1 Introduction

It is an interesting problem to determine the structure of rings of Siegel modular forms with respect to congruence subgroups. A famous theorem of Igusa [8] shows that every Siegel modular form of degree two and even weight for the full modular group \(\mathrm {Sp}_4(\mathbb {Z})\) can be written uniquely as a polynomial in forms \(\phi _4, \phi _6, \phi _{10}, \phi _{12}\) of weights 4, 6, 10, 12, and that odd-weight Siegel modular forms are precisely the products of even weight Siegel modular forms with a distinguished cusp form \(\psi _{35}\) of weight 35. It was proved by Aoki and Ibukiyama [1] that the rings of modular forms for the congruence subgroups:

$$\begin{aligned} \Gamma _{0, 1}^{(2)}(N) = \Big \{ \left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \in \mathrm {Sp}_4(\mathbb {Z}): \; c \equiv 0 \, (N), \; \mathrm {det}(a) \equiv \mathrm {det}(d) \equiv 1 \, (N)\Big \}, \; N = 2, 3, 4 \end{aligned}$$

have an analogous structure: they are generated by four algebraic independent modular forms together with their Jacobian (or first Rankin–Cohen–Ibukiyama bracket). The rings \(M_*(\Gamma _0^{(2)}(N))\) where

$$\begin{aligned} \Gamma _0^{(2)}(N) = \{\left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \in \mathrm {Sp}_4(\mathbb {Z}): \, c \equiv 0 \, (N)\}, \end{aligned}$$

therefore, have a simple structure as well.

The goal in this paper is to extend the methods of Aoki and Ibukiyama to level \(N=5\). This is not quite straightforward, as the natural underlying ring is no longer \(M_*(\Gamma _{0,1}^{(2)}(5))\) but rather a ring \(M_*^!(\Gamma _{0,1}^{(2)}(5))\) of meromorphic Siegel modular forms with singularities on Humbert surfaces. We will define a hyperplane arrangement \(\mathcal {H}\) as the \(\Gamma _0^{(2)}(5)\)-orbit of the Humbert surface

$$\begin{aligned} \{Z = \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \in \mathbb {H}_2: \; \mathrm {det}(Z) = 1 - 5z\}, \end{aligned}$$

which, if one views points in \(\mathbb {H}_2\) as parameterizing abelian surfaces, is a locus of principally polarized abelian surfaces with real multiplication that respects a \(\Gamma _0^{(2)}(5)\) level structure. We then investigate the ring \(M_*^!(\Gamma _{0, 1}^{(2)}(5))\) of meromorphic Siegel modular forms on \(\Gamma _{0, 1}^{(2)}(5)\) with singularities supported on \(\mathcal {H}\). Using a generalization of the modular Jacobian approach of [12], we prove in Theorem 3.6 that \(M_*^!(\Gamma _{0, 1}^{(2)}(5))\) is generated by four algebraically independent singular additive lifts \(f_1, f_2, g_1, g_2\) of weights 1, 1, 2, 2 and by their Jacobian; in particular, the associated threefold \(X_{0, 1}^{(2)}(5)\) is rational. The local isomorphism from \(\mathrm {Sp}_4\) to \(\mathrm {SO}(3,2)\) and Borcherds’ theory of orthogonal modular forms with singularities are essential. \(\mathrm {Proj}(M_*^!(\Gamma _{0, 1}^{(2)}(5)))\) is the Looijenga compactification [9] of the complement \((\mathbb {H}_2 \setminus \mathcal {H}) / \Gamma _{0, 1}^{(2)}(5)\), which plays a similar role to the Satake–Baily–Borel compactification of \(Y_{0,1}^{(2)}(5)\).

It follows from the above that every Siegel modular form of level \(\Gamma _0^{(2)}(5)\) can be expressed uniquely in terms of the basic forms \(f_1, f_2, g_1, g_2\). It is not clear to the authors how to compute the ring \(M_*(\Gamma _0^{(2)}(5))\) of (holomorphic) Siegel modular forms from this information alone; however, allowing a formula of Hashimoto [6] for the dimensions of cusp forms (itself an application of the Selberg trace formula), the ring structure becomes a straightforward Gröbner basis computation. We will prove that \(M_*(\Gamma _0^{(2)}(5))\) is minimally generated by eighteen modular forms of weights 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 10, 11, 11, 11, 13, 13, 13, 15 in Theorem 4.2.

This paper is organized as follows. In Sect. 2, we review the realization of Siegel modular groups as orthogonal groups and the theory of Borcherds lifts. In Sect. 3, we determine two rings of meromorphic Siegel modular forms. In Sect. 4, we use this to determine the ring of holomorphic Siegel modular forms for \(\Gamma _{0}^{(2)}(5)\).

2 Theta lifts to Siegel modular forms of degree two

2.1 \(\Gamma _0^{(2)}(N)\) as an orthogonal group

Recall that the Pfaffian of an antisymmetric \((4 \times 4)\)-matrix M is

$$\begin{aligned} \mathrm {pf}(M) = \mathrm {pf}\left( {\begin{matrix} 0 &{} a &{} b &{} c \\ -a &{} 0 &{} d &{} e \\ -b &{} -d &{} 0 &{} f \\ -c &{} -e &{} -f &{} 0 \end{matrix}}\right) = af - be + cd. \end{aligned}$$

We view \(\mathrm {pf}\) as a quadratic form and define the associated bilinear form:

$$\begin{aligned} \langle x, y \rangle = \mathrm {pf}(x + y) - \mathrm {pf}(x) - \mathrm {pf}(y). \end{aligned}$$

The Pfaffian is invariant under conjugation \(M \mapsto A^T M A\) by \(A \in \mathrm {SL}_4(\mathbb {R})\), and this action identifies \(\mathrm {SL}_4(\mathbb {R})\) with the Spin group \(\mathrm {Spin}(\mathrm {pf})\). The symplectic group \(\mathrm {Sp}_4(\mathbb {R})\), by definition, preserves

$$\begin{aligned} \mathcal {J} = \left( {\begin{matrix} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{matrix}}\right) \end{aligned}$$

under conjugation, so it also preserves the orthogonal complement \(\mathcal {J}^{\perp }\), and indeed, it is exactly the Spin group of \(\mathrm {pf}\) restricted to \(\mathcal {J}^{\perp }\). If the entries of M are labelled as above, then \(M \in \mathcal {J}^{\perp }\) if and only if \(b + e = 0\).

For any \(N \in \mathbb {N}\), the group \(\Gamma _0^{(2)}(N) = \{\left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \in \mathrm {Sp}_4(\mathbb {Z}): \; c \equiv 0 \, (N)\}\) stabilizes the lattice

$$\begin{aligned} L = \left\{ M = \left( {\begin{matrix} 0 &{} a &{} b &{} c \\ -a &{} 0 &{} d &{} -b \\ -b &{} -d &{} 0 &{} f \\ -c &{} b &{} -f &{} 0 \end{matrix}}\right) : \; a, b, c, d, f \in \mathbb {Z}, \; a \equiv 0 \, (N)\right\} , \end{aligned}$$

which is of type \(U \oplus U(N) \oplus A_1\). By [7, Sect. 2] the special discriminant kernel \(\widetilde{\mathrm {SO}}(L)\) of L is exactly the projective modular group \(\Gamma _{0, 1}^{(2)}(N) / \{\pm I\}\) under this identification, where

$$\begin{aligned} \widetilde{\mathrm {SO}}(L)&=\{ g \in \mathrm {SO}(L):\; g(v)-v\in L \; \text {for all}\; v \in L' \},\\ \Gamma _{0, 1}^{(2)}(N)&= \{\left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \in \Gamma _0^{(2)}(N): \; \mathrm {det}(a) \equiv 1 \, (N)\}. \end{aligned}$$

It follows that

$$\begin{aligned} \Gamma _0^{(2)}(N) / \{\pm I\} = \langle \widetilde{\mathrm {SO}}(L), \; \varepsilon _u: \; u \in (\mathbb {Z} / N\mathbb {Z})^{\times } \rangle , \end{aligned}$$

where \(\varepsilon _u\) is the matrix

$$\begin{aligned} \varepsilon _u = \begin{pmatrix} u &{} 0 &{} b &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ N &{} 0 &{} u^* &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} \in \Gamma _0^{(2)}(N) \end{aligned}$$

for any integer solutions \(u^*, b\) to \(uu^* - Nb = 1\) (the choice does not matter). The map induced by \(\varepsilon _u\) on \(L'/L \cong A_1'/A_1 \oplus U(N)'/U(N)\) acts trivially on \(A_1'/A_1\) and acts on \(U(N)'/U(N) \cong \mathbb {Z}/N\mathbb {Z} \oplus \mathbb {Z}/N\mathbb {Z}\) as the map

$$\begin{aligned} \varepsilon _u : \mathbb {Z}/N\mathbb {Z} \oplus \mathbb {Z}/N\mathbb {Z} \rightarrow \mathbb {Z}/N\mathbb {Z} \oplus \mathbb {Z}/N\mathbb {Z}, \; (x, y) \mapsto (ux, u^{-1}y). \end{aligned}$$

The symplectic group \(\mathrm {Sp}_4(\mathbb {R})\) acts on the Siegel upper half-space \(\mathbb {H}_2\) by Möbius transformations:

$$\begin{aligned} M \cdot Z = \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \cdot Z = (aZ + b)(cZ + d)^{-1}. \end{aligned}$$

Let \(j(M; Z) = \mathrm {det}(c Z + d)\) be the usual automorphy factor. We embed the Siegel upper half-space into \(L \otimes \mathbb {C}\) as follows:

$$\begin{aligned} Z = \begin{pmatrix} \tau &{} z \\ z &{} w \end{pmatrix} \mapsto \mathcal {Z} := \phi (Z) := \begin{pmatrix} 0 &{} 1 &{} z &{} w \\ -1 &{} 0 &{} -\tau &{} -z \\ -z &{} \tau &{} 0 &{} \tau w - z^2 \\ -w &{} z &{} z^2 - \tau w &{} 0 \end{pmatrix}. \end{aligned}$$

Then one has the relation

$$\begin{aligned} M^T \mathcal {Z} M = j(M; Z) \phi (M \cdot Z), \; M \in \mathrm {Sp}_4(\mathbb {R}) \end{aligned}$$

as one can check on any system of generators.

For any \(\lambda \in L'\) of positive norm \(D = Q(\lambda )\), the space

$$\begin{aligned} \{Z \in \mathbb {H}_2: \; \mathcal {Z} \in \lambda ^{\perp }\} \end{aligned}$$

is known as a Humbert surface \(H(D, \lambda )\) of discriminant D. If \(\lambda \) is written in the form \(\left( {\begin{matrix} 0 &{} a &{} b &{} c \\ -a &{} 0 &{} d &{} -b \\ -b &{} -d &{} 0 &{} f \\ -c &{} b &{} -f &{} 0 \end{matrix}}\right) \) then

$$\begin{aligned} H(D, \lambda ) = \{Z = \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \in \mathbb {H}_2: \; a \mathrm {det}(Z) -c \tau + 2bz + dw + f = 0 \}. \end{aligned}$$

If \(\gamma \) instead is a coset of \(L'/L\), then we define

$$\begin{aligned} H(D, \gamma ) = \sum _{\begin{array}{c} \lambda \in \gamma \\ \lambda \; \text {primitive in}\; L' \\ Q(\lambda ) = D \end{array}} H(D, \lambda ). \end{aligned}$$

These unions are locally finite and, therefore, descend to well-defined divisors on \(\widetilde{\mathrm {O}}(L) \backslash \mathbb {H}_2.\) We will use the notation \(H(D, \pm \gamma )\) because \(\lambda ^{\perp } = (-\lambda )^{\perp }\) implies \(H(D, \gamma ) = H(D, -\gamma )\). Note that many references omit the condition that \(\lambda \) is primitive in \(L'\), so \(H(D, \pm \gamma )\) satisfy inclusions; our divisors \(H(D, \pm \gamma )\) do not.

2.2 Theta lifts

Let L be the lattice in the space of \((4 \times 4)\) antisymmetric matrices from the previous subsection. The weight k theta kernel is

$$\begin{aligned} \Theta _k(\tau ; Z) = \frac{\pi ^k}{\mathrm {det}(V)^k \Gamma (k)} \sum _{\lambda \in L'} \langle \lambda , \mathcal {Z} \rangle ^k \text {e}^{-\frac{\pi y}{\mathrm {det}(V)} | \langle \lambda , \mathcal {Z} \rangle |^2} \text {e}^{2\pi i \overline{\tau } \mathrm {pf}(\lambda )} \mathfrak {e}_{\lambda }, \end{aligned}$$

where \(\tau = x+iy \in \mathbb {H}\) and \(Z = U+iV \in \mathbb {H}_2\); and \(\mathcal {Z}\) is the image of Z in \(L \otimes \mathbb {C}\). By applying a theorem of Vignéras on indefinite theta series [11] one can deduce the behaviour of \(\Theta _k\) under the action of \(\mathrm {SL}_2(\mathbb {Z})\) on \(\tau \): it transforms like a modular form of weight \(\kappa := k -1/2\) with respect to the Weil representation \(\rho _L\) (see Definition 2.1). On the other hand, for any \(M = \left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \in \Gamma _0^{(2)}(N)\),

$$\begin{aligned} \Theta _k(\tau ; M \cdot Z)&= \frac{\pi ^k}{\mathrm {det} \, \mathrm {im}(M \cdot Z)^k \Gamma (k)} \sum _{\lambda \in L'} \mathrm {det}(cZ + d)^{-k} \langle \lambda , M^T \mathcal {Z} M \rangle \\&\quad \times \text {e}^{-\frac{\pi y}{\mathrm {det}(V)} | \langle \lambda , M^T \mathcal {Z} M \rangle |^2} \text {e}^{2\pi i \overline{\tau } \mathrm {pf}(\lambda )} \mathfrak {e}_{\lambda } \\&= \frac{\pi ^k}{\mathrm {det}(V)^k \Gamma (k)} \overline{\mathrm {det}(cZ + d)^k} \sum _{\lambda \in L'} \langle M^{-T} \lambda M^{-1}, \mathcal {Z} \rangle ^k\\&\quad \times \text {e}^{-\frac{\pi y}{\mathrm {det}(V)} | \langle M^{-T} \lambda M^{-1}, \mathcal {Z} \rangle |^2} \text {e}^{2\pi i \overline{\tau } \mathrm {pf}(M^{-T} \lambda M^{-1})} \mathfrak {e}_{\lambda } \\&= \overline{\mathrm {det}(cZ + d)^k} \sigma (M) \Theta _k(\tau ; Z), \end{aligned}$$

where \(\sigma \) is the map

$$\begin{aligned} \sigma : \Gamma _0^{(2)}(N) \longrightarrow \mathrm {Aut} \, \mathbb {C}[L'/L], \; \; \sigma (M) \mathfrak {e}_{\lambda } := \mathfrak {e}_{M^T \lambda M}. \end{aligned}$$

Following Borcherds [3], one defines the theta lift of a vector-valued modular form F with a pole at \(\infty \) as the regularized integral of F against the kernel \(\Theta _k\):

Definition 2.1

(i) The Weil representation \(\rho _L\) associated to an even lattice (L, Q) is the representation \(\rho : \mathrm {Mp}_2(\mathbb {Z}) \rightarrow \mathrm {GL}\, \mathbb {C}[L'/L]\) defined by

$$\begin{aligned} \rho \left( \left( {\begin{matrix} 1 &{} 1 \\ 0 &{} 1 \end{matrix}}\right) , 1 \right) \mathfrak {e}_{\gamma }= & {} \text {e}^{-2\pi i Q(\gamma )}\mathfrak {e}_{\gamma };\\ \rho \left( \left( {\begin{matrix} 0 &{} -1 \\ 1 &{} 0 \end{matrix}}\right) , \sqrt{\tau } \right) \mathfrak {e}_{\gamma }= & {} \text {e}^{\pi i \mathrm {sig}(L)/4} |L'/L|^{-1/2} \sum _{\beta \in L'/L} \text {e}^{2\pi i \langle \gamma , \beta \rangle } \mathfrak {e}_{\beta }, \end{aligned}$$

where \(\mathrm {Mp}_2(\mathbb {Z})\) is the metaplectic group of pairs \((M, \phi )\) where \(M = \left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \in \mathrm {SL}_2(\mathbb {Z})\) and \(\phi \) is a square root of \(c\tau + d\), and \(\mathfrak {e}_{\gamma }\), \(\gamma \in L'/L\) is the standard basis of the group ring \(\mathbb {C}[L'/L]\).

(ii) A weakly holomorphic vector-valued modular form for \(\rho _L\) is a holomorphic function \(F : \mathbb {H} \rightarrow \mathbb {C}[L'/L]\) which satisfies

$$\begin{aligned}F((M, \phi ) \cdot \tau ) = \phi (\tau )^{2k} \rho _L(M) F(\tau ), \; \; (M, \phi ) \in \mathrm {Mp}_2(\mathbb {Z})\end{aligned}$$

and which is meromorphic at the cusp \(\infty \), i.e. its Fourier series has only finitely many negative exponents.

(iii) Let \(k \ge 1\) and let \(F \in M_{\kappa }^!(\rho _L)\) be a weakly holomorphic modular form. The (singular) theta lift of F is

$$\begin{aligned} \Phi _F(Z) = \int ^{\mathrm {reg}}_{\mathrm {SL}_2(\mathbb {Z}) \backslash \mathbb {H}} \langle F(\tau ), \Theta _k(\tau ; Z) \rangle y^{\kappa } \frac{\mathrm {d}x \, \mathrm {d}y}{y^2}. \end{aligned}$$

Here, the regularization means one takes the limit as \(w \rightarrow \infty \) of the integral over \(\mathcal {F}_w = \{\tau = x+iy \in \mathbb {H}: \; x^2 + y^2 \ge 1, \; |x| \le 1/2, \; y \le w\}\); in effect, it means one integrates first with respect to x, which mollifies the contribution of the principal part of F to the integral; and then secondly with respect to y. The behaviour of the theta lift under Möbius transformations is

$$\begin{aligned} \Phi _F(M \cdot Z)&= \int ^{\mathrm {reg}} \langle F(\tau ), \Theta _k(\tau ; M \cdot Z) \rangle \; y^{\kappa - 2} \, \mathrm {d}x \, \mathrm {d}y \\&= \mathrm {det}(cZ + d)^k \int ^{\mathrm {reg}} \sum _{\gamma \in L'/L} F_{\gamma }(\tau ) \overline{\Theta _{k; M^{-T} \gamma M^{-1}}(\tau ; Z)} \, y^{\kappa - 2} \, \mathrm {d}x \, \mathrm {d}y \\&= \mathrm {det}(cZ+d)^k \Phi _{\sigma (M)^{-1} F}(Z). \end{aligned}$$

Therefore, the singular theta lift \(\Phi _F\) transforms like a Siegel modular form of weight k on the subgroup of \(\Gamma _0^{(2)}(N)\) that fixes F. Borcherds’ results [3, Theorem 14.3] show that \(\Phi _F\) is meromorphic, with singularities of multiplicity k along Humbert surfaces associated to the principal part of the input F. This is the so-called Borcherds additive lift. Since the Borcherds additive lift is a generalization of the Gritsenko lift [5], we will also call it the singular Gritsenko lift. When the input F has weight \(\kappa =-\frac{1}{2}\) (i.e. \(k=0\)), the modified exponential of \(\Phi _F\) defines a remarkable modular form which has an infinite product expansion (cf. [3, Theorem 13.3]), called a Borcherds product, or more specifically the Borcherds lift of F. In this paper we will need both types of singular theta lifts.

Remark 2.2

It is often useful to consider the pullback or restriction of a Siegel modular form to a Humbert surface. The result is traditionally interpreted as a Hilbert modular form attached to a real-quadratic field. From the point of view of orthogonal modular forms, this is very simple: to restrict a form \(\Phi (Z)\) to the sublattice \(v^{\perp }\) (with \(v \in L'\)), one simply restricts to Z satisfying \(\langle Z, v \rangle = 0\).

The pullback of a theta lift \(\Phi _F\) as above is again a theta lift, \(\Phi _{\vartheta F}\), where \(\vartheta F \in M_{\kappa + 1/2}(\rho _{v^{\perp }})\) is the theta contraction; this is the vector-valued modular form associated to the Weil representation of \(L_v = L \cap v^{\perp }\) characterized by

$$\begin{aligned}\langle \vartheta F, \Theta _{k; L_v}(\tau ; Z) \rangle = \langle F, \Theta _{k; L}(\tau ; Z) \rangle , \quad Z \in v^{\perp },\end{aligned}$$

where \(\Theta _{k; L_v}\) and \(\Theta _{k;L}\) are the weight k theta kernels attached to \(L_v\) and L, respectively. More explicitly one can define \(\vartheta F\) as the zero-value of a vector-valued Jacobi form for the Weil representation attached to \(L_v\) whose associated theta decomposition is F itself [14]. The important point is that one can check rigorously whether a theta lift \(\Phi _F\) vanishes identically on a Heegner divisor, with the computations taking place only on the level of vector-valued modular forms.

3 The ring of meromorphic Siegel modular forms of level 5

We consider the ring \(M_*^!(\Gamma _0^{(2)}(5))\) of meromorphic Siegel modular forms of level \(\Gamma _0^{(2)}(5)\) whose poles may lie only on the orbit \(\mathcal {H}\) of the Humbert surface:

$$\begin{aligned} \{Z = \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \in \mathbb {H}_2: \; \mathrm {det}(Z) = 1 - 5z\}, \end{aligned}$$

which is a locus of principally polarized RM abelian surfaces with \(\Gamma _0^{(2)}(5)\) level structure. In view of the discussion of Sect. 2.1, \(\mathcal {H}\) splits as the union of two irreducible \(\Gamma _{0, 1}^{(2)}(5)\)-orbits of Humbert surfaces:

$$\begin{aligned} \mathcal {H} = H(1/20, \pm \gamma _1) + H(1/20, \pm \gamma _2), \end{aligned}$$

each invariant under the discriminant kernel of \(L = U\oplus U(5)\oplus A_1\), where we have fixed any coset \(\gamma _1 \in L'/L\) of norm \(1/20 + \mathbb {Z}\) and define \(\gamma _2 = \varepsilon _2(\gamma _1)\). The Humbert surface \(H_{1/5}\) of discriminant 1/5, the orbit of \(\{\left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \in \mathbb {H}_2: \; \tau = 2z\}\) under \(\Gamma _0^{(2)}(5)\), also splits into two \(\Gamma _{0, 1}^{(2)}(5)\)-invariant divisors:

$$\begin{aligned} H_{1/5} = H(1/5, \pm \delta _1) + H(1/5, \pm \delta _2), \end{aligned}$$

where \(\delta _n = 2\gamma _n \in L'/L\).

For a finite-index subgroup \(\Gamma \le \Gamma _0^{(2)}(5)\) or \(\Gamma \le \mathrm {O}(L)\), we define \(M_*^!(\Gamma ,\chi )\) to be the ring of meromorphic forms, holomorphic away from \(\mathcal {H}\), which are modular under \(\Gamma \) with character \(\chi \).

We first prove a form of Koecher’s principle for meromorphic modular forms with poles supported on \(\mathcal {H}\).

Lemma 3.1

Let \(f\in M_k^!(\Gamma _{0,1}^{(2)}(5),\chi )\). If k is negative, then f is identically zero. If \(k=0\), then f is constant.

Proof

We prove the lemma in the context of \(\mathrm {O}(3, 2)\). Let v and \(u\ne \pm v\) be primitive vectors of norm 1/20 in \(L'\), such that \(v^\perp , u^\perp \in \mathcal {H}\). Suppose that f is not identically zero and has poles of multiplicity \(c_v\) along \(v^\perp \). We denote the intersection of \(v^\perp \) and the symmetric domain \(\mathbb {H}_2\) (resp. the lattice L) by \(v^\perp \cap \mathbb {H}_2\) (resp. \(L_v\)). Then \(L_v\) is a lattice of signature (2, 2) and discriminant 5, equivalent to the lattice \(U + \mathbb {Z}[(1 + \sqrt{5}) / 2]\) where the quadratic form is the field norm. It is easy to see that the space \(L_v\otimes \mathbb {Q}\) contains no isotropic planes, so the Koecher principle holds for modular forms on \(\widetilde{\mathrm {O}}(L_v)\). We find that the projection of u in \(L_v\) has non-positive norm, which implies that the intersection of \(u^\perp \) and \(v^\perp \cap \mathbb {H}_2\cong \mathbb {H}\times \mathbb {H}\) is empty. Thus, the quasi-pullback of f to \(v^\perp \cap \mathbb {H}_2\), i.e. the leading term in the power series expansion about that hyperplane, is a nonzero holomorphic modular form of weight \(k-c_v\). By Koecher’s principle, we conclude \(k-c_v\ge 0\) and, therefore, \(k \ge 0\), and when \(k=0\), we must have \(c_v=0\), and thus, f is holomorphic and must be constant (by Koecher’s principle on \(\widetilde{\mathrm {O}}(L)\)).

We now construct some basic modular forms using Borcherds additive lifts (singular Gritsenko lifts) and Borcherds products.

Lemma 3.2

There are singular Gritsenko lifts \(f_1, f_2\) of weight one on \(\widetilde{\mathrm {O}}(L)\) whose divisors are exactly

$$\begin{aligned} \mathrm {div}(f_1) = -H(1/20, \pm \gamma _1) + 4H(1/20, \pm \gamma _2) + H(1/5, \pm \delta _1) \end{aligned}$$

and

$$\begin{aligned} \mathrm {div}(f_2) = 4H(1/20, \pm \gamma _1) - H(1/20, \pm \gamma _2) + H(1/5, \pm \delta _2). \end{aligned}$$

Proof

Using the algorithm of [13], we find a weakly holomorphic modular form of weight 1/2 for the Weil representation associated to L for which the Fourier expansion takes the form:

$$\begin{aligned} 2 q^{-1/20}(\mathfrak {e}_{\gamma _1} - \mathfrak {e}_{-\gamma _1}) + O(q^{1/20}), \end{aligned}$$

which is mapped under the Gritsenko lift to a meromorphic form \(f_1\) with simple poles only on \(H(1/20, \pm \gamma _1)\) and \(H(1/20, \pm \gamma _4)\). Applying the automorphism \(\varepsilon _2\) on \(L'/L\) to the input into \(f_1\) yields the input into \(f_2\).

On the other hand, we found a weakly holomorphic modular form of weight \(-1/2\) for which the principal part at \(\infty \) is

$$\begin{aligned} 2 \mathfrak {e}_0 - 2q^{-1/20} (\mathfrak {e}_{\gamma _1} + \mathfrak {e}_{-\gamma _1}) + 4q^{-1/20} (\mathfrak {e}_{\gamma _2} + \mathfrak {e}_{-\gamma _2}) + q^{-1/5} (\mathfrak {e}_{\delta _1} + \mathfrak {e}_{-\delta _1}), \end{aligned}$$

which is mapped under the Borcherds lift to a meromorphic modular form \(F_1\) (possibly with character) of weight one and the claimed divisor. By taking theta contractions of the input form, one finds that \(f_1\) vanishes on \(H(1/5, \pm \delta _1)\). Then the quotient \(f_1 / F_1\) lies in \(M_0^!(\widetilde{\mathrm {O}}(L), \chi )\) so it is constant by Lemma 3.1.

Remark 3.3

The Fourier expansions of \(f_1\) and \(f_2\) begin

$$\begin{aligned} f_1 \left( \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \right)&= 1 + 3q + 3s + 4q^2 + (2r^{-1} + 6 + 2r) qs + 4s^2 + O(q,s)^3;\\ f_2 \left( \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \right)&= q - s - 2q^2 + 2s^2 + 4q^3 + (4r^2 + 2 + 4r) qs (q - s) \\&\quad - 4s^3 + O(q, s)^4, \end{aligned}$$

where as usual \(q = \text {e}^{2\pi i \tau }\), \(r = \text {e}^{2\pi i z}\), \(s = \text {e}^{2\pi i w}\). For more coefficients, see Fig. 1. Setting \(s = 0\), one obtains the (holomorphic) modular forms:

$$\begin{aligned} \Phi (f_1) = 1 + 3q + 4q^2 \pm ..., \; \Phi (f_2) = q - 2q^2 + 4q^3 \pm ... \end{aligned}$$

of weight one and level \(\Gamma _1(5)\) which freely generate the ring \(M_*(\Gamma _1(5))\).

There are nine Heegner divisors of discriminant 1/4. One is the mirror of the reflective vector \(r = 1/2 \in A_1'\), represented by the diagonal in \(\mathbb {H}_2\), and the other eight are of the form \(H(1/4, r + \gamma )\) where \(\gamma \) are the isotropic cosets of \(U(5)'/U(5)\). It will be convenient to fix concrete representatives. We take the Gram matrix \(\mathbf {S} = \left( {\begin{matrix} 0 &{} 0 &{} 5 \\ 0 &{} 2 &{} 0 \\ 5 &{} 0 &{} 0 \end{matrix}}\right) \) for \(U(5) \oplus A_1\), such that \(L'/L \cong \mathbf {S}^{-1} \mathbb {Z}^3 / \mathbb {Z}^3\) and fix the cosets

$$\begin{aligned}&\gamma _1 = (1/5, 1/2, 4/5) + L,&\gamma _2 = (2/5, 1/2, 2/5) + L,&\\&\gamma _3 = (3/5, 1/2, 3/5) + L,&\gamma _4 = (4/5, 1/2, 1/5) + L,&\end{aligned}$$

of norm \(1/20 + \mathbb {Z}\). The norm 1/4 cosets other than r are labelled

$$\begin{aligned} \alpha _n = (n/5, 1/2, 0) + L, \; \beta _n = (0, 1/2, n/5) + L, \; n \in \{1, 2, 3, 4\}. \end{aligned}$$

Lemma 3.4

There are singular Gritsenko lifts \(g_1, g_2, h_1, h_2\) of weight two on \(\widetilde{\mathrm {O}}(L)\) whose divisors are exactly

$$\begin{aligned} \mathrm {div}\, g_1&= 3 H(1/20, \pm \gamma _1) - 2 H(1/20, \pm \gamma _2) + H(1/4, \pm \alpha _2); \\ \mathrm {div}\, g_2&= -2H(1/20, \pm \gamma _1) + 3H(1/20, \pm \gamma _2) + H(1/4, \pm \alpha _1); \\ \mathrm {div}\, h_1&= 3 H(1/20, \pm \gamma _1) - 2 H(1/20, \pm \gamma _2)+ H(1/4, \pm \beta _2); \\ \mathrm {div}\, h_2&= -2H(1/20, \pm \gamma _1) + 3H(1/20, \pm \gamma _2) + H(1/4, \pm \beta _1). \end{aligned}$$

Proof

The proof is essentially the same argument as Lemma 3.2. Using the pullback trick, one constructs weight two Gritsenko lifts which vanish on the claimed discriminant 1/4 Heegner divisors. Then one constructs Borcherds products of weight two with the claimed divisors. The respective quotients lie in \(M_0^!(\widetilde{\mathrm {O}}(L), \chi )\) and are, therefore, constant by Lemma 3.1. To determine the precise (weakly holomorphic) vector-valued modular forms which lift to \(g_1, g_2, h_1, h_2\), one only needs to compute the four-dimensional space of weakly holomorphic forms of weight 3/2 for \(\rho _L\) with a pole of order at most 1/20 at \(\infty \) and identify the unique (up to scalar) forms whose pullback to \(\alpha _n^{\perp }\) or \(\beta _n^{\perp }\) is respectively zero. The input forms \(G_1, G_2, H_1, H_2\) can be chosen such that their Fourier expansions begin as follows:

$$\begin{aligned} G_1:&\quad q^{-1/20} (\mathfrak {e}_{\gamma _2} + \mathfrak {e}_{\gamma _3}) + (\mathfrak {e}_{(0, 0, 1/5)} + \mathfrak {e}_{(0, 0, 4/5)} - \mathfrak {e}_{(0, 0, 2/5)} + \mathfrak {e}_{(0, 0, 3/5)} ) + O(q^{1/20}), \\ G_2:&\quad q^{-1/20} (\mathfrak {e}_{\gamma _1} + \mathfrak {e}_{\gamma _4}) + (\mathfrak {e}_{(0, 0, 2/5)} + \mathfrak {e}_{(0, 0, 3/5)} - \mathfrak {e}_{(0, 0, 1/5)} + \mathfrak {e}_{(0, 0, 4/5)} ) + O(q^{1/20}), \\ H_1:&\quad q^{-1/20} (\mathfrak {e}_{\gamma _2} + \mathfrak {e}_{\gamma _3}) + (\mathfrak {e}_{(1/5, 0, 0)} + \mathfrak {e}_{(4/5, 0, 0)} - \mathfrak {e}_{(2/5, 0, 0)} + \mathfrak {e}_{(3/5, 0, 0)} ) + O(q^{1/20}), \\ H_2:&\quad q^{-1/20} (\mathfrak {e}_{\gamma _1} + \mathfrak {e}_{\gamma _4}) + (\mathfrak {e}_{(2/5, 0, 0)} + \mathfrak {e}_{(3/5, 0, 0)} - \mathfrak {e}_{(1/5, 0, 0)} + \mathfrak {e}_{(4/5, 0, 0)} ) + O(q^{1/20}). \end{aligned}$$

These expansions determine \(G_1,G_2,H_1,H_2\) uniquely because there are no vector-valued cusp forms of weight 3/2 for \(\rho _L\).

Remark 3.5

The Fourier expansions of \(g_1, g_2, h_1, h_2\) begin as follows:

$$\begin{aligned} g_1\left( \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \right)&= q + q^2 - 5qs + 2q^3 + (-3r^{-1} + 1 - 3r) q^2 s \\&\quad + (-r^{-1} + 7 - r) qs^2 + O(q, s)^4;\\ g_2\left( \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \right)&= -q - q^2 + (r^{-1} + 3 + r)qs - 2q^3 + (-r^{-1} + 7 - r) q^2 s \\&\quad + (-3r^{-1} + 1 - 3r) qs^2 + O(q, s)^4;\\ h_1 \left( \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \right)&= s - 5qs + s^2 + (-r^{-1} +7 - r) q^2 s \\&\quad + (-3r^{-1} + 1 - 3r) qs^2 + 2s^3 + O(q, s)^4;\\ h_2 \left( \left( {\begin{matrix} \tau &{} z \\ z &{} w \end{matrix}}\right) \right)&= -s + (r^{-1} + 3 + r) qs - s^3 + (-3r^{-1} + 1 - 3r) q^2 s \\&\quad + (-r^{-1} + 7 - r) qs^2 - 2s^3 + O(q, s)^4. \end{aligned}$$

We can now determine the structure of \(M_*^!(\Gamma _{0, 1}^{(2)}(5))\). Recall that \(\Gamma _{0, 1}^{(2)}(5) /\{\pm I\} \cong \widetilde{\mathrm {SO}}(L)\). The decomposition

$$\begin{aligned} M_k^!(\widetilde{\mathrm {SO}}(L))= M_k^!(\widetilde{\mathrm {O}}(L)) \oplus M_k^!(\widetilde{\mathrm {O}}(L), \det ) \end{aligned}$$

suggests that we first consider the ring of modular forms for the discriminant kernel \(\widetilde{\mathrm {O}}(L)\). We will show that \(M_*^!(\widetilde{\mathrm {O}}(L))\) is freely generated using a generalization of the modular Jacobian approach of [12, Theorem 5.1]. We briefly introduce the main objects of this approach. For any four \(\psi _i\in M_{k_i}^!(\widetilde{\mathrm {O}}(L))\) with \(1\le i\le 4\), their Jacobian (see [12, Theorem 2.5] and [1, Proposition 2.1])

$$\begin{aligned} J(\psi _1,\psi _2,\psi _3,\psi _4)=\left|\begin{array}{cccc} k_1\psi _1 &{} k_2\psi _2 &{} k_3\psi _3 &{} k_4\psi _4 \\ \frac{\partial \psi _1}{\partial \tau } &{} \frac{\partial \psi _2}{\partial \tau } &{} \frac{\partial \psi _3}{\partial \tau } &{} \frac{\partial \psi _4}{\partial \tau } \\ \frac{\partial \psi _1}{\partial z} &{} \frac{\partial \psi _2}{\partial z} &{} \frac{\partial \psi _3}{\partial z} &{} \frac{\partial \psi _4}{\partial z} \\ \frac{\partial \psi _1}{\partial w} &{} \frac{\partial \psi _2}{\partial w} &{} \frac{\partial \psi _3}{\partial w} &{} \frac{\partial \psi _4}{\partial w} \end{array} \right|\end{aligned}$$

lies in \(M_{k_1+k_2+k_3+k_4+3}^!(\widetilde{\mathrm {O}}(L),\det )\). The Jacobian \(J(\psi _1,\psi _2,\psi _3,\psi _4)\) is not identically zero if and only if the four forms \(\psi _i\) are algebraically independent over \(\mathbb {C}\).

The discriminant kernel \(\widetilde{\mathrm {O}}(L)\) contains reflections associated to vectors of norm 1 in L (the so-called 2-reflections)

$$\begin{aligned} \sigma _v: x \mapsto x - (x,v)v. \end{aligned}$$

The hyperplane \(v^\perp \) is called the mirror of the reflection \(\sigma _v\). Since \(\det (\sigma _v)=-1\), the chain rule implies that the above Jacobian vanishes on all mirrors of 2-reflections. Conversely, the main theorem of [12], and its generalization to meromorphic modular forms with constrained poles, implies that

Theorem 3.6

The ring \(M_*^!(\widetilde{\mathrm {O}}(L))\) is a free algebra:

$$\begin{aligned} M_*^!(\widetilde{\mathrm {O}}(L)) = \mathbb {C}[f_1, f_2, g_1, g_2]. \end{aligned}$$

Define \(J:=J(f_1,f_2,g_1,g_2)\). Then

$$\begin{aligned} M_*^!(\Gamma _{0, 1}^{(2)}(5)) = \mathbb {C}[f_1, f_2, g_1, g_2, J]. \end{aligned}$$

Proof

The Jacobian J of \(f_1, f_2, g_1, g_2\) has weight 9 and vanishes on the mirrors of 2-reflections, which form a union of Heegner divisors of discriminants 1/4 and 1 denoted \(\Delta \). Using the Fourier expansions of the forms it is easy to check that J is not identically zero. Using the algorithm of [13], we find a Borcherds product \(J_0\) with divisor

$$\begin{aligned}\mathrm {div}\, J_0 = \Delta + 6 H(1/20, \pm \gamma _1) + 6 H(1/20, \pm \gamma _2).\end{aligned}$$

The quotient \(J / J_0\) lies in \(M_0^!(\Gamma _{0, 1}^{(2)}(5),\chi )\) and is, therefore, a constant denoted c by Lemma 3.1. We will now prove the claim by an argument which appeared essentially in [12]. Suppose that \(M_*^!(\widetilde{\mathrm {O}}(L))\) was not generated by \(h_1 := f_1\), \(h_2 := f_2\), \(h_3 := g_1\) and \(h_4 := g_2\), and let \(h_5 \in M_{k_5}^!(\widetilde{\mathrm {O}}(L))\) be a modular form of minimal weight which is not contained in \(\mathbb {C}[f_1,f_2,g_1,g_2]\). Set \(k_1 = k_2 = 1\) and \(k_3 = k_4 = 2\), such that \(k_i\) is the weight of \(h_i\). For \(1\le j \le 5\), we define \(J_j\) as the Jacobian of the four modular forms \(h_i\) omitting \(h_j\), such that \(c J_0=J=J_{5}\). It is clear that \(g_j:=J_j/J\) is a modular form on \(\widetilde{\mathrm {O}}(L)\) with poles supported on \(\mathcal {H}\). We compute the determinant and find the identity:

$$\begin{aligned} 0 = \mathrm {det} \begin{pmatrix} k_1 h_1 &{} ... &{} k_4 h_4 &{} k_5 h_5 \\ k_1 h_1 &{} ... &{} k_4 h_4 &{} k_5 h_5 \\ \nabla h_1 &{} ... &{} \nabla h_4 &{} \nabla h_5 \end{pmatrix} = \sum _{i=1}^5 (-1)^{i+1} k_i h_i J_i. \end{aligned}$$

Since \(J_i=Jg_i\) and \(g_5=1\), we have

$$\begin{aligned} \sum _{i=1}^{5} (-1)^{i+1} k_t h_t g_t = 0, \quad \text {i.e} \quad k_5 h_5 = \sum _{i=1}^4 (-1)^i h_i g_i. \end{aligned}$$

Since \(h_5\) was chosen to have minimal weight, \(g_i \in \mathbb {C}[h_1,h_2,h_3,h_4]\) for all i, and thus, \(h_5\in \mathbb {C}[h_1,h_2,h_3,h_4]\), which is a contradiction.

Now any \(h\in M_k^!(\widetilde{\mathrm {O}}(L),\det )\) vanishes on all mirrors of 2-reflections, which implies that \(h/J \in M_{k-9}^!(\widetilde{\mathrm {O}}(L))\). Therefore,

$$\begin{aligned} M_*^!(\Gamma _{0,1}^{(2)}(5)) = M_*^!(\widetilde{\mathrm {SO}}(L)) = M_*^!(\widetilde{\mathrm {O}}(L)) \oplus M_*^!(\widetilde{\mathrm {O}}(L), \det ) \end{aligned}$$

is generated by \(f_1\), \(f_2\), \(g_1\), \(g_2\), and J.

Remark 3.7

The weight two singular Gritsenko lifts satisfy the relations:

$$\begin{aligned} g_1 - h_1 = h_2 - g_2 = f_1 f_2. \end{aligned}$$

The product \(f_1 f_2\) is holomorphic and in fact itself a Gritsenko lift, but it has a quadratic character under \(\Gamma _0^{(2)}(5)\). There is a unique normalized Siegel modular form \(e_2\) of weight two for \(\Gamma _0^{(2)}(5)\), which can be constructed as the Gritsenko lift of the unique vector-valued modular form of weight 3/2 for \(\rho _L\) invariant under all automorphisms of the discriminant form. (The uniqueness follows from Corollary 3.8.) In terms of the generators of \(M_*^!(\Gamma _{0, 1}^{(2)}(5))\), a computation shows

$$\begin{aligned} e_2 = f_1^2 + f_2^2 - 4(g_1 + g_2). \end{aligned}$$

Corollary 3.8

The ring \(M_*^!(\Gamma _0^{(2)}(5))\) is minimally generated in weights 2, 2, 4, 4, 4, 4, 4, 11, 11, 11 by the ten forms

$$\begin{aligned}&f_1^2 + f_2^2,&e_2,&f_1^2 g_1 + f_2^2 g_2,&f_1 f_2 (g_1 - g_2),&f_1 f_2 (f_1 - f_2)(f_1 + f_2),&\\&f_1^2 f_2^2,&g_1 g_2,&J f_1 f_2,&J (f_1^2 - f_2^2),&J (g_1 - g_2).&\end{aligned}$$

Proof

The group \(\Gamma _0^{(2)}(5)\) is generated by the special discriminant kernel of L and by the order four automorphism \(\varepsilon _2\) which acts on the generators of \(M_*^!(\Gamma _{0, 1}^{(2)}(5))\) by

$$\begin{aligned} \varepsilon _2 : f_1 \mapsto f_2, \; f_2 \mapsto -f_1, \; g_1 \mapsto g_2, \; g_2 \mapsto g_1, \; J \mapsto -J \end{aligned}$$

as one can see on the input functions into the Gritsenko lifts. We conclude the action of \(\varepsilon _2\) on J (as a Jacobian) from the actions of \(\varepsilon _2\) on \(f_1\), \(f_2\), \(g_1\), and \(g_2\). The expressions in \(f_1, f_2, g_1, g_2, J\) in the claim generate the ring of invariants under this action.

Remark 3.9

The same argument shows that the kernel of \(\varepsilon _2^2 = \varepsilon _4\) is generated by J and by the weight two forms:

$$\begin{aligned} f_1^2, f_2^2, f_1 f_2, g_1, g_2. \end{aligned}$$

This corresponds to the quadratic Nebentypus \(\chi \left( \left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \right) = \left( \frac{5}{\mathrm {det}\, d}\right) \) on \(\Gamma _0^{(2)}(5)\). Note that \(f_1 f_2\) is the Siegel Eisenstein series of weight two for the character \(\chi \), and that the Jacobian J is the unique cusp form of weight nine for \(\chi \) up to scalars.

Remark 3.10

There is a seven-dimensional space of modular forms of weight 7/2 for \(\rho _L\), and a four-dimensional subspace on which \(\varepsilon _2\) acts trivially, so the weight four Maass space for \(\Gamma _0^{(2)}(5)\) is four dimensional. Using the structure theorem above, we can identify it by comparing only a few Fourier coefficients:

$$\begin{aligned} \mathrm {Maass}_4 = \mathrm {Span}(\phi _1, \phi _2, \phi _3, \phi _4), \end{aligned}$$

where

$$\begin{aligned} \phi _1&= e_2^2 + f_1^2 f_2^2; \\ \phi _2&= f_1^2 g_1 + f_2^2 g_2 - 2g_1 g_2; \\ \phi _3&= f_1 f_2 (f_1^2 - 2 f_1 f_2 - f_2^2 + 2 g_1 - 2g_2); \\ \phi _4&= 2 g_1 g_2 + f_1 f_2 (g_1 - g_2). \end{aligned}$$

The form \(\phi _4\) is a cusp form and indeed spans \(S_4(\Gamma _0^{(2)}(5))\), which was shown to be one dimensional by Poor and Yuen [10].

4 The ring of holomorphic Siegel modular forms of level 5

In this section, we investigate the ring \(M_*(\Gamma _0^{(2)}(5))\) of holomorphic Siegel modular forms for \(\Gamma _0^{(2)}(5)\). We will need the Hilbert–Poincaré series for this ring, which can be derived from dimension formulas available in the literature.

Theorem 4.1

The Hilbert–Poincaré series of dimensions of modular forms for \(\Gamma _0^{(2)}(5)\) is

$$\begin{aligned} \sum _{k = 0}^{\infty } \mathrm {dim}\, M_k(\Gamma _0^{(2)}(5)) t^k = \frac{(1 - t)^2 (1 + t^7) P(t)}{(1 - t^2)^2 (1 - t^3)(1 - t^4)^2 (1 - t^5)}, \end{aligned}$$

where P(t) is the irreducible palindromic polynomial

$$\begin{aligned} P(t)= & {} 1 + 2 t + 2 t^{2} + t^{3} + 3 t^{4} + 5 t^{5} + 8 t^{6} + 8 t^{7} + 8 t^{8} + 5 t^{9}\nonumber \\&\quad + 3 t^{10} + t^{11} + 2 t^{12} + 2 t^{13} + t^{14}. \end{aligned}$$

The first values of \(\dim M_k(\Gamma _0^{(2)}(5))\) are given in Table 1.

Table 1 \(\dim M_k(\Gamma _0^{(2)}(5))\)
Full size table

Proof

The dimensions of the spaces of cusp forms of weight \(k \ge 5\) have been computed in closed form by Hashimoto by means of the Selberg trace formula and in lower weights by Poor and Yuen [10]: we have \(\mathrm {dim}\, S_4(\Gamma _0^{(2)}(5)) = 1\) and \(\mathrm {dim}\, S_k(\Gamma _0^{(2)}(5)) = 0\) for \(k \le 3\). All odd-weight modular forms are cusp forms, and by a more general theorem of Böcherer–Ibukiyama [2], for even \(k > 2\),

$$\begin{aligned}\mathrm {dim}\, M_k(\Gamma _0^{(2)}(5)) = \mathrm {dim}\, S_k(\Gamma _0^{(2)}(5)) + 2 \cdot \mathrm {dim}\, S_k(\Gamma _0(5)) + 3. \end{aligned}$$

We can now determine the generators of \(M_*(\Gamma _0^{(2)}(5))\) using Corollary 3.8 together with the above generating series.

Theorem 4.2

The ring of Siegel modular forms of level \(\Gamma _0^{(2)}(5)\) is minimally generated by the weight two form

$$\begin{aligned} e_2 = f_1^2 + f_2^2 - 4 g_1 - 4g_2, \end{aligned}$$

five weight four forms

$$\begin{aligned} f_1^2 g_1 + f_2^2 g_2, \quad f_1 f_2 (g_1 - g_2), \quad f_1 f_2 (f_1^2 - f_2^2), \quad f_1^2 f_2^2, \quad g_1 g_2, \end{aligned}$$

four weight six forms

$$\begin{aligned} f_1^2 f_2^2 (g_1 + g_2), \quad f_1^3 f_2 g_1 - f_2^3 f_1 g_2, \quad f_1^2 g_1^2 + f_2^2 g_2^2, \quad g_1 g_2 (f_1^2 + f_2^2), \end{aligned}$$

the weight ten form

$$\begin{aligned} f_1^2 f_2^2 (f_1^2 + f_2^2)^3, \end{aligned}$$

three weight eleven forms

$$\begin{aligned} f_1 f_2 J, \quad (f_1^2 - f_2^2) J, \quad (g_1 - g_2) J, \end{aligned}$$

three weight thirteen forms

$$\begin{aligned} (f_1^2 + f_2^2) f_1 f_2 J, \quad (f_1^4 - f_2^4) J, \quad (f_1^2 + f_2^2)(g_1 - g_2) J, \end{aligned}$$

and the weight fifteen form

$$\begin{aligned} (f_1^2 - f_2^2)^3 J. \end{aligned}$$

Proof

From the divisors of \(f_1, f_2, g_1, g_2\), and J, it is easy to see that all of the forms above (except for \(e_2\), which was discussed in the previous section) are holomorphic and \(\varepsilon _2\)-invariant. The Hilbert series of this ring was computed in Macaulay2 [4] and coincides exactly with the series predicted by Theorem 4.1, so we can conclude that these forms are sufficient to generate all holomorphic Siegel modular forms.