The Burkhardt Group and Modular Forms

Eberhard Freitag Riccardo Salvati Manni^{1} Mathematisches Institut Dipartimento di Matematica, Im Neuenheimer Feld 288 Piazzale Aldo Moro, 2 D69120 Heidelberg I-00185 Roma, Italy. freitag@mathi.uni-heidelberg.de salvati@mat.uniroma1.it

1. Introduction

In this paper we investigate the ring of Siegel modular forms of genus two and level 3. We determine the structure of this ring. It is generated by 10 modular forms (5 of weight 1 and 5 of weight 3) and there are 20 relations (5 in weight 5 and 15 in weight 6). The proof consists of two steps. In a first step we prove that the Satake compactification of the modular variety of genus 2 and level 3 is the normalization of the dual of the Burkhardt quartic.

This is a hypersurface of degree 18 which we describe explicitly. We give also a description of the normalization map and we prove that it is bijective.

The second part consists in the normalization of the Burkhardt dual. Our basic tool is the Burkhardt group G = G_{25 920}, which acts on our varieties. The representation theory of the Burkhardt group was a strong leading guide.

In fact, we first construct an element in the normalization, then using the action of G, we are able to construct other elements, at the end we get a ring contained in the field of fractions that satisfies Serre’s criterion of normality. This ring will result also to be Gorenstein.

Several complicated polynomial identities will occur. It is easy to verify them by means of a computer, but it is very tedious to verify them by hand. But we want to point out that in principle this is possible, because no really expensive algorithms as Groebner bases have to be used. (We used them to find identities, but after one has them they can be easily verified.) Similarly, we used for representation theoretic questions and for polynomial computations respectively the Computer algebra systems GAP, cf. [Ga] and SINGULAR, cf. [Si] . But as before it would be possible to manage the calculations by hand with help of the known character tables ATLAS, cf. [At].

We didn’t reproduce any program in this paper. A reader, who does not rely in our statements could write his own programs or can consult [Fr3].

The ring of modular forms with respect to the group Γ_{0}[3] has been determined by Ibukiyama, cf. [Ib]. His ring is a subring of our ring. We do not need results

1 Research partially supported by Universität Heidelberg while the author was visiting it.

from Ibukiyama’ s paper and can reprove his main results. But we want to point out that his paper was extremely useful for us.

2. A five dimensional space of Siegel modular forms of genus two, weight one and level three

Let Γ_{n} = Sp(2n, ℤ) be the full Siegel modular group of genus n. We denote by

Γ_{n}[q] = Kernel (Sp(2n, ℤ) ⟶ Sp(2n, ℤ/qℤ))

the principal congruence group of level q and by

Γ_{n,0}[q] := \{M ∈ Γ_{n}; M ≡ ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) mod q\}  (1)

the subgroup of Γ_{n} defined by C ≡ 0 mod q. The group

Γ_{2}/Γ_{2}[3] ≅ Sp(4, ℤ/3ℤ)

contains the negative unit matrix −E in its center. The so-called Burkhardt group

G := Sp(4, ℤ/3ℤ)/{±E}

is the finite simple group of order 25, 920.

For a subgroup of finite index Γ ⊂ Γ_{n}, an integer r and a character χ, we consider the space of modular forms [Γ, r, χ]. Its elements are holomorphic functions on the Siegel upper half plane

H_{n} = {Z ∈ M_{n}(ℤ); Z = Z′, Im(Z) > 0}

with the property

f(MZ) = χ(M) det(CZ + D)^{r}f(Z), (MZ = (AZ + B)(CZ + D)^{−1}).

Here M is divided into 4 n × n-blocks as usual.

In the case n = 1 the standard regularity condition at the cusps has to be added.

We write [Γ , r], if the character is trivial.

Let Γ_{0} ⊂ Γ be a normal subgroup of finite index. The group Γ acts on [Γ_{0}, r] by means of

f ⟶ f|M, (f|M)(Z) := det(CZ + D)^{−r}f(MZ)

and this action factors through Γ/Γ_{0}.

We notice that for even n the negative unit-matrix acts trivially.

We consider the graded algebra

A(Γ) := ⨁_{r∈ℤ} [Γ, r].

which is related to the Satake compactification

X(Γ) := \overline{H_{n}/Γ} ≅ Proj(A(Γ)).

Here we use the fundamental theorem of Baily, cf. [BB], which states that A(Γ) is finitely generated and that its associated projective variety is biholomorphic equivalent to the complex space X(Γ).

We are interested in the ring A(Γ_{2}[3]). The Burkhardt group G acts on this ring as well as on X(Γ_{2}[3]). We recall that set theoretically we have

X(Γ_{2}[3]) = H_{2}/Γ_{2}[3] ∪ ⋃_{i=1}^{40} C_{i} ∪ ⋃_{i=1}^{40} P_{i}.  (2)

Here the union is disjoint and we denote with C_{i}, P_{i}, respectively the 1-dimensional and the 0-dimensional cusps. We recall that each C_{i} is biholomorphic to H_{1}/Γ_{1}[3] and we have a (40 , 40)_{4} relation among the cusps. This means that each copy of \overline{C_{i}} ≅ X(Γ_{1}[3]) ≅ ℙ^{1} contains 4 cusps P_{i} and each P_{i} is contained in 4 copies of X(Γ_{1}[3]) .

We introduce the group

Γ_{1,1}[3] := \{M ∈ Γ_{2}; M ≡ ( \begin{matrix}∗ & 0 & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ ∗ & 0 & ∗ & ∗ \\ 0 & 0 & 0 & ∗ \end{matrix} ) mod 3\} .  (3)

Both are subgroups of index 40 in Γ_{2} and we have that the 0-dimensional cusps are stabilized by subgroups conjugate to Γ_{2,0}[3] and the 1-dimensional cusps are stabilized by subgroups conjugate to Γ_{1,1}[3].

We want to introduce certain theta series with respect to the root lattice A_{2}. We use the Gram matrix

S := ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ) .

Next we introduce 5 characteristics

P_{1} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) P_{2} = ( \begin{matrix}0 & 1 \\ 0 & 1 \end{matrix} ) P_{3} = ( \begin{matrix}1 & 0 \\ 1 & 0 \end{matrix} ) P_{4} = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) P_{5} = ( \begin{matrix}1 & −1 \\ 1 & −1 \end{matrix} ) .

They are isotropic in the sense SP ≡ 0 mod 3. We define the theta series

ϑ_{i}(Z) := ∑_{G integral} e (tr(S[(G + P_{i})/3]Z))  (4)

(using the notations S[P] = P′SP and e(t) = exp(πi t)). Easily, from [An] p.24, we deduce the following

Proposition 1. The theta series ϑ_{1}, …, ϑ_{5} are contained in [Γ_{2}[3], 1]. They span a 5-dimensional space V which is invariant under the Burkhardt group and with irreducible action. The action is given by means of the following 4 matrices which generate Γ_{2}:

I = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ −1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) S_{0} = ( \begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

S_{1} = ( \begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) S_{2} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

These four matrices act on the ϑ_{i} by means of the following 5 × 5-matrices in the same ordering:

Ĩ:= ( \begin{matrix}−1/3 & −2/3 & −2/3 & −2/3 & −2/3 \\ −1/3 & 1/3 & −2/3 & 1/3 & 1/3 \\ −1/3 & −2/3 & 1/3 & 1/3 & 1/3 \\ −1/3 & 1/3 & 1/3 & 1/3 & −2/3 \\ −1/3 & 1/3 & 1/3 & −2/3 & 1/3 \end{matrix} ) ,  S̃_{0} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & ω & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & ω & 0 \\ 0 & 0 & 0 & 0 & ω \end{matrix} ) ,

S̃_{1} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & ω^{2} & 0 \\ 0 & 0 & 0 & 0 & ω \end{matrix} ) ,  S̃_{2} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & ω & 0 & 0 \\ 0 & 0 & 0 & ω & 0 \\ 0 & 0 & 0 & 0 & ω \end{matrix} )

with ω := exp(2πi/3). The four tilde-matrices generate a copy G̃ ⊂ GL(5, ℂ) of the Burkardt group. We mention that this group acts on V from the right: To apply an element g ∈ G̃ to α_{1}ϑ_{1} + ⋯ + α_{5}ϑ_{5} ∈ V means to multiply the row (α_{1}, …, α_{5}) from the right with g.

We mention some simple properties of G̃. Complex conjugation defines an automorphism of G̃ which is no inner one. It transforms a representation into its dual representation. We need the formula

g ⟶ ḡ′^{−1} = T gT^{−1} for g ∈ G̃, T = ( \begin{matrix}2 \\ & 1 \\ & & 1 \\ & & & 1 \\ & & & & 1 \end{matrix} ) .

3. The Burkhardt-Coble invariants

We consider the polynomial ring ℂ[A_{1}, …, A_{5}] in five indeterminates. We define the action of the Burkhardt group such that the natural homomorphism

ℂ[A_{1}, …, A_{5}] ⟶ ℂ[ϑ_{1}, …, ϑ_{5}], A_{i} ⟶ ϑ_{i}

is equivariant.

The ring of invariants ℂ[A_{1}, …, A_{5}]^{G} has been determined by Burkhardt. We use Coble’s construction for these invariants, cf. [Co].

The form ϑ_{1} belongs to [Γ_{2,0}[3] , 1 , χ_{S}]. Here χ_{S} is the real Dirichlet character associated to the quadratic form S, we remind, cf. [An] p.26, that χ_{S}^{2} = 1 and χ_{S} = 1 if we restrict to Γ_{2}[3]. This means that the variable A_{1} is ”quasi-invariant” under the image of B ⊂ G of Γ_{2,0}[3]. Following Coble, we define the invariants g_{m} ∈ ℂ[A_{1}, …, A_{5}] by

g_{m} = γ_{m} ∑_{g∈B\G}(A_{1}^{m}) | g (m = 2, 4, 6, …).

The normalizing constants which we use here have the effect that the constant Fourier coefficients of the corresponding modular forms are 1 if it is different from 0. (Otherwise we need no normalizing constant).

The invariants g_{4}, g_{6}, g_{10}, g_{12}, g_{18} are algebraically independent. They are socalled primary invariants, i.e. the full ring of invariants is a finitely generated free module over ℂ[g_{4}, g_{6}, g_{10}, g_{12}, g_{18}]. We are interested in the corresponding modular forms G_{4}, G_{6}, G_{10}, G_{12}, G_{18}.

With some patience or the aid of a computer, some Fourier coefficients can be computed. We write the Fourier coefficient of a modular form as

f(Z) = ∑_{T} a_{f}(T)e(tr(T Z)).

Here T runs through a certain lattice of rational symmetric matrices. In the case of the full modular group or Γ_{2,0}[3] this lattice consists of all integral matrices with even diagonal (called even matrices). Recall that a(T) vanishes if T is not semi-positive.

We start with a list of Fourier coefficients of the thetas ϑ_{i}, 1 ≤ i ≤ 5. Here the matrices T have the property that 3T is even. For the following table we define

a_{i}(T) = a_{ϑ_{i}}(T/3), (T even).

The following Fourier coefficients are in the range

3t_{0} ≤ 12,  t_{2} ≤ 6,  T = ( \begin{matrix}t_{0} & t_{1} \\ t_{1} & t_{2} \end{matrix} ) .

Fourier coefficients in this range vanish when they do not occur in the following list:

a_{1}\binom{00}{00} = 1, a_{1}\binom{00}{06} = 6, a_{1}\binom{60}{00} = 6, a_{1}\binom{6−6}{−6 6} = 6, a_{1}\binom{6−3}{−3 6} = 12, a_{1}\binom{63}{36} = 12, a_{1}\binom{66}{66} = 6.

a_{2}\binom{00}{02} = 3, a_{2}\binom{6−3}{−3 2} = 6, a_{2}\binom{60}{02} = 6, a_{2}\binom{63}{32} = 6.

a_{3}\binom{20}{00} = 3, a_{3}\binom{2−3}{−3 6} = 6, a_{3}\binom{20}{06} = 6, a_{3}\binom{23}{36} = 6, a_{3}\binom{80}{00} = 3, a_{3}\binom{8−6}{−6 6} = 6, a_{3}\binom{80}{06} = 6, a_{3}\binom{86}{66} = 6.

a_{4}\binom{2−1}{−1 2} = 6, a_{4}\binom{22}{22} = 3, a_{4}\binom{8−4}{−4 2} = 3, a_{4}\binom{82}{26} = 6.

a_{5}\binom{2−2}{−2 2} = 3, a_{5}\binom{21}{12} = 6, a_{5}\binom{8−2}{−2 2} = 6, a_{5}\binom{84}{42} = 3.

This table can be used to compute some Fourier coefficients a_{g_{i}}(T) of the invariants:

T  \binom{00}{00} \binom{00}{02}  \binom{2−1}{−1 2}  \binom{20}{02}  \binom{40}{00}  \binom{4−1}{−1 2}
G_{4}  1 240  13440  30240  2160  138240
G_{6}  1 −504  44352  166320  −16632  2128896
G_{10}  1 −264 −\frac{26304}{13}  \frac{965520}{13}  −135432  \frac{92233728}{13}
G_{12}  1 \frac{16272}{61}  \frac{16873344}{2501}  \frac{199422432}{2501}  \frac{11599632}{61}  \frac{20120196096}{2501}
G_{18}  1 \frac{16632}{205} −\frac{105153984}{224065} \frac{1446026256}{224065} −\frac{26449416}{205} −\frac{60302112768}{44813}

4. The Burkhardt dual

The Fourier coefficients of G_{4} and G_{6} are integral. Actually they are Eisenstein series. Inspection of the Fourier coefficients shows that the forms G_{4}G_{6}, G_{10} as well as G_{4}^{3}, G_{6}^{2}, G_{12} are linearly independent. By a well-known theorem of Igusa, cf. [Ig] or [Fr2], the ring of even-weight modular forms of genus two with respect to the full modular group is generated by the forms of weight 4,6,10,12. This means that this ring is contained in ℂ[ϑ_{1}, …, ϑ_{5}]. As a consequence the ring A(Γ_{2}[3]) is integral over ℂ[ϑ_{1}, …, ϑ_{5}]. We claim that both rings have the same field of fractions. It is sufficient to show that they have the same homogeneous field of fractions (=the field generated by quotients of homogeneous elements of the same degree). The homogeneous field of fractions K(Γ) of A(Γ) is nothing else but the field of modular functions. We have the inclusions

K(Γ_{2}) ⊂ K(ϑ_{i}/ϑ_{j}) ⊂ K(Γ_{2}[3]).

It is known that the degree of K(Γ_{2}[3]) over K(Γ_{2}) equals the index of Γ_{2}[3] in Γ_{2} which is the order of the Burkhardt group (25,920). The Burkhardt group also acts non-trivially on K(ϑ_{i}/ϑ_{j}) and this action is faithful because the Burkhardt group is simple. We obtain that the degree of K(ϑ_{i}/ϑ_{j}) over K(Γ_{2}) is greater or equal to 25, 920. This implies the claimed equality. We have proved:

Proposition 2. The graded algebra A(Γ_{2}[3]) is the normalization of the algebra

ℂ[ϑ_{1}, …, ϑ_{5}].

We want to construct this normalization. The first task in this direction is to determine the kernel of

ψ : ℂ[A_{1}, …, A_{5}] ⟶ ℂ[ϑ_{1}, …, ϑ_{5}].  (5)

This is a principal ideal. The modular form G_{18} must be expressible as polynomial in the forms G_{4}, G_{6}, G_{10}, G_{12}. This gives a non-trivial relation and there is no relation of smaller degree. Using the Fourier coefficients above one can compute this relation and obtains:

Proposition 3. The kernel of the natural homomorphism

ψ : ℂ[A_{1}, …, A_{5}] ⟶ ℂ[ϑ_{1}, …, ϑ_{5}]

is generated by the following polynomial of degree 18.

\begin{matrix}-716696985600 & P_{18} := \\ 173650375g_{18} − 67489485g_{6}g_{12} − 205937433g_{4}^{2}g_{10} + 4148960g_{6}^{3} + 95627583g_{4}^{3}g_{6}. \end{matrix}

This polynomial is normalized such that it has integral coprime coefficients. This will turn out to be a very interesting polynomial, a good reason to write it in expanded form:

P_{18}(A_{1}, …, A_{5})  =  2A_{4}^{3}A_{5}^{9}A_{3}^{6} − 2A_{1}^{3}A_{4}^{9}A_{2}^{6} − 2A_{1}^{3}A_{5}^{6}A_{4}^{9} −

18A_{1}A_{5}^{2}A_{4}^{8}A_{2}^{5}A_{3}^{2} − 18A_{1}A_{2}^{5}A_{5}^{8}A_{4}^{2}A_{3}^{2} + 2A_{3}^{3}A_{4}^{9}A_{2}^{6} + A_{5}^{12}A_{2}^{6} + A_{2}^{12}A_{4}^{6} + 2A_{2}^{3}A_{4}^{9}A_{3}^{6} − 2A_{1}^{3}A_{5}^{9}A_{4}^{6}−6A_{1}^{2}A_{4}^{10}A_{5}A_{2}^{4}A_{3}−2A_{1}^{3}A_{2}^{6}A_{5}^{9}−2A_{5}^{9}A_{4}^{9}−2A_{2}^{3}A_{5}^{3}A_{3}^{12}−6A_{5}^{6}A_{2}^{6}A_{3}^{6}+2A_{5}^{3}A_{4}^{6}A_{3}^{9}+ 2A_{2}^{6}A_{5}^{3}A_{3}^{9} + 2A_{2}^{3}A_{5}^{9}A_{3}^{6} + 2A_{5}^{3}A_{4}^{9}A_{3}^{6} + 8A_{1}^{3}A_{2}^{3}A_{5}^{9}A_{3}^{3} + 4A_{5}^{3}A_{4}^{3}A_{1}^{9}A_{3}^{3} − 18A_{2}^{5}A_{5}^{2}A_{4}^{2}A_{3}^{8}A_{1} − 10A_{5}^{3}A_{4}^{3}A_{1}^{6}A_{3}^{6} + A_{1}^{6}A_{4}^{6}A_{5}^{6} + 12A_{1}A_{4}^{2}A_{5}^{2}A_{2}^{11}A_{3}^{2} − 2A_{5}^{3}A_{4}^{3}A_{3}^{12} − 6A_{4}^{6}A_{5}^{6}A_{3}^{6} − 6A_{2}^{6}A_{5}^{6}A_{4}^{6} + 2A_{5}^{6}A_{4}^{3}A_{3}^{9} − 8A_{3}^{3}A_{2}^{3}A_{4}^{6}A_{5}^{6} − 6A_{1}^{2}A_{4}A_{5}A_{2}^{10}A_{3}^{4} − 10A_{2}^{3}A_{5}^{3}A_{1}^{6}A_{3}^{6} − 2A_{3}^{3}A_{5}^{3}A_{4}^{12}+ 4A_{2}^{3}A_{5}^{3}A_{1}^{9}A_{3}^{3} + 2A_{3}^{3}A_{5}^{9}A_{4}^{6} + 8A_{2}^{3}A_{5}^{3}A_{3}^{9}A_{1}^{3} − 12A_{1}^{2}A_{2}A_{5}^{4}A_{4}^{7}A_{3}^{4} − 108A_{1}^{4}A_{4}^{5}A_{5}^{2}A_{2}^{5}A_{3}^{2}+ 8A_{5}^{3}A_{4}^{3}A_{3}^{9}A_{1}^{3} − 12A_{1}^{2}A_{5}^{7}A_{4}A_{2}^{4}A_{3}^{4} − 18A_{1}A_{2}^{8}A_{5}^{5}A_{4}^{2}A_{3}^{2} − 12A_{5}A_{2}^{4}A_{4}^{7}A_{1}^{2}A_{3}^{4} + 18A_{1}^{7}A_{2}^{2}A_{4}^{2}A_{5}^{5}A_{3}^{2} + 72A_{5}^{5}A_{4}^{5}A_{2}^{2}A_{3}^{5}A_{1} − 2A_{5}^{9}A_{3}^{6}A_{1}^{3} − 12A_{1}^{2}A_{4}A_{5}^{4}A_{2}^{7}A_{3}^{4} + 2A_{3}^{3}A_{2}^{9}A_{5}^{6} + 168A_{5}^{3}A_{4}^{6}A_{2}^{3}A_{1}^{3}A_{3}^{3} − 8A_{5}^{6}A_{4}^{3}A_{2}^{3}A_{3}^{6} − 8A_{4}^{3}A_{5}^{3}A_{2}^{6}A_{3}^{6} − 8A_{5}^{3}A_{4}^{6}A_{2}^{3}A_{3}^{6} − 6A_{2}^{6}A_{4}^{6}A_{3}^{6} + 168A_{5}^{6}A_{4}^{3}A_{2}^{3}A_{1}^{3}A_{3}^{3} + 168A_{4}^{3}A_{5}^{3}A_{2}^{6}A_{1}^{3}A_{3}^{3} + A_{5}^{6}A_{3}^{12} − 2A_{2}^{9}A_{1}^{3}A_{3}^{6} − 18A_{4}^{2}A_{5}^{2}A_{2}^{8}A_{1}A_{3}^{5} − 18A_{4}^{2}A_{2}^{2}A_{5}^{8}A_{3}^{5}A_{1} + 72A_{4}^{2}A_{5}^{5}A_{2}^{5}A_{3}^{5}A_{1} − 315A_{4}^{4}A_{5}^{4}A_{2}^{4}A_{1}^{2}A_{3}^{4} − 108A_{1}^{4}A_{5}^{5}A_{4}^{5}A_{2}^{2}A_{3}^{2} − 2A_{4}^{6}A_{1}^{3}A_{3}^{9} + 8A_{3}^{3}A_{4}^{9}A_{2}^{3}A_{5}^{3} − 8A_{3}^{3}A_{4}^{6}A_{5}^{3}A_{2}^{6} + A_{4}^{6}A_{1}^{6}A_{3}^{6} − 2A_{3}^{3}A_{2}^{3}A_{5}^{12} − 6A_{4}A_{1}^{2}A_{2}^{10}A_{5}^{4}A_{3} + 2A_{5}^{3}A_{2}^{9}A_{3}^{6} − 2A_{4}^{9}A_{3}^{9} − 2A_{2}^{3}A_{4}^{3}A_{3}^{12} + 12A_{1}A_{4}^{11}A_{5}^{2}A_{2}^{2}A_{3}^{2} + A_{2}^{12}A_{3}^{6} + A_{4}^{6}A_{3}^{12} + A_{4}^{12}A_{3}^{6} + 8A_{2}^{9}A_{5}^{3}A_{4}^{3}A_{3}^{3} − 18A_{2}^{2}A_{1}A_{4}^{8}A_{5}^{5}A_{3}^{2} − 2A_{3}^{3}A_{2}^{12}A_{5}^{3} + 24A_{2}A_{1}^{2}A_{5}^{7}A_{4}^{7}A_{3} − 12A_{1}^{2}A_{4}^{4}A_{5}^{4}A_{2}^{7}A_{3} − 108A_{2}^{5}A_{5}^{2}A_{4}^{2}A_{1}^{4}A_{3}^{5} + 90A_{1}^{5}A_{4}^{4}A_{5}^{4}A_{2}^{4}A_{3} − 4A_{5}^{6}A_{2}^{6}A_{1}^{3}A_{3}^{3} − 6A_{1}^{2}A_{5}A_{2}^{10}A_{4}^{4}A_{3} + 24A_{1}^{2}A_{4}A_{2}^{7}A_{5}^{7}A_{3} − 6A_{4}^{4}A_{2}A_{5}A_{3}^{10}A_{1}^{2} + 8A_{2}^{3}A_{4}^{9}A_{1}^{3}A_{3}^{3} − 6A_{2}^{4}A_{4}A_{5}A_{1}^{8}A_{3}^{4} + 72A_{4}^{5}A_{5}^{2}A_{2}^{5}A_{3}^{5}A_{1} + 8A_{1}^{3}A_{5}^{9}A_{4}^{3}A_{2}^{3} + 12A_{4}^{4}A_{2}A_{5}A_{1}^{5}A_{3}^{7} + 12A_{1}A_{5}^{11}A_{4}^{2}A_{2}^{2}A_{3}^{2} + 2A_{5}^{6}A_{2}^{3}A_{3}^{9} + 4A_{1}^{9}A_{2}^{3}A_{4}^{3}A_{5}^{3} + 12A_{1}^{5}A_{5}A_{2}^{4}A_{4}^{7}A_{3} + 2A_{4}^{3}A_{5}^{9}A_{2}^{6} − 4A_{2}^{6}A_{4}^{6}A_{1}^{3}A_{3}^{3} − 27A_{1}^{4}A_{4}^{2}A_{5}^{2}A_{2}^{8}A_{3}^{2} − 27A_{1}^{4}A_{4}^{2}A_{2}^{2}A_{5}^{8}A_{3}^{2} − 108A_{1}^{4}A_{4}^{2}A_{5}^{5}A_{2}^{5}A_{3}^{2} + 8A_{1}^{3}A_{4}^{9}A_{2}^{3}A_{5}^{3} + 2A_{4}^{6}A_{2}^{3}A_{3}^{9} − 2A_{2}^{3}A_{4}^{12}A_{3}^{3} − 2A_{5}^{9}A_{3}^{9} − 2A_{2}^{9}A_{3}^{9} + A_{2}^{6}A_{1}^{6}A_{3}^{6} + 8A_{3}^{3}A_{5}^{9}A_{4}^{3}A_{2}^{3} − 2A_{5}^{6}A_{3}^{9}A_{1}^{3}+A_{5}^{12}A_{3}^{6}+8A_{1}^{3}A_{2}^{3}A_{4}^{3}A_{3}^{9}+18A_{1}^{7}A_{2}^{5}A_{5}^{2}A_{4}^{2}A_{3}^{2}−18A_{1}A_{2}^{2}A_{4}^{5}A_{5}^{8}A_{3}^{2}+8A_{4}^{3}A_{2}^{9}A_{1}^{3}A_{3}^{3}− 12A_{1}^{2}A_{5}^{7}A_{4}^{4}A_{2}^{4}A_{3} −10A_{2}^{3}A_{4}^{3}A_{1}^{6}A_{3}^{6} −6A_{1}^{8}A_{2}^{4}A_{4}A_{5}^{4}A_{3} +8A_{2}^{3}A_{4}^{3}A_{5}^{3}A_{3}^{9} −18A_{3}^{2}A_{5}^{2}A_{2}^{8}A_{4}^{5}A_{1} + A_{5}^{6}A_{4}^{12} − 6A_{4}A_{2}A_{5}^{10}A_{1}^{2}A_{3}^{4} + 12A_{1}^{5}A_{4}A_{5}^{4}A_{2}^{7}A_{3} + 12A_{1}^{5}A_{5}^{7}A_{4}A_{2}A_{3}^{4}+90A_{1}^{5}A_{2}^{4}A_{4}A_{5}^{4}A_{3}^{4}+12A_{1}^{5}A_{5}^{7}A_{4}A_{2}^{4}A_{3}+90A_{1}^{5}A_{5}^{4}A_{4}^{4}A_{2}A_{3}^{4}+2A_{5}^{3}A_{4}^{9}A_{2}^{6}− 2A_{5}^{3}A_{4}^{12}A_{2}^{3}−6A_{1}^{8}A_{5}^{4}A_{4}^{4}A_{2}A_{3}+A_{1}^{6}A_{2}^{6}A_{4}^{6}−2A_{5}^{12}A_{4}^{3}A_{3}^{3}−4A_{1}^{3}A_{5}^{6}A_{4}^{3}A_{3}^{6}+12A_{1}^{5}A_{2}A_{5}^{4}A_{4}^{7}A_{3}+ 2A_{2}^{9}A_{5}^{6}A_{4}^{3} + 24A_{5}A_{1}^{2}A_{2}^{7}A_{4}^{7}A_{3} + 2A_{3}^{3}A_{2}^{9}A_{4}^{6} − 6A_{1}^{2}A_{3}A_{2}A_{4}^{10}A_{5}^{4} − 10A_{1}^{6}A_{2}^{6}A_{4}^{3}A_{3}^{3} + 18A_{1}^{7}A_{4}^{5}A_{5}^{2}A_{2}^{2}A_{3}^{2} − 10A_{1}^{6}A_{5}^{3}A_{4}^{6}A_{3}^{3} + 12A_{1}^{5}A_{4}^{4}A_{5}A_{2}^{7}A_{3} + 12A_{1}^{5}A_{2}A_{4}^{4}A_{5}^{7}A_{3} + 12A_{1}^{5}A_{2}^{7}A_{5}A_{3}^{4}A_{4} −2A_{1}^{3}A_{2}^{9}A_{4}^{6} −3A_{1}^{10}A_{5}^{2}A_{2}^{2}A_{4}^{2}A_{3}^{2} −6A_{1}^{8}A_{4}^{4}A_{2}^{4}A_{5}A_{3} −6A_{1}^{2}A_{5}^{10}A_{2}^{4}A_{4}A_{3} − 27A_{1}^{4}A_{5}^{2}A_{2}^{2}A_{4}^{2}A_{3}^{8} − 6A_{1}^{8}A_{5}^{4}A_{2}A_{4}A_{3}^{4} + 2A_{2}^{9}A_{5}^{3}A_{4}^{6} − 4A_{5}^{6}A_{2}^{3}A_{3}^{6}A_{1}^{3} − 12A_{5}^{4}A_{4}^{4}A_{2}A_{3}^{7}A_{1}^{2} + A_{5}^{6}A_{1}^{6}A_{3}^{6} − 4A_{2}^{6}A_{5}^{3}A_{3}^{6}A_{1}^{3} − 2A_{4}^{9}A_{3}^{6}A_{1}^{3} + 18A_{5}^{2}A_{2}^{2}A_{4}^{2}A_{1}^{7}A_{3}^{5} − 4A_{5}^{3}A_{4}^{6}A_{1}^{3}A_{3}^{6} + A_{2}^{6}A_{3}^{12} + 8A_{1}^{3}A_{2}^{9}A_{5}^{3}A_{4}^{3} − 8A_{2}^{6}A_{5}^{6}A_{4}^{3}A_{3}^{3} − 10A_{1}^{6}A_{5}^{6}A_{4}^{3}A_{3}^{3} − 2A_{2}^{12}A_{4}^{3}A_{3}^{3} + 8A_{1}^{3}A_{5}^{3}A_{2}^{9}A_{3}^{3} + 24A_{2}^{7}A_{5}A_{3}^{7}A_{4}A_{1}^{2} − 18A_{2}^{2}A_{5}^{2}A_{4}^{8}A_{3}^{5}A_{1} − 12A_{2}A_{4}^{4}A_{5}^{7}A_{1}^{2}A_{3}^{4} − 6A_{2}A_{5}A_{4}^{10}A_{1}^{2}A_{3}^{4} + 12A_{5}^{4}A_{2}A_{4}A_{1}^{5}A_{3}^{7} + 12A_{2}^{4}A_{4}A_{5}A_{1}^{5}A_{3}^{7} − 18A_{5}^{5}A_{2}^{2}A_{4}^{2}A_{3}^{8}A_{1} − 10A_{1}^{6}A_{2}^{6}A_{5}^{3}A_{3}^{3} + 24A_{4}^{7}A_{5}A_{2}A_{3}^{7}A_{1}^{2} − 4A_{4}^{6}A_{2}^{3}A_{3}^{6}A_{1}^{3} + 24A_{5}^{7}A_{4}A_{2}A_{3}^{7}A_{1}^{2} − 12A_{2}^{4}A_{4}A_{5}^{4}A_{3}^{7}A_{1}^{2} + 2A_{5}^{6}A_{4}^{9}A_{3}^{3} − 6A_{2}^{4}A_{4}A_{5}A_{3}^{10}A_{1}^{2} − 18A_{4}^{5}A_{5}^{2}A_{2}^{2}A_{1}A_{3}^{8} − 108A_{4}^{5}A_{5}^{2}A_{2}^{2}A_{1}^{4}A_{3}^{5} − 6A_{5}^{4}A_{2}A_{4}A_{1}^{2}A_{3}^{10} + 168A_{2}^{3}A_{4}^{3}A_{5}^{3}A_{1}^{3}A_{3}^{6} − 10A_{1}^{6}A_{5}^{6}A_{2}^{3}A_{3}^{3} − 12A_{4}^{4}A_{2}^{4}A_{5}A_{1}^{2}A_{3}^{7} + 90A_{4}^{4}A_{2}^{4}A_{5}A_{1}^{5}A_{3}^{4} − 10A_{1}^{6}A_{4}^{6}A_{2}^{3}A_{3}^{3} + 4A_{1}^{9}A_{2}^{3}A_{4}^{3}A_{3}^{3} − 4A_{2}^{6}A_{4}^{3}A_{3}^{6}A_{1}^{3} + A_{5}^{6}A_{2}^{12} + A_{1}^{6}A_{5}^{6}A_{2}^{6} + 8A_{1}^{3}A_{5}^{3}A_{4}^{9}A_{3}^{3} − 12A_{4}^{4}A_{5}A_{2}^{7}A_{1}^{2}A_{3}^{4} − 108A_{1}^{4}A_{5}^{5}A_{2}^{2}A_{4}^{2}A_{3}^{5} + 12A_{5}^{2}A_{2}^{2}A_{4}^{2}A_{3}^{11}A_{1} + 12A_{1}^{5}A_{4}^{7}A_{5}A_{2}A_{3}^{4} − 2A_{4}^{3}A_{5}^{3}A_{2}^{12} − 4A_{1}^{3}A_{4}^{6}A_{5}^{3}A_{2}^{6} − 2A_{2}^{9}A_{5}^{9} − 4A_{1}^{3}A_{2}^{3}A_{4}^{6}A_{5}^{6} + A_{4}^{12}A_{2}^{6} − 2A_{2}^{9}A_{4}^{9} + A_{5}^{12}A_{4}^{6} − 4A_{1}^{3}A_{4}^{3}A_{2}^{6}A_{5}^{6} − 4A_{1}^{3}A_{4}^{6}A_{5}^{6}A_{3}^{3} + 2A_{2}^{3}A_{4}^{9}A_{5}^{6} + 2A_{5}^{9}A_{4}^{6}A_{2}^{3} − 2A_{1}^{3}A_{2}^{9}A_{5}^{6} − 2A_{4}^{3}A_{5}^{12}A_{2}^{3} − 10A_{1}^{6}A_{4}^{3}A_{5}^{3}A_{2}^{6} − 10A_{1}^{6}A_{5}^{6}A_{4}^{3}A_{2}^{3} − 10A_{1}^{6}A_{5}^{3}A_{4}^{6}A_{2}^{3} − 6A_{1}^{2}A_{3}A_{2}A_{5}^{10}A_{4}^{4} − 96A_{1}^{6}A_{2}^{3}A_{4}^{3}A_{5}^{3}A_{3}^{3} + 2A_{2}^{6}A_{4}^{3}A_{3}^{9}+2A_{3}^{3}A_{2}^{6}A_{5}^{9}−27A_{1}^{4}A_{2}^{2}A_{5}^{2}A_{4}^{8}A_{3}^{2}−2A_{1}^{3}A_{2}^{6}A_{3}^{9}−6A_{1}^{8}A_{4}^{4}A_{2}A_{5}A_{3}^{4}+8A_{1}^{3}A_{4}^{3}A_{5}^{9}A_{3}^{3}+ 2A_{4}^{3}A_{2}^{9}A_{3}^{6} + 72A_{1}A_{2}^{5}A_{5}^{5}A_{4}^{5}A_{3}^{2} − 12A_{1}^{2}A_{3}A_{2}^{4}A_{4}^{7}A_{5}^{4}.

There is a much better description of the polynomial P_{18}. It is connected with the famous Burkhardt’s quartic polynomial

g_{4} = A_{1}^{4} + 8A_{1}A_{2}^{3} + 8A_{1}A_{3}^{3} + 8A_{1}A_{4}^{3} + 8A_{1}A_{5}^{3} + 48A_{2}A_{3}A_{4}A_{5}  (6)

defining the Burkhardt quartic B, we recall that this variety has 45 double points, it is rational and is birational to X(Γ_{2}[3]), for these and other data we refer to the monographies [Ba], and [Hu] and to the paper [Ge].

The polynomial P_{18} is essentially the dual of the quartic. Recall that the dual Q of an irreducible homogeneous polynomial P (X_{1}, …, X_{n}) is defined by

Q(grad P) ≡ 0 mod P.

Geometrically it describes the set of all tangent hyperplanes. Up to some exceptional cases, the dual of a polynomial is unique up to a constant factor and the dual of the dual is the polynomial itself. The dual of g_{4} is also an invariant polynomial, but invariant under the dual representation, i.e.invariant under the group of all (g′)^{−1}, g ∈ G. It is a real polynomial. Therefore it is also invariant under all ḡ′^{−1}. But as we mentioned, this group is conjugate to G, where the conjugation map is given by doubling the first variable A_{1}.

Proposition 4. If one takes the dual polynomial of Burkhardt’s quartic polynomial g_{4} and replaces A_{1} by 2A_{1} one obtains (up to a constant factor) the polynomial P_{18}. The proof is done by straightforward calculation. We set

A := ℂ[A_{1}, …, A_{5}]/(P_{18})  (7)

and we shall denote the projective variety defined by the vanishing of P_{18} with B^{∨} and we call it simply the Burkhardt dual,

B^{∨} := Proj(A).  (8)

Hunt [Hu] predicted, using general results of invariant theory, that the degree of the dual of the Burkhardt quartic is 18. He also made some comments about the singular locus of the dual. Actually this singular locus can be determined completely.

5. The singular locus of the Burkhardt dual

We denote by Δ the diagonal in H_{2}, i.e.

Δ := \{Z ∈ H_{2}; Z = ( \begin{matrix}τ_{1} & 0 \\ 0 & τ_{2} \end{matrix} )\} .

For a subgroup Γ ⊂ Γ_{2} of finite index denote the natural projection by

π_{Γ} : H_{2} ⟶ H_{2}/Γ.

The image π_{Γ}(Δ) is an irreducible subvariety of H_{2}/Γ. We use also the notations

Δ[3] := π_{Γ_{2}[3]}(Δ) and Δ_{0}[3] := π_{Γ_{2,0}[3]}(Δ).

We are interested in

GΔ[3] = ⋃_{g∈G} g(Δ[3])

and in its closure G\overline{Δ[3]}.

We have to determine the normalizer of Δ[3] in G. This normalizer N_{G}(Δ[3]) contains the direct product of two copies of SL(2ℤ/3ℤ/{±E}. The whole normalizer is an extension of index two of the direct product of the two copies. This shows

[G : N_{G}(Δ[3])] = 45.

Therefore GΔ[3] ⊂ X(Γ_{2}[3]) has 45 irreducible components, that result to be disjoint.

We denote by H the image of Γ_{1,1}[3] in G. The subgroups N_{G}(Δ[3]), B, H are maximal in G , their orders are respectively 576, 648, 648 and they have been extensively studied by Burkhardt, cf. [Bu].

We recall that \overline{Δ[3]} contains [N_{G}(Δ[3]) : (N_{G}(Δ[3]) ∩ H)] = 8 boundary components of the type C_{i}, viceversa each C_{i} is contained in [H : (N_{G}(Δ[3]) ∩ H)] = 9 irreducible boundary components. Similarly \overline{Δ[3]} contains [N_{G}(Δ[3]) : (N_{G}(Δ[3]) ∩ B)] = 16, viceversa each cusps P_{i} is contained in B : (N_{G}(Δ[3])∩B)] = 18 irreducible boundary components.

We know that the intersection of two closed components of the diagonal locus consists either of two of the 45 ℙ^{1} or of the four intersection points of two boundary components. (There are no intersection points away from the boundary.)

Proposition 5. The singular locus of the Burkhardt dual has 45 irreducible components. Each of them has codimension one. The Burkhardt group permutes the components transitively. One of the components is defined (as set) by the zeros of the ideal

𝔭 := (A_{5} − A_{4}, A_{1}A_{4} − A_{2}A_{3}) ⊂ A.  (9)

This component equals the closure of the image of the diagonal under the natural maps

H_{2} ⟶ X(Γ_{2}[3]) ⟶ Proj(A).

The whole singular locus can be described also (as set) as the zero locus of the principal ideal

(P_{10}) ⊂ A, where P_{10} := \frac{13}{777600} (g_{4}g_{6} − g_{10}).

The normalizing constant has been chosen such that the Fourier coefficient of the corresponding modular form with respect to \binom{21}{12} is 1.

Proof. By a classical result, cf. [Ig1] or [Fr1] the modular form χ_{10}, corresponding to P_{10}, vanishes at the 45 diagonal components and has no other zeros. Under the normalization map

ϕ : X(Γ_{2}[3]) ⟶ Proj(A)  (10)

the 45 diagonal components map to 45 different irreducible subvarieties in the variety Proj(A). This follows from the maximality of B in the Burkhardt group. (The only alternative that all 45 collapse to one can be excluded easily).

A direct computation shows that P_{10} and P_{18} are contained in the ideal 𝔭 and, consequently, also in the 45 conjugate ideals. They all are prime ideals. Hence they define in A the minimal prime ideals containing P_{10}.

Another direct computation shows that the squares of the partial derivatives of P_{18} are contained in the ideal (P_{18} , P_{10}) and finally that P_{10}^{3} is contained in the ideal generated by P_{18} and the first of the partial derivatives.. This proves the proposition.

The proof showed that the first partial derivative of P_{18} has the same zero sets (but possibly with different multiplicities) as P_{10} (considered in A). We will use this later.

So the map ϕ results to be biholomorphic outside the locus in the variety Proj(A) defined by P_{10} = 0. We want to study in details the map along this locus.

The maximality of the subgroups N_{G}(Δ[3]), B, H and the G-equivariance of the map ϕ imply that the singular locus of the Burkhardt’s dual not only has 45 irreducible components, but it has also 40 1-dimensional subvarieties corresponding to the boundary components C_{i} and 40 points corresponding to the cusps P_{i} ( also in these cases, the only alternatives would be that all components collapse to one, but this can be excluded easily).

In a first step we claim that the restriction of ϕ to one of the 45 irreducible components is injective. For example we restrict to \overline{Δ[3]}, the closure of the image of the diagonal in the Satake compactification.

For an accurate description of the map ϕ, we need to recall some fact about modular forms of genus 1. We introduce 2 characteristics

v_{1} = ( \begin{matrix}0 \\ 0 \end{matrix} ) v_{2} = ( \begin{matrix}1 \\ 1 \end{matrix} )

and we define the theta series of genus 1

θ_{i}(τ) := ∑_{G integral} e (tr(S[(G + v_{i})/3]τ)) .

We recall that θ_{1} and θ_{2} generate the ring of modular forms A(Γ_{1}[3]), this is a consequence of the dimension formula given, for example, in [Mi] p.60. If we denote by ϕ̃ the restriction of ϕ to \overline{Δ[3]}, we get that

ϕ̃: \overline{Δ[3]} ⟶ ℙ^{1} × ℙ^{1} ⟶ ℙ^{4}  (11)

(τ_{1}, τ_{2}) ⟶ (θ_{1}(τ_{1})θ_{1}(τ_{2}), θ_{1}(τ_{1})θ_{2}(τ_{2}), θ_{2}(τ_{1})θ_{1}(τ_{2}), θ_{2}(τ_{1})θ_{2})(τ_{2}), θ_{2}(τ_{1})θ_{2})(τ_{2}))

is bijective.

We have proved now that \overline{Δ[3]} maps bijectively to the zero locus of 𝔭 in the Burkhardt dual. To get the bijectivity we need some information about there intersection behavior.

Let us consider the standard 1-dimensional boundary component C, which is the one containing

lim_{y→∞} \binom{τ 0}{0 iy},

we have the equation of its image is A_{3} = A_{4} = A_{5} = 0.

Moreover the equation of the image of the so called cusp

∞ := lim_{y→∞} \binom{iy 0}{0 iy}

is A_{2} = A_{3} = A_{4} = A_{5} = 0.

We have to prove that the 45 components of the singular locus of the Burkhardt dual have the same intersection behavior as the corresponding components in the Satake compactification. It is easy to compute the 45 ideals conjugate to 𝔭 and to verify the intersection behavior, since the images of the boundary components in the Burkhardt dual are given by linear equations. We will not reproduce this calculation here. We only remind that the polynomial P_{18} is contained already in (A_{4} − A_{5}, A_{1}A_{4} − A_{2}A_{3}) ⊂ ℂ[A_{1}, …, A_{5}]. Hence the calculations have to be performed in the polynomial ring ℂ[A_{1}, …, A_{5}] (and not in the factor ring A).

Theorem 1. The normalization map

ϕ : X(Γ_{2}[3]) ⟶ Proj(A)

is bijective.

6. A basic modular Form of weight three

We have to investigate the action of B in the Burkhardt group G on the 45 diagonal components. From the results of the previous section, it doesn’t make a difference whether we consider it in Proj(A) or in the normalization.

Proposition 6. The set of the 45 components of GΔ[3] decomposes into two orbits under B. The orbit BΔ[3] contains 18 components. Consequently the other orbit contains 27 components. Proof. This is an immediate consequence of the following well known facts

G = B · N_{G}(Δ[3] ∪ B · I · N_{G}(Δ[3],  and  [B : N_{G}(Δ[3]) ∩ B] = 18.

Thus BΔ[3] contains 18 components and other orbit contains 27 components.

We now consider the polynomial of degree 17

\frac{∂P_{18}}{∂A_{1}} ∈ ℂ[A_{1}, …, A_{5}].

This is a B-invariant polynomial. As we mentioned after the proof of proposition 5 this polynomial has the same zero set as P_{10}. This leads us to consider

F := \frac{P_{10}^{2}}{∂P_{18}/∂A_{1}}

(in the field of fractions of A). What we found is an element of the normalization! Theorem 2. One has the following integral equations in A

F^{2} = P_{6}, F^{3} = P_{9}

with the polynomials P_{6}(A_{1}, …, A_{5}) := 2^{−4}3^{−18}· (A_{1}^{6}−2A_{1}^{3}A_{2}^{3}+A_{2}^{6}−2A_{1}^{3}A_{3}^{3}−A_{2}^{3}A_{3}^{3}+A_{3}^{6}−2A_{1}^{3}A_{4}^{3}−A_{2}^{3}A_{4}^{3}−A_{3}^{3}A_{4}^{3}+A_{4}^{6}+9A_{1}^{2}A_{2}A_{3}A_{4}A_{5}− 2A_{1}^{3}A_{5}^{3} − A_{2}^{3}A_{5}^{3} − A_{3}^{3}A_{5}^{3} − A_{4}^{3}A_{5}^{3} + A_{5}^{6})

P_{9}(A_{1}, …, A_{5}) := −2^{−7}3^{−27}· (2A_{1}^{9} − 6A_{1}^{6}A_{2}^{3} + 6A_{1}^{3}A_{2}^{6} − 2A_{2}^{9} − 6A_{1}^{6}A_{3}^{3} + 3A_{1}^{3}A_{2}^{3}A_{3}^{3} + 3A_{2}^{6}A_{3}^{3} + 6A_{1}^{3}A_{3}^{6} + 3A_{2}^{3}A_{3}^{6} − 2A_{3}^{9} − 6A_{1}^{6}A_{4}^{3} + 3A_{1}^{3}A_{2}^{3}A_{4}^{3} + 3A_{2}^{6}A_{4}^{3} + 3A_{1}^{3}A_{3}^{3}A_{4}^{3} − 12A_{2}^{3}A_{3}^{3}A_{4}^{3} + 3A_{3}^{6}A_{4}^{3} + 6A_{1}^{3}A_{4}^{6} + 3A_{2}^{3}A_{4}^{6} + 3A_{3}^{3}A_{4}^{6} − 2A_{4}^{9} + 27A_{1}^{5}A_{2}A_{3}A_{4}A_{5} − 27A_{1}^{2}A_{2}^{4}A_{3}A_{4}A_{5} − 27A_{1}^{2}A_{2}A_{3}^{4}A_{4}A_{5} − 27A_{1}^{2}A_{2}A_{3}A_{4}^{4}A_{5} + 81A_{1}A_{2}^{2}A_{3}^{2}A_{4}^{2}A_{5}^{2} − 6A_{1}^{6}A_{5}^{3} + 3A_{1}^{3}A_{2}^{3}A_{5}^{3} + 3A_{2}^{6}A_{5}^{3} + 3A_{1}^{3}A_{3}^{3}A_{5}^{3} − 12A_{2}^{3}A_{3}^{3}A_{5}^{3} + 3A_{3}^{6}A_{5}^{3} + 3A_{1}^{3}A_{4}^{3}A_{5}^{3} − 12A_{2}^{3}A_{4}^{3}A_{5}^{3} − 12A_{3}^{3}A_{4}^{3}A_{5}^{3} + 3A_{4}^{6}A_{5}^{3} − 27A_{1}^{2}A_{2}A_{3}A_{4}A_{5}^{4} + 6A_{1}^{3}A_{5}^{6} + 3A_{2}^{3}A_{5}^{6} + 3A_{3}^{3}A_{5}^{6} + 3A_{4}^{3}A_{5}^{6} − 2A_{5}^{9}). There is another representation of F with the G-invariant denominator P_{10}, namely

F = \frac{P_{13}}{P_{10}} where

P_{13}(A_{1}, …, A_{5}) := 2^{−3}3^{−13}· (A_{1}^{7}A_{2}^{3}A_{3}^{3} − 2A_{1}^{4}A_{2}^{6}A_{3}^{3} + A_{1}A_{2}^{9}A_{3}^{3} − 2A_{1}^{4}A_{2}^{3}A_{3}^{6} − 4A_{1}A_{2}^{6}A_{3}^{6} + A_{1}A_{2}^{3}A_{3}^{9} + A_{1}^{7}A_{2}^{3}A_{4}^{3} − 2A_{1}^{4}A_{2}^{6}A_{4}^{3} + A_{1}A_{2}^{9}A_{4}^{3} + A_{1}^{7}A_{3}^{3}A_{4}^{3} − 15A_{1}^{4}A_{2}^{3}A_{3}^{3}A_{4}^{3} + 2A_{1}A_{2}^{6}A_{3}^{3}A_{4}^{3} − 2A_{1}^{4}A_{3}^{6}A_{4}^{3} + 2A_{1}A_{2}^{3}A_{3}^{6}A_{4}^{3}+A_{1}A_{3}^{9}A_{4}^{3}−2A_{1}^{4}A_{2}^{3}A_{4}^{6}−4A_{1}A_{2}^{6}A_{4}^{6}−2A_{1}^{4}A_{3}^{3}A_{4}^{6}+2A_{1}A_{2}^{3}A_{3}^{3}A_{4}^{6}−4A_{1}A_{3}^{6}A_{4}^{6}+ A_{1}A_{2}^{3}A_{4}^{9} + A_{1}A_{3}^{3}A_{4}^{9} − A_{1}^{9}A_{2}A_{3}A_{4}A_{5} + 3A_{1}^{3}A_{2}^{7}A_{3}A_{4}A_{5} − 2A_{2}^{10}A_{3}A_{4}A_{5} + 33A_{1}^{3}A_{2}^{4}A_{3}^{4}A_{4}A_{5} + 3A_{2}^{7}A_{3}^{4}A_{4}A_{5} + 3A_{1}^{3}A_{2}A_{3}^{7}A_{4}A_{5} + 3A_{2}^{4}A_{3}^{7}A_{4}A_{5} − 2A_{2}A_{3}^{10}A_{4}A_{5} + 33A_{1}^{3}A_{2}^{4}A_{3}A_{4}^{4}A_{5} + 3A_{2}^{7}A_{3}A_{4}^{4}A_{5} + 33A_{1}^{3}A_{2}A_{3}^{4}A_{4}^{4}A_{5} − 12A_{2}^{4}A_{3}^{4}A_{4}^{4}A_{5} + 3A_{2}A_{3}^{7}A_{4}^{4}A_{5} + 3A_{1}^{3}A_{2}A_{3}A_{4}^{7}A_{5} + 3A_{2}^{4}A_{3}A_{4}^{7}A_{5} + 3A_{2}A_{3}^{4}A_{4}^{7}A_{5} − 2A_{2}A_{3}A_{4}^{10}A_{5} − 54A_{1}^{2}A_{2}^{5}A_{3}^{2}A_{4}^{2}A_{5}^{2} − 54A_{1}^{2}A_{2}^{2}A_{3}^{5}A_{4}^{2}A_{5}^{2} − 54A_{1}^{2}A_{2}^{2}A_{3}^{2}A_{4}^{5}A_{5}^{2} + A_{1}^{7}A_{2}^{3}A_{5}^{3} − 2A_{1}^{4}A_{2}^{6}A_{5}^{3} + A_{1}A_{2}^{9}A_{5}^{3} + A_{1}^{7}A_{3}^{3}A_{5}^{3} − 15A_{1}^{4}A_{2}^{3}A_{3}^{3}A_{5}^{3} + 2A_{1}A_{2}^{6}A_{3}^{3}A_{5}^{3} − 2A_{1}^{4}A_{3}^{6}A_{5}^{3} + 2A_{1}A_{2}^{3}A_{3}^{6}A_{5}^{3} + A_{1}A_{3}^{9}A_{5}^{3} + A_{1}^{7}A_{4}^{3}A_{5}^{3} − 15A_{1}^{4}A_{2}^{3}A_{4}^{3}A_{5}^{3} + 2A_{1}A_{2}^{6}A_{4}^{3}A_{5}^{3} − 15A_{1}^{4}A_{3}^{3}A_{4}^{3}A_{5}^{3} + 93A_{1}A_{2}^{3}A_{3}^{3}A_{4}^{3}A_{5}^{3} + 2A_{1}A_{3}^{6}A_{4}^{3}A_{5}^{3} − 2A_{1}^{4}A_{4}^{6}A_{5}^{3} +2A_{1}A_{2}^{3}A_{4}^{6}A_{5}^{3} +2A_{1}A_{3}^{3}A_{4}^{6}A_{5}^{3} +A_{1}A_{4}^{9}A_{5}^{3} +33A_{1}^{3}A_{2}^{4}A_{3}A_{4}A_{5}^{4} +3A_{2}^{7}A_{3}A_{4}A_{5}^{4} + 33A_{1}^{3}A_{2}A_{3}^{4}A_{4}A_{5}^{4} − 12A_{2}^{4}A_{3}^{4}A_{4}A_{5}^{4} + 3A_{2}A_{3}^{7}A_{4}A_{5}^{4} + 33A_{1}^{3}A_{2}A_{3}A_{4}^{4}A_{5}^{4} − 12A_{2}^{4}A_{3}A_{4}^{4}A_{5}^{4} − 12A_{2}A_{3}^{4}A_{4}^{4}A_{5}^{4} +3A_{2}A_{3}A_{4}^{7}A_{5}^{4} −54A_{1}^{2}A_{2}^{2}A_{3}^{2}A_{4}^{2}A_{5}^{5} −2A_{1}^{4}A_{2}^{3}A_{5}^{6} −4A_{1}A_{2}^{6}A_{5}^{6} −2A_{1}^{4}A_{3}^{3}A_{5}^{6} + 2A_{1}A_{2}^{3}A_{3}^{3}A_{5}^{6} − 4A_{1}A_{3}^{6}A_{5}^{6} − 2A_{1}^{4}A_{4}^{3}A_{5}^{6} + 2A_{1}A_{2}^{3}A_{4}^{3}A_{5}^{6} + 2A_{1}A_{3}^{3}A_{4}^{3}A_{5}^{6} − 4A_{1}A_{4}^{6}A_{5}^{6} + 3A_{1}^{3}A_{2}A_{3}A_{4}A_{5}^{7} + 3A_{2}^{4}A_{3}A_{4}A_{5}^{7} + 3A_{2}A_{3}^{4}A_{4}A_{5}^{7} + 3A_{2}A_{3}A_{4}^{4}A_{5}^{7} + A_{1}A_{2}^{3}A_{5}^{9} + A_{1}A_{3}^{3}A_{5}^{9} + A_{1}A_{4}^{3}A_{5}^{9} − 2A_{2}A_{3}A_{4}A_{5}^{10}).

An easy computation tells us

Proposition 7. The zero locus of F consists of the B-orbit of 27 diagonal components. The multiplicities —counted in H_{2}— are one. Proof. Since we have only 2 possibilities, it is enough to check that the point (1, 0, …, 0) is not contained in the zero locus of F. This is easily verified taking the representation F = P_{9}/P_{6}.

We obtain more elements of the normalization, when we apply the Burkhardt group to F (and we will show that they generate the normalization). Because F is invariant under B it is one of 40 conjugate elements. It can be shown that they are linearly independent. Computations of this type can be easily managed be means of a list of special points of the Burkhardt dual. Such a list can be constructed using the classical unirationalization of the Burkhardt quartic (zero locus of g_{4}), cf. [Co],

\begin{aligned}y_{1} & = 3x_{1}x_{2}x_{3}x_{4} \\ y_{2} & = x_{1}(x_{2}^{3} + x_{3}^{3} − x_{4}^{3}) \\ y_{3} & = −x_{2}(x_{1}^{3} + x_{3}^{3} + x_{4}^{3}) \\ y_{4} & = x_{3}(−x_{1}^{3} + x_{2}^{3} + x_{4}^{3}) \\ y_{5} & = x_{4}(x_{1}^{3} + x_{2}^{3} − x_{3}^{3}). \end{aligned}  (12)

The partial derivatives of g_{4} evaluated at points (2y_{1}, y_{2}, y_{3}, y_{4}, y_{5}) define points of the Burkhardt dual (zero locus of P_{18}). It is not difficult to compute the values of the 40 conjugate forms at such a special points. It is possible to find 40 different points such that the corresponding 40 × 40-matrix is non-singular. Hence the conjugates of F span a 40-dimensional space which is invariant under G. But this representation is not irreducible. Its character can be computed and it turns out that it contains a 5-dimensional sub-representation. Both 5-dimensional representations of the Burkhardt group contain a B-invariant element. This element defines a modular form in [Γ_{2,0}[3], 3, χ_{S}]. It is known that the space of these forms has dimension 3. Fortunately we have already such three forms, namely ψ_{1} := F (ϑ_{1}, …, ϑ_{5}), ϑ_{1}^{3} and ϑ_{2}^{3} + ϑ_{3}^{3} + ϑ_{4}^{3} + ϑ_{5}^{3}.

What we have seen is that there exist constants α, β such that the forty conjugate forms of

F + αA_{1}^{3} + β(A_{2}^{3} + A_{3}^{4} + A_{4}^{3} + A_{5}^{3})

span a five dimensional space. A numerical calculation shows that there is exactly one pair of numbers α, β with this property. In this way we obtain:

Proposition 8. The element

C_{1} := −2^{5}3^{9}F − 7A_{1}^{3} + 4(A_{2}^{3} + A_{3}^{3} + A_{4}^{3} + A_{5}^{3})  (13)

is contained in a five dimensional representation space of G. It is the unique (up to a constant factor) B-invariant element of this space. Here are some Fourier coefficients of C_{1}(ϑ_{1}, …, ϑ_{5}):

T  \binom{00}{00} \binom{00}{02} \binom{2−1}{−1 2} \binom{20}{02}  \binom{40}{00} \binom{4−1}{−1 2}
C_{1}  1 −90 3060 2160 −216 4320

We conclude this section observing that we can express the above modular forms as linear combinations of theta series. In fact we can consider the theta series related to the lattices E_{6} and E_{6}^{∗}(3).Looking at the Fourier coefficients we get

ϑ_{E_{6}} = ϑ_{1}^{3} + 2(ϑ_{2}^{3} + ϑ_{3}^{3} + ϑ_{4}^{3} + ϑ_{5}^{3})

and

27ϑ_{E_{6}^{∗}(3)} = 25ϑ_{1}^{3} + 2C_{1}(ϑ_{1}, …, ϑ_{5}) − 10(ϑ_{2}^{3} + ϑ_{3}^{3} + ϑ_{4}^{3} + ϑ_{5}^{3}).

7. A five dimensional space of modular forms of weight 3

We mentioned already that G has two 5-dimensional representations, and that are dual. Hence the five dimensional space, which contains C_{1} could be isomorphic to the space generated by the A-s or isomorphic to its dual. One computes that the first case happens and obtains:

Proposition 9. The element C_{1} is part of a tuple (C_{1}, …, C_{5}) with the same transformation law as the elements (A_{1}, …, A_{5}). This defines C_{2}, …, C_{5}. We derive an explicit representation for the C_{i} as linear combinations of transformed of F. Using the notations of the above theorem, we introduce 5 elements of the Burkhardt group

g_{1} = unit-matrix, g_{2} = Ĩ, g_{3} = ĨS̃_{0}, g_{4} = ĨS̃_{1}, g_{5} = ĨS̃_{2}.

We use

C := ( C_{1} ; ⋮ ; C_{5} )  ,  G = ( C_{1}^{g_{1}} ; ⋮ ; C_{1}^{g_{5}} ) .

Finally we introduce the matrix

T = ( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ −1/2 & 1/2ω − 1/2 & 0 & −1 − 1/2ω & 0 \\ −1/2 & 1/2ω − 1/2 & −1 − 1/2ω & 0 & 0 \\ 1/2 + 1/2ω & 1 + 1/2ω & 1/2 + ω & 1/2 + ω & −1/2 − ω \\ −1/2ω & −3/2 − 3/2ω & −1/2ω + 1/2 & −1/2ω + 1/2 & 1/2 + ω \end{matrix} )

Remark. One has

C = T · G.  (14)

8. Relations

We have to treat the relations between A_{1}, …, C_{5}. There must be a B- invariant relation in weight 5 as follows already from the dimension formula, dim[Γ_{2,0}[3], 5, χ_{S}] = 4.

This relation can be computed by means of the technique of special points and it can be transformed under the Burkhardt group. It turns out that it generates a five dimensional G-invariant space of relations.

Here is a basis:

Proposition 10. The space of relations in weight 5 is generated by

( \begin{matrix}A_{1}^{2} & 2A_{2}^{2} & 2A_{3}^{2} & 2A_{4}^{2} & 2A_{5}^{2} \\ A_{2}^{2} & 2A_{1}A_{2} & 2A_{4}A_{5} & 2A_{3}A_{5} & 2A_{3}A_{4} \\ A_{3}^{2} & 2A_{4}A_{5} & 2A_{1}A_{3} & 2A_{2}A_{5} & 2A_{2}A_{4} \\ A_{4}^{2} & 2A_{3}A_{5} & 2A_{2}A_{5} & 2A_{1}A_{4} & 2A_{2}A_{3} \\ A_{5}^{2} & 2A_{3}A_{4} & 2A_{2}A_{4} & 2A_{2}A_{3} & 2A_{1}A_{5} \end{matrix} ) ( \begin{matrix}C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \\ C_{5} \end{matrix} ) =

( \begin{matrix}A_{1}^{5} + 120A_{1}A_{2}A_{3}A_{4}A_{5} − 10A_{1}^{2}A_{2}^{3} − 10A_{1}^{2}A_{3}^{3} − 10A_{1}^{2}A_{4}^{3} − 10A_{1}^{2}A_{5}^{3} \\ −4A_{2}^{5} − 5A_{1}^{3}A_{2}^{2} + 20A_{2}^{2}A_{3}^{3} + 20A_{2}^{2}A_{5}^{3} + 20A_{2}^{2}A_{4}^{3} + 30A_{1}^{2}A_{3}A_{4}A_{5} \\ −4A_{3}^{5} − 5A_{1}^{3}A_{3}^{2} + 20A_{3}^{2}A_{4}^{3} + 20A_{3}^{2}A_{5}^{3} + 20A_{2}^{3}A_{3}^{2} + 30A_{1}^{2}A_{2}A_{4}A_{5} \\ −4A_{4}^{5} − 5A_{1}^{3}A_{4}^{2} + 20A_{3}^{3}A_{4}^{2} + 20A_{2}^{3}A_{4}^{2} + 20A_{4}^{2}A_{5}^{3} + 30A_{1}^{2}A_{2}A_{3}A_{5} \\ −5A_{1}^{3}A_{5}^{2} + 20A_{3}^{3}A_{5}^{2} + 20A_{2}^{3}A_{5}^{2} + 20A_{4}^{3}A_{5}^{2} − 4A_{5}^{5} + 30A_{1}^{2}A_{2}A_{3}A_{4} \end{matrix} )  (15)

It is worthwhile to mention that the determinant of the above matrix up to a constant equals P_{10}. This is another explanation for the formula F = P_{13}/P_{10} from section 6. We remark that, if we express C_{1}, …, C_{5} as rational function in the A-s, using (13) and F = P_{13}/P_{10}, the above equality gives an identity in the polynomial ring ℂ[A_{1}, A_{2}, A_{3}, A_{4}, A_{5}] and not only in A.

We describe now relations in weight 6. The relation F^{2} = P_{6} is one. If we apply the Burkhardt group to this relation we obtain 40 linearly independent relations. The 40-dimensional space generated by these relations splits into a 25-dimensional and an irreducible 15-dimensional G-invariant space of relations. Multiplying the 5 relations in weight 5 with A_{1}, …, A_{5} we obtain 25 relations. Actually they span the mentioned 25-dimensional space. We mention that this 25-dimensional representation contains the trivial representation and a 24-dimensional irreducible one. We describe the remaining 15-dimensional space:

Proposition 11. There are 15 relations of the form

C_{i}C_{j} = Q_{1}^{(ij)}C_{1} + Q_{2}^{(ij)}C_{2} + Q_{3}^{(ij)}C_{3} + Q_{4}^{(ij)}C_{4} + Q_{5}^{(ij)}C_{5} + Q^{(ij)}, 1 ≤ i ≤ j ≤ 5

with polynomials Q_{k}^{(ij)} of degree three and Q^{(ij)} of degree six. They are given explicitly in the following tables.

Q^{(11)}= 32A_{5}^{6} + 5A_{1}^{6} + 32A_{4}^{6} + 32A_{2}^{6} + 32A_{3}^{6} + 8A_{1}^{3}A_{5}^{3} + 64A_{2}^{3}A_{5}^{3} + 8A_{1}^{3}A_{2}^{3} + 64A_{5}^{3}A_{4}^{3} +

8A_{1}^{3}A_{3}^{3} + 64A_{3}^{3}A_{4}^{3} + 8A_{1}^{3}A_{4}^{3} + 64A_{2}^{3}A_{4}^{3} + 64A_{2}^{3}A_{3}^{3} + 64A_{3}^{3}A_{5}^{3} − 144A_{1}^{2}A_{2}A_{4}A_{5}A_{3} Q^{(12)}= 288A_{3}^{2}A_{4}^{2}A_{5}^{2} − 24A_{5}^{3}A_{1}^{2}A_{2} − 24A_{4}^{3}A_{1}^{2}A_{2} − 3A_{1}^{5}A_{2} + 48A_{1}^{2}A_{2}^{4} − 24A_{3}^{3}A_{1}^{2}A_{2} +

144A_{3}A_{4}A_{5}A_{1}A_{2}^{2} Q^{(13)}= 48A_{3}^{4}A_{1}^{2} + 288A_{4}^{2}A_{5}^{2}A_{2}^{2} − 24A_{3}A_{5}^{3}A_{1}^{2} − 24A_{3}A_{1}^{2}A_{2}^{3} − 24A_{3}A_{4}^{3}A_{1}^{2} − 3A_{3}A_{1}^{5} +

144A_{3}^{2}A_{4}A_{5}A_{1}A_{2} Q^{(14)}= −24A_{4}A_{5}^{3}A_{1}^{2} − 24A_{4}A_{1}^{2}A_{2}^{3} − 24A_{3}^{3}A_{4}A_{1}^{2} + 288A_{3}^{2}A_{5}^{2}A_{2}^{2} − 3A_{4}A_{1}^{5} + 48A_{4}^{4}A_{1}^{2} +

144A_{3}A_{4}^{2}A_{5}A_{1}A_{2} Q^{(15)}= 48A_{5}^{4}A_{1}^{2} − 3A_{5}A_{1}^{5} − 24A_{4}^{3}A_{5}A_{1}^{2} − 24A_{5}A_{1}^{2}A_{2}^{3} − 24A_{3}^{3}A_{5}A_{1}^{2} + 288A_{3}^{2}A_{4}^{2}A_{2}^{2} +

144A_{3}A_{4}A_{5}^{2}A_{1}A_{2} Q^{(22)}= 21A_{1}^{4}A_{2}^{2} + 96A_{3}A_{5}^{4}A_{4} + 96A_{3}A_{4}^{4}A_{5} + 24A_{4}^{3}A_{1}A_{2}^{2} + 24A_{5}^{3}A_{1}A_{2}^{2} + 24A_{1}A_{2}^{5} +

48A_{3}A_{4}A_{5}A_{1}^{3} − 48A_{3}A_{4}A_{5}A_{2}^{3} + 96A_{3}^{4}A_{4}A_{5} + 24A_{3}^{3}A_{1}A_{2}^{2} Q^{(23)}= −24A_{3}A_{1}A_{2}^{4} − 3A_{3}A_{1}^{4}A_{2} + 72A_{4}^{2}A_{5}^{2}A_{1}^{2} − 24A_{3}^{4}A_{1}A_{2} + 120A_{3}A_{4}^{3}A_{1}A_{2} +

120A_{3}A_{5}^{3}A_{1}A_{2} + 144A_{3}^{2}A_{4}A_{5}A_{2}^{2} Q^{(24)}= −24A_{4}A_{1}A_{2}^{4} − 3A_{4}A_{1}^{4}A_{2} + 72A_{3}^{2}A_{5}^{2}A_{1}^{2} − 24A_{4}^{4}A_{1}A_{2} + 120A_{4}A_{5}^{3}A_{1}A_{2} +

120A_{3}^{3}A_{4}A_{1}A_{2} + 144A_{3}A_{4}^{2}A_{5}A_{2}^{2} Q^{(25)}= −24A_{5}A_{1}A_{2}^{4} + 72A_{3}^{2}A_{4}^{2}A_{1}^{2} − 24A_{5}^{4}A_{1}A_{2} − 3A_{5}A_{1}^{4}A_{2} + 120A_{4}^{3}A_{5}A_{1}A_{2} +

120A_{3}^{3}A_{5}A_{1}A_{2} + 144A_{3}A_{4}A_{5}^{2}A_{2}^{2} Q^{(33)}= 21A_{3}^{2}A_{1}^{4} + 96A_{4}^{4}A_{5}A_{2} + 24A_{3}^{2}A_{4}^{3}A_{1} + 96A_{4}A_{5}^{4}A_{2} + 96A_{4}A_{5}A_{2}^{4} + 24A_{3}^{5}A_{1} −

48A_{3}^{3}A_{4}A_{5}A_{2} + 24A_{3}^{2}A_{5}^{3}A_{1} + 24A_{3}^{2}A_{1}A_{2}^{3} + 48A_{4}A_{5}A_{1}^{3}A_{2} Q^{(34)}= −24A_{3}^{4}A_{4}A_{1} − 3A_{3}A_{4}A_{1}^{4} + 72A_{5}^{2}A_{1}^{2}A_{2}^{2} − 24A_{3}A_{4}^{4}A_{1} + 120A_{3}A_{4}A_{1}A_{2}^{3} +

120A_{4}A_{5}^{3}A_{3}A_{1} + 144A_{3}^{2}A_{4}^{2}A_{5}A_{2} Q^{(35)}= −3A_{3}A_{5}A_{1}^{4} + 72A_{4}^{2}A_{1}^{2}A_{2}^{2} − 24A_{3}A_{5}^{4}A_{1} + 120A_{3}A_{5}A_{1}A_{2}^{3} + 120A_{4}^{3}A_{5}A_{3}A_{1} +

144A_{3}^{2}A_{4}A_{5}^{2}A_{2} − 24A_{3}^{4}A_{5}A_{1} Q^{(44)}= 21A_{4}^{2}A_{1}^{4} + 96A_{3}A_{5}^{4}A_{2} + 24A_{4}^{2}A_{1}A_{2}^{3} + 24A_{3}^{3}A_{4}^{2}A_{1} + 96A_{3}A_{5}A_{2}^{4} + 24A_{4}^{5}A_{1} −

48A_{4}^{3}A_{5}A_{3}A_{2} + 96A_{3}^{4}A_{5}A_{2} + 24A_{4}^{2}A_{5}^{3}A_{1} + 48A_{3}A_{5}A_{1}^{3}A_{2} Q^{(45)}= 72A_{3}^{2}A_{1}^{2}A_{2}^{2} − 3A_{4}A_{5}A_{1}^{4} − 24A_{4}^{4}A_{5}A_{1} + 120A_{4}A_{5}A_{1}A_{2}^{3} + 144A_{4}^{2}A_{5}^{2}A_{3}A_{2}

+ 120A_{3}^{3}A_{4}A_{5}A_{1} − 24A_{4}A_{5}^{4}A_{1} Q^{(55)}= 24A_{5}^{5}A_{1} + 21A_{5}^{2}A_{1}^{4} + 24A_{5}^{2}A_{1}A_{2}^{3} + 96A_{3}A_{4}A_{2}^{4} + 24A_{4}^{3}A_{5}^{2}A_{1} + 96A_{3}A_{4}^{4}A_{2} +

96A_{3}^{4}A_{4}A_{2} + 24A_{3}^{3}A_{5}^{2}A_{1} − 48A_{4}A_{5}^{3}A_{3}A_{2} + 48A_{3}A_{4}A_{1}^{3}A_{2}

The polynomials Q_{k}^{(ij)} are given in the same ordering in the following table:

\begin{matrix}{4A_{2}^{3}−4A_{1}^{3}+4A_{4}^{3} \atop +4A_{3}^{3}+4A_{5}^{3}} & {12A_{1}A_{2}^{2}− \atop 24A_{3}A_{4}A_{5}} & {12A_{1}A_{3}^{2} \atop −24A_{2}A_{4}A_{5}} & {12A_{1}A_{4}^{2} \atop −24A_{2}A_{3}A_{5}} & {12A_{1}A_{5}^{2} \atop −24A_{2}A_{3}A_{4}} \\ 3A_{1}^{2}A_{2} & {−4A_{3}^{3}−4A_{4}^{3}−4A_{5}^{3} \atop +A_{1}^{3}+8A_{2}^{3}} & −12A_{2}A_{3}^{2} & −12A_{2}A_{4}^{2} & −12A_{2}A_{5}^{2} \\ 3A_{1}^{2}A_{3} & −12A_{2}^{2}A_{3} & {−4A_{5}^{3}−4A_{2}^{3}−4A_{4}^{3} \atop +A_{1}^{3}+8A_{3}^{3}} & −12A_{3}A_{4}^{2} & −12A_{3}A_{5}^{2} \\ 3A_{1}^{2}A_{4} & −12A_{2}^{2}A_{4} & −12A_{3}^{2}A_{4} & {−4A_{5}^{3}−4A_{3}^{3}−4A_{2}^{3} \atop +A_{1}^{3}+8A_{4}^{3}} & −12A_{4}A_{5}^{2} \\ 3A_{1}^{2}A_{5} & −12A_{2}^{2}A_{5} & −12A_{3}^{2}A_{5} & −12A_{4}^{2}A_{5} & {−4A_{2}^{3}−4A_{3}^{3}−4A_{4}^{3} \atop +A_{1}^{3}+8A_{5}^{3}} \\ {−12A_{3}A_{4}A_{5} \atop +6A_{1}A_{2}^{2}} & 6A_{1}^{2}A_{2} & −12A_{1}A_{4}A_{5} & −12A_{1}A_{3}A_{5} & −12A_{1}A_{3}A_{4} \\ −6A_{1}A_{2}A_{3} & −3A_{1}^{2}A_{3} & −3A_{1}^{2}A_{2} & −12A_{4}A_{5}^{2} & −12A_{4}^{2}A_{5} \\ −6A_{1}A_{2}A_{4} & −3A_{1}^{2}A_{4} & −12A_{3}A_{5}^{2} & −3A_{1}^{2}A_{2} & −12A_{3}^{2}A_{5} \\ −6A_{1}A_{2}A_{5} & −3A_{1}^{2}A_{5} & −12A_{3}A_{4}^{2} & −12A_{3}^{2}A_{4} & −3A_{1}^{2}A_{2} \\ {6A_{1}A_{3}^{2}− \atop 12A_{2}A_{4}A_{5}} & −12A_{1}A_{4}A_{5} & 6A_{1}^{2}A_{3} & −12A_{1}A_{2}A_{5} & −12A_{1}A_{2}A_{4} \\ −6A_{1}A_{3}A_{4} & −12A_{2}A_{5}^{2} & −3A_{1}^{2}A_{4} & −3A_{1}^{2}A_{3} & −12A_{2}^{2}A_{5} \\ −6A_{1}A_{3}A_{5} & −12A_{2}A_{4}^{2} & −3A_{1}^{2}A_{5} & −12A_{2}^{2}A_{4} & −3A_{1}^{2}A_{3} \\ {6A_{1}A_{4}^{2}− \atop 12A_{2}A_{3}A_{5}} & −12A_{1}A_{3}A_{5} & −12A_{1}A_{2}A_{5} & 6A_{1}^{2}A_{4} & −12A_{1}A_{2}A_{3} \\ −6A_{1}A_{4}A_{5} & −12A_{2}A_{3}^{2} & −12A_{2}^{2}A_{3} & −3A_{1}^{2}A_{5} & −3A_{1}^{2}A_{4} \\ {6A_{1}A_{5}^{2}− \atop 12A_{2}A_{3}A_{4}} & −12A_{1}A_{3}A_{4} & −12A_{1}A_{2}A_{4} & −12A_{1}A_{2}A_{3} & 6A_{1}^{2}A_{5} \end{matrix}

We shall need a certain relation of weight 8, which is a consequence of the relations of weight 5 and 6. One obtains this relation if one multiplies the first of the 5 relations in weight 5 by C_{1} and replaces then all the occurring C_{1}C_{i} by linear functions in the C-s (relations of weight 6). The result is

Proposition 12. The polynomial

R_{6} := 5C_{1}A_{1}^{5} − 5A_{1}^{8} − 20C_{1}A_{1}^{2}A_{2}^{3} − 20C_{1}A_{1}^{2}A_{3}^{3} − 20C_{1}A_{1}^{2}A_{4}^{3} − 20C_{1}A_{1}^{2}A_{5}^{3} − 14A_{1}^{3}A_{2}^{2}C_{2} − 14A_{1}^{3}A_{3}^{2}C_{3} − 14A_{1}^{3}A_{5}^{2}C_{5} − 14A_{1}^{3}A_{4}^{2}C_{4} + 32A_{1}^{2}A_{2}^{3}A_{5}^{3} + 32A_{1}^{2}A_{5}^{3}A_{4}^{3} + 32A_{1}^{2}A_{3}^{3}A_{4}^{3} + 32A_{1}^{2}A_{2}^{3}A_{4}^{3} + 32A_{1}^{2}A_{2}^{3}A_{3}^{3} + 32A_{1}^{2}A_{3}^{3}A_{5}^{3} + 32A_{2}^{3}A_{5}^{2}C_{5} + 32A_{2}^{3}A_{3}^{2}C_{3} + 32A_{2}^{3}A_{4}^{2}C_{4} + 32A_{2}^{2}A_{3}^{3}C_{2} + 32A_{2}^{2}A_{4}^{3}C_{2} + 32A_{2}^{2}A_{5}^{3}C_{2} + 32A_{3}^{2}A_{5}^{3}C_{3} + 32A_{3}^{3}A_{5}^{2}C_{5} + 32A_{3}^{2}A_{4}^{3}C_{3} +32A_{3}^{3}A_{4}^{2}C_{4} +32A_{4}^{2}A_{5}^{3}C_{4} +32A_{4}^{3}A_{5}^{2}C_{5} −128A_{1}^{2}A_{5}^{6} +120C_{1}A_{1}A_{2}A_{3}A_{4}A_{5} − 128A_{1}^{2}A_{4}^{6} − 128A_{1}^{2}A_{2}^{6} − 128A_{1}^{2}A_{3}^{6} − 2A_{1}^{5}A_{5}^{3} − 2A_{1}^{5}A_{2}^{3} − 2A_{1}^{5}A_{3}^{3} − 2A_{1}^{5}A_{4}^{3} − 16A_{2}^{5}C_{2} − 16A_{3}^{5}C_{3}−16A_{4}^{5}C_{4}−16A_{5}^{5}C_{5}+24A_{1}^{2}A_{2}A_{4}A_{5}C_{3}+144A_{1}^{4}A_{2}A_{4}A_{5}A_{3}+24A_{1}^{2}A_{2}A_{3}A_{5}C_{4}+ 24A_{1}^{2}A_{2}A_{3}A_{4}C_{5} + 24A_{1}^{2}A_{3}A_{4}A_{5}C_{2} − 2304A_{2}^{2}A_{3}^{2}A_{4}^{2}A_{5}^{2} − 288A_{2}^{4}A_{3}A_{4}A_{5}A_{1} − 288A_{3}^{4}A_{4}A_{5}A_{1}A_{2} − 288A_{4}^{4}A_{3}A_{5}A_{1}A_{2} − 288A_{5}^{4}A_{3}A_{4}A_{1}A_{2}

is contained in the ideal generated by the relations of weight 5 and 6.

9. The ring of modular forms

In this section we will see that the elements and relations, which we constructed in the previous sections are enough to describe the ring of modular forms, in fact, in several steps , we shall prove:

Theorem 3. The graded algebra A(Γ_{2}[3]) is generated by the forms A_{1}, …, A_{5} of weight one and forms C_{1}, …, C_{5} of weight three.

Both span five dimensional irreducible representations of the Burkhardt group. There are 5 relations in weight 5 and 15 relations in weight 6 . These relations generate the ideal of all relations. The Hilbert function is

∑_{r=0}^{∞} dim[Γ_{2}(3), r]t^{r} = \frac{(1 + t + t^{2} + 6t^{3} + 6t^{4} + t^{5} + t^{6} + t^{7})}{(1 − t)^{4}} =

1 + 5t + 15t^{2} + 40t^{3} + 95t^{4} + 196t^{5} + 360t^{6} + 605t^{7} + 949t^{8} + 1410t^{9} + · · · .

Proof. We take A_{1}, …, C_{5} as ten independent variables and consider the polynomial ring

ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}].

Let ℑ be the ideal generated by the 20 relations . We denote the factor ring by

R := ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}]/ℑ.

The relations of weight 5 are polynomials of degree ≤ 1 in the C-s. We denote them by R_{1}, …, R_{5}. Recall that there is an additional relation R_{6} of weight 8. We also consider the polynomial ring

ℂ[A_{1}, ⋯, A_{5}]

and its free module of rank 6

F := ℂ[A_{1}, ⋯, A_{5}] ⊕ ℂ[A_{1}, ⋯, A_{5}]C_{1} + ⋯ + ℂ[A_{1}, ⋯, A_{5}]C_{5} ≅ ℂ[A_{1}, ⋯, A_{5}]^{6}.

We consider the free submodule

G = ℂ[A_{1}, ⋯, A_{5}] · R_{1} ⊕ ⋯ ⊕ ℂ[A_{1}, ⋯, A_{5}] · R_{6} ≅ ℂ[A_{1}, ⋯, A_{5}]^{6} ⊂ F,

where R_{1}, …, R_{6} are embedded into F in an obvious way.

Proposition 13. The sequence

0 ⟶ G ⟶ F ⟶ R ⟶ 0  (16)

is exact. Proof. The map F → R is surjective because of the relations of weight 6. The module G is in the kernel of this map. We have to show that it is the full kernel. For this reason it is convenient to equip F with a ring structure. We can consider F as the factor ring of ℂ[A_{1}, …, C_{5}] by the ideal which is generated by the relations of weight 6. Then the map F → R is a ring homomorphism and the kernel is generated by G as ideal. All we have to show is that G is an ideal (and not only a module over ℂ[A_{1}, …, A_{5}]). To prove this, one has to consider the thirty products C_{i}R_{j}, i = 1, …, 5, j = 1, …, 6. After replacing the occurring C_{i}C_{j} by means of the relations of degree 6, one gets elements of F. One has to prove that these are contained in R · G. This is done by explicit calculations which we will not reproduce here.

This proposition shows that R has a free resolution of lenght one, thus the ”Auslander-Buchsbaum formula” implies that is a Cohen-Macaulay module over ℂ[A_{1}, …, A_{5}], cf.[BH].^{2} A well know criterion, cf. [Ei], states: Let A → B a homomorphism of noetherian rings such that B is a finitely generated A-module and let M be a finitely generated B-module. Then M is a Cohen-Macaulay B-module if and only if it is a Cohen-Macaulay A-module. If especially B is a Cohen-Macaulay A-module then B is a Cohen-Macaulay ring (i.e. a Cohen-Macaulay B-module). We obtain

Corollary 1. The ring R = ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}]/ℑ is a Cohen-Macau-lay ring. An important consequence of this fact and of Theorem 1 is that the map

Proj(R) ⟶ Proj(A)

is a bijective, thus we obtain:

The variety Proj(R) is irreducible (but possibly not reduced).

This argument shows even more. The localizations of the rings A and R by the multiplicative set generated by P_{10} are isomorphic. Especially we obtain:

The variety Proj(R) is nonsingular outside the locus of the ideal P_{10} · R.

We know that the locus P_{10} = 0 considered in Proj(A(Γ_{2}[3]) consists of 45 irreducible components. We obtain that the locus of the ideal P_{10} · R in Proj(R) also consists of 45 irreducible components. They are transitively permuted under the Burkhardt group.

2 A finitely generated module M over a noetherian ring R is called a Cohen-Macaulay module if M_{𝔭} is a Cohen-Macaulay module for all prime ideals 𝔭 in R on the sense of [Se].

Lemma 1. The point [0, 0, 0, 1, 1, −16, 0, 0, 0, 0] is in the zero set of the ideal ℑ. Moreover it is in the locus of P_{10} · R. It is a regular point of Proj(R).

It is easy to check the equations for the point and for an individual point it is easy to check that it is regular. We omit the computations.

It follows from the above lemma that the singular locus of Proj(R) has codimension ≥ 2. The same then is true for Spec(R). From Serre’s normality criterion, cf. [Se] now it follows that R is normal ring (and especially an integral domain).

The dimension formula is an easy consequence of the exact sequence 0 → G → F → R → 0 .

We mention that our dimension formula agrees with the formulas obtained by means of the Trace formula or by Riemann-Roch (for r big enough), cf. [Ch1], [Ch2] (there is a numerical error even in the second corrected form), [Ha], [Ya]. Moreover, independently, Gunji has found the same expression for the Hilbert function, cf. [Gu], studying in great detais the low dimensional cases. This completes the proof of the theorem .

The nature of the singular locus can be described in more detail:

Proposition 14. Let A_{𝔭} be the homogeneous localization (local ring of the variety Proj(A) at the generic point of

𝔭 ⊂ A.

There is an isomorphism of its formal completion

Â_{𝔭} ≅ K[[X^{2}, X^{3}, …]].

Here K denotes the field of rational functions of the zero locus of 𝔭.

Proof. We denote by 𝔮 the prime ideal of the corresponding locus in ProjR. We know that R is regular at the generic point of 𝔮. Hence

R̂𝔮 ≅ K[[X, X^{2}, X^{3}, …]]

is isomorphic to the ring of formal power series. As variable X we can take a suitable conjugate F′ of F. (The latter does not vanish along the diagonal but only along of conjugates of the diagonal). We know that F^{2} and F^{3} are in A. The same is true for F′. Hence K[[X^{2}, X^{3}, …]] is contained in Â_{𝔭}. Equality must hold, because this ring is not regular.

We recall that a Cohen-Macaulay graded algebra A is said to be Gorenstein if its canonical module is isomorphic to A. As an immediate consequence of the results of [St] we have:

Corollary 2. The ring A(Γ_{2}[3]) is a Gorenstein ring.

10. Final Remark

The central role in our picture plays the dual of the Burkhardt quartic. In the usual algebro-geometric approaches the quartic itself plays a central role. Classically it is known that the modular variety H_{2}/Γ_{2}[3] is birational equivalent to the Burkhardt quartic. This is in accordance with our result, since a hypersurface and its dual always are birational equivalent.

Nevertheless one may ask whether the classical birational map is visible in our picture. The answer is yes.

There is modular form X_{5} of weight 5 with respect to the full modular group but which picks up the non-trivial character of this group. From the zero locus it is clear that B_{1} := X_{5}/F is a holomorphic modular form of weight two. It can be shown that this form is contained in a five dimensional G-invariant space of modular forms on Γ_{2}[3], but all with the same non-trivial character. This five dimensional space defines a Burkhardt quartic and yields the classical birational map between the moduli space and the Burkhardt quartic.

The modular form B_{1} can be expressed by means of theta nullwerte and also as additive lift in the sense of Borcherds. We intend to come back to this subject in a separate paper.

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