An improved spectral large sieve inequality for SL_3(\mathbb{Z})
aa r X i v : . [ m a t h . N T ] F e b AN IMPROVED SPECTRAL LARGE SIEVE INEQUALITY FOR SL ( Z ) MATTHEW P. YOUNG
Abstract.
We prove an improved spectral large sieve inequality for the family of SL ( Z )Hecke-Maass cusp forms. The method of proof uses duality and its structure reveals unex-pected connections to Heath-Brown’s large sieve for cubic characters. Introduction
A large sieve inequality for a family of automorphic forms is a flexible and versatile toolthat represents quantitative orthogonality properties of the family. Strong results are knownfor GL and GL (e.g. see [M] and [IK, Chapter 7] for some surveys), but progress has beenmore elusive in higher rank. The main focus of this article, the SL ( Z ) spectral large sieve,has seen some recent attention in a series of papers [B, Y, BB]. For some other notablehigher rank examples, see [DK, V, TZ].We set some notation before continuing the discussion on the large sieve. Let F cusp denote the family of Hecke-Maass cusp forms on SL ( Z ). Similarly, let F Eis denote thefamily of SL ( Z ) Eisenstein series induced by SL ( Z ) cusp forms (see [G, Section 10.5] for adefinition). Let F = F cusp ∪F Eis . For F ∈ F , let µ F = µ = ( µ , µ , µ ) ∈ a ∗ C be its Langlandsparameters, so the Ramanujan conjecture predicts that µ ∈ i R . Let λ F ( m, n ) denote theHecke eigenvalues of F . Let Ω ⊂ a ∗ be compact, Weyl group invariant, and disjoint fromthe Weyl chamber walls. Let B = B V be a box of sidelength V / ≤ V ≤ T and B V ⊂ T Ω, and let F V ⊂ F denote the set of Hecke-Maass cusp forms and Eisenstein serieswith µ F ∈ W ( B ), where W is the Weyl group. Write(1.1) F cusp V = F cusp ∩ F V , and F Eis V = F Eis ∩ F V . For F ∈ F cusp , let ω F = Res s =1 L ( F ⊗ F , s ). The Weyl law proved by Lapid and M¨uller [LM]gives that the cardinality of F cusp V is V T o (1) . Any potential violation to the Ramanujanconjecture must occur near the Weyl chamber walls (e.g. see [B, p.678]), so automatically anycusp form with µ F ∈ T Ω satisfies Ramanujan. For a ∈ ℓ , we use the notation | a | = k a k .The culmination of [B, Y, BB] is the following. Theorem 1.1 ([BB]) . We have (1.2) X ∗∈{ cusp , Eis } X F ∈F ∗ T ω F (cid:12)(cid:12)(cid:12) X N ≤ n ≤ N a n λ F (1 , n ) (cid:12)(cid:12)(cid:12) ≪ ( T + T N ) ε | a | , (1.3) X ∗∈{ cusp , Eis } X F ∈F ∗ ω F (cid:12)(cid:12)(cid:12) X N ≤ n ≤ N a n λ F (1 , n ) (cid:12)(cid:12)(cid:12) ≪ ( T + T N ) ε | a | . This material is based upon work supported by the National Science Foundation under agreement No.DMS-2001306. 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.
The formulations of (1.2) and (1.3) are imprecise because we have not fully described themeaning of P F ∈F Eis ω − F for F an Eisenstein series; see e.g. [BB, Section 4] for the correctnormalizing factor. The proof of Theorem 1.1 notably relies on the GL Kuznetsov formula.The spectral side of the Kuznetsov formula includes both the cusp forms as well as Eisensteinseries, which explains why Theorem 1.1 covers both types of automorphic forms.By general principles of bilinear forms (cf. [IK, Chapter 7]), the optimal bound one couldexpect on the right hand side of (1.2) would be ( T + N ) ε , while that of (1.3) would be( T + N ) ε . However, Blomer and Buttcane showed that the term T N in (1.2) cannotbe reduced in size, by constructing a choice of vector a so that the contribution from theEisenstein series is at least T N | a | , for N ≫ T δ . An examination of the proof of [BB,Proposition 1.3] shows their method leads to a lower bound of size T N | a | for the left handside of (1.3). In Section 7 we sketch an alternative method to produce this lower bound.A natural question is if the bounds (1.2)-(1.3) can be improved when the family is restrictedto cusp forms. The main result of this article affirms this. Theorem 1.2.
We have (1.4) X F ∈F cusp ω F (cid:12)(cid:12)(cid:12) X N ≤ n ≤ N a n λ F (1 , n ) (cid:12)(cid:12)(cid:12) ≪ ( T + N + T N / ) ε | a | . Note that the right hand side of (1.4) is smaller than the right hand side of (1.3) for N ≫ T ε , and is also just as good as the “ N + T ” theoretically optimal bound for N ≫ T .The starting point of our proof is to use the duality principle and the functional equationof Rankin-Selberg L -functions on GL × GL . This method is most effective when N is large,since this makes the dual length of summation relatively shorter. The final step in our proofis an application of the Buttcane-Blomer bound (1.3), which is strongest for relatively smallvalues of N .A curious aspect of the proof is that is reveals that certain aspects of the family F arein analogy with the family of cubic Hecke characters. A large sieve inequality for this latterfamily was proved by Heath-Brown [H-B] with an application to the problem of estimatingsums of cubic Gauss sums of prime arguments. See Section 6 below for a more thoroughdiscussion of Heath-Brown’s work and its connections to our proof of Theorem 1.2.For simplicity, Theorem 1.2 is stated for the localized family F but in principle one coulduse the same method to study F T as well. Typically, small families are more difficult thanlarge families, so one might expect that a bound on F T would be even easier to prove thanthat for F . However, our proof of Theorem 1.2 using duality requires the conductor of L (1 / , F ⊗ G ) for F, G ∈ F which is a bit simpler to express for
F, G ∈ F than for general F, G ∈ F T . Generically, for F, G ∈ F T , the conductor of L (1 / , F ⊗ G ) is of size T but thereare various conductor-dropping ranges to consider. Indeed, for F, G ∈ F , the conductor of L (1 / , F ⊗ G ) is of size T . As an aside, this discussion indicates that the approach via dualityis beneficial when N ≫ T δ (since T is the square-root of the conductor), consistent withthe remark above that (1.4) is an improvement in this range.2. Preliminaries
Maass forms on SL ( Z ) . N IMPROVED SPECTRAL LARGE SIEVE INEQUALITY FOR SL ( Z ) 3 Lemma 2.1 (Hecke relations) . Let F ∈ F . Then (2.1) λ F ( m, λ F (1 , n ) = X d | ( m,n ) λ F (cid:16) md , nd (cid:17) , and (2.2) λ F ( m, n ) = X d | ( m,n ) µ ( d ) λ F (cid:16) md , (cid:17) λ F (cid:16) , nd (cid:17) . Moreover, (2.3) λ F ( m, n ) = λ F ( n, m ) . The relation (2.1) appears in [G, Theorem 6.4.11], from which (2.2) follows from M¨obiusinversion. For (2.3), see [G, Theorem 9.3.11 Addendum].
Lemma 2.2 (Convexity bound) . For any F ∈ F T and any X ≥ we have (2.4) X m n ≤ X | λ F ( m, n ) | ≪ ε X ( XT ) ε . This follows from work of Xiannan Li [L].
Lemma 2.3.
Let
F, G ∈ F cusp . The Rankin-Selberg L -function L ( s, F ⊗ G ) is defined by (2.5) L ( s, F ⊗ G ) = X d,m,n ≥ λ F ( m, n ) λ G ( m, n )( d m n ) s . It has meromorphic continuation to s ∈ C with a possible pole at s = 1 only, and satisfiesthe functional equation (2.6) γ ( s, F ⊗ G ) L ( s, F ⊗ G ) = γ (1 − s, G ⊗ F ) L (1 − s, G ⊗ F ) , where (2.7) γ ( s, F ⊗ G ) = γ ( s, µ F , µ G ) = Y i,j =1 Γ R ( s + µ i ( F ) + µ j ( G )) . The pole at s = 1 exists if and only if F = G . For a reference, see [G, Theorem 7.4.9, Proposition 11.6.17].2.2.
Separation of variables.Lemma 2.4.
Suppose f is Schwartz-class. Then (2.8) f ( x ) = Z ∞−∞ b f ( y ) e ( xy ) dy, where k b f k ≪ k f k + k f ′′ k . The implied constant is absolute. This follows from Fourier inversion and integration by parts. To give an idea of how wewish to use Lemma 2.4 to separate variables, consider the following example.
MATTHEW P. YOUNG
Example 2.5.
Suppose that f is Schwartz-class. Moreover suppose that γ m and δ n are somesequences of real numbers and that I is some finite set of integers. Then (2.9) max | b | =1 (cid:12)(cid:12)(cid:12) X m,n ∈ I b m b n f ( γ m + δ n ) (cid:12)(cid:12)(cid:12) ≤ k b f k max | b | =1 (cid:12)(cid:12)(cid:12) X m ∈ I b m (cid:12)(cid:12)(cid:12) . where | b | = ( P n ∈ I | b n | ) / .Proof. By Lemma 2.4, we have (cid:12)(cid:12)(cid:12) X m,n b m b n f ( γ m + δ n ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z ∞−∞ b f ( y ) (cid:16) X m b m e ( γ m y ) (cid:17)(cid:16) X n b n e ( δ n y ) (cid:17)(cid:12)(cid:12)(cid:12) dy (2.10) ≤ k b f k max y ∈ R (cid:12)(cid:12)(cid:12) X m b m e ( γ m y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n b n e ( δ n y ) (cid:12)(cid:12)(cid:12) . (2.11)Taking the maximum over | b | = 1 immediately gives the result. (cid:3) Definitions of norms and some relations between them
We begin by defining the basic norm that appears (implicitly) in Theorem 1.2:(3.1) ∆ ( F V , N ) = max | a | =1 X F ∈F cusp V ω F (cid:12)(cid:12)(cid:12) X N ≤ n ≤ N a n λ F (1 , n ) (cid:12)(cid:12)(cid:12) . By the duality principle (cf. [IK, p.170]), we have ∆ ( F V , N ) = ∆ (1) ( F V , N ), where(3.2) ∆ (1) ( F V , N ) = max | b | =1 X N ≤ n ≤ N (cid:12)(cid:12)(cid:12) X F ∈F cusp V b F ω − / F λ F (1 , n ) (cid:12)(cid:12)(cid:12) . We also define a related norm ∆ ( F V , N ) = ∆ (2) ( F V , N ) by(3.3) ∆ ( F V , N ) = max | a | =1 X F ∈F cusp V ω F (cid:12)(cid:12)(cid:12) X N ≤ m n ≤ N a m,n λ F ( m, n ) (cid:12)(cid:12)(cid:12) , and where(3.4) ∆ (2) ( F V , N ) = max | b | =1 X N ≤ m n ≤ N (cid:12)(cid:12)(cid:12) X F ∈F cusp V b F ω − / F λ F ( m, n ) (cid:12)(cid:12)(cid:12) . Finally we define a third norm ∆ ( F V , N ) = ∆ (3) ( F V , N ) by(3.5) ∆ ( F V , N ) = max | a | =1 X F ∈F cusp V ω F (cid:12)(cid:12)(cid:12) X N ≤ d m n ≤ N a d,m,n λ F ( m, n ) (cid:12)(cid:12)(cid:12) , and where(3.6) ∆ (3) ( F V , N ) = max | b | =1 X N ≤ d m n ≤ N (cid:12)(cid:12)(cid:12) X F ∈F cusp V b F ω − / F λ F ( m, n ) (cid:12)(cid:12)(cid:12) . We obviously have ∆ ( F V , N ) ≤ ∆ ( F V , N ) ≤ ∆ ( F V , N ). We also want relations in theother direction. Lemma 3.1.
We have (3.7) ∆ ( F V , N ) ≪ (log N ) max R ≪ N (cid:16) NR (cid:17) / ∆ ( F V , R ) . N IMPROVED SPECTRAL LARGE SIEVE INEQUALITY FOR SL ( Z ) 5 Proof.
We prove this on the dual side, using (3.6) and (3.4). By breaking the sum up so R ≤ m n ≤ R and summing R over dyadic segments, we obtain(3.8) ∆ (3) ( F V , N ) ≪ (log N ) max ≪ R ≪ N (cid:16) NR (cid:17) / max | b | =1 X R ≤ m n ≤ R (cid:12)(cid:12)(cid:12) X F ∈F cusp V b F ω − / F λ F ( m, n ) (cid:12)(cid:12)(cid:12) . The result follows immediately. (cid:3)
Lemma 3.2.
We have (3.9) ∆ ( F V , N ) ≪ ( N T ) ε max Y X ≪ N min (cid:16) Y ∆ ( F V , X ) , X ∆ ( F V , Y ) (cid:17) . Proof.
Again, we prove this on the dual side, using (3.4) and (3.2). By the Hecke relation(2.2), we deduce(3.10) ∆ (2) ( F V , N ) = max | b | =1 X N ≤ m n ≤ N (cid:12)(cid:12)(cid:12) X d | ( m,n ) µ ( d ) X F ∈F cusp V b F ω − / F λ F (cid:16) md , (cid:17) λ F (cid:16) , nd (cid:17)(cid:12)(cid:12)(cid:12) . Applying Cauchy’s inequality and a divisor function bound to take the sum over d to theoutside, we deduce(3.11) ∆ (2) ( F V , N ) ≪ N ε max | b | =1 X N ≤ m n ≤ N X d | ( m,n ) (cid:12)(cid:12)(cid:12) X F ∈F cusp V b F ω − / F λ F (cid:16) md , (cid:17) λ F (cid:16) , nd (cid:17)(cid:12)(cid:12)(cid:12) . Interchanging the order of summation and changing variables m → dm and n → dn , weobtain(3.12) ∆ (2) ( F V , N ) ≪ N ε max | b | =1 X N ≤ d m n ≤ N (cid:12)(cid:12)(cid:12) X F ∈F cusp V b F ω − / F λ F ( m, λ F (1 , n ) (cid:12)(cid:12)(cid:12) . Now we further restrict d , m and n so dm ≍ Y and n ≍ X , and let b ′ F = b F λ F ( m, (2) ( F V , N ) ≪ N ε max XY ≪ N max | b | =1 X N ≤ d m n ≤ Ndm ≍ Yn ≍ X (cid:12)(cid:12)(cid:12) X F ∈F cusp V b ′ F ω − / F λ F (1 , n ) (cid:12)(cid:12)(cid:12) ≪ N ε max XY ≪ N max | b | =1 X dm ≍ Y ∆ (1) ( F V , X ) X F ∈F cusp V | b F | | λ F ( m, | . From Lemma 2.2 we deduce P dm ≍ Y | λ F ( m, | ≪ Y ( N T ) ε , uniformly in F , leading to∆ (2) ( F V , N ) ≪ ( N T ) ε max Y X ≪ N Y ∆ (1) ( F V , X ) . It remains to show that a similar bound holds but with Y ∆ (1) ( F V , X ) replaced by X ∆ (1) ( F V , Y ).This follows by going through the same proof but reversing the roles of m and n , and using(2.3) along the way. (cid:3) Chaining together Lemmas 3.1 and 3.2, we immediately deduce the following.
Lemma 3.3.
We have (3.13) ∆ ( F V , N ) ≪ ( N T ) ε max Y X ≪ N (cid:16) NXY (cid:17) / min (cid:16) Y ∆ ( F V , X ) , X ∆ ( F V , Y ) (cid:17) . See Section 6 for a comparison of Lemma 3.3 with [H-B, Lemma 6].We also observe that the analogs of Lemmas 3.1–3.3 hold equally well for Eisenstein series.
MATTHEW P. YOUNG Functional equation
In this section we use the functional equation of the Rankin-Selberg L -function to deducethe following estimate. Lemma 4.1.
We have (4.1) ∆ (3) ( F , N ) ≪ N + NT ( N T ) ε max ≤ Z ≪ T N ( T N ) ε ∆ (3) (cid:16) F , Z (cid:17) . The proof of Lemma 4.1 crucially uses that the family is restricted to cusp forms. Thereader may examine the proof of Proposition 7.1 below to see how the family of Eisensteinseries exhibits different behavior than the cusp forms.
Proof.
Select a smooth nonnegative bump function w with compact support on the positivereals, satisfying w ( x ) ≥ ≤ x ≤
2. Then(4.2) ∆ (3) ( F V , N ) ≤ max | b | =1 X d,m,n w (cid:16) d m nN (cid:17)(cid:12)(cid:12)(cid:12) X F ∈F cusp V b F ω − / F λ F ( m, n ) (cid:12)(cid:12)(cid:12) . Next open the square, apply Mellin inversion, and evaluate the resulting Dirichlet seriesusing (2.5), giving(4.3) ∆ (3) ( F V , N ) ≤ max | b | =1 X F,G ∈F cusp V b F b G ω / F ω / G πi Z (3 / N s e w ( s ) L ( s, F ⊗ G ) ds. Next we shift the contour of integration to the line Re( s ) = − ε , change variables s → − s ,and apply the functional equation (2.6). In this process we cross a potential pole at s = 1only, which exists if and only if F = G . This pole contributes the term of size O ( N ) to theright hand side of (4.1). In all we obtain ∆ (3) ( F V , N ) is at most O ( N ) plus(4.4) max | b | =1 (cid:12)(cid:12)(cid:12) X F,G ∈F cusp V b F b G ω / F ω / G πi Z (3 / N − s e w (1 − s ) γ ( s, µ G , µ F ) γ (1 − s, µ F , µ G ) L ( s, G ⊗ F ) ds (cid:12)(cid:12)(cid:12) . Now we examine the ratio of gamma factors appearing in (4.4). Six out of the nine gammafactors in (2.7) have | µ i ( F ) + µ j ( G ) | large, of size T (the precise size determined up to O (1)by the location of the box B ). The remaining three gamma factors have | µ i ( F ) + µ j ( G ) | ofsize O (1). Moreover, since F and G automatically satisfy Ramanujan by the location of thebox B , then µ i ( F ) + µ j ( G ) ∈ i R . This means that for Re( s ) >
0, we have that the ratio ofgamma factors appearing in (4.4) is analytic, and satisfies the bound(4.5) (cid:12)(cid:12)(cid:12) Q − s e w (1 − s ) γ ( s, µ G , µ F ) γ (1 − s, µ F , µ G ) (cid:12)(cid:12)(cid:12) ≪ Re( s ) ,A (1 + | s | ) − A , for any A >
0, where Q = T . Now in (4.4) we open up the Dirichlet series, obtaining(4.6) max | b | =1 (cid:12)(cid:12)(cid:12) X d,m,n X F,G ∈F cusp V b F b G Nω / F ω / G πi Z (3 / e w (1 − s ) γ ( s, µ G , µ F ) γ (1 − s, µ F , µ G ) λ G ( m, n ) λ F ( m, n )( d m nN ) s ds (cid:12)(cid:12)(cid:12) . We may truncate the Dirichlet series at d m n ≪ QN ( T N ) ε with a very small error term(certainly smaller than the O ( N ) term already accounted for), by shifting contours far to theright. Having imposed this truncation, we may then shift the contour to the line Re( s ) = ε .We may also truncate the integral at | Im( s ) | ≪ ( N T ) ε , without producing a new error term. N IMPROVED SPECTRAL LARGE SIEVE INEQUALITY FOR SL ( Z ) 7 Then (4.6) is reduced to(4.7) max | b | =1 (cid:12)(cid:12)(cid:12) X d m n ≪ T N ( NT ) ε X F,G ∈F cusp V b F b G Nω / F ω / G πi Z Re( s )= ε | Im( s ) |≪ ( NT ) ε e w (1 − s ) γ ( s, µ G , µ F ) γ (1 − s, µ F , µ G ) λ G ( m, n ) λ F ( m, n )( d m nN ) s ds (cid:12)(cid:12)(cid:12) . At a first pass, the reader is encouraged to “pretend” that γ ( s,µ G ,µ F ) γ (1 − s,µ F ,µ G ) equals Q s − (which isa good first-order approximation) and continue with (4.10) to finish the proof. Unfortunately,a rigorous argument is a bit more technical. The plan is to separate the variables µ F and µ G in the ratio of gamma factors. The basic idea is encoded in Example 2.5. To this end, let µ i ( B ), i = 1 , , B (the choice of point is irrelevant). Then(4.8) Γ R ( s + µ i ( F ) + µ j ( G ))Γ R (1 − s + µ i ( F ) + µ j ( G )) = Γ R ( s + µ i ( B ) + µ j ( B ) + i ( δ i + ν j ))Γ R (1 − s + µ i ( B ) + µ j ( B ) + i ( δ i + ν j )) , where iδ i = µ i ( F ) − µ i ( B ) and iν j = µ j ( G ) − µ j ( B ). Here δ i , ν j = O (1) and are real. Thegoal is to separate δ i from ν j . Let(4.9) f ( x ) = Γ R ( s + µ i ( B ) + µ j ( B ) + ix )Γ R ( s + µ i ( B ) + µ j ( B )) Γ R (1 − s + µ i ( B ) + µ j ( B ))Γ R (1 − s + µ i ( B ) + µ j ( B ) + ix ) . By Stirling, for x ∈ R and | x | ≪
1, we have | f ( x ) | + | f ′′ ( x ) | ≪ T ε . By Lemma 2.4 and (2.9),in effect this means we can separate the variables δ i , ν j at “cost” at most T ε . Applying thiswith each of the gamma factors, we obtain that (4.7) is bounded by(4.10) NT ( N T ) ε max | b | =1 X d m n ≪ T N ( NT ) ε (cid:12)(cid:12)(cid:12) X F,G ∈F cusp V b F b G ω / F ω / G λ G ( m, n ) λ F ( m, n ) (cid:12)(cid:12)(cid:12) ≪ NT ( N T ) ε max ≤ Z ≪ T N ( T N ) ε ∆ (3) (cid:16) F , Z (cid:17) . (cid:3) Completion of the proof
Now we prove Theorem 1.2. We chain together the results from Section 3 as well as Lemma4.1, giving(5.1) ∆ ( F , N ) ≤ ∆ (3) ( F , N ) ≪ N + NT ( N T ) ε max ≤ Z ≪ T N ( T N ) ε ∆ (3) (cid:16) F , Z (cid:17) ≪ N + NT ( N T ) ε max Y X ≪ T N ( NT ) ε (cid:16) T /NXY (cid:17) / min (cid:16) Y ∆ ( F , X ) , X ∆ ( F , Y ) (cid:17) . This sequence of inequalities is reminiscent of (and somewhat inspired by) [H-B, Section8]. Finally we insert the Blomer-Buttcane bound ∆ ( F , M ) ≪ ( T + T M )( T M ) ε from MATTHEW P. YOUNG
Theorem 1.1. In all, we obtain∆ ( F , N ) ≪ N + NT ( N T ) ε max Y X ≤ T N (cid:16) T /NXY (cid:17) / min (cid:16) Y ( T + T X ) , X ( T + T Y ) (cid:17) ≪ N + NT ( N T ) ε max Y X ≤ T N (cid:16) T /NXY (cid:17) / T (cid:16) XY + T min( X, Y ) (cid:17) ≪ N + NT T (cid:16) T N + T (cid:16) T N (cid:17) / (cid:17) ( N T ) ε ≪ N + ( N T ) ε (( T N ) / + T ) , completing the proof of Theorem 1.2.6. Cubic characters
In this section we briefly recall the large sieve inequality of Heath-Brown for cubic char-acters [H-B] for the purpose of developing an analogy with the SL ( Z ) cusp form familyconsidered in this paper.Let θ = exp(2 πi/ m, n ∈ Z [ θ ] let ( m/n ) denote the cubic residuesymbol. The cubic reciprocity law gives that ( m/n ) = ( n/m ) . Let N ( · ) denote the normmap of Q [ ω ] / Q . Let(6.1) ∆ ( M, Q ) = max | a | =1 X ∗ n ∈ Z [ ω ] N ( n ) ≤ M (cid:12)(cid:12)(cid:12) X ∗ q ∈ Z [ θ ] N ( q ) ≤ Q a q (cid:16) nq (cid:17) (cid:12)(cid:12)(cid:12) , where the symbol P ∗ means the sums are restricted to (nonzero) square-free integers. Heath-Brown’s cubic large sieve is the bound(6.2) ∆ ( M, Q ) ≪ ( M + Q + ( M Q ) / )( M Q ) ε . To make the notation appear more similar to the SL ( Z ) family, define (for m, n, q ∈ Z [ θ ])(6.3) λ q ( m, n ) = (cid:16) nq (cid:17) (cid:16) mq (cid:17) . Note the simple identities which the reader is invited to compare with Lemma 2.1: λ q ( m, n ) = λ q ( n, m ) = λ q ( mn ,
1) = λ q (1 , nm ) = λ q ( m, λ q (1 , n ) . Also observe λ q ( d ,
1) = 1 for ( d, q ) = 1.Heath-Brown’s first step is to drop the condition that n is square-free in (6.2), leading tothe definition(6.4) ∆ ( M, Q ) = max | a | =1 X d,m,n ∈ Z [ θ ] N ( d m n ) ≤ M | µ ( mn ) | (cid:12)(cid:12)(cid:12) X ∗ q ∈ Z [ θ ] N ( q ) ≤ Q a q λ q ( m, n ) (cid:12)(cid:12)(cid:12) Obviously ∆ ( M, Q ) ≤ ∆ ( M, Q ), which parallels our relation ∆ ( F V , N ) ≤ ∆ ( F V , N ).The same steps used to prove Lemma 3.3 can be applied here to show(6.5) ∆ ( M, Q ) ≪ ( M Q ) ε max XY ≪ M (cid:16) MXY (cid:17) / min( X ∆ ( Y, Q ) , Y ∆ ( X, Q )) , which is essentially [H-B, Lemma 6]. N IMPROVED SPECTRAL LARGE SIEVE INEQUALITY FOR SL ( Z ) 9 Heath-Brown also gives a relationship between ∆ ( M, Q ) and ∆ ( Q /M, Q ) (see [H-B,Lemmas 7 and 8] for the precise statement) which arises from the functional equation andis analogous to Lemma 4.1. 7. Lower bound via duality
Proposition 7.1.
There exists a choice of vector a so that (7.1) X F ∈F Eis ω F (cid:12)(cid:12)(cid:12) X N ≤ n ≤ N a n λ F (1 , n ) (cid:12)(cid:12)(cid:12) ≫ ( T N ) − ε | a | , for N ≫ T / δ . Since a lower bound of this type was already proved in [BB], for brevity we only give asketch which could be made rigorous with more work. Blomer and Buttcane [BB, Section4] showed that the lower bound of size “ T N ” in (1.2) comes from the Eisenstein series E ( z, / it, u j ) induced from SL ( Z ) cusp forms u j . This Eisenstein series E has Heckeeigenvalues λ E (1 , n ) = λ ( n ) = X d d = n λ j ( d ) d − it d it , and Langlands parameters µ = (2 it, − it + it j , − it − it j ), where t j is the spectral parameterof u j . Moreover, ω F = T o (1) , so we will drop this aspect in the proof. Proof.
To simplify notation, say that B is the spectral ball of size O (1) centered at i (2 T, T, − T ).This means t = T + O (1) and t j = T + O (1). The contribution of this family of Eisensteinseries, on the dual side, takes the form (after smoothing)(7.2) S := X n w ( n/N ) (cid:12)(cid:12)(cid:12) Z t,t j = T + O (1) β t,t j λ ( n ) (cid:12)(cid:12)(cid:12) . Expanding the square, we obtain(7.3) S = Z t,t ′ ,t j ,t ′ j = T + O (1) ββ ′ πi Z (1+ ε ) N s e w ( s ) ∞ X n =1 λ ( n ) λ ′ ( n ) n s | {z } Z uj,u ′ j,t,t ′ ( s ) ds. Note(7.4) Z u j ,u ′ j ,t,t ′ ( s ) = X d d = e e λ j ( d ) d − it d it λ ′ j ( e ) e it ′ e − it ′ ( d d ) s = Y p (1 + p − s [ λ j ( p ) λ ′ j ( p ) p − it + it ′ + λ j ( p ) p − it − it ′ + λ ′ j ( p ) p it + it ′ + p it − it ′ ] + O u j ,u ′ j ( p − s )) . With some care, including use of the convexity bound for GL × GL L -functions due toIwaniec [I], one may then derive(7.5) Z u j ,u ′ j ,t,t ′ ( s ) = ζ ( s − it +2 it ′ ) L ( s + it +2 it ′ , u j ) L ( s − it − it ′ , u ′ j ) L ( s + it − it ′ , u j ⊗ u ′ j ) A ( s ) , where A ( s ) = A u j ,u ′ j ,t,t ′ ( s ) is given by an absolutely convergent Euler product for Re( s ) > / | A ( σ + iy ) | ≪ σ T ε , for σ > / Returning to (7.3), we shift contours to the line Re( s ) = 1 / ε . Note the pole of zetaat s = 1 + 2 it − it ′ which occurs for all pairs u j , u ′ j . This polar term, say denoted S ,contributes (roughly)(7.6) Z t,t ′ ,t j ,t ′ j = T + O (1) ββ ′ N it − it ′ e w (1+2 it − it ′ ) L (1+3 it, u j ) L (1 − it ′ , u ′ j ) L (1+3 it − it ′ , u j ⊗ u ′ j ) . If we choose β t,t j = L (1 + 3 it, u j ) − (alternatively, one could take β t,t j = L (1 + 3 it, u j )) thenthis polar term becomes approximately(7.7) N X t j ,t ′ j = T + O (1) L (1 , u j ⊗ u ′ j ) ≈ N T . With a bit more care, one can derive |S | ≫ ( N T ) − ε for this choice of β .Next we estimate the contribution to S from the new line of integration at Re( s ) = 1 / ε ;call this S ′ . Jutila and Motohashi [JM] showed a Weyl bound for the SL ( Z ) cusp forms,namely(7.8) L (1 / it, u j ) ≪ (1 + | t | + | t j | ) / ε . Combining this with the convexity bound for the GL × GL factor in (7.5) (which hasconductor of size T ) gives | Z ( σ + iy ) | ≪ T / ε . Note that the ζ -factor is evaluated at σ + iy with | y | ≪ T ε , so it practically gives no contribution here. Hence, for this choice of β , we have(7.9) S ′ ≪ N / ε T / ε Z t,t ′ ,t j ,t ′ j = T + O (1) | ββ ′ | ≪ N / ε T / ε T . Thus(7.10) |S| ≫ ( N T ) − ε + O ( N / ε T / ε T ) . Note the polar term dominates the error term provided N ≫ T / δ . Finally, we observethat(7.11) Z t,t j = T + O (1) | β t,t j | dt = T o (1) , whence |S| ≫ ( N T ) − ε R | β | with this choice of β . (cid:3) Loose ends
We list a few possible directions for future work.(1) Extend Theorem 1.2 to cover the family F V for more general V , with 1 ≪ V ≪ T .(2) Extend Theorem 1.1, which gives a bound on the norm ∆ using the Kuznetsovformula, to directly give a bound on the norm ∆ (modified to include the Eisensteinseries as well as the cusp forms). The point would be to bypass the use of Lemma3.2 in (5.1), though it is unclear if any improvement is possible this way.(3) Is it possible to use the SL ( Z ) Kuznetsov formula to directly bound the cuspidalpart of the spectrum in the large sieve inequality? (By subtracting off the Eisensteinparts, which one would presumably then control with lower-rank tools such as the GL Kuznetsov formula.)
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Department of Mathematics, Texas A&M University, College Station, TX 77843-3368,U.S.A.
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