Bilinear forms with G L 3 Kloosterman sums and the spectral large sieve
aa r X i v : . [ m a t h . N T ] S e p BILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THESPECTRAL LARGE SIEVE
MATTHEW P. YOUNG
Abstract.
We analyze certain bilinear forms involving GL Kloosterman sums. As anapplication, we obtain an improved estimate for the GL spectral large sieve inequality. Introduction
Given a family of L -functions, { L ( s, f ) : f ∈ F } , one of the most basic questions onecan study is its orthogonality properties. More precisely, if L ( s, f ) = P ∞ n =1 λ f ( n ) n − s , thenone wishes to understand ∆ F ( m, n ) := P f ∈F λ f ( m ) λ f ( n ). For instance, when the familyconsists of Dirichlet characters, a formula for ∆ F is given by orthogonality of characters.For families of GL forms, ∆ F can be expanded into a sum of Kloosterman sums, by thePetersson/Bruggeman-Kuznetsov trace formula, which has seen extensive applications innumber theory.A large sieve inequality takes this analysis even futher, by bounding(1.1) X f ∈F (cid:12)(cid:12)(cid:12) X n ≤ N a n λ f ( n ) (cid:12)(cid:12)(cid:12) , where a n are arbitrary complex coefficients. By general principles, the best one may hopefor is a bound of the form ( |F | + N ) P n ≤ N | a n | . One can view this as a much more robustform of orthogonality, probing the sequence of values of λ f ( n ) by correlations with arbitrarysequences a n . Large sieve inequalities are flexible and powerful estimates for bilinear formshaving many applications. For instance, the classical large sieve inequality for Dirichletcharacters plays a key role in proving the Bombieri-Vinogradov theorem. The GL spectrallarge sieve has been valuable in understanding mean values of L -functions (in particular, tothe fourth moment of the zeta function, which was Iwaniec’s original application [I]). Thereader is referred to [IK, Chapter 7] for a good introduction to large sieve inequalities.The corresponding studies of higher rank families are still in their infancy. Bump, Fried-berg, and Goldfeld [BFG] developed many of the foundational properties of the GL Poincareseries, and in particular discovered the analogous sums to the GL Kloosterman sums. Re-cently, Blomer [Bl] succeeded in formulating a GL Bruggeman-Kuznetsov formula withsmooth bump functions appearing on the spectral side. Blomer also derived a form of thespectral GL large sieve inequality, but without a focus on obtaining a sharp result. In prin-ciple, one may also derive a large sieve inequality from Goldfeld-Kontorovich’s work [GK],but again this was not the focus of the authors and the result would not be numericallystrong. This material is based upon work supported by the National Science Foundation under agreement No.DMS-1401008. Any opinions, findings and conclusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the views of the National Science Foundation.
One of our main goals here is to obtain a stronger form of the GL spectral large sieveinequality. To state the results, we set up some of the necessary notation as in [BFG] [G] [Bl].Consider the family of Hecke-Maass cusp forms φ j for SL ( Z ) \H , with spectral parameters ν , ν . The Langlands parameters associated to φ j are α = 2 ν + ν , α = − ν + ν ,and α = − ν − ν . Blomer has shown that the number of φ j with ν = iT + O (1), ν = iT + O (1), weighted by R − j , where(1.2) R j = Res s =1 L ( s, φ j × φ j ) , is ≍ T T ( T + T ) (also see [Bl, (1.4)]). This is a natural weighting from the point of view ofthe Bruggeman-Kuznetsov formula. Let λ j ( m, n ) denote the Hecke eigenvalues of φ j , with λ j (1 ,
1) = 1. With an appropriate choice of scaling of Whittaker functions, then k φ j k ≍ R j (e.g., see [Bl, Lemma 1]). Theorem 1.1.
For an arbitrary complex sequence a n , we have (1.3) X ν = iT + O (1) ν = iT + O (1) R j (cid:12)(cid:12)(cid:12) X n ≤ N a n λ j ( n, (cid:12)(cid:12)(cid:12) ≪ (cid:16) T T ( T + T ) + T T N / (cid:17) ε X n ≤ N | a n | . For comparison, Blomer’s proof of the spectral large sieve (implicitly) shows(1.4) X ν = iT + O (1) ν = iT + O (1) R j (cid:12)(cid:12)(cid:12) X n ≤ N a n λ j ( n, (cid:12)(cid:12)(cid:12) ≪ (cid:16) T T ( T + T ) + T T N (cid:17) ε X n ≤ N | a n | , so Theorem 1.2 saves a potentially rather large factor N / . In fact, Blomer shows a dyadicbound:(1.5) X T ≤| ν |≤ T T ≤| ν |≤ T R j (cid:12)(cid:12)(cid:12) X n ≤ N a n λ j ( n, (cid:12)(cid:12)(cid:12) ≪ (cid:16) T T ( T + T ) + T T N (cid:17) ε X n ≤ N | a n | , which saves a factor T T in the second, “off-diagonal,” term compared to (1.4), via anoscillatory integral. The proof of Theorem 1.1 also uses an oscillatory integral for an extrasavings, but it is a technical challenge to combine these two sources of savings and convertTheorem 1.1 into a dyadic version with a secondary term of the same size. It should be notedthat Blomer’s estimate arises by applying absolute values to the GL Kloosterman sum, andestimating everything trivially (analogously to applying the Weil bound for Kloostermansums). One can view the quality of a large sieve inequality for a family F as a measure ofhow well one may average with the family. As such, it is desirable to have strong results.There are also large sieve-type results in higher rank due to Duke and Kowalski [DK],Venkatesh [V], and Blomer-Buttcane-Maga [BBM], but these study the conductor (or level)aspect. By adapting the method of [DK, Theorem 4], one could use duality and the convexitybound for Rankin-Selberg L -functions on GL × GL to attempt to obtain estimates on theleft hand side of (1.3). However, this method requires N to be very large compared to T + T to give a strong bound.The GL Bruggeman-Kuznetsov formula relates these spectral sums to a sum of GL Kloosterman sums. The main technical contribution of this paper is to analyze multilinearforms with these Kloosterman sums. We will be using the Bruggeman-Kuznetsov formulain the form of [Bl, Proposition 4] . The geometric side of this formula involves the GL A corrected version of the formula can be found in [BBM, Theorem 6]
ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 3
Kloosterman sums, which we now define. The (long element) Kloosterman sum is(1.6) S ( m , m , n , n ; D , D ) = X X B ,C (mod D ) B ,C (mod D )( B ,C ,D )=( B ,C ,D )=1 D C + B B + C D ≡ D D ) e (cid:16) m B + n ( Y D − Z B ) D (cid:17) e (cid:16) m B + n ( Y D − Z B ) D (cid:17) , where Y , Y , Z , Z are defined (chosen) so that(1.7) Y B + Z C ≡ D ) , Y B + Z C ≡ D ) . Bump, Friedberg, and Goldfeld [BFG, Lemmas 4.1 and 4.2] have shown that the above sumis well-defined, meaning that the value of the sum is independent of the choices of the Y i and Z i , and the coset representatives of the B i and C i .Define(1.8) S = S ( α, β, γ ) = X D ,D ,m,n γ D ,D α m β n S (1 , m, n, , D , D ) , where α m , β n , γ D ,D are finite sequences. For our application to the spectral large sieve, weare most interested in the case where | γ D ,D | ≤
1. Especially in light of its connections tothe large sieve, it is fundamental to estimate S , but it is also of independent interest. Ourmain result is Theorem 1.2.
Suppose that α m , β n , and γ D ,D are complex sequences supported on m, n ≤ N , D ≤ X , and D ≤ X . Furthermore suppose that | γ D ,D | ≤ . For an arbitrary finitelysupported sequence β = ( β n ) , let (1.9) M ( β ) = X q ≤ min( X ,X ) X d | q d q X c ≤ X q ( c,q )=1 X ∗ t (mod c ) (cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) tnc (cid:17)(cid:12)(cid:12)(cid:12) where Σ ∗ denotes that t is restricted by ( t, c ) = 1 . Then (1.10) |S| ≪ ( X X ) ε M ( α ) / M ( β ) / . For some special choices of coefficients α m , β n (e.g. Dirichlet series coefficients of an L -function), one could potentially use alternative techniques to handle small c , which explainswhy we have stated Theorem 1.2 in this form. For arbitrary coefficients, one cannot dobetter than the large sieve inequality (see [IK, Theorem 7.11]), which implies(1.11) M ( β ) ≪ ( X + N ) X ε k β k , where here and throughout the paper we use the notation (for an arbitrary sequence β offinite support)(1.12) k β k = (cid:16) X n ∈ Z | β n | (cid:17) / . Hence we immediately derive
MATTHEW P. YOUNG
Corollary 1.3.
With the same conditions and notation as Theorem 1.2, we have (1.13) |S| ≪ ( X X ) ε ( X X )( X + N ) / ( X + N ) / k α k k β k . For the applications to the GL spectral large sieve inequality, the formulation in Theorem1.2 is better, because one can obtain additional savings using a hybrid large sieve inequality,which includes an archimedean integral.For some ranges of the parameters, the following result is superior to Theorem 1.2: Theorem 1.4.
Let ≤ H ≤ X , ≤ H ≤ X . Then (1.14) |S| ≪ ( X H + X H )( X X ) ε M ∗ ( α ) / M ∗ ( β ) / + ( X X ) / ε N ε k α kk β k ( H − + H − ) , where M ∗ ( β ) is defined as in (1.9) , but with q restricted by q ≤ min( H , H ) . Remarks. In case H = X , H = X the first term in (1.14) reduces to Theorem 1.2 (andthe second term may be dropped). For the opposite extreme H = H = 1, the latter termcorresponds to the “Weil bound” (see (3.2) below) while the first term may be dropped. Therestrictions 1 ≤ H i ≤ X i may be dropped from the statement of Theorem 1.4, however thenthe result is worse than Theorem 1.2 or (3.2) below.Again, the large sieve implies Corollary 1.5.
Let ≤ H ≤ X , ≤ H ≤ X . Then (1.15) |S| ≪ h ( X H + X H )( X + N ) / ( X + N ) / + ( X X ) / NH + ( X X ) / NH i ( X X N ) ε k α k k β k . Remarks. For N large, say N ≫ X + X , Corollary 1.5 is optimized with H = X / X / , H = X / X / , and reduces to a bound that can be seen to be inferior to Corollary1.3. On the other hand, if N ≪ min( X , X ), then the optimal bound occurs with H = N / X / X − / , H = N / X / X − / , and gives(1.16) |S| ≪ ( X X ) / N / ( X / + X / )( X X N ) ε k α k k β k . Recently, Buttcane [Bu1] [Bu2] has developed Mellin-Barnes integral representations forthe weight functions occuring on the Kloosterman sum side of the Bruggeman-Kuznetsovformula. Blomer and Buttcane [BB] have used this formulation, with additional ideas, to ob-tain a subconvexity result for GL Maass forms in the spectral aspect. It could be interestingto investigate if these alternative integral representations lead to additional savings in thespectral large sieve. Our preliminary calculations indicate this could be rather complicated,and since our main focus here is on the arithmetical aspects of the problem (rather than thearchimedean integrals), we leave this for another occasion.2.
Acknowledgments
I thank Valentin Blomer, Jack Buttcane, and the referee for numerous suggestions andcorrections that improved the quality of the paper.
ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 5 Heuristic remarks
On theorem 1.2.
We include a few remarks of an informal nature indicating thatTheorem 1.2 is in a somewhat robust form, at least, under the assumption that X and X are not highly asymmetrical in size.The Weil-type bound of Steven [S] (see [Bl, Lemma 3]) implies(3.1) X D ≤ X X D ≤ X | S (1 , m, n, , D , D ) | ≪ ( X X ) / ε ( mn ) ε , and therefore the trivial bound applied to S along with Cauchy’s inequality gives(3.2) |S| ≪ ( X X ) / ε N ε k α kk β k . Therefore, for large N , Corollary 1.3 saves an additional factor ( X X ) / over (3.2).In case ( D , D ) = 1, then from [BFG, Property 4.9], we have(3.3) S ( m , m , n , n , D , D ) = S ( D m , n , D ) S ( D m , n , D )so the contribution to S from ( D , D ) = 1, say S ′ , is(3.4) S ′ = X m,n ( D ,D )=1 γ D ,D α m β n S ( D , n, D ) S ( D , m, D ) . It could so happen that γ D ,D always has the same sign as P m,n α m β n S ( D , n, D ) S ( D , m, D ),so it should be essentially impossible to do better than bounding S ′ as follows:(3.5) |S ′ | ≤ X ( D ,D )=1 (cid:12)(cid:12)(cid:12) X n β n S ( D , n, D ) X m α m S ( D , m, D ) (cid:12)(cid:12)(cid:12) . By an application of Cauchy’s inequality, we have(3.6) |S ′ | ≤ (cid:16) X ( D ,D )=1 (cid:12)(cid:12)(cid:12) X n β n S ( D , n, D ) (cid:12)(cid:12)(cid:12) (cid:17) / ( . . . ) / , with the dots representing a similar term. Next we drop the condition ( D , D ) = 1 andextend the sum over D to D ≤ M D where M is the unique integer satisfying X ≤ M D < X + D (this extension is presumably rather wasteful in case X is much smallerthan X ). Then we have(3.7) X D ≤ MD S ( n , D , D ) S ( n , D , D ) = M D X ∗ x (mod D ) e (cid:16) x ( n − n ) D (cid:17) , so the first expression in parentheses on the right hand side of (3.6) satisfies(3.8) ( . . . ) ≤ ( X + X ) X D ≤ X X ∗ x (mod D ) (cid:12)(cid:12)(cid:12) X n β n e (cid:16) xnD (cid:17)(cid:12)(cid:12)(cid:12) . A similar bound holds for the second factor in (3.6), of course. Therefore, by the large sieveinequality, we have(3.9) |S ′ | ≤ ( X + X )( X + N ) / ( X + N ) / k α kk β k . This gives a limitation to the final estimates we wish to obtain for S . One observes that thebound (3.9) is superior to that of Corollary 1.3 by a factor min( X , X ), which arises in theproof from considering D and D with a common factor. MATTHEW P. YOUNG
The opposite extreme of ( D , D ) = 1 is D = D . For simplicity consider D = D = p ,prime. In this case, we have (see [BFG, Property 4.10] or Lemma 4.2 below)(3.10) S (1 , m, n, , p, p ) = S ( m, p ) S ( n, p ) + p. Therefore, if p | ( m, n ) the Kloosterman sum is of order p , while if p ∤ m , p ∤ n , it is of order p .The term p gives the dominant contribution, because in the situation when the Kloostermansum has order p (i.e., p | ( m, n )), the rarity in m and n has relatively frequency p − , whichis a net saving by a factor p . These terms give to S an amount, say S ′′ , given by(3.11) S ′′ = X p ≤ min( X ,X ) ( p + 1) γ p,p X ( m,p )=1 α m X ( n,p )=1 β n . If say X = X = X , then(3.12) S ′′ ≪ X (cid:12)(cid:12)(cid:12) X m α m (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) X n β n (cid:12)(cid:12)(cid:12) , which is best-possible since the sum of ( p + 1) γ p,p may have the same sign as P m α m P n β n .A bound of this magnitude is included with c = q = d = 1 in (1.9) and (1.10). Cauchy’sinequality applied to S ′′ gives(3.13) S ′′ ≪ X N k α kk β k . This matches the bound in Corollary 1.3 for N large and X = X .Of course, in actual practice it is necessary to treat all possible values of gcd( D , D ) that“interpolate” the two extremes ( D , D ) = 1, and D = D , and indeed this is accomplishedin the proof of Theorem 1.2. In fact, this is the main difficulty in the proof.The above remarks indicate that the quality of Theorem 1.2 comes largely from termswhere ( D , D ) is large. This might be surprising in light of the relative rarity of such terms.3.2. The GL spectral large sieve. The spectral large sieve for SL ( Z ) \ H was originallyproved by Iwaniec [I], while the case of congruence subgroups was extensively developed byDeshouillers and Iwaniec [DI]. Here we sketch a proof inspired by Jutila [J, Section 3], sincewe shall use this method as a motivating guide for the more challenging GL case. Recallthat the GL spectral large sieve states(3.14) X T ≤ t j ≤ T +∆ R j (cid:12)(cid:12)(cid:12) X n ≤ N a n λ j ( n ) (cid:12)(cid:12)(cid:12) ≪ ( N T ) ε (∆ T + N ) X n | a n | , where R j is given by (1.2) (but for φ j a Hecke-Maass cusp form on SL ( Z )), and 1 ≤ ∆ ≤ T .The GL Bruggeman-Kuznetsov formula gives(3.15) X T ≤ t j ≤ T +∆ R j (cid:12)(cid:12)(cid:12) X N/ 1. Here b depends on X , but not on x . Applying this formula to K , we derive(3.18) K / X C dyadic ∆ TCX Z | t |≪ X b (2 t ) X c ≍ C c − it X ∗ a (mod c ) (cid:16) X m ≍ N a m m it e (cid:16) amc (cid:17)(cid:17)(cid:16) X n ≍ N a n n it e (cid:16) anc (cid:17)(cid:17) dt. The hybrid large sieve inequality of Gallagher [Ga] states(3.19) Z | t |≤ X X c ≤ C X ∗ a (mod c ) (cid:12)(cid:12)(cid:12) X n ≤ N a n n − it e (cid:16) anc (cid:17)(cid:12)(cid:12)(cid:12) dt ≪ ( C X + N ) X n | a n | . Applying this to K after a use of Cauchy-Schwarz, and using X ≍ NC , and C ≪ N ∆ T ( N T ) ε ,we derive(3.20) K / ∆ T X C dyadic ( CX ) − ( C X + N ) X n | a n | ≪ N ( N T ) ε X n | a n | . The main observation is that the GL hybrid large sieve inequality drives the final esti-mations, and only rather crude information is required on B , namely its truncation and sizeof its Mellin transform. The hybrid aspect of the large sieve is able to recover the loss inseparation of variables in B .For later use, we shall require a different version (though morally equivalent) of the hybridlarge sieve than that given by Gallagher. The following is a special case of [Y, Lemma 6.1]. Lemma 3.1. Let b m be arbitrary complex numbers, and suppose Y ≫ . Then (3.21) Z X b ≤ B X ∗ x (mod b ) (cid:12)(cid:12)(cid:12) X N ≤ m For ease of reference, we collect here some results. First we need an individual “Weil-type”bound. This estimate was proved by Stevens [S] but without explicit dependence on the m i and n i , which was subsequently investigated by Buttcane [Bu1, Theorem 4]. Lemma 4.1. For m , m , n , n ∈ Z \ { } , we have (4.1) S ( m , m , n , n , D , D ) ≪ ( D D ) / ε (( D , D )( m n , [ D , D ])( m n , [ D , D ])) / . This estimate is not sharp for ( D , D ) > 1, but it is difficult to extract clean results fromthe literature (see [DF, Theorem 3.7]). We may obtain some easy improvements by way ofexplicit computations in some important special cases: Lemma 4.2 ([BFG]) . Suppose l ≥ . Then (4.2) S ( m , m , n , n , p, p l ) = S ( n , p ) S ( m , n p, p l ) + S ( m , p ) S ( n , m p ; p l ) + δ l =1 ( p − . Lemma 4.3. Suppose b ≥ , c ≥ , and ( αβ, p ) = 1 . Then (4.3) S ( α, βp b , p c ) = 0 . MATTHEW P. YOUNG Proof. If b ≥ c , then S ( α, βp b , p c ) = S (1 , , p c ) = 0 since c ≥ 2, so suppose c ≥ b +1. Openingthe Kloosterman sum as a sum over x (mod p c ), we change variables x = x (1 + p c − b x ),where x runs modulo p c − b (coprime to p ) and x runs modulo p b . Then x ≡ x (mod p c − b ),and so(4.4) S ( α, βp b , p c ) = X ∗ x (mod p c − b ) X x (mod p b ) e (cid:16) αx (1 + p c − b x ) + βp b x p c (cid:17) . The sum over x then vanishes since b ≥ αx , p ) = 1. (cid:3) Consider S ( a, y, x, b, D , D ) with ( a, D ) = ( b, D ) = 1. Then define its (partial, middletwo-variable) Fourier transform by(4.5) b S ( a, u, t, b, D , D ) = 1 D D X x (mod D ) X y (mod D ) S ( a, y, x, b, D , D ) e (cid:16) − xtD (cid:17) e (cid:16) − yuD (cid:17) , so that the Fourier inversion formula reads(4.6) S ( a, m, n, b, D , D ) = X t (mod D ) X u (mod D ) e (cid:16) tnD + umD (cid:17) b S ( a, u, t, b, D , D ) . Define(4.7) R ( t, D , D ) = max ( ab,D )=1 X u (mod D ) | b S ( a, u, bt, , D , D ) | . Remark. Using elementary properties of the Kloosterman sums, we may alternatively usethe definition(4.8) R ( t, D , D ) = max ( b,D )=1 X u (mod D ) | b S (1 , u, bt, , D , D ) | . This follows by using that S ( a, y, x, , D , D ) = S (1 , y, ax, , D , D ) (see [BFG, Property4.3]), so that after a change of variables we derive(4.9) b S ( a, u, bt, , D , D ) = b S (1 , u, abt, , D , D ) . The presence of the maximum in (4.7) is to facilitate the use of the Chinese RemainderTheorem which leads to a more pleasant multiplicative structure for R : Lemma 4.4. The function R ( t, D , D ) is jointly multiplicative in t, D , D .Proof. Say D = C E and D = C E with ( C C , E E ) = 1. Also write x = x C E E + x E C C , and similarly y = y C E E + y E C C , where x C , y C , x E , y E run modulo C , C , E , E ,respectively. Then using [BFG, Property 4.15], we have(4.10) b S ( a, u, t, , D , D ) = X x C ,y C S ( E E a, E E y C , x C , , C , C ) e (cid:16) − x C E tC (cid:17) e (cid:16) − y C E uC (cid:17) D D X x E ,y E S ( C C a, C C y E , x E , , E , E ) e (cid:16) − x E C tE (cid:17) e (cid:16) − y E C uE (cid:17) . ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 9 Changing variables y C → E E y C , y E → C C y E , we derive(4.11) b S ( a, u, t, , D , D ) = 1 C C X x C ,y C S ( E E a, y C , x C , , C , C ) e (cid:16) − x C E tC (cid:17) e (cid:16) − y C E E uC (cid:17) E E X x E ,y E S ( C C a, y E , x E , , E , E ) e (cid:16) − x E C tE (cid:17) e (cid:16) − y E C C uE (cid:17) . Therefore,(4.12) b S ( a, u, t, , C E , C E ) = b S ( E E a, uE E , tE , , C , C ) b S ( C C a, uC C , tC , , E , E ) . Using (4.12), we derive that(4.13) R ( t, C E , C E ) = max ( ab,C E )=1 X u (mod C E ) | b S ( E E a, uE E , btE , , C , C ) || b S ( C C a, uC C , btC , , E , E ) | . In the right hand side of (4.13), the first line only depends on u modulo C , and a, b modulo C , while the second line only depends on u modulo E , and a, b modulo E . Therefore, wehave R ( t, C E , C E ) = max ( a C b C ,C )=1 X u C (mod C ) | b S ( E E a C , u C E E , b C tE , , C , C ) | max ( a E b E ,E )=1 X u E (mod E ) | b S ( C C a E , u E C C , b E tC , , E , E ) | . (4.14)Changing variables a C → E E a C , u C → E E u C , b C → E b C , and similarly for a E , u E , and b E , we derive R ( t, C E , C E ) = R ( t, C , C ) R ( t, E , E ), as desired. (cid:3) Lemma 4.5. Let (4.15) R ′ ( u, D , D ) = max ( a,D )=1( b,D )=1 X t (mod D ) | b S ( a, bu, t, , D , D ) | . Then (4.16) R ′ ( u, D , D ) = R ( u, D , D ) . Proof. Using S ( a, y, x, , D , D ) = S ( x, , a, y, D , D ) = S (1 , x, y, a, D , D ) (see Properties4.5 and 4.4 of [BFG]), along with S (1 , x, y, a, D , D ) = S (1 , ax, y, , D , D ) ([BFG, Property4.3]), we derive that(4.17) b S ( a, bu, t, , D , D ) = 1 D D X x (mod D ) X y (mod D ) S (1 , x, y, , D , D ) e (cid:16) − xatD (cid:17) e (cid:16) − ybuD (cid:17) = b S (1 , at, bu, , D , D ) . From this, and using (4.8), we complete the proof. (cid:3) Lemma 4.6. We have (4.18) b S ( a, u, t, b, D , D ) = b S ( b, t, u, a, D , D ) . Proof. A minor variation of the proof of Lemma 4.5 gives the result. (cid:3) Definition 4.7 (Definition of ν ) . Suppose p is a prime. If n ∈ Z , we define ν p ( n ) to be thestandard p -adic valuation of n . If k ≥ and t ∈ Z /p k Z we define ν p ( t ) to be the largest j ≤ k such that t ≡ p j ) . Remark. One may easily check that ν p ( t ) is well-defined for t ∈ Z /p k Z ; without therestriction j ≤ k , two coset representatives may have different p -adic valuations. Lemma 4.8. Suppose ( ab, p ) = 1 , and set ν = ν p ( t ) . Then (4.19) R ( t, p k , p l ) ≤ ( k + 1) p l + p ν + l δ ( ν ≤ min( k, l )) . Remark. For k = l , our proof shows that we can replace p ν + l by p ν + l , and restrict ν ≤ min( k, l ). It is plausible one can save this factor p ν/ for k = l , but since this wouldnot improve Theorem 1.2, and since our proof is already quite long, we avoided this line ofinquiry. The key point in Lemma 4.8 is that the “loss” from the factor p ν is countered bythe condition p ν | t . This has the practical effect that large values of ν give essentially thesame bound as for ν = 0.The proof of Lemma 4.8 is given in Section 6. Corollary 4.9. We have (4.20) R ( t, D , D ) ≪ D ( D D ) ε X d | td | ( D ,D ) d. Proof. Since both sides are multiplicative, it suffices to check on prime powers, in which caseit follows immediately from Lemma 4.8. (cid:3) Lemma 4.10. Let q ≤ X . The number of integers n ≤ X that share the same set of primedivisors as q (that is, such that ν p ( n ) ≥ iff ν p ( q ) ≥ for all primes p ) is ≪ ε X ε , for any ε > .Proof. This is similar to a divisor-type bound. Suppose that the prime factors occuring in q are p , . . . , p r . Then by Rankin’s trick, we have(4.21) X n = p a ...p arr ≤ Xa i ≥ , all i ≤ ∞ X a =1 · · · ∞ X a r =1 (cid:16) Xp a . . . p a r r (cid:17) ε = X ε ( p ε − . . . ( p εr − . Given ε > 0, there are finitely many primes such that p ε ≤ 2. Then with C ( ε ) = Q p : p ε ≤ ( p ε − − , we may bound the right hand side of (4.21) by C ( ε ) X ε . (cid:3) Proof of Theorems 1.2 and 1.4 Initial decomposition. Our first steps involve factoring D and D in appropriateways and using the Chinese remainder theorem to correspondingly factor the Kloostermansum.First we extract the largest divisors of D and D that are coprime to each other. Precisely,write D = g E , D = g E , where ( E E , g g ) = 1, ( E , E ) = 1, and g and g have the ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 11 same set of prime divisors (meaning, ν p ( g ) ≥ ν p ( g ) ≥ S (1 , m, n, , g E , g E ) = S ( g g , g g m, n, , E , E ) S ( E E , E E m, n, , g , g ) . By (3.3), we have(5.2) S ( g g , g g m, n, , E , E ) = S ( E g g , n, E ) S ( E g g , m, E ) . Therefore,(5.3) |S| ≤ X ′ g ,g ,E ,E (cid:12)(cid:12)(cid:12) X m,n α m β n S ( E g g , n, E ) S ( E g g , m, E ) S ( E E , E E m, n, , g , g ) (cid:12)(cid:12)(cid:12) , where the prime represents the conditions:(5.4) g E ≤ X , g E ≤ X , ( E E , g g ) = 1 , ( E , E ) = 1 , ν p ( g ) ≥ ν p ( g ) ≥ . We factor the moduli further by extracting the prime factors of g and g such that ν p ( g ) = ν p ( g ) = 1. Precisely, write g = qh , g = qh where q is the product of primessuch that ν p ( g ) = ν p ( g ) = 1, so that for all p | h h , ν p ( h ) ≥ ν p ( h ) ≥ 2, and( q, h h ) = 1. Then we have(5.5) S ( E E , E E m, n, , qh , qh )= S (( h E ) h E , ( h E ) h E m, n, , q, q ) S ( qE E , qE E m, n, , h , h ) . By (3.10), and using ( ab, p ) = 1, we have(5.6) S ( a, bm, n, , p, p ) = S ( m, , p ) S ( n, , p ) + p = p − p + 1 , p | ( m, n ) ,p + 1 , p ∤ m, p ∤ n, , p | m, p ∤ n , p | n, p ∤ m, and so by the Chinese remainder theorem, if q is squarefree and ( ab, q ) = 1, then(5.7) S ( a, bm, n, , q, q ) = Y p | qp ∤ m,p ∤ n ( p + 1) Y p | ( m,n,q ) ( p − p + 1) . Set d = ( n, q ) and d = ( m, q ), and define(5.8) A ( d , d , q ) = Y p | d ,d ( p − p + 1) Y p | q,p ∤ d ,p ∤ d ( p + 1) . Then the above calculations show(5.9) S (( h E ) h E , ( h E ) h E m, n, , q, q ) = A ( d , d , q ) . One easily checks(5.10) A ( d , d , q ) ≪ q ε ( d , d ) d d . Summarizing the above discussion, we have shown(5.11) |S| ≤ X ′ h ,h ,q,E ,E X d ,d | q A ( d , d , q ) (cid:12)(cid:12)(cid:12) X ( n,q )= d X ( m,q )= d α m β n S ( qE E , qE E m, n, , h , h ) S ( E qh h , n, E ) S ( E qh h , m, E ) (cid:12)(cid:12)(cid:12) , where the prime on the sum is updated to represent the conditions: qh E ≤ X , qh E ≤ X , ( E E , qh h ) = 1 , ( E , E ) = 1 , ν p ( q ) ∈ { , } , ( q, h h ) = 1 , ν p ( h ) = 0 iff ν p ( h ) = 0 , p | h h ⇒ ν p ( h ) ≥ ν p ( h ) ≥ . (5.12)Remark. Heuristically, the sum over h and h is somewhat small since both integers sharethe same prime divisors, and for each prime p | h h , p divides at least one of h , h . If welet S ′′′ denote the terms on the right hand side of (5.11) with h = h = 1, then followingthe arguments of Section 3.1, one can derive(5.13) S ′′′ ≪ ( X + X ) ε ( X + min( X , X ) N ) / ( X + min( X , X ) N ) / k α kk β k . This is better than our final bound on S given by Corollary 1.3 for large X , X , so perhapsa more careful analysis of h and h could lead to a modest improvement.If either h or h is large, then it can be beneficial to estimate the sum with absolutevalues, exploiting the reduced number of moduli under consideration. Define S qh i ≤ H i to bethe sum on the right hand side of (5.11) with qh ≤ H and qh ≤ H , and similarly define S qh >H and S qh >H corresponding to the terms with qh > H and qh > H , respectively.Then we have the decomposition |S| ≤ S qh >H + S qh >H + S qh i ≤ H i . In the proof of Theorem1.2, we may set H = X , H = X , and then S qh i >H i = 0, for i = 1 , 2, so these terms maybe discarded.5.2. Large h i . In this subsection we estimate S qh >H and S qh >H . Lemma 5.1. We have (5.14) S qh >H ≪ H − ( X X ) / ε N k α kk β k , and (5.15) S qh >H ≪ H − ( X X ) / ε N k α kk β k . Proof. Define(5.16) T ( m, n, D , D ) = max ( a,D )=1( b,D )=1 | S ( a, bm, n, , D , D ) | . By the Weil bound, we have(5.17) S qh >H ≤ X ′ h ,E ,E X ′ qh >H X d ,d | q A ( d , d , q ) X ( n,q )= d X ( m,q )= d | α m β n | τ ( E ) τ ( E )( E E ) / T ( m, n, h , h ) . ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 13 Trivially summing over E and E and using (5.10), we obtain(5.18) S qh >H ≪ ( X X ) / ε X q ≤ min( X ,X ) X d ,d | q ( d , d ) q X ′ h >q − H X ′ h h h ) / X ( n,q )= d X ( m,q )= d | α m β n | T ( m, n, h , h ) . Write the prime factorizations of h and h as follows: h = j k l , j = p . . . p r , k = q b . . . q b s s , l = ρ c . . . ρ c t t h = j k l , j = p a . . . p a r r , k = q . . . q s , l = ρ γ . . . ρ γ t t , (5.19)where a i , b i , c i , γ i ≥ 2, for all i , ( p i , q j ρ k ) = ( q j , ρ k ) = 1 for all i, j, k , and all p i , q j , ρ k areprime.We first estimate T ( m, n, j , j ). Suppose l ≥ 2. By Lemmas 4.2 and 4.3, we have(5.20) | S ( a, bm, n, , p, p l ) | = | S ( n, p ) S ( m, p ; p l ) | . If p ∤ m then this vanishes, while if p | m then we have(5.21) | S ( a, bm, n, , p, p l ) | = p | S ( n, p ) S ( mp , p l − ) | ≤ p ( n, p ) δ ( p | m ) lp l − . Therefore,(5.22) T ( m, n, j , j ) ≪ ( j j ) / ε ( n, j ) δ ( j | m ) , and by symmetry,(5.23) T ( m, n, k , k ) ≪ ( k k ) / ε ( m, k ) δ ( k | n ) . Finally, by Lemma 4.1, we have(5.24) T ( m, n, l , l ) ≤ ( l l ) / ε (( l , l )( mn, [ l , l ])) / . Therefore, we have(5.25) S qh >H ≪ ( X X ) / ε X q ≤ min( X ,X ) X d ,d | q ( d , d ) q X ′ h >q − H h X ′ h ( l , l ) / h X n ≡ k )( n,q )= d X m ≡ j )( m,q )= d | α m β n | ( mn, [ l , l ]) / ( n, j )( m, k ) . We claim(5.26) X n ≡ k )( n,q )= d X m ≡ j )( m,q )= d | α m β n | ( mn, [ l , l ]) / ( n, j )( m, k ) ≪ N ( X X ) ε ( d d ) / k α kk β k . Toward this, we first observe the simple bound(5.27) X n ≤ N | α n | ( n, q ) / ≤ τ ( q ) N / k α k . Using the trivial inequalities ( mn, [ l , l ]) ≤ ( m, l l )( n, l l ), ( n, j ) ≤ ( n, j ) / j / , ( m, k ) ≤ ( m, k ) / k / , and the fact that ( j , k ) = 1, we derive the claim. Inserting (5.26) into (5.25), we conclude(5.28) S qh >H ≪ ( X X ) / ε N k α kk β k X q ≤ min( X ,X ) X d ,d | q ( d , d )( d d ) / q X ′ h >q − H h X ′ h ( l , l ) / h , and to complete the proof of Lemma 5.1 it now suffices to show(5.29) X ′ h >H h X ′ h ( l , l ) / h ≪ H − ( X X ) ε , where H > 0, using the easy estimate ( d , d ) ≤ ( d d ) / , and trivially summing over q (itis essentially a harmonic series).We now prove (5.29). First we examine the inner sum over h . Writing the expression interms of the prime factorizations (5.19), we have(5.30) X ′ h ( l , l ) / h = X a ,...,a r ,γ ,...,γ t ≥ ρ min( c ,γ . . . ρ min( ct,γt )2 t p a . . . p a r r q . . . q s ρ γ . . . ρ γ t t . The reader may recall that once h is fixed, the prime divisors of h are already determined,which explains why the sum is only over the exponents a i , γ i . It is easily noted that(5.31) X a ,...,a r ≥ p a . . . p a r r ≤ r ( p . . . p r ) , X γ ,...,γ t ≥ ρ min( c ,γ . . . ρ min( ct,γt )2 t ρ γ . . . ρ γ t t ≪ t ρ . . . ρ t , with an absolute implied constant, and so,(5.32) 1 h X ′ h ( l , l ) / h ≪ r + t ( p . . . p r ) q b +11 . . . q b s +1 s ρ c +11 . . . ρ c t +1 t . Inserting this into the left hand side of (5.29), and recalling the implicit condition h ≤ X ,we have(5.33) X ′ h >H h X ′ h ( l , l ) / h ≪ X p ...p r q b ...q bss ρ c ...ρ ctt >H r + t ( p . . . p r ) q b +11 . . . q b s +1 s ρ c +11 . . . ρ c t +1 t ≤ H X p ...p r q b ...q bss ρ c ...ρ ctt >H r + t ( p . . . p r ) q . . . q s ρ . . . ρ t ≪ H − X ε . This shows (5.29), and concludes the proof of (5.14). The other estimate (5.15) follows from(5.14) by symmetry. (cid:3) Small h i . The main goal of this subsection is Lemma 5.2. We have (5.34) S qh i ≤ H i ≪ ( X H + X H )( X X N ) ε M ∗ ( α ) / M ∗ ( β ) / . Choosing H = X , H = X , we have M ∗ ( β ) = M ( β ) and M ∗ ( α ) = M ( α ), and S qh >X = S qh >X = 0, and we obtain Theorem 1.2. More generally, combining Lemmas 5.1 and 5.2proves Theorem 1.4. ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 15 Proof. We begin by inserting the formula(5.35) S ( qE E , qE E m, n, , h , h )= X t (mod h ) X u (mod h ) b S ( qE E , qE E u, t, , h , h ) e (cid:16) tnh (cid:17) e (cid:16) umh (cid:17) , into (5.11), getting(5.36) S qh i ≤ H i ≤ X ′ h ,h ,q,E ,E X d ,d | q A ( d , d , q ) X t (mod h ) X u (mod h ) | b S ( qE E , qE E u, t, , h , h ) | (cid:12)(cid:12)(cid:12) X ( n,q )= d β n S ( E qh h , n, E ) e (cid:16) tnh (cid:17) X ( m,q )= d α m S ( E qh h , m, E ) e (cid:16) umh (cid:17)(cid:12)(cid:12)(cid:12) . We shall occasionally leave the conditions qh ≤ H , qh ≤ H implicit in the notation. ByCauchy’s inequality, we write S qh i ≤ H i ≤ S / S / where(5.37) S = X ′ h ,h ,q,E ,E X d ,d | q A ( d , d , q ) X t (mod h ) X u (mod h ) | b S ( qE E , qE E u, t, , h , h ) | (cid:12)(cid:12)(cid:12) X ( n,q )= d β n S ( E qh h , n, E ) e (cid:16) tnh (cid:17)(cid:12)(cid:12)(cid:12) , and S is given by a similar formula. Write this as(5.38) S = X ′ q,E X d ,d | q A ( d , d , q ) S ′ , where(5.39) S ′ = X ′ h ,h ,E X t (mod h ) X u (mod h ) | b S ( qE E , qE E u, t, , h , h ) | (cid:12)(cid:12)(cid:12) X ( n,q )= d β n S ( E qh h , n, E ) e (cid:16) tnh (cid:17)(cid:12)(cid:12)(cid:12) . Recalling the definition of R from (4.7), we have(5.40) S ′ ≤ X ′ h ,h ,E X t (mod h ) R ( t, h , h ) (cid:12)(cid:12)(cid:12) X ( n,q )= d β n S ( E qh h , n, E ) e (cid:16) tnh (cid:17)(cid:12)(cid:12)(cid:12) . Using the trick described surrounding (3.7), we extend the sum over E to a complete summodulo E ≤ X qh (forgetting the various coprimality conditions on E by positivity), giving(5.41) S ′ ≤ X ∗ x (mod E ) X ′ h ,h X t (mod h ) R ( t, h , h ) (cid:16) X qh + X qh (cid:17)(cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) xnE (cid:17) e (cid:16) tnh (cid:17)(cid:12)(cid:12)(cid:12) . Next write g = ( t, h ), and change variables h = g h ′ , t = g t ′ , so that ( t ′ , h ′ ) = 1. Thisgives(5.42) S ′ ≤ X ∗ x (mod E ) X ′ g ,h ′ ,h X ∗ t ′ (mod h ′ ) R ( t ′ g , g h ′ , h ) (cid:16) X qg h ′ + X qh (cid:17)(cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) xnE (cid:17) e (cid:16) t ′ nh ′ (cid:17)(cid:12)(cid:12)(cid:12) . Here the primes on the sums refer to the conditions (5.12) with h replaced by g h ′ . Observe R ( t ′ g , h ′ g , h ) = R ( g , h ′ g , h ), by definition, and re-arrange this in the form(5.43) S ′ ≤ X ∗ x (mod E ) X ′ h ′ ≤ H q X ∗ t ′ (mod h ′ ) (cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) xnE (cid:17) e (cid:16) t ′ nh ′ (cid:17)(cid:12)(cid:12)(cid:12) X ′ g ≤ H qh ′ X ′ h ≤ H q R ( g , g h ′ , h ) (cid:16) X qg h ′ + X qh (cid:17) . We claim(5.44) X ′ g ≤ H qh ′ X ′ h ≤ H q R ( g , g h ′ , h ) (cid:16) X qg h ′ + X qh (cid:17) ≪ ( X X ) ε q h ′ ( X H + X H ) . Proof of claim. We first bound the sum with the factor X qg h ′ . By Corollary 4.9, we have(5.45) X ′ g ≤ H qh ′ X ′ h ≤ H q R ( g , g h ′ , h ) X qg h ′ ≪ ( X X ) ε X ′ g ≤ H qh ′ X ′ h ≤ H q h X qg h ′ X d | g d | ( g h ′ ,h ) d. Reversing the order of summation, and estimating the sum over h by Lemma 4.10 (one maysafely drop the condition d | h when summing over h ), this is(5.46) ≪ ( X X ) ε X H q h ′ X d ≤ X d X g ≤ H h ′ d | g , d | ( g h ′ ) g . Next we write g = dr , where now d | r h ′ , so we may execute the sum over d first as adivisor sum, and finally the sum over r satisfies P r ≤ X r − ε ≪ X ε . This immediately givesa bound consistent with (5.44).For the second sum with X qh , we have by Corollary 4.9 that(5.47) X ′ g ≤ H qh ′ X ′ h ≤ H q R ( g , g h ′ , h ) X qh ≪ ( X X ) ε X ′ g ≤ H qh ′ X ′ h ≤ H q X q X d | g d | ( g h ′ ,h ) d. We shall reverse the order of summation and execute the sum over h first. The sum over h is bounded by O ( X ε ) using Lemma 4.10, because one of the summation conditions is that h and g h ′ share the same prime factors. Then the right hand side of (5.47) is(5.48) ≪ ( X X ) ε X q X g ≤ H qh ′ X d | g d | ( g h ′ ) d. ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 17 Write g = dr , whence this is(5.49) ≪ ( X X ) ε X q X d ≤ X d X r ≤ H dqh ′ r h ′ ≡ d ) . Suppose that d = d ′ f where d ′ consists of the prime powers corresponding to the primesthat divide h ′ , so that f := d/d ′ is then coprime to h ′ . Then we have r ≡ f ). Let f ∗ be the integer such that the congruence r ≡ f ) is equivalent to r ≡ f ∗ ).Then the above expression is bounded by(5.50) ≪ ( X X ) ε X H q h ′ X d ′ X f f ∗ ≪ ( X X ) ε X H q h ′ , since Lemma 4.10 shows the sum over d ′ is ≪ ( X X ) ε , and the sum over f is ≪ ( X X ) ε by elementary reasoning (e.g. Rankin’s trick). Thus we arrive at a bound consistent with(5.44). (cid:3) The claim (5.44) applied to (5.43) implies(5.51) S ′ ≪ ( X X ) ε q ( X H + X H ) X ∗ x (mod E ) X ′ h ′ ≤ H h ′ X ∗ t ′ (mod h ′ ) (cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) xnE (cid:17) e (cid:16) t ′ nh ′ (cid:17)(cid:12)(cid:12)(cid:12) . Inserting this into S , and using the Chinese remainder theorem to combine the sums modulo E and h ′ to the single modulus E h ′ , we derive(5.52) S ≪ ( X H + X H )( X X ) ε X q ≤ min( H ,H ) X d ,d | q A ( d , d , q ) q X E h ′ ≤ X q h ′ ≤ H h ′ X ∗ x (mod E h ′ ) (cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) xnE h ′ (cid:17)(cid:12)(cid:12)(cid:12) . We group together E h ′ into a single variable c , use h ′− ≤ S ≪ ( X H + X H )( X X ) ε X q ≤ min( H ,H ) X d ,d | q ( d , d ) qd d X c ≤ X q ( c,q )=1 X ∗ x (mod c ) (cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) xnc (cid:17)(cid:12)(cid:12)(cid:12) . Using the crude bound ( d d ) − ( d , d ) ≤ d , and trivially summing over d , we obtain(5.54) S ≪ ( X H + X H )( X X ) ε X q ≤ min( H ,H ) X d | q d q X c ≤ X q ( c,q )=1 X ∗ x (mod c ) (cid:12)(cid:12)(cid:12) X ( n,q )= d β n e (cid:16) xnc (cid:17)(cid:12)(cid:12)(cid:12) , which is ( X H + X H )( X X ) ε M ∗ ( β ) where recall M ( β ) was defined by (1.9) and M ∗ ( β )has the same definition but with q ≤ min( H , H ).We also need to estimate S . It is given by a similar formula to S , except with h and h switched, E and E switched, β n replaced by α m , d and d switched, and we need towork with R ′ ( u, h , h ) instead of R (for which see Lemma 4.5). Therefore, by a symmetryargument, we have S ≪ ( X H + X H )( X X ) ε M ∗ ( α ). (cid:3) Remark 5.3. The proof given above works equally well if we replace S (1 , m, n, , D , D ) by S (1 , ǫ m, ǫ n, , D , D ) for ǫ , ǫ ∈ {− , } . Bounds on R This section is devoted to the long proof of Lemma 4.8. The overarching idea of the proofis to evaluate b S ( a, u, t, b, p k , p l ) in explicit terms (as much as possible), and to trivially sumover u . Lemma 4.6 will allow us to focus almost entirely on the case k < l . Except for thecases k = l ≥ 2, we have evaluated b S exactly. It is a pleasant fact that this is much easierthan evaluating the Kloosterman sum itself (compare to Theorem 0.3 of [DF]).In the proof of Theorem 1.2 we only needed estimates on R ( t, p k , p l ) when k, l ≥ k, l ) ≥ 2, but since the small values of k and l are easily treated, we shall cover all thecases as stated in Lemma 4.8.6.1. The case k = 0 , or l = 0 . By a direct calculation, and using (3.3), we have(6.1) b S ( a, u, t, b, , p l ) = e (cid:16) ubp l (cid:17) δ ( p ∤ u ) , and by symmetry (that is, Lemma 4.6),(6.2) b S ( a, u, t, b, p k , 1) = e (cid:16) tap k (cid:17) δ ( p ∤ t ) . Trivially summing over u , we easily derive R ( t, p k , p l ) ≤ p l in case k = 0 or l = 0.6.2. The case k = l = 1 . By Lemma 4.2,(6.3) S ( a, y, x, b, p, p l ) = S ( x, p ) S ( y, bp ; p l ) + S ( a, p ) S ( b, yp ; p l ) + ( p − δ ( l = 1) . Therefore, recalling ( ab, p ) = 1 and S ( a, p ) = − S ( b, p ), we have(6.4) b S ( a, u, t, b, p, p ) = 1 p X x (mod p ) X y (mod p ) e (cid:16) − xtp (cid:17) e (cid:16) − yup (cid:17) [ S ( x, p ) S ( y, p ) + p ]= δ ( p ∤ t ) δ ( p ∤ u ) + pδ ( p | t ) δ ( p | u ) . We immediately deduce R ( t, p, p ) ≤ p .6.3. The case k = 1 , l ≥ . Using (6.3) and the fact that S ( b, yp ; p l ) = 0 following fromLemma 4.3, we derive(6.5) b S ( a, u, t, b, p, p l ) = 1 p l +1 X x (mod p ) X y (mod p l ) S ( x, p ) S ( y, bp ; p l ) e (cid:16) − xtp (cid:17) e (cid:16) − yup l (cid:17) = δ ( p ∤ t ) δ ( p ∤ u ) e (cid:16) ubpp l (cid:17) . We conclude(6.6) R ( t, p, p l ) ≤ δ ( p ∤ t ) p l . ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 19 The case l = 1 , k ≥ . By (6.5) and Lemma 4.6, we have(6.7) b S ( a, u, t, b, p k , p ) = δ ( p ∤ t ) δ ( p ∤ u ) e (cid:16) tapp k (cid:17) , and so R ( t, p k , p ) ≤ p .Remark. For the remaining cases we do not use direct evaluations of S and insteadcalculate b S from the definition. We have(6.8) b S ( a, u, t, b, p k , p l ) = 1 p k + l X x (mod p k ) X y (mod p l ) S ( a, y, x, b, p k , p l ) e (cid:16) − xtp k (cid:17) e (cid:16) − yup l (cid:17) = h p k + l X x (mod p k ) X y (mod p l ) X B ,C (mod p k )( B ,C ,p k )=1 X B ,C (mod p l )( B ,C ,p l )=1 p k C + B B + p l C ≡ p k + l ) Y B + Z C ≡ p k ) Y B + Z C ≡ p l ) e (cid:16) aB + x ( Y p l − Z B ) p k (cid:17) e (cid:16) yB + b ( Y p k − Z B ) p l (cid:17) e (cid:16) − xtp k (cid:17) e (cid:16) − yup l (cid:17)i . This simplifies as(6.9) b S ( a, u, t, b, p k , p l ) = X B ,C (mod p k )( B ,C ,p k )=1 X C (mod p l )( u,C ,p l )=1 p k C + B u + p l C ≡ p k + l ) Y p l − Z u ≡ t (mod p k ) Y B + Z C ≡ p k ) Y u + Z C ≡ p l ) e (cid:16) aB p k (cid:17) e (cid:16) b ( Y p k − Z B ) p l (cid:17) . Although a large expression, we found it helpful to have all the conditions written in thesummation sign.6.5. The case l > k ≥ . Suppose that p ν || t , and write t = p ν t ′ . Here we will show(6.10) b S ( a, u, t, b, p k , p l ) = p ν e (cid:16) bu ′ p l − k + ν (cid:17) e (cid:16) at ′ p l − k p ν (cid:17) S ( a, bt ′ u ′ , p ν ) δ ( ν ≤ k/ , where the sum vanishes unless p ν || u , in which case we write u = p ν u ′ .Using only the trivial bound (not even the Weil bound) for the Kloosterman sum, weconclude(6.11) R ( t, p k , p l ) ≤ p ν X u (mod p l ) ,p ν | u p ν = p ν + l , and in addition we have ν ≤ k/ 2. This estimate is consistent with Lemma 4.8.The congruence p k C + B u + p l C ≡ p k + l ) implies p k | B u , and is equivalent to(6.12) C ≡ − B up k − p l − k C (mod p l ) . Suppose p k || B , and write b S = P kk =0 V k correspondingly. We first evaluate the terms with1 ≤ k ≤ k − 1. We write B = p k R where R runs mod p k , with k + k = k . We alsohave p k | u . Since p | B and p | u , the coprimality conditions now require p ∤ C and p ∤ C . If p k +1 | u then (6.12) would imply p | C , a contradiction. So p k || u , and we write u = p k u ′ .We set Y = Y = 0, and Z i = C i . Then we have(6.13) V k = X ∗ R (mod p k ) X ∗ C (mod p k ) X ∗ C (mod p l ) C ≡− R u ′ − p l − k C (mod p l ) − C u ′ p k ≡ t (mod p k ) e (cid:16) aR p k (cid:17) e (cid:16) − bC R p l − k (cid:17) . Next we observe that p k || t , so we write t = p k t ′ , and then we have C ≡ − u ′ t ′ (mod p k ).With these evaluations, we have(6.14) V k = X ∗ R (mod p k ) X ∗ C (mod p k ) C ≡− u ′ t ′ (mod p k ) e (cid:16) aR p k (cid:17) e (cid:16) bR ( R u ′ + p l − k C ) p l − k (cid:17) . To help simplify this expression, we expand as follows:(6.15) e (cid:16) bR ( R u ′ + p l − k C ) p l − k (cid:17) = e (cid:16) bu ′ ( R u ′ + p l − k C − p l − k C )( R u ′ + p l − k C ) p l − k (cid:17) . After simplification, this gives(6.16) V k = e (cid:16) bu ′ p l − k (cid:17) X ∗ R (mod p k ) X ∗ C (mod p k ) C ≡− u ′ t ′ (mod p k ) e (cid:16) aR p k (cid:17) e (cid:16) − bu ′ C ( R u ′ + p l − k C ) p k (cid:17) . We claim that if 1 ≤ k < k ≤ k − 1, then V k = 0. For this, write C = f + p k − f with f running mod p k − and f mod p k +1 . Since k < k , the congruence C ≡ − u ′ t ′ (mod p k ) gives no condition on f . Then note that since l − k + k − ≥ k , we have(6.17) e (cid:16) − bu ′ ( f + p k − f )( R u ′ + p l − k ( f + p k − f )) p k (cid:17) = e (cid:16) − bu ′ ( f + p k − f )( R u ′ + p l − k f ) p k (cid:17) , and so the sum over f will cause V k to vanish.Now suppose k ≥ k . Then the congruence on C mod p k determines C mod p k , andhence(6.18) V k = e (cid:16) bu ′ p l − k (cid:17) X ∗ R (mod p k ) X ∗ C (mod p k ) C ≡− u ′ t ′ (mod p k ) e (cid:16) aR p k (cid:17) e (cid:16) bt ′ ( R u ′ + p l − k ( − u ′ t ′ )) p k (cid:17) , which simplifies as(6.19) V k = p k e (cid:16) bu ′ p l − k (cid:17) X ∗ R (mod p k ) e (cid:16) aR p k (cid:17) e (cid:16) bt ′ u ′ ( R − p l − k t ′ ) p k (cid:17) . Changing variables R → R + p l − k t ′ , we have(6.20) V k = p k e (cid:16) bu ′ p l − k (cid:17) e (cid:16) at ′ p l − k p k (cid:17) S ( a, bt ′ u ′ , p k ) , ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 21 and we recollect that p k || u , p k || t , and 1 ≤ k ≤ k ≤ k − 1. If we define ν by p ν || t , then wehave(6.21) k − X k =1 V k = p ν e (cid:16) bu ′ p l − k + ν (cid:17) e (cid:16) at ′ p l − k p ν (cid:17) S ( a, bt ′ u ′ , p ν ) δ (1 ≤ ν ≤ k/ . So far we have left the cases with k = 0 and k = k unevaluated, so we next turn to this.We claim V = 0, which takes some calculation. We have p ∤ B so we can set Y = B and Z = 0. If p k +1 | u then (6.12) would mean p | C , a contradiction. So p k || u and we write u = p k u ′ . We must have p ∤ C so we set Y = 0, Z = C . With these evaluations, we have p k | t and(6.22) V = X ∗ B (mod p k ) X C (mod p k ) X C (mod p l ) C ≡− B u ′ − p l − k C (mod p l ) e (cid:16) aB p k (cid:17) e (cid:16) − bC B p l (cid:17) . Now we can write C = f + p k − f , and C = − B u ′ − p l − k f − p l − f , and so(6.23) V = X ∗ B (mod p k ) e (cid:16) aB p k (cid:17) X f (mod p k − ) X f (mod p ) e (cid:16) − b ( − B u ′ − p l − k f − p l − f ) B p l (cid:17) . Note 1 + p l − f ≡ − p l − f (mod p l ) since l ≥ 2. This shows that the sum over f vanishes,as desired.Finally we evaluate V k . Then we have p k | B so may set B = 0 (that is, we choose theinteger 0 for the coset representative of 0 (mod p k )), p ∤ C , p | C so p ∤ u , and we may set Y = 0, Z = C , Y = u , Z = 0. Then(6.24) V k = e (cid:16) bp k up l (cid:17) X ∗ C (mod p k ) − C u ≡ t (mod p k ) X C (mod p l ) C ≡− p l − k C (mod p l ) . This means p ∤ t and both C and C are uniquely determined. Therefore,(6.25) V k = δ ( p ∤ t ) δ ( p ∤ u ) e (cid:16) bp k up l (cid:17) , which coincidentally agrees with the right hand side of (6.20), except with k = ν = 0.Therefore, by adding (6.21) and (6.25) we obtain (6.10), as desired.6.6. The case k > l ≥ . By (6.10) and Lemma 4.6, we have(6.26) b S ( a, u, t, b, p k , p l ) = p ν e (cid:16) at ′ p k − l + ν (cid:17) e (cid:16) bu ′ p k − l p ν (cid:17) S ( b, at ′ u ′ , p ν ) δ ( ν ≤ l/ , where again ν is defined by p ν || u and p ν || t . Using only the trivial bound for the Kloostermansum, we derive(6.27) R ( t, p k , p l ) ≤ p ν X u (mod p l ) ,p ν | u p ν ≤ p l + ν , and in addition we have ν ≤ l/ 2. Again, this is consistent with Lemma 4.8. The case k = l ≥ . The case k = l follows somewhat similar lines to the k = l case,but there are some significant differences that require careful scrutiny. We do not have aclean formula for b S analogous to (6.10).Performing some mild simplifications in (6.9), we obtain that p k | B u and then(6.28) b S ( a, u, t, b, p k , p k ) = X B ,C (mod p k )( B ,C ,p k )=1 X C (mod p k )( u,C ,p k )=1 C + C ≡− B upk (mod p k ) − Z u ≡ t (mod p k ) Y B + Z C ≡ p k ) Y u + Z C ≡ p k ) e (cid:16) aB p k (cid:17) e (cid:16) − bZ B p k (cid:17) . As before, let V k denote the subsum with p k || B . We have p k | u , where k + k = k , butunlike the case l > k ≥ 2, we cannot conclude that p k || u . We first estimate the cases with k = 0 and k = k .We claim V = 0. With these terms, we have p ∤ B , and so we may set u = 0 (that is,we choose u = 0 as the coset representative of 0 (mod p k )). Then p ∤ C , and C ≡ − C (mod p k ). We also have t = 0. We may set Y = 0 , Z = C , and Y = B , Z = 0. Withthese evaluations, we derive(6.29) V = X ∗ B (mod p k ) X ∗ C (mod p k ) e (cid:16) aB p k (cid:17) e (cid:16) − bC B p k (cid:17) . The sum over C vanishes since it is a Ramanujan sum with modulus p k , k ≥ V k , we have B = 0. Then p ∤ C and we set Y = 0, Z = C . Then (6.12) becomes C ≡ − C (mod p k ), so we have p ∤ C and we are free to set Y = 0, Z = C . Thus(6.30) V k = X ∗ C (mod p k ) − C u ≡ t (mod p k ) . Since u is uniquely determined from C , we have(6.31) X u (mod p k ) | V k | = φ ( p k ) . Now consider V k with 1 ≤ k ≤ k − 1. Then p | B and p | u so p ∤ C , p ∤ C , and we set Y = Y = 0, Z i = C i . We write B = p k R . Suppose ν p ( t ) = ν , and write t = p ν t ′′ . Thenwe must have ν ≥ k , from the congruence − C u ≡ t (mod p k ), and we can write u = p ν u ′′ instead of u = p k u ′ . Then(6.32) V k = X ∗ R (mod p k ) X ∗ C (mod p k ) C ≡− u ′′ t ′′ (mod p k − ν ) e (cid:16) aR p k (cid:17) e (cid:16) bR ( C + R p ν − k u ′′ ) p k (cid:17) . We claim that V k = 0 if k < ν < k , as we now argue. Observing that k − ν < k (since k − ν = k + k − ν ), we can write C = − u ′′ t ′′ + p k − ν f , with f running mod p ν . Then we ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 23 can write f = f + f p ν − k − where f runs mods p ν − k − , and f runs mod p k +1 . Then wearrive at a sum over f of the form(6.33) X f (mod p k ) e (cid:16) α (1 + βp k − f ) p k (cid:17) = X f (mod p k ) e (cid:16) α (1 − βp k − f ) p k (cid:17) , where ( αβ, p ) = 1, using k ≥ k > k − ν ≥ 1. Since this sum over f vanishes, this means V k = 0.If ν = k , we have from (6.32), after changing variables C → C − R p ν − k u ′′ , that(6.34) V k = X ∗ R (mod p k ) X ∗ C (mod p k ) e (cid:16) aR p k (cid:17) e (cid:16) bR C p k (cid:17) = p k S (1 , p k ) = ( , k ≥ ,p k , k = 1 . For ν = k , we have(6.35) k − X k =1 X u (mod p k ) | V k | = p k − δ ( ν = k ) . Now suppose ν ≤ k < k . This condition implies k − ν ≥ k , so the congruence C ≡ − u ′′ t ′′ (mod p k − ν ) determines C modulo p k , and so (6.32) simplifies as(6.36) V k = p ν X ∗ R (mod p k ) e (cid:16) aR p k (cid:17) e (cid:16) bR u ′′ ( − t ′′ + R p ν − k ) p k (cid:17) . If, in addition, ν − k ≥ k , then this is simply given by(6.37) V k = p ν S ( u ′′ a − t ′′ b, p k ) δ ( k ≤ ν/ . It follows easily that for these values of k and ν that(6.38) X u (mod p k ) | V k | ≤ p ν p k p k − ν p k = 2 p k , and so,(6.39) X ≤ k ≤ ν/ X u (mod p k ) | V k | ≤ νp k . On the other hand, if ν − k < k (we continue to assume ν ≤ k < k ), then we can write R = f + p k − ν f , with f mod p k − ν , and f mod p ν − k . Then (6.36) becomes(6.40) V k = p ν X ∗ f (mod p k − ν ) X f (mod p ν − k ) e (cid:16) a ( f + p k − ν f ) p k (cid:17) e (cid:16) bu ′′ ( f + p k − ν f )( − t ′′ + f p ν − k ) p k (cid:17) . The inner sum over f simplifies, and detects u ′′ a ≡ bt ′′ (mod p ν − k ). Hence(6.41) V k = p ν − k X ∗ f (mod p k − ν ) e (cid:16) af p k (cid:17) e (cid:16) bu ′′ f ( − t ′′ + f p ν − k ) p k (cid:17) δ ( u ′′ a ≡ t ′′ b (mod p ν − k )) . Therefore, by a trivial bound on the f -sum, we have(6.42) | V k | ≤ p ν − k φ ( p k − ν ) δ ( u ′′ a ≡ t ′′ b (mod p ν − k )) , which implies(6.43) X u (mod p k ) | V k | ≤ p ν − k φ ( p k − ν ) p k − ν p ν − k ≤ φ ( p k + k − ν ) . Here we certainly have k ≤ ν (from the paragraph following (6.31)), but we also have that ν is restricted by ν ≤ min( k − k , k ) ≤ k/ 3. Therefore, these values of k give(6.44) X k X u (mod p k ) | V k | ≤ p k + ν δ ( ν ≤ k/ . Combining (6.31), (6.35), (6.39), and (6.44), we derive(6.45) R ( t, p k , p k ) ≤ ( k + 1) p k + p k + ν δ ( ν ≤ k/ . Spectral summation formula The remaining sections of the paper contain the proof of Theorems 1.1.We shall use the GL Bruggeman-Kuznetsov formula in the form given by Blomer [Bl].Suppose that T , T ≫ 1, and consider the sum(7.1) X ν = iT + O (1) ν = iT + O (1) R j (cid:12)(cid:12)(cid:12) X N/ We have (7.9) X ν = iT + O (1) ν = iT + O (1) R j (cid:12)(cid:12)(cid:12) X N/ 1. Here we have used the shorthand a ′′ m = a ′ m m i (2 τ + τ ) , b ′′ n = a ′ n n i ( τ +2 τ ) . Proposition 8.1. We have (8.7) | U | ≪ X X (cid:16) X + NT + T (cid:17) / (cid:16) X + NT + T (cid:17) / ( N T T ) ε X n ≤ N | a n | . The estimate is uniform in terms of γ, X , X , x , x , x . ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 27 Assuming Proposition 8.1, we may quickly show the following variant of Theorem 1.1:(8.8) X ν = iT + O (1) ν = iT + O (1) R j (cid:12)(cid:12)(cid:12) X n ≤ N a n λ j ( n, (cid:12)(cid:12)(cid:12) ≪ (cid:16) T T ( T + T ) + T T N T + T (cid:17) ε X n ≤ N | a n | , as we now explain. By inserting (8.7) into (8.5), we obtain(8.9) (cid:12)(cid:12)(cid:12) X n,m a n a m Σ (cid:12)(cid:12)(cid:12) ≪ X X ,X ≪ N ( T T − ε dyadic T T ( T + T )( T T N ) ε X X X X (cid:16) X + NT + T (cid:17) / (cid:16) X + NT + T (cid:17) / X n ≤ N | a n | , plus a small error term from the truncation on X , X . By a direct calculation, this gives(8.10) (cid:12)(cid:12)(cid:12) X n,m a n a m Σ (cid:12)(cid:12)(cid:12) ≪ T T ( T + T )( T T N ) ε (cid:16) NT + T (cid:17) X n | a n | , which proves (8.8). Proof of Proposition 8.1. The main difficulty in the proof is exploiting cancellation in the y , y integrals. For point of reference, if we apply Corollary 1.3 directly to (8.6), triviallyintegrating over y and y , we obtain(8.11) | U | ≪ X X ( X + N ) / ( X + N ) / ( N T T ) ε X n ≤ N | a n | . One easily observes that if T + T ≪ N ε , then (8.11) implies (8.7), so for the rest of theproof we assume(8.12) T + T ≫ N ε . We will first show the bound (8.7) under the assumptions(8.13) | x | , | x | ≤ δ ( T + T ) , where δ > / Y = X X x + x ′ x x ′ + x , and Y = X X x + x x x + x . With this definition, we have that the y -integral takes the form(8.15) Z h ( y ) e iφ ( y ) dy , φ ( y ) = c ( T + T ) log y + c x y + c Ny Y , where each c i ≍ h is a weight function with bounded derivatives. Under the assumption(8.13), repeated integration by parts (see [BKY, Lemma 8.1]) shows the integral is smallerthan an arbitrarily large negative power of max( T + T , NY ) (and hence, using (8.12), anarbitrarily large power of T T N ), unless NY ≍ ( T + T ). If (8.13) does not hold then there ispotentially cancellation between the first two terms in the phase in which case this argumentbreaks down. A similar argument holds for y also. Thus we may assume(8.16) Y , Y ≪ NT + T . Moving the integrals to the outside, we derive(8.17) | U | ≪ Z Z (cid:12)(cid:12)(cid:12) X D ≍ X D ≍ X X n,m b ′′ n a ′′ m S (1 , ǫ m, ǫ n, , D , D ) γ D ,D e (cid:16) − nD y D x x + x x + x (cid:17) f (cid:16) nD y D ξ / ξ (cid:17) e (cid:16) − mD y D x x ′ + x x + x ′ (cid:17) f (cid:16) mD y D ξ / ξ (cid:17)(cid:12)(cid:12)(cid:12) dy y dy y . Now we can change variables y → y − D X X D , and y → y − D X X D , giving that(8.18) | U | ≪ Z y ≍ Z y ≍ (cid:12)(cid:12)(cid:12) X D ≍ X D ≍ X X n,m b ′′ n a ′′ m S (1 , ǫ m, ǫ n, , D , D ) γ D ,D e (cid:16) − my Y (cid:17) e (cid:16) − ny Y (cid:17) f (cid:16) ny X X ξ / ξ (cid:17) f (cid:16) my X X ξ / ξ (cid:17)(cid:12)(cid:12)(cid:12) dy y dy y . Now we can apply Mellin inversion to f ( ny X X ξ / ξ ) (and the other f ), showing now(8.19) | U | ≪ Z ∞−∞ 11 + r Z ∞−∞ 11 + r Z y ≍ Z y ≍ (cid:12)(cid:12)(cid:12) X D ≍ X D ≍ X γ D ,D X n,m b ′′ n n ir a ′′ m m ir S (1 , ǫ m, ǫ n, , D , D ) e (cid:16) − my Y (cid:17) e (cid:16) − ny Y (cid:17)(cid:12)(cid:12)(cid:12) dy y dy y dr dr . Remark. The r and r integrals are practically harmless because our bound will be in termsof the L norms of the sequences ( b ′′ n ) and ( a ′′ m ), which are then independent of r , r .At this point we can apply Theorem 1.2 (see also Remark 5.3), showing(8.20) | U | ≪ ( X X ) ε h Z ∞−∞ 11 + r Z y ≍ X q ≤ min( X ,X ) X d | q d q X c ≤ X q ( c,q )=1 X ∗ t (mod c ) (cid:12)(cid:12)(cid:12) X ( n,q )= d b ′′ n n ir e (cid:16) tnc (cid:17) e (cid:16) − ny Y (cid:17)(cid:12)(cid:12)(cid:12) dy dr i / [ . . . ] / , with [ . . . ] representing a similar term. Using the hybrid large sieve (Lemma 3.1) shows thatthe first expression in brackets is bounded by(8.21) X q ≤ min( X ,X ) X d | q d q (cid:16) X q + Y d (cid:17) X ( n,q )= d | β n | ≪ ( X X ) ε (cid:16) X + NT + T (cid:17) X n ≤ N | b n | . The second expression in brackets in (8.20) is bounded in a similar way, which completesthe proof under the assumption (8.13). ILINEAR FORMS WITH GL KLOOSTERMAN SUMS AND THE SPECTRAL LARGE SIEVE 29 Now we show how to modify the proof in case (8.13) does not hold. Say that | x | ≥ δ ( T + T ), and | x | ≤ δ ( T + T ). The y -analysis is unchanged while in (8.6), we changevariables y → y mN . Following the calculations above, in place of (8.18), we obtain(8.22) | U | ≪ Z y ≍ Z y ≍ (cid:12)(cid:12)(cid:12) X D ≍ X D ≍ X X n,m b ′′ n a ′′′ m S (1 , ǫ m, ǫ n, , D , D ) γ D ,D e (cid:16) − my x N (cid:17) e (cid:16) − ny Y (cid:17) f (cid:16) ny X X ξ / ξ (cid:17) f (cid:16) my N (cid:17)(cid:12)(cid:12)(cid:12) dy y dy y , where | a ′′′ m | = | a ′′ m | . This has the same essential form as (8.18) but with Y replaced by N | x | ≪ NT + T . Thus we arrive at the same bound in this case. By symmetry, the same boundholds in case | x | ≤ δ ( T + T ) and | x | ≥ δ ( T + T ). A simple modification covers the case | x | , | x | ≥ δ ( T + T ), where we apply the change of variables in both y , y . (cid:3) Proof of Theorem 1.1 If the X i are large, then we can obtain an improved version of Proposition 8.1, namely Proposition 9.1. We have (9.1) | U | ≪ h ( X H + X H ) (cid:16) X + NT + T (cid:17) / (cid:16) X + NT + T (cid:17) / + ( X X ) / NH + ( X X ) / NH i ( X X ) ε X n ≤ N | a n | . Here the proof is identical to that of Proposition 8.1 except at (8.20) we apply Theorem1.4 instead of Theorem 1.2, so we omit the details.We continue with bounding (8.5). We shall use the bound implied by (8.9) for certainranges of X i . Specifically, for the values of X with X ≤ NT + T , the bound (8.9) simplifiesas(9.2) ≪ X X ≪ NT T X ≪ N ( T T − ε T T ( T + T ) (cid:16) NT + T (cid:17) / (cid:16) X + NT + T (cid:17) / ( T T ) ε X n ≤ N | a n | ≪ T T N / ( T + T ) / ( T T ) ε X n ≤ N | a n | , which is stronger than required for Theorem 1.1. By symmetry, the same bound holds if X ≤ NT + T . For the complementary terms with X > NT + T and X > NT + T , Proposition9.1 simplifies to give(9.3) | U | ≪ ( X X ) h ( X H + X H ) + ( X X ) / NH + ( X X ) / NH i ( X X ) ε X n ≤ N | a n | . 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