On distinguishing Siegel cusp forms of degree two

One of the fundamental problems in the theory of automorphic forms is whether we can distinguish them by a set of eigenvalues. It is well known that in the elliptic modular forms case, this question is equivalent to asking how many Fourier coefficients are sufficient to determine a normalized eigenform. This question is answered first by the classical result of Sturm [25]. In 2011, Ghitza [6] obtains a result by considering two cuspidal Hecke eigenforms of distinct weights, which improves a result of Ram Murty [18]. Later, Vilardi and Xue [26] give a much stronger result that two normalized eigenforms of full level can be determined by their second coefficients under the assumption of Maeda’s conjecture for the Hecke operator $T_{2}$. Recently, Xue and Zhu [28] generalize this result in terms of their third coefficients under a similar assumption.

However, distinguishing Siegel cusp forms is a long-standing unanswered problem and only recently Schmidt [24], in a remarkable paper, gives an affirmative answer to this question for normalized eigenvalues of a Siegel cuspidal eigenform of degree two. This result has been improved by Kumar, Meher and Shankhadhar [13] in the full level case, in which they essentially show that any set of eigenvalues (normalized or non-normalized) at primes $p$ of positive upper density are sufficient to determine the Siegel cuspidal eigenform. In this work we further investigate the question on distinguishing Siegel cusp forms of degree two from various aspects with several improved results. We point out that we have also discussed the similar question for paramodular forms with the combination of the methods from both of automorphic side and Galois side in [27].

Let $\mathcal{S}_{k}(\Gamma_{0}(N))$ be the space of Siegel cusp form of level $\Gamma_{0}(N)$ and weight $k$, where $\Gamma_{0}(N)$ is the Siegel congruence subgroup of level $N$ defined as in (11). Let $F\in\mathcal{S}_{k}(\Gamma_{0}(N))$ be a Hecke eigenform with eigenvalue $\lambda_{F}(n)$ for $(n,N)=1$. Then our first main result is as follows.

Next, we assume that $N=1$, and let $\Gamma_{2}={\rm Sp}(4,{\mathbb{Z}})$. It is well known that the space $\mathcal{S}_{k}(\Gamma_{2})$ has a natural decomposition into orthogonal subspaces

$$\mathcal{S}_{k}(\Gamma_{2})=\mathcal{S}_{k}^{\mathbf{(P)}}(\Gamma_{2})\oplus\mathcal{S}_{k}^{\mathbf{(G)}}(\Gamma_{2})$$

(2)

with respect to the Petersson inner product. Here, $\mathcal{S}_{k}^{\mathbf{(P)}}(\Gamma_{2})$ is the subspace of Saito-Kurokawa liftings, and $\mathcal{S}_{k}^{\mathbf{(G)}}(\Gamma_{2})$ is the subspace of non-liftings. We refer the reader to [24, § 2.1] for more details about this type decomposition. For our purpose, let

$$\mathcal{K}^{\mathbf{(*)}}(2)=\{k\in{\mathbb{Z}}\colon\text{The characteristic polynomial of }T_{k}(2)\text{ for }F\in\mathcal{S}_{k}^{\mathbf{(*)}}(\Gamma_{2})\text{ is irreducible}\},$$

(3)

where $\mathcal{S}_{k}^{\mathbf{(*)}}(\Gamma_{2})$ is the set of those $F\in\mathcal{S}_{k}(\Gamma_{2})$ of type $\mathbf{(*)}$ as in (2). Then we can prove the following result.

In addition, we can also distinguish Hecke eigenforms in each type by using $L$-functions with different methods. First, recall that Saito-Kurokawa liftings of level one and weight $k\in 2{\mathbb{Z}}_{>0}$ can be obtained from elliptic cusp forms of level one and weight $2k-2$. More precisely, let $f\in\mathcal{S}_{2k-2}(\Gamma_{1})$ be a Hecke eigenform, and let $\pi_{f}$ be the cuspidal automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ associated to $f$. Here, ${\mathbb{A}}$ is the ring of adeles of ${\mathbb{Q}}$. Then the resulting Saito-Kurokawa lifting is in $\mathcal{S}_{k}(\Gamma_{2})$, denoted by $F_{f}$. It is well known that $F_{f}$ is also a Hecke eigenform; see [15, 17] for more details about the classical Saito-Kurokawa liftings. The normalized spinor $L$-function of $F_{f}$ and the normalized $L$-function of $f$ are connected by the following relation:

$$L(s,\pi_{F_{f}},\rho_{4})=\zeta(s+1/2)\zeta(s-1/2)L(s,\pi_{f}),$$

(4)

where $\rho_{4}$ is the $4$-dimensional irreducible representation of ${\rm Sp}(4,{\mathbb{C}})$, and $\pi_{F_{f}}$ is the cuspidal automorphic representation of ${\rm GSp}(4,{\mathbb{A}})$ corresponding to $F_{f}$. Let $\xi$ be a primitive Dirichlet character, and let $\chi$ be the corresponding Hecke character of ${\mathbb{Q}}^{\times}\backslash{\mathbb{A}}^{\times}$. Let $\sigma_{1}$ be the standard representation of the dual group ${\rm GL}(1,{\mathbb{C}})={\mathbb{C}}^{\times}$. Then we can define the twisted $L$-function by

$$L(s,\pi_{F_{f}}\times\chi,\rho_{4}\otimes\sigma_{1})=L(s+1/2,\chi)L(s-1/2,\chi)L(s,\pi_{f}\times\chi).$$

(5)

To prove this proposition, it suffices to show that $f=g$, which is due to [16, Theorem B].

Finally, we will distinguish Hecke eigenforms of type $\mathbf{(G)}$ by using Rankin-Selberg $L$-functions under the Generalized Riemann hypothesis. Let $k_{1},k_{2}$ be even integers. Let $F\in\mathcal{S}_{k_{1}}^{\mathbf{(G)}}(\Gamma_{2})$ and $G\in\mathcal{S}_{k_{2}}^{\mathbf{(G)}}(\Gamma_{2})$ be Hecke eigenforms, and let $\pi_{F}$ (resp. $\pi_{G}$) be the cuspidal automorphic representation of ${\rm GSp}(4,{\mathbb{A}})$ corresponding to $F$ (resp. $G$). Then we can define the Rankin-Selberg $L$-function of $F$ and $G$, denoted by $L(s,\pi_{F}\times\pi_{G},\rho_{i}\otimes\rho_{j})$ with $i,j\in\{4,5\}$; see [20, (271)]. Note that the $L$-function here is actually the finite part of $L$-functions in [20]. Moreover, $L(s,\pi_{F}\times\pi_{G},\rho_{i}\otimes\rho_{j})$ has a simple pole at $s=1$ if and only if $i=j,k_{1}=k_{2}$ and $F=c\cdot G$ for some non-zero constant $c$; see [20, Theorem 5.2.3]. In this case, we can prove the following result:

To prove above theorem, we will apply the method in [5]. Then combine with Lemma 5.1 and we will conclude Theorem 1.4.

We thank Wenzhi Luo, Kimball Martin, Ralf Schmidt, Biao Wang, Pan Yan and Liyang Yang for their helpful discussions and comments. We thank Ariel Weiss for drawing our attention to the low weights case ($k_{i}=1,2$) in Theorem 1.1. We thank Biplab Paul for forwarding us their paper [7]. Shaoyun Yi is supported by the National Natural Science Foundation of China (No. 12301016) and the Fundamental Research Funds for the Central Universities (No. 20720230025).

We consider the symplectic similitude group

$${\rm GSp}(4)\coloneqq\{g\in{\rm GL}(4)\colon\>^{t}gJg=\mu(g)J,\>\mu(g)\in{\rm GL}(1)\},$$

(8)

which is an algebraic ${\mathbb{Q}}$-group. Here, $J=\begin{bmatrix}&&&1\\ &&1&\\ &-1&&\\ -1&&&\end{bmatrix}$. The function $\mu$ is called the multiplier homomorphism. The kernel of this function is the symplectic group ${\rm Sp}(4)$. Let $\mathrm{Z}$ be the center of ${\rm GSp}(4)$ and ${\rm PGSp}(4)={\rm GSp}(4)/\mathrm{Z}$. When speaking about Siegel modular forms of degree two, it is more convenient to realize symplectic groups using the symplectic form $J=\begin{bmatrix}0&1_{2}\\ -1_{2}&0\end{bmatrix}$. The Siegel upper half plane of degree 2 is defined by

$$\mathbb{H}_{2}\coloneqq\{Z\in\mathrm{Mat}_{2}({\mathbb{C}})\colon\,^{t}Z=Z,\mathrm{Im}(Z)>0\}.$$

(9)

The group ${\rm GSp}(4,\mathbb{R})^{+}\coloneqq\{g\in{\rm GSp}(4,{\mathbb{R}})\colon\mu(g)>0\}$ acts on $\mathbb{H}_{2}$ by

$$g\langle Z\rangle\coloneqq(AZ+B)(CZ+D)^{-1}\quad\text{for }g=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\in{\rm GSp}(4,\mathbb{R})^{+}\text{ and }Z\in\mathbb{H}_{2}.$$

(10)

Let $\Gamma_{2}={\rm Sp}(4,{\mathbb{Z}})$. In general, for a positive integer $N$ we let

$$\Gamma_{0}(N)\coloneqq\left\{\begin{bmatrix}A&B\\ C&D\end{bmatrix}\in{\rm Sp}(4,{\mathbb{Z}})\colon C\equiv 0\pmod{N}\right\}$$

(11)

be the Siegel congruence subgroup of level $N$. It is clear that $\Gamma_{2}=\Gamma_{0}(1)$.

Let $\mathcal{M}_{k}(\Gamma_{0}(N))$ be the space of Siegel modular form of weight $k$ with respect to $\Gamma_{0}(N)$, and let $\mathcal{S}_{k}(\Gamma_{0}(N))$ be the subspace of cusp forms. That is to say, for any function $F\in\mathcal{M}_{k}(\Gamma_{0}(N))$, it is a holomorphic ${\mathbb{C}}$-valued function on $\mathbb{H}_{2}$ satisfying $\big{(}F|_{k}\gamma\big{)}(Z)=F(Z)$ for all $\gamma\in\Gamma_{0}(N)$. Here,

$$\big{(}F|_{k}g\big{)}(Z)\coloneqq\mu(g)^{k}j(g,Z)^{-k}F(g\langle Z\rangle)\quad\text{for }g=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\in{\rm GSp}(4,\mathbb{R})^{+}\text{ and }Z\in\mathbb{H}_{2},$$

(12)

where $j(g,Z)\coloneqq\operatorname{det}(CZ+D)$ is the automorphy factor. We remark that this operator differs from the classical one used in [1] by a factor. We do so to make the center of ${\rm GSp}(4,\mathbb{R})^{+}$ act trivially.

Let $F\in\mathcal{S}_{k}(\Gamma_{0}(N))$ be a Hecke eigenform, i.e., it is an eigenvector for all the Hecke operator $T(n),(n,N)=1$. Denote by $\lambda_{F}(n)$ the eigenvalue of $F$ under $T(n)$ when $(n,N)=1$. For any prime $p\nmid N$, we let $\alpha_{p,0},\alpha_{p,1},\alpha_{p,2}$ be the classical Satake parameters of $F$ at $p$. It is well known that

$$\alpha_{p,0}^{2}\alpha_{p,1}\alpha_{p,2}=p^{2k-3}.$$

(13)

In particular, let $N=1$ and $F\in\mathcal{S}_{k}(\Gamma_{2})$ be a Hecke eigenform, we can define the $L$-series

$$H(s)=\sum_{n=1}^{\infty}\frac{\lambda_{F}(n)}{n^{s}}.$$

(14)

This can be written as a Euler product

$$H(s)=\prod_{p}H_{p}(s)=\prod_{p}\left(1+\frac{\lambda_{F}(p)}{p^{s}}+\frac{\lambda_{F}(p^{2})}{p^{2s}}+\cdots\right)$$

(15)

provided $\Re(s)>k$. Moreover, one can show that

$$H_{p}(s)=\big{(}1-p^{2k-4-2s}\big{)}L_{p}(s,F,\operatorname{spin}),$$

(16)

where $L_{p}(s,F,\operatorname{spin})$ is the local spinor $L$-factor of $F$ at $p$ and it can be given by

$$L_{p}(s,F,\operatorname{spin})^{-1}=(1-\alpha_{p,0}p^{-s})(1-\alpha_{p,0}\alpha_{p,1}p^{-s})(1-\alpha_{p,0}\alpha_{p,2}p^{-s})(1-\alpha_{p,0}\alpha_{p,1}\alpha_{p,2}p^{-s}).$$

(17)

On the other hand, by [1, pp. 62, 69] one can see that

$$L_{p}(s,F,\operatorname{spin})^{-1}=1-\lambda_{F}(p)p^{-s}+(\lambda_{F}(p)^{2}-\lambda_{F}(p^{2})-p^{2k-4})p^{-2s}-\lambda_{F}(p)p^{2k-3-3s}+p^{4k-6-4s}.$$

(18)

In this case, we can define the spinor $L$-function

$$L(s,F,\operatorname{spin})=\prod_{p}L_{p}(s,F,\operatorname{spin}).$$

(19)

Let $\alpha_{p}=p^{3/2-k}\alpha_{p,0}$ and $\beta_{p}=\alpha_{p}\alpha_{p,1}$. By comparing (17) with (18), we obtain (also see [19, Proposition 4.1])

	$\displaystyle\lambda_{F}(p)$	$\displaystyle=p^{k-3/2}(\alpha_{p}+\alpha_{p}^{-1}+\beta_{p}+\beta_{p}^{-1}),$		(20)
	$\displaystyle\lambda_{F}(p^{2})$	$\displaystyle=p^{2k-3}\left((\alpha_{p}+\alpha_{p}^{-1})^{2}+(\alpha_{p}+\alpha_{p}^{-1})(\beta_{p}+\beta_{p}^{-1})+(\beta_{p}+\beta_{p}^{-1})^{2}-2-1/p\right).$		(21)

Let $\rho_{4}$ be the $4$-dimensional irreducible representation of ${\rm Sp}(4,{\mathbb{C}})$. In fact, $\rho_{4}$ is the natural representation of ${\rm Sp}(4,{\mathbb{C}})$ on ${\mathbb{C}}^{4}$, which is also called the spin representation. For later use, we would normalize the spinor $L$-function. More precisely, the normalized spinor $L$-function $L(s,\pi_{F},\rho_{4})$ is defined as follows

$$L(s,\pi_{F},\rho_{4})=L(s+k-3/2,F,\operatorname{spin})=\sum_{n=1}^{\infty}\frac{a_{F}(n)}{n^{s}}.$$

(22)

Note that this is the finite part of the completed $L$-function of $\pi_{F}$, where $\pi_{F}$ is the cuspidal automorphic representation of ${\rm GSp}(4,{\mathbb{A}})$ associated to $F$. For more details about the connection between Siegel modular forms of degree two and automorphic representations of ${\rm GSp}(4,{\mathbb{A}})$; for example see [2] and [23, Section 4.2]. Moreover, let $\tilde{\lambda}_{F}(n)=n^{3/2-k}\lambda_{F}(n)$ be the normalized eigenvalues. It follows that

$$\sum_{n=1}^{\infty}\frac{\tilde{\lambda}_{F}(n)}{n^{s}}=\zeta(2s+1)^{-1}L(s,\pi_{F},\rho_{4}).$$

(23)

Here, $\zeta$ is the Riemann zeta function. Note that if $F\in\mathcal{S}_{k}(\Gamma_{0}(N))$ with $N>1$, we still can define the partial spinor $L$-functions by Euler products for all primes $p$ not dividing $N$. In particular, the local factor at $p$ with $(p,N)=1$ is defined in the same way as above.

Similarly, let $\rho_{5}$ be the $5$-dimensional irreducible representation of ${\rm Sp}(4,{\mathbb{C}})$. An explicit formula for the representation $\rho_{5}$ as a map ${\rm Sp}(4,{\mathbb{C}})\to{\rm SO}(5,{\mathbb{C}})$ is given in [21, Appendix A.7]. The standard $L$-function associated to $F$ is defined as

$$L(s,\pi_{F},\rho_{5})=\prod_{p}L_{p}(s,F,\mathrm{std})=\sum_{n=1}^{\infty}\frac{b_{F}(n)}{n^{s}},$$

(24)

where

$$L_{p}(s,F,\mathrm{std})^{-1}=(1-p^{-s})(1-\alpha_{p,1}p^{-s})(1-\alpha_{p,2}p^{-s})(1-\alpha_{p,1}^{-1}p^{-s})(1-\alpha_{p,2}^{-1}p^{-s}).$$

(25)

Again, for $F\in\mathcal{S}_{k}(\Gamma_{0}(N))$ with $N>1$, we can define the partial standard $L$-functions by Euler products for all primes $p$ not dividing $N$ in the same way.

First, by (16) and (18) we have

$$\lambda_{F}(p^{3})=2\lambda_{F}(p)\lambda_{F}(p^{2})-\lambda_{F}(p)^{3}+\lambda_{F}(p)(p^{2k-3}+p^{2k-4}),$$

(26)

and

$$\lambda_{F}(p^{4})=-\lambda_{F}(p)^{4}+\lambda_{F}(p)^{2}\lambda_{F}(p^{2})+\lambda_{F}(p^{2})^{2}+\lambda_{F}(p)^{2}p^{2k-4}+\lambda_{F}(p^{2})p^{2k-4}+2\lambda_{F}(p)^{2}p^{2k-3}-p^{4k-6}.$$

(27)

Then we can prove the following result:

To prove this theorem, we need the following lemma.

Then Theorem 1.1 immediately follows from Theorem 3.1 and Lemma 3.3 below.

In this section, we only consider $\Gamma_{2}={\rm Sp}(4,{\mathbb{Z}})$. Let $m_{k}=\dim_{{\mathbb{C}}}\mathcal{S}_{k}(\Gamma_{2})$. For $\mathbf{(*)}\in\{\mathbf{(G)},\mathbf{(P)}\}$, recall that the set $\mathcal{K}^{\mathbf{(*)}}(2)$ of integers is defined as in (3). Moreover, we let $m_{k}^{\mathbf{(*)}}=\dim_{\mathbb{C}}\mathcal{S}_{k}^{\mathbf{(*)}}(\Gamma_{2})$. Evidently, $m_{k}=m_{k}^{\mathbf{(P)}}+m_{k}^{\mathbf{(G)}}$. It is well known that $m_{k}^{\mathbf{(P)}}>0$ only if $k\in{\mathbb{Z}}_{\geq 10}$ is even and $m_{k}^{\mathbf{(G)}}>0$ only if $k\in{\mathbb{Z}}_{\geq 20}$; for example see [22, Theorem 3.1]. We also note that $m_{k}^{(\mathbf{P})}=\dim_{\mathbb{C}}\mathcal{S}_{2k-2}(\Gamma_{1})$.