On the graded ring of modular forms of the Siegel paramodular group of level 2

Tomoyoshi Ibukiyama and Fumie Onodera

In this paper, we shall describe the concrete ring structure of the graded rings of modular forms belonging to the Siegel paramodular group Γ^{para}(2) of degree two with polarization diag(1, 2). We also show that the Satake compactification of the quotient variety by this group is rational. Here, for each prime p, we define the group Γ^{para}(p) by

Γ^{para}(p) = {g ∈ M_{4}(ℤ); ^{t}gJ_{2}(p)g = J_{2}(p)},

where for any number d, we put

J_{2}(d) = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & d \\ −1 & 0 & 0 & 0 \\ 0 & −d & 0 & 0 \end{matrix} ) .

The main results will be given in section 1.

Historically, Freitag [1] has obtained the ring structure for a certain group which contains our group Γ^{para}(2) with index 2. He used some geometrical method. Since the dimension formula for Γ^{para}(p) has been known by Ibukiyama [9], we can use more direct method here, and his result is also obtained as a corollary of our result. Various generators of the ring have been considered by various approach. (cf. Gritsenko [2], [3], Gritsenko and Nikulin [4], [5], or Runge [15]). But the ring structure was not known as far as the authors know.

Actually we treat the discrete subgroup K(p) of Sp(2, ℝ) which is GL_{4}(ℚ)- conjugate to Γ^{para}(p) and defined by

K(p) = Sp(2, ℚ) ∩ ( \begin{matrix}ℤ & ℤ & p^{−1}ℤ & ℤ \\ pℤ & ℤ & ℤ & ℤ \\ pℤ & pℤ & ℤ & pℤ \\ pℤ & ℤ & ℤ & ℤ \end{matrix} ) .

(The fact that K(p) is conjugate to Γ^{para}(p) is well known and was remarked also in the introduction of Ibukiyama [9] without proof. As for the written proof, see e.g. Hulek, Kahn and Weintraub [7] or Gritsenko [2].) This group K(p) has been treated in Ibukiyama [8], [9] as one of standard parahoric subgroups in some different context and several results there are applicable here. For example, the dimension formula for Siegel cusp forms of weight k ≥ 5 belonging to K(p) has been known in [9] for each prime p, and some forms of small weights belonging to K(2) have been given explicitly with their L functions in [8]. If we take the Iwahori subgroup B(2) of level 2 defined e.g. in [8], then K(2) contains B(2) (cf. [6]) and the ring structure of modular forms belonging to B(2) has been known in Ibukiyama [10]. We shall use these facts. By the way, the structure of A(Γ(2)) for the principal congruence subgroup of degree 2 is well known by Igusa [12] and our K(2) contains Γ(2). But Γ(2) is not a normal subgroup of K(2), and we need some work to get A(K(2)). In section 1, we shall state the main result, and the proof will be given in section 2.

The first author is grateful to Max Planck Institut für Mathematik, Bonn, during his visit when this work was finished.

1 Main results

1.1 Preliminary definitions

For any ring S, we denote by Sp(n, S) the usual symplectic group of size 2n defined by

Sp(n, S) = {g ∈ M_{2n}(S); ^{t}gJg = J},

where

J = ( \begin{matrix}0 & −1_{n} \\ 1_{n} & 0 \end{matrix} ) .

We denote by H_{n} the Siegel upper half space of degree n defined by

H_{n} = {Z ∈ M_{n}(ℂ); ^{t}Z = Z, Im(Z) > 0}.

Let Γ be a discrete subgroup of Sp(n, ℝ) with covolume finite. We denote by A_{k}(Γ) or S_{k}(Γ) the space of modular forms, or cusp forms, of weight k belonging to Γ, respectively. We define two graded rings as follows.

A(Γ) = ⊕_{k=0}^{∞}A_{k}(Γ)  and  A_{even}(Γ) = ⊕_{k=0}^{∞}A_{2k}(Γ).

For any F (Z) ∈ A_{k}(Γ), we write

F |[g] = F |_{k}[g] = F (gZ) det(CZ + D)^{−k}, for g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(n, ℝ).

Next, we define several discrete subgroups of Sp(2, ℝ). For each prime p, we define “Iwahori subgroup” B(p) by

B(p) = Sp(2, ℤ) ∩ ( \begin{matrix}ℤ & ℤ & ℤ & ℤ \\ pℤ & ℤ & ℤ & ℤ \\ pℤ & pℤ & ℤ & pℤ \\ pℤ & pℤ & ℤ & ℤ \end{matrix} ) .

We need several groups which contain B(2). Put

s_{0} = ( \begin{matrix}0 & 0 & −p^{−1} & 0 \\ 0 & 1 & 0 & 0 \\ p & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , s_{1} = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) , s_{2} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) .

We also put Γ0^{′}(p) = B(p) ∪ B(p)s_{2}B(p), and Γ0^{′′}(p) = B(p) ∪ B(p)s_{0}B(p). These are groups. As for more explicit description, cf.[8] p.601. By the general theory of Bruhat-Tits, we get K(p) = B(p) ∪ B(p)s_{0}B(p) ∪ B(p)s_{2}B(p) ∪ B(p)s_{0}s_{2}B(p) and the group K(p) is generated by Γ0^{′}(p) and Γ0^{′′}(p). Now, put

ρ = \frac{1}{\sqrt{p}} ( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & p & 0 & 0 \\ p & 0 & 0 & 0 \end{matrix} ) .

Then we get Γ0^{′′}(p) = ρΓ0^{′}(p)ρ^{−1} and ρK(p)ρ^{−1} = K(p). We denote by K^{∗}(p) the group generated by K(p) and ρ. We have [K^{∗}(p) : K(p)] = 2.

1.2 Generaters of Modular forms

For any m = (m′, m′′) ∈ ℤ^{2n} (m′, m′′ ∈ ℤ^{n}), we define a theta constant θ_{m′,m′′} = θ_{m′,m′′}(Z) by the following function of Z ∈ H_{n}.

θ_{m′,m′′}(Z) = ∑_{p∈Z^{n}} exp(2πi(^{t}(p + m′/2)Z(p + m′/2)/2 + ^{t}(p + m′/2)m′′/2)).

We also put

X = (θ_{0000}^{4} + θ_{0001}^{4} + θ_{0010}^{4} + θ_{0011}^{4})/4,
Y = (θ_{0000}θ_{0001}θ_{0010}θ_{0011})^{2},
Z = (θ_{0100}^{4} − θ_{0110}^{4})^{2}/16384,
K = (θ_{0100}θ_{0110}θ_{1000}θ_{1001}θ_{1100}θ_{1111})^{2}/4096,
T = (θ_{0100}θ_{0110})^{4}/256,

where the theta constants are for n = 2. Each of the above is a modular form which belongs to B(2) of weight 2, 4, 4, 6, or 4, respectively. We define the following functions.

F_{4} = X^{2} + 3Y + 3072Z + 960T,
F_{6} = X^{3} − 9XY − 9216XZ + 27648K + 4032T X,
F_{8} = 16Y Z − 16XK + 64T^{2} − T X^{2} + 1024T Z + T Y,
F_{12} = 32X^{3}K + 64X^{2}Y Z − 96XY K − 98304XZK + 5X^{4}T − 14X^{2}Y T
−14336X^{2}ZT − 6144XKT + 9Y^{2}T + 18432Y ZT + 9437184Z^{2}T
−896X^{2}T^{2} + 1152Y T^{2} + 1179648ZT^{2} + 36864T^{3}
G_{10} = 4X^{2}K − 16XY Z + 12Y K + 12288ZK + X^{3}T − XY T − 1024XZT
+768KT − 64XT^{2}
G_{12} = 3014656T X^{2}Z − 2944T X^{2}Y + 12582912KXZ − 12288KXY

+184320T^{2}Y − 188743680T^{2}Z − 1152T Y^{2} + 1207959552T Z^{2} − 1024X^{4}Z +2097152X^{2}Z^{2} + 3145728Y Z^{2} − 1073741824Z^{3} + X^{4}Y − 2X^{2}Y^{2} − 3072Y^{2}Z + Y^{3}

G_{11} = θ_{0000}θ_{0001}θ_{0010}θ_{0011}θ_{0100}θ_{0110}θ_{1000}θ_{1001}θ_{1100}θ_{1111}
×(θ_{1000}^{12} − θ_{1001}^{12} − θ_{1100}^{12} + θ_{1111}^{12})/1536( = χ_{11} in [10].)

Proposition 1 The function F_{4}, F_{6}, F_{8}, F_{12}, G_{10}, G_{12}, or G_{11} defined above is a modular form which belongs to K(2) and of weight 4, 6, 8, 12, 10, 12 or 11, respectively. The first 5 forms belong also to K^{∗}(2), and we get G_{11}|[ρ] = −G_{11} and G_{12}|[ρ] = −G_{12}. Besides, F_{8}, F_{12}, G_{10} and G_{11} are cusp forms.

The proof of this Proposition will be given in section 2.

1.3 Main results

We denote by B the following subring of A(K(2))

B = ℂ[F_{4}, F_{6}, F_{8}, F_{12}].

Theorem 1 The modular forms F_{4}, F_{6}, F_{8}, F_{12} are algebraically independent and B is a weighted polynomial ring. The graded ring A_{even}(K(2)) is given by

A_{even}(K(2)) = B ⊕ (G_{12})B ⊕ (G_{10})B ⊕ (G_{10}G_{12})B,

and we get

A(K(2)) = A_{even}(K(2)) ⊕ (G_{11})A_{even}(K(2)),

where ⊕ means the direct sum as modules. The ideal of cusp forms of A(K(2)) is spanned by F_{8}, F_{12}, G_{10}, and G_{11}.

The fundamental relations of the generators of the above graded ring are given as follows.

G_{10}^{2} = F_{8}F_{12}/4,
729G_{12}^{2} = 26873856F_{12}^{2} − 10368F_{4}^{3}F_{12} − 71663616F_{4}F_{8}F_{12} − 10368F_{6}^{2}F_{12}
+F_{4}^{6} − 6912F_{4}^{4}F_{8} − 2F_{4}^{3}F_{6}^{2} + 15925248F_{4}^{2}F_{8}^{2}
−13824F_{4}F_{6}^{2}F_{8} + F_{6}^{4} − 12230590464F_{8}^{3}
+G_{10}(82944F_{4}^{2}F_{6} + 63700992F_{6}F_{8}),
G_{11}^{2} = 3^{−3} · 2^{6}(−F_{6}F_{8}^{2} + 3F_{4}F_{8}G_{10} − F_{12}G_{10}).

Next Corollary is first proved by Freitag [1] for even weights.

Corollary 1 We get

A(K^{∗}(2)) = B ⊕ (G_{10})B ⊕ (G_{11}G_{12})B ⊕ (G_{11}G_{12}G_{10})B.

Now, we will give a result of the structure of the variety. For the sake of simplicity, we put

α = G_{10}/F_{4}F_{6}
β = F_{6}^{2}/F_{4}^{3}
γ = F_{8}/F_{4}^{2}

Further, we define the automorphic functions A, B, C belonging to K(2) by

A = 27γ(20736α^{2} − γ)(G_{12}/F_{4}^{3})
B = β(20736α^{2} − γ)^{2} − 20736α^{2}γ − 143327232α^{2}γ^{2} − γ^{2}
−6912γ^{3} + 41472αγ^{2} + 31850496αγ^{3}
C = γ(768γ^{2} + γ − 13824αγ + 15925248α^{2}γ − 2α + 20736α^{2})

As an application of the above theorem, we get

Corollary 2 The Satake compactification S(K(2)\H_{2}) = P roj(A(K(2))) is a rational variety. The function field is given by

ℂ(\frac{G_{10}}{F_{4}F_{6}} , \frac{A}{C}, \frac{B}{C}).

2 Proofs

2.1

We first review on dimension formulas.

Proposition 2 (cf. [9]) We get

∑_{k=0}^{∞} dim A_{k}(K(2))t^{k} = \frac{(1 + t^{10})(1 + t^{12})(1 + t^{11})}{(1 − t^{4})(1 − t^{6})(1 − t^{8})(1 − t^{12})},
∑_{k=0}^{∞} dim S_{k}(K(2))t^{k} = \frac{(t^{8} + t^{10} + t^{12} − t^{20})(1 + t^{12})}{(1 − t^{4})(1 − t^{6})(1 − t^{8})(1 − t^{12})},
∑_{k=0}^{∞} dim A_{k}(Γ0^{′′}(2))t^{k} = \frac{(1 + t^{11})(1 + t^{6} + t^{8} + t^{10} + t^{12} + t^{18})}{(1 − t^{4})^{2}(1 − t^{6})(1 − t^{12})}.

Proof. As for the formula for Γ0^{′}(2), this is an easy corollary to Igusa [12]. When the weight k ≤ 4, the results for K(2) is obtained from explicit structure of A_{k}(B(2)) very easily. For general k with k ≥ 5, as for dim S_{k}(K(2)), the above formula is the special case of [9] Theorem 4. As for dim A_{k}(K(2)), it is easily obtained by the surjectivity of Φ operator by Satake [16] and the explicit description of cusps of K(p) given e.g. in [11] for each prime p. We omit the details. q.e.d.

It has been written in [8] how to obtain forms in A_{k}(K(2)). We review this shortly for readers convenience. For automorphic form F ∈ A_{k}(B(2)), write the Fourier expansion as

F (Z) = ∑_{T} a(T) exp(2πitr(T Z)),

where T runs over positive semi-definite half integral matrices. The subspace A_{k}(Γ0^{′′}(2)) is characterized by those forms F ∈ A_{k}(B(2)) such that a(T) = 0 for all T with odd (1,1) component. (As for the proof, see [8].) Since we know dim A_{k}(Γ0^{′′}(2)) and the generators of A(B(2)) (cf. [10]), we can determine A_{k}(Γ0^{′′}(2)) if k is given explicitly and enough Fourier coefficients are known. Now, we have A_{k}(K(2)) = A_{k}(Γ0^{′′}(2)) ∩ A_{k}(Γ0^{′}(2)) and A_{k}(Γ0^{′}(2)) = A_{k}(Γ0^{′′}(2))|ρ. So we can get A_{k}(K(2)) for given small k. We know that X, K, T are invariant by the action of ρ and Y |_{4}[ρ] = 1024Z, Z|_{4}[ρ] = Y/1024. Hence, we can also show that F_{4}, F_{6}, F_{8}, F_{12}, G_{10} ∈ A_{k}(K^{∗}(2)). We can show G_{12}|[ρ] = −G_{12} easily from the above. If we put G = G_{11}|[ρ_{1}] for

ρ_{1} = ( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) ,

then obviously (G_{11}|[ρ])(Z) = 2^{11}G(2Z). By the theta transformation formula (cf. [13]), we get

G(Z) = −(θ_{0001}^{12} − θ_{1001}^{12} − θ_{0011}^{12} + θ_{1111}^{12}) ∏_{m} θ_{m}/1536,

where m runs over ten even characteristics mod 2. Since dim A_{11}(K(2)) = 1 , obviously G_{11}|[ρ] is G_{11} or −G_{11}, hence comparing one non vanishing Fourier coefficient, we can show it is −G_{11}. If you prefer more theoretical proof, you can prove this by using the following relations (cf. [13] p.232)

θ_{0000}(2Z)θ_{0100}(2Z) = (θ_{0100}(Z)^{2} + θ_{0110}(Z)^{2})/4,
θ_{1000}(2Z)θ_{1100}(2Z) = (θ_{0100}(Z)^{2} − θ_{0110}(Z)^{2})/4,
θ_{0011}(2Z)θ_{1111}(2Z) = (θ_{1100}(Z)θ_{1111}(Z))/2,
θ_{0010}(2Z)θ_{0110}(2Z) = (θ_{0100}(Z)θ_{0110}(Z))/2,
θ_{0001}(2Z)θ_{1001}(2Z) = (θ_{1000}(Z)θ_{1001}(Z))/2,
θ_{0001}(2Z)^{2} = (θ_{0000}(Z)θ_{0001}(Z) + θ_{0010}(Z)θ_{0011}(Z))/2,
θ_{1001}(2Z)^{2} = (θ_{0000}(Z)θ_{0001}(Z) − θ_{0010}(Z)θ_{0011}(Z))/2,
θ_{0011}(2Z)^{2} = (θ_{0000}(Z)θ_{0011}(Z) + θ_{0010}(Z)θ_{0001}(Z))/2,
θ_{1111}(2Z)^{2} = (θ_{0000}(Z)θ_{0011}(Z) − θ_{0010}(Z)θ_{0001}(Z))/2,

and Riemann’s theta formula (cf. [12].) Since K, Y Z, and T (X^{2} − Y − 1024Z − 64T) are cusp forms (cf. [10]), it is easy to see F_{8}, F_{12} and G_{10} are cusp forms. Hence, Proposition 1 is proved.

Now we introduce the Witt operator. For any function F (Z) on H_{2}, we put

(W F)(z_{1}, z_{2}) = F ( \begin{matrix}z_{1} & 0 \\ 0 & z_{2} \end{matrix} ) (z_{1}, z_{2} ∈ H_{1}).

We denote by E_{k}(z) (z ∈ H_{1}) the Eisenstein series of weight k belonging to SL_{2}(ℤ) having 1 as the constant term. For short we write E_{k} = E_{k}(z_{2}) and Ek^{′} = E_{k}(2z_{1}) for mutually independent variables z_{1}, z_{2} ∈ H_{1}. The image of theta constants by W is again easily expressed by theta constants. Also, it is easy to see the following relations.

E_{4} = θ_{01}^{8}(z_{2}) + θ_{01}^{4}(z_{2})θ_{10}^{4}(z_{2}) + θ_{10}^{8}(z_{2}),
E4^{′} = (16θ_{01}^{8}(z_{1}) + 16θ_{01}^{4}(z_{1})θ_{10}^{4}(z_{1}) + θ_{10}^{8}(z_{1}))/16,
E_{6} = (2θ_{01}^{4}(z_{2}) + θ_{10}^{4}(z_{2}))(θ_{01}^{8}(z_{2}) + θ_{01}^{4}(z_{2})θ_{10}^{4}(z_{2}) − 2θ_{10}^{8}(z_{2}))/2,
E6^{′} = (2θ_{01}^{4}(z_{1}) + θ_{10}^{4}(z_{1}))(32θ_{01}^{8}(z_{1}) + 32θ_{01}^{4}(z_{1})θ_{10}^{4}(z_{1}) − θ_{10}^{8}(z_{1}))/64,

Hence, it is easy to show that W (F_{8}) = W (G_{10}) = 0 and

W (F_{4}) = 4E4^{′}E_{4},
W (F_{6}) = −8E6^{′}E_{6},
W (F_{12}) = (E_{4}^{′3} − E_{6}^{′2})(E_{4}^{3} − E_{6}^{2})/81,
W (G_{12}) = 3^{−3} · 2^{6}(E_{4}^{′3}E_{6}^{2} − E_{4}^{3}E_{6}^{′2}).

Lemma 1 Let P and Q be polynomials of three variables which satisfy the following relation

P (W (F_{4}), W (F_{6}), W (F_{12})) + W (G_{12})Q(W (F_{4}), W (F_{6}), W (F_{12})) = 0.

Then we get P = Q = 0.

Proof. It is clear that E_{4}, E4^{′}, E_{6}, E6^{′} are algebraically independent. The forms W (F_{4}), W (F_{6}) and W (F_{12}) are invariant under the exchange between E_{k} and Ek^{′} (k = 4, 6) and G_{12} becomes −G_{12} by this exchange. So, we get P (W (F_{4}), W (F_{6}), W (F_{12})) = 0. But obviously, W (F_{4}), W (F_{6}) and W (F_{12}) are algebraically independent. Hence P = Q = 0 as polynomials.

Now we prove Theorem 1. First, it is easy to show the formula for G_{k}^{2} for k = 10, 11, 12 in Theorem, since we know the relation

64K^{2} = −16XT K − T (−16Y Z + X^{2}T − Y T − 1024ZT − 64T^{2}),

and the formula to express G_{11}^{2} by X, Y, Z, K and T in Appendix of [10]. All we must do is to express both sides of the formulas as polynomials of X, Y, Z, K, T of degree one with respect to K to find out they are the same. In particular we get

G_{10}^{2} = F_{8}F_{12}/4.

Proposition 3 Let P_{i}(x_{1}, x_{2}, x_{3}, x_{4}) (1 ≤ i ≤ 4) be polynomials of four variables which satisfiy the following relation

P_{1}(F_{4}, F_{6}, F_{8}, F_{12}) + G_{12}P_{2}(F_{4}, F_{6}, F_{8}, F_{12})
+ G_{10}P_{3}(F_{4}, F_{6}, F_{8}, F_{12}) + G_{10}G_{12}P_{4}(F_{4}, F_{6}, F_{8}, F_{12}) = 0.

Then P_{i} = 0 (i = 1, …, 4) as polynomials. In particular, F_{4}, F_{6}, F_{8}, F_{12} are algebraically independent.

Proof. We take the image under the Witt operator of the both sides of the above relation. Since W (G_{10}) = 0, by the above lemma we get

P_{1}(x_{1}, x_{2}, 0, x_{4}) = P_{2}(x_{1}, x_{2}, 0, x_{4}) = 0.

That is, for i = 1, 2, we have P_{i} = x_{3}Q_{i} for some polynomials Q_{i}. Now, multiplying G_{10} to both sides, we get

F_{8}G_{10}Q_{1}(F_{4}, F_{6}, F_{8}, F_{12}) + F_{8}G_{10}G_{12}Q_{2}(F_{4}, F_{6}, F_{8}, F_{12})
+ 4^{−1}F_{8}F_{12}P_{3}(F_{4}, F_{6}, F_{8}, F_{12}) + 4^{−1}F_{8}F_{12}G_{12}P_{4}(F_{4}, F_{6}, F_{8}, F_{12}) = 0.

Now dividing both sides by F_{8}, then applying W on both sides, and dividing by W (F_{12}) ≠ 0, we can see as before that P_{i} = x_{3}Q_{i} for i = 3, 4, for some polynomials Q_{i}. Repeating this process, we can conclude that P_{i} = 0 for all i = 1,…,4. q.e.d.

By the above proposition, we can calculate the dimensions of (B + G_{10}B + G_{12}B + G_{10}G_{12}B) ∩ A_{k}(K(2)) for each k to find that it is equal to dim A_{k}(K(2)). Hence our main theorem is proved. Corollary 1 is clear from Theorem. We now prove Corollary 2. We denote by K the function field of P roj(A(K(2))). We define elements A, B, C, α, β, γ ∈ K as in section 1. By the formula for G_{10}^{2}, we get F_{12}/F_{4}^{3} = 4α^{2}β/γ. By Theorem 1, it is easy to see that K is generated by α, β, γ, and G_{12}/F_{4}^{3}. Now, we define the field K′ by K′ = ℂ(α, A/C, B/C). By modifying the formula for G_{12}^{2}, we can show that A^{2} = B^{2} − γC^{2}. Hence γ = (B/C)^{2} − (A/C)^{2} ∈ K′. Since C is a polynomial of α and γ, we get C ∈ K′. Hence A, B ∈ K′ and also G_{12}/F_{4}^{3}, β ∈ K′. Thus we get K′ = K and Corollary 2.

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Department of Mathematics Graduate School of Science Osaka University Tonoyoka, Osaka, 560 Japan ibukiyam@math.wani.osaka-u.ac.jp fonodera@diana.math.wani.osaka-u.ac.jp