aa r X i v : . [ m a t h . N T ] J a n A DENSITY THEOREM FOR
Sp(4)
SIU HANG MAN
Abstract.
Strong bounds are obtained for the number of automorphic forms for the group Γ ( q ) ⊆ Sp(4 , Z ) violating the Ramanujan conjecture at any given unramified place, which go beyond Sar-nak’s density hypothesis. The proof is based on a relative trace formula of Kuznetsov type, andbest-possible bounds for certain Kloosterman sums for Sp(4) . Introduction
In number theory, we often study families of objects, and this is especially the case in the contextof automorphic forms, and this methodology has been described formally in [Sar08, SST16]. On onehand, this can smooth out irregularities of individual members (which may exist, or whose proofof non-existence is beyond our reach) and allows the use of statistical concepts and deformationtechniques to investigate properties within a given family. On the other hand, strong analytic tools,such as various types of trace formulae, are available, to give non-trivial statements to the familiesas a whole.One of the key conjectures in the theory of automorphic forms is the Ramanujan conjecture,which states that cuspidal automorphic representations of the group
GL( n ) over a number field F are tempered. While the conjecture is still far out of reach even for the GL(2) case, one can thinkof generalisations to automorphic forms of other reductive groups, such as
Sp(2 n ) . It is well-knownthat the naive generalisation of the Ramanujan conjecture is false for the group Sp(4) , because of thepresence of Saito-Kurokawa lifts, which are not tempered. This is not the end of the investigation,however, because it is still an open question whether generic cuspidal automorphic representationsof
Sp(4) are tempered. While this is still far out of reach, we can consider approximations tothe conjecture as a substitute, and try to bound the number of members in a family violatingthe conjecture relative to the amount by which they violate the conjecture. This gives a densityresult, which is analogous to the ones given in [Blo19] for the
GL( n ) case. This does not prove theconjecture, but such density results often suffice in applications.In this paper we consider the family of generic cuspidal automorphic representations for the group Γ ( q ) ⊆ Sp(4 , Z ) of matrices whose lower left × block is divisible by q . Fix a place v of Q . For anautomorphic representation π = N v π v , we denote by µ π ( v ) = ( µ π ( v, , µ π ( v, its local Langlandsspectral parameter (see (2.2)), each entry viewed modulo πi log p Z if v = p is a prime. We write σ π ( v ) = max {|ℜ µ π ( v, | , |ℜ µ π ( v, |} . (1.1)The representation π is tempered at v if σ π ( v ) = 0 , and the size of σ π ( v ) gives a measure on howfar π is from being tempered at v . An example of a non-tempered representation is the trivialrepresentation, which satisfies σ triv ( v ) = 3 / for all places v .For a finite family F of automorphic representations of Sp(4) and σ ≥ we define N v ( σ, F ) = |{ π ∈ F | σ π ( v ) ≥ σ }| . Date : January 26, 2021.2020
Mathematics Subject Classification.
Key words and phrases. exceptional eigenvalues, Kloosterman sums, Kuznetsov formula, Ramanujan conjecture.The author is supported in part by DAAD Graduate School Scholarship Programme, and by the German ResearchFoundation under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.
Trivially, we have N v (0 , F ) = |F | , and if F contains the trivial representation, then we have N v (3 / , F ) ≥ . One may hope to interpolate linearly between the two extreme cases, and obtaina bound of the form N v ( σ, F ) ≪ v,ε |F | − σa + ε (1.2)with a = 3 / . In the context of groups G of real rank 1, for the principal congruence subgroup Γ( q ) = { γ ∈ G ( Z ) | γ = id (mod q ) } and v = ∞ , this is known as Sarnak’s density hypothesis[Sar90, p. 465].Density theorems have attracted much attention in the history, and many strong density resultsare known for various automorphic families on GL(2) with different settings [Hux86, Sar87, Iwa90,BM98, BM03, BBR14]. Via Kuznetsov-type trace formulae on
GL(3) , strong density results on
GL(3) were obtained in [Blo13, BBR14, BBM17]. Blomer [Blo19] further generalised the techniqueto obtain results in
GL( n ) beyond Sarnak’s density hypothesis. However, relatively little is knownfor general reductive groups. Finis-Matz [FM19] gives as by-products some density results for thefamily of Maaß forms of Laplace eigenvalue up to a height T and fixed level. However, the value of a is large, and is at least quadratic in r , the rank of the group, so that even the “convexity bound”cannot be obtained.More concretely, in this paper we consider the family F I ( q ) of generic cuspidal automorphicrepresentations for the group Γ ( q ) ⊆ Sp(4 , Z ) for a large prime q , and Laplace eigenvalue λ in afixed interval I . When the size of I is sufficiently large, we have |F I ( q ) | ≍ I q . For this family andany place v = q of Q , we go beyond the density hypothesis and obtain a = 3 / , which is halfwaybetween the density hypothesis and the Ramanujan conjecture. Theorem 1.1.
Let q be a prime, and v a place of Q different from q , I ⊆ [0 , ∞ ) a fixed interval, ε > , and σ ≥ . Then N v ( σ, F I ( q )) ≪ I,v,n,ε q − σ + ε . The proof is based on a careful analysis on the arithmetic side of the Kuznetsov formula, and on thespectral side through a relation of Fourier coefficients of automorphic forms and Hecke eigenvalues.Let λ ( m, π ) be the Hecke eigenvalue of π ∈ F I ( q ) for the m -th standard Hecke operator T ( m ) . Itis convenient to adopt the normalisation λ ′ ( m, π ) := m − / λ ( m, π ) . Theorem 1.2.
Keep the notations as above. Let m ∈ N be coprime to q and Z ≥ . Then X π ∈F I ( q ) (cid:12)(cid:12) λ ′ ( m, π ) (cid:12)(cid:12) Z σ π ( ∞ ) ≪ I,ε q ε uniformly in mZ ≪ q for a sufficiently small implied constant depending on I . Let us roughly sketch the proof of Theorem 1.2. We denote by { ̟ } an orthonormal basis of right K -invariant automorphic forms for Γ ( q ) , cuspidal or Eisenstein series, where K is the maximalcompact subgroup of Sp(4 , R ) . We denote by R ( q ) d̟ the integral over the complete spectrum of L (Γ ( q ) \ Sp(4 , R ) /K ) . Very roughly, the Kuznetsov formula takes the form Z ( q ) | A ̟ ( M ) | Z σ π ( ∞ ) δ λ ̟ ∈ I d̟ “ ≈ ” X id = w ∈ W X c ,c Kl q,w ( c, M, M ) c c , (1.3)where M = (1 , m ) ∈ Z , A ̟ ( M ) is the M -th Fourier coefficient of ̟ , defined in (5.1), W is theWeyl group of Sp(4) , and Kl q,w ( c, M, M ) is a generalised Kloosterman sum, defined in (4.2) below,associated with the Weyl element w , and moduli c = ( c , c ) . Note that the Kuznetsov formula onlyextracts the generic spectrum.However, the situation here is very different from GL( n ) case found in [Blo19]. In the symplecticcase, there are no simple relations between the Fourier coefficients A ̟ ( M ) of a cuspidal newform ̟ and Hecke eigenvalues λ ′ ( m, π ) of the corresponding automorphic representation (i.e. ̟ ∈ V π ). This DENSITY THEOREM FOR
Sp(4) is in stark contrast with the GL( n ) case, where the Fourier coefficients and Hecke eigenvalues areproportional [Gol06, Theorem 9.3.11]. It is because of this obstacle that the Kuznetsov formula isnot yet a standard tool for the group GSp(4) , and the present paper seems to be the first applicationof the Kuznetsov formula that is seen in action for a group other than
GL( n ) .While the Fourier coefficients in principle contain the information on Hecke eigenvalues, it is notobvious how to extract it. A detailed analysis of the relations between them is found in Section 6.In Theorem 6.2 we establish a recursive formula of λ ( p r , π ) in terms of Fourier coefficients. Wealso outline an algorithm for computing arbitrary Fourier coefficients of a cuspidal form in termsof its Hecke eigenvalues in the appendix. While this is not needed for the proof of the theorems,such results serve an independent interest in number theory, in laying the groundwork for furtherapplications of the Kuznetsov formula on Sp(4) , as well as Fourier analysis of automorphic formson
Sp(4) in general.Using Theorem 6.2, we deduce from Lemma 6.4 that for a prime p ∤ q and r ∈ N , the size of Fouriercoefficients A ̟ (1 , p r ) of an L -normalised generic cuspidal form ̟ is often as big as q − / − ε p rσ π ( p ) .Through this relation, we are able to use the Kuznetsov formula to derive information on σ π ( p ) froman analysis of the Kloosterman sums. Meanwhile, the factor Z σ π ( ∞ ) deals with the infinite place,so the test function | A ̟ ( M ) | Z σ π ( ∞ ) treats the finite places and the infinite place essentially onthe same footing.When mZ ≪ q , the Kloosterman sums associated to non-trivial Weyl elements are empty, hencethe off-diagonal terms vanish completely. We will use this observation to prove Theorem 1.3 below.To obtain stronger density results, we have to deal with the Kloosterman sums appearing in theoff-diagonal term, and improve the trivial bound | S q,w ( c, M, N ) | ≤ c c . Obtaining such bounds forgeneral groups remains a major open problem.Finally, we give an application of Theorem 1.2, for a large sieve inequality analogous to the GL( n ) case [Blo19]. Theorem 1.3.
Let q be prime and { α ( m ) } m ∈ N any sequence of complex numbers. Then X π ∈F I ( q ) (cid:12)(cid:12)(cid:12) X m ≤ x α ( m ) λ ′ ( m, π ) (cid:12)(cid:12)(cid:12) ≪ I,ε q X m ≤ x | α ( m ) | uniformly in x ≪ q for a sufficiently small implied constant depending on I . And as a corollary, we establish a bound for the second moment of spinor L-functions on thecritical line. Precisely, let L ( s, π ) be the spinor L-function associated to π , normalised such that itscritical strip is < ℜ s < . Corollary 1.4.
For q prime and t ∈ R , we have X π ∈F I ( q ) | L (1 / it, π ) | ≪ I,t,ε q ε . Acknowledgement
The author would like to thank Valentin Blomer for his guidance, and Edgar Assing for his helpfulexplanation on the subject. 2.
Preliminaries
Let U ⊆ Sp(4) be the standard unipotent subgroup U = x x x x x x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ij ∈ G a ,x = − x ,x = x + x x . SIU HANG MAN
Let V ⊆ Sp(4) be the group of diagonal matrices with entries ± . Let W be the Weyl group of Sp(4) , which is generated by the matrices s α = − − , s β = − as the elements in Sp(4) /V . We denote the long element of the Weyl group by w := s α s β s α s β .For w ∈ W , we define U w = w − U ⊤ w ∩ U . For N ∈ R we define a character θ N : U ( R ) → S by θ N ( x ) = e ( N x + N x ) . (2.1)Note that if N ∈ Z , this defines a character U ( R ) /U ( Z ) → S . If N = (1 , , we drop it from thenotation of the character.Let T ⊆ Sp(4) be the diagonal torus. The standard minimal parabolic subgroup is given by P = T U . We embed y = ( y , y ) ∈ R into T ( R ) via the map ι ( y ) = ( y y / , y / , /y y / , /y / ) . Wedenote the image of R in T ( R ) by T ( R + ) . An element g ∈ Sp(4 , R ) admits Iwasawa decomposition g = xyk , with x = U ( R ) , y ∈ T ( R + ) and k ∈ K , where K = SO(4 , R ) ∩ Sp(4 , R ) is the maximalcompact subgroup of Sp(4 , R ) . We denote by y( g ) = ι − ( y ) the Iwasawa y -coordinates of g . For w ∈ W , y ∈ R we write y w = y( wι ( y ) − w − ) .For α ∈ C , y ∈ R , we write y α = y α y α . Let η = (2 , / . We define measures dx = dx dx dx dx , d ∗ y = y − η dy y dy y on U ( R ) and R respectively. We denote the pushforward of d ∗ y to T ( R + ) by ι also by d ∗ y . Then dx is the Haar measure on U ( R ) , and dxd ∗ y is a left Sp(4 , R ) -invariant measure on Sp(4 , R ) /K .We define another embedding of R into T ( R + ) by c = ( c , c ) c ∗ = diag(1 /c , c /c , c , c /c ) . A simple calculation shows that y( c ∗ ) η = ( c c ) − .Let π = N π v be a globally generic irreducible spherical representation of GSp(4) with trivialcentral character. Using notations in [RS07], π v is induced from the character χ × χ ⋊ σ , given by diag (cid:0) t , t , t − v, t − v (cid:1) χ ( t ) χ ( t ) σ ( v ) . As π v is right K v -invariant, we may assume that χ , χ , σ are unramified, and we may write χ = | · | α , χ = | · | α and σ = | · | β . As π v has trivial central character, we have α + α + 2 β = 0 . Sothe L-parameter is given by ( χ χ σ, χ σ, χ σ, σ ) = (cid:18) α + α , α − α , − α + α , − α − α (cid:19) . We then take µ π ( v ) = (cid:18) α + α , α − α (cid:19) , (2.2)so the L-parameter becomes ( µ π ( v, , µ π ( v, , − µ π ( v, , − µ π ( v, . When π v is lifted to a self-dualrepresentation of GL(4) , this is precisely the natural Langlands parameter of the lift.
DENSITY THEOREM FOR
Sp(4) Auxiliary results
Since the Iwasawa decomposition
Sp(4 , R ) = U ( R ) T ( R + ) K is actually the Gram-Schmidt orthog-onalisation of rows, we can compute y( g ) explicitly. Let ∆ be the norm of the third row of g , and ∆ be the area of the parallelogram spanned by the bottom two rows of g . Then we have g ≡ / ∆ ∗ ∗ ∗ ∆ / ∆ ∗ ∗ ∆ ∗ ∆ / ∆ (mod K ) . In particular, we have y( g ) = (∆ / ∆ , ∆ / ∆ ) . Conversely, if y( g ) = ( Y , Y ) , then ∆ ( g ) = Y − Y − / and ∆ ( g ) = Y − Y − . Lemma 3.1.
Let w ∈ W , x ∈ U w ( R ) , and y, c, B ∈ R . Write y( ι ( B ) c ∗ wxι ( y )) = Y ∈ R and A = ι ( B ) c ∗ . Then we have c ≪ y,Y B B / , c ≪ y,Y B B , and ≤ ∆ ( wx ) ≪ y,Y y( A ) y( A ) / , ≤ ∆ ( wx ) ≪ y,Y y( A ) y( A ) . Proof.
Note that ∆ i ( wx ) ≥ as one of its minors is always 1. For the first statement, we compute ∆ ( ι ( B ) c ∗ wxι ( y )) = c B B / ∆ ( wxι ( y )) , ∆ ( ι ( B ) c ∗ wxι ( y )) = c B B ∆ ( wxι ( y )) . Then we obtain c ≤ c ∆ ( wx ) ≪ y c ∆ ( wxι ( y )) = ∆ ( ι ( B ) c ∗ wxι ( y )) B B / ≪ Y B B / ,c ≤ c ∆ ( wx ) ≪ y c ∆ ( wxι ( y )) = ∆ ( ι ( B ) c ∗ wxι ( y )) B B ≪ Y B B . For the second statement, we observe that c − = y( c ∗ ) y( c ∗ ) / , c − = y( c ∗ ) y( c ∗ ) . Hence ∆ ( wx ) ≪ y,Y B B / c − = ( B y( c ∗ )) ( B y( c ∗ )) / = y( A ) y( A ) / , ∆ ( wx ) ≪ y,Y B B c − = ( B y( c ∗ )) ( B y( c ∗ )) = y( A ) y( A ) , finishing the proof. (cid:3) Lemma 3.2.
Let N ∈ N and w ∈ W . For x ∈ U w ( R ) , define x ′ = ι ( N ) xι ( N ) − . Then dx ′ dx = ( N w ) η N η . Proof.
By direct computation. (cid:3)
Lemma 3.3.
Let B ∈ R , and w = s β s α s β . Then vol { x ∈ U w ( R ) | ∆ j ( wx ) ≤ B j , j = 1 , } ≪ ( B B ) ε for any ε > .Proof. We can assume without loss of generality that B j ≥ , otherwise we deduce from Lemma 3.1that the volume is 0. We have x = x x x x ∈ U w ( R ) wx = − x x − − x − x . SIU HANG MAN
Then we obtain bounds | x | ≤ B , (cid:12)(cid:12) x x − x (cid:12)(cid:12) ≤ B . We also have | x | , | x | ≤ b := 1 + max { B , B } . If I ⊆ R is any interval of length | I | ≥ , then vol (cid:8) ( x, y ) ∈ [ − b, b ] (cid:12)(cid:12) xy ∈ I (cid:9) ≤ Z | y |≤ b min (cid:26) | I || y | , b (cid:27) dy ≤ | I | (1 + log b ) . Hence, if | x | ≤ B is fixed, the volume of ( x , x ) is O ( B log b ) . This establishes the bound. (cid:3) Sp(4)
Kloosterman sums
Properties of Kloosterman sums for
Sp(4 , Z ) were given in [Man20]. They generalise in a naturalway to the congruence subgroup Γ ( q ) . The Bruhat decomposition gives Sp(4 , Q ) = ` w ∈ W G w ( Q ) with G w = U wT U w as a disjoint union. Let M, N, c ∈ N , w ∈ W . Then if θ M ( wc ∗ x ( c ∗ ) − w − ) = θ N ( x ) (4.1)for all x ∈ U ∩ w − U w , the Kloosterman sum Kl q,w ( c, M, N ) := X xwc ∗ x ′ ∈ U ( Z ) \ G w ( Q ) ∩ Γ ( q ) /U w ( Z ) θ M ( x ) θ N ( x ′ ) (4.2)is well-defined [Man20, Proposition 5.1]. If (4.1) does not hold, we set Kl q,w ( c, M, N ) = 0 . TheKloosterman sum Kl q,w ( c, M, N ) is nonzero only if w = id , s α s β s α , s β s α s β , w [Man20].Now suppose the entries of M = ( M , M ) and N = ( N , N ) are coprime to q . Considering theBruhat decomposition of Γ ( q ) , we deduce that the Kloosterman sum Kl q,w ( c, M, N ) is nonemptyonly if q | c for w = s α s β s α , q | c and q | c for w = s β s α s β , w . (4.3)Meanwhile, the well-definedness condition (4.1) says that the Kloosterman sums are well-definedprecisely if N = M c c if w = s α s β s α , N = M c c if w = s β s α s β . Hence the Kloosterman sums are well-defined only if v q ( c ) = v q ( c ) if w = s α s β s α , v q ( c ) = 2 v q ( c ) if w = s β s α s β . (4.4)From the abstract definition [Ste87, Man20], the Kloosterman sums Kl q,w ( c, M, N ) also enjoycertain multiplicativity in the moduli. We state one particular case. Let q be prime. For c =( c , c ) ∈ N , let c ′ j = q − v q ( c j ) c j , j = 1 , , and c ′ = ( c ′ , c ′ ) . Then we have Kl q,w ( c, M, N ) = Kl q,w (cid:16) ( q v q ( c ) , q v q ( c ) (cid:17) , M ′ , N ′ ) Kl ,w ( c ′ , M ′′ , N ′′ ) (4.5)for some M ′ , N ′ , M ′′ , N ′′ ∈ N . Moreover, if the entries of M , N are coprime to q , then so are M ′ , N ′ . From [DR98], we have a trivial bound Kl ,w ( c ′ , M ′′ , N ′′ ) ≤ | U ( Z ) \ G w ( Q ) ∩ Sp(4 , Z ) /U w ( Z ) | ≤ c ′ c ′ . (4.6) DENSITY THEOREM FOR
Sp(4) Evaluations of Kloosterman sums.
For the proof of the theorems in Section 8, we computethe following Kloosterman sums: Kl q,s α s β s α (( q, q ) , M, N ) , Kl q,s β s α s β (cid:0) ( q, q ) , M, N (cid:1) , Kl q,w (cid:0) ( q, q ) , M, N (cid:1) , Kl q,w (cid:0) ( q, q ) , M, N (cid:1) . (i) Consider the Bruhat decomposition for summands in Kl q,s α s β s α (( q, q ) , M, N ) : γ = β β β β β − β − q − q v /q v /q v /q v /q − v /q = β q β v + β β v − β v /q + β v /q − /q β v + β β q β v + 1 β v − β v /q + v /q β v + β q v v v − β q − β v − β v − v /q − β v + 1 ∈ Γ ( q ) , with v , v , v (mod q ) chosen such that ( v , v , ( q, v )) = 1 , and (cid:0) ( q, v ) , qv + v v (cid:1) = q .As γ ∈ Γ ( q ) , by considering the lower left block, we deduce that v = 0 , and solve β ≡ . The conditions on v , v then simplify as ( q, v ) = 1 . Considering the second row,we solve β ≡ − v v q (mod 1) , β ≡ v v q (mod 1) . So the Kloosterman sum is given by Kl q,s α s β s α (( q, q ) , M, N ) = X v (mod q )( v ,q )=1 X v (mod q ) e (cid:18) M v v q (cid:19) = 0 . (ii) Consider the Bruhat decomposition for summands in Kl q,s β s α s β (cid:0) ( q, q ) , M, N (cid:1) : γ = β β β β β − β − q − q − q − q − v /q v /q v /q v /q = − β q β q β v /q + β /q + β v /q β v /q − β v /q − /q − β q β q β v /q + β v /q + 1 /q β v /q − β v /q q v /q v /q − q − β q − β v /q + v /q − β v /q − v /q ∈ Γ ( q ) , with v , v , v (mod q ) chosen such that ( q , v , v ) = q , and ( q, v , v ) = 1 , where v = − v + v v q . As γ ∈ Γ ( q ) , by considering the lower left block, we solve β ≡ . Then, − β v /q + v /q being an integer implies q | v , so ( q, v ) = 1 . Write v = qv ′ , v = qv ′ , and β = β ′ /q , β = β ′ /q for some β ′ , β ′ ∈ Z . By considering thesecond row, we deduce that β ′ v ′ + β ′ v /q + 1 , β ′ v ′ − β ′ v ′ ∈ q Z , from which we deduce β ′ ≡ v ′ v (mod q ) , and β ≡ v ′ v q (mod 1) . Writing v = qv ′ ,the Kloosterman sum is given by Kl q,s β s α s β (cid:0) ( q, q ) , M, N (cid:1) = X v ′ ,v ′ ,v ′ (mod q )( q,v ′ ,v ′ )=1( q,v )=1 e (cid:18) M v ′ v + N v ′ q (cid:19) , SIU HANG MAN where v = − ( v ′ + v ′ v ′ ) . We evaluate X v ′ ,v ′ (mod q )( q,v ′ )=1 X v ′ ,v ′ ,v ′ (mod q )( q,v ′ )=1 , ( q,v )=1 e (cid:18) M v ′ v + N v ′ q (cid:19) = q ( q − − X v ′ ,v ′ (mod q )( q,v ′ )=1 e (cid:18) N v ′ q (cid:19) = q . (iii) Consider the Bruhat decomposition for summands in Kl q,w (cid:0) ( q, q ) , M, N (cid:1) : γ = β β β β β − β − q − − q − q q v /q v /q v /q v /q v /q − v /q = β q β v + β q β v + β v /q + β v /q − /q β v − β /q + β v /qβ q β v + β q β v + β v /q + v /q β v + β v /q − /qq v v v − β q − β v + q − β v + v /q − β v + v /q ∈ Γ ( q ) , with v , v , v (mod q ) , v , v (mod q ) chosen such that v q + v v − v q = 0 , ( q, v , v , v ) =1 , and ( q , v , v , v , v ) = 1 , where v = v v − v q q and v = v v − v v q . As γ ∈ Γ ( q ) ,by considering the lower left block, we deduce that v = 0 , and solve β ≡ . Thelast row being integers implies that q | v , v . Write v = qv ′ , v = qv ′ . The relation v q + v v − v q = 0 says v ′ = v . We check that q | v as well, so ( q, v ) = 1 . Write β = β ′ /q , β = β ′ /q for some β ′ , β ′ ∈ Z . By considering the second row, we deduce that β ′ v + β ′ v ′ , β ′ v + β ′ v ′ − ∈ q Z , from which we deduce β ′ ≡ v v (mod q ) , and β = v v q (mod 1) . The Kloosterman sumis given by Kl q,w (cid:0) ( q, q ) , M, N (cid:1) = X v ,v ,v ′ (mod q )( q,v ,v )=1( q,v )=1 e (cid:18) M v v + N v ′ q (cid:19) , where v = v v ′ − v .Fix v , v ′ = 0 . As v = 0 varies, v v ≡ v ′ − v v runs through nonzero residuesexcept v ′ modulo q ; hence, as v varies, v v runs through all residues except v ′ modulo q . Hence X v ,v ,v ′ (mod q )( q,v )=1 , ( q,v ′ )=1( q,v )=1 e (cid:18) M v v + N v ′ q (cid:19) = − X v ,v ′ (mod q )( q,v )=1( q,v ′ )=1 e M v ′ + N v ′ q ! = − ( q − S ( M , N ; q ) . If v = 0 and v ′ = 0 , then v = − v . The corresponding part of the sum becomes X v ,v (mod q )( q,v )=1 e (cid:18) − M v v q (cid:19) = 0 . Meanwhile, for v = 0 , we have v ′ = 0 , and v = v v ′ , so v v = v ′ . Hence this part ofthe sum is X v ,v ′ (mod q )( q,v )=1( q,v ′ )=1 e M v ′ + N v ′ q ! = ( q − S ( M , N ; q ) . Combining the parts above, we conclude that Kl q,w (cid:0) ( q, q ) , M, N (cid:1) = 0 . DENSITY THEOREM FOR
Sp(4) (iv) Consider the Bruhat decomposition for summands in Kl q,w (cid:0) ( q, q ) , M, N (cid:1) : γ = β β β β β − β − q − − q − q q v /q v /q v /q v /q v /q − v /q = β q β v + β q β v + β v /q + β v /q − /q β v − β /q + β v /qβ q β v + β q β v + β v /q + v /q β v + β v /q − /q q v v v − β q − β v + q − β v + v /q − β v + v /q ∈ Γ ( q ) , with v , v , v (mod q ) , v , v (mod q ) chosen such that v q + v v − v q = 0 , ( q, v , v , v ) =1 , and ( q , v , v , v , v ) = 1 , where v = v v − v q q and v = v v − v v q . As γ ∈ Γ ( q ) ,by considering the lower left block, we deduce that v = 0 , and solve β ≡ . Thelast row being integers implies that q | v , v . Write v = qv ′ , v = qv ′ . The relation v q + v v − v q = 0 says v ′ = v q . We check that q | v as well, so ( q, v ) = 1 . Write β = β ′ /q , β = β ′ /q for some β ′ , β ′ ∈ Z . By considering the second row, we deduce that β ′ v q + β ′ v ′ , β ′ v q + β ′ v ′ − ∈ q Z , from which we deduce β ′ ≡ v v (mod q ) , and β = v v q (mod 1) . The Kloostermansum is given by Kl q,w (cid:0) ( q, q ) , M, N (cid:1) = X v ,v (mod q ) , v ′ (mod q )( q,v ,v )=1 , ( q,v )=1 e (cid:18) M v v + N v ′ q (cid:19) , where v = v v ′ − v q .Fix v = 0 . Then from ( q, v ) = 1 we have ( q, v ′ ) = 1 , and v = 0 . For a fixed v ′ , wesee that as v varies, v v ≡ v ′ − v v q runs through nonzero residues modulo q that arecongruent to v ′ (mod q ) , except v ′ ; hence, as v varies, v v runs through all residuesmodulo q that are congruent to v ′ (mod q ) , except v ′ . Hence X v ,v (mod q ) v ′ (mod q )( q,v )=1 , ( q,v )=1( q,v ′ )=1 , ( q,v )=1 e (cid:18) M v v + N v ′ q (cid:19) = − X v (mod q ) v ′ (mod q )( q,v )=1 , ( q,v ′ )=1 e M v ′ + N v ′ q ! = − ( q − S ( M , N ; q ) . Meanwhile, for v = 0 , we have ( q, v ′ ) = 1 , and v = v v ′ , so v v = v ′ . Hence thispart of the sum is X v (mod q ) v ′ (mod q )( q,v )=1 , ( q,v ′ )=1 e M v ′ + N v ′ q ! = ( q − S ( M , N ; q ) . Combining the parts above, we conclude that Kl q,w (cid:0) ( q, q ) , M, N (cid:1) = 0 .5. Automorphic forms and Whittaker functions
We denote by { ̟ } an orthonormal basis of right K -invariant automorphic forms for Γ ( q ) , cuspidalor Eisenstein series. The space L (Γ ( q ) \ Sp(4 , R ) /K ) is equipped with the standard inner product h f, g i = Z Γ ( q ) \ Sp(4 , R ) /K f ( xy ) g ( xy ) dxd ∗ y. An integral over the complete spectrum of L (Γ ( q ) \ Sp(4 , R ) /K ) is denoted by R ( q ) d̟ . All theautomorphic forms ̟ belong to representations π of level q ′ | q , and we assume that { ̟ } containsall cuspidal newvectors of level q ′ | q . For simplicity in notations, we denote the local archimedeanspectral parameter µ π ( ∞ ) by µ = ( µ , µ ) .Let ̟ be an automorphic form for Γ ( q ) , with spectral parameter µ . We suppose ̟ is genericthroughout the section. For M = ( M , M ) ∈ Z , the M -th Fourier coefficient of ̟ is given by ̟ M ( g ) = Z U ( Z ) \ U ( R ) ̟ ( xg ) θ M ( x ) dx. The Fourier coefficients ̟ M ( g ) are actually Whittaker functions. For g = xyk ∈ Sp(4 , R ) , we have ̟ M ( g ) = A ̟ ( M ) M η θ M ( x ) · W µ ( ι ( M ) y ) , (5.1)where A ̟ ( M ) ∈ C is a constant, called the M -th Fourier coefficient of ̟ , W µ : R → C is thestandard Whittaker function on Sp(4 , R ) , with detailed descriptions found in [Ish05]. As in the GL( n ) case, the size of σ π ( ∞ ) captures the growth of W µ near the origin. Precisely, for a function E on R and X ∈ R , we define E ( X ) ( y , y ) = E ( X y , X y ) . (5.2) Lemma 5.1.
Assume that µ = ( µ , µ ) varies in some compact set Ω , and let Z ≥ . There exists r ∈ N and a compact set S ⊆ R depending only on Ω (independent of Z ), and a finite collectionof functions E , · · · , E r : R → R depending on Ω and Z that are uniformly bounded and supportedin a compact subset of S such that r X j =1 (cid:12)(cid:12)(cid:12)D E (1 ,Z ) j , W µ E(cid:12)(cid:12)(cid:12) ≫ Ω Z η +2 σ π ( ∞ ) = Z σ π ( ∞ ) . Proof.
The case Z ≪ is proved in [BBM17, Blo19]. 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 . Now choose open neighbourhoods U µ about µ such that ℜ W µ ∗ ( y ) = 0 for all y ∈ S µ and µ ∗ ∈ U µ , or ℑ W µ ∗ ( y ) = 0 for all y ∈ S µ and µ ∗ ∈ U µ . By compactness, Ω is covered by a finite collection of neighbourhoods U µ , · · · , U µ r , andwe may pick corresponding E j to be real-valued functions with supports on S µ j and non-vanishingon the interior S ◦ µ j .Now suppose Z ≫ is sufficiently large. Consider the following renormalisation of the Whittakerfunction: W ∗ µ ( y ) := y − η W µ ( y ) . (5.3)The Mellin transform M ∗ µ ( s ) = R R W ∗ µ ( y ) y s dy y dy y is given by [Ish05] (where ν , ν in [Ish05] are µ + µ and µ − µ in our notation) M ∗ µ ( s ) =2 − Γ (cid:18) s + µ + µ (cid:19) Γ (cid:18) s + µ − µ (cid:19) Γ (cid:18) s − µ + µ (cid:19) Γ (cid:18) s − µ − µ (cid:19) Γ (cid:18) s + µ (cid:19) Γ (cid:18) s − µ (cid:19) Γ (cid:18) s + µ (cid:19) Γ (cid:18) s − µ (cid:19)(cid:26) Γ (cid:18) s + s + µ (cid:19) Γ (cid:18) s + s − µ (cid:19)(cid:27) − F (cid:18) s , s + µ , s − µ s + s + µ , s + s − µ (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . By Weyl group symmetry, we may assume without loss of generality that σ π ( ∞ ) = ℜ µ . For ℜ ( s ) sufficiently large, M ∗ µ ( s ) is holomorphic for ℜ ( s ) > σ π ( ∞ ) , as poles can only occur at s = ± µ − k , ± µ − k for k ∈ N . Hence, for ℜ ( s ) sufficiently large, the function M † µ ( s ) := M ∗ µ ( s )( s + µ )( s − µ )( s + µ )( s − µ ) DENSITY THEOREM FOR
Sp(4) is holomorphic for ℜ ( s ) > σ π ( ∞ ) − .For β ∈ C , let D β = − y ∂ y + β . This is a commutative family of differential operators, whichcorrespond to multiplication by s + β under Mellin transform. Now let ˆ M µ ( s ) := M † µ ( s ) s − µ = M ∗ µ ( s )( s + µ )( s + µ )( s − µ ) . Taking inverse Mellin transforms, we get ˆ W µ ( y ) = D µ D µ D − µ W ∗ µ ( y ) . On the other hand, we compute the inverse Mellin transform directly, and by shifting the contourto ℜ ( s ) = σ π ( ∞ ) − , we obtain the estimate ˆ W ∗ µ ( y ) = y − µ W ∗∗ µ ( y ) + O y ,µ ( y − σ π ( ∞ )+ ) for y → , where W ∗∗ µ ( y ) = Γ ( µ + 1) Γ (cid:18) µ + µ (cid:19) Γ (cid:18) µ − µ (cid:19) W ∗ µ ( y ) y − µ , where W ∗ µ ( y ) = y − / W µ ( y ) is a normalised GL(2) -Whittaker function.The rest of the proof follows the argument in [Blo19]. We see that if D β w ( y ) ∼ cy − a as y → for some constants a ≥ , β, c ∈ C and Z ≥ is sufficiently large, then there exist constants < γ < γ < (depending on all parameters, but uniformly bounded away from 0 when β, c, a vary in a fixed compact set and Z is sufficiently large) such that | w ( y ) | ≥ cy − a for y ∈ [ γ /Z, γ /Z ] .Iterating this argument, and adjust the constants γ , γ if necessary, we deduce that (cid:12)(cid:12) W ∗ µ ( y ) (cid:12)(cid:12) ≫ y − σ π ( ∞ )2 (cid:12)(cid:12) W ∗∗ µ ( y ) (cid:12)(cid:12) for y ∈ [ γ /Z, γ /Z ] , when y and µ vary in some fixed compact domain. Now choose functions E ∗ j : R + → C , depending on Ω but not Z , such that P j (cid:12)(cid:12)(cid:12)D E ∗ j , W ∗∗ µ E(cid:12)(cid:12)(cid:12) ≫ for µ ∈ Ω . Now define E j ( y , y ) = δ γ ≤ y ≤ γ E ∗ j ( y ) . This choice depends on Z , but the support of E varies inside someinterval depending only on Ω . We then obtain X j (cid:12)(cid:12)(cid:12)D E (1 ,Z ) j , W ∗ µ E(cid:12)(cid:12)(cid:12) ≫ Z σ π ( ∞ ) . Using the relation (5.3), we obtain the lemma. (cid:3) Hecke eigenvalues and Fourier coefficients
Let M be a set of matrices in GSp(4 , Q ) + that is left- and right-invariant under Γ = Sp(4 , Z ) and is a finite union S j Γ M j of left cosets. Then M defines a Hecke operator T M on the space ofcuspidal automorphic forms by T M ̟ ( g ) = X j ̟ ( M j g ) . For a matrix g ∈ GSp(4 , Q ) + , we denote by T g the Hecke operator T Γ g Γ . For m ∈ N , let S ( m ) : = n M ∈ GSp(4 , Z ) + (cid:12)(cid:12)(cid:12) M ⊤ J M = mJ o , J = (cid:18) I − I (cid:19) . The m -th standard Hecke operator is then given by T ( m ) = T S ( m ) . The set of matrices H ( m ) = ((cid:18) A m − BDD (cid:19) ∈ S ( m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A = (cid:18) a aa (cid:19) , B = (cid:18) b b b b (cid:19) , a , a > , ≤ a < a , ≤ b i < m, AD ⊤ = mI ,BD ≡ m ) ) . (6.1)gives a complete system of left coset representatives for Γ \ S ( m ) [Spe72]. For r ∈ N , ≤ a ≤ b ≤ r/ and any prime p , define T ( r ) a,b ( p ) := T diag( p a ,p b ,p r − a ,p r − b ) . When the context is clear, we suppress p from the notation, and write T ra,b instead. Then T ( p r ) admits a decomposition T ( p r ) = X ≤ a ≤ b ≤ r/ T ( r ) a,b ( p ) . It is well-known that the Hecke algebra H of Sp(4 , R ) is generated by T ( p ) = T (1)0 , ( p ) and T (2)0 , ( p ) for primes p , along with the identity.We also define involutions T ε , T ε on the space of cuspidal automorphic forms by T ε ̟ ( g ) = ̟ ( ε g ) , T ε ̟ (cid:18)(cid:18) Y X ( Y − ) ⊤ (cid:19)(cid:19) = ̟ (cid:18)(cid:18) Y − X ( Y − ) ⊤ (cid:19)(cid:19) , where ε = diag( − , , − , . It is clear that ( T ε ̟ ) ( M ,M ) ( g ) = ̟ ( − M ,M ) ( g ) , ( T ε ̟ ) ( M ,M ) ( g ) = ̟ ( M , − M ) ( g ) . (6.2)It is also straightforward to check that T ε , T ε commute with the Hecke operators and the invariantdifferential operators. So we may assume a cuspidal automorphic form ̟ is also an eigenfunctionof T ε , T ε .Let π be the irreducible automorphic representation corresponding to ̟ . We write λ ( m, π ) and λ ( r ) a,b ( p, π ) to denote the eigenvalue of ̟ with respect to T ( m ) and T ( r ) a,b respectively, and write λ ′ ( m, π ) := m − / λ ( m, π ) . Again, we omit π from the notation when the context is clear.It is known that if ̟ is generic and L -normalised, then by [CI19, Theorem 1.1] and [Li10,Theorem 3], we have | A ̟ (1 , | ≍ µ , Z ) : Γ ( q )] L (1 , π, Ad) ≫ µ q − − ε . (6.3)In particular, A ̟ (1 , = 0 . Notation.
Let ̟ be an L -normalised generic cuspidal newform. For the rest of the section, it ishowever instructive to have an alternative normalisation, such that the (1 , -st Fourier coefficient is1. To avoid confusion, we always denote by ̟ an L -normalised form, and by ̟ a scalar multipleof ̟ such that A ̟ (1 ,
1) = 1 . From (6.3), we see that ̟ = k̟ for some | k | ≪ q ε .Now fix a prime p ∤ q . Let M = ( M , M ) , and ≤ c, d ≤ r such that p d − c | M and p r − d | M .Write Γ diag( p a , p b , p r − a , p r − b )Γ = [ i Γ h i as a finite union of left cosets. We can assume that h i ∈ U ( Q ) T ( Q + ) . Consider the decomposition h i = ˆ y i ˆ x i , with ˆ y i ∈ T ( Q + ) , ˆ x i ∈ U ( Q + ) . We define exponential sums S ( r ) a,b,M ( c, d ) := X Γ h i ⊆ Γ diag( p a ,p b ,p r − a ,p r − b )Γˆ y i =diag( p c ,p d ,p r − c ,p r − d ) θ M (ˆ x i ) , DENSITY THEOREM FOR
Sp(4) and S ( r ) ( c, d ) := X ≤ a,b ≤ r/ S ( r ) a,b, (1 , ( c, d ) = X Γ h i ⊆ S ( p r )ˆ y i =diag( p c ,p d ,p r − c ,p r − d ) θ (ˆ x i ) . Proposition 6.1.
We have λ ( r ) a,b ( p ) A ̟ ( M ) = X ≤ c,d ≤ rp c − d | M ,p d − r | M S ( r ) a,b,M ( c, d ) p c + d − r A ̟ ( M p d − c , M p r − d ) . Proof.
We compute the Fourier coefficient of T ( r ) a,b ̟ in two ways. On one hand, we have Z U ( Z ) \ U ( R ) T h ̟ ( xy ) θ M ( x ) dx = λ ( r ) a,b ( p ) A ̟ ( M ) M η W µ ( ι ( M ) y ) . (6.4)On the other hand, we expand the Hecke operator Z U ( Z ) \ U ( R ) T h ̟ ( xy ) θ M ( x ) dx = X Γ h i Z U ( Z ) \ U ( R ) ̟ ( h i xy ) θ M ( x ) dx = p − r X Γ h i Z U ( p r Z ) \ U ( R ) ̟ ( h i xy ) θ M ( x ) dx. Write h i x = x ′ ˆ y i , with x ′ ∈ U ( R ) , and ˆ y i = diag( c , · · · , c ) . A simple calculation shows that x ′ k,l = c l X j ( h i ) kj x jl . In particular, we have x = c c x ′ − ( h i ) c = c c x ′ − (ˆ x i ) , x = c c x ′ − ( h i ) c = c c x ′ − (ˆ x i ) . Making this substitution, the expression becomes p − r X Γ h i Y k,l Z ckcl p r + P j ( h i ) kj x jl P j ( h i ) kj x jl ̟ ( x ′ ˆ y i y ) e ( M (ˆ x i ) + M (ˆ x i ) ) e (cid:18) − c c M x ′ − c c M x ′ (cid:19) c l c k dx ′ k,l , where ( k, l ) runs through the indices (1 , , (1 , , (2 , , (2 , . By periodicity, we shift the integraland get p − r X Γ h i Y k,l Z ckcl p r ̟ ( x ′ ˆ y i y ) e ( M (ˆ x i ) + M (ˆ x i ) ) e (cid:18) − c c M x ′ − c c M x ′ (cid:19) c l c k dx ′ k,l , Since ̟ ( x ′ ˆ y i y ) is 1-periodic with respect to x ′ kl , this integral vanishes unless c | c M and c | c M .We sum over the terms with the same ˆ y i = ( p c , p d , p r − c , p r − d ) and get X ≤ c,d ≤ rp c − d | M ,p d − r | M S ( r ) a,b,M ( c, d ) Z U ( Z ) \ U ( R ) ̟ ( x ′ ˆ y i y ) e (cid:16) − p d − c M x ′ − p r − d M x ′ (cid:17) dx ′ . Evaluating the integral gives X ≤ c,d ≤ rp c − d | M ,p d − r | M S ( r ) a,b,M ( c, d ) p c + d − r A ̟ ( p d − c M , p r − d M ) M η W µ ( ι ( M ) y ) . (6.5)Comparing (6.4) and (6.5) gives the result. (cid:3) Theorem 6.2.
Let ̟ ∈ V π be a cuspidal newform such that A ̟ (1 ,
1) = 1 , and p ∤ q a prime. TheHecke eigenvalues λ ( p r , π ) of π with respect to T ( p r ) are given by λ ( p, π ) = p / A ̟ (1 , p ) ,λ ( p r , π ) = p r/ (cid:0) A ̟ (1 , p r ) − p − A ̟ (1 , p r − ) (cid:1) , r ≥ . Proof.
Plugging in M = (1 , to Proposition 6.1 gives λ ( p r ) A ̟ (1 ,
1) = X ≤ a,b ≤ r/ λ ( r ) a,b ( p ) A ̟ (1 ,
1) = X ≤ c ≤ d ≤ r/ S ( r ) ( c, d ) p c + d − r A ̟ ( p d − c , p r − d ) . (6.6)We evaluate S ( r ) ( c, d ) explicitly. We set A a := (cid:18) p c ap d (cid:19) , and partition the sum S ( r ) ( c, d ) = X ≤ a
Sp(4) Hence we conclude S ( r ) ( c, d ) = p r if ( c, d ) = (0 , , − p r − if ( c, d ) = (1 , , otherwise.Putting this back into (6.6) gives the statement. (cid:3) Hecke eigenvalues can also be expressed in terms of local Satake parameters α p , β p associated to π .Without loss of generality, assume | α p | ≥ | β p | ≥ . Then up to some ordering we have p µ π ( p, = α p , p µ π ( p, = β p , and σ π ( p ) = µ π ( p, . By an identity of Shimura [Shi63, Theorem 2], we have ∞ X r =0 λ ( p r ) x r = (1 − p x )(1 − p / α p x ) − (1 − p / α − p x ) − (1 − p / β p x ) − (1 − p / β − p x ) − . (6.8)For convenience, we define σ + π ( p ) = + σ π ( p ) . Lemma 6.3.
For a prime p ∤ q and r ≥ we have max ≤ j ≤ (cid:12)(cid:12) λ ( p r − j ) (cid:12)(cid:12) ≥ p rσ + π ( p ) . Proof.
We derive from (6.8) that (1 − p / α − p x )(1 − p / β p x )(1 − p / β − p x ) ∞ X r =0 λ ( p r ) x r = (1 − p x ) ∞ X r =0 (cid:0) p / α p (cid:1) r x r . Comparing coefficients gives λ ( p r ) − λ ( p r − ) p / (cid:0) α − p + β p + β − p (cid:1) + λ ( p r − ) p ( α − p β p + α − p β − p + 1) + λ ( p r − ) p / α − p = p r/ ( α rp − p − α r − p ) . Assume the contrary. Then the left hand side is bounded by p rσ + π ( p ) ≤ p rσ + π ( p ) − p r − σ + π ( p ) ≤ p r/ (cid:12)(cid:12) α rp − p − α r − p (cid:12)(cid:12) , a contradiction. (cid:3) Lemma 6.4.
Let ̟ ∈ V π be a cuspidal newform such that A ̟ (1 ,
1) = 1 , p ∤ q a prime, and r ∈ N . Then the inequality | A ̟ (1 , p r ) | ≥ p rσ π ( p ) holds for some r ≤ r ≤ r + 5 .Proof. By Lemma 6.3, we have | λ ( p r ) | ≥ p rσ + π ( p ) for some r + 2 ≤ r ≤ r + 5 . By Theorem 6.2, we have p r/ (cid:0) | A ̟ (1 , p r ) | + p − (cid:12)(cid:12) A ̟ (1 , p r − ) (cid:12)(cid:12)(cid:1) ≥ p rσ + π ( p ) , and the statement follows. (cid:3) Poincaré series and the Kuznetsov formula
Let E : R → C be a fixed function with compact support, X ∈ R a “parameter”. We define E ( X ) ( y , y ) = E ( X y , X y ) , and a right K -invariant function F ( X ) : Sp(4 , R ) → C by F ( X ) ( xy ) = θ ( x ) E ( X ) (y( y )) (7.1)for x ∈ U ( R ) and y ∈ T ( R + ) , where θ = θ (1 , is as in (2.1). For N ∈ N , we define the Poincaréseries of level q to be P ( X ) N ( xy ) = X γ ∈ U ( Z ) \ Γ ( q ) F ( X ) ( ι ( N ) γxy ) . Note that F ( X ) ( ι ( N ) xy ) = θ N ( x ) E ( X ) ( N y( y )) = θ N ( x ) E ( XN y( y )) . For w ∈ W , let G w = U wT U ,and Γ w := U ( Z ) ∩ w − U ( Z ) ⊤ w . Let R w ( q ) be a complete system of coset representatives for P ∩ Γ ( q ) \ Γ ( q ) ∩ G w / Γ w .We compute the Fourier coefficients of the Poincaré series: Z U ( Z ) \ U ( R ) P ( X ) M ( x, y ) θ N ( x ) dx = X γ ∈ P ∩ Γ ( q ) \ Γ ( q ) Z U ( Z ) \ U ( R ) F ( X ) ( ι ( M ) γxy ) θ N ( x ) dx = X w ∈ W X γ ∈ R w ( q ) X ℓ ∈ Γ w Z U ( Z ) \ U ( R ) F ( X ) ( ι ( M ) γℓxy ) θ N ( x ) dx = X w ∈ W X c ∈ N Kl q,w ( c, M, N ) Z U w ( R ) F ( X ) ( ι ( M ) c ∗ wxy ) θ N ( x ) dx. For fixed y , it follows from Lemma 3.1 and E having compact support that the c -sum runs over afinite set, and U w ( R ) runs over a compact domain. In particular, the right hand side is absolutelyconvergent.Now let ̟ be an automorphic form, not necessarily cuspidal, in the spectrum of L (Γ ( q ) \ Sp(4 , R ) /K ) .By unfolding, (5.1) and a change of variables ι ( N ) y y , we obtain D ̟, P ( X ) N E = Z T ( R + ) Z U ( Z ) \ U ( R ) ̟ ( xy ) θ N ( − x ) E ( X ) ( N · y( y )) dxd ∗ y = N η A ̟ ( N ) D W µ , E ( X ) E . By Parseval, we obtain D P ( X ) M , P ( X ) N E = M η N η Z ( q ) A ̟ ( M ) A ̟ ( N ) (cid:12)(cid:12)(cid:12)D W µ , E ( X ) E(cid:12)(cid:12)(cid:12) d̟. Meanwhile, unfolding the inner product directly gives D P ( X ) M , P ( X ) N E = Z T ( R + ) Z U ( Z ) \ U ( R ) P ( X ) M θ N ( − x ) E ( X ) ( N · y( y )) dxd ∗ y = X w ∈ W X c ∈ N Kl q,w ( c, M, N ) Z T ( R + ) Z U w ( R ) F ( X ) ( ι ( M ) c ∗ wxy ) θ N ( − x ) E ( XN · y( y )) dxd ∗ y. Now define A = ι ( XM ) c ∗ wι ( XN ) − w − = ι (( XM ) · ( XN ) w ) c ∗ ∈ T ( R + ) . (7.2) DENSITY THEOREM FOR
Sp(4) Then y( A ) η c c = (( XM ) · ( XN ) w ) η . By change of variables ι ( XN ) y y , ι ( XN ) xι ( XN ) − x ,we can express D P ( X ) M , P ( X ) N E as X w ∈ W X c ∈ N Kl q,w ( c, M, N ) ( XM ) η ( XN ) η c c y( A ) η Z T ( R + ) Z U w ( R ) F ( X ) ( ι ( X ) − Awxy ) θ X − ( − x ) E (y( y )) dxd ∗ y. We then conclude a Kuznetsov-type trace formula.
Lemma 7.1.
Let
M, N ∈ N , X ∈ R , E a function on R with compact support, and define F ( X ) as in (7.1) . Then Z ( q ) A ̟ ( M ) A ̟ ( N ) (cid:12)(cid:12)(cid:12)D W µ , E ( X ) E(cid:12)(cid:12)(cid:12) d̟ (7.3) = X w ∈ W X c ∈ N Kl q,w ( c, M, N ) X η c c y( A ) η Z T ( R + ) Z U w ( R ) F ( X ) ( ι ( X ) − Awxy ) θ X − ( − x ) E (y( y )) dxd ∗ y, with A as in (7.2) . Proof of theorems
We establish the following proposition, from which the other theorems are proved.
Proposition 8.1.
Keep the notations as above. Let m ∈ N be coprime to q and Z ≥ . Then Z ( q ) | A ̟ (1 , m ) | Z σ π ( ∞ ) δ λ ̟ ∈ I d̟ ≪ I,ε q ε uniformly in mZ ≪ q for a sufficiently small implied constant depending on I .Proof. We take X = (1 , Z ) , M = N = (1 , m ) , and apply Lemma 7.1. By Lemma 5.1, there is afinite set of compactly supported functions E j such that Z η +2 σ π ( ∞ ) δ λ ̟ ∈ I ≪ I X j (cid:12)(cid:12)(cid:12)D W µ , E ( X ) j E(cid:12)(cid:12)(cid:12) . (8.1)Now we consider the arithmetic side of the Kuznetsov formula for a fixed E ( X ) = E ( X ) j . It suffices toconsider the Weyl elements w ∈ W for which the Kloosterman sum Kl q,w ( c, M, N ) does not vanish,namely, w = id , s α s β s α , s β s α s β , w .For w = id , we have c = c = 1 , and hence the contribution is O ( Z η ) = O ( Z ) .Now let w ∈ { s α s β s α , s β s α s β , w } . Apply Lemma 3.1 with B = ( XM ) · ( XN ) w . Concretely, weset ( B , B ) = ( ( mZ, if w = s α s β s α , (1 , ( mZ ) ) if w = s β s α s β , w . Then we obtain c ≪ B B / = mZ, c ≪ B B = ( mZ if w = s α s β s α , ( mZ ) if w = s β s α s β , w . We assume mZ ≪ q with a sufficiently small implied constant, such that c , c < q for w = s α s β s α , and c < q , c < q for w = s β s α s β , w (8.2)always hold. Now we consider the Kloosterman sums Kl q,w ( c, M, N ) = X xc ∗ wx ′ ∈ U ( Z ) \ G w ( Q ) ∩ Γ ( q ) /U w ( Z ) θ M ( x ) θ N ( x ′ ) , where the entries of M = ( M , M ) and N = ( N , N ) are coprime to q . The Kloosterman sumsare nonzero only when (4.3) and (4.4) are satisfied. By (4.5), (4.6) and (8.2), the problem reducesto computing the Kloosterman sums Kl q,s α s β s α (( q, q ) , M, N ) , Kl q,s β s α s β (cid:0) ( q, q ) , M, N (cid:1) , Kl q,w (cid:0) ( q, q ) , M, N (cid:1) , Kl q,w (cid:0) ( q, q ) , M, N (cid:1) . From Section 4.1, we see that only Kl q,s β s α s β (cid:0) ( q, q ) , M, N (cid:1) does not vanish. So only w = s β s α s β contributes.The next step is to estimate for w = s β s α s β the integral (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ( R + ) Z U w ( R ) F ( X ) ( ι ( X ) − Awxy ) θ X − ( − x ) E (y( y )) dxd ∗ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T ( R + ) Z U w ( R ) | E (y( Awxy )) E (y( y )) | . This integral is bounded by the size of the set of x ∈ U w ( R ) such that y( Awxy ) lies in the supportof E . Using Lemma 3.1 and Lemma 3.3, we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ( R + ) Z U w ( R ) F ( X ) ( ι ( X ) − Awxy ) θ X − ( − x ) E (y( y )) dxd ∗ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ E vol n x ∈ U w ( R ) (cid:12)(cid:12)(cid:12) ∆ ( wx ) ≪ E y( A ) y( A ) / , ∆ ( wx ) ≪ E y( A ) y( A ) o ≪ E y( A ) η (1+ ε ) . So the contribution from w = s β s α s β is given by X c ∈ N Kl q,w ( c, M, N ) X η c c y( A ) η Z T ( R + ) Z U w ( R ) F ( X ) ( ι ( X ) − Awxy ) θ X − ( − x ) E (y( y )) dxd ∗ y ≪ E X c ′ ≪ mZ/q Z η y( A ) ε q ≪ Z q ε . Combining the estimates with (8.1), we obtain Z ( q ) | A ̟ (1 , m ) | Z σ π ( ∞ ) δ λ ̟ ∈ I d̟ ≪ I Z ( q ) | A ̟ (1 , m ) | (cid:12)(cid:12)(cid:12)D W µ , E ( X ) E(cid:12)(cid:12)(cid:12) d̟ ≪ ε Z q ε . Dividing both sides by Z yields the theorem. (cid:3) Proof of Theorem 1.2.
It follows easily from Proposition 8.1, Theorem 6.2 and the estimate (6.3)that X π ∈F I ( q ) (cid:12)(cid:12) λ ′ ( m, π ) (cid:12)(cid:12) Z σ π ( ∞ ) ≪ ε q ε Z ( q ) | A ̟ (1 , m ) | Z σ π ( ∞ ) δ λ ̟ ∈ I ≪ I,ε q ε . (cid:3) Proof of Theorem 1.3.
This is just a simple variation of the proofs above. Again we have X π ∈F I ( q ) (cid:12)(cid:12)(cid:12) X m ≤ x ( m,q )=1 α ( m ) λ ′ ( m, π ) (cid:12)(cid:12)(cid:12) ≪ ε q ε Z ( q ) (cid:12)(cid:12)(cid:12) X m ≤ x ( m,q )=1 α ( m ) A ̟ ( M ) (cid:12)(cid:12)(cid:12) δ λ ̟ ∈ I d̟ = q ε X m ,m ≤ x ( m m ,q )=1 α ( m ) α ( m ) Z ( q ) A ̟ ( M ) A ̟ ( M ) δ λ ̟ ∈ I d̟, DENSITY THEOREM FOR
Sp(4) where M = (1 , m ) , M = (1 , m ) , M = (1 , m ) . Now we apply Lemma 7.1 and evaluate theKloosterman sums on the arithmetic side. For w = id , apply Lemma 3.1 with B = M · M w . Weget c ≪ ( m m ) / ≤ x, c ≪ m ≤ x for w = s α s β s α ,c ≪ ( m m ) / ≤ x, c ≪ m m ≤ x for w = s β s α s β , w . Note that when x ≪ q with a sufficiently small implied constant, the condition ( m, q ) = 1 is void,and we deduce from (4.3) that the Kloosterman sums Kl q,w ( c, M, N ) are empty for w = id . Henceonly the trivial Weyl element contributes, and we obtain the desired bound. (cid:3) Proof of Corollary 1.4.
Observe that for π ∈ F I ( q ) an approximate functional equation has length q / (see [IK04, Section 5]). So, for all but O (1) cuspidal representations π ∈ F I ( q ) (and ε < / )we have | L (1 / it, π ) | ≪ I,t,ε q ε X j = M ≤ q / ε M (cid:12)(cid:12)(cid:12) X M ≤ m ≤ M λ ′ π ( m ) (cid:12)(cid:12)(cid:12) . The statement then follows from Theorem 1.3. (cid:3)
Proof of Theorem 1.1.
We first assume v = p = q is a finite place. We choose ν maximal such that p ν ≪ q with an implied constant that is admissible to Proposition 8.1. Then by Lemma 6.4 andthe estimate (6.3), there exists ν − ≤ ν π ≤ ν such that | A ̟ (1 , p ν π ) | ≫ q − − ε p ν π σ π ( p ) . Note that p ν π ≍ q . We apply Proposition 8.1 with m = p ν π , Z = 1 , and conclude that N p ( σ, F I ( q )) ≤ X π ∈F I ( q ) p ν π σ π ( p ) p ν π σ ≪ q − σ + ε Z ( q ) X ν − ≤ ν ≤ ν | A ̟ (1 , p ν ) | δ λ ̟ ∈ I ≪ I,ε q − σ + ε . For v = ∞ , we use the estimate (6.3), apply Proposition 8.1 with m = 1 , Z ≪ q , and concludethat N ∞ ( σ, F I ( q )) ≤ X π ∈F I ( q ) Z σ π ( ∞ ) − σ ≪ q − σ + ε Z ( q ) | A ̟ (1 , | Z σ π ( ∞ ) ≪ I,ε q − σ + ε . (cid:3) Appendix: Computation of Fourier coefficients
In this appendix, we outline an algorithm for computing arbitrary Fourier coefficients of a cuspidalnewform ̟ ∈ V π with A ̟ (1 ,
1) = 1 . For this purpose, it suffices to compute the actions of T ( p ) and T (2)0 , ( p ) , which generate the Hecke algebra. By Proposition 6.1, we compute λ ( p, π ) A ̟ ( M , M ) = p / (cid:0) A ̟ ( M , pM ) + A ̟ ( p − M , pM ) | {z } if p | M + A ̟ ( pM , p − M ) + A ̟ ( M , p − M ) | {z } if p | M (cid:1) , (9.1) and if p ∤ M , (cid:0) λ (2)0 , ( p, π ) + 1 (cid:1) A ̟ ( M , M ) = p (cid:0) A ̟ ( pM , M ) + A ̟ ( p − M , p M ) + A ̟ ( p − M , M ) | {z } if p | M (cid:1) . (9.2) We proceed to show how the Fourier coefficients A ̟ ( p k , p k ) are obtained. Starting from A ̟ (1 ,
1) = 1 , we apply (9.1) and (9.2) with M = (1 , and solve the coefficients A ̟ ( p,
1) = p − (cid:16) λ (2)0 , ( p ) + 1 (cid:17) , A ̟ (1 , p ) = p − / λ ( p ) . Inductively, suppose the Fourier coefficients A ̟ ( p k , p k ) are known for all k + k ≤ r . For ≤ k ≤ r , applying (9.1) with M = ( p k , p r − k ) yields the coefficient A ̟ ( p k , p r − k +1 ) . Then, applying(9.2) with M = ( p r , yields the coefficient A ̟ ( p k +1 , , since the coefficient A ̟ ( p k − , p ) hasalready been determined. This shows that the Fourier coefficients A ̟ ( p k , p k ) with k + k ≤ r + 1 can be expressed in terms of λ ( p ) and λ (2)0 , ( p ) , finishing the induction.Writing X := p − / λ ( p, π ) and Y := p − (cid:16) λ (2)0 , ( p, π ) + 1 (cid:17) , the Fourier coefficients A ̟ ( p k , p k ) for small k i are computed in the following table: A ̟ ( p k , p k ) k = 0 k = 1 k = 2 k = 3 k = 0 1 X X − Y − X − XY − Xk = 1 Y XY − X X Y − X − Y − Y + 1 X Y − X − XY + 2 Xk = 2 − X + Y + Y − X Y + Y + X + 2 Y − − X + X Y + X Y − Y + 2 X − Y − X + X Y + 2 X Y − XY + 2 X − XY − Xk = 3 − X Y + Y + X +2 Y − − X Y + XY + 2 X + XY − X − X Y + X Y + 2 X +3 X Y − Y + X Y − Y − X − Y + 2 Y + 1 − X Y + X Y + 2 X +5 X Y − XY − X Y − XY − X + 4 XY + 2 X From Theorem 6.2, we obtain λ ( p , π ) = p ( X − Y − − p . Hence the Fourier coefficients canalso be expressed in terms of eigenvalues λ ( p r , π ) of standard Hecke operators.It is evident from the Proposition 6.1 that Fourier coefficients are multiplicative, that is, A ̟ ( M N , M N ) = A ̟ ( M , M ) A ̟ ( N , N ) if ( M M , N N ) = 1 . (9.3)Using (9.3), and (6.2) for negative coefficients, we are able to compute A ̟ ( M ) for every M ∈ Z . References [BBM17] V. Blomer, J. Buttcane, and P. Maga. Applications of the Kuznetsov formula on
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Siu Hang Man, Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn,Germany
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