EQUIDISTRIBUTION THEOREMS for HOLOMORPHIC SIEGEL MODULAR FORMS for Gsp4; HECKE FIELDS and N-LEVEL DENSITY

EQUIDISTRIBUTION THEOREMS FOR HOLOMORPHIC SIEGEL MODULAR FORMS FOR GSp4; HECKE FIELDS AND n-LEVEL DENSITY

HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

Abstract. This paper is a continuation of [21]. We supplement four results on a family of

holomorphic Siegel cusp forms for GSp4/Q. First, we improve the result on Hecke ﬁelds. Namely, we prove that the degree of Hecke ﬁelds is unbounded on the subspace of genuine forms which

do not come from functorial lift of smaller subgroups of GSp4 under a conjecture in local-global

compatibility and Arthur’s classiﬁcation for GSp4. Second, we prove simultaneous vertical Sato- Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at ﬁnitely many primes. Third, we compute the n-level density of degree 4 spinor L-functions, and thus we can distinguish the symmetry type depending on the root numbers. This is conditional on certain conjecture on root numbers. Fourth, we consider equidistribution of paramodular forms. In this case, we can prove a result on root numbers. Main tools are the equidistribution theorem in our previous work and Shin-Templier’s work [45].

1. Introduction

This paper is a continuation of [21]. We use the same notations throughout this paper. We study Hecke ﬁelds, simultaneous vertical Sato-Tate theorem, and the n-level density on a family of holomorphic Siegel cusp forms for GSp4/Q. First, Hecke ﬁelds. Let S1(Γ (N)) be the space of elliptic cusp forms of weight k 2 with k 0 ≥ respect to a congruent subgroup Γ (N). The Hecke operators T acting on the space and it 0 { p}p-N has a basis consisting of simultaneous eigenforms in Hecke operators which are called normalized

Hecke eigenforms. Let f be such an eigenform and ap(f) be the Hecke eigenvalue of Tp for p - N, i.e. T f = a (f)f. The ﬁeld Q := Q(a (f) p - N) generated by such eigenvalues over Q is called p p f p | 1 the Hecke ﬁeld of f. Since Sk(Γ0(N)) has an integral structure preserved by Hecke operators,

1991 Mathematics Subject Classification. 11F46, 11F70, 22E55, 11R45. Key words and phrases. trace formula, Hecke fields, Siegel modular forms. The first author is partially supported by NSERC. The second author is partially supported by JSPS Grant-in- Aid for Scientific Research (No. 26800006, 25247001, 15K04795). The third author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) No.15K04787. 1 2 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI the eigenvalues are algebraic numbers and it turns out that Qf is a finite extension over Q. The Hecke field of f reflects various arithmetic properties of f and has been studied by many people

[43], [41]. For example Q-simple factors of the Jacobian J0(N) of the modular curve X0(N) can be described in terms of the degree of the Hecke ﬁelds and one can ask the maximal dimension of Q-simple factors for J0(N) (see [28], [41], [33],[53] [30]).

Let Sk(Γ(N), χ) be the space of classical holomorphic Siegel cusp forms of degree 2 with the level Γ(N), a central character χ :(Z/NZ) C , and weight k = (k ,k ),k k 3 (cf. ×−→ × 1 2 1 ≥ 2 ≥ Section 2 of [21]). For a prime p - N, let T (pn) be the Hecke operator with the similitude pn. Any eigenform with respect to T (pn) for any non-negative integer n and any prime p - N is called a Hecke eigen cusp form. We denote by HEk(Γ(N), χ) the set of all such eigenforms in

Sk(Γ(N), χ). n n Let F be a Hecke eigen cusp form and λF (p ) be the Hecke eigenvalue of F for T (p ), i.e. n n n T (p )F = λ(p )F . It is known that λF (p ) is an algebraic integer (cf. Lemma 2.1 of [47]). We consider the Hecke ﬁeld Q := Q(λ (pn), χ(p), χ (p) p - N) which turns out to be a ﬁnite F F 2 | extension over Q (see (2.12) of [21] for χ2). We call F a genuine form if it never comes from any functorial lift from a smaller subgroup of GSp4, hence, it is neither a CAP form, an endoscopic lift, a base change lift, an Asai lift, nor a symmetric cubic lift (see Section 2 for the details).

Theorem 1.1. Assume Conjecture 1 of [13] and Arthur’s classiﬁcation for GSp4 Fix a weight k =(k ,k ) with k k 3 and a prime p. Then 1 2 1 ≥ 2 ≥

lim sup [QF : Q] F HEk(Γ(N), χ): genuine = . N→∞,p-N, (N,11!)=1 { | ∈ } ∞ ord`(N)≥4 if `|N Theorem 1.2. Let the notation and assumptions be as above. Let p be a fixed prime. Put d (χ) = HE (Γ(N), χ) Then k,N | k | d (χ) N 9+ε F HE (Γ(N), χ) [Q : Q] A = O k,N = O |{ ∈ k | F ≤ }| log d (χ) log N k,N as N goes to infinity satisfying either of the following conditions: (1) ord (N) 4 if ` N and (N,p)=1 ; or ` ≥ | (2) ord (N) as N . p → ∞ → ∞ The claim is still true even if we replace HE (Γ(N), χ) with F HE (Γ(N), χ) F : genuine k { ∈ k | } but keeping the additional condition (N, 11!) = 1. Note that the cardinality of that subspace is approximately equal to dk,χ(N) when N, (N, 11!)= 1 goes to infinity. HECKE FIELDS AND n-LEVEL DENSITY 3

1 Theorem 1.3. Keep the condition in Theorem 1.2. Put f(N) = (loglog N) 2 if N is in the ﬁrst 1 case of the two conditions in Theorem 1.2 and f(N) = (log N) 4 otherwise. Then

inf [Q : Q] F HE (Γ(N), χ): genuine f(N) { F | ∈ k } as N goes to inﬁnity with (N, 11!)= 1.

tm Second, simultaneous vertical Sato-Tate theorem. Let Sk(Γ(N), χ) be the subspace of

Sk(Γ(N), χ) generated by Hecke eigen forms F outside N so that πF,p is tempered for any p - N. Let HE (Γ(N), χ)tm = S (Γ(N), χ)tm HE (Γ(N), χ). For a prime p - N, let a ,b [ 2, 2] k k ∩ k F,p F,p ∈ − be Hecke eigenvalues as in [21]. Then in Section 4, we generalize Theorem 1.4 of [21] to finitely many primes. Namely, given finitely many distinct primes p1,...,pr,((aF,p1,bF,p1),..., (aF,pr,bF,pr )) is equidistributed with respect to a suitable measure (Theorem 4.1). Third, n-level density of degree 4 spinor L-functions. In [21], we studied the one-level density of degree 4 spinor L-functions of a family of holomorphic Siegel cusp forms for GSp4/Q, and we showed that its symmetry type is SO(even), SO(odd), or O type, as predicted by [14]. However, we could not distinguish the symmetry type among SO(even), SO(odd), and O type since the support of φˆ is smaller than ( 1, 1). In order to distinguish them, we need to compute the − n-level density. For the degree 4 spinor L-function L(s, πF , Spin), we denote the non-trivial zeros of L(s, π , Spin) by 1 + √ 1γ . Let φ(x ,...,x ) = n φ (x ) be an even Schwartz class F 2 − j 1 n i=1 i i ˆ function in each variables whose Fourier transform φ(u1Q,...,un) is compactly supported. We (n) define D (πF , φ, Spin) as in Section 5.

We ﬁrst prove the following theorem which may be of independent interest. Let L(s, πF , Spin) = ˜ s vp(m) m∞=1 λF (m)m− . Let m = p m p . For simplicity, denote Sk(Γ(N), 1),dk,N (1) by Sk(N),dk,N , | Presp. Q

Theorem 1.4. Put k =(k ,k ), k k 3. 1 2 1 ≥ 2 ≥ (1) (level-aspect) Fix k ,k . Then as N , 1 2 → ∞

1 1 2 vp(m) 2 c λ˜F (m) = δm− 2 (1 + p− + + p− ) + O(N − m ), dk,N ··· F HE (N) p m ∈ Xk Y|

1, if m is a square where δ = .  0, otherwise  4 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

(2) (weight-aspect) Fix N. Then as k + k , 1 2 → ∞ 1 1 2 vp(m) λ˜F (m) = δm− 2 (1 + p− + + p− ) dk,N ··· F HE (N) p m ∈ Xk Y| mc m d + O + O − , (k 1)(k 2) (k k + 1)(k + k 3) 1 − 2 − 1 − 2 1 2 − for some constants c,d> 0.

For one-level density, the root number (πF ) did not play a role. However, higher level density depends on the root number. When N = 1 (i.e., level one case), we have (π )=( 1)k2 ([39]). (In this case, k k should F − 1 − 2 be even.) Then we have the following n-level density in the weight aspect. Let HEk = HEk(1) and dk = dk,1.

Theorem 1.5. Let φ(x ,...,x ) = φ (x ) φ (x ), where each φ is an even Schwartz function 1 n 1 1 ··· n n i and φˆ(u ,...,u ) = φˆ (u ) φˆ (u ). Assume the Fourier transform φˆ of φ is supported in 1 n 1 1 ··· n n i i ( β ,β ) for i =1, , n.(β < 1 can be explicitly determined.) Then − n n ··· n 1 n φ(x)W (SO(even))(x) dx + O , if k2 is even 1 (n) R log ck D (πF , φ, Spin) =  , dk 1 F HEk R n φ(x)W (SO(odd))(x) dx + O , if k2 is odd ∈X  R log ck R where W (SO(even)) and W (SO(even)) are the n-level density functions deﬁned in Section 5, and ck = ck,1 is the analytic conductor deﬁned in Section 5.

Let S±(N) be the subspace of S (N) with the root number (π ) = 1. Let HE±(N)be a k k F ± k basis of S±(N) consisting of Hecke eigenforms outside N, and denote HE±(N) = d± . When k | k | k,N 1 9 N is large, we expect dk±,N = 2dk,N + O(N − ), and the analogue of Theorem 1.4 holds when we replace HEk(N) by HEk±(N). We assume it as Conjecture 5.1. Then we can prove the n-level density result for HEk±(N) (Theorem 5.9 and Theorem 5.10). In Section 6, we study the n-level density of the degree 5 standard L-functions of holomorphic Siegel cusp forms. We show that the root number is always one. Hence the symmetry type of the n-level density should be Sp. Under Conjecture 6.1 which is an analogue of Conjecture 5.1, we show

Theorem 1.6. Let φ(x ,...,x ) = φ (x ) φ (x ), where each φ is an even Schwartz function 1 n 1 1 ··· n n i and φˆ(u ,...,u ) = φˆ (u ) φˆ (u ). Assume the Fourier transform φˆ of φ is supported in 1 n 1 1 ··· n n i i HECKE FIELDS AND n-LEVEL DENSITY 5

( β ,β ) for i =1, , n. Then − n n ··· 1 (n) ω(N) D (πF , φ, St) = φ(x)W (Sp)(x) dx + O . dk,N Rn log ck,N F HEk,N Z ∈X Fourth, we consider paramodular forms. In our previous paper [21], we considered only principal congruence subgroup Γ(N). Here we can deal with the paramodular group Kpara(N). We prove equidistribution results on paramodular forms. In particular we can show Conjecture 5.1 for paramodular forms. Hence n-level density for spinor L-functions of paramodular forms for weight aspect (analogues of Theorem 5.9 and Theorem 5.10) hold. In a similar way, we can show simultaneous vertical Sato-Tate theorem for paramodular forms (analogue of Theorem 4.1). Acknowledgments. We would like to thank K. Morimoto, R. Schmidt and S-W. Shin for helpful discussions.

2. Genuine forms

In this section, we follow the notation and the contents in Section 2 of [21] for holomorphic Siegel modular forms of genus 2. The readers should consult the references there if necessary.

2.1. Classical Siegel modular forms and Hecke ﬁelds. For a pair of non-negative integers k = (k ,k ), k k 3 and N 1, we denote by S (Γ(N), χ) the space of cusp forms of the 1 2 1 ≥ 2 ≥ ≥ k weight k with the character χ :(Z/NZ) C for a principal congruence subgroup Γ(N). Here ×−→ × the weight corresponds to the algebraic representation λk of GL2 with the highest weight k by

k k k V = Sym 1− 2 St det 2 St , k 2 ⊗ 2 where St2 is the standard representation of dimension 2. For each prime p - N and n 1 one can define the Hecke operator T (pn) acting on S (Γ(N), χ). ≥ k There exists a basis of the space consisting of eigenforms for all such T (pn) which are called Hecke eigen cusp forms. Let HEk(Γ(N), χ) be the set of all Hecke eigen cusp forms for Sk(Γ(N), χ). For F HE (Γ(N), χ) we have T (pn)F = λ (pn)F, λ (pn) C. Since S (Γ(N), χ) has an ∈ k F F ∈ k n integral structure and is of finite dimensional, the eigenvalue λF (p ) is an algebraic integer once we fix an embedding Q , C. We define the Hecke field Q of F by Definition 2.14 in [21]. → F 2.2. CAP forms and endoscopic lifts. For F HE (Γ(N), χ) let π be the corresponding ∈ k F cuspidal representation of GSp4(AQ). We say F is a CAP form (resp. an endoscopic lift) if πF CAP is a CAP (resp. endoscopic) representation. We denote by Sk(Γ(N), χ) the space generated 6 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

EN by all CAP forms in Sk(Γ(N), χ). We also deﬁne Sk(Γ(N), χ) for endoscopic lifts similarly. Under mind condition on the level N we have the following estimation on the dimension of the space as above.

Proposition 2.1. If k = k or χ is not the square of a character, then S (Γ(N), χ)CAP =0. If 1 6 2 k k := k1 = k2 and χ is the square of a character, then

CAP 7+ε dimSk(Γ(N), χ) = O(kN ) as N + k and (N, 11!)= 1 → ∞ Proof. It follows from Section 4.4 and Theorem 4.3 in [21].

Next we consider endoscopic lifts.

Proposition 2.2. It holds that

dimS (Γ(N), χ)EN = O((k k + 1)(k + k 3)N 8+ε) k 1 − 2 1 2 − as N + k + k and (N, 11!)= 1 1 2 → ∞ Proof. It follows from Theorem 4.2 in [21].

2.3. Local-global compatibility for holomorphic Siegel modular forms. In this subsection we discuss some results which follow from Conjecture 1 of [13] concerning local-global compatibility for holomorphic Siegel modular forms. That conjecture is satisﬁed for Siegel modular forms with Iwahori level structure in [13] and for Siegel paramodular forms by [39] and [46]. Stable Let Sk (Γ(N)) be the space generated by all non-endoscopic, non-CAP Hecke eigen cusp forms in Sk(Γ(N)). Such a form is called a stable form by Arthur. For a Hecke eigenform F in SStable(Γ(N)) let us consider the corresponding cuspidal representation π = π of GSp (A ). k F ⊗p0 p 4 Q Proposition 2.3. Let the notations be as above. Assume Conjecture 1 of [13]. Then there exists a unique globally generic cuspidal representation π = π of GSp (A ) such that π and π 0 ⊗p0 p0 4 Q p p0 belong to the same L-packet and they are both tempered.

Proof. For such πF , by Weissauer [51], there exists a globally generic cohomological cuspidal representation Π of GSp4(AQ) so that πF is weakly equivalent to Π. By [46], for any ﬁxed prime `, it gives rise to a Galois representation ρ : G GSp (Q ) which satisﬁes local- Π,` Q−→ 4 ` global compatibility. Under Conjecture 1 of [13] we also have a Galois representation ρπF ,` : HECKE FIELDS AND n-LEVEL DENSITY 7

G GSp (Q ) which satisﬁes local-global compatibility and ρ ρ by Chebotarev density Q−→ 4 ` Π,` ∼ π,` theorem. Therefore πp and Πp belong to the same L-packet. The temperedness of Πp which is proved in [46] implies that of πp by [8].

Corollary 2.4. Keep the notations as in the previous proposition and assume Conjecture 1 of [13].

Then πp is generic if its L-packet is a singleton. Otherwise the L-packet of πp has two members which are given in the notation of [31] by VIa, VIb , VIIIa, VIIIb , Va, θ((σSt )JL, (σξSt )JL) , { } { } { 2 2 } XIa, θ((σSt)JL, (σπsc)JL) , or θ(π ,π ), θ((π )JL, (π )JL) where πsc,π , and π stand { 2 } { p,1 p,2 p,1 p,2 } 1,p 2,p for unitary supercuspidal representations of GL (Q ) such that π π , σ is a quasi charac- 2 p 1,p 6∼ 2,p ter of Qp× and ξ is a quadratic character of Qp×. Here θ is the local theta correspondence from

GSO(4) or GSO(2, 2) to GSp4 (cf. Section 4.3 of [21]).

Proof. First we remark that the L-packet of any generic supercuspidal representation which is not in the image of the local theta correspondence from GSO(2, 2) is a singleton since it has an irreducible L-parameter. Such a representation never appears in this setting. Then by using Table A.1 of [31], Proposition 2.3, and local Langlands classiﬁcation [8], one can list every non-generic representation whose L-packet is not a singleton (see also Section 2.4 of [31]).

Put K(pr) :=(1+ prM (Z )) GSp (Z ) for a non-negative integer r. 4 p ∩ 4 p

Proposition 2.5. Let πp be the representation as in Proposition 2.3. Assume Conjecture 1 of

[13] and that the L-packet of πp consists of two members. Assume further that (p, 11!)= 1. Then r K(p ) 9r r dim πp = O(p ) when p goes to inﬁnity.

Proof. Under Conjecture 1 of [13], one can see that πp is one of representations in Corollary 2.4. Except for representations obtained by the local theta correspondence it is a constituent of a normalized induced representation IndGSp4(Qp)χ where χ is a quasi character of B(Q ). Let B(Qp) p r γ1,...,γt be a complete system of the double coset representatives of B(Qp) GSp4(Qp)/K(p ). r \ Then (IndGSp4(Qp)χ)K(p ) is generated by the following functions: B(Qp)

1 2 r δ (b)χ(b), if g = bγik B(Qp)γiK(p ), f (g) = B ∈ i  0, otherwise.

r 1 so that χ is trivial on B(Q ) γ K(p )γ− . Therefore we roughly estimate p ∩ i i K(pr) GSp4(Qp) K(pr) r r 4r dim π dim (Ind χ) t = ]B(Z/p Z) GSp4(Z/p Z) = O(p ). ≤ B(Qp) ≤ \ 8 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

JL JL JL sc JL Let us suppose that πp = θ((σSt2) , (σξSt2) ), or πp = θ((σSt)2 , (σπ ) ). Put τ = σSt σξSt or τ = σSt σπsc, respectively. Recall that (p, 11!) = 1. Then one can apply the 2 × 2 2 × argument in Theorem 4.2 of [21] which gives us a similar estimation: r r vol(KH(p )) 1 r dim πK(p ) = dim τ KH(p ) = O(p9r) vol(K(pr)) p2r where K (pr) := Ker(H(Z ) H(Z /prZ )) for a unique endoscopic subgroup H = GSO(2, 2) H p −→ p p in GSp4 (see Section 4.3 of [21]). The remaining case is similar to this case.

The following lemma will be used to deﬁne a non-canonical map (2.1).

Lemma 2.6. Let πp be as in Corollary 2.4. Assume that L-packet of πp is not a singleton and let g g π ,πp be the L-packet where π is non-generic and πp is generic. Then for a positive integer r, { p } p g K(pr) g K(pmax{4,r}) K(pr) (πp) =0 (resp. (πp) =0) if (π ) =0 except for the case of π = VIb (resp. 6 6 p 6 p for the case of πp = VIb).

Proof. We prove this lemma case by case and all cases are given in Corollary 2.4. Let us ﬁrst g 1 1 consider the type VIa and VIb. Then πp = τ(S, ν− 2 σ) and πp = τ(T, ν− 2 σ). By deﬁnition 1 1 1 1 τ(T, ν− 2 σ) = σ τ(T, ν− 2 ) and τ(S, ν− 2 σ) = σ τ(S, ν− 2 ). Therefore the character σ should ⊗ ⊗ r 1 1 be trivial on 1+ p Zp. Note that τ(S, ν− 2 ) (resp. τ(S, ν− 2 )) is of paramodular level 4 (resp. 2). Since the paramodular subgroup Kpara(pt) contains K(pt) for any positive integer t, the claim follows. In the case of VIIIa and VIIIb, by Table A.4 of [31], both of their Jacquet modules along the Klingen parabolic subgroup are 1 o π where π is a unitary supercuspidal representation of

GL2(Qp). The claim follows from this. g 1 Next we consider the case πp = V a = δ([ξ,νξ], ν−2 σ) where ξ is a quadratic character. Clearly r 1 1 σ is trivial on 1+ p Z . If ξσ is ramiﬁed, then δ([ξ,νξ], ν−2 σ) = σ δ([ξ,νξ], ν−2 ). The claim p ⊗ is now easy to follow. The other remaining case is done similarly. For the level of theta correspondence we can apply the argument in Theorem 4.2 of [21] regarding the transfer of some Hecke elements with respect to congruence subgroups and the claim follows from this directly.

2.4. A non-canonical map between stable forms. Throughout this subsection we assume

Conjecture 1 of [13] as in Proposition 2.3 and Arthur’s classification for GSp4 (which will be completed soon). Under these assumptions we relate holomorphic stable forms with globally generic stable forms. According to Lemma 2.6 we also assume that ord (N) 4 when a prime p ≥ HECKE FIELDS AND n-LEVEL DENSITY 9 p divides N. Let π = π be a cuspidal representation which comes from a Hecke eigen form ⊗p p Stable g large in Sk (Γ(N)). Let π := Dl , l be the large discrete series with Harish-Chandra parameter ∞ 1 − 2 (l ,l )=(k 1,k 2) (see Section 2.3 of [49]) and we denote by Slarge(Γ(N)) the space of 1 2 1 − 2 − k C Hecke eigen automorphic forms whose representation of GSp (R) is isomorphic to Dlarge ∞ 4 l1,l2 and which give rise to globally generic representations. We say an element of this space a globally generic form. Let Sgg(Γ(N)) be the space of globally generic forms. For π = π k ⊗p p ∈ Stable Sk (Γ(N)), we define the finite set S of primes p so that the L-packet of πp is not a singleton g and πp is non-generic. We denote by πp the generic representation in the same L-packet as πp for p S. Since π is stable, by Arthur’s classification for GSp (cf. line 5 in p.11 of [40]), the ∈ 4 admissible representation g large g π := Dl , l πp πp 1 − 2 ⊗ ⊗ p S p S O∈ O6∈ is an automorphic cuspidal representation which is generic everywhere. Then by Proposition 2.3, this is a globally generic representation so there is a unique distinguished vector F g in πg. The uniqueness follows from [12]. Therefore we have a non-canonical map

(2.1) T g : SStable(Γ(N)) Sgg(Γ(N)), F F g. k −→ k 7→ Note that T g takes the forms with respect to Γ(N) by Lemma 2.6. For any subset S S 0 ⊂ ∪ {∞} g one can also consider π(S0) := p S πp p S0 πp which is also automorphic. Similarly we have ∈ ⊗ 6∈ Stable S a non-canonical map T (S0) fromNSk (Γ(NN)). There exists 2| | maps for T (S0). We will exploit this map to estimate the number of functorial lifts with the following proposition.

Proposition 2.7. Under the assumptions in this section, dim KerT g = O(N 8+ε).

Proof. Let Stable(Γ(N)) be the space of automorphic forms on GSp (A) which corresponds to Ak 4 SStable(Γ(N)). Then by [21, (2.17)], dim Stable(Γ(N)) = ϕ(N)dimSStable(Γ(N)). Hence the k Ak k contribution from adelic forms to classical forms is diﬀer by ϕ(N). Proposition 2.5 implies that g 9 S ω(N) ε the dimension of the adelic version of T is O(N ). Since 2| | = O(2 ) = O(N ), the claim follows.

2.5. Symmetric cube lifts. There is a functorial lift from GL2 to GL4 which is called symmetric cube lift constructed by Kim and Shahidi [20]. For an elliptic cusp form f of weight greater than

2, let Π0 be the image of πf under the symmetric cube lift. Then it descends to a unique globally generic cuspidal representation Π := Sym3π of GSp (A ). We say F HE (Γ(N)) is a f f 4 Q ∈ k 10 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI symmetric cube lift if π Π for almost all prime p = and an elliptic cusp form f. We F,p ∼ f,p 6 ∞ Cube denote by Sk(Γ(N), χ) the space generated by all symmetric cube lifts in Sk(Γ(N), χ). In what follows we will try to estimate the dimension of this space. We can deﬁne the following map as in (2.1):

(2.2) T g := T g(N, χ): SStable(Γ(N), χ) Snon-E,gg(Γ(N), χ),T (F ) = F g k k −→ k

Stable where Sk (Γ(N), χ) is the space generated by all Hecke eigen stable forms in Sk(Γ(N), χ) and non-E,gg Sk (Γ(N), χ) is the space generated by all non-endoscopic globally generic Hecke eigen forms gg in Sk (Γ(N), χ). Cube Cube, gg Let Sk (Γ(N), χ) (resp. Sk (Γ(N), χ)) be the space generated by all symmetric cube Cube gg lifts (resp. all generic symmetric cube lifts) in Sk (Γ(N), χ) (resp. Sk (Γ(N), χ)). It is easy to see that SCube(Γ(N), χ) SStable(Γ(N), χ) and SCube, gg(Γ(N), χ) Snon-E,gg(Γ(N), χ). Fur- k ⊂ k k ⊂ k thermore T g induces a surjective linear map

T g : SCube(Γ(N), χ) SCube,gg(Γ(N), χ). k −→ k

Then we have

Theorem 2.8. Fix a weight k. Then it holds that

Cube,gg 4+ε dimSk (Γ(N), χ) = O(N ) as N with (N, 11!)= 1. → ∞

Cube,gg Proof. Let F be a globally generic Hecke eigen cusp form in Sk (Γ(N), χ). By Lemma 9.1 of 4 3 [21], the conductor of πF is bounded by N . On the other hand, if πF = Sym πf for an elliptic 2 cusp form f, by applying Theorem in Section 6.5 of [2] to Sym πf and πf , the lower bound of the conductor of πF is given by

2 c(πf )(c(πf ) c(πf )) 2 − = c(πf) c(πf ) c(πf ) − since Sym2π π = Sym3π π ω . Hence we have that c(π ) = O(N 2). The ﬁrst claim f f f f ⊗ πf f follows from this with the dimension formula for the space of elliptic cusp forms with respect to 2 Γ1(N ) for a ﬁxed weight. HECKE FIELDS AND n-LEVEL DENSITY 11

2.6. Automorphic induction. In this section we are concerned with Automorphic induction.

For a quadratic field K/Q, there is a functorial lift from GL2/K to GL4/Q which is called Automorphic induction (some of people say an Asai lift). To descend it to a globally generic representation of GSp4, the central character ωπ should be invariant under the non-trivial element σ in Gal(K/Q). Hence ω = ω N for a character ω : Q A C . π ◦ K/Q ×\ ×−→ × When K is a real quadratic field, then for any Hilbert modular cusp form f of weight (k1 + k 2,k k + 2) one can construct a generic cuspidal automorphic representation whose 2 − 1 − 2 large representation Π at infinity is isomorphic to D with (l1,l2)= (k1 1,k2 2). We say l1,l2 − − F S (Γ(N)) is an automorphic induction if π is weakly equivalent to such a Π. ∈ k F Similarly one can consider an automorphic induction for an imaginary quadratic field K. However by [9] if F S (Γ(N)) is given by such a way, then k has to be 2 which contradicts ∈ k 2 with our condition k 3. Therefore we have only to consider only contributions from Hilbert 2 ≥ modular forms for real quadratic fields. AI AI,gg Let Sk (Γ(N), χ) (resp. Sk (Γ(N), χ)) be the space generated by all automorphic induction gg (resp. all generic automorphic induction) in Sk(Γ(N), χ) (resp. Sk (Γ(N), χ)). It is easy to see that SAI(Γ(N), χ) Sstable(Γ(N), χ) and SAI, gg(Γ(N), χ) Sgg,non-E(Γ(N), χ). Furthermore k ⊂ k k ⊂ k T g induces a surjective linear map

T g : SAI(Γ(N), χ) SAI,gg(Γ(N), χ). k −→ k Then we have

Theorem 2.9. Fix a weight k. Then it holds that

AI,gg 11 2 +ε dimSk (Γ(N), χ) = O(N ) as N with (N, 11!)= 1. → ∞ AI,large Proof. Let F be a globally generic Hecke eigen cusp form in Sk (Γ(N), χ). By Lemma 9.1 4 of [21] again the conductor of πF is bounded by N . Assume that πF comes from a unique Hilbert cusp form for a quadratic ﬁeld K with the weight (k + k 2,k k + 2) and the 1 2 − 1 − 2 level with a character χ so that χ σχ = χ. By comparing the conductor we have A ⊂ OK 0 0 0 D N ( ) = O(N 4) Then we have K K/Q A dimSAI,large(Γ(N), χ) = O(N 4 ζ ( 1)) k K − DK|N K:realX quadratic 12 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI by Shimizu’s dimension formula in [42]. ζ (2) 3 Now for a real quadratic ﬁeld K, ζ ( 1) = K D 2 by the functional equation. Since K 3 1 2 K − π Γ( 2 ) 2 − ζK(2) ζ(2) , ≤ 3 3 + ζ ( 1)) D 2 N 2 . K − K D |N K DK N K:realX quadratic X|

2.7. Asai transfer. In this section we are concerned with the Asai transfer which is a transfer from GL /K to GL /Q for an etale algebra K of degree 2 over Q. When F = Q Q, it is the 2 4 ⊕ Rankin-Selberg convolution product. When F is a ﬁeld, it is called the Asai transfer. In the former case, given a pair (π1,π2) of two cuspidal representations of GL2(AQ), the automorphic product π1 π2 descend to a unique globally generic representation of GSp4 only when one of

πi’s should be dihedral. (See [15].) When K is a ﬁeld, the Asai transfer As(π) of a cuspidal representation π of GL /K descend to GSp only when π is dihedral. We say F S (Γ(N), χ) 2 4 ∈ k an Asai lift if πF is weakly equivalent to such a representation of GSp4. Asai, non-E We denote by Sk (Γ(N), χ) the space generated by all non-endoscopic Asai lifts in stable Asai, non-E, gg Sk (Γ(N), χ). Similarly we can deﬁne Sk (Γ(N), χ). As in the previous section we have a surjective linear map

g Asai,non E Asai, non-E, gg T : S − (Γ(N), χ) S (Γ(N), χ). k −→ k By using this map we have

Theorem 2.10. Fix a weight k. Then it holds that

Asai,non-E,gg 6+ε dim Sk (Γ(N), χ) = O(N ) as N with (N, 11!)= 1. → ∞

Proof. Let us ﬁrst consider the case when K is not a ﬁeld. Let (f1, f2) be a pair of two elliptic cusp forms of level N1,N2 with characters χ1, χ2 so that χ1χ2 = χ but one of them is a CM form

(say, f2). Note that once we fix an imaginary quadratic field defining f2, then the character of f2 is fixed and so is f . As in the previous proposition, we have N N N 4 by Theorem in Section 1 1 2 ≤ 6.5 of [2]. Therefore we have only to consider

2 1 4 1 h(a DK) dimSk1 (Γ0(N N1− ), ), 4 2 · ∗ N1 N a DK |N1 X| K:imaginaryX quadratic HECKE FIELDS AND n-LEVEL DENSITY 13 where a2 is the square factor dividing N , and is a ﬁxed character and h(a2D ) stands for the 1 ∗ K 2 class number of the binary forms with discriminant a DK. 1 4 1 4 1 Now for a ﬁxed k and N , dimS (Γ (N N − ) = O(N N − ), and 1 1 k1 0 1 1

2 DK 1 1 + 1+ h(a D ) = h a (1 ( )p− ) D 2 a . K K − p | K| p a Y|

Hence

1 1 Asai, non-E, gg 4 1 2 +ε 4 2 + 4+ dim S (Γ(N), χ) N N − (a d) 2 N N − N . k 1 1 N N 4 a2d N N N 4 X1| X| 1 X1|

When K is a field, then there exists a quartic field L/Q which contains K such that πF is L weakly equivalent to AI (τ) for some Hecke character τ : A×/L C . By comparing the Q L ×−→ × 4 conductor we see that DLNL/Q(c(τ)) = O(N ) where c(τ) is the conductor of τ. Hence we need to count the number of such Hecke characters. Given a quartic field of discriminant d, 4 1 N (c(τ)) = O(N d − ). Given a positive integer n, there are O(τ(n)) integral ideals of norm L/Q | | n, where τ(n) is the number of divisors of n. Given an ideal a, there are at most N(a)hL Hecke characters with the conductor a. Therefore, given a positive integer n, there are at most 1 + nτ(n)h n1+D 2 Hecke characters. Hence the number of Hecke characters in question is L L

1 1 4 1 1+ 2 + 4+ + (N d − ) D N D − 2 . | L| L | L| D

Let N(d) be the number of quartic ﬁelds of discriminant d which contains a quadratic subﬁeld. | | 6+ Then d x N(d) x ([6]). Hence by partial summation, the last estimation becomes O(N ). | |≤ HenceP our result follows.

2.8. Proofs of main theorems for Hecke ﬁelds. We are now ready to prove main theorems. Before going into proofs we give a precise deﬁnition of genuine forms.

Deﬁnition 2.3. Let F be a Hecke eigen Siegel cusp form of weight (k ,k ), k k 3 which 1 2 1 ≥ 2 ≥ is neither a CAP form nor an endoscopic lift. We say F is a genuine form if the corresponding automorphic representation πF is not weakly equivalent to any of a base change lift, an Asai lift, and a symmetric cube lift. 14 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

By using results in the previous section, Theorem 1.1 and 1.3 of [21] hold if we replace HE (N, χ) with the subset F HE (N, χ) F : genuine , since k { ∈ k | } F HE (N, χ) F : non-genuine = O(dim Ker(T g)) +dim SCube,gg(Γ(N), χ) { ∈ k | } k AI,gg Asai,non-E,gg 8+ε + dim Sk (Γ(N), χ)+ dim Sk (Γ(N), χ) = O(N ).

This proves Theorem 1.1. Theorem 1.2,1.3 follow from the argument in Section 6.2 and 6.3 of [45] with Theorem 1.1 of [21].

3. Galois representations for genuine forms

In this section we characterize a genuine form in terms of Galois representations. Let F be a

Hecke eigen Siegel cusp form in Sk(Γ(N)) which is neither CAP nor endoscopic. By Laumon- Weissauer, for any prime ` there exists a unique irreducible Galois representation

ρ : G := Gal(Q/Q) GSp (Q ) F,` Q −→ 4 ` such that det(I Xρ (Frob )) coincides with the Hecke polynomial at p for any prime p - `N 4 − F,` p (see (2.13) of [21] for Hecke polynomials). Since πF is weakly equivalent to a generic cuspidal representation of GSp4, it can be transfered to a cuspidal representation of GL4. The irreducibility of ρF,` follows from this fact and the main result of [3].

Theorem 3.1. The notation being as above. Then F is genuine if and only if the Zariski closure of the image of ρF,` contains Sp4 for all but ﬁnitely many `.

Proof. Let G` be the Zariski closure of the image of ρF,`. By Corollary 4.4 and Proposition 4.5 of

[3] one of the following cases happens: (1) G` contains Sp4; (2) G` is contained in the subgroup GSO(2, 2) = GL GL /GL of GSp ; (3) G is contained in the subgroup Sym3GL of GL ; (4) 2 × 2 1 4 ` 2 4 ρF,` is an induced representation of 2-dimensional Galois representation associated to a Hilbert modular form for a quadratic real ﬁeld; (5) ρF,` is an induced representation of a character.

By Theorem 8.7 for all but ﬁnitely many `, the reduction ρF,` is irreducible. In the case (2), there exists a 2-dimensional irreducible Galois representation ρ : G GL (Q ) such that ` Q−→ 2 ` 3 ρF,` = Sym ρ`. The irreducibility of ρF,` implies that of ρ`. Therefore ρ` is modular by Serre conjecture due to Khare-Wintenberger [17]. It follows that its image contains SL2(F`) otherwise 3 Sym ρ` is reducible. Hence ρ is absolute irreducible even if we restrict it to GQ(ζ`). Then by HECKE FIELDS AND n-LEVEL DENSITY 15

Kisin [18] for the ordinary case and Emerton [7] for the non-ordinary case, ρp is modular. Hence the case (2) corresponds to a symmetric cubic lift.

We now assume that F is genuine and ρF,` does not contain Sp4 for inﬁnitely many `. We denote by Σ the set of such primes `. The cases (3), (4) and (5) are excluded since F is genuine with the argument above in the case (3). Then by applying Theorem 1.0.1 of [27] for a suﬃciently large ` Σ (note that they used the notation SGO(4) instead of GSO(4)), we see that F is a ∈ Rankin-Selberg convolution which contradicts with the assumption on F .

4. Simultaneous vertical Sato-Tate theorem

Let F HE (Γ(N), χ)tm. For a prime p - N, let a ,b [ 2, 2] be Hecke eigenvalues as in ∈ k F,p F,p ∈ − [21]. Recall also the measure

+ ST µp = fp(x, y)gp (x, y)gp−(x, y) µ · ∞ on Ω:=[ 2, 2]2/S (the non-trivial element in S acts by (x, y) (y,x)on [ 2, 2]2), where − 2 2 7→ − 2 2 2 2 (p + 1) ST (x y) x y fp(x, y) = , µ = − 1 1 , 2 2 ∞ π2 − 4 − 4 √p + 1 x2 √p + 1 y2 r r √p − √p − p +1 gp±(x, y) = . 2 2 2 √p + 1 2 1 + xy 1 x 1 y √p − 4 ± − 4 − 4 q q Let C0(Ω, R) be the space of R-valued continuous functions on Ω. Then we can generalize Theorem 1.4 of [21] to ﬁnitely many primes.

Theorem 4.1. Let p ,...,p be distinct primes. Then for f C0(Ωr, R), 1 r ∈ r 1 tm f((aF,p1 ,bF,p1),..., (aF,pr,bF,pr )) = f((x1, y1),..., (xr, yr)) dµpi dk,N (χ) Ωr F HEk(Γ(N),χ) Z i=1 ∈ X Y O((p p )aφ(N)N 2), level aspect 1 ··· r − + b c  (p pr) (p pr) O 1··· + O 1··· , weight aspect  ((k1 k2+1)(k1 1)(k2 2) ((k1 k2+1)(k1+k2 3) − − − − − for some constants a, b, c > 0.

Proof. Take S0 = p ,...,p in Proposition 5.3 of [21]. Then we can follow along the proof of { 1 r} Theorem 1.3 of [21] by using Theorem 6.4 and 6.5 of [21]. 16 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

5. n-level density of degree 4 spinor L-functions

Katz and Sarnak [14] proposed a conjecture on low-lying zeros of L-functions in natural families, which says that the distributions of the low-lying zeros of L-functions in a family F is predicted by a symmetry type G(F) attached to F: For a given entire L-function L(s, π), we denote the non-trivial zeros of L(s, π) by 1 + √ 1γ . Since we don’t assume GRH for L(s, π), γ can be 2 − j j n a complex number. Let φ(x1,...,xn) = i=1 φi(xi) be an even Schwartz class function in each ˆ variables whose Fourier transform φ(u1,...,uQ n) is compactly supported. We deﬁne

(n) ∗ log cπ log cπ log cπ D (π, φ) = φ γj1 ,γj2 ,...,γjn j , ,jn 1 ··· 2π 2π 2π X where ∗ is over ji Z (if the root number is 1) or Z 0 with ja = jb for a = b, and j1,...,jn ∈ − \{ } 6 ± 6 cπ is theP analytic conductor of L(s, π). Let F(X)be the set of L-functions in F such that X

W (U)(x) = det(K0(xj,xk)) 1≤j≤n , 1≤k≤n

W (SO(even))(x) = det(K1(xj,xk)) 1≤j≤n , 1≤k≤n n

W (SO(odd))(x) = det(K 1(xj,xk)) 1≤j≤n + δ(xν)det(K 1(xj,xk)) 1≤j6=ν≤n , − 1≤k≤n − 1≤k6=ν≤n ν=1 X 1 W (Sp)(x) = det(K 1(xj,xk)) 1≤j≤n , W (O)(x) = (W (SO(even))(x) + W (SO(odd))(x)), − 1≤k≤n 2

sin π(x y) sin π(x + y) where K (x, y) = − + , 1, 0 . π(x y) π(x + y) ∈ {± } We will study the family− of degree 4 spinor L-functions L(s, π , Spin) for F HE (N). We F ∈ k 1 2 may assume that π is not a CAP form. (For a CAP form, a (p) 4p 2 and a (p ) 4p in F | F | ≤ | F | ≤ (5.2). Hence if the support of φ is smaller than ( 1, 1), then the sum over p in (5.2) is O(N). − But the dimension of the space of CAP forms is O(N 7+). Hence it is negligible.) HECKE FIELDS AND n-LEVEL DENSITY 17

Let L(s, πF , Spin) be the degree 4 spinor L-function. Let

∞ s L(s, πF , Spin) = λ˜F (n)n− . n=1 X It satisﬁes the functional equation:

s k1 + k2 3 k1 k2 +1 Λ(s, π , Spin) = q(F ) 2 Γ (s + − )Γ (s + − )L(s, π , Spin), F C 2 C 2 F Λ(s, π , Spin) = (π )Λ(1 s, π , Spin), F F − F where (π ) 1 and N q(F ) N 4. F ∈ {± } ≤ ≤ Let φ be a Schwartz function which is even and whose Fourier transform has a compact support. Deﬁne γ D(π , φ, Spin) = φ F log c , F 2π k,N γ XF 1 2 2 where log ck,N = d F HE (N) log c(F ) for k =(k1,k2), and c(F )=(k1+k2) (k1 k2+1) q(F ) k,N ∈ k − is the analytic conductor.P We showed in [21] that

1 1 ω(N) D(πF , φ, Spin) = φ(0) + φ(0) + O , dk,N 2 log ck,N F HEk(N) ∈ X where ω(N) is the number of distinct prime factorsb of N. So the possible symmetry type could be O, SO(even) or SO(odd) but we cannot distinguish them because the support of φˆ is too small. In order to distinguish them, we need to compute the n-level density. n n n Let λF (p ) be the eigenvalue of the Hecke operator T (p ) for p - N. Here T (p ) is the sum of a a a +κ a +κ Hecke operators Tm for m = text(p 1 ,p 2,p− 1 ,p− 2 ), where 0 a2 a1 κ. k +k −3 ≤ ≤ ≤ n n n 1 2 Let T 0(p ) = T (p )p− 2 . Then

k +k 3 L(s, π , Spin) = L(s 1 2− ,F, Spin), F − 2 s 1 and L(s,F, Spin) = QF,p(p− )− ,

Q 1 k +k 4 2 1 ∞ n n Q (t)− = (1 p 1 2− t )− λ (p )t . F,p − F n=0 X Therefore s 1 L(s, πF , Spin) = QF,p0 (p− )− , where Y 1 1 2 1 ∞ n n Q0 (t)− = (1 p− t )− λ0 (p )t . F,p − F n=0 X 18 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

˜ Hence λF0 (p) = λF0 (p), and n 2 n n +i 2i p− 2 + p− 2 λ0 (p ), if n is even  F ˜ n i=1 λF0 (p ) =  n−1 X  2  n−1 +i 2i+1 p− 2 λF0 (p ), if n is odd Xi=0  2i Therefore by [21, Theorem 8.1], we have the following. Note that T 0(p ) is a sum of Hecke 3i operators and only T 0i contributes the main term p− . p E4

Theorem 5.1. Put k =(k ,k ), k k 3. 1 2 1 ≥ 2 ≥ (1) (level-aspect) Fix k ,k . Then as N , 1 2 → ∞ n n 2 2i 2 c p 2 + O(N p ), if n is even 1 n i=0 − − − λ˜F (p ) = , dk,N  2 c F HE (N) OP(N − p ), if n is odd ∈ Xk  for some constant c > 0.  (2) (weight-aspect) Fix N. Then as k + k , 1 2 → ∞ n n c −d 2 2i p p p− 2 − + O + O , if n is even 1 n i=0 (k1 1)(k2 2) (k1 k2+1)(k1+k2 3) λ˜ (p ) = − − − − , F  c −d dk,N p p F HE (N) OP + O , if n is odd k  (k1 1)(k2 2) (k1 k2+1)(k1+k2 3) ∈ X − − − − for some constantsc,d> 0.

Since λ˜F is multiplicative, we have proved Theorem 1.4.

For one-level density, the root number (πF ) did not play a role. However, higher level density depends on the root number.

Let S±(N) be the subspace of S (N) with the root number (π ) = 1. Let HE±(N)be a k k F ± k basis of S±(N) consisting of Hecke eigenforms outside N, and denote HE±(N) = d± . k | k | k,N When N = 1 (i.e., level one case), we have (π )=( 1)k2 ([39]). (In this case, k k should F − 1 − 2 + be even.) Hence HEk (1) = HEk(1) when k2 is even, and HEk−(1) = HEk(1) when k2 is odd. However, when N is large, we expect

1 9 Conjecture 5.1. dk±,N = 2 dk,N +O(N − ), and Theorem 1.4 is true for each subspace HEk±(N), vp(m) namely, for example, in level aspect, (m = p p )

1 1 Q 2 vp(m) 2 c λ˜F (m) = δm− 2 (1 + p− + + p− ) + O(N − m ), d+ ··· k,N F HE+(N) p m ∈ Xk Y| HECKE FIELDS AND n-LEVEL DENSITY 19

We assume Conjecture 5.1 in this paper. The calculation of the n-level density is well-known. But for the sake of completeness, we give an outline. We follow closely [5], [34].

1 5.1. The case (πF )=1. We denote the non-trivial zeros of L(s, πF ,Spin) by σF,i = 2 + √ 1γ . Without assuming the GRH for L(s, π , Spin), we can order them as − F,i F

Re(γF, 2) Re(γF, 1) 0 Re(γF,1) Re(γF,2) . ···≤ − ≤ − ≤ ≤ ≤ ≤··· Let φ(x ,...,x ) = φ (x ) φ (x ), where each φ is an even Schwartz function and φˆ(u ,...,u ) = 1 n 1 1 ··· n n i 1 n φˆ (u ) φˆ (u ). For a ﬁxed n > 0, assume the Fourier transform φˆ of φ is supported in 1 1 ··· n n i i ( β ,β ) for i =1,...,n.(β < 1 can be explicitly determined.) The n-level density function is − n n n log c log c log c (n) ∗ k,N k,N k,N (5.1) D (πF , φ, Spin) = φ γj1 ,γj2 ,...,γjn j , ,jn 1 ··· 2π 2π 2π X 1 where ∗ is over ji = 1, 2,... with ja = jb for a = b, and log ck,N = + + log c(F ) j1,...,jn d F HE (N) ± ± 6 ± 6 k,N ∈ k 2 2 for k =(Pk1,k2), and c(F )=(k1 + k2) (k1 k2 + 1) q(F ) is the analytic conductor.P Let −

L0 ∞ s (s, π , Spin) = Λ(n)a (n)n− . − L F F n=1 X Recall the one-level density function ([21], page 68):

1 (1) 2 aF (p) log p (5.2) D (πF , φ, Spin) = φ(0) φ dk,N − dk,N log ck,N √p log ck,N F HEk(N) F HEk(N) p ∈ X ∈ X X b b 2 a (p2) 2log p 1 F φ + O . − dk,N log ck,N p log ck,N log ck,N F HEk(N) p ∈ X X b Let L run over all ways of decomposing 1,...,n into disjoint subsets [L ,...,L ]. Let ν = ν(L). { } 1 ν For each l =1,...,ν, let Φl = i S φi. ∈ l By Rubinstein [34], Q

1 (n) 1 n ν(L) (5.3) D (πF , φ, Spin) = ( 2) − d+ d+ − k,N F HE+ k,N F HE+ L ∈Xk,N ∈Xk,N X ν(L) 2 2 1 ( Ll 1)! Φl(x)dx Σl,1(Φl) Σl,2(Φl) + O , · | | − R − log R − log ck,N log ck,N Yl=1 Z 20 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI where a (p)log p log p Σ (Φ ) = F Φ , l,1 l p l log c p √ k,N X a (p2)log pb 2log p Σ (Φ ) = F Φ . l,2 l p l log c p k,N X b Note that 2 2 1 Φ (x)dx Σ (Φ ) Σ (Φ ) = D(1)(π , Φ ) + O . l − log c l,1 l − log c l,2 l F l log c ZR k,N k,N k,N We show, following Lemma 2 in [34], that we can ignore O 1 terms in (5.3). log ck,N Lemma 5.2.

ν(L) 1 2 2 + Φl(x)dx Σl,1(Φl) Σl,2(Φl) d R − log ck,N − log ck,N k,N F HE+ l=1 Z ∈Xk,N Y ν(L) 1 (1) 1 = + D (πF , Φl) + O . d log ck,N k,N F HE+ l=1 ∈Xk,N Y We prove two lemmas analogous to Claim 2 and Claim 3 in Rubinstein [34].

Lemma 5.3. (Claim 2 of [34])

a Λ(mj)aF (mj) log mj + a 1 Φlj dk,N (log ck,N ) − . √mj log ck,N F HE+ mi≥1 j=1 Xk,N m mX···m 6= Y ∈ 1 2 a b where mi’s are a prime or a square of a prime.

Proof. By changing the order of the sums, we need to consider, for p ,p , ,p , q , q , , q , 1 2 ··· r 1 2 ··· t distinct primes,

(5.4) a (p )e1 a (p )er a (q2)γ1 a (q2)γt, F 1 ··· F r F 1 ··· F t F HE+ ∈Xk,N where e + + e + γ + + γ = a. 1 ··· r 1 ··· t Now, by rearranging, if there exists e = 1 for some i, we can assume that e = = e = 1 and i 1 ··· b e e i eb+1 > 1, ...er > 1. Then use the fact that λF (p) = i=0 ciλF (p ) for some constants ci. Hence P HECKE FIELDS AND n-LEVEL DENSITY 21 a (p )e1 a (p )er a (q2)γ1 a (q2)γt is the sum of λ (p p n), where n is not divisible by F 1 ··· F r F 1 ··· F t F 1 ··· b p1,...,pb. Hence by Theorem 1.4, (5.4) gives rise to an error term. So we have the result. Suppose e > 1 for all i. Then e 3 for some i by assumption. Then we have the result by i i ≥ using the bounds ([5, (4.10) and (4.11)]) :

βn R 2 , if i =1 a (p) i(log p)i F  2 (5.5) | | (log R) , if i =2 , pi  p Rβn  ≤X O(1), if i 3. p ≥   a (p2) j(log p)j  O(log R), if j =1 (5.6) | F | pj  p Rβn/2 O(1), if j 2. ≤X  ≥ 

Lemma 5.4. (Claim 3 of [34])

a l 1 2 Λ(mj)aF (mj) log mj + − Φlj d log ck,N  √mj log ck,N  k,N + mi≥1 j=1 F HEk,N ∈X m1m2X···ma= Y  b  c S2 /2 S2 | | δπ | | S /2 1 = Φ (u)du 2| 2| u Φ (u)Φ (u)du + O ,  2 l   | | aj bj  log c S ⊆S c R R k,N 2 l S2 Z P2 j=1 Z |SX| even Y∈ X Y 2  b   b b  where mi’s are a prime or a square of a prime, and S = l1,...,la , S2⊆S is over all subsets { } |S2| even S of S whose size is even, and is over all ways of pairing up the elements of S . 2 P2 P 2 P Proof. First of all, in (5.4), if e 4 or γ 2 for some i,j, then by (5.5) and (5.6), those terms i ≥ j ≥ + a 1 are majorized by dk,N (log ck,N ) − . Hence we only need to consider the sum

a (p )2 a (p )2a (q2) a (q2), F 1 ··· F r F 1 ··· F t F HE+ ∈Xk,N where 2r + t = a. In this case, S2 = l1,l2,...,l2r 1,l2r , and S = l1,...,l2r,l2r+1,...,la . { − } { } 2 1 1 We use the fact that a (p) = 1 + p− (polynomials in p− )+sum of Hecke operators, and F × a (p2) = 1 + p 1 (polynomials in p 1)+sum of Hecke operators. F − − × − We deﬁne 1 ∆ (Φ )=Σ (Φ ) + Φ (0) log c . l,2 l l,2 l 4 l k,N 22 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

Since R Φl(x)dx = Φl(0), R ν(L) 1 b n ν(L) (5.3) = ( 2) − ( Fl 1)! d+ − | | − k,N F HE+ L l=1 ∈Xk,N X Y 1 2 2 1 Φ (0) + Φ (0) Σ (Φ ) ∆ (Φ ) + O . · l 2 l − log c l,1 l − log c l,2 l log c k,N k,N k,N The following three lemmasb enable us to ﬁnd an explicit expression for (5.3).

Lemma 5.5. 1 2 a 1 + − ∆l1,2(Φl1) ∆la,2(Φla) . d log ck,N ··· log ck,N k,N F HE+ ∈Xk,N Lemma 5.6. 1 2 a 1 + − Σl1,1(Φl1) Σlr,1(Φlr )∆lr+1,2(Φlr+1 ) ∆la,2(Φla) = O . d log ck,N ··· ··· log ck,N k,N F HE+ ∈Xk,N Lemma 5.7. 1 2 a + − Σl1,1(Φl1) Σla,1(Φla) d log ck,N ··· k,N F HE+ ∈Xk,N a a 1 2 2 u Φ (u)Φ (u) du + O , if a is even lai lbi  R | | log ck,N = XP2 Yi=1 Z  1 O , b b otherwise. log c k,N  Here is over all ways of paring up the elements of 1, 2,...,a . If a =2r, it runs through P2 { } (2r)! productsP of 2-cycles of the form (l1l2) (l2r 1l2r). There are of them. ··· − 2rr! Therefore, we have proved

Theorem 5.8.

ν(L) 1 (n) n ν(L) (5.7) D (πF , φ, Spin) = ( 2) − ( Fl 1)! d+ − | | − k,N F HE+ L l=1 ∈Xk,N X Y |S| 2 1 |S| 1 Φl(0) + Φl(0) 2 2 u Φl (u)Φl (u) du + O ,  ai bi  c 2 R | | log ck,N S even l S ! P2 i=1 Z X Y∈ X Y  c  b b  HECKE FIELDS AND n-LEVEL DENSITY 23 where S runs through the subset of even cardinality in 1,...,ν(L) , and is over all ways of { } P2 pairing up the elements of S. P

We summarize it as

Theorem 5.9. Let φ(x ,...,x ) = φ (x ) φ (x ), where each φ is an even Schwartz function 1 n 1 1 ··· n n i and φˆ(u ,...,u ) = φˆ (u ) φˆ (u ). Assume the Fourier transform φˆ of φ is supported in 1 n 1 1 ··· n n i i ( β ,β ) for i =1, , n. Then − n n ··· 1 (n) ω(N) + D (πF , φ, Spin) = φ(x)W (SO(even))(x) dx + O . d Rn log ck,N k,N F HE+ Z ∈Xk,N 5.2. The case (π ) = 1. If (π ) = 1, L(s, π , Spin) always has a family zero at s = 1 . By F − F − F 2 Rubinstein [34], the n-level density function is

log c log c log c (n) ∗ k,N k,N k,N D (πF , φ, Spin) = φ γj1 ,γj2 ,...,γjn j1, ,jn 2π 2π 2π ··· Xlog c log c log c ∗ k,N k,N k,N = φ γj1 ,γj2 ,...,γjn j =0,...,jn=0 16 6 2π 2π 2π X n log c log c log c log c log c ∗ k,N k,N k,N k,N k,N + φ γj1 ,γj2 ,...,γjν−1 , 0,γjν+1 ,...,γjn . jν =0,jk=0,k=ν 2π 2π 2π 2π 2π ν=1 6 6 X X By Rubinstein [34], the ﬁrst term gives rise to log c log c log c 1 ∗ k,N k,N k,N φ γj1 ,γj2 ,...,γjn d− j1=0,...,jn=0 2π 2π 2π k,N F HE− 6 6 ∈Xk,N X ν(L) n ν(L) 2Σl,1(Φl) 2Σl,2(Φl) 1 = ( 2) − ( Fl 1)! Φl(0) Φl(0) + O − | | − − log ck,N − log ck,N − log ck,N XL Yl=1 ν(L) c n ν(L) 1 2Σl,1(Φl) 2∆l,2(Φl) 1 = ( 2) − ( Fl 1)! Φl(0) Φl(0) + O . − | | − − 2 − log ck,N − log ck,N log ck,N XL Yl=1 It is equal to c ν(L) n ν(L) 1 ( 2) − ( F 1)! Φ (0 Φ (0) − | l| − l − 2 l L l=1 S even l Sc ! X Y X Y∈ |S| c 2 |S| 1 2 2 u Φl (u)Φl (u) du + O ,  ai bi  · R | | log ck,N P2 i=1 Z X Y   d d  24 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI which equals the n-level density of the symplectic type. We summarize it as

Theorem 5.10. Let the notations be as in Theorem 5.9. Then

1 (n) ω(N) D (πF , φ, Spin) = φ(x)W (SO(odd))(x) dx + O . d− Rn log ck,N k,N F HE− Z ∈Xk,N 6. n-level density of degree 5 standard L-functions

Let L(s, πF , St) be the degree 5 standard L-function. Let

∞ s L(s, πF , St) = µF (n)n− . n=1 X s It satisﬁes the functional equation: Let Λ(s, π , St) = q(F, St) 2 Γ (s)Γ (s + k 1)Γ (s + k F R C 1 − C 2 − 2)L(s, πF , St). Then Λ(s, π , St) = (π , St)Λ(1 s, π , St), F F − F where (π , St) 1 and N q(F, St) N 28. Lapid [24] showed F ∈ {± } ≤ ≤

Proposition 6.1. Let πF be as above. Then (πF , St) = 1.

In fact, Lapid proved it only for globally generic cusp forms. However, holomorphic cusp forms are always in the same L-packet with a globally generic cusp form. Hence the result follows. Because of the above proposition, we expect that the symmetry type of L(s, πF , St) is Sp. However, we need the following conjecture. We showed it in [21, Proposition 9.5] when m is of the form m = pa1 par , where p ’s are distinct primes, and a =1, 2 for each i. 1 ··· r i i Conjecture 6.1. Put k =(k ,k ), k k 3. 1 2 1 ≥ 2 ≥ vp(m) (1) (level-aspect) Fix k1,k2. Let m = p m p . Then as N , | → ∞ Q 1 1 1 2 c µF (m) = (δp,m + p− h(p− )) + O(N − m ), dk,N F HE (N) p m ∈ Xk Y|

1, if vp(m) is even where δ = , p,m  0, otherwise and h is a polynomial with integer coeﬃcients and c > 0 is a constant.  HECKE FIELDS AND n-LEVEL DENSITY 25

(2) (weight-aspect) Fix N. Then as k + k , 1 2 → ∞ c d 1 1 1 m m µF (m) = (δp,m+p− h(p− ))+O +O . dk,N (k1 1)(k2 2) (k1 k2 + 1)(k1 + k2 3) F HEk(N) p m ∈ X Y| − − − − Let

L0 ∞ s (s, π , St) = Λ(n)b (n)n− . − L F F n=1 X 2 1 1 We use the fact that b (p) = 1+p− (polynomials in p− )+sum of eigenvalues of Hecke operators, F × and b (p2)=1+ p 1 (polynomials in p 1)+sum of eigenvalues of Hecke operators. F − × − Let φ be a Schwartz function which is even and whose Fourier transform has a compact support. Deﬁne γ D(π , φ, St) = φ F log c , F 2π k,st,N γ XF 1 2 where log ck,st,N = d F HE (N) log c(F, St), and c(F, St) = (k1k2) q(F,St) is the analytic k,N ∈ k conductor. P We showed in [21] that

1 1 1 ω(N) D(πF , St, φ) = φ(0) φ(0)+O = φ(x)W (Sp)(x) dx++O . dk,N −2 log ck,st,N R log ck,st,N F HEk(N) Z ∈ X b Let φ(x ,...,x ) = φ (x ) φ (x ), where each φ is an even Schwartz function and φˆ(u ,...,u ) = 1 n 1 1 ··· n n i 1 n φˆ (u ) φˆ (u ). For a ﬁxed n > 0, assume the Fourier transform φˆ of φ is supported in 1 1 ··· n n i i ( β ,β ) for i =1,...,n.(β < 1 can be explicitly determined.) The n-level density function is − n n n log c log c log c (n) ∗ k,st,N k,st,N k,st,N D (πF , φ, St) = φ γj1 ,γj2 ,...,γjn j , ,jn 1 ··· 2π 2π 2π X where ∗ is over ji = 1, 2,... with ja = jb for a = b. Then as in the degree 4 spinor j1,...,jn ± ± 6 ± 6 L-functions,P we can show Theorem 1.6. (We can prove the analogues of (5.5) and (5.6) for i bF (p ) from [21, Appendix]: Let Π be the cuspidal representation of GL5/Q such that L(s, Π) = L(s, π , St). Then (5.5) follows from the fact that L(s, Π Π) converges absolutely for Re(s) > 1 F × and has a simple pole at s = 1; (5.6) follows from the fact that b (p2) b (p) 2 for any p, and | F || F | 2 1 2 the Ramanujan bound b (p ) 5p − 26 .) | F | ≤ 26 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

7. Paramodular forms

In this section, we ﬁx a square free positive integer N. We deal with a compact subgroup para Kp (N) of level N in G(Qp), which is deﬁned by

para 1 K (N) = xM (Z )x− G(Q ), x = diag(1, 1,N, 1), p 4 p ∩ p para where G means GSp4 as in [21, Section 2]. We have an open compact subgroup K (N) = para para p Kp (N) in G(Aﬁn). Furthermore, an arithmetic subgroup Γ (N) of G(Q) is given by Γpara(N) = G(Q) (G(R)Kpara(N)), called the paramodular subgroup. An element w in Q ∩ N,p G(Qp) is given by 0 0 01  0 0 1 0 wN,p = − G(Qp).  0 N 0 0 ∈  −    N 0 00   Then, the Atkin-Lehner involution onG(Qp) is provided by the double coset

para para para para Kp (N)wN,pKp (N) = wN,pKp (N) = Kp (N)wN,p.

Let k =(k ,k ), k k 4, and S (Γpara(N)) denote the space of paramodular forms of weight 1 2 1 ≥ 2 ≥ k para para k with the trivial central character. Set dk = dim Sk(Γ (N)). Then by [11],

para 7 3 1 2 d =2− 3− 5− (k 1)(k 2)(k k + 1)(k + k 3) (p + 1) k 1 − 2 − 1 − 2 1 2 − p N Y| + O(N(k k + 1)(k + k 3)) + O(N(k 1)(k 2)). 1 − 2 1 2 − 1 − 2 − 2 2 Here note that p N (p +1)= cN N for some constant 1

0 ˜ h = fS ,α f charKp p S S0 v N ∈ \ Ot{ | ∞} ˜ where f denotes p N,p-M charKpara(N) p M charw Kpara(N) . By the same argument as ⊗ | p ⊗ | N,p p the proof of [21, Proposition 6.3], one can N show that there exist positive constants a0 and b0 such that k1+k2 para 1 2 κ I (f) vol(K (N))− ν(α) −0 = O(p 0 (k k + 1)(k + k 3)) 2 × ×| |S S 1 − 2 1 2 − k1+k2 para 1 2 κ I (f) vol(K (N))− ν(α) −0 = O(p 0 (k 1)(k 2)), 3 × ×| |S S 1 − 2 − k +k 1 2 0 0 para 1 2 a κ+b I (f) + I (f) vol(K (N))− ν(α) −0 = O(p 0 (k + k 3)), { 4 6 } × ×| |S S 1 2 − k +k 1 2 0 0 para 1 2 a κ+b I (f) vol(K (N))− ν(α) −0 = O(p 0 (k + k 3)) 5 × ×| |S S 1 2 − for any (k ,k ), κ 1, S0, and f 0 , which satisfy the conditions k k 3, f 0 1 2 ≥ S ,α 1 ≥ 2 ≥ S ,α ∈ ur κ H (G(QS0 )) , and N is prime to p S0 p. Notice that an important result [44, Proposition ∈ 8.7] was used for the proof. Thus, thisQ theorem is derived from the above estimates.

non-C para Let Sk (Γ (N)) denote the space of paramodular forms whose associated representations CAP para are not related to the Saito-Kurokawa representations. We write Sk (Γ (N)) for the sub- para space consisting of Saito-Kurokawa liftings in Sk(Γ (N)) (i.e., it is called the Maass space). CAP para non-C para para Hence, Sk (Γ (N)) is the orthogonal complement of Sk (Γ (N)) in Sk(Γ (N)) by the Petersson inner product. Namely, we have an isomorphism

non-C para para Sk (Γ (N)) ∼= Nπ(K (N)) π=π∞ π M⊗ fin where π moves over automorphic representations of G(A) such that π has the trivial central character, π is not a Saito-Kurokawa representation, π is isomorphic to the holomorphic discrete ∞ series of G(R) with the Harish-Chandra parameter (k 1,k 2), and the subspace N (Kpara(N)) 1− 2− π para of K (N)-fixed vectors in πfin is not trivial. By [52, Theorem 3.3] or [22, a comment for n = 2 after Theorem A], for each above πfin = v< πv, if πv is spherical, then πv satisfies the ⊗ ∞ Ramanujan conjecture. Hence, all spherical representations πv belong to the class I in [31, Table A.13 in p.293] (see also [37, Table 3]). Hence, by [37, 31], one has an isomorphism

N (Kpara(N)) N (Kpara(N)) π ∼= πv v v< O∞ 28 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

para para where Nπv (Kp (N)) denotes the subspace of Kv (N)-fixed vectors in πv, and for each prime p N, π satisfies (i) or (ii); | p para (i) πp is spherical and dim Nπp (Kp (N))=2 , para (ii) πp is non-spherical and dim Nπp (Kp (N))= 1. para As for the case (i), the trace of the Atkin-Lehner involution on Nπp (Kp (N)) is zero. When πp satisfies (ii), the eigenvalue of the Atkin-Lehner involution means the ε-factor of its local spinor L-factor (see [31, Theorem 5.7.3 in p.185] or [37, Proposition 1.3.1]). 1 CAP para By [38], it is obvious that dpara tr(Tm0 wN,M Sk (Γ (N))) is negligible. Furthermore, let k,N | non-C,new para non-C para Sk (Γ (N)) denote the subspace of newforms in Sk (Γ (N)). Here, newform means that its associated automorphic representation π has no Kpara(M)-fixed vectors for any natural number M such that M N and M

non-C,new para para Sk (Γ (N)) ∼= Nπ(K (N)) π= v πv M⊗ where π moves over all automorphic representations satisfying the same conditions as above and π is of the case (ii) for each p N. Therefore, one has p |

non-C,new para ω(M) non-C para tr(T 0 S (Γ (N))) = ( 2) tr(T 0 S (Γ (N/M))) m| k − m| k M N X| where ω(M) is the number of distinct prime factors of M. non-C,new,+ para non-C,new, para Let Sk (Γ (N)) (resp. Sk −(Γ (N))) denote the subspace of newforms whose ε-factors are 1 (resp. 1). By the above mentioned arguments, one gets −

non C,new, para tr(T 0 S − ±(Γ (N))) m| k 1 non-C,new para k non-C para = tr(T 0 S (Γ (N))) ( 1) 2 tr(T 0 w S (Γ (N))) , 2 m| k ± − m N,N | k h i Therefore, if we set

para,new, non-C,new, para dk ± = dim Sk ±(Γ (N))), then by [11] and Theorem 7.1,

para,new, 8 3 1 2 d ± =2− 3− 5− (k 1)(k 2)(k k + 1)(k + k 3) (p 1) k 1 − 2 − 1 − 2 1 2 − − p N Y| + O(N(k k + 1)(k + k 3)) + O(N(k 1)(k 2)). 1 − 2 1 2 − 1 − 2 − HECKE FIELDS AND n-LEVEL DENSITY 29

Furthermore, by Theorem 7.1, there exist absolute constants a and b such that for each prime a a a +κ a +κ p - N and m = diag(p 1 ,p 2,p− 1 ,p− 2 ), a , a , κ Z satisfying 0 a a κ and 1 2 ∈ ≤ 2 ≤ 1 ≤ m Z (Q), 6∈ G aκ+b 1 non C,new, para p para,new, tr(Tm0 Sk − ±(Γ (N))) = B1 + B2 + O( ). d ± | (k1 k2 + 1)(k1 1)(k2 2) k,N − − − Hence we have proved Conjecture 5.1 for paramodular newforms. Therefore, we have proved n-level density for spinor L-functions of paramodular newforms in weight aspect (analogues of Theorem 5.9 and Theorem 5.10 for paramodular forms). In a similar way, we can show a simultaneous vertical Sato-Tate theorem for paramodular forms (analogue of Theorem 4.1).

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Henry H. Kim, Department of mathematics, University of Toronto, Toronto, Ontario M5S 2E4, CANADA, and Korea Institute for Advanced Study, Seoul, KOREA E-mail address: [email protected] 32 HENRY H. KIM, SATOSHI WAKATSUKI AND TAKUYA YAMAUCHI

Satoshi Wakatsuki, Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa, 920-1192, JAPAN E-mail address: [email protected]

Takuya Yamauchi, Mathematical Inst. Tohoku Univ., 6-3,Aoba, Aramaki, Aoba-Ku, Sendai 980- 8578, JAPAN E-mail address: [email protected]