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W. Kohnen Res Math Sci (2022) 9:50 https://doi.org/10.1007/s40687-022-00348-w RESEARCH On the growth of Fourier coeﬃcients of Siegel cusp forms of degree two Winfried Kohnen Correspondence: winfried@mathi.uni- heidelberg.de Mathematisches Institut der Universität, INF 205, 69120 Heidelberg, Germany 1 Introduction A very important and fundamental subject in the theory of modular forms is the study of their Fourier coeﬃcients. While for the corresponding problem for Hecke eigenvalues, there are deep tools available, e.g., representation theory or arithmetic algebraic geometry, the situation for the Fourier coeﬃcients in general is quite diﬀerent. One way (often stony, but sometimes fruitful) to proceed is to look at Fourier–Jacobi coeﬃcients and then (eventually after a number of steps) get all the way down to half-integral or integral weight forms in one variable. In this way, one sometimes may be able to deduce results for the higher-dimensional case from the one-dimensional theory. In this paper, we will study growth properties of the Fourier coeﬃcients of Siegel modular forms of degree two via genus theory of binary quadratic forms and the theory of Jacobi forms. More precisely, let F be a Siegel cusp form of even integral weight k for the Siegel modular group := Sp (Z) of degree two and denote by A(T) its Fourier coeﬃcients. 2 2 Here, T runs over positive deﬁnite symmetric half-integral (2, 2)-matrices. We let D := ab/2 −4det T be the discriminant of T.If T = with a, b and c integers, a > 0and b/2 c D = b − 4ac < 0, then we usually identify T with the positive deﬁnite integral binary 2 2 quadratic form ax + bxy + cy of discriminant D. We recall that T and T are called 1 2 equivalent if there exists U ∈ := SL (Z) such that T = T [U]:= U T U where U is 1 2 2 1 1 the transpose of U. The Fourier coeﬃcients A(T) depend only on the equivalence class of T. A well-known conjecture of Resnikoﬀ and Saldaña [22] says that k/2−3/4+ A(T) |D| (> 0), (1.1) ,F where the constant implied in only depends on and F.Estimate(1.1) can be viewed as a formal generalization of the famous Ramanujan–Petersson conjecture for cusp forms in one variable, proved by Deligne. Some motivation for (1.1) was given in [17], where in particular it was shown that it holds “on average.” On the other hand, one knows that (1.1) in general fails to hold for © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 50 Page 2 of 15 W. Kohnen Res Math Sci (2022) 9:50 sequences (A(nT )) where T is a ﬁxed primitive form (i.e., (a, b, c) = 1) and F is in the 0 n≥1 0 Maass subspace ([16], sect. 4.2) and [18]). For full level so far no other counter examples seem to be known. Conjecture (1.1) has not yet been proved for a single F = 0, in fact a full proof (where it should be true) seems to be far out of reach. The best result known so far to our knowledge was given in [14,15] and says that k/2−13/36+ A(T) |D| (> 0). (1.2) ,F The proof uses Jacobi forms. Let us brieﬂy mention that in [22] a more general estimate than (1.1) conjecturally valid for all degrees n ≥ 2 was given. One has to replace the exponent k/2 − 3/4 + by k/2 − (n + 1)/4 + . For some comments and some weak results, we refer again to [17] and to [2]. One can hope to improve upon the bound (1.2) by using widely believed conjectures on bounds for L-functions. Indeed, assuming the generalized Riemann hypothesis (GRH) and the reﬁned Gan–Gross–Prasad conjectures for Bessel periods, it was shown in [10] that (1.2) is true with the exponent on the right-hand side replaced by k/2 − 1/2 + . The purpose of this paper is to improve upon (1.2) in a slightly diﬀerent direction, for certain T, e.g., under the generalized Lindelöf hypothesis (GLH) or (GRH). For example, our main result (Theorem 3, Sect. 4)willimply that (1.1) for any given D < 0 holds for at least one primitive T, under the assumption of GRH for L-functions attached to Dirichlet characters and the assumption of a suitable growth condition for quadratic twists of Hecke L-functions associated with cuspidal newforms which are normalized Hecke eigenforms of weight 2k − 2 and level 1 or a prime p, at the central point. Here, “suitable” means polynomial growth in the level and growth à la GLH with respect to the conductor of the quadratic character. The paper is organized as follows. In Sect. 2, we will brieﬂy recall GLH and GRH for the L-functions relevant for us. In Sect. 3, we will shortly remind the reader of some basic facts on genus theory of binary quadratic forms of arbitrary negative (not necessarily fundamental) discriminants D. Section 4 contains a precise statement of our results. In Sect. 5, we will recall several facts on Jacobi forms, while in Sect. 6, we shall give the proof of our results. We remark that some parts of the paper (including the proofs of some more elementary Lemmas) intentionally have been written rather broadly, for the convenience of the reader. 2 Conditional bounds on L-functions GLH and GRH—originally formulated for the Riemann zeta function—can be stated much more generally and in a uniform way for arbitrary “L-functions”, see ([9], chap. 5). Here by an L-function, we understand a function L(s) of a complex variable s given by an absolutely convergent Dirichlet series in (s) > 1 and satisfying some important additional condi- tions (e.g., existence of an Euler product, meromorphic continuation to the entire plane and functional equation under s → 1 − s when completed with an appropriate -factor). Then, GLH states that L(s) (q(s)) ((s) = ) 2 W. Kohnen Res Math Sci (2022) 9:50 Page 3 of 15 50 for any > 0 where q(s)(s ∈ C) is the so-called analytic conductor of L(s). The implied constant in depends only on . We do not need here the explicit value of q(s)inthe most general case. Furthermore, GRH states that all zeroes of L(s) in the critical strip 0 < (s) < 1lie on the critical line (s) = . It is known that GRH for L(s) implies GLH for L(s). The cases relevant for us are the following. Conjecture A (GLH) Let L(s, χ) be the Dirichlet L-function attached to a primitive Dirich- let character χ modulo M. Then for any > 0, L( + it, χ) M(1 +|t|) (t ∈ R). Conjecture B (GRH) Let L(s, χ) be as above. Then, all zeroes of L(s, χ)in0 < (s) < 1 are on the line (s) = . Now, let g be a normalized cuspidal Hecke eigenform which is a newform of even integral weight k and level N.Let χ be a primitive Dirichlet character modulo M with (M, N) = 1. Let χ(n)a(n) k + 1 L(g, χ,s):= (s) > n 2 n≥1 be the Hecke L-function of g twisted with χ, where a(n)(n ≥ 1) are the Fourier coeﬃcients of g.Itiswell-knownthat −s 2 s/2 (2π) (M N) (s)L(g, χ,s) has homomorphic continuation to C and satisﬁes a functional equation under s → k − s relating χ to χ¯ . Note that we have preferred here to keep the classical normalization, so in particular k k the critical line is (s) = . We need GLH in this case only for the central point s = 2 2 of the critical line and do not need a bound in k-aspect. Also we require only polynomial growth in the level N. Conjecture C (GLH) With the above notations, for any > 0 L(g, χ, ) M · Q(N) k, where Q is a polynomial (not depending on g, N, χ or ). 3 Genus theory We collect several facts from genus theory of binary quadratic forms. For more details, we refer the reader to ([7], chap. 6) or to [11]. Recall that a discriminant is a nonzero integer D with D ≡ 0, 1 (mod 4). By a funda- mental discriminant, we understand either 1 or the discriminant of a quadratic extension of Q. Every discriminant D has a unique expression D = D f where D is a fundamental 0 0 discriminant and f ∈ N. Attached to D there is the quadratic character χ = of modulus |D| (Kronecker symbol). It is primitive if and only if D is fundamental. We will usually suppose that D < 0. Let K = Q( D ) and denote by O = 0 D a+b D { | a, b ∈ Z,a ≡ bD (mod 2)} the order of K of conductor f . A fractional ideal a of O is called O -invertible if there is a fractional ideal b of O such that ab = O . D D D D Recall that two fractional ideals a and b of O are called (properly) equivalent if a = (λ)b ∗ 2 for some λ ∈ K . Note that N(λ) =|λ| > 0since D < 0. 50 Page 4 of 15 W. Kohnen Res Math Sci (2022) 9:50 The set of equivalence classes of O -invertible fractional O -ideals will be denoted by D D CL . It is a group in a natural way (class group). The class number h(D) = #CL is ﬁnite. D D There is a bijective correspondence between CL and the set of -equivalence classes D 1 of primitive positive deﬁnite integral binary quadratic forms with discriminant D.Itis αβ −βα induced from the map which associates with the ideal a = Zα ⊕ Zβ (with > 0) N(λ) the function a = Z → Z, λ → (viewed as an integral quadratic form). N(a) By deﬁnition, two fractional ideals a and b belong to the same genus if there is λ ∈ K such that N(a) = N(λ)N(b). It is well-known that this is equivalent to saying that the associated integral primitive binary quadratic forms of discriminant D are equivalent under SL (Q). The set of genera forms a group G which is naturally isomorphic to the 2 D character group of the ﬁnite abelian group CL /CL . Every genus contains the same number of ideal classes (respectively, classes of quadratic forms). The number of genera μ(D)−1 is given by #G = 2 where r, if D ≡ 1 (mod 4) or D ≡ 4 (mod 16) μ(D):= r + 1, if D ≡ 8 (mod 16),D ≡ 12 (mod 16) or D ≡ 16 (mod 32) r + 2, if D ≡ 0 (mod 32) (3.1) (see [7], p. 222). Here, r is the number of diﬀerent odd prime factors of D.If D is funda- mental, one checks that μ(D) is equal to the number of prime factors of D. There is another well-known and useful description of the genera. Two primitive integral binary quadratic forms f (x, y)and g(x, y) of discriminant D belong to the same genus if and only if they represent the same values in (Z/|D|Z) . More precisely, the values in ∗ 2 2 D 2 (Z/|D|Z) represented by the principal genus (given by CL , respectively, by x − y if D 4 2 1−D 2 D is even and x + xy + y if D is odd) form a subgroup H of ker χ (for example, if D is odd, then H is the subgroup of nonzero squares). The values in (Z/|D|Z) represented by (the genus) of any other primitive form of discriminant D form a coset of H in ker χ . A typical application of these facts is the following result which is used in Sect. 6. Lemma 1 With the above notations, assume that p is a prime not dividing D and assume that p is represented modulo |D| by g(x, y). Then, there is a form in the genus of g(x, y) which represents p over Z. Proof Since p is represented modulo |D|,wemusthave = 1. Hence, by the well- known formula for the total number of representations of p by (classes) of forms of discriminant D, there exists a form f (x, y) of discriminant D that represents p over Z. However, then both f and g represent p modulo |D|. Since diﬀerent cosets are disjoint, it follows that f and g belong to the same coset, i.e., to the same genus. We ﬁnish this section with some remarks on genus characters. Suppose D isafunda- mental discriminant with D |D and the quotient again is a discriminant. Then, we can write D = D D f where D is a fundamental discriminant and f ∈ N. 1 2 2 0 Attached to this decomposition, there is a quadratic character χ of Cl (called a D ,D D 1 2 genus character) which essentially is given as products of Jacobi symbols evaluated at norms of ideals. In the language of quadratic forms, one has χ ([f ]) = χ (m) where D ,D D 1 2 1 m is any integer prime to D and represented by f and where [f ] denotes the equivalence class of f . One has χ = χ if and only if (D ,D ) ∈{(D ,D ), (D ,D )} and each D ,D 1 2 2 1 D ,D 1 2 1 2 1 2 W. Kohnen Res Math Sci (2022) 9:50 Page 5 of 15 50 quadratic character of Cl occurs as one of the χ . Associated with χ , there is a D D ,D D ,D 1 2 1 2 Dirichlet series χ (a) D ,D 1 2 L(s, χ ):= ((s) > 1), D ,D 1 2 s N(a) where a runs over O -invertible ideals of O . This series has meromorphic continuation D D to the entire plane and satisﬁes a functional equation under s → 1 − s.Accordingto[11], one has the identity L(s, χ ) = L(s, χ )L(s, χ ) · e (s, χ ), (3.2) D ,D D D p D ,D 1 2 1 2 1 2 p|f where L(s, χ )and L(s, χ ) are the Dirichlet series attached to χ and χ , respectively, D D D D 1 2 1 2 and where e (s, χ ) p D ,D 1 2 −s −s m −1−2m s 1−s 1−s p p (1 − χ (p)p )(1 − χ (p)p ) − p (p − χ (p))(p − χ (p)) D D D D 1 2 1 2 = . −2s 1 − p Here, m is the highest power of p dividing f . (Of course, if f = 1, i.e., D is a fundamental p 0 0 discriminant this is classical.) The following result will be needed in Sect. 6. Lemma 2 On (s) = ,one has e (s, χ ) f (> 0). p D ,D 1 2 p|f −it Proof Writing s = + it and X = p with t real one ﬁnds q(X) e (s, χ ) = (3.3) p D ,D 1 2 1 − X where −1/2 −1/2 q(X) = (1 − χ (p)p X)(1 − χ (p)p X) D D 1 2 2m −1/2 −1/2 −X (X − χ (p)p )(X − χ (p)p ). D D 1 2 We observe that the numerator in (3.3)has azeroat X =±1, so the denominator in (3.2) cancels. More precisely, we can write −1 2 q(X) = (1 − χ (p)χ (p)p )(1 − X ) D D 1 2 −1/2 −1/2 2m +(1 − χ (p)p )(1 − χ (p)p )(1 − X ), D D 1 2 and hence using the ﬁnite geometric series, we obtain q(X) −1 = (1 − χ (p)χ (p)p ) D D 1 2 1 − X −1/2 −1/2 2 2m −2 +(1 − χ (p)p )(1 − χ (p)p )(1 + X + ··· + X ). D D 1 2 Therefore, |e (s, χ )|≤ 2(1 + 2m ) ≤ 6m . p D ,D p p 1 2 50 Page 6 of 15 W. Kohnen Res Math Sci (2022) 9:50 Our assertion now follows easily. Indeed, let > 0 and choose C = C() > 0 such that 6x ≤ Ce (x ≥ 1). It follows that |e (s, χ )|≤ (6m ) ≤ C · e . p D ,D p 1 2 p|f p|f p|f p|f 0 0 0 0 Choose c = c() such that C ≤ 2 . Then with σ (n) denoting the number of positive divisors of n and bearing in mind that σ (n) n (> 0),wesee that c c c C ≤ 2 = σ (p) ≤ σ (f ) 0 0 0 p|f p|f p|f 0 0 0 f . Furthermore, m m 2 m log p 2 log f p p p 0 e = e ≤ e = e p|f = f . This proves our assertion. 4 Statement of results We let S ( ) be the space of Siegel cusp forms of even integral weight k for the Siegel modular group .For F ∈ S ( ), we denote by A(T)the Fouriercoeﬃcients of F.Inthe 2 2 statements below, we identify T with its -class. For a discriminant D < 0, recall that G 1 D denotes the group of genera of positive deﬁnite primitive integral binary quadratic forms of discriminant D. Theorem 1 Let F ∈ S ( ).Let > 0. Then for any genus in G , there exists T in this k 2 D genus such that k/2−31/72+ A(T) |D| . (4.1) ,F Theorem 2 Suppose that Conjecture A (GLH, Sect. 3)holds forL(s, χ ) for all fundamen- tal discriminants .Let F ∈ S ( ) and let > 0. Then for any genus in G , there exists T k 2 D in this genus such that k/2−1/2+ A(T) |D| . (4.2) ,F Our main result is Theorem 3 Suppose that Conjecture B (GRH, Sect. 3)holds forL(s, χ) for all primitive Dirichlet characters χ. Assume further that Conjecture C (GLH, Sect. 3)holds forall cuspidal normalized Hecke eigennewforms of weight 2k − 2 and level N where N is 1 or a prime, and for all Dirichlet characters χ where is a negative fundamental discriminant with ( ,N) = 1.Let F ∈ S ( ) and let > 0. Then for any genus in G , there exists T in k 2 D this genus such that k/2−3/4+ A(T) |D| . (4.3) ,F μ(D)−1 Recall that #G = 2 with μ(D) given by (3.1). As an immediate consequence, we then obtain: W. Kohnen Res Math Sci (2022) 9:50 Page 7 of 15 50 Corollary Under the given conditions of Theorems 1,2 and 3, respectively, for any D < 0, μ(D)−1 there are at least 2 pairwise -inequivalent T that satisfy (4.1), (4.2) and (4.3), respectively. The proofs will be given in Sect. 6, after recalling several facts from the theory of Jacobi forms. The proofs of (4.1)and (4.2) follow more or less the same lines as the proofs of the corresponding statements in [14,15]. The proof of (4.3)ismoreelaborate andwilluse in particular the theory of Jacobi newforms (which is based on the trace formula) and an explicit Waldspurger formula in the context of Jacobi forms. 5 Jacobi forms For basic deﬁnitions, we refer the reader to [5]. For k an integer and m a positive integer, cusp cusp we denote by J the space of Jacobi cusp forms of weight k and index m.Each φ in J k,m k,m has a Fourier expansion n r φ(τ,z) = c(n, r)q ζ , n,r∈Z,r <4mn 2πiτ 2πiz where c(n, r) ∈ C and τ ∈ H (= complex upper half-plane), z ∈ C,q = e , ζ = e . The Fourier coeﬃcients c(n, r) depend only on the discriminant D = r − 4mn < 0and the residue class of r (mod 2m). We therefore often write the Fourier expansion of φ as r −D 4m φ(τ,z) = C(D, r) q ζ . D<0,r,D≡r (mod 4m) cusp The space J becomes a ﬁnite-dimensional Hilbert space under the Petersson scalar k,m product −4πmy /v k−3 φ, ψ = φ(τ,z)ψ(τ,z)e v du dv dx dy (τ = u + iv, z = x + iy), k,m where F is a fundamental domain for the action of the Jacobi group = Z on H × C. We usually write φ, ψ for φ, ψ . k,m It is a fundamental fact that if F ∈ S ( ), then F has a Fourier Jacobi expansion 2πimτ F(Z) = φ (τ,z)e , m≥1 τ z where Z = is in the Siegel upper half-space of degree two (i.e., τ, τ ∈ H,z ∈ z τ nr/2 cusp C, (z) < τ ·τ )and φ ∈ J . In particular, then the coeﬃcient A k,m r/2 m equals the (n, r)-th Fourier coeﬃcient of φ . We recall from [14,15] cusp Proposition 1 Let φ ∈ J and k > 2.Then, k,m k/2−3/4 |D| 1/2 1/2+ c(n, r) m +|D| · ·||φ|| (> 0). ,k (k−1)/2 Proposition 2 Let F ∈ S ( ) with Fourier–Jacobi coeﬃcients given by (5.1).Then, k/2−2/9+ ||φ || m (> 0). m ,F 50 Page 8 of 15 W. Kohnen Res Math Sci (2022) 9:50 The following results will be used in the proof of Theorem 3. We will suppose that mis squarefree. (We will actually only need the cases m = 1or m aprime.) We denote by cusp,old cusp J = J |V k,m k, |m,>1 the space of cuspidal Jacobi oldforms of weight k and index m. Here, n r cusp cusp n r k−1 n r V : J → J , c(n, r)q ζ → a c( , ) q ζ k, k,m a a 2 2 a|(n,r,) n,r∈Z,r <4mn n,r∈Z,r <4mn cusp,new is the usual shift operator ([5], p. 43). We let J be the space of cuspidal Jacobi k,m cusp,old cusp newforms of weight k and index m (orthogonal complement of J in J ). k,m k,m The following results hold true. i) Let S (m) be the space of cusp forms of weight 2k − 2 and level m and S (m) 2k−2 2k−2 the subspace of forms which have eigenvalue (−1) under the Fricke involution. We −,new cusp,new new let S (m) ⊂ S (m) be the subspace of newforms. Then, the spaces J and 2k−2 2k−2 k,m cusp,new −,new S (m) are isomorphic as Hecke modules. More precisely, the space J has an 2k−2 k,m orthogonal basis of Hecke eigenforms for all Hecke operators T (n ∈ N, (n, m) = 1) and all the Atkin–Lehner involutions W (|m), uniquely determined up to multiplication with nonzero scalars and up to permutation. If φ is such an eigenform, then there exists −,new a unique normalized Hecke eigenform g in S (m) which has the same eigenvalues as 2k−2 φ under the Hecke operators T(n)(n ∈ N, (n, m) = 1) and the Atkin–Lehner involutions W (|m). Moreover, if r is an integer and D is a negative fundamental discriminant with D ≡ r 0 0 0 (mod 4m), then there is a map cusp S : J → S (m), D ,r 0 0 k,m 2k−2 r −D D n n r k−2 2πinw 4m C(D, r) q ζ → ( )d C(D ,r ) e 0 0 d d d n≥1 d|n D<0,r,D≡r (mod 4m) (w ∈ H) which preserves newforms and commutes with all Hecke operators and Atkin–Lehner involutions. For more details, see [24]. In connection with the above lifting map, the following elementary result is useful. We will state it here only in the case where will need it, namely for m = p a prime. cusp Lemma 3 Suppose that φ ∈ J where p is a prime. Let D < 0 and r modulo 2pwith k,p 2 2 D ≡ r (mod 4p) be given and suppose that p does not divide D. Write D = D f with D a fundamental discriminant and f ∈ N. Then, there is an integer r satisfying D ≡ r 0 0 0 (mod 4p) and r f ≡ r (mod 2p). In particular, C(D, r) = C(D f ,r f ). 0 0 0 2 2 Proof Note that p does not divide f by assumption and that D f ≡ r (mod 4p). First suppose that f is odd. We then set r ≡ rf (mod 2p) where f ∈ Z is an inverse of f modulo 2p. By deﬁnition, we then have r f ≡ r (mod 2p)and D ≡ r (mod 4p). 0 0 Next suppose that f is even, f = 2f .Then, r must be even, r = 2r , and we have 1 1 2 2 2 D f ≡ r (mod p). Observe that p does not divide f . It follows that D ≡ r f (mod p) 0 1 0 1 1 1 1 where f ∈ Z is an inverse of f modulo p.Since p is odd, we can change r and f modulo 1 1 1 1 W. Kohnen Res Math Sci (2022) 9:50 Page 9 of 15 50 p such that r ≡ f ≡ 1 (mod 2) if D is odd and r ≡ 0 (mod 2) if D is even. We then 1 0 1 0 have D ≡ r f (mod 4p), and hence set r ≡ r f (mod 2p). Then, by construction, 0 0 1 1 1 our claim follows. ii) Let φ and g be as above. Then, we have the identity |C(D ,r )| (k − 2)! L(g, χ ,k − 1) 0 0 D k−3/2 0 = |D | , 2k−3 k−1 k−2 φ, φ g, g 2 π m where D is a fundamental discriminant with (D ,m) = 1and r is an integer with D ≡ r 0 0 0 0 (mod 4m). The left-hand side does not depend on the choice of r . For more details, see ([6], chap. II). iii) One has a direct sum decomposition cusp cusp,new J =⊕ J |V |m k,m k, (see [24], p. 138). cusp Lemma 4 The space J is the orthogonal sum of the one-dimensional subspaces gener- k,m ated by φ|V ,where runs over all divisors of m and where φ for each ﬁxed runs over an cusp,new orthogonal basis of Hecke eigenforms of J . k, cusp,new cusp,new Proof We ﬁrst claim that if φ ∈ J and ψ ∈ J are two Hecke eigenforms, m m k, k, then φ|V , ψ |V = 0unless = and φ and ψ are proportional. Indeed, suppose that J J φ|T (n) = λ(n)φ and ψ |T (n) = μ(n)ψ for all n with (n, m) = 1. Then, J J λ(n)φ|V , ψ |V = φ|V T (n), ψ |V =φ|V , ψ |V T (n) = μ(n)φ|V , ψ |V , since the T (n) are Hermitian and commute with the operators V for (n, m) = 1. If φ|V , ψ |V = 0 it follows that λ(n) = μ(n) for all n with (n, m) = 1, hence the Hecke eigenforms g and h corresponding to φ and ψ, respectively, have the same eigenvalues under all T(n) for (n, m) = 1. Since g and h are newforms, it follows that = and that g = h, by Atkin-Lehner theory. This proves the claim. cusp,new We still have to show that if φ ∈ J is a Hecke eigenform, then φ|V is not the zero k, function. To see this, note that according to ([24], p. 139) one has the commutation rule cusp ψ |V |S = ψ |S |B (ψ ∈ J ), (5.1) D ,r D ,r m 0 0 0 0 k, where B is the operator S ( ) → S (m) which sends a function h(z)to h(z) + 2k−2 2k−2 k−1 h(z). By Atkin–Lehner theory, (say) B is injective. Also, by ([24], p. 137), there is a linear combination in the S which is injective. It now easily follows that φ|V = 0. D ,r 0 0 Remark For arbitrary index m, the theory of Jacobi oldforms is more complicated than the squarefree case, e.g., the operators U ( |m)sending φ(τ,z)to φ(τ, z) now come into play. In particular, diﬀerent types of spaces of oldforms need not necessarily be orthogonal to each other. The latter phenomenon is well-known from the theory of elliptic modular forms, cf. [19]. Nevertheless, we think that there should exist a natural decomposition of cusp,old J into an orthogonal direct sum of one-dimensional Hecke eigenspaces. For the one k,m variable case, see e.g., [3,23]. 50 Page 10 of 15 W. Kohnen Res Math Sci (2022) 9:50 iv) We ﬁnally will need the following elementary Lemma 5 Let V be a ﬁnite dimensional complex Hilbert space, with scalar product , . Let λ : V → C be a functional. Let |λ(v )| K := v ,v ν ν ν=1 where {v , ... ,v } is any orthogonal basis of V (the deﬁnition does not depend on the choice 1 d of the basis). Then, 2 2 |λ(v)| ≤ K ||v|| (∀v ∈ V ). The assertion is essentially obvious. Indeed, there is v ∈ V such that λ(v) =v, v for all v ∈ V , hence by the Cauchy–Schwartz inequality, it follows that 2 2 2 |λ(v)| ≤||v|| ||v || . On the other hand, since the basis is orthogonal, one has v ,v 0 ν v = v 0 ν v ,v ν ν ν=1 and hence v ,v = K . 0 0 λ Remark Lemma 5 is applied in the proof of Theorem 3, with V a space of Jacobi forms, the v being Hecke eigenforms and with λ the map sending φ ∈ V to a ﬁxed Fourier coeﬃcient (so v is a Poincaré series). We note that a similar reasoning was implicitly applied in the proof of Proposition 1; however, in the latter case, the scalar product v ,v was estimated 0 0 directly using the fact that the Fourier coeﬃcients of Jacobi Poincaré series can be given explicitly in terms of Bessel functions and Kloosterman–Salié sums. 6 Proof of results We may suppose that F = 0 which as is well-known implies that k ≥ 10. We start with the proof of Theorem 1. Fix a genus in G . According to [1,8], there is T in the genus 1/4+ which represents a positive integer m prime to D and such that m m (> 0). Clearly, we can suppose that the representation is primitive, T = m with (γ , δ) = 1. Replacing T by T[U] where U is in with second column equal to , we can suppose nr/2 that T = .Let φ be the m-th Fourier–Jacobi coeﬃcient of F,so A(T)isthe r/2 m (n, r)-th Fourier coeﬃcient of φ . Applying Propositions 1 and 2, we ﬁnd that k/2−3/4 |D| 1/2 1/2+ A(T) m +|D| · ·||φ || (> 0) ,k m (k−1)/2 1/2 1/2+ k/2−3/4 5/18 m +|D| ·|D| · m , ,F 1/4+ and hence using m m ,weobtain k/2−31/72+ A(T) |D| . ,F W. Kohnen Res Math Sci (2022) 9:50 Page 11 of 15 50 This proves our assertion. The basic idea for the proof of Theorem 2 is the same; however, in addition, we want to have m very small with respect to |D|, i.e., m |D| (> 0). This can be achieved using GLH for genus character L-series which in turn follows from GLH for Dirichlet series (as requested in the hypothesis) together with the decomposition (3.2) and Lemma 2. In fact, to deduce the existence of m, one essentially imitates the proof in [8]; however, in the Perron-type integrals of a product of genus character L-series over the line (s) = 2([8], p. 205), one shifts the line of integration to the line (s) = and then estimates using GLH. Once the existence of m is established one argues as before. Note that one does not need here the assertion of Proposition 2 which is more diﬃcult to prove but instead can apply k/2 the “trivial” bound ||φ || m ([13], p. 543). We now turn to the Proof of Theorem m F 3 which is more lengthy. We ﬁrst claim that under GRH for Dirichlet L-functions in the above argument of the proof of Theorem 2 we can choose m as a prime p, i.e., there exists T in the given genus which represents a prime p with (D, p) = 1and p |D| . This follows from a more general result given in ([20], Theorem 1.4), which says the following. Assume GRH and let q be a positive integer with q ≥ 20000. Let H be a subgroup of G = (Z/qZ) with index h = [G : H] > 1. Let p be the smallest prime lying in a given coset aH. Then, either p < 10 ,or p ≤ (h − 1) log q + 3(h + 1) + (log log q) . For our situation, we take q =|D| and let H be the subgroup corresponding to the μ(D) principal genus, see Sect. 4. Note that h = 2 in this case with μ(D) given by (3.1). Using σ (n) n (n ≥ 1,> 0), one immediately sees that h |D| . Applying Lemma 1, the result follows for |D|≥ 20000. Note that we can always omit ﬁnitely many D in the proof of (4.3), so in the following, we will always tacitly understand that |D|≥ 20000. Changing T by an element U ∈ if necessary as before we can assume that T = nr/2 ,so A(T) is the (n, r)-th Fourier coeﬃcient of the p-th Fourier–Jacobi coeﬃcient r/2 p cusp φ ∈ J of F.Since p is a prime have k,p cusp cusp cusp,new J = J |V ⊕ J . k,p k,1 k,p (1) (d) (1) (e) We let {φ , ... , φ } and {ψ , ... , ψ }, be an orthogonal basis of Hecke eigenforms cusp cusp,new for J and J , respectively. By Lemma 4, then k,1 k,p (1) (d) (1) (e) {φ |V , ... , φ |V , ψ , ... , ψ } p p cusp cusp is an orthogonal basis for J . We apply Lemma 5 with V = J and λ the functional k,p k,p sending φ to its (D, r)-th Fourier coeﬃcient C (D, r). We then obtain 2 2 |A(T)| ≤ K ||φ || , (6.1) p;D,r p where (1) (2) K := K + K , p;D,r p;D,r p;D,r and where |C (D, r)| (ν) φ |V (1) p K := (6.2) p;D,r (ν) (ν) φ |V , φ |V p p ν=1 50 Page 12 of 15 W. Kohnen Res Math Sci (2022) 9:50 and e 2 |C (D, r)| (μ) (2) ψ K := . (6.3) p;D,r (μ) (μ) ψ , ψ μ=1 We have k/2 ||φ || p (6.4) p F as used before. (μ) We now estimate the sum (6.3)ﬁrst. Letusﬁx μ and simply write ψ for ψ .By 2 2 Lemma 3, we have C (D, r) = C (D f ,r f )) where D = D f with D a fundamental ψ ψ 0 0 0 0 discriminant, f ∈ N and where r is an integer with D ≡ r (mod 4p)and r ≡ r f 0 0 0 (mod 2p). Applying S to the Hecke eigenform ψ and using multiplicity 1, we ﬁnd that D ,r 0 0 D n n k−2 d C (D ,r ) = C (D ,r )a(n)(∀n ≥ 1), (6.5) ψ 0 0 ψ 0 0 d d d d|n −,new 2πinw where g(w) = a(n)e ∈ S is a normalized eigenform. Inverting (6.5), we n≥1 2k−2 see that D n 2 k−2 C (D n ,r n) = C (D ,r ) μ(d)d a( )(∀n ≥ 1) ψ 0 0 ψ 0 0 d d d|n where μ is the Möbius function. In particular, with n = f we ﬁnd that D f k−2 C (D, r) = C (D ,r ) μ(d)d a( ). (6.6) ψ ψ 0 0 d d d|f For > 0, the sum over f in absolute value can be estimated from above by f f k−2 k−3/2 k−3/2+ −1/2− d σ ( )( ) f d d d d|f d|f k−3/2+ f σ (f ) k−3/2+2 f . Here, we have used Deligne’s theorem for g (observe that p does not divide f ) and the usual bound for σ (n) frequently applied before. We also have by ii), Sect. 5 |C (D ,r )| (k − 2)! L(g, χ ,k − 1) ψ 0 0 D k−3/2 0 = |D | . 2k−3 k−1 k−2 ψ, ψ 2 π p g, g Therefore, we ﬁnd from (6.6) that 2 k−3/2+2 |C (D, r)| |D| L(g, χ ,k − 1) ψ D · (> 0). ,k k−2 ψ, ψ p g, g By ([21], p. 42, (2.19)), see also ([4], Propos. 4), we have −1 p (> 0). ,k g, g Also by our assumption on GLH for any > 0, we have L(g, χ ,k − 1) |D | Q(p) |D| Q(p) D ,k 0 ,k 0 W. Kohnen Res Math Sci (2022) 9:50 Page 13 of 15 50 where Q is a polynomial. Combining we therefore ﬁnd that 2 k−3/2+3 |C (D ,r )| |D| ψ 0 0 Q(p). (6.7) ,k k−1− ψ, ψ p The number of Hecke eigenforms occurring in (6.3)isbounded by dim S (p) ≤ [ : 2k−2 1 (p)] p. Therefore from (6.4)and (6.7), we deduce that 0 k (2) 2 k−3/2+3 2+ K ||φ || |D| · p Q(p). p ,F p;D,r Since by asumption p |D| , we ﬁnally obtain (re-deﬁning ) that (2) 2 k−3/2+ K ||φ || |D| . (6.8) p ,F p;D,r We now look at the sum (6.2) which essentially can be estimated in a similar way. Let (ν) # us write φ for φ and denote the Fourier coeﬃcients of φ|V by C (D, r). First we want to relate the norm of φ|V to the norm of φ. We observe that φ|V , φ|V =φ|V V , φ, p p p cusp cusp where V : J → J is the operator adjoint to V .Accordingto([13], Proposition ii) k,p k,1 (where a more general case is treated), one has ∗ J k−1 k−2 V V = T (p) + (p + p )id, so ∗ k−1 k−2 φ|V V = (λ + p + p )g, p p where λ is the eigenvalue of the eigenform g ∈ S (1) corresponding to φ. Therefore, p 2k−2 k−1 k−2 φ|V , φ|V = (λ + p + p )φ, φ. (6.9) p p p We note that the pre-factor on the right-hand side of (6.9) is nonzero, either by Lemma k−3/2 4 or by a simple direct computation, cf. [12]. In fact, using Deligne’s bound |λ |≤ 2p , one sees immediately that k−1 k−2 k−2 |λ + p + p |≥ p . (6.10) By Lemma 3, with the same notation as used before, we can write C (D, r) = # 2 C (D f ,r f ). By multiplicity 1, 0 0 φ|S = C(D ,r )g, D ,r 0 0 0 0 hence recalling the commutation rule (5.1) φ|S |B = φ|V |S D ,r p p D ,r 0 0 0 0 (cf. (5.1)) we ﬁnd D n n n k−2 # k−1 d C D ,r = C (D ,r ) a(n) + p a( ) (∀n ≥ 1), 0 0 φ 0 0 d d d p d|n where a(n) are the Fourier coeﬃcients of g, and we understand that a( ) = 0if p does not divide n. Inverting gives D n # 2 k−2 C (D n ,r n) = C (D ,r ) μ(d)d a 0 0 φ 0 0 d d d|n D n k−2 + μ(d)d a . d pd d|n 50 Page 14 of 15 W. Kohnen Res Math Sci (2022) 9:50 We choose n = f . Since by p does not divide f , the second sum is zero, and we have D f # k−2 C (D, r) = C (D ,r ) μ(d)d a . φ 0 0 d d d|f k−3/2+2 The absolute value of the sum over d is bounded from above by f as already proved above. Therefore using ii) in Sect. 5 with m = 1, (6.9), (6.10)and (6.4)and our hypothesis on GLH, we ﬁnd that (1) 2 k−3/2+3 2 K ||φ || |D| · p Q(p). p ,k p;D,r Using our assumption on p, we therefore obtain (1) 2 k−3/2+ K ||φ || |D| . (6.11) p ,F p;D,r From (6.1), (6.8)and (6.11), now our claim follows. Remark Very probably the proof of Theorem 3—at the cost of more technical diﬃculties—would go through if one would merely require that m is squarefree rather than m = p being a prime. On the other hand, to deduce from GRH the existence of a squarefree integer instead of a prime with the required appropriate growth with respect to the discriminant does not seem really easier. However, the situation would be diﬀerent if the arguments on the side of the theory of Jacobi forms could be carried over to an arbi- trary m ≥ 1, which as was pointed out in the remark after the proof of Lemma 4 would be some non-trivial work. Instead then the assumption of GRH could be completely dropped, and it would suﬃce to merely suppose GLH (for Dirichlet L-functions as in Theorem 2, for quadratic twists of Hecke L-functions of eigenforms at the central point as above). Acknowledgements We would like to thank V. Blomer and K. Soundararajan for some useful discussions. Funding Open Access funding enabled and organized by Projekt DEAL. Data availability statement My manuscript has no associated data. Received: 13 April 2022 Accepted: 28 June 2022 Published online: 25 July 2022 References 1. Baker, A., Schinzel, A.: On the least integer represented by the genera of binary quadratic forms. Acta Arithmetica XVII I, 137–144 (1971) 2. Böcherer, S., Kohnen, W.: Estimates for Fourier coeﬃcients of Siegel cusp forms. Math. Ann. 297, 499–517 (1993) 3. Choie, Y.J., Kohnen, W.: The ﬁrst sign change of Fourier coeﬃcients of cusp forms. Am. J. Math. 131, 517–543 (2009) 4. Duke, W.: The critical order of vanishing of automorphic L-functions with large level. Invent. Math. 119, 165–174 (1995) 5. Eichler, M., Zagier, D.: The Theory of Jacobi Forms, Progress in Mathematics, vol. 55. Birkhäuser (1985) 6. Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives of L-series. II. Math. Ann. 278, 497–562 (1987) 7. 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Published: Sep 1, 2022
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