There are two well-known L-functions attached to Siegel modular forms of degree two. These are the Spin and the standard L-function. They have been studied by Andrianov, Böcherer, Shimura, and others. Other L-functions associated to degree two modular forms include the adjoint L-function (degree 10), and two others (degree 14 and degree 16).

The Spin L-function is of particular interest in the cases of paramodular forms of level t and weight k. Examples of paramodular forms can be made using the Gritsenko lift, a generalization of the Saito-Kurokawa lifting. The Gritsenko lift is actually a map defined on Jacobi forms, but it may also be considered a lift of classical newforms \mathcal{S}_{2k-2}^{new-}(\Gamma_0(t)), where the space \mathcal{S}_{2k-2}^{new-}(\Gamma_0(t)) consists of newforms such that the sign in the functional equation of their L-function is -1. Skoruppa and Zagier defined an isomorphism from the space J_{k,t}^{cusp, new} of cuspidal Jacobi newforms to \mathcal{S}_{2k-2}^{new-}(\Gamma_0(t)). So one takes a classical newform in the “minus space,” maps it to the space of Jacobi forms using the inverse of the Skoruppa-Zagier map, then lifts it with the Gritsenko lift to the space of paramodular forms S_k(K(t)). We will also call the composition of these two maps the Gritsenko lift.

Euler factors of the Spin L-function for paramodular forms of square-free level t=N were computed by Schmidt using representation theoretic methods. At the time of his paper, the local Langlands correspondence for GSp(4) was unverified, but Gan and Takeda have since proven its existence and so Schmidt’s results hold. He had shown that for f\in \mathcal{S}_{2k-2}^{new-}(\Gamma_0(N)), the lifting produces a cusp form F\in S_k(K(N)) of degree 2 whose completed Spin L-function is given by

L(s,F, {\rm spin})=\dfrac{1}{4\pi}\left(s-\dfrac{1}{2}\right)Z\left(s+\dfrac{1}{2}\right)Z\left(s-\dfrac{1}{2}\right)L(s,f)

where Z is the completed Riemann zeta function. Moreover, this lifting preserves the Atkin-Lehner eigenvalues, i.e., \eta_pF=\varepsilon_pF for every prime p.

Schmidt also determined the local components of the automorphic representation \pi_f associated to a classical newform f\in\mathcal{S}_{2k-2}^{new-}(\Gamma_0(N)). Moreover, he explained that the Gritsenko lift and similar lifts are predicted by Langlands functoriality. Schmidt also computed the Euler factors of the Spin L-function for the space of paramodular forms of square-free level that are lifts using representation theoretic methods since the lift is functorial. Schmidt and Roberts later determined the possible Euler factors of any paramodular representation, but precise information for the non-square-free level case remained unknown.

The Jacobi group is defined as

G^J={\rm SL}(2,\mathbb{Z})\ltimes H,

where H denotes the integral Heisenberg group. This group is isomorphic to the maximal parabolic subgroup of {\rm Sp}(4,\mathbb{Z})

\Gamma_\infty=\Gamma_\infty(\mathbb{Z})=\left\{\begin{pmatrix}  *&0&*&*\\  *&*&*&*\\  *& 0&*&*\\  0&0&0&*\\  \end{pmatrix}\in{\rm Sp}(4,\mathbb{Z})\right\},

where * denotes an arbitrary entry.
We also define the groups

G_R^J=\left\{\begin{pmatrix}  *&0&*&*\\  *&*&*&*\\  *&0&*&*\\  0&0&0&*\\  \end{pmatrix}\in{\rm GSp}(4,R)^+\right\}

and

\Gamma_\infty(R)=\left\{\begin{pmatrix}  *&0&*&*\\  *&*&*&*\\  *&0&*&*\\  0&0&0&*\\  \end{pmatrix}\in{\rm Sp}(4,R)\right\},

where R is a ring.

Definition. A holomorphic function \phi(\tau,z):\mathbb{H}_1\times\mathbb{C}\longrightarrow\mathbb{C} is called a Jacobi form of index t and weight k if the function

\hat{\phi}(Z)=\phi(\tau,z)e(t\omega),

called a completed Jacobi form, is a Siegel modular form of degree 2 and weight k with respect to the parabolic group \Gamma_\infty. That is
\hat{\phi}|_k M=\hat{\phi} for any M\in\Gamma_\infty and the function has the Fourier expansion

\phi(\tau,z)=\sum_{n,l\in\mathbb{Z}, n\geq 0, 4nt\geq r^2}c(n,r)e(n\tau+rz).

We call \phi a Jacobi cusp form if there is a strict inequality in the summation. The space of Jacobi forms of index t and weight k will be denoted by J_{k,t} and its subspace of cusp forms by J_{k,t}^{\rm cusp}. The space of completed Jacobi forms of index t and weight k and its subspace of completed Jacobi cusp forms will be denoted by \hat{J}_{k,t} and \hat{J}_{k,t}^{\rm cusp}, respectively.

Now we define the Hecke-Jacobi ring and its action of the space of Jacobi forms. This is used to define a Hecke-Jacobi newform. This section is based on the results of Gritsenko.

Definition. Let \Gamma be a subgroup of a semigroup G. A pair (\Gamma, G) is called a Hecke pair if for any g\in G the double coset \Gamma g\Gamma is the union of a finite number of left and right cosets relative to \Gamma. The Hecke ring \mathcal{H}(\Gamma, G) of the pair (\Gamma, G) is the \Gamma-invariant subspace of the \mathbb{Q}-vector space consisting of all formal finite linear combinations X=\sum a_i\Gamma g_i, where a_i\in\mathbb{Q}, g_i\in G. A representation of \Gamma on this space is defined by right multiplication

X\cdot\gamma=\sum a_i\Gamma(g_i\gamma).

For any X, Y\in\mathcal{H}(\Gamma, G) with X=\sum a_i\Gamma g_i and Y=\sum b_j\Gamma h_j, their product is defined by

X\cdot Y=\sum a_ib_j\Gamma(g_i h_j).

This is well-defined since the product is independent of the choices of representatives g_i and h_j.

To the pair (\Gamma_\infty, G^J(\mathbb{Q})) we have the associated Hecke ring \mathcal{H}^J, called the (global) Hecke-Jacobi ring. The Hecke ring associated to the pair (\Gamma_\infty, G^J(\mathbb{Z}_{[\frac{1}{p}]})) will be called the local Hecke-Jacobi ring and denoted by \mathbb{H}_p^J. The Hecke-Jacobi ring can be written as a restricted tensor product

\mathcal{H}^J=\otimes_p' \mathcal{H}_p^J.

Note that the global and local Hecke-Jacobi rings are not commutative. The Hecke-Jacobi ring acts on \hat{J}_{k,t}. Let X\in\mathcal{H}^J, say X=\sum a_j(\Gamma_\infty g_j). The action of X on \hat{J}_{k,t} is given by

\hat{\phi}|_k X=\sum_j a_j\hat{\phi}|_k g_j.

The subring \mathcal{H}_0^J is defined as the set of elements of \mathcal{H}^J of determinant 1, or, equivalently, the Hecke ring for the pair (\Gamma_\infty, \Gamma_\infty(\mathbb{Q})). This ring has the form

\mathcal{H}_0^J=\left\{\sum a_i\Gamma_\infty\begin{pmatrix}  *&0&*&*\\  *&1&*&*\\  *&0&*&*\\  0&0&0&*\\  \end{pmatrix}\in\mathcal{H}^J\right\}.

Similarly, the ring \mathcal{H}_{p,0}^J is defined as the set of elements of \mathcal{H}_p^J of determinant 1.

Certain distinguished elements in these Hecke rings will be useful. Define

T^J(n)=\Gamma_\infty[1,n^{-1},1,n]\Gamma_\infty,\qquad \Delta_n=\Gamma_\infty[n,n,n,n]\Gamma_\infty

\nabla_n=\sum_{\kappa\in \mathbb{Z}/n\mathbb{Z}}\Gamma_\infty\begin{pmatrix}  1&&&\\  &1&&\frac{\kappa}{n}\\  &&1&\\  &&&1\\  \end{pmatrix}  \Gamma_\infty,

and

\Xi_n=\sum_{\lambda,\mu\,\kappa\in \mathbb{Z}/n\mathbb{Z}}\Gamma_\infty\begin{pmatrix}  1&&&\frac{\mu}{n}\\  -\frac{\lambda}{n}&1&\frac{\mu}{n}&\frac{\kappa}{n}\\  &&1&-\frac{\lambda}{n}\\  &&&1\\  \end{pmatrix},

where [a_1,a_2,a_3,a_4]={\rm diag}[a_1,a_2,a_3,a_4]. Note that the local Hecke-Jacobi ring \mathcal{H}_p^J can also be described as the subring of \mathcal{H}_{p,0}^J generated by the elements T^J(p^n), \nabla_p, and \Xi_p.

Definition. Let \hat{\phi}\in \hat{J}_{k,t}. Then \hat{\phi} is called a Hecke-Jacobi eigenform if \hat{\phi} is an eigenform for all operators T^J(n). That is,

\hat{\phi}|_k T^J(n)=\lambda_{\phi,n}\hat{\phi},

for some \lambda_{\phi,n}\in\mathbb{C}.

Let \tilde{\mathcal{H}}_0^J be the subring of \mathcal{H}^J generated by T^J, \Delta, \nabla, and \Xi. Let \tilde{\mathcal{H}}_{p,0}^J=\tilde{\mathcal{H}}_0^J\cap\mathcal{H}_p^J. If \hat{\phi}\in \hat{J}_{k,t} is a Hecke-Jacobi eigenform, then \hat{\phi} is an eigenform with respect to all elements of \tilde{\mathcal{H}}_0^J. This follows from the identity

T^J(p^2)=\left(T^J(p)\right)^2-(p-1)\nabla_p T^J(p)-p\left(\Xi_p-\nabla_p+p^2+p\right).

Combining some results of Gritsenko on the local factors of the Spin L-function of Gritsenko lifts with Schmidt’s local factor results, we obtain the following theorem.

Theorem. Let N\in\mathbb{N} be square-free. Let F=F_\phi\in S_k^{\rm new}(K(N)) be a paramodular newform in the span of Gritsenko lifts with {\rm Gr}(\phi)=F_\phi. Let \pi=\otimes\pi_p be the automorphic representation of {\rm GSp}(4,\mathbb{A}) associated to F.

Case 1: p\nmid N. The local component \pi_p is a spherical representation with Satake parameters (\alpha_0,\alpha_1,\alpha_2) satisfying the equations

p^{k-3}(p^2+p+\lambda_{\hat{\phi},p}) = \alpha_0(\alpha_1+1)(\alpha_2+1)

p^{2k-5}(2p^2+p\lambda_{\hat{\phi},p}+\lambda_{\hat{\phi},p}) = \alpha_0^2(\alpha_1+\alpha_2+\alpha_1\alpha_2(\alpha_1+\alpha_2+2))

p^{3k-6}(p^2+p+\lambda_{\hat{\phi},p}) = \alpha_0^3\alpha_1\alpha_2(\alpha_1+1)(\alpha_2+1)

p^{4k-6} = \alpha_0^4\alpha_1^2\alpha_2^2.

Case 2: p|N. If \epsilon_{\hat{\phi},p}=0, then the local factor of the associated automorphic representation is \Pi({\rm St}\otimes{\rm St}). If \epsilon_{\hat{\phi},p}\neq0, the local component \pi_p is a functorial lift of either the Steinberg St or the twisted Steinberg representation of {\rm GL}(2). It is \Pi(\xi{\rm St}\otimes 1) if one of the following equations holds:

\lambda_{\hat{\phi},p}^2 = 4p

\lambda_{\hat{\phi},p} = p^{k-2}+\dfrac{1}{p^{k-3}}

\lambda_{\hat{\phi},p} = p^{k-1}+\dfrac{1}{p^{k-2}}.

Otherwise it is \Pi({\rm St}\otimes 1).

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