An explicit bound for the first sign change of the Fourier coefficients

An explicit bound for the first sign change of the Fourier coefficients of a Siegel cusp form arXiv:1403.4712v1 [math.NT] 19 Mar 2014 YoungJu Choie, Sanoli Gun and Winfried Kohnen Abstract We give an explicit upper bound for the first sign change of the Fourier coefficients of an arbitrary non-zero Siegel cusp form F of even integral weight on the Siegel modular group of arbitrary genus g ≥ 2. Keynote : Siegel cusp form, Jacobi expansion, Fourier expansion, sign change 1991 Mathematics Subject Classification :11F30, 11F46, 11F50 1. Introduction Fourier coefficients of cusp forms in general are quite mysterious objects. In particular, when real this applies to the distribution of their signs. Over the past years various aspects of the latter problem have been studied by several authors. For example, in [9] it was shown that the Fourier coefficients when real of a non-zero elliptic cusp form f on a congruence subgroup of the full modular group Γ1 := SL2 (Z) have infinitely many sign changes. The proof which is rather straightforward uses the analytic properties of the Hecke L-function and the Rankin-Selberg zeta function of f . The above result was generalized in [8] to the case of a non-zero Siegel cusp form F of even integral weight on the symplectic group Γg := Spg (Z) ⊂ GL2g (Z) of arbitrary genus g ≥ 2. In fact, in [8] it was proved that if the Fourier coefficients a(T ) of F are real (T > 0 a positive definite half-integral matrix of size g), then there exist infinitely many T > 0 (modulo the usual action of GLg (Z)) such that a(T ) > 0, and similarly such that a(T ) < 0. The proof uses similar arguments as for g = 1, with e.g. the Hecke L-function of f replaced by the Koecher-Maass series of F . A more subtle problem is to give explicit upper bounds for the first sign change in terms of the weight and the level. If f is a Hecke eigenform on the Hecke congruence group Γ0 (N ) of level N this was done e.g. in [6,7,13]. The corresponding problem for arbitrary forms (not necessarily Hecke eigenforms) f of squarefree level N was studied in [3]. In fact, in [3] the question was reduced to the case of Hecke eigenforms by writing f as a linear combination of the latter and then using Chebyshev’s inequality in combination with uniform lower bounds for the Petersson norms of those Hecke eigenforms. The technical details were rather complicated. In this paper we will give an explicit upper bound for the first sign change of the Fourier coefficients of an arbitrary non-zero Siegel cusp form F of even integral weight on Γg (g ≥ 2). To the best of our knowledge, we think that this is the first general result in 1 this direction. The main idea will be to look at the Fourier-Jacobi expansion of F where the coefficients are Jacobi forms on the generalized Jacobi group Γ1 ∝ (Zg−1 × Zg−1 ) . Using Taylor expansions of these coefficients we will reduce the question to the case of elliptic modular forms and then will apply the results of [3]. Though this strategy appears rather simple, the actual technical details are somewhat involved. On the way we will also obtain a bound for the first non-vanishing Taylor coefficient function of a generalized Jacobi form φ(τ, z) (τ ∈ H= upper half-plane, z ∈ Cg−1 ) around z = 0. This generalizes a basic result of [4] in the case of classical Jacobi forms (Thm. 1.2) and eventually may be of independent interest. Although we do not have any direct and immediate application of our main result, we do think that the main steps in the proof may highlight again the theory of Jacobi forms, as an important bridge between Siegel modular forms and classical elliptic modular forms. Note that in the same spirit Jacobi forms played an important role beforehand in the proof of the Saito-Kurokawa conjecture [4] and in estimating Fourier coefficients of Siegel cusp forms [2]. 2. Statement of main result We will always suppose that g ≥ 2. For k ∈ N we let Sk (Γg ) be the space of Siegel cusp forms of weight k on Γg . Recall that this is the space of complex valued holomorphic functions F (Z) on the Siegel upper half-space Hg (consisting of symmetric complex matrices of size g with positive definite imaginary part) such that F ((AZ + B)(CZ + D)−1 ) = det(CZ + D)k F (Z) for all  A C B D  ∈ Γg and such that F has a Fourier expansion F (Z) = X a(T ) e2πitr(T Z) T >0 (Z ∈ Hg ), where T runs over all positive definite half-integral matrices of size g. We note that a(T [U ]) = (det U )k a(T ) for all T > 0 and all U ∈ GLg (Z), where we have used the standard abbreviation A[B] := B t AB for complex matrices A and B of appropriate sizes. In particular, if k is even and the a(T ) are real, then a(T [U ]) is of the same sign as a(T ) for all unimodular U . 2 Theorem. Let k be even and F ∈ Sk (Γg ), F 6= 0. Suppose that the Fourier coefficients a(T ) (T > 0) of F are real. Then there exist T1 > 0, T2 > 0 with tr T1 , tr T2 ≪ (k · cg )5 log26 (k · cg ) such that a(T1 ) > 0, a(T2 ) < 0. Here cg := g · 2g−1 · (4/3)g(g−1)/2 and the constant involved in ≪ is absolute and effective. Remarks. i) There is an obvious reformulation of the Theorem for F ∈ Sk (Γg ) a Siegel cusp form with arbitrary complex Fourier coefficients, with a(T1 ), a(T2 ) replaced by Re(a(T1 )), Re(a(T2 )) (and Im(a(T1 )), Im(a(T2 ))). This follows from the well-known fact that for F ∈ Sk (Γg ) the Fourier series with general coefficients Re(a(T )) (resp. Im(a(T ))) are again in Sk (Γg ). ii) Any improvement of the bound obtained in [3] for elliptic cusp forms valid for all even integral weights would lead to a corresponding improvement of the bound in the Theorem, as will be clear from the proof. iii) One may ask more generally for the distribution of signs of the coefficients a(T ) where T > 0 runs over primitive matrices only, or (oppositely) of the ”radial” coefficients a(nT ) (n ≥ 1), with T > 0 fixed. The latter coefficients in the special case g = 2 are related to the corresponding Hecke eigenvalues (in case F is a Hecke eigenform), see [1]. For some results regarding sign changes of eigenvalues we refer to [10,11,14]. The method used here, however does not seem to give any new insights into the questions addressed above. The proof of the Theorem will be given in the next section. 3. Proof It is sufficient to show the existence of T > 0 with tr T in the given range such that a(T ) < 0, since we can always replace F by −F . Let us write Z=  τ zt z τ′  (τ ∈ H, τ ′ ∈ Hg−1 , z ∈ Cg−1 ). Then F has a Fourier-Jacobi expansion X ′ (1) F (Z) = φM (τ, z)e2πitr(M τ ) M >0 where M runs over all positive definite half-integral matrices of size g − 1 and the functions φM are Jacobi cusp forms of weight k and index M on the generalized Jacobi group Γ1 ∝ (Zg−1 × Zg−1 ). 3 Those are complex-valued holomorphic functions φ(τ, z) on H × Cg−1 with the transformation laws   2πicM [z t ] aτ + b z a b k (cτ +d) φ(τ, z) (∀ ∈ Γ1 ) (2) φ( , ) = (cτ + d) e c d cτ + d cτ + d and t φ(τ, z + λτ + µ) = e−2πi(M [λ (3) ]τ +2λz t ) φ(τ, z) (∀λ, µ ∈ Zg−1 ), and having a Fourier expansion X φ(τ, z) = c(n, r) e2πi(nτ +rz t ) n≥1,r∈Zg−1 ,4n>M −1 [rt ] (we use the notation A > B for symmetric real matrices A and B to indicate that A − B > 0)(see [17]). Note that the (n, r)-th coefficient of φM (τ, z) in (1) is equal to   n r/2 a r t /2 M   n r/2 (the condition > 0 is equivalent to n ≥ 1, 4n > M −1 [r t ] as follows from the r t /2 M usual Jacobi decomposition of the latter matrix). Since F 6= 0 there exists T0 =  n0 r0 /2 r0t /2 M0  >0 such that tr T0 ≪g k, a(T0 ) 6= 0 as is well-known. In fact, one can find such a T0 whose trace satisfies the explicit bound (4). tr T0 ≤ 2 k · √ · g · (4/3)g(g−1)/2. 4π 3 As was kindly communicated to the authors by C. Poor, this is an easy consequence of basic reduction theory [5,12] and results proved in [15,16]. Since a(T0 ) 6= 0 the function φM0 (τ, z) is not identically zero. We define Y ΦM0 (τ, z) := φM0 (τ, ǫz) ǫ where ǫ = (ǫ1 , . . . , ǫg−1 ) runs over all elements of {−1, 1}g−1 and ǫz := (ǫ1 z1 , . . . , ǫg−1 zg−1 ). 4 We will simply write Φ(τ, z) instead of ΦM0 (τ, z). Note that φM0 (τ, ǫz) = φM0 [Dǫ ] (τ, z) where Dǫ is the diagonal matrix with entries on the diagonal in the given order ǫ1 , . . . , ǫg−1 . This follows immediately if one acts on F with the matrix 1 0  0 Dǫ  0 0 0 0  0 0 0 0  . 1 0 0 Dǫ  Therefore φM0 (τ, ǫz) is a non-zero Jacobi cusp form of weight k and index M0 [Dǫ ] (which of course can also be checked by direct inspection). Hence we conclude that Φ(τ, z) is a non-zero Jacobi cusp form of weight 2g−1 k and index X M0 [Dǫ ]. (5) M0 := ǫ By construction this function is even w.r.t. each of the variables zν , ν ∈ {1, . . . , g − 1}. We observe that tr M0 = 2g−1 · tr M0 . We now develop Φ(τ, z) in a Taylor series around z = 0, i.e. write X Φ(τ, z) = ν g−1 χν1 ,...,νg−1 (τ )z1ν1 . . . zg−1 . ν1 ,...,νg−1 ≥0 Let us choose αg−1 , αg−2 , . . . , α2 , α1 in a “minimal” way such that χα1 ,α2 ,...,αg−2 ,αg−1 (τ ) is not the zero function. Here “minimal” means that for all τ we have χν1 ,...,νg−1 (τ ) = 0 (0 ≤ ∀νg−1 < αg−1 ; ∀νg−2 , . . . , ν1 ≥ 0), χν1 ,...,νg−2 ,αg−1 (τ ) = 0 (0 ≤ ∀νg−2 < αg−2 ; ∀νg−3 , . . . , ν1 ≥ 0), . . . . . . , χν1 ,α2 ,...,αg−1 (τ ) = 0 (0 ≤ ∀ν1 < α1 ). In the following, we will denote the diagonal elements of M0 by m011 , m022 , . . . , m0g−1,g−1 . 5 Lemma. The function (6) X χν1 ,α2 ,...,αg−1 (τ )z1ν1 ν1 ≥0 is a (classical) non-zero Jacobi cusp form of weight 2g−1 k + α2 + . . . + αg−1 and index 2g−1 m011 . Proof. Up to a non-zero universal factor the function in (6) is equal to   g−1 ∂zα22 . . . ∂zαg−1 Φ(τ, z1 , . . . , zg−1 ) . |(z2 ,...,zg−1 )=(0,...,0) We differentiate equation (2) successively w.r.t. z2 , . . . , zg−1 up to the orders α2 , . . . , αg−1 , respectively and use Leibniz rule together with the “minimality” of αg−1 , . . . , α2 . Then we see that indeed (6) behaves like a Jacobi form of weight 2g−1 k + α2 + . . . + αg−1 and index 2g−1 m011 w.r.t. to the action of Γ1 . Likewise in (3) we take λ = (λ1 , 0, . . . , 0) and µ = (µ1 , 0, . . . , 0) and differentiate successively to see the correct behavior of (6) under the action of Z2 . t The conditions n ≥ 1, 4n > M−1 0 [r ] in the Fourier expansion of Φ are equivalent to   n r/2 > 0. r t /2 M0 Hence taking into account (5) (which implies that the (1, 1)-entry of M0 is 2g−1 m011 ), it follows that   n r1 /2 > 0. r1 /2 2g−1 m011 Therefore 4n · 2g−1 m011 > r12 and so the Fourier expansion of (6) is indeed as required. Finally we note that the function in (6) is not identically zero since χα1 ,α2 ,...,αg−1 (τ ) is not the zero function by hypothesis. This proves the Lemma. We now observe that χα1 ,...,αg−1 (τ ) is a non-zero cusp form of weight (7) k1 := 2g−1 k + α1 + α2 + . . . + αg−1 on Γ1 , by the “minimality” of α1 and a similar argument as above. If g = 2, full details are given in [4, p. 31]. By Thm. 1.2, p. 10 in [4] it follows that α1 ≪ 2g−1 m011 . We now proceed inductively regarding the other variables z2 , z3 . . .. Thus we successively choose βg−1 , βg−2 , . . . , β3 , β1 , β2 ≥ 0 in a “minimal” way such that χβ1 ,β2 ,...,βg−1 (τ ) 6 is not the zero function, then successively γg−1 , γg−2 , . . . , γ4 , γ2 , γ1 , γ3 ≥ 0 in a “minimal” way such that χγ1 ,γ2 ,γ3 ...,γg−1 (τ ) is not the zero function and so on. In the same way as above, using “minimality” one then shows that X χβ1 ,ν2 ,β3 ...,βg−1 (τ )z2ν2 ν2 ≥0 is a non-zero Jacobi cusp form of weight 2g−1 k + β1 + β3 + . . . + βg−1 and index 2g−1 m022 , the function X χγ1 ,γ2 ,ν3 ,γ4 ,...,γg−1 (τ )z3ν3 ν3 ≥0 is a non-zero Jacobi cusp form of weight 2g−1 k + γ1 + γ2 + γ4 + . . . + γg−1 and index 2g−1 m033 , and so on. Likewise as above, the functions χβ1 ,β2 ,...,βg−1 (τ ) resp. χγ1 ,γ2 ,...,γg−1 (τ ) and so on are non-zero cusp forms on Γ1 of weights 2g−1 k + β1 + . . . + βg−1 resp. 2g−1 k + γ1 + . . . + γg−1 and so on, and β2 ≪ 2g−1 m022 , γ3 ≪ 2g−1 m033 , . . .. Since both the α’s and the β’s are “minimal”, we conclude immediately from these conditions that βg−1 = αg−1 , βg−2 = αg−2 , . . . , β3 = α3 , in a successive way. It then follows that α2 ≤ β2 , for otherwise we had 0 = χβ1 ,β2 ,α3 ,...,αg−1 (τ ) = χβ1 ,β2 ,β3 ,...,βg−1 (τ ), a contradiction. We therefore conclude that α2 ≪ 2g−1 m022 . Proceeding in the same way with the α’s and the γ’s, we infer that α3 ≪ 2g−1 m033 . Working on inductively, we finally conclude that α1 ≪ 2g−1 m011 , . . . , αg−1 ≪ 2g−1 m0g−1,g−1 , hence (8) α1 + . . . + αg−1 ≪ 2g−1 tr M0 . 7 We note that the arguments used above more generally show the following: Proposition. Let φ(τ, z) (τ ∈ H, z ∈ Cg−1 ) be a generalized Jacobi form of weight k and index M > 0 on Γ1 ∝ (Zg−1 × Zg−1 ) and let χν1 ,...,νg−1 (τ ) (ν1 , . . . , νg−1 ≥ 0) be its Taylor coefficients around z = 0. Then there exists (α1 , . . . , αg−1 ) ∈ N0g−1 such that α1 + . . . + αg−1 ≪ tr M and χα1 ,...,αg−1 (τ ) is a non-zero cusp form. This result generalizes Thm. 1.1 in [4] in the classical case Γ1 ∝ Z2 and may be of independent interest. With the definition (7) it now follows from (8) that k1 ≪ 2g−1 (k + tr M0 ). (9) Let us write a(n) (n ≥ 1) for the Fourier coefficients of χα1 ,...,αg−1 (τ ). Then by [3] (in the case of level 1) there exists ñ ≥ 1 such that ñ ≪ k15 log26 k1 , a(ñ) < 0. By (9) it follows that (10) ñ ≪ 2g−1 (k + tr M0 )
5
log26 2g−1 (k + tr M0 ) . Observe that χα1 ,...,αg−1 (τ ) up to a non-zero universal scalar equals  g−1 ∂zα11 . . . ∂zαg−1 Φ(τ, z)  |z=0 . Therefore if the Fourier coefficients of Φ(τ, z) are denoted by C(n, r), it follows that up to a non-zero scalar a(ñ) is equal to α X g−1 C(ñ, r)r1α1 . . . rg−1 . t r∈Zg−1 ,4ñ>M−1 0 [r ] Since Φ(τ, z) is an even function w.r.t. each of the variables z1 , . . . , zg−1 , the α1 , . . . , αg−1 are all even integers. Hence it follows that there exists r̃ ∈ Zg−1 such that C(ñ, r̃) < 0. However, C(ñ, r̃) is a finite sum of products of Fourier coefficients   nǫ ∗ a ∗ M0 [Dǫ ] where ǫ runs over {−1, 1}g−1 as before, and with X nǫ = ñ. ǫ 8 Hence at least one of the coefficients a  nǫ ∗ ∗ M0 [Dǫ ]  must be negative. Since nǫ ≤ ñ and tr M0 ≤ tr T0 we infer from (10) tr  nǫ ∗ ∗ M0 [Dǫ ]  = nǫ + tr M0
5
log26 2g−1 (k + tr M0 ) + tr T0
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≪ 2g−1 (k + tr T0 ) log26 2g−1 (k + tr T0 ) . ≪ 2g−1 (k + tr M0 ) Inserting from (4) we then obtain our assertion. Acknowledgements: This work was partially supported by NRF 2013053914, NRF2011-0008928 and NRF-2013R1A2A2A01068676. References [1] A.N. Andrianov: Euler products corresponding to Siegel modular forms of genus 2, Russian Math. Surveys 29 (1974), 45-116. [2] S. Böcherer and W. Kohnen: Estimates for Fourier coefficients of Siegel cusp forms, Math. Ann. 297 (1993), 499-517. [3] Y. Choie and W. Kohnen: The first sign change of Fourier coefficients of cusp forms, Amer. J. Math. 131 (2009), no. 2, 517-543. [4] M. Eichler and D. Zagier: The theory of Jacobi forms. Progress in Math. vol 55, Birkhäuser 1985. [5] E. Freitag: Siegelsche Modulformen. Grundl. d. Math. Wiss. 254, Springer 1983. [6] A. Ghosh and P. Sarnak: Real zeros of holomorphic Hecke cusp forms, J. Eur. Math. Soc. 14 (2012), 465-487. [7] H. Iwaniec, W. Kohnen and J. Sengupta: The first negative Hecke eigenvalue, Int. J. Number Theory 3 (2007), 355-363. [8] S. Jesgarz: Vorzeichenwechsel von Fourierkoeffizienten von Siegelschen Spitzenformen, Diploma Thesis (unpublished), Univ. of Heidelberg 2008. [9] M. Knopp, W. Kohnen and W. Pribitkin: On the signs of Fourier coefficients of cusp forms, Ramanujan J. 7 (2003), no. 1-3, 269-277. [10] W. Kohnen: Sign changes of Hecke eigenvalues of Siegel cusp forms of genus two, Proc. Amer. Math. Soc. 135 (2007), 997-999. 9 [11] W. Kohnen and J. Sengupta: The first negative Hecke eigenvalue of a Siegel cusp form of genus two, Acta Arithmetica 129.1 (2007), 53-62. [12] J. Martinet: Perfect lattices in Euclidean spaces, Grundl. d. Math. Wiss. 327, Springer 2003. [13] K. Matomäki: On signs of Fourier coefficients of cusp forms, Math. Proc. Camb. Phil. Soc. 152 (2012), 2007-2022. [14] A. Pitale and R. Schmidt: Sign changes of Hecke eigenvalues of Siegel cusp forms of degree two, Proc. Amer. Math. Soc. 136 (2008), 3831-3838. [15] C. Poor and D.S. Yuen: Linear dependence among Siegel modular forms, Math. Ann. 318 (2000), 205-234. [16] C. Poor and D.S. Yuen: Dimensions of cusp forms for Γ0 (p) in degree two and small weights, Abh. Math. Sem. Univ. Hamburg 77 (2007), 59-80. [17] C. Ziegler: Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (1989), 191-224. YoungJu Choie, Department of Mathematics, Pohang University of Science and Technology, and PMI (Pohang Mathematical Institute), Pohang 790-784, Korea E-mail: yjc@postech.ac.kr Sanoli Gun, The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, India E-mail: sanoli@imsc.res.in Winfried Kohnen, Mathematisches Institut der Universität, INF 288, D-69120 Heidelberg, Germany E-mail: winfried@mathi.uni-heidelberg.de 10