On sign changes of Fourier coefficients of Hermitian cusp forms of degree two

We prove a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form of degree 2. In addition, we prove a quantitative result for the number of sign changes of the primitive Fourier coefficients. We give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2 over certain imaginary quadratic extensions.


Introduction and statement of the main results
The distribution of signs of the Fourier coefficients of a non-zero elliptic cusp form has been a subject of study for several mathematicians over the past years. One aspect of this problem is the study of number of sign changes of the Fourier coefficients. Knopp, Kohnen, and Pribitkin in [14] proved that the Fourier coefficients of a nonzero elliptic cusp form f on a congruence subgroup of the full modular group SL 2 (Z) have infinitely many sign changes. They use the Landau's theorem on Dirichlet series with non-negative coefficients and the finiteness of the Hecke L-function attached to the elliptic cusp form f to prove their result. In addition, one can see that [16] and [18] are devoted to the study of sign changes of the Fourier coefficients of an elliptic Hecke eigenform. A more subtle problem is to give an explicit upper bound for the first sign change. This has been studied for elliptic cusp forms of square-free level by Choie and Kohnen [2]. Later, their result has been improved by He and Zhao [11]. For elliptic Hecke eigenforms of level N the problem has been dealt in [13,16].
The theory of elliptic modular forms has been generalized to several variables. Hermitian modular forms over an imaginary quadratic field K are one of those generalizations. In this article, we give a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form F of degree 2. Moreover, we also give a quantitative result for the number of sign changes of the primitive Fourier coefficients. Note that Yamana [22] has established that F is determined by its primitive Fourier coefficients. Also, we provide an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form over certain imaginary quadratic fields. To the best of our knowledge, this is the first attempt to study the distribution of signs of the Fourier coefficients of a Hermitian cusp form. Now, we introduce the necessary notations to state our results.
Let d > 0 be a square free integer. Throughout the article, let K = Q( √ −d) be a fixed imaginary quadratic field. Let be the discriminant of K . Let O K be the ring of integers of K and O # K = i √ |D K | O K be the inverse different of K over Q. The Hermitian modular group of degree 2 over K is given by where J 2 = 0 2 −I 2 I 2 0 2 , I 2 and 0 2 are the 2 × 2 identity matrix and zero matrix respectively. The subgroup coincides with the full modular group U 2 (O K ) if D K = −3, −4. We denote by S k (SU 2 (O K )) the space of Hermitian cusp forms of degree 2 on SU 2 (O K ) (defined in Sect. 2.1). Any F ∈ S k (SU 2 (O K )) has a Fourier series expansion of the form: The first result of this article gives a quantitative result for the sign changes of the Fourier coefficients of F.
) be a non-zero Hermitian cusp form with real Fourier coefficients A F (T ). Then A F (T ) changes sign at least once for |D K |det(T ) ∈ (X , X + X 3/5 ] for X 1.
For any T ∈ + 2 , we define We say that T is primitive if μ(T ) = 1. The Fourier coefficient of F at a primitive T is known as primitive Fourier coefficient. The second result of this article gives the following quantitative result on the number of sign changes of the primitive Fourier coefficients.

non-zero with real Fourier coefficients A F (T ). Then the primitive Fourier coefficients A F (T ) changes sign at least once
Theorem 1.1 implies that there are infinitely many sign changes of the Fourier coefficients of F ∈ S k (SU 2 (O K )). Next, we focus our attention on establishing an explicit upper bound for the first sign of any F ∈ S k (SU 2 (O K )). To accomplish this, we first establish a Sturm bound for Hermitian modular forms of degree 2.
Finally, using Theorem 1.3, we give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2.
for any real > 0, where such that Therefore, there is an obvious reformulation of Theorems 1.1, 1.2 and 1.4 for arbitrary The article is organized as follows: In the next section we recall the definition of three concepts used in this paper; Hermitian modular forms of degree 2, Hermitian Jacobi forms and Jacobi forms with matrix index. We show that Hermitian Jacobi forms occur as the coefficients in the Fourier-Jacobi expansion of a Hermitian modular form of degree 2. In Sects. 3 and 4, we give the proof of Theorems 1.1 and 1.2 respectively. Section 5 is the largest, and contains the proof of Theorem 1.3. We prove Proposition 5.1, Theorems 5.2, and 5.3 in this section, which may be of interest on their own. Finally, in Sect. 6, we prove Theorem 1.4.
Notation For any ring R ⊂ C, we write by R n = {(α 1 , · · · , α n ) | α i ∈ R} the set of row matrices of size 1 × n with entries in R. We denote by M n (R) the set of all n × n matrices with entries in R. Let G L n (R) be the group of matrices in M n (R) with non-zero determinant and let SL n (R) be the group of matrices with determinant 1.

Hermitian modular forms of degree two
The Hermitian upper-half space of degree 2 is defined by The Hermitian modular group U 2 (O K ) acts on H 2 by For any non-negative integer k, we define the action of U 2 (O K ) on the set of functions from H 2 to C by For a positive integer N , we define the congruence subgroup (2) Note that if N = 1 then (2) Definition 2.1 A holomorphic function F : H 2 → C is called a Hermitian modular form of weight k on (2) for all M ∈ (2) 0 (N ).
We have the following lemma.

Lemma 2.3 Let T ∈ +
2 be a primitive matrix. Then there exists g ∈ SL 2 (O K ) such that g t T g = * * * p for some odd prime p.
We know that det(g) = , where ∈ O × K . Therefore, we can take , we get the following relation on the Fourier coefficients of F Now using Theorem 2.2 and Lemma 2.3 we get the following.

Lemma 2.4 Suppose F ∈ S k (SU 2 (O K )) is non-zero with Fourier coefficients A F (T ).
Then, for some odd prime, p there exists a primitive T 0 = * * * p ∈ + 2 such that A F (T 0 ) = 0.

Hermitian Jacobi forms
K be the Hermitian Jacobi group over O K . The Jacobi group G acts on H × C 2 as follows:

Definition 2.5 A holomorphic function
and φ has a Fourier series expansion of the form We denote by J k,m ( (1) 0 (N )) the vector space of all Hermitian Jacobi forms of weight k and index m on (1) 0 (N ).

Theta decomposition
The invariance of φ under the action of (λ, 0) in (5) yields that the Fourier coefficient c(n, r ) is completely determined by r (mod mO K ) and nm − N (r ). We define The theta decomposition of φ ∈ J k,m ( (1) 0 (N )) is given by The theta components h s of φ are elliptic modular forms on the prinicipal congruence subgroup (1) (|D K |N m) (see [9,10]).

Fourier-Jacobi expansion
Let F ∈ S k ( (2) 0 (N )) has Fourier series expansion of the form (3). We write the Fourier series expansion of F as For are in (2) 0 (N ). These matrices act on H 2 by respectively. Because F satisfies the transformation law (2), we can deduce the two transformation laws of Hermitian Jacobi forms for φ m , and therefore, 0 (N )). We call (6) the Fourier-Jacobi expansion of F and φ m 's the Fourier-Jacobi coefficients of F.

Jacobi form with matrix index
The Jacobi group = SL 2 (Z) (Z × Z ) acts on H × C as follows: and ψ has a Fourier series expansion of the form where τ ∈ H, z = (z 1 , · · · , z ) ∈ C , q = e 2πiτ , ζ r = e 2πir z t and M # is the adjugate of M.

Proof of Theorem 1.1
Since F = 0, there exists a m 0 such that the Fourier-Jacobi coefficient φ m 0 = 0 in the Fourier-Jacobi expansion of F. Therefore, there exists s 0 ∈ O # K /m 0 O K such that the theta component h s 0 = 0 in the theta decomposition of φ m 0 . The Fourier series expansion of h s 0 is given by where the direct sum is over all Dirichlet characters modulo Suppose the Fourier series expansion of f ψ is given by f ψ = Let λ = k − 1 andâ Putting these values in (10), we get n≥1â (n)n (λ−1)/2 e(nτ ) = ψ n≥1â ψ (n)n (λ−1)/2 e(nτ ).
From the above we also haveâ where c is a constant depending on h s 0 (|D K |m 0 τ ) and is any real number greater than 0. Now applying [12, Theorem 2.1] we get thatâ(n) changes sign at least once for n ∈ (X , X + X 3/5 ] for X 1. This implies that a(n) and hence for X 1.

Proof of Theorem 1.2
The ring of integers We define the following set We first prove the following proposition which will be required to prove Theorem 1.2.
0 ( p 2 )) such that the Fourier coefficients of G p is given by (tr(T Z)).

Proof Let
We claim that G ∈ S k ( 0 ( p 2 )). It is enough to show that for any Y ∈ J , we have . It is easy to check that This implies that G | k M = G , which asserts our claim. Now the Fourier series expansion of G is given by

Now for any
Therefore, the Fourier series expansion of G is given by Thus, we get the required G p .

Proof of Theorem 1.2
Since F = 0, by Lemma 2.4 there exists a primitive T 0 = n 0 r 0 r 0 p ∈ + 2 for some odd prime p such that A F (T 0 ) = 0. Applying Proposition 4.1, we construct G p from F such that G p ∈ S k ( (2) 0 ( p 2 )) and the Fourier series expansion of G p is given by

A F (T )e(tr(T Z)).
Let H = F − G p . We observe that H ∈ S k ( (2) 0 ( p 2 )) and the Fourier series expansion of H is given by

A F (T )e(tr(T Z)).
Since T 0 is primitive H = 0. We consider the Fourier-Jacobi coefficient φ p in the Fourier-Jacobi expansion of H whose Fourier series expansion is given by We have φ p ∈ J k, p ( . We consider the theta component h s 0 = 0 in the theta decomposition of φ p . The Fourier series expansion of h s 0 is given by We have h s 0 ∈ S k−1 ( (1) (|D K | p 3 )). Now doing the similar calculation as we have done in the proof of Theorem 1.1, we get that , changes sign atleast once for |D K |det(T ) ∈ (X , X + X 3/5 ] for X 1.

Sturm bound
Sturm [21] proved that an elliptic modular form is determined by its first few Fourier series coefficients. The number of these first few Fourier coefficients is known as Sturm bound. Sturm's result has had a significant impact on the study of elliptic modular forms. In this section we first develop a Sturm bound for Jacobi form with matrix index. Following this we establish a relation between Jacobi form with matrix index and Hermitian Jacobi forms. We use this relation to derive a Sturm bound for Hermitian Jacobi forms. Finally we prove Theorem 1.3 using the Fourier-Jacobi expansion of Hermitian modular form and Sturm bound of Hermitian Jacobi forms..

Sturm bound for Jacobi form with matrix index
a(n, r )q n ζ r ∈ J k,M ( (1) where α i j ∈ Z for i < j and α ii ≥ 0. We consider the Taylor series expansion of φ at z 1 = z 2 = · · · = z = 0, with Taylor coefficients For each a b c d ∈ (1) 0 (N ), using the transformation property (7) of φ and above equation we get This implies that Following Eichler and Zagier [6, p. 31], we define We further define where the map from J k,M ( (1) 0 (N )) to M k+v 1 ,··· ,v ( (1) 0 (N )) is given by
In the following theorem we establish a Sturm bound for Jacobi form with matrix index.

Relation between Hermitian Jacobi and Jacobi form with matrix index
In [17, Theorem 2.3], Meher and the second author proved a relation between Hermitian Jacobi form over Q(i) and Jacobi form with matrix index. In the following theorem, we generalize their result for an arbitrary imaginary quadratic field.
Then the space J k,m ( (1) 0 (N )) is isomorphic to J k,A ( (1) 0 (N )) as a vector space over C.
Proof First let us consider the case −d ≡ 2, 3 (mod 4). We define a map η 1 : J k.m ( (1) Using the transformation property of φ mentioned in (4) and (5) we can verify thatφ satisfies (7) and (8). Suppose φ has Fourier series expansion Any r ∈ O # K can be written as r = i We now consider an element ρ ∈ Z 2 , where ρ = (−β, −α). Then the correspondence r → ρ is clearly a bijection from O # K to Z 2 . Now the above Fourier series expansion ofφ can be expressed asφ which is of the form given in (9). This implies that η 1 is a well defined linear map. In a similar manner, one can show that the map is also a well-defined linear map. Now consider the composition map η 2 • η 1 , Similarly we can also check that (η 1 • η 2 )(ψ(τ, z 1 , z 2 )) = ψ(τ, z 1 , z 2 ) and hence η 2 • η 1 = I 1 , η 1 • η 2 = I 2 , where I 1 and I 2 are identity maps on the vector spaces J k,m ( (O K )) and J k,A ( 2 ) respectively. Now consider the case −d ≡ 1 (mod 4). Here we define the maps Approaching as above we can easily verify that η 1 and η 2 are well defined linear maps and also they satisfy

Sturm bound for Hermitian Jacobi forms
Using Theorems 5.2 and 5.3, we establish a Sturm bound for Hermitian Jacobi forms.
If we put N = 1 in the above we get the following.
If a(n, r ) = 0 for all n ≤ β, where We are now ready to prove Theorem 1.3. We first define some necessary terms. For a(n, r )q n ζ r 1 ζ r 2 ∈ J k,m (SL 2 (Z)), we define ord(φ) = min{n | a(n, r ) = 0}.
From Corollary 5.5, Sturm bound for Hermtian Jacobi form when −d ≡ 2, 3 (mod 4) is We put β = m = t in the above and see that it is possible to get a positive value of Similarly, when −d ≡ 1 (mod 4), we will get a positive value of t = 2k 19−d , if d ∈ {3, 7, 11, 15}. We will use the Fourier-Jacobi expansion and a transformation of Hermitian modular form F to show that if the Fourier coefficients A F (n, r , m) = 0 for all n ≤ t and m ≤ t then F = 0.

Proof of Theorem 1.3
We will prove the result when d ∈ {1, 2}. The case d ∈ {3, 7, 11, 15} will follow similarly. We consider the Fourier-Jacobi expansion of F We will show that φ m = 0 for all m ≥ 0. We first consider that m ≤ k 2(5−d) . Then Therefore, by Corollary 5.5, we have φ m = 0. Assume that m > k 2(5−d) . We use induction on m to show that φ m = 0. Suppose that φ m = 0 for all m < m. Now we consider Using the transformation property (2) of F for M, we get F(τ, z 1 , z 2 , τ ) = (−1) k F(τ , z 2 , z 1 , τ ), which shows that A F (n, r , m) = (−1) k A F (m, r , n) = 0 for all n < m. Therefore, Again by Corollary 5.5, we have φ m = 0. This completes the proof.
Then from the proof of Proposition 5.1 we get that a 1 ≤ 8m 0 and a 2 ≤ 8m 0 d. Also D a 1 ,a 2 (ϕ) = α X a 1 ,a 2 (τ ) in Proposition 5.1, for some non-zero α ∈ C. This implies that X a 1 ,a 2 (τ ) is a non-zero elliptic modular form of weight k 1 = 4k + a 1 + a 2 . Therefore, we have Suppose the Fourier series expansion of ϕ is given by d(n, s)e(nτ + αz 1 + βz 2 ).

Remark 6.1
We note that Theorem 1.1 is true for Hermitian modular forms on the congruence subgroup (2) 0 (N ) of SU 2 (O K ) with Dirichlet character χ (mod N ), where χ acts on (2)  Moreover, if the conductor of χ is N then Theorem 1.2 also holds.