Applications of the Kuznetsov formula on GL(3): the level aspect
aa r X i v : . [ m a t h . N T ] J u l APPLICATIONS OF THE KUZNETSOV FORMULA ON
GL(3)
II: THE LEVELASPECT
VALENTIN BLOMER, JACK BUTTCANE, AND P´ETER MAGA
Abstract.
We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applicationsinclude a Lindel¨of on average type bound for the sixth moment of GL(3) L -functions in the level aspect, anautomorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maaßforms violating the Ramanujan conjecture at finite places. Introduction
While the toolbox of analytic number theory for classical automorphic forms for congruence subgroupsof SL ( Z ) is well developed, much less is known in the case of higher rank groups. It is therefore verydesirable to extend the collection of available methods by genuine higher rank tools, such as explicit andfor the purpose of analytic number theory user-friendly spectral summation formulae. In this paper wewill introduce a version of the powerful Bruggeman-Kuznetsov formula for congruence subgroups of SL ( Z )and see it in action.In the situation of the group SL ( Z ) this versatile formula was first developed independently by Brugge-man [Br] and Kuznetsov [Ku]. Starting with the groundbreaking work of Deshouillers and Iwaniec [DI1], ithas become a very attractive tool, among other things because it provides a method for studying averagesof Kloosterman sums by automorphic techniques, and has shown itself capable of going sometimes beyondthe powerful bounds known for individual Kloosterman sums by the Riemann Hypothesis over finite fields.Classical applications include, among others, density results on exceptional eigenvalues, a proof of Selberg’s3/16 theorem, the best known results on the proportion of critical zeros of the Riemann zeta function, andequidistribution of integral points on spheres.While such a formula exists in great generality, using the theory of general automorphic forms, a mainissue for the purpose of analytic number theory is to make the resulting expression analytically useful.This requires a very good understanding of the integral transforms which relate the test functions on bothsides of the formulas at all places which turns out to be a problem both in real and p -adic analysis. In thispaper the focus is on the level aspect, and a good deal of work is devoted to the investigation of the fineproperties of GL(3) Kloosterman sums with prime power moduli, but some of the applications also requiremore precise information on the archimedean test function than those developed in [Bl].We proceed to describe the new applications that we group into three sections.1.1. Moments of L -functions. While individual L -functions remain rather elusive objects, statisticalinformation in families F of L -functions is often more easily available. The archetypical result in thisdirection is a statement on the order of magnitude (asymptotic formulae, upper bounds, or sometimeslower bounds) of some moment of a family of L -functions X f ∈F | L (1 / , f ) | k . Mathematics Subject Classification.
Primary 11F72, 11F66.
Key words and phrases.
Kuznetsov formula, Kloosterman sums, moments of L -functions, Lindel¨of hypothesis, exceptionaleigenvalues, large sieve, Ramanujan conjecture.The first author was supported by the Volkswagen Foundation and a Starting Grant of the European Research Council.The second author was supported by a Starting Grant of the European Research Council. The third author was supportedby a Starting Grant of the European Research Council and OTKA grant no. NK104183. at the central point. Trace formulae are particularly suitable to evaluate such moments if the family isgiven by “spectral properties”. Here we consider the GL(3) L -functions of large (prime) level N for thecongruence subgroup Γ ( N ), the subgroup of matrices in SL ( Z ) with bottom row congruent to (0 , , ∗ )modulo N acting on the generalized upper half plane H . This is a subgroup of index N o (1) in SL ( Z ).For an SO(3)-invariant subspace of a spherical cuspidal automorphic representation π ⊆ L (Γ ( N ) \ H )let µ π denote the spectral parameter (Langlands parameter at infinity) of π , normalized so that it ispurely imaginary if the Ramanujan conjecture holds. We fix once and for all a compact set Ω ⊆ a ∗ C , thecomplexified dual of the Lie algebra of the maximal torus in PGL ( R ). If Ω is not too small, there areroughly ≍ Ω N o (1) such representations π with µ π ∈ Ω.It is a fairly straightforward exercise with Kuznetsov formula to prove the following best-possible (“Lin-del¨of-on-average”) bound for the fourth moment: X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω | L (1 / , π ) | ≪ N ε for any ε >
0. Here and henceforth in this paper we will apply the usual ε -convention: the letter ε denotesan arbitrarily small real number, not necessarily the same on each occurrence.The main work of the paper is devoted to bound a sixth moment in a best-possible fashion. This givesus the possibility to highlight the finer details of the Kloosterman side of the Kuznetsov formula and togive a sample argument how to combine this formula with the rest of the machinery of analytic numbertheory, such as multiple Poisson summation, estimation of multiple character sums and stationary phasetype arguments for the archimedean weight functions.One of the technical problems is that the Kuznetsov formula – as any spectral summation formula –requires a spectrally complete expression. Therefore we have to artificially add the continuous spectrumwhich unfortunately produces a term of larger of magnitude (the maximal Eisenstein series contribute N / , see Section 5.2). This problem was already faced in the GL(2) situation in [DFI] and [BHM]; in[DFI], a delicate analysis identified a term on the arithmetic side of the Kuznetsov formula that cancelledthe continuous spectrum contribution, while in [BHM] this problem was solved by introducing extra zerosin the Mellin transform of the weight function in the approximate functional equation. Both approachesrequire extremely subtle and precise information on the archimedean test functions in the Kuznetsovformula that is not easily available in higher rank situations. Here we deal with this problem by twistingthe automorphic forms in question with a fixed character in order to kill the unwanted poles incurred bythe continuous spectrum. Our first main theorem is as follows. Theorem 1.
Let N be a large prime and let p be a fixed prime. Let χ be a primitive character modulo p of order > and let Ω ⊆ a ∗ C . Then X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω | L (1 / , π × χ ) | ≪ p, Ω ,ε N ε for every ε > . Spectral mean values and a large sieve.
Many applications call for an estimate of Fourier coeffi-cients, averaged over the automorphic spectrum. For each π ⊆ L (Γ ( N ) \ H ) we choose a newvector ̟ that we normalize such that its Fourier coefficients, defined in (2.7) and (2.8) below, satisfy A ̟ (1 ,
1) = 1.The following useful result is the level analogue of [Bl, Theorem 5]:
Theorem 2.
Let n, m, N ∈ N , ( mn, N ) = 1 , Ω ⊆ a ∗ C . Then we have X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω | A ̟ ( n, m ) | ≪ Ω ,ε ( N mn ) ε ( N + N / nm ) for every ε > . PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 3
The first term on the right hand side is (up to ε ) the number of terms in the sum which dominates thesecond term provided nm ≪ N / . In particular, in this region the result is best possible and can often beused as a substitute for the Ramanujan conjecture.A more refined estimate of this type is the following large sieve inequality for the unramified Heckeeigenvalues λ π ( n ) = A ̟ ( n, n, N ) = 1. Theorem 3.
Let N ∈ N , Ω ⊆ a ∗ C , X > , and let α ( n ) be a sequence of complex numbers supported on X n X . Then X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω (cid:12)(cid:12)(cid:12) X X n X ( n,N )=1 λ π ( n ) α ( n ) (cid:12)(cid:12)(cid:12) ≪ Ω ,ε ( N X ) ε ( N + X N / ) k α k for every ε > . This is in the spirit of the celebrated large sieve inequalities of [DI1]. It should be compared with thecase n = 3 of [Ve] which requires X N / to be optimal whereas our result covers the much largerrange X N / (cf. also [DK] for a different large sieve inequality). This shows the advantage of using apowerful tool like the Kuznetsov formula as opposed to the soft methods in [Ve] which on the other handgeneralize directly to GL( n ).1.3. Exceptional eigenvalues and the Ramanujan conjecture.
The Ramanujan conjecture is one ofthe central open problems in the theory of automorphic forms, known only for cohomological forms. Inanalytic number theory it is often important to control the degree to which the Ramanujan conjecture isviolated, and to show that this cannot happen too frequently. The following theorems provide bounds forthe density of forms violating the Ramanujan conjecture at a given place, and we will show in particularthat in a quantitative sense almost all Maaß forms satisfy the Ramanujan conjecture at a given place. Inthe eigenvalue aspect this has been investigated in [BBR]. We start with the archimedean place and showthe following density result for exceptional Maaß forms of large level:
Theorem 4.
Let N be a prime and Ω ⊆ a ∗ C . Then X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω N kℜ µ π k ≪ Ω ,ε N ε for every ε > . Here and in the following, k . k denotes the maximum norm. The Jacquet-Shalika [JS] bounds imply kℜ µ π k / kℜ µ π k /
14. Our result recovers (essentially) theJacquet-Shalika bounds, but it shows much more: exceptional Maaß forms occur less and less frequent,the more the Ramanujan conjecture at infinity is violated.A similar result can be obtained for a fixed finite place. Let α π ( p ) denote the Satake parameter of arepresentation π at p . The Ramanujan conjecture states that all three entries of α π ( p ) have absolute valueone. Theorem 5.
Let N ∈ N , fix a prime p ∤ N and let δ > . Let Ω ⊆ a ∗ C . Then there exists η > (dependingon δ and p ) such that (cid:8) π ⊆ L (Γ ( N ) \ H ) : µ π ∈ Ω , k α π ( p ) k > δ (cid:9) ≪ Ω ,δ,p ( N ) − η . The Kuznetsov formula for congruence subgroups of SL ( Z )In this section we state and prove the Kuznetsov formula and correct a small error in the statement ofthe formula in [Bl]. This requires a bit of notational preparation. Let N ∈ N be the level. We follow the VALENTIN BLOMER, JACK BUTTCANE, AND P´ETER MAGA approach in [Bl] and compute the inner product of two Poincar´e series in two ways. Let F : (0 , ∞ ) → C be a smooth compactly supported function. Let(2.1) F ∗ ( y , y ) := F ( y , y ) . For two positive integers m , m and z = (cid:16) x x x (cid:17) (cid:16) y y y (cid:17) ∈ H let F m ,m ( z ) := e ( m x + m x ) F ( m y , m y ) . Then we consider the following Poincar´e series: P m ,m ( z ) := X γ ∈ Γ ∞ \ Γ ( N ) F m ,m ( γz )where Γ ∞ is the subgroup of unipotent upper triangular matrices. The Fourier expansion of these functionsfeatures Kloosterman sums and their archimedean analogues, certain special functions given by an integralrepresentation. The three non-trivial terms in the Kuznetsov formula are attached to the elements w = (cid:16) (cid:17) , w = (cid:16)
11 1 (cid:17) and w = (cid:16) − (cid:17) in the Weyl group. Correspondingly, for m , m , n , n ∈ Z \ { } we define(2.2) ˜ S ( m , n , n ; D , D ) := X C (mod D ) ,C (mod D )( C ,D )=( C ,D /D )=1 e (cid:18) n ¯ C C D + n ¯ C D /D + m C D (cid:19) , for D | D and S ( N ) ( m , m , n , n ; D , D )= X B ,C (mod D ) B ,C (mod D ) D C + B B + D C ≡ D D )( B j ,C j ,D j )=1 ,N | B e (cid:18) m B + n ( Y D − Z B ) D + m B + n ( Y D − Z B ) D (cid:19) (2.3)for N | D , N | D , where Y j B j + Z j C j ≡ D j ) for j = 1 ,
2. The latter is almost the same sum asin [BFG, Section 4] for level 1 except for the additional divisibility condition N | B . Note, however, that N | B B is automatic (since N | D , D ), so the additional condition N | B is relatively minor.The archimedean functions don’t see the additional level and are identical to the level 1 case. For ǫ ∈ {± } or {± } , F as above and A , A > J ǫ ; F ( A ) = A − Z ∞ Z ∞ Z ∞−∞ Z ∞−∞ e ( − ǫAx y ) e (cid:18) y · x x x + 1 (cid:19) e (cid:18) Ay y · x x + x + 1 (cid:19) × F y · p x + x + 1 x + 1 , Ay y · p x + 1 x + x + 1 ! F ( Ay , y ) dx dx dy dy y y , (2.4) J ǫ ; F ( A , A ) = ( A A ) − Z ∞ Z ∞ Z ∞−∞ Z ∞−∞ Z ∞−∞ e ( − ǫ A x y − ǫ A x y ) × e (cid:18) − A y · x x + x x + x + 1 (cid:19) e (cid:18) − A y · x ( x x − x ) + x ( x x − x ) + x + 1 (cid:19) F ( A y , A y ) × F A y · p ( x x − x ) + x + 1 x + x + 1 , A y · p x + x + 1( x x − x ) + x + 1 ! dx dx dx dy dy y y . (2.5) PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 5
Next we define for µ ∈ a ∗ C and y , y > W µ ( y , y ) = y y π | Γ( (1 + i ℑ ( µ + 2 µ )))Γ( (1 + i ℑ ( µ − µ )))Γ( (1 + i ℑ (2 µ + µ ))) |× πi ) Z (1) Z (1) Q j =1 Γ( ( s + µ j )) Q j =1 Γ( ( s − µ j ))4 π s + s Γ( ( s + s )) y − s y − s ds ds . (2.6)For a (not necessarily cuspidal) automorphic form ̟ of level N and spectral parameter µ we define theFourier coefficient ˜ A ̟ ( m , m ) ( m , m = 0) by(2.7) Z Z Z ̟ ( z ) e ( − m x − m x ) dx dx dx = ˜ A ̟ ( m , m ) | m m | ˜ W µ ( | m | y , | m | y ) . To ease notation, we will denote by { ̟ } an orthonormal basis of automorphic forms of level N , cuspidal orEisenstein series, containing all cuspidal newvectors, and we denote by R ( N ) d̟ a combined sum/integralover the complete spectrum of level N . The relevant spectral decomposition is a special case of Langlands’general theory, see e.g. [Ar] for a convenient summary in adelic language. By Hecke theory, we can andwill assume that all ̟ are eigenfunctions of the Hecke algebra coprime to N . SinceΓ ( N )diag( m m m , m m , m )Γ ( N ) = Γ (1)diag( m m m , m m , m )Γ (1)for ( m m m , N ) = 1, this is just the unramified Hecke algebra that satisfies the usual GL(3) Heckerelations as in [Go, Theorem 6.4.11]. The proof of [Go, Theorem 6.4.11] also shows that if ˜ A ̟ (1 ,
1) = 0,then ˜ A ̟ ( m , m ) = 0 whenever ( m m , N ) = 1. If ˜ A ̟ (1 , = 0, which is the case in particular fornewvectors ̟ , we write(2.8) A ̟ ( m , m ) = ˜ A ̟ ( m , m ) / ˜ A ̟ (1 , , in which case the normalized Fourier coefficients A ̟ ( m , m ) satisfy the multiplicativity relations of [Go,Theorem 6.4.11]. If ˜ A ̟ (1 ,
1) = 0, we simply write A ̟ ( m , m ) = ˜ A ̟ ( m , m ) and remark already at thisplace that for such ̟ only vanishing Fourier coefficients will come up in our analysis (which, in a trivialway, satisfy the Hecke relations), so that the normalization is irrelevant. For notational consistency wewrite N ( ̟ ) = ˜ A ̟ (1 , if ˜ A ̟ (1 , = 0 and N ( ̟ ) = 1 otherwise.Rankin-Selberg theory shows (see e.g. [Bl, Lemma 1]) that for a cuspidal newform ̟ ∈ π one has N ( ̟ ) ≍ [SL ( Z ) : Γ ( N )] · res s =1 X m ,m | A ̟ ( m , m ) | m s m s , and it follows from [Li, Theorem 2] that(2.9) N ( ̟ ) ≪ N ( N (1 + | µ π | )) ε . We define an inner product on (0 , ∞ ) by h f, g i := Z ∞ Z ∞ f ( y , y ) g ( y , y ) dy dy ( y y ) . With this notation we are ready to state our version of the Kuznetsov formula.
Theorem 6.
Let F be a compactly supported test function with F ∗ as in (2.1) . Let N, n , n , m , m ∈ N .Then (2.10) Z ( N ) A ̟ ( n , n ) A ̟ ( m , m ) N ( ̟ ) |h ˜ W µ π , F i| d̟ = ∆ + Σ + Σ + Σ where ∆ = δ n ,m δ n ,m k F k , Σ = X ǫ = ± X ND | D n D = m D ˜ S ( ǫm , n , n , D , D ) D D ˜ J ǫ ; F ∗ (cid:18)r n n m D D (cid:19) , Σ = X ǫ = ± X N | D | D n D = m D ˜ S ( ǫm , n , n , D , D ) D D ˜ J ǫ ; F (cid:18)r n n m D D (cid:19) , Σ = X ǫ ,ǫ = ± X N | D ,N | D S ( N ) ( ǫ m , ǫ m , n , n , D , D ) D D J ǫ ; F (cid:18) √ n m D D , √ n m D D (cid:19) . (2.11) Remarks: (1) In [Bl], the first two entries in the long Weyl element Kloosterman sum are mistakenlyinterchanged, cf. [BFG, p. 64].(2) The Fourier coefficients of Eisenstein series for Γ ( N ) ⊆ SL ( Z ) for all indices are computed in detailin [Ba].(3) Note that there is a small asymmetry in the definition of Σ and Σ . If ( n m , N ) = 1, then thesummation condition in Σ is equivalent to D = N d d , D = N d , n d = m d , while the summationcondition in Σ is equivalent to D = N d , D = N d d , n d = m d , so complete symmetry betweenΣ and Σ is restored. Proof.
This is exactly as in [Bl, Proposition 4] by computing the inner product of two level N Poincar´eseries by unfolding and by spectral decomposition and then comparing both expressions. We only haveto verify that the definition of our Kloosterman sums agrees with the Fourier expansion of the level N Poincar´e series. The exponential sums appearing in the latter are most easily, but abstractly, definedin terms of the Bruhat decomposition, so the procedure is to enumerate the terms in the sum using thePl¨ucker coordinates, determine the summand as a function of the Pl¨ucker coordinates by writing out theBruhat decomposition of each term, and then verify that the summand only depends on the residue classesof the Pl¨ucker coordinates.Let U ( R ) be the group of upper-triangular matrices with ones on the diagonal and entries in the ring R , W the Weyl group and V the diagonal orthogonal matrices of SL ( Z ). We also need the decompositionof U ( R ) by w ∈ W , so set U w ( R ) = ( w − U ( R ) w ) ∩ U ( R ) , ¯ U w ( R ) = ( w − U ( R ) t w ) ∩ U ( R ) . Define characters of U ( R ) by ψ n ,n x ∗ x = e ( n x + n x )where we assume n , n ∈ Z . Then the Bruhat decomposition of some γ ∈ SL ( Z ) takes the form γ = bcvwb ′ with w ∈ W , v ∈ V , b, b ′ ∈ U ( R ) and c = diag(1 /c , c /c , c ) for some c , c ∈ N . The Bruhatdecomposition is only defined up to an element of U w ( R ).Now let w ∈ W , n , n , m , m ∈ Z and c = diag(1 /c , c /c , c ) as before. If the compatibility condition ψ n ,n (( cw ) u ( cw ) − ) ψ m ,m ( u − ) = 1 for all u ∈ U w ( R )holds, we define the Kloosterman sums(2.12) S w ( ψ n ,n , ψ m ,m ; c ) = X γ = bcwb ′ ∈ U ( Z ) \ Γ ( N ) /V ¯ U w ( Z ) ψ n ,n ( b ) ψ m ,m ( b ′ ) . PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 7
The sum is over representatives γ in the quotient space having the prescribed components c and w in theirBruhat decomposition, which is well-defined by the compatibility condition. The quotient by V simplyallows us to restrict to positive moduli c and c by conjugating the v matrix, which contains the signs ofthe moduli, to the right. If the compatibility relation fails, we simply define S w ( ψ n ,n , ψ m ,m ; c ) = 0.By a computation of Friedberg [Fr, pp. 173-174], only sums satisfying the compatibility condition occurin the Fourier expansion of a Poincar´e series. In particular, for n n m m = 0, only the I , w , w , and w Weyl elements contribute, since otherwise the compatibility relation is never satisfied.We now wish to show that the concrete expressions for the Kloosterman sums given in (2.2) and (2.3)match the abstract definition (2.12).We may parameterize representatives of U ( Z ) \ Γ ( N ) by the Pl¨ucker coordinates A , B , C and A , B , C satisfying ( A , B , C ) = ( A , B , C ) = 1 ,A C + B B + C A = 0 ,N | A , N | B . (2.13)For a matrix γ = (cid:18) g h id e fa b c (cid:19) ∈ Γ ( N ), these are computed by A = a, B = b, C = c, A = bd − ae, B = af − cd, C = ce − bf. Our computation now essentially follows [BFG], but we must keep track of the level condition N | A , N | B . The auxiliary parameters Z = g , Y = h , X = i , Z = ge − dh , Y = di − gf , and X = f h − ei aresolutions to the equations Z C + Y B + X A = 1 ,Z C + Y B + X A = 1 . (2.14)These equations do not completely determine the auxiliary parameters, but we will only require that theauxiliary parameters are some solution, as the final expression for the Kloosterman sum will be independentof the choice. The right-translation action of x ∈ U ( Z ) on the Pl¨ucker coordinates gives the new values( A , B , C ) ( A , B + x A , C + x B + x A ) , ( A , B , C ) ( A , B − x A , C − x B + ( x x − x ) A ) . Now the Bruhat decomposition for elements of the long element Weyl cell in Γ ( N ) may be written as γ = Z B − Y A A Z A Y A − Z B A A A A A w B A C A − B A . Note that we put our decompositions in the form bcwb ′ with b ′ ∈ ¯ U w ( Q ), as opposed to [BFG], who puttheir decompositions in the form bwcb ′ with b ∈ ¯ U w − ( Q ). The decompositons are equivalent, but theformer is more standardized.Restricting to a fundamental domain for the action of ¯ U w ( Z ) = U ( Z ), we may write the sum (2.12) for w = w as S w ( ψ n ,n , ψ m ,m ; ( A , A )) = X B ,C ,B ,C e (cid:18) n Y A − Z B A + n Z B − Y A A − m B A + m B A (cid:19) , where the sum is taken over 0 ≤ B , C < A and 0 ≤ B < A subject to (2.13) and (2.14). Now [BFG,Lemma 4.1] shows that the summand is independent of the choice of auxiliary parameters. Note that thecompatibility condition is trivially true for the long element as U w ( R ) = { I } . VALENTIN BLOMER, JACK BUTTCANE, AND P´ETER MAGA
We conclude the long element analysis by mentioning that the proofs of [BFG, Lemmas 4.1, 4.2] showthat the sum is well-defined if we replace the summation conditions with their modular equivalents B , C (mod A ) , B (mod A ) , N | A , N | B A C + B B + C A ≡ A A ) , ( A , B , C ) = 1 , ( A , B , C ) = 1 ,Z C + Y B = 1 (mod A ) , Z C + Y B = 1 (mod A ) , (2.15)and the sum is empty unless N | A . This matches the previous definition with(2.16) S w ( ψ n ,n , ψ m ,m ; ( A , A )) = δ N | A S ( N ) ( m , − m , n , − n ; A , A ) . As noted in the proof of [BFG, Theorem 5.1], replacing m , n , B , C , C , Y , Z , Z by their negativesleaves the sum invariant, so we may drop the negatives on m and n .Elements of the w cell necessarily have A = 0 and B , A = 0, so that B | A . With the Pl¨ucker andauxiliary coordinates as before, for γ having A = 0 we have γ = Z B A Y B Z C − X A B A A B B w − C A C B . The compatibility condition becomes n A = m B . Now the conditions (2.13) and (2.14) simplify to( B , C ) = 1 , ( A /B , C ) = 1 , B = − C A B , N | B ,Z C ≡ B ) , Z C ≡ A /B ) , and the space ¯ U w ( Z ) ⊂ U ( Z ) is defined by x = 0, so we may write the Kloosterman sum as S w ( ψ n ,n , ψ m ,m ; ( B , A )) = δ n A = m B N | B | A ˜ S ( m , n , n ; B , A ) . Unlike the long element case, no extra work is needed to justify our use of the modular summationconditions.For the w cell, we have A = 0 and A , B = 0, so that B | A , and the compatibility condition is n A = m B . With the Pl¨ucker and auxiliary coordinates as before, we have γ = X A − Z C B Z A − Z B A B B A A w − C B C A . Now the conditions (2.13) and (2.14) simplify to( A /B , C ) = 1 , ( B , C ) = 1 , B = − C A B , N | A , N | B ,Z C ≡ A /B ) , Z C ≡ B ) , but we may take this one step further: we have B B N = − C A N and the condition ( B , C ) = 1 implies B | A N . Conversely, the condition N B | A implies N | ( − C A B ) = B , so we may write the Kloostermansum as S w ( ψ n ,n , ψ m ,m ; ( A , B )) = δ n A = m B NB | A ˜ S ( − m , − n , − n ; B , A ) . Note that changing the sign of both n and n leaves ˜ S invariant by C
7→ − C . PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 9 Weight functions in the Kuznetsov formula
In order to use the Kuznetsov formula for a spectral average, we need a function F such that |h F, ˜ W µ i| appearing on the left hand side of (2.10) is bounded away from zero for µ ∈ Ω ⊆ a ∗ C . For our purposes thefollowing slightly weaker statement suffices. Lemma 1.
For a fixed compact Ω ⊆ a ∗ C there is a finite collection of smooth compactly supported functions F , . . . , F J such that P j |h F j , ˜ W µ i| ≫ for µ ∈ Ω . Proof.
This follows from a simple compactness argument: for each µ ∈ Ω choose an open set S µ ⊆ R > such that ℜ ˜ W µ ( y ) = 0 for all y ∈ S µ or ℑ ˜ W µ ( y ) = 0 for all y ∈ S µ . This is possible by continuity of˜ W µ ( y ) in y . By continuity in µ , we can choose open neighbourhoods U µ about µ such that ℜ ˜ W µ ∗ ( y ) = 0for all y ∈ S µ and all µ ∗ ∈ U µ or ℑ ˜ W µ ( y ) = 0 for all y ∈ S µ and all µ ∗ ∈ U µ . By compactness we picka finite collection of such neighbourhoods U µ , . . . , U µ J covering Ω, and define the corresponding F j to bereal-valued functions with support on S µ j and non-vanishing on the interior ˚ S µ j .For the proof of Theorem 4 we will need a function that blows up on the exceptional spectrum. Werecall that by unitarity the exceptional spectrum is parametrized by(3.1) µ = ( ρ + iγ, − ρ + iγ, − iγ )for γ ∈ R , ρ ∈ [ − / , /
2] (by the Jacquet-Shalika bounds) and its translates under the Weyl group. Fora fixed smooth compactly supported function F and two parameters X , X > F ( X ,X ) ( y , y ) := F ( X y , X y )so that F = F (1 , . Lemma 2.
Fix Ω ⊆ a ∗ C , and let X , X > , ε > . Assume that F is non-negative and supported in a(depending on Ω ) sufficiently small neighbourhood about (1 , and that X , X are sufficiently large. Thenfor exceptional µ ∈ Ω of the form (3.1) with | ρ | > ε we have h F ( X ,X ) , ˜ W µ i ≍ ( X X ) | ρ | . Proof.
We have by (2.6) that h F ( X ,X ) , ˜ W µ i = Z (1) Z (1) cosh( πγ ) Q j =1 Γ( ( s + µ j )) Q j =1 Γ( ( s − µ j ))4 π s + s Γ( ( s + s )) F ( − − s , − − s ) X s X s ds ds (2 πi ) where F is the double Mellin transform of F , an entire function in both variables. If without loss ofgenerality ρ > µ , µ , µ are sufficiently distinct), we shift contours to the leftand obtain h F ( X ,X ) , ˜ W µ i = c µ F ( − − ρ + iγ, − − ρ − iγ )( X ρ − iγ + O ( X ))( X ρ + iγ + O ( X ))for some constant c µ = 0. If F is non-negative and supported in a sufficiently small neighbourhood about(1 , F ( − − ρ + iγ, − − ρ − iγ ) = 0 for all µ ∈ Ω. This proves the lemma.Next we provide bounds for the functions ˜ J ǫ ; F ( A ) and J ǫ ; F ( A , A ) defined in (2.4) and (2.5). Here F will always be a fixed compactly supported function and all implied constants may depend on F . Forbounds in the case of certain highly oscillating functions F see [Bl, Proposition 5]. We define F ( X ,X ) asin (3.2). The following basic bound suffices in many cases. Lemma 3.
Let X , X > . (a) We have ˜ J ǫ ; F ( X ,X ( A ) = 0 unless A ≫ X − / + X − / , in which case ˜ J ǫ ; F ( X ,X ( A ) ≪ ( X X ) . (b) We have J ǫ ; F ( X ,X ( A , A ) = 0 unless min( A A , A A ) ≫ ( X X ) − / , in which case d i dA i d j dA j J ǫ ; F ( X ,X ( A , A ) ≪ i,j ( X X ) ( A A ) ε (cid:16) A / A / (cid:17) i (cid:16) A / A / (cid:17) j for all i, j ∈ N = { , , , . . . } . Remark:
Except for one instance in the proof of Theorem 4 we will always apply this lemma with X = X = 1, so for most of the paper the variables X , X can be ignored. Proof. (a) This is straightforward from the definition and uses only trivial bounds, noting that thesupport of F ( X ,X ) restricts the variables to y ≍ ( X A ) − , y ≍ X − , x ≍ A / X , x + x ≍ A / ( X X ) . This forces A ≫ X − / + X − / . The upper bound follows now from trivial estimates .(b) The support of F ( X ,X ) restricts the variables to y ≍ ( X A ) − , y ≍ ( X A ) − , ( x x − x ) + x + 1 =: ξ ≍ Ξ := A / A / ( X X ) , x + x + 1 =: ξ ≍ Ξ := A / A / ( X X ) which implies that both A A and A A must be at least of order ( X X ) − / . We recall from [Bl, Lemma4] that(3.3) Z ξ ≍ Ξ ξ ≍ Ξ dx dx dx ≪ (Ξ Ξ ) / ε = ( A A X X ) ε for Ξ , Ξ ≫
1. For the derivatives we differentiate under the integral sign and estimate trivially, see also[Bl, (8.16)].For one application we need a more refined estimate of a certain 6-fold Fourier transform involving J ǫ ; F . Lemma 4.
Let W : (0 , ∞ ) → C be a fixed smooth compactly supported function. Let A , A > and define A := exp(max( | log A | , | log A | )) . Let P > , and let α , α , β , β , γ , γ ∈ R be such that min( | α | , | α | , | β | , | β | , | γ | , | γ | ) P . Then the six-fold Fourier transform b J := Z R J ǫ ; F ( A √ t u v , A √ t u v ) W ( t , t , u , u , v , v ) × e ( − t α − t α − u β − u β − v γ − v γ ) dt dt du du dv dv is bounded by (3.4) O C (cid:16) ( P A ) ε ( P max (cid:16) A − / A − / , A − / A − / (cid:17) + P − C ) (cid:17) for any constant C > . In addition, it is bounded by (3.5) A ε max( | α | , | β | , | γ | ) − / max( | α | , | β | , | γ | ) − / , as long as both maxima are non-zero. Proof.
We recall the notation( x x − x ) + x + 1 =: ξ ≍ Ξ := A / A / , x + x + 1 =: ξ ≍ Ξ := A / A / from the previous proof. We will frequently use (3.3) with X = X = 1. We also write x = x x − x and η = x x + x , η = x x + x . This corrects an error [Bl, (8.7)] where X X should be replaced with X X . PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 11
We express J ǫ ; F by its defining 5-fold integral (2.5) and write J ǫ ; F ( A , A ) = 1( A A ) Z ξ ≍ Ξ ξ ≍ Ξ K ( A , A ; x , x , x ) dx dx dx where K is the double y , y -integral, i.e. K ( A , A ; x , x , x ) = Z ∞ Z ∞ e (cid:18) − ǫ A x y − ǫ A x y − A η y ξ − A η y ξ (cid:19) × F ( A y , A y ) F A ξ / y ξ , A ξ / y ξ ! dy dy y y . (3.6)We start with the proof of (3.4). Suppose that | α | is the smallest of the variables (possibly | α | = 0).Choose a sufficiently large constant c and a sufficiently large constant c > c . We split the x , x , x -integration in four pieces(i) | x | , A | η | /ξ c P, (ii) | x | c P, A | η | /ξ > c P, (iii) | x | > c P, A | η | /ξ c P and the remaining portion (iv), which is contained in | x | , A | η | /ξ > c P . The conditions (i) imply | x | ≪ P , x x ≪ P A / A / (note that we may assume by Lemma 3 that this is ≫ P ). The area of thisregion is ≪ P Z x x ≪ P A / A / x ,x ≪ A O (1) dx dx ≪ P A / A / ( AP ) ε . As K ( A , A ; x , x , x ) ≪
1, the total contribution of this case to b J is P A − / A − / ( AP ) ε .To deal with the region (ii), we note that the phase in the y -integral in (3.6) is given by e (cid:18) − ǫ A x y − A η y ξ (cid:19) . If c is sufficiently large compared to c (or if ǫ x and η have different signs), the phase has no stationarypoint, and after sufficiently many integrations by parts, using for instance [BKY, Lemma 8.1] with X = A , U = Q = 1 A , Y = P, R = A Y, we bound after trivial estimation in all other variables this portion of b J by ≪ A ε P − C . The same argumentworks for the region (iii).In order to analyze the region (iv), we consider the expression (3.6) in more detail, first without anyrestrictions on the x -variables. We could run a careful stationary phase argument as in [BKY, Proposition8.2], but we can also proceed in a completely elementary way. Applying the stationary phase method onlyon a formal basis shows that the oscillation of the y -integral is given by e (cid:16) − sgn( η ) 2 p | x η | A √ ξ (cid:17) , coming from the stationary point at y = ( η / ( ǫ x ξ )) / . With this in mind let us define˜ K ( A , A ; x , x , x ) := e sgn( η ) 2 p | x η | A √ ξ + sgn( η ) 2 p | x η | A √ ξ ! K ( A , A ; x , x , x )= Z ∞ Z ∞ e (cid:16) g ( A , y ) + h ( A , y ) (cid:17) F ( A y , A y ) F A ξ / y ξ , A ξ / y ξ ! dy dy y y with g ( A , y ) = g ǫ x ,η ,ξ ( A , y ) = − ǫ A x y − A η y ξ + sgn( η ) 2 p | x η | A √ ξ
12 VALENTIN BLOMER, JACK BUTTCANE, AND P´ETER MAGA and h ( A , y ) = h ǫ x ,η ,ξ ( A , y ) = − ǫ A x y − A η y ξ + sgn( η ) 2 p | x η | A √ ξ . We show the uniform bound(3.7) ∂ i ∂A i ∂ j ∂A j ˜ K ( A , A ; x , x , x ) ≪ i,j A − i A − j . Indeed, one checks by direct computation that ∂∂A g ( A , y ) ∂∂y g ( A , y ) = ± y A · p | η | − p | x | ξ y p | η | + p | x | ξ y so that ∂ i ∂y i ∂ j ∂A j (cid:16) ∂∂A g ( A , y ) ∂∂y g ( A , y ) (cid:17) ≪ i,j y A y − i A − j for i, j ∈ N . Hence combining each differentiation with respect to A with an integration by parts in y ,we obtain the desired bound (3.7) in A , and the bound in A follows similarly.Having proved (3.7), we return to the estimation of b J . The phase of the Fourier integral in question isgiven by(3.8) e ± p | x η | A √ t u v √ ξ ± p | x η | A √ t u v √ ξ − t α − t α − u β − u β − v γ − v γ ! , which needs to be integrated against the non-oscillating functions˜ K ( A √ t u v , A √ t u v ; x , x , x ) W ( t , t , u , u , v , v )with respect to x , x , x and t , t , u , u , v , v . If we are in the region (iv), then in particular | x | , A | η | /ξ > c P , as mentioned above. Since | α | P , the phase has no stationary point if c is sufficiently large, and byrepeated partial integration in any of the variables t we obtain again the bound A ε P − C . This completesthe proof of (3.4) if | α | is minimal. If any of the other variables is minimal, we can run the same argument,possibly with interchanged indices.For the bound (3.5) we return to (3.8) for an arbitrary choice of x , x , x . The simple stationary phasetype bound Z e ( at + b √ t ) W ( t ) dt ≪ | a | − / , a = 0for a fixed smooth function W with compact support in (0 , ∞ ) applied twice, followed by trivial estimations,yields readily the bound (3.5). This completes the proof.4. Kloosterman sums
In this section we collect some results about the Kloosterman sums defined in (2.2) and (2.3). We startwith useful upper bounds.
Lemma 5.
Let
N, D , D ∈ N , m , m , n , n ∈ Z \ { } . We have ˜ S ( m , m , n ; D , D ) ≪ (cid:16) ( n , D /D ) D , ( m , n , D ) D (cid:17) ( D D ) ε and (4.1) S ( N ) ( m , m , n , n ; D , D ) ≪ ( D D ) / ε (cid:0) ( D , D )( m n , [ D , D ])( m n , [ D , D ]) (cid:1) / for any ε > . PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 13
Proof.
The bound for ˜ S is Larsen’s bound [BFG, Appendix]. The bound for S ( N ) is Stevens’ bound [St,Theorem 5.1] in its uniform version given in [Bu, p. 39]. Note that for the level N Kloosterman sum, onlythose S a,b ( n, ψ, ψ ′ ) (in the notation of [St, Section 5]) contribute to the Kloosterman sum where a s − k , b r with s, r > k whenever p k k N (cf. also [DF, Remark 2.5]). In particular, Stevens’ bound holds afortiori for level N Kloosterman sums.As in [Bl, Lemma 3] we conclude from (4.1) that X N | D X N | D X | S ( N ) ( m , m , n , n ; D , D ) |≪ ( X X ) / ε N / X δδ X /Nδδ X /N δ / (cid:0) ( m n , δ )( m n , δ )( m n , δ )( m n , δ )( m n , δ )( m n , δ ) (cid:1) / ≪ ( X X ) / ε ( m n m n ) ε N / X δ X ( m n , δ ) / ( m n , δ ) / δ / ≪ ( X X ) / ε ( m n m n ) ε N / (4.2)if ( m m n n , N ) = 1. Lemma 6.
For N ∈ N the following holds. (a) The sum S ( N ) ( m , m , n , n ; D , D ) depends only on m j , n j modulo D j for j = 1 , . (b) If ( t t , u u ) = 1 for j = 1 , and N | t u , t u , then S ( N ) ( m , m , n , n ; t u , t u )= S (gcd( N,t )) (¯ u u m , ¯ u u m , n , n ; t , t ) S (gcd( N,u )) (¯ t t m , ¯ t t m , n , n ; u , u ) . (c) Let N be prime and let r q ( n ) denote the Ramanujan sum. Then S ( N ) ( m , m , n , n ; N, N ) = N − r N ( n ) r N ( m ) = N ( N − , N | n , N | m ,N, N ∤ n m , else. Proof.
This is proved as in [BFG, Section 4], cf. Properties 4.6, 4.7 and 4.10, respectively.Part (a) is trivial.For part (b) we observe that the assumptions ( t , u ) = 1 and N | t u imply that(4.3) (cid:0) ( N, t ) u , ( N, u ) t (cid:1) = N. We now follow verbatim the proof of [BFG, Property 4.7]. Given two sets of summation variables B j , C j and B ′ j , C ′ j such that ( B j , C j , t j ) = ( B ′ j , C ′ j , u j ) = 1 for j = 1 ,
2, (
N, t ) | B , ( N, u ) | B ′ , t t | t C + B B + C t and u u | u C ′ + B ′ B ′ + C ′ u we choose r, r ′ ∈ Z with rt t + r ′ u u = 1. We define newvariables d = t u , b = r ′ u u B + rt t B ′ , c = ( r ′ ) u u C + r t t C ′ ,d = t u , b = r ′ u u B + rt t B ′ , c = ( r ′ ) u u C + r t t C ′ , and observe that b runs through all numbers modulo d = t u that are divisible by (4.3), which is thedesired extra divisibility condition N | b for S ( N ) ( m , m , n , n ; t u , t u ). Now we continue verbatim Notice that in [Bl, Lemma 3] the indices should be exchanged and read as in (4.1) above as a consequence of Remark 1after Theorem 6. as in [BFG].To prove (c) we observe that S ( N ) ( m , m , n , n ; N, N ) for N prime equals X C ,B ,C (mod N ) C + C ≡ N )( C ,N )=( B ,C ,N )=1 e (cid:18) − n ¯ C B N + m B N (cid:19) = X B,C (mod N )( C,N )=1 e (cid:18) − n CBN + m BN (cid:19) = N − X B,C (mod N )( BC,N )=1 e (cid:18) − n CBN + m BN (cid:19) = N − r N ( n ) r N ( m ) . This completes the proof of the lemma.
Remark:
For completeness we also state the following two properties of the level N Kloosterman sumsthat can be proved as in Property 4.3 and Property 4.4+4.5 of [BFG]:(d) For ( D D , ab ) = 1 we have S ( N ) ( am , bm , n , n ; D , D ) = S ( N ) ( m , m , an , bn ; D , D ) . (e) We have S ( N ) ( m , m , n , n ; D , D ) = S ( N ) ( n , n , m , m ; D , D ) . Note, however, that an analogue of Property 4.4 or Property 4.5 alone does not exist due to asymmetry ofthe summation condition N | B . We do not need the statements (d) and (e) in this paper.For later purposes we study a certain 6-fold Fourier transform of the long Weyl element Kloostermansum. Let d, D , D ∈ N , and let a ∈ ( Z /D Z ) ∗ and b ∈ ( Z /D Z ) ∗ . For integers x , x , y , y , z , z wedefine b S a,b,d ( x , x , y , y , z , z ; D , D ) := 1 D D X n ,m ,l (mod D ) n ,m ,l (mod D ) S (1) ( am d, bn l , n l , m d ; D , D ) × e (cid:18) − n x + m y + l z D (cid:19) e (cid:18) − n x + m y + l z D (cid:19) . (4.4)This is the non-archimedean analogue of the function studied in Lemma 4.We also need a twisted version. Let χ be a primitive character modulo a prime p such that ( d, p ) = 1.Assume that p | D and p | D . Then we define b S χa,b,d ( x , x , y , y , z , z ; D , D ) := 1 D D X n ,m ,l (mod D ) n ,m ,l (mod D ) ¯ χ ( n l m ) χ ( m n l ) × S ( p ) ( am d, bn l , n l , m d ; D , D ) e (cid:18) − n x + m y + l z D (cid:19) e (cid:18) − n x + m y + l z D (cid:19) . (4.5)By the Chinese remainder theorem and Lemma 6(b) we have the following multiplicativity formulae b S χa,b,d ( x , x , y , y , z , z ; t p α , t p α ) = b S χa,b,d (¯ t x , t ¯ t x , t ¯ t y , ¯ t y , ¯ t z , ¯ t z ; p α , p α ) × b S a,b,d ( p α x , p α p α x , p α p α y , p α y , p α z , p α z ; t , t )(4.6)whenever α , α > p ∤ t t , as well as b S a,b,d ( x , x , y , y , z , z ; t u , t u ) = b S a,b,d (¯ t x , t ¯ t x , t ¯ t y , ¯ t y , ¯ t z , ¯ t z ; u , u ) × b S a,b,d (¯ u x , u ¯ u x , u ¯ u y , ¯ u y , ¯ u z , ¯ u z ; t , t )(4.7)whenever ( t t , u u ) = 1. Here a and b on the right hand sides are understood as primitive residue classesin the respective smaller residue rings. PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 15
Let q be a prime. By (2.3) we have S ( N ) ( am d, bn l , n l , m d ; q α , q α )= X B ,C (mod q α ) B ,C (mod q α ) e (cid:18) am dB + n l ( Y q α − Z B ) q α + bn l B + m d ( Y q α − Z B ) q α (cid:19) (4.8)for N | q min( α ,α ) , where the sum is subject to(4.9) q α C + B B + q α C ≡ q α + α ) , ( B j , C j , q ) = 1 , N | B and(4.10) Y j B j + Z j C j ≡ q α j ) for j = 1 , . We keep in mind that q ∤ ab . This sum is well-defined as shown in [BFG, Lemma 4.2], and does not dependon the choice of the representatives B , B . In particular, we can and will always assume1 B j q α j for j = 1 , . For future purposes we notice that (4.9) implies(4.11) v q ( B ) α , v q ( B ) α where v q denotes the q -adic valuation. Indeed, the first inequality is trivial if α α . If α > α , the firstcondition in (4.9) implies q α C + B B ≡ q α ). If q ∤ B , there is nothing to prove, otherwise wehave q ∤ C , so v q ( B ) v q ( B B ) = α . Similarly one shows the second inequality. Lemma 7.
Let q be a prime, and let α , α ∈ N . For x , y , z , x , y , z ∈ Z and d ∈ N define γ := min (cid:0) v q ( x ) , v q ( x ) , v q ( y ) , v q ( y ) , v q ( z ) , v q ( z ) (cid:1) , δ = v q ( d ) . Then we have | b S a,b,d ( x , x , y , y , z , z ; q α , q α ) | q α ,α ) − ( α + α )+2( γ + δ ) . Here we apply the usual convention min( ∞ , n ) = n for n ∈ N . The bound is meaningless for x = x = y = y = z = z = 0. This case will be considered in Lemma 9. We defer the lengthy proof of Lemma 7to the end of this section. A similar result holds for the twisted transform. Lemma 8.
Let χ be a primitive character modulo a prime p , and let α , α ∈ N . For x , x , y , y , z , z ∈ Z define ρ := min (cid:0) v p ( x ) , v p ( x ) , v p ( y ) , v p ( y ) , v p ( z ) , v p ( z ) (cid:1) . Assume that ( d, p ) = 1 . Then we have | b S χa,b,d ( x , x , y , y , z , z ; p α , p α ) | p α ,α ) − ( α + α )+2 ρ +5 . Proof.
This follows from the previous lemma and the following simple observation. Let χ be a primitivecharacter modulo p , and let S be a p α -periodic function with α >
1. Then(4.12) 1 p α X n (mod p α ) χ ( n ) S ( n ) e (cid:18) − nxp α (cid:19) = 1 τ ( ¯ χ ) p − X β =1 ¯ χ ( β ) 1 p α X n (mod p α ) S ( n ) e (cid:18) − n ( x + p α − β ) p α (cid:19) where as usual τ ( χ ) denotes the Gauß sum (a complex number of absolute value p / ). We apply thisformula for all six summation variables in (4.5) and estimate the various β -sums trivially (this producesan extra factor of p / = p ). Then we apply Lemma 7 with γ ρ + 1 and δ = 0. Lemma 9.
Let χ be a primitive non-quadratic character modulo a prime p , α , α > , ( d, p ) = 1 . Then b S χa,b,d (0 , , , , , p α , p α ) = 0 . Proof.
We start with the observation that a Gauß sum X r (mod p λ ) χ ( r ) e (cid:18) Krp λ (cid:19) vanishes unless v p ( K ) = λ − p | B , hence p ∤ C , and we can choose Y = 0, Z = ¯ C in(4.10). Hence the n , n -sum becomes X n (mod p α ) X n (mod p α ) ¯ χ ( n ) χ ( n ) e (cid:18) − n l ¯ C B p α (cid:19) e (cid:18) bn l B p α (cid:19) . Since ( l l , p ) = 1 by the presence of the character and ( b, p ) = 1 by assumption, this is only non-zero if α = α = α >
3, say, which we assume from now on. Moreover, v p ( B ) = α −
1. Next, the sum over m equals X m (mod p α ) χ ( m ) e (cid:18) am dB p α (cid:19) which implies v p ( B ) = α −
1. We write B = p α − β , B = p α − β with p ∤ β , β . Then C ≡− C − β β p α − (mod p α ), and the B , B , C , C -sum becomes X β ,β (mod p )( β β ,p )=1 X C (mod p α )( C ,p )=1 e am dβ − n l ¯ C β p + bn l β + m d ( C + β β p α − ) β p ! = p α − X β ,β ,C (mod p )( β β C ,p )=1 e (cid:18) am dβ − n l ¯ C β p + bn l β + m d ¯ C β p (cid:19) . Here we use that α >
3. The character implies that all variables in the numerator are coprime to p .Changing variables β ¯ m β , β n l β , C n l n l C , we see that this expression depends only onthe product m n l m n l , not on the six variables individually. Calling this expression T ( m n l m n l ),we obtain finally by another change of variables (e.g. m m n ) that X n ,n ,m ,m ,l ,l (mod p α ) ¯ χ ( n l m ) χ ( n l m ) T ( m n l m n l )= X n ,n ,m ,m ,l ,l (mod p α ) ¯ χ ( n l m ) χ ( n l m ) T ( m l m n l )and the n -sum vanishes since χ is not quadratic.We combine the previous computations to the following useful result: Corollary 10.
Let χ be a primitive non-quadratic character modulo a prime p . Let D , D , d ∈ N satisfying p | D , p | D and ( d, p ) = 1 . Let x , x , y , y , z , z ∈ Z . Then | b S χa,b,d ( x , x , y , y , z , z ; D , D ) | p τ (( D , D )) ( D , D ) D D ( x , x , y , y , z , z , D , D ) ( d, D , D ) where τ is the ternary divisor function. Moreover, b S χa,b,d (0 , , , , , D , D ) = 0 . Indeed, the last statement is a direct consequence of (4.6) and Lemma 9. The first bound follows fromthe Chinese remainder theorem together with (4.6), (4.7) and Lemmas 7 and 8, noting that τ ( q ) = 3 fora prime q and q α ,α ) − ( α + α ) = ( q α , q α ) / ( q α q α ). PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 17
Finally we give the proof of Lemma 7. We will frequently use the following simple result(4.13) (cid:12)(cid:12)(cid:12) X n,l (mod q α ) e (cid:18) nlBq α (cid:19) e (cid:18) − nx − lzq α (cid:19)(cid:12)(cid:12)(cid:12) = ( q α + v q ( B ) , v q ( B ) min( v q ( x ) , v q ( z ))0 , otherwisefor integers x, z and v q ( B ) α . Indeed, the n -sum vanishes unless lB − x ≡ q α ) in which caseit equals q α . This implies in particular v q ( B ) v q ( x ) and defines l modulo q α − v q ( B ) . Then the l -sumvanishes unless v q ( z ) > v q ( B ), and the result follows easily.We will now distinguish several cases to estimate b S a,b,d , but the overall strategy is always the same. Weopen the Kloosterman sum and pull the B , B , C , C -sums outside. Then we sum over n , n , l , l , m , m using orthogonality of additive characters or (4.13). At this point we estimate trivially and just count howmany quadruples ( B , B , C , C ) survive in the outer sum.In order to avoid pathological cases, we treat the case α α = 0 separately. The case α = α = 0 istrivial, so let us assume α > α = 0. In this case the w -Kloosterman sum degenerates to an ordinaryKloosterman sum S (1) ( am d, bn l , n l , m d, q α ,
1) = S ( am d, n l , q α )by [BFG, Property 4.9], so that we need to bound1 q α X n ,l ,m (mod q α ) S ( am d, n l , q α ) e (cid:18) − n x − m y − l z q α (cid:19) . We open the Kloosterman sum, sum over n , l by (4.13) and over m by orthogonality of characters gettingthe bound q − α + δ . The case α = 0, α > α > α > q ∤ C or q | C . This gives a decomposition b S a,b,d = T a,b,d + U a,b,d . The second term where q | C is easier. In this case q ∤ B by (4.9), and the first condition there implies q | B (since α , α > q ∤ C . Hence we can choose Y = ¯ B , Z = ¯ C , Z = Y = 0 . Considering the q -powers in the first condition in (4.9) again, we see that v q ( B B ) α , but v q ( q α C + q α C ) > min( α , α + 1), and we conclude α α . Thus U a,b,d = 1 q α + α ) X B ,C (mod q α ) B ,C (mod q α )(4.9) holds with N = 1 q ∤ B C ,q | C ,q | B X n ,m ,l (mod q α ) n ,m ,l (mod q α ) e (cid:18) am dB q α + bn l B − m d ¯ C B q α (cid:19) × e (cid:18) − n x + m y + l z q α (cid:19) e (cid:18) − n x + m y + l z q α (cid:19) . We sum trivially over n , l , we use (4.13) in combination with (4.11) for the sum over n , l , and wesum over m , m using orthogonality of characters. The latter two sums leave q δ choices for B and C respectively (recall that q ∤ B ). Now the first condition in (4.9) determines B modulo q α which thendetermines C . Altogether we obtain(4.14) | U a,b,d | q α − α +2 δ = q α ,α ) − α − α +2 δ . Now we turn to the estimation of T a,b,d where q ∤ C , so that Z = ¯ C , Y = 0 , and we obtain T a,b,d = 1 q α + α ) X B ,C (mod q α ) B ,C (mod q α )(4.9) holds with N = 1 q ∤ C X n ,m ,l (mod q α ) n ,m ,l (mod q α ) e (cid:18) − n x + m y + l z q α (cid:19) × e (cid:18) − n x + m y + l z q α (cid:19) e (cid:18) am dB − n l ¯ C B q α + bn l B + m d ( Y q α − Z B ) q α (cid:19) . (4.15)By (4.13), the sum over n , l contributes q α + v q ( B ) if v q ( B ) min( v q ( x ) , v q ( z )) and is zero otherwise.Similarly the sum over n , l contributes q α + v q ( B ) if v q ( B ) min( v q ( x ) , v q ( z )) and is zero otherwise(note that by (4.11) we have v q ( B ) α , so that (4.13) is applicable). We conclude that the combinedsum over n , n , l , l contributes(4.16) q α + α +2 min( v q ( x ) ,v q ( x ) ,v q ( z ) ,v q ( z )) . As before the m -sum leaves at most q δ choices for B , and we fix one of them.Let us first assume that q ∤ B , so that(4.17) Y = ¯ B , Z = 0 . Then the m -sum leaves at most q α + δ choices for B (and trivially there are at most q α ) choices for B ). Again we fix one of them. If α α fix a choice for C , otherwise fix a choice for C . Ineither case, the other C -variable is determined by (4.9). We conclude that there are in total at most q δ +min( α + δ,α )+min( α ,α ) q α ,α )+2 δ choices for the quadruples ( B , B , C , C ) satisfying q ∤ B .Let us now assume q | B , so that q ∤ C and Y = 0 , Z = ¯ C . The m -sum leaves at most q δ + v q ( B ) q δ + α choices for C (and trivially there are at most q α choicesfor C ). If α α fix a choice for C , otherwise fix a choice for B . In either case, the other variable isdetermined by (4.9). As above we conclude that there are in total at most q α ,α )+2 δ choices for thequadruples ( B , B , C , C ) satisfying q | B .We conclude from the previous discussion that the sum over m , m together with the sum over B , B , C , C contributes 2 q α + α +2 min( α ,α )+2 δ , and we obtain by (4.16) the total bound(4.18) | T a,b,d | q α ,α ) − α − α +2 min( v q ( x ) ,v q ( x ) ,v q ( z ) ,v q ( z ))+2 δ . This is not quite sufficient to substantiate the claim of Lemma 7, so we proceed to prove an alternativebound for T a,b,d as defined in (4.15). Let us first assume that q ∤ B , so that (4.17) holds. Then by (4.13),the n , l , n , l -sums contribute q α + α , while the m , m -sums leave as above q δ +min( α + δ,α ) choices forthe pair ( B , B ). Fix a choice for C if α α , otherwise fix a choice for C ; in either case the othervariable is determined by (4.9). In total we obtain at most q δ +2 min( α ,α ) choices for the quadruples( B , B , C , C ), so that together with the m , m -sum we obtain a total contribution of(4.19) q α ,α ) − α − α +2 δ for the terms q ∤ B .Let us now consider the terms with q | B , so that q ∤ C and Y = 0 , Z = ¯ C , PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 19 so that (4.15) simplifies to1 q α + α ) X B ,C (mod q α ) B ,C (mod q α )(4.9) holds with N = 1 q ∤ C C ,q | B X n ,m ,l (mod q α ) n ,m ,l (mod q α ) e (cid:18) am dB − n l ¯ C B q α + bn l B − m d ¯ C B q α (cid:19) × e (cid:18) − n x + m y + l z q α (cid:19) e (cid:18) − n x + m y + l z q α (cid:19) . (4.20)In the following we assume without loss of generality 1 y q α and 1 y q α , so that v q ( y ) α and v q ( y ) α . It is convenient to first dispense with the case min( v q ( y ) , v q ( y )) > α (and hence = α ).Here the n , n , l , l -sums contribute by (4.13) and (4.11) at most q α + α +2 min( α ,α ) while there are atmost q δ choices for B and trivially at most q α q v q ( y ) ,v q ( y )) choices for ( B , C ) which determines C . This gives the total bound(4.21) q α ,α ) − α − α +2 min( v q ( y ) ,v q ( y ))+ δ for (4.20) under the present assumption min( v q ( y ) , v q ( y )) > α . From now on we assume(4.22) min( v q ( y ) , v q ( y )) < α . We distinguish two cases.Let us first assume v q ( y ) v q ( y ). By (4.13) and (4.11) the sum over n , l , n , l contributes at most q α + α +2 min( α ,α ) . The m -sum leaves at most q δ choices for B and each of them satisfies v q ( B ) v q ( y ).We fix one of them. Similarly then the sum over m leaves at most q δ + v q ( B ) q δ + v q ( y ) choices for C .If B , C are fixed, then the first condition in (4.9) leaves at most q v q ( B ) q v q ( y ) choices for B modulo q α , and the triple B , B , C determines C . We conclude that there are at most q v q ( y )+2 δ choices forthe quadruple ( B , B , C , C ), and we obtain the total bound(4.23) q α ,α ) − α − α +2 min( v q ( y ) ,v q ( y ))+2 δ . for (4.20) under the present assumption (4.22) and v q ( y ) v q ( y ). This bound dominates (4.21).The other case v q ( y ) > v q ( y ) cannot happen: first we observe that the m , m -sum vanishes unless dB ≡ ¯ ay (mod q α ) and dB ≡ − C y (mod q α ). Together with (4.22) this leads to a contradictionunless v q ( y ) > α . But this is impossible since α > v q ( y ) > v q ( y ). We summarize that (4.23) is anupper bound for (4.20) in all cases, and together with (4.19) we conclude(4.24) | T a,b,d | q α ,α ) − α − α +2 min( v q ( y ) ,v q ( y ))+2 δ . Combining (4.18) and (4.24) with (4.14) completes the proof of Lemma 7.5.
The sixth moment
Setting up the Kuznetsov formula.
We prepare now for the proof of Theorem 1. We recall thesetup that N is a large prime, p is a fixed prime and χ is a primitive non-quadratic character modulo p . Allimplied constants may depend on p . Let π ⊆ L (Γ ( N ) \ H ) be a cuspidal automorphic representation.Its L -function has conductor dividing N [JPSS, Th´eor`eme]. The contribution in the moment estimate ofTheorem 1 of those π with conductor 1 is independent of N , hence O (1), therefore it suffices to consider π of conductor N . Fix a newvector ̟ ∈ π and denote its normalized Fourier coefficients, defined in (2.7)and (2.8), with A ̟ (1 ,
1) = 1. By an approximate functional equation [IK, Theorem 5.2] we have | L (1 / , π × χ ) | = (cid:12)(cid:12)(cid:12)X n a π × χ ( n ) n / V (cid:16) nN / (cid:17) + η X n a π × χ ( n ) n / V (cid:16) nN / (cid:17)(cid:12)(cid:12)(cid:12) where a π × χ ( n ) are the Dirichlet coefficients of L ( s, π × χ ), V is a smooth, bounded, rapidly decayingfunction depending on π and p , and η is a complex number of absolute value 1 depending on π and p . The coefficients a π × χ ( n ) are multiplicative and satisfy a π × χ ( n ) = A ̟ ( n, χ ( n ) for ( n, N p ) = 1, and a π × χ ( n ) ≪ n / ε by known bounds towards the Ramanujan conjecture on GL(3) (although much weakerbounds would suffice for our purpose). Thus we have L ( s, π × χ ) = X n A ̟ ( n, χ ( n ) n s ∞ X ν =0 a π × χ ( p ν ) p νs L N ( s )for a certain Euler factor L N ( s ).We truncate the sums at n N / ε at the cost of a negligible error. Writing V as its inverse Mellintransform, moving the contour to real part ε and pulling the rapidly converging integral outside the absolutevalues, we obtain | L (1 / , π × χ ) | ≪ N ε Z | t | N ε (cid:12)(cid:12)(cid:12) X n N / ε a π × χ ( n ) n / ε + it (cid:12)(cid:12)(cid:12) dt. Coupled with a smooth partition of unity and the Cauchy-Schwarz inequality we have for some compactlysupported weight functions W j (independent of π ) that | L (1 / , π × χ ) | ≪ N ε Z | t | N ε (cid:12)(cid:12)(cid:12) X j N / ε X n a π × χ ( n ) n / it W j (cid:16) n j (cid:17)(cid:12)(cid:12)(cid:12) dt, up to a negligible error. Since N is prime, the coefficients of the Euler factor L N ( s ) are irrelevant (for ε < / a π × χ ( p ν ) trivially, we conclude | L (1 / , π × χ ) | ≪ N ε X ν p ν ( − − ε ) Z | t | N ε (cid:12)(cid:12)(cid:12) X j N / ε X n A ̟ ( n, χ ( n ) n / it W j (cid:18) np ν j (cid:19)(cid:12)(cid:12)(cid:12) dt ≪ N ε X ν p ν/ Z | t | N ε X j N / ε X n ,n A ̟ ( n , A ̟ ( n , χ ( n ) ¯ χ ( n )( n n ) / (cid:18) n n (cid:19) it W j (cid:18) n p ν j (cid:19) W j (cid:18) n p ν j (cid:19) dt. We observe that the n , n -sum is non-negative and that (for ε < /
2) the variables n , n are coprime to N , so that the Fourier coefficients satisfy the unramified Hecke relations, as discussed prior to the statementof Theorem 6 (recall that for ̟ with A ̟ (1 ,
1) = 0 all coefficients coming up in the previous sum vanish).We multiply three such expressions together. Applying H¨older’s inequality to the combined ν -sum and t -integral with exponents 2 / / X ν Z | t | N ε (cid:16) p ν/ (cid:17) / ! / ≪ N ε , we obtain | L (1 / , π × χ ) | ≪ N ε X ν p ν/ Z | t | N ε X j N / ε X n ,m ,l n ,m ,l χ ( n m l ) ¯ χ ( n m l ) (cid:18) n m l n m l (cid:19) it × A ̟ ( n , A ̟ ( n , A ̟ ( m , A ̟ ( m , A ̟ ( l , A ̟ ( l , n n m m l l ) / × W j (cid:18) n p ν j (cid:19) W j (cid:18) n p ν j (cid:19) W j (cid:18) m p ν j (cid:19) W j (cid:18) m p ν j (cid:19) W j (cid:18) l p ν j (cid:19) W j (cid:18) l p ν j (cid:19) dt. PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 21
Finally we multiply this with P i |h F i , ˜ W µ π i| where F i is a collection of functions as in Lemma 1 and sumover π . This gives X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω | L (1 / , π × χ ) | ≪ N ε max i max | t | N ε max M N / ε X π ⊆ L (Γ ( N ) \ H ) |h F i , ˜ W µ π i| × (cid:12)(cid:12)(cid:12) X n ,m ,l A ̟ ( n , A ̟ ( m , A ̟ ( l , χ ( n l ) ¯ χ ( m )( n m l ) / W (cid:16) n M (cid:17) W (cid:16) m M (cid:17) W (cid:18) l M (cid:19)(cid:12)(cid:12)(cid:12) for some smooth compactly supported weight function W . In the interest of readable and compact notationlet us introduce X m ∼ M f ( m ) := X m f ( m ) W (cid:16) mM (cid:17) for some unspecified smooth compactly supported weight function satisfying W ( j ) ≪ ε,j N ε for all j ∈ N . In other words, ∼ has the same meaning as ≍ except that an additional smooth weightfunction is attached to the sum which comes in handy when one applies Poisson summation.We now use the Hecke relation [Go, Theorem 6.4.11] A ̟ ( n, A ̟ ( m, A ̟ ( l,
1) = X d d | nd d | md d | l A ̟ (cid:18) nld d d , md d d (cid:19) . By (2.9) and another application of the Cauchy-Schwarz inequality applied to the d , d , d -sum, we obtain X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω | L (1 / , π × χ ) | ≪ max i max M N / ε N ε M X π ⊆ L (Γ ( N ) \ H ) |h F i , ˜ W µ π i| N ( ̟ ) × X d ,d ,d ( d d d ,p )=1 (cid:12)(cid:12)(cid:12) X n ∼ M/d d m ∼ M/d d l ∼ M/d d A ̟ ( nl, md ) χ ( nl ) ¯ χ ( m ) (cid:12)(cid:12)(cid:12) . By positivity, we can add the rest of the spectrum. For technical reasons it convenient to sum over thespectrum of L (Γ ( p N ) \ H ) which contains the sum in the preceding display as oldforms. We open thesquare and exchange summations. This gives finally our basic inequality X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω | L (1 / , π × χ ) | ≪ max i max M N / ε N ε M X d ,d ,d ( d d d ,p )=1 X n ,n ∼ M/d d m ,m ∼ M/d d l ,l ∼ M/d d ¯ χ ( n l m ) χ ( m n l ) × Z ( p N ) A ̟ ( n l , m d ) A ̟ ( n l , m d ) |h F i , ˜ W µ π i| N ( ̟ ) d̟. (5.1)5.2. Bounding the Kloosterman terms.
The spectral term is now in shape for an application of theKuznetsov formula (Theorem 6), and accordingly we write the right hand side as a sum of four terms ∆ + Σ + Σ + Σ , where∆ = max i max M N / ε N ε M X d ,d ,d ( d d d ,p )=1 X n ,n ∼ M/d d m ,m ∼ M/d d l ,l ∼ M/d d ¯ χ ( n l m ) χ ( m n l ) δ n l = n l m = m k F i k , Σ = max i max M N / ε N ε M X d ,d ,d ( d d d ,p )=1 X n ,n ∼ M/d d m ,m ∼ M/d d l ,l ∼ M/d d ¯ χ ( n l m ) χ ( m n l ) X ǫ = ± X p ND | D d m D = l n D ˜ S ( ǫd m , d m , l n , D , D ) D D ˜ J ǫ ; F ∗ i s l n d m m D D , Σ = max i max M N / ε N ε M X d ,d ,d ( d d d ,p )=1 X n ,n ∼ M/d d m ,m ∼ M/d d l ,l ∼ M/d d ¯ χ ( n l m ) χ ( m n l ) X ǫ = ± X p N | D | D l n D = d m D ˜ S ( ǫl n , l n , d m , D , D ) D D ˜ J ǫ ; F i r l n l n d m D D ! , Σ = max i max M N / ε N ε M X d ,d ,d ( d d d ,p )=1 X n ,n ∼ M/d d m ,m ∼ M/d d l ,l ∼ M/d d ¯ χ ( n l m ) χ ( m n l ) X ǫ ,ǫ = ± X p N | D ,p N | D S ( p N ) ( ǫ d m , ǫ l n , l n , d m , D , D ) D D J ǫ ; F i (cid:18) √ l n d m D D , √ l n d m D D (cid:19) . It is easy to see that | ∆ | + | Σ | + | Σ | ≪ N ε : indeed, by a divisor argument we have∆ ≪ max M N / ε N ε M X d ,d ,d X n ,n ∼ M/d d m ,m ∼ M/d d l ,l ∼ M/d d δ n l = n l m = m ≪ max M N / ε N ε M X d ,d ,d M ( d d d ) ≪ N ε as desired. For Σ we simply observe that the conditions p N | D | D and n l D = m d D implies N | D and N | D since N ∤ n l is prime, so that D D > N , but n l m d n l ≪ M ≪ N / ε ,so that (for sufficiently small ε > N ) we have Σ = 0 by Lemma 3(a) (with X = X = 1). Similarly one shows Σ = 0.The term corresponding to the long Weyl element can be bounded byΣ ≪ max M N / ε N ε M X d ,d ,d ( d ,p )=1 X ǫ ∈{± } (cid:12)(cid:12)(cid:12) X n ,n ∼ M/d d m ,m ∼ M/d d l ,l ∼ M/d d ¯ χ ( n l m ) χ ( m n l ) × X p | D p | D S ( p ) ( ǫ ¯ N m d , ǫ ¯ N n l , n l , m d ; D , D ) D D J ǫ ; F (cid:18) √ n l m d D N / D , √ n l m d D N / D (cid:19)(cid:12)(cid:12)(cid:12) . (5.2) PPLICATIONS OF THE KUZNETSOV FORMULA ON GL(3) II: THE LEVEL ASPECT 23 where J ǫ ; F satisfies the properties of Lemma 3(b). Here we used Lemma 6(b) and (c) and note that thesupport of J given in Lemma 3(b) implies(5.3) D , D ≪ M d N ( d d d ) ≪ N / ε . so that automatically ( N, D D ) = 1. Remark:
We pause for a moment and observe that the contribution of the terms M = N / , d = d = d = 1, D = D ≍ N / without the character χ exhibits no essential cancellation and is of size N / as predicted by the contribution of the maximal Eisenstein series: indeed, the maximal Eisensteinseries are parametrized by GL(2) cusp forms f for Γ ( N ), and a typical Fourier coefficient is given by A ( n,
1) = P d | n λ f ( n ); thus the maximal Eisenstein contribution is very roughly of the form X f (cid:12)(cid:12)(cid:12) X n ≪ N / √ n X d | n λ f ( d ) (cid:12)(cid:12)(cid:12) ≈ X f (cid:12)(cid:12)(cid:12) N / L (1 , f ) (cid:12)(cid:12)(cid:12) ≈ N / . In order to obtain the targeted bound N we will have to use the extra oscillation of the character.We apply Poisson summation in the 6 variables n , n , m , m , l , l . We call the dual variables x , x , y , y , z , z ,respectively. By Lemma 3(b) the function J : n
7→ J ǫ ; F (cid:18) √ n l m d D N / D , √ n l m d D N / D (cid:19) satisfies n i J ( i ) ( n ) ≪ i (cid:18) M d N ( d d d ) D (cid:19) i for all i ∈ N under the present size conditions of the variables. We conclude that the dual variable x canbe bounded by | x | N ε · D M/d d · M d N ( d d d ) D = N ε M N d d ≪ N ε up to a negligible error. By a similar argument, the same bound holds for x , and we also have | y | , | y | N ε M N d d d ≪ N ε , | z | , | z | N ε M N d d ≪ N ε . Now we can apply (3.5) with α = x Md d D , β = y Md d D , γ = z Md d D , α = x Md d D , β = y Md d D , γ = z Md d D unless x x y y z z = 0, in which case we apply (3.4) with P = N ε and A = √ M d D d d d N / D , A = √ M d D d d d N / D . In this way we conclude by trivial estimates thatΣ ≪ max M N / ε N ε M X d ,d ,d ( d ,p )=1 X ǫ ∈{± } X p | D ,D ≪ M d N ( d d d D D × min( d d , d d , d d )( D D ) / M + ( d d d ) N ( D + D ) M d ! × M ( d d d ) X | x | , | x |≪ N ε | y | , | y |≪ N ε | z | , | z |≪ N ε (cid:12)(cid:12)(cid:12) b S χǫ ¯ N,ǫ ¯ N,d ( x , x , y , y , z , z ; D , D ) (cid:12)(cid:12)(cid:12) where b S was defined in (4.5). Notice that the D , D -sum restricts the d -variable to d ≪ N / ε . ByCorollary 10 we can bound the innermost sum by N ε ( D , D ) d ( D D ) − , and obtainΣ ≪ N ε X d ,d ,d d ≪ N / ε d d d X D ,D ≪ N / ε ( D , D ) ( D + D )( D D ) ≪ N ε as desired. This completes the proof of Theorem 1.6. Proofs of Theorems 2 - 5
For the proof of Theorems 2, 3 and 5 we choose functions F , . . . , F J as in Lemma 1, and we applyLemma 3 with X = X = 1.For the proof of Theorem 3 we proceed as follows. The outer sum is over cuspidal automorphicrepresentations that we interpret as a sum over newvectors. We add artificially the oldforms and theEisenstein spectrum and bound the mean value in question by N ε J X j =1 Z ( N ) (cid:12)(cid:12)(cid:12) X n ≍ X ( n,N )=1 A ̟ ( n, α ( n ) (cid:12)(cid:12)(cid:12) |h F j , ˜ W µ π i| N ( ̟ ) − d̟. Here we used also (2.9). We open the square and apply the Kuznetsov formula. The diagonal termcontributes ≪ N ε k α k .The contribution of the long Weyl element is bounded by N ε X n,m ≍ X ( nm,N )=1 | α ( n ) α ( m ) | X N | D ,D D ,D ≪ X | S ( N ) ( ± , ± m, n, D , D ) | D D ≪ ( N X ) ε X N / k α k by (4.2), since the support condition in Lemma 3(b) with X = X = 1 restricts D , D ≪ X .The contribution of the w element is bounded by N ε X n,m ≍ X ( nm,N )=1 | α ( n ) α ( m ) | X N | D | D nD = D D D ≪ X | ˜ S ( ± m, n, D , D ) | D D , again by the support condition in Lemma 3(a). The summation condition implies D = ndN , D = nd N ,which is only possible if N ≪
1, so that we obtain X n,m ≍ X | α ( n ) α ( m ) | X d ≪ | ˜ S ( ± m, n, ndN, nd N ) | n d ≪ X n,m ≍ X | α ( n ) α ( m ) | ≪ X k α k by Lemma 5. This term is dominated by the w contribution. A similar argument works for the w con-tribution. This completes the proof.A similar, but simpler, argument shows the bound in Theorem 2.The proof of Theorem 5 follows along the lines of [BBR, Theorem 2]. If k α π ( p ) k ∞ > δ , then λ π ( p l ) > ( l + 1)( l + 2) for some sufficiently large l = l ( δ ), see [BBR, (24)]. Hence for any k > (cid:8) π ⊆ L (Γ ( N ) \ H ) : µ π ∈ Ω , k α π ( p ) k > δ (cid:9) X π ⊆ L (Γ ( N ) \ H ) µ π ∈ Ω | A ̟ ( p l , | k (( l + 1)( l + 2)) k . (6.1)By [BBR, (14)] we have | A ̟ ( p l , | k = X r + s lk α r,s,l,k A ̟ ( p r , p s ) A ̟ (1 , , X r + s + lk | α r,s,l,k | (cid:18) ( l + 1)( l + 2)2 (cid:19) k so that by Cauchy-Schwarz and Theorem 2 the right hand side of (6.1) is bounded by ≪ ( N p lk ) ε − k ( N + N / p kl ) / N ≪ ( N p lk ) ε − k ( N + N / p kl ) . Choosing k = ⌊ N l log p ⌋ >
1, we obtain (cid:8) π ⊆ L (Γ ( N ) \ H ) : µ π ∈ Ω , k α π ( p ) k > δ (cid:9) ≪ N ε − l log p . Finally we prove Theorem 4. Here we choose sufficiently large parameters X = X = X to be determinedlater and apply the Kuznetsov formula with the function F ( X,X ) as in (3.2). Then by Lemma 2 and (2.9)we have X π ⊆ L (Γ ( N ) \ H ) µ π =( ρ + iγ,ρ − iγ, − iγ ) ∈ Ω | ρ | > ε X |ℜ µ π | ≪ N ε Z ( N ) | A ̟ (1 , | |h F ( X,X ) , ˜ W µ π i| N ( ̟ ) d̟. The diagonal term contributes N ε X . By Lemma 3(b), the long Weyl element contributes( N X ) ε N X X N | D ,D ≪ X | S ( N ) ( ± , ± , , D , D ) | D D . Assuming X N − ε and recalling that N is prime, we have N ∤ D , D . Combining Lemma 6(b), Lemma6(c) and Lemma 5 (with N = 1), we obtain the bound( XN ) ε N X X D ′ ,D ′ ≪ X /N N ( D ′ D ′ ) / ε ( D ′ , D ′ ) / D ′ D ′ N ≪ ( N X ) ε X . By Lemma 3(a) the w element contributes( N X ) ε N X X N | D | D D = D D D ≪ X ˜ S ( ± , , D , D ) D D = 0if X N − ε . Similarly, the w contribution vanishes. Choosing X = N − ε , we obtain the result. Acknowledgement
The authors would like to thank the referee for a careful reading of the manuscript.
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