CUSPIDALITY AND THE GROWTH OF FOURIER COEFFICIENTS: SMALL WEIGHTS ¨ SIEGFRIED BOCHERER AND SOUMYA DAS

Abstract. We characterize Siegel cusp forms in the space of Siegel modular forms of small weight k ≥ n + 4 on the congruence subgroups Γn0 (N ) of any degree n and any level N , by a suitable growth of their Fourier coefficients (eg., by the well known Hecke bound) at any one of the cusps. For this, we use the formalism of Jacobi forms and the ‘Witt-operator’on modular forms.

1. Introduction The question of characterizing cusp forms by information on the growth of their Fourier coefficients at any cusp for congruence subgroups of the integral symplectic group has received quite a bit of attention. This question seems to have become fascinating in recent times following a proposal by W. Kohnen [9] who considered this question in the context of elliptic modular forms (see [11]). In the context of Jacobi, Siegel or Hilbert modular forms, see the works [3], [12], [14], [15], [13]; all of which answer this converse question assuming a suitable growth of the Fourier coefficients either in the case of level one or for the congruence subgroup Γn0 (N ) (N square–free) at the cusp ∞. It is desirable to answer the above question in any degree and for arbitrary congruence subgroups. Even more interestingly, one should be able to answer the question based on information at any one cusp, i.e., the role of the cusps should be uniform. This was considered recently by the authors in [4], where, by using a ‘local’ version of the Andrianov’s identity, they were able to show that if a Siegel modular form F of weight k on any congruence subgroup Γ ⊂ Sp(n, Z) satisfies the property that a(F, T ) F det(T )α (T a positive definite ‘half-integral’ matrix) at any one of cusps of Γ, with α < k − n and k ≥ 2n, then F must be a cusp form. The previous mentioned result may be considered as a statement valid only for the so-called “large weights” k > 2n, a common feature in the theory of Siegel modular forms. This restriction can not be removed by the local method adopted 2000 Mathematics Subject Classification. Primary 11F30; Secondary 11F46. Key words and phrases. Small Weights, Siegel modular forms, Growth of Fourier coefficients, Hecke bound. 1

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

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in [4], as it depends on the availability of robust estimates for eigenvalues of cusp forms on Siegel congruence subgroups in higher degrees; and this was also mentioned in loc. cit.. It is interesting and certainly desirable to obtain results for the so-called “small weights” which are roughly half of the size of the large weights. The aim of this paper is to provide an answer to the above-mentioned result in the case of small weights for the important class of subgroups of the form Γn0 (N ). This improvement is due to the use of either the Fourier-Jacobi expansion or the so called Witt-operator on modular forms. In this way we do not have to encounter the restriction on the weights due to estimates of eigenvalues. Let us now state our main result more precisely. Let Mkn (N ) denote the space of Siegel modular forms of weight k on the group Γn0 (N ). Theorem 1.1. Let the Fourier coefficients a(F, T ) of F ∈ Mkn (N ) satisfy a(F, T ) F (det T )α

(T ∈ Λ+ n ),

for some α ∈ R in some cusp. Suppose that (i) α < k − (ii) α < k −

n+3 , 2 n+1 , 2

k≥

n 2

+ 1 and N ≥ 1, or

k ≥ max{2, n−1 } and N is square-free or equals 1. 2

Then F is cuspidal. The proof is given in section 4.2. Important ingredients in the proof of Theorem 1.1 are the certain ‘nice’ explicit representatives of the zero and onedimensional cusps for Γn0 (N ) for any N . We believe that these are not available in the literature and section 4.1 has been devoted to this end. For a detailed discussion on the application of this description of cusps to our question, see section 4. We use these representatives to reduce the question to that of Siegel modular forms of small degrees (actually degrees 1 or 2) on principal congruence subgroups and then appeal to the ‘local method’ in [4]. In fact, implicit in the proof of Theorem 1.1 is a method to deal with the case of Jacobi forms (e.g., as in [13]) of higher degrees, see Remark 4.6. A corollary to our main result is stated in section 4.5. It calculates the constant βkn (Γ), which was originally defined in [3], for certain congruence subgroups Γ. See Corollary 4.9. This constant measures the amount of cancellation in the positivedefinite Fourier coefficients of elements in the space of Eisenstein series. Part (ii) of Theorem 1.1 implies that for Γ = Γn0 (N ) with N square–free and k ≥ 2n + 2, βkn (Γn0 (N )) = k −

n+1 . 2

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This settles the question on cancellations of Fourier coefficients of Eisenstein series asked in [3]; and implies that there can be no uniform cancellations. This also lead us to make a precise conjecture about its value in general. See Conjecture 4.10. 2. Acknowledgements We thank the Department of Mathematics, University of Tokyo, Indian Institute of Science, Bangalore and HRI, Allahabad, where parts of this work was done, for providing a very pleasant working atmosphere. Finally, the second author acknowledges financial support from UGC-DST India and IISc., Bangalore during this work. 3. Notation and setup For basic facts about Siegel modular forms we refer to [1] or [6]. The symplectic similitude group GSp+ (n, R) defined as GSp+ (n, R) := {g t Jg = µ(g)J,

µ(g) ∈ R>0 },

acts on Siegel’s half-space Hn defined as Hn := {Z = Z t ∈ M (n, C) | (Z − Z)/2i is positive definite}, in the usual way by ghZi = (AZ + B)(CZ + D)−1 . For a function f : Hn −→ C and k an integer, we define the “slash” operator of weight k by (f |k g)(Z) = det(CZ + D)−k f (ghZi)

B ) ∈ GSp+ (n, R)). (g = ( CA D

Further, the symplectic group Sp(n, R) ⊂ GSp+ (n, R) consists of all g with µ(g) = 1 and Sp(n, Z) denotes the subset of Sp(n, R) with integral entries. A subgroup Γ ⊂ Sp(n, Z) is called a congruence subgroup if it contains the Siegel principal subgroup Γn (N ) for some N . A holomorphic function f on Hn is called a modular form for Γ of weight k if it satisfies the transformation law f |k γ = f

B ) ∈ Γ), (γ = ( CA D

with the additional condition of being holomorphic at the cusps when n = 1. We denote the space of all such functions by Mkn (Γ). If Γ = Γn0 (N ), we denote Mkn (Γn0 (N )) simply by Mkn (N ). Further, we set Mkn = Mkn (1). For a congruence subgroup Γ, we define the set of 0 and 1-dimensional cusps to be the double coset spaces C := Γ\Γn /Γn,∞ ,

C1 := Γ\Γn /Cn,n−1

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respectively; where Γn,∞ (resp. Cn,n−1 ) denotes the Siegel (resp. Klingen) parabolic subgroup of Γn defined as: B) ∈ Γ | C = 0 Γn,∞ := {( CA D n n,n }, ∗ ∗ Cn,n−1 := {g ∈ Γn | g = ( 01,2n−1 ∗ )},

where 0r,s denotes the r × s zero-matrix. It is known that both C and C1 are the union of finitely many double cosets represented by, say, (ωj ) and (γi ) respectively. By abuse of notation, we refer to the ωj and the γi s’ as the ‘0-dimensional (resp. 1-dimensional) cusps’. More information on the cusps can be obtained from section 4.1. At each cusp ωj as above, an element f ∈ Mkn (Γ) has a Fourier expansion (for some positive integer M depending on Γ and the ωj ’s): X (3.1) (f |k ωj )(Z) = a(f |k ωj , T )e(T Z/M ), T ∈Λn

where tr A denotes the trace of the matrix 2πiA and e(T Z) := exp(tr T Z). Further, 1 Λn := {S = S t ∈ M (n, Z) | Si,i ∈ Z, and S is positive semi-definite}; 2 + and Λn denotes the subset of Λn consisting of positive definite matrices. We next define the Siegel’s Φ-operator on Mkn by the following: Definition 3.1. Let f ∈ Mkn and t > 0. (3.2)

Φf (Z) := lim f ( Z0 it0 ) t→∞

Then it is well known [6] that Φf ∈ Mkn−1 . Moreover, f ∈ Mkn is a cusp form if and only if f is in the kernel of the Φ operator. In the case of an arbitrary congruence subgroup Γ, one defines by the same way operators Φγi for each 1dimensional cusp γi of Γ, and g ∈ Mkn (Γ) is a cusp form if and only if it is in the kernel of all such operators, see e.g., [1]. Alternatively, one knows that (see [1]) the space of cusp forms, denoted by consists of those f ∈ Mkn (Γ) for which the Fourier expansion (3.1) is supported on elements of Λ+ n at all the cusps of Γ. We end this section by recalling the main result from [4], which we refer to as the ‘local method’. Skn (Γ),

Theorem 3.2 ([4]). Let Γ be a congruence subgroup of Γn such that the Fourier coefficients a(F, T ) of F ∈ Mkn (Γ) satisfy a(F, T ) F (det T )α

(T ∈ Λ+ n ),

for some α ∈ R in some cusp of Γ. Suppose that k ≥ 2n and α < k − n. Then F is a cusp form.

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In the rest of the paper, we say that a modular form f ∈ Mkn (Γ) satisfies the condition K(α) (α ∈ R) at some cusp if the Fourier coefficients a(f, T ) of f at that cusp satisfies a(f, T ) f det(T )α for all T ∈ Λ+ n. 4. Smaller weights for arbitrary degree: an application of the local method in small degrees From the introduction, we see that it is desirable to have a result which is uniform in terms of the weight k in Theorem 1.1, at least in the range where the modular form is not singular: n k≥ . 2 It was also noticed in the introduction that, by the local method, any improvement in the range of the weights for which Theorem 3.2 holds will involve some nontrivial estimates for eigenvalues of cusp forms. In this section we would like to improve on the range of applicabilty of the weights of our results for the groups Γn0 (N ) using either (a) the Fourier-Jacobi expansion of a particular type for N arbitrary; or, (b) the Witt-operator for square–free N , along with an application of Theorem 3.2 in the case of degree 2 or 1. As will be seen in section 4.2, even though the result implied by (a) is more general than that of (b), the weights that are allowed in the growth of the Fourier coefficients is a little more relaxed in (b) as compared to (a). See Theorem 1.1 in this regard. More importantly, we feel that each of the methods (a) and (b) have their own merits and might be useful while dealing with different kinds of modular forms, as the situation demands. 4.1. Zero and one dimensional cusps in degree n. In the proof of Theorem 1.1, we will need convenient representatives of the 0 and 1-dimensional cusps in degree n. To describe the representatives (or equivalence classes), we recall the real parabolic subgroups B ) ∈ Sp(n, R) | C = 0 } Cn,0 (R) := {M = ( CA D n

Cn,n−1 (R) := {M ∈ Sp(n, R), | last row is of form (0(1,2n−1) , ∗)} from section 3. We have put Cn,r (Z) = Cn,r (R) ∩ Sp(n, Z), r = 0, n − 1. The 0-dimensional cusps has been defined in section 3, note that Cn,0 = Γn,∞ . It is easy to see that in order to check the cuspidality of a modular form F on Γn0 (N ), it is enough to have Φ(F | γi ) = 0, where Γn0 (N )\Sp(n, Z)/Cn,n−1 (Z) = ∪i Γn0 (N )γi Cn,n−1 (Z).

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¨ SIEGFRIED BOCHERER AND SOUMYA DAS

This follows from the following commutation relation: Φ(F |k M ) = det(D4 )k Φ(F ) |k π(M ), where π(M ) is the projection homomorphism A 1 Cn,n−1 (R) → Spn−1 (R), M = C∗1

0 ∗ 0 0 0

∗ ∗ ∗ 0

∗ ∗ ∗ D4

 7→

A1 B1 C1 D1




.

4.1.1. Representatives for the 0-dimensional cusps. In an unpublished Saarbruecken thesis [10] and also in [5], explicit representatives of the Γn0 (N )-inequivalent 0dimensional cusps C = Γn0 (N )\Sp(n, Z)/Cn,0 (Z) = ∪i Γn0 (N )γi Cn,0 (Z) (at least if N is odd; in general, only a possibly redundant set) was given. We only need a very weak version of these results: we can choose the representatives of these double cosets inside the subgroup Sp(n − 2, Z) × Sp(2, Z) ,→ Sp(n, Z). Proposition 4.1. We may choose representatives of the double cosets in C of the form γi = ωi × δi , where ωi ∈ Sp(n − 2, Z) and δi ∈ Sp(2, Z). If N is square–free, we can choose representatives inside SL(2, Z)n ,→ Sp(n, Z). Proof. The first part follows from the above discussion. For N square–free, it is a consequence of the Bruhat decomposition for finite fields that if N = p1 p2 . . . pt , one can choose the requisite representatives of the form R(`1 , `2 , . . . , `n ) ≡ (1n−`i × J`i ) mod pi where the `i vary between 0 to n and Jr = for example.

0 1r


−1r 0

(i = 1, 2, . . . , t). ∈ Sp(r, Z). See [2, Lem. 8.1]

But Jr = Ja × Jb , if r = a + b and thus we can write R(`1 , `2 , . . . , `n ) ≡ αi (1) × αi (2) · · · αi (n) mod pi , where each αi (j) is in SL(2, Z). Now, by the Chinese remainder theorem, we can simultaneously find ω(j) ∈ SL(2, Z) for each 1 ≤ j ≤ n such that ωj ≡ αi (j) mod pi ,

(i = 1, 2, . . . , t).

Then it follows easily that one can choose R(`1 , `2 , . . . , `n ) = ω1 × ω2 · · · × ωn . This completes the proof.  Remark 4.2. If N is odd, the representatives can actually be chosen inside SL(2, Z)n ,→ Sp(n, Z). We refer the reader to [5].

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4.1.2. Representatives of the 1-dimensional cusps. For the 1-dimensional cusps, we would need representatives for the double cosets C1 (N ) = Γn0 (N )\Sp(n, Z)/Cn,n−1 (Z) = ∪j Γn0 (N )γj Cn,n−1 (Z), similar to the 0-dimensional ones. This is the content of the next proposition, where we choose representatives in the form Sp(n − 2, Z) × Sp(2, Z). We believe that this is not available in the existing literature. Proposition 4.3. We can choose representatives of the double cosets C1 (N ) of the form γj = 12n−4 × βj , where βj ∈ Sp(2, Z). Proof. For square–free levels this is well-known, see e.g. [2, Lem. 8.1] and we will reduce the statement to this case. We write N = M R with M square–free M = p1 p2 . . . pt , and R | M ∞ (i.e., R divides a power of M ). Then representatives for C1 (M ) can be chosen in the form: (4.1)

12n−2 × γ

with γ = ( ac db ) ∈ SL(2, Z).

Indeed, this follows again from [2, Lem. 9.1]. We do not repeat the same proof as in Proposition 4.1. Now, we note that the set of left coset representatives of Γn0 (N ) in Γn0 (M ) can be chosen in lower triangular form:   1n 0n n n Γ0 (M ) = ∪j Γ0 (N ) , (M | Lj , Lj = Ltj ). Lj 1n For convenience, we ˜ for L = M L  p Mr

recall the elementary proof. We need to solve the equation ∗ s



1n 0n ˜ 0n ML



 =

 ∗ ∗ n ˜ ∗ ∈ Γ0 (N ). M r + M sL

This amounts to solving the equation ˜ ≡ 0 mod R. r + sL Now pt s ≡ 1n mod M and so s is invertible in the rings Mn (Z/pt Z), for all t. Thus, since det s is an unit in the rings Z/pm t Z for all m ≥ 1, s is invertible in the rings Mn (Z/pm Z) and hence in the ring Mn (Z/RZ) as R | M ∞ . L can be t chosen to be symmetric, since (r, s) is a symmetric pair: rt s = st r implies that ˜ mod R. st r ≡ −st sL Representatives of Γn0 (N )\Sp(n, Z)/Cn,n−1 (Z) may then (possibly redundantly) be obtained in the form   1n 0n · (12n−2 × γ), L 1n

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where L is a symmetric n × n matrix whose entries are divisible by M and γ is L L as in (4.1). We decompose L as L = L1t2 L24 with L4 ∈ Z. Then we get

(4.2)

  1n−1 0 0n−1 0    0 1n 0n a 0 b   · (12n−2 × γ) =   L1 L 1n L2 a 1n−1 L2 b  0 L4 b + d Lt2 L4 a + c   1n−1 0 0n−1 0    0 a 0 b  1n 0n   = · −L1 0 0n−1 L2 a 1n−1 L2 b  1n 0 0 Lt2 L4 a + c 0 L4 b + d

The right factor above can be ignored, because it is in Cn,n−1 (Z). Now we choose a V ∈ GL(n − 1, Z) such that (see [1, p. 17]) ˜ 2 = (0, . . . , 0, h)t (h ∈ Z). V · L2 = L  −t  V 0 0n 0 1 The matrix W := lies in the intersection of the groups Γn0 (N ) and V 0 0n 0 1 Cn,n−1 (Z), we may therefore conjugate the first matrix in (4.2) by W to obtain the representative  1n−1 0 0n−1 0  0 a 0 b   = 12n−4 × β,  ˜ 0n−1 L2 a 1n−1 L2 b  ˜ t L4 a + c 0 L4 b + d L 2 

B ) ∈ Sp(2, Z) with where β = ( CA D         1 0 0 0 0 h 1 hb A= ,B = ,C = ,D = . 0 a 0 b h L4 a + c 0 L4 b + d

This completes the proof of the proposition.



4.2. Proof of Theorem 1.1. We are now in a position to prove Theorem 1.1 with a quite better range of applicability in terms of the weights. We shall use the Fourier-Jacobi expansion corresponding to the decomposition n = (n − 2) + 2, and the Witt operator corresponding to the decomposition n = (n − 1) + 1 to reduce the question to the case of degree 2 (resp. 1) using the above description of the cusps and apply the local method to these cases of small degrees. This allows for the better range of the weights.

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Remark 4.4. We note here that if a non-zero cusp form F in Mkn (N ) has the . This is because an argument K(α) property, then necessarily α > 0 if k > n+1 2 with Rankin-Selberg convolution (see [3, Remark 5.3]) shows that α ≥ k2 − n+1 . 4 Therefore, there are no cusp forms with bounded Fourier coefficients when k > n+1 . Further, the condition k ≥ n2 is necessary, as F can not be singular. 2 Proof of Theorem 1.1 (i). Let γ ∈ C represent the cusp of Γn0 (N ) at which the growth property is given in the above theorem. By Proposition 4.1, we can assume that γ = ω × δ, where ω ∈ Sp(n − 2, Z) and δ ∈ Sp(2, Z). Let us write down the Fourier expansion of F at the cusp γ: X (4.3) (F |k γ)(Z) = a(F, T )e(T Z/N ). T ∈Λn

Let us introduce an auxillary function g defined by g(Z) := F |k (ω × 14 )(Z). Now g is certainly a modular form for some congruence subgroup of Γn , and so we can look at its Fourier-Jacobi expansion of type (2, n − 2) (see [16, Prop. 6.1, p. 65]), i.e.,   X 0(n−2) z (4.4) g(Z) = φM (τ, z)e(Mτ 0 ), (Z = τ zt τ (2) ). M∈Λn−2

Please note that our Fourier-Jacobi expansion is a bit different from the usual t one; we expand g(Z) along the parabolic Cn,2 , i.e., with respect to the upper left corner of Z. Let the Jacobi form φM (M > 0) have the following theta-decompostion, whose existence is well known for the full Jacobi group [18, p. 210]; in the case of arbitrary congruence subgroup, see [16, Prop. 3.5, p. 52]: X (4.5) φM (τ, z) = hµ (τ )θµ,M (τ, z) µ∈H/K

where K = Z(n−2,2) ⊂ H = (2M)−1 Z(n−2,2) are lattices in Q(n−2,2) . Moreover 2 hµ ∈ Mk− n−2 (Γ), for a certain congruence subgroup Γ with the usual convention 2 to be adopted for the half-integral weight modular forms as in [16, p. 42] if n is odd. θµ,M (τ, z) are the so-called Jacobi-theta series of type (n − 2, 2) of characteristic (µ, 0) attached to M, defined by

θµ,M (τ, z) =

X λ∈Z (n−2,2)

 e

 1 t M · [(λ + µ)τ (λ + µ) + 2(λ + µ)z] 2

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Now, notice that g | (12n−4 × δ) = F | γ. Also one has with δ = ( ac db ),  0  +d)−1 cz t z(cτ +d)−1 g | (12n−4 × δ) = det(cτ + d)−k g( τ (c−z(cτ ). t τ t +dt )−1 z t (aτ +b)(cτ +d)−1 From this is easily follows that the Fourier-Jacobi expansion of f | γ is X (4.6) (F | γ)(Z) = (φM |k,M [δ, 0])e(Mτ 0 ), M∈Λn−2 (2,n−2)

here [δ, 0] ∈ Sp(2, Z) n HZ , the Jacobi group of higher degree. We refer the reader to [18] for a detailed exposition of the theory of Jacobi forms of higher degree. – The upshot of this is that the theta-components hµ of a Fourier-Jacobi coefficient φM of g with M > 0 satisfy the K(α) property at the cusp represented by δ. This can be seen as follows. From (4.6), one gets the following formula for the transformed Fourier-Jacobi coefficients: X φM |k,M [δ, 0] = (hµ |k− n−2 δ) · (θµ,M | n−2 [δ, 0])(τ, z); (4.7) 2

2

µ∈H/K

so that the Fourier coefficients of hµ | δ being those of φM | [δ, 0], in turn occur as positive-definite Fourier coefficients of those of F | γ . From [18, Lem. 3.2], for all δ ∈ Sp(2, Z) we get the existence of a (n−2)×(n−2) unitary matrix UM (δ) such that
(4.8) (θµ,M )µ∈H/K | n−2 [δ, 0] = UM (δ) · (θµ,M )µ∈H/K . 2

Accordingly, putting (4.8) into (4.7) and using the uniqueness of the thetadecomposition, we find that for each µ, the hµ | δ can be written as linear com(δ) binations of the theta-components hµ of φM | [δ, 0]: (4.9)

−1 (hµ | δ)µ∈H/K = (h(δ) µ )µ∈H/K · UM (δ) . (δ)

By our hypothesis, each of the (hµ )µ satisfy the K(α) property on their respective Fourier coefficients which are indexed by positive-definite matrices. It follows that each hµ | δ likewise satisfy such K(α) property. At this point it would have been convenient to appeal to a version of Theorem 1.1 for half-integral weight modular forms, but we have avoided that in this paper. Therefore we consider the integral weight modular forms (4.10)

Hµ := h2µ | δ,

(µ ∈ H/K).

Then Hµ has weight 2k − (n − 2) > 0 (since k ≥ n2 + 1) on some congruence subgroup of Sp(2, Z), which satisfy the K(2α + 3) property at the cusp ∞; see

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[3, Lem. 3.3]. Please note that in [3, Lem. 3.3], there is a typographical error; the exponent should read K(α + β + n + 1) instead of K(α + β + n − 1). Now, Theorem 1.1 implies that Hµ (and hence hµ ) is a cusp form if 2k − (n − 2) ≥ 4, i.e., k ≥ n/2 + 1, which is true, and n+3 2k − (n − 2) > 2α + 3 + 2 i.e., if α < k − ; 2 which also is assured by the assumption in Theorem 1.1 (i). – Our next goal is to show that the cuspidality of the theta-components hµ for all µ implies that for F . To this end, let γ 0 ∈ C1 be a 1-dimensional cusp. We need to prove that Φ(F | γ 0 ) = 0. Proposition 4.3 implies that we can assume γ 0 = ω × β, where ω is as in the beginning of the proof (recall γ = ω × δ), and β ∈ Sp(2, Z). Hence, we have F | γ 0 = g | (12n−4 × β), and as in (4.6), the Fourier-Jacobi expansion of F | γ 0 with respect to τ 0 is X (F | γ 0 )(Z) = (4.11) (φM |k,M [β, 0])e(Mτ 0 ). M∈Λn−2 0 (n−1) Next, we observe that the Fourier-Jacobi  expansion of ) of  Φ(F | γ )(Z 0(n−2) type (n − 2, 1) with respect to τ 0 , where Z = τ ∗ τ1∗(1) becomes X (4.12) φeM (τ1 , ∗)e(Mτ 0 ), Φ(F | γ 0 )(Z) = M singular

where φeM (τ1 , ∗) = lim (φM | [β, 0])(( τ01 it0 ) , (∗, 0)) t→∞

This follows since the Φ-operator annihilates the cuspidal theta-components of the Fourier-Jacobi coefficients φM |k,M [β, 0] in the Fourier-Jacobi expansion of F | γ 0 in (4.9). This follows in turn from (4.9) with δ replaced by β. (β) hµ

Therefore the Φ-operator annihilates φM | [β, 0]; with M > 0 and we get the validity of (4.12). Furthermore, (4.12) implies that Φ(F | γ 0 ) is a singular modular form. We refer the reader to [17] for basic facts on this topic. However since our assumption implies that k ≥ n−1 , we must have Φ(F | γ 0 ) = 0 by the 2 theory of singular modular forms, see e.g., [17, Thm. 4.3]. Since the cusp γ 0 was arbitrary, it follows that the Φ-operator annihilates F at all the cusps, and hence F is cuspidal.  Remark 4.5. The reason for choosing the decomposition n = (n − 2) + 2 for the Fourier-Jacobi method over the more natural choice n = 1 + (n − 1) is due to

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the description of the 1-dimensional cusps in Proposition 4.3. For N arbitrary, it is not true that every 1-dimensional cusp γ of Γn0 (N ) can be written in the form ω × δ with ω ∈ Γn−1 (N ) and δ ∈ Γ10 (N ). This can be seen from the following 0 example in the case n = 2 and for the group Γ20 (p2 ) (p prime): the cusp   12 0 2 2 Γ0 (p ) 0 p C (Z) 12 2,1 p 0 does not have a diagonal representative. For square–free levels, of course this is possible, by Proposition 4.1. This explains our choice of decomposition n = (n − 1) + 1 while arguing via the Wittoperator, which is presented next. Remark 4.6. Implicit in the proof of Theorem 1.1 is a method to deal with the analogous question in the case of Jacobi forms of higher degrees. The idea is to use the theta-decomposition of such forms and to apply Theorem 1.1 in appropriate degrees. Remark 4.7. We mention two cases related to the previous theorem, where a slightly better result is possible. Namely could have obtained the result that “if F is a Siegel modular form of integral weight k and degree n which satisfies the n+2 n K(α) property on Λ+ n , then it is cuspidal when α < k − 2 and k ≥ 2 + 3” if, either (1) an analogue of Theorem 1.1 holds (at least in degree 2) for the half-integral case; or (2) when n is even (and we do not need to resort to “squaring” as in (4.10)). 4.3. The Witt operator. In this section we will prove Theorem 1.1 (ii) for arbitrary degree n for the congruence subgroups Γn0 (N ). The idea is to use the Witt operator defined by (W F )(τ, τ 0 ) := F (( τ0 τ00 )),

τ ∈ Hr , τ 0 ∈ Hs , r + s = n.

If F ∈ Mkn (Γ0 (N )) then (W F )(τ, τ 0 ) is a modular form in each of the variables τ, τ 0 of degree r and s respectively. More generally, we recall the following easily verified identity concerning the action of embedded copies of subgroups and restriction arising from W . Writing Z = ( xτt τx0 ) ∈ Hn , where τ ∈ H and τ 0 ∈ Hn−1 : W (F |k ω × δ) (Z) = j(ω, τ )−k j(δ, τ 0 )−k W F (ωhτ i, δhτ 0 i) ,

where ω = ac11 db11 ∈ Γr , δ = ac22 db22 ∈ Γs and ω×δ denotes the diagonal emedding  a1 0 b1 0  ω × δ = c01 a02 d01 b02 . (4.13)

0 c2 0 d2

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We next prove a lemma which essentially allows one to estimate the Fourier coefficients of the image of the Witt operator. 4.4. A counting lemma. Let us define, for m ∈ Λn , N ≥ 0, the set   m r/2 A (m, N ) := {r ∈ Zn | rt /2 N ≥ 0}. Lemma 4.8. For m ∈ Λ+ n , N ≥ 1, #A (m, N ) n,m N n/2 , where the implied constant depends only on m and n. Proof. Let us recall the Jacobi identity on positive definite matrices:       m r/2 N 0 1 0 (4.14) = . rt /2 N 0 N − m−1 [rt /2] m−1 rt /2 In By the Jacobi identity, we see that it is sufficient to estimate the number of integral vectors r such that N > m−1 [r/2] where m is positive half integral of size n. Equivalently we need to have 4N det m > (det m · m−1 )[r], now everything is integral. Let M := det m · m−1 , y := 4N det m. We first diagonalize M by an orthogonal matrix P = (Pi,j )1≤i,j≤n : M = P t diag(λ1 , . . . , λn )P. Let us further define the quantities z = (zi )1≤i≤n := P r and note that X rt M r = z t diag(λ1 , . . . , λn )z = λi · zi2 . 1≤i≤n

Therefore, from the above we have |zi | ≤ (y/λi )1/2 ,

(1 ≤ i ≤ n).

Since r = P t z, we get |ri | = |

X i

Pi,j zi | ≤

X

|zi | ≤ n · (y/λi )1/2 .

i

Thus the number of possibilities for r is at most n Y

(2n · (y/λi )1/2 + 1) ≤ cn,m y n/2 ≤ cn,m · N n/2 ,

i=1

where cn,m is a positive constant depending only on n and m.



¨ SIEGFRIED BOCHERER AND SOUMYA DAS

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Proof of Theorem 1.1 (ii). Here we assume that N is either 1 or square–free. By (4.1), we can assume that the cusp γ where the growth property of the Fourier coefficients is given, is of the form ω ×δ, where ω ∈ Sp(n−1, Z) and δ ∈ SL(2, Z). From the equivariance property of the Witt-operator under ω × δ in (4.13), we have for τ ∈ Hn−1 and τ 0 ∈ H   X X N r/2 0 a(F, rt /2 m ) e(N τ /M ) · e(mτ 0 /M ), (4.15) W (F |k γ)(τ, τ ) = N ≥0 r∈A (m,N ) m∈Λn−1

where A (m, N ) is as defined in Lemma 4.8. Since (W F )(τ, τ 0 ) is a modular form of weight k in each of the variables τ and τ , it has an expression of the form: X X (4.16) W F (τ, τ 0 ) = λ(i, j) f˜i (τ ) · g˜j (τ 0 ) + µ(i, r) f˜i (τ ) · e˜r (τ 0 ); 0

i,j

i,r

where the f˜i (i = 1, . . . , dn−1 ) and the g˜j (j = 1, . . . , d1 ) run over some basis of Mkn−1 (N ) and Skn (N ) respectively. Further, e˜r run over a basis of Ek (N ) (r = 1, . . . , κ). Then in view of (4.13), we also have W (F |k γ)(τ, τ 0 ) equals X X (4.17) λ(i, j) (f˜i |k ω)(τ )(˜ gj |k δ)(τ 0 ) + µ(i, r)(f˜i |k ω)(τ )(˜ er |k ω)(τ 0 ), i,j

i,r

for some λ(i, j), µ(r, i) ∈ C. For ease of notation, let us call fi := f˜i |k ω,

er := e˜r |k ω,

gj := g˜j |k δ;

and write down their Fourier expansions: X X fi (τ ) = a(fi , m)e(mτ ), er (τ ) = c(er , m)e(mτ ), m∈Λn−1

m∈Λn−1

gj (τ 0 ) =

X

b(gj , N )e(N τ 0 ).

N ≥1

From (4.15) and (4.17), we get by comparing the ‘positive definite’ Fourier coefficients: for m ∈ Λ+ n−1 and N > 1,   X m r/2 a(F, rt /2 N ) r∈A (m,N )

(4.18)

=

X i,j

λ(i, j) a(fi , m) · b(gj , N ) +

X µ(i, r) a(fi , m) · c(er , N ). i,r

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The Fourier coefficients occurring on left hand side of (4.18) satisfy the K(α) property from the hypothesis of Theorem 1.1. Thus Lemma 4.8 allows us to estimate   X X m r/2 (4.19) | a(F, r/2 )|  (det m)α (N − m−1 [r/2])α N r∈A (m,N )

r∈A (m,N )

 #{r ∈ A (m, N )}(det m · det N )α  (det N )α+

n−1 2

,

by Lemma 4.8. Note that in the first inequality above, we have used the Jacobi decomposition as in (4.14). Moreover, the implied constants in the above inequalities depend only on m, n, F and not on N . Now for any fixed m ∈ Λ+ n−1 the modular form Fm (τ 0 ) :=

X X λ(i, j)a(fi , m) · gj (τ 0 ) + µ(i, r)a(fi , m) · er (τ 0 ) i,j

i,r

satisfies the K(α + (n − 1)/2) property on the Fourier coefficients; this follows from (4.18) and (4.19). Invoking Theorem 1.1 for degree 1, we conclude that Fm is a cusp form for each m > 0, using the fact that α + n−1 < k − 1 and k ≥ 2. 2 This implies in particular that for all m > 0 and r X (4.20) µ(i, r) a(fi , m) = 0, 1≤i≤dn−1

which in turn implies that the degree n − 1 modular form Hr defined by X Hr (τ ) := µ(i, r)fi (τ ) 1≤i≤dn−1

is singular. However, since we have assumed k ≥ n−1 , we must have Hr = 0 for 2 all r. Since the fi ’s were linearly independent (recall that fi = f˜i | ω), this shows that µ(i, r) = 0 for all i, r. Going back to (4.16), this implies that X (W F )(τ, τ 0 ) = λ(i, j)f˜i (τ ) · g˜j (τ 0 ). i,j

In order to check the cuspidality of F , we now need the representatives of C1 (N ). Since N is square–free, by (4.1), we can choose these as γ 0 = 12n−2 × β, where β ∈ SL(2, Z). We have to prove thata Φ(F | γ 0 ) = 0 for all such γ 0 . To this

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

16

end, we again appeal to (4.17) and find that, Φ(F | 12n−2 × β)(w) = lim W (F | 12n−2 × β)(τ, it) t→∞ X = lim λ(i, j)f˜i (τ ) · (˜ gj | β)(it) = 0, t→∞

i,j

since the g˜j ’s are cuspidal. Since γ 0 is arbitrary, this shows that F is cuspidal. This completes the proof.  4.5. A corollary. Theorem 1.1 has an interesting corollary. Before we state it, let us recall the definition of the constant βkn (Γ) in [3, Def. 5.4]: βkn (Γ) := inf{α ∈ R, F ∈ Ekn (Γ) \ {0} | F has the K(α) property on Λ+ n }, where Ekn (Γ) is the space of Eisenstein series for a congruence subgroup Γ ⊂ Γn . Then it was asked in [3] to determine the constant βkn (Γ). It measures the amount of cancellation, if possible, in the positive-definite Fourier coefficients of elements of Ekn (Γ). The reason for such a question was that the results of [3] (akin to Theorem 1.1, but only at the cusp ∞) showed that if k > 2n and F satisfied the K(α) property with α < k − n; then F is a cusp form. Whereas, a result due to Kitaoka [8] says that for any modular form F on Γ, one has the K(k − n+1 ) property on Λ+ n if k ≥ 2n + 2. In fact Kitaoka [7, 8] also 2 shows that for the usual Siegel Eisenstein series E := Enk (N ), one has a(E, T )  det(T )k−

n+1 2

,

where  means ‘the same order of magnitude’. Since k − n ≤ k − n+1 , it was not 2 clear from the main results in [3] whether there could exist an element in Ekn (Γ) with the K(β) property on Λ+ n such that n+1 2 for n > 1. Thanks to Theorem 1.1 and Remark 4.7, we can conclude, k−n≤β
Corollary 4.9. n n+3 for k ≥ + 1, N ≥ 1; 2 2 n+2 n n n βk (Γ0 (N )) ≥ k − for k ≥ + 3, N ≥ 1, n even; 2 2 n + 1 βkn (Γn0 (N )) = k − for k ≥ 2n + 2, N square–free. 2 βkn (Γn0 (N )) ≥ k −

Thus, in the last case above, there can not be any uniform cancellation in the positive-definite Fourier coefficients of elements in the space of Eisenstein series.

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17

In this regard, note that for an arbitrary congruence subgroup Γ, the main result in [4] only shows that βkn (Γ) ≥ k − n and by virtue of Corollary 4.9 we are inevitably led to make the following conjecture: Conjecture 4.10. The following equivalent statements are true for any Γ. n+1 2

n . 2 n+1 n . (ii) T heorem 1.1 is true for all k ≥ and any α < k − 2 2

(i) βkn (Γ) = k −

for all k ≥

References [1] A. N. Andrianov, V. G. Zuravlev, Modular Forms and Hecke operators, Translations of Mathematical Monographs, 145, American Mathematical Society, Providence, RI, 1995. [2] S. B¨ ocherer, R. Schulze-Pillot, Siegel modular forms and theta series attached to Quaternion algebras. Nagoya Math. J. 121, 1991, 35–96. [3] S. B¨ ocherer, S. Das, Characterization of Siegel cusp forms by the growth of their Fourier coefficients, Math. Ann., 359, Issue 1-2, (2014), 169–188. [4] S. B¨ ocherer, S. Das, Cuspidality and the growth of Fourier coefficients of modular forms, J. Reine Angew. Math., to appear. [5] S. B¨ ocherer, Y. Hironaka, F. Sato, Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms. Quadratic forms–algebra, arithmetic, and geometry, Contemp. Math., 493, Amer. Math. Soc., Providence, RI, 2009, 51–82. [6] E. Freitag, Siegelsche Modulfunktionen. Grundl. Math. Wiss., 254 Springer–Verlag, (1983). [7] Y. Kitaoka, Dirichlet series in the theory of Siegel modular forms. Nagoya Math. J. 95, 1984, 73–84. [8] Y. Kitaoka, Siegel Modular Forms and Representations by Quadratic Forms. Lecture Notes, Tata Institute of Fundamental Research, (1986). [9] W. Kohnen, On certain generalized modular forms. Funct. Approx. Comment. Math. 43, 2010, 23–29. [10] M. Klein, Verschwindungss¨ atze f¨ ur Hermitesche Modulformen sowie Siegelsche Modulforucken, 2004, men zu den Kongruenzuntergruppen Γn0 (N ) sowie Γn (N ). Ph.D. thesis Saarbr¨ available at http://www.math.uni-sb.de/ag/schulze/mklein/DissMKlein.pdf [11] W. Kohnen, On certain generalized modular forms, Funct. Approx. Comment. Math. 43 (2010), 23–29. [12] W. Kohnen, Y. Martin, A characterization of degree two cusp forms by the growth of their Fourier coefficients, Forum Math., DOI 10.1515/forum-2011-0142. [13] J. Lim, A characterization of Jacobi cusp forms of certain types, J. Number Theory, 141, 2014, 278–287. [14] B. Linowitz, Characterizing Hilbert modular cusp forms by coefficients size, Kyushu Math. J., 68 no. 1, 2014, 105–111. [15] Y. Mizuno, On characterisation of Siegel cusp forms by the Hecke bound, Mathematika, 61, np. 1, 2015, 89–100. [16] G. Shimura, On certain reciprocity-laws for Theta functions and Modular forms, Acta Mathematica, 141, Issue 1, (1978), 35–71. [17] R. Weissauer, Stabile Modulformen und Eisensteinreihen, Lecture Notes in Mathematics, 1219, (1986), Springer-Verlag.

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[18] C. Ziegler, Jacobi Forms of Higher Degree, Abh. Math. Sem. Univ. Hamburg, 59, (1989), 191–224. ¨ r Mathematik, Universita ¨ t Mannheim, 68131 Mannheim (Germany). Institut fu E-mail address: [email protected] Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India. E-mail address: [email protected], [email protected]