma is lemma 73 of [4, 7] Lemma 6 If g is any fnction of m variables then g m x j,, x jm = j,,j m s= g x iσ,, x iσm σ Pm,s i,,i s