The Burkhardt Group and Modular Forms 2

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 the paper [F SM] we determined the graded algebra A(Γ_{2}[3]) of Siegel modular forms of genus two and level three. The Burkhardt group G = Sp(2, ℤ/3ℤ)/{±E} acts on this algebra. The algebra is generated by five forms A_{1}, …, A_{5} of weight one and five forms C_{1}, …, C_{5} of weight three and the relations between them could be described. The ring A(Γ_{2}[3]) turned out to be the normalization of the algebra ℂ[A_{1}, …, A_{5}] where the defining relation is a certain polynomial of degree 18. This polynomial turned out to be the dual of the Burkhardt quartic polynomial. In other approaches to the Siegel modular variety of genus two and level three the Burkhardt quartic itself comes into the play. The reason that this was not directly visible in our approach is that the modular forms which define the less more wellknown birational map from the modular variety to the Burkhardt quartic pick up a non-trivial character.

From our point of view this picture must be visible if we consider the kernel of this character, which is a subgroup of index two

Γ′_{2}[3] ⊂ Γ_{2}[3].

In this paper we will see that the graded algebra A(Γ′_{2}[3]) can be determined. It is generated by the A-s, the C-s and five more modular forms B_{1}, …, B_{5} of weight two. These forms belong to Γ_{2}[3] but they pick up the mentioned character. They satisfy the Burkhardt quartic relation and hence define a rational map

H_{2}/Γ_{2}[3] ⟶ B (Burkhardt quartic)

which turns out to be exactly the same map as in the mentioned other approaches occur.

So our main result states

A(Γ′_{2}[3]) = ℂ[A_{1}, …, A_{5}, B_{1}, …, B_{5}, C_{1}, …, C_{5}]

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

and we also will determine generators of the ideal of relations as well as the Hilbert function.

As in the first paper the action of the Burkhardt group is one of the tools which allows us to get hold of this rather complicated ring. From the first paper we know already that five dimensional spaces generated by the A-s as well as by the C-s are isomorphic five dimensional representations of the Burkardt group. One knows that the Burkhardt group admits two isomorphy classes of five dimensional representations, which are dual to each other. It is basic for us that the B-s also span a five dimensional representation but it is dual to the to the A- or C-representation.

The modular forms B_{1}, …, B_{5} are basic for this paper. The structure theorem shows that the space [Γ_{2}[3], 2, χ] is five dimensional and containes a unique (up to constant factor) element which belongs to the Hecke- group Γ_{2,0}[3] (defined by C ≡ 0 mod 3). We assume that B_{1} is this element. The other B-s are in the span of the G-orbit of B_{1}. Hence we can concentrate on the definition of B_{1}. There are several different possibilities to do this:

1. B_{1} can be obtained as a product of 4 theta nullwerte with characteristics with denominator 3. This description is closely related to the mentioned older descriptions of the relation between the modular variety and the Burkhardt quartic.

2. There is a Siegel modular form of weight 5 due to Igusa X_{5}, which belongs to the full modular group but which has a non-trivial character. The product of the B-s with X_{5}^{3} is a modular form which can be expressed as a polynomials of degree 17 in the A-s. These polynomials are derivatives of the Burkhardt dual polynomial.

3. In [F SM] a certain modular form F of weight three has been constructed. F B_{1} is a constant multiple of X_{5}.

4. The forms B can be described as additive lifts of elliptic modular forms in the sense of Borcherds. (This is a generalization of the Saito-Kurokawa lift, which has been treated by several authors). To apply the framework of Borcherds one needs a profound translation of the symplectic group Sp(2) int the orthogonal group O(2, 3). These constructions have been done in an unpublished paper of Klaus Merz. He kindly allowed us to include some of his results in this paper. Another contribution is:

5. B_{1} is a Borcherds product.

We will use the descriptions two and three to derive our structure theorem. The other constructions of the B-s will be postponed to the appendix of this paper. In this appendix we will explain also the relation of the B-s to the mentioned older approaches.

2. Modular varieties and maps

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}[q, 2q] the subgroup of Γ_{n}[q] defined by diag(CD′) ≡ diag(AB′) ≡ 0 mod 2q. The group

Γ_{2}/Γ_{2}[3] ≅ Γ_{2}[1, 2]/Γ_{2}[3, 6] ≅ 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}, a half integer r and a multiplier 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 four 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 multiplier is trivial.

Let Γ_{0} ⊂ Γ be a normal subgroup of finite index. We assume that the multiplier system χ is defined on Γ. Then the group Γ acts on [Γ_{0}, r, χ] by means of

f → f|M, (f|M)(Z) := χ(M)^{−1}det(CZ + D)^{−r}f(MZ)

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

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 some details about the Satake compactification. Set theoretically we have

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

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]) .

In the paper [F SM] the structure of A(Γ_{2}[3]) has been determined:

A(Γ_{2}[3]) ≅ R := ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}]/ℑ.

The modular forms which correspond to the A-s are of weight 1 and those which correspond to the C-s are of weight 3.

We recall the action of the Burkhardt group G on the vector space ℂA_{1} + ⋯ + ℂA_{5}. This is an action from the right, which we denote by P ↦ P^{g}. It is given

( A_{1} ; ⋮ ; A_{5} )^{g} = π(g) ( A_{1} ; ⋮ ; A_{5} )

where

π : G ⟶ GL(5, ℂ), π(gh) = π(h)π(g)

is an antihomomorphism. The definition of π(g) on the level of generators is given in proposition 1 in the first part of our paper. The same map defines the action on ℂC_{1} + ⋯ + ℂC_{5}. We extend this action to an action from the right on the polynomial ring ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}]. The natural homomorphism

ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}] ⟶ A(Γ_{2}[3])

is G-equivariant. Warning. Sometimes we will denote the modular form corresponding to A_{i} by the same letter. It will always be clear from the context, whether A_{i} means an indeterminate or a modular form (similarly for C_{i} and later also for B_{i}). In doubtful cases we use the notation

P (A_{1}, …, C_{5}) ≡ 0

for a polynomial P in 15 variables to indicate that the associated modular from vanishes.

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

We recall that the full Siegel modular group of genus two admits a unique nontrivial quadratic character χ, cf. [Ig]. This character is trivial on Γ_{2}[2] and comes from the fact that

Sp(2, ℤ/2ℤ) ≅ S_{6}.

Especially

χ(M_{0}) = 1 where M_{0} = ( \begin{matrix}U & 0 \\ 0 & U \end{matrix} ) with U = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

The kernel of χ is the commutator subgroup Γ′_{2} of Γ_{2} = Γ_{2}[1]. The restriction of this character to Γ_{2}[3] is non-trivial. We denote the kernel by Γ′_{2}[3]. This is a normal subgroup of the full modular group. The group

Ĝ := (Γ_{2}/Γ′_{2}[3])/±

is a two fold covering of the Burkhardt group

Ĝ ⟶ G.

There is a natural isomorphism

G ≅ (Γ′_{2}/Γ′_{2}[3])/±,

hence the extension splits,

Ĝ ≅ G × ℤ/2ℤ

and we can consider G as a subgroup of Ĝ.

G ⊂ Ĝ.

We are interested in the space [Γ_{2}[3], 2, χ]. The full modular group acts on this space because χ is defined on the full modular group and this action factors through the Burkhardt group G because χ is one on the negative unit matrix.

Remark. There are two different actions of Ĝ on [Γ_{2}[3], 2, χ], namely

det(CZ + D)^{−r}f(MZ) resp. χ(M)^{−1}det(CZ + D)^{−r}f(MZ).

The restrictions to the subgroup G ⊂ Ĝ are the same. Whenever we talk about the action of the Burkhardt group on [Γ_{2}[3], 2, χ], we of course have to take this action.

We also use the Hecke group Γ_{2,0}[3] which is defined inside Γ_{2} by C ≡ 0 mod 3.

We write the Fourier coefficient of a modular form as

f(Z) = ∑_{T} a(T)e(tr(T Z)).

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

We need some results of [F SM]. Let B be the image of Γ_{2,0}[3] in the Burkhardt group G. We also need the diagonal

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

The image of Δ in H_{2}/Γ_{2}[3] is denoted by Δ[3]. We take from [F SM] the following result (proposition 6). The G-orbit GΔ[3] ⊂ H_{2}/Γ_{2}[3] consists of 45 irreducible components. Under B this set splits into two orbits. The first orbit containing Δ[3] consist of 18 the second of 27 components. We also have to use that there exists a modular form

F = −\frac{1}{2^{5}3^{9}} (C_{1} + 7A_{1}^{2} − 4(A_{2}^{2} + ⋯ + A_{5}^{2}))

of weigth three on Γ_{2,0}[3] whose character is trivial on Γ_{2}[3] and whose zero divisor consists of the second B-orbit (proposition 7 in [F SM]).

We recall also that due to Igusa there exists a cusp form X_{5} of weight 5 with respect to the full modular group but the non-trivial character. This form is normalized by the condition

a_{X_{5}} ( \begin{matrix}1 & 1/2 \\ 1/2 & 1 \end{matrix} ) = 1.

The zero divisor of X_{5} is precisely Γ_{2}Δ. In the first part we introduced a form of weight 10, namely P_{10} = \frac{13}{777600}(g_{4}g_{6} − g_{10}), where g_{k} are the Coble invariants ([F SM], sec. 3) normalized such that the zero Fourier coefficient is one. The relation between X_{5} and P_{10} is simply

P_{10} = X_{5}^{2}.

The location of the zeros shows that

B_{1} := −2^{−2}3^{−9} · \frac{X_{5}}{F}

is a holomorphic modular form and we obtain:

Proposition 1. The exists a character χ′ on Γ_{2,0}[3] and a unique modular form

B_{1} ∈ [Γ_{2,0}[3], 2, χ′], a_{B_{1}} ( \begin{matrix}1 & 1/2 \\ 1/2 & 1 \end{matrix} ) = 1,

whose zero divisor consists of the 18 components of GΔ[3] in the first orbit. The divisor of the modular form −2^{5}3^{9} F = C_{1} + 7A_{1}^{3} − 4(A_{2}^{3} + ⋯ + A_{5}^{3}) (see proposition 8 in [F SM])) consists of the other 27 components and

B_{1}(C_{1} + 7A_{1}^{2} − 4(A_{2}^{2} + ⋯ + A_{5}^{2}))= 8 X_{5}.

The characters χ and χ′ agree on Γ_{2}[3]. We are interested in the G-orbit of B_{1}. For this we need a second description of B_{1}. In [F SM] a certain G-invariant polynomial P_{18} of degree 18 in 5 variables has been defined. It is essentially the dual of the Burkhardt quartic polynomial and it describes the generating relations between the modular forms A_{1}, …, A_{5}. The partial derivatives

P_{18}^{(k)} := \frac{∂P_{18}}{∂A_{k}} (A_{1}, …, A_{5})

are different from 0. From the definition of F in section 6 of [F SM] we derive: Proposition 2. The modular form B_{1} equals

B_{1} = −2^{−2}3^{−9} P_{18}^{(1)}/X_{5}^{3}.

The linear span of the G-orbit of B_{1} is a five dimensional space [Γ_{2}[3], 2, χ]. A basis is given by

B_{k} = −2^{−2}3^{−9} P_{18}^{(k)}/X_{5}^{3}, k = 1 … 5.

This representation of the Burkhardt group G is isomorphic to the dual of the representation π, more precisely: The action of G is given by

( B_{1} ; ⋮ ; B_{5} ) ⟼ π(g)′^{−1} ( B_{1} ; ⋮ ; B_{5} ) .

The forms B_{1}, …, B_{5} are cusp forms. But they have no common zero in the interior. Later we will see that [Γ_{2}[3], 2, χ] is five dimensional. Proof. Only the part about the zeros has to be proved. From the explicit description of the Satake compactification one can derive, that the orbit of the 18 diagonal components runs into each of the 40 boundary components. We should mention that this is not true for the other orbit. The form F is no cusp form. Finally we have to show that the B-s have no common zero in the interior. Here on has to use a well-known fact: The intersection of two irreducible components of GΔ in the upper half plane is empty. The rest follows from the explixit determination of the zero locus of B_{1} and hence for the conjugates B_{1}^{g}.

The partial derivatives of P are coordinates of the dual of the zero locus of P, hence of the dual of the dual of the Burkhardt quartic. It is known that the double dual of a hypersurface is the hypersurface itself. We obtain that the five dimensional space containing B_{1} gives a birational map from H_{2}/Γ_{2}[3] onto the Burkhardt quartic:

Proposition 3. One has Q(B_{1}, …, B_{5}) = 0, where Q is the unique (up to a constant factor) G-invariant quartic polynomial. The forms B_{1}, …, B_{5} define a birational map from X(Γ_{2}[3]) onto the Burkhardt quartic (=zero locus of Q). So we have a commutative diagram

X(Γ_{2}[3]) ⟶^{B_{1},...,B_{5}} B
∥  ↕
X(Γ_{2}[3]) ⟶^{A_{1},...,A_{5}} B^{∨}

That the B-s are annihilated by the invariant quartic polynomial also follows from the fact that there is no cusp form of weight 8 with respect to the full Siegel modular group.

We will see later that this birational map is the same as considered in the older literature.

4. Relations

We consider the two fold covering Ĝ of the Burkhardt group. This group acts on the polynomial ring in the variables A_{1}, …, C_{5} as follows. The action on the A-s and C-s factors through G but the action on the B-s is defined by

π̂(g)(B_{i}) = χ(g)π(g)′^{−1}(B_{i}) g ∈ Ĝ,

where π denotes the action which factors through G. This definition implies that the homomorphism

ℂ[A_{1}, …, B_{1}, …, C_{5}] ⟶ A(Γ′_{2}[3])

is Ĝ-invariant.

We describe now relations: From the Euler relation

A_{1} \frac{∂P_{18}}{∂A_{1}} + ⋯ + A_{5} \frac{∂P_{18}}{∂A_{5}} = P_{18}

we obtain that the element A_{1}B_{1} + ⋯ + A_{5}B_{5} is zero as modular form. Another way to see this is to use that there is no non-zero Siegel modular form of weight three with respect to the full Siegel modular group. We obtain:

Proposition 4. One has the relation

A_{1}B_{1} + ⋯ + A_{5}B_{5} ≡ 0.

Another relation comes from the fact that the forms B_{i}B_{j} have trivial character and hence must be expressible by the A-s and C-s. The space generated by the B_{i}B_{j} is an irreducible 15-dimensional representation of the Burkhardt group G. Hence it is sufficient to know one of the relations. We use the dimension formula, dim[Γ_{2,0}[3], 4] = 4. A basis is given by the forms

A_{1}^{4}, A_{1}C_{1}, A_{1} ∑_{i=2}^{5} A_{i}^{3}, A_{2}A_{3}A_{4}A_{5}.

Hence B_{1}^{2} must be a linear combination of them. An explicit calculation with Fourier coefficients shows:

Proposition 5. One has the following relation:

2^{3}3^{5} B_{1}^{2} ≡ −A_{1}^{4} + A_{1}C_{1} + 4A_{1}(∑_{i=2}^{5} A_{i}^{3}) − 24A_{2}A_{3}A_{4}A_{5}.

If one applies the Burkhardt group G to this relation one obtains more of them. They can be described in the following manner. We use the notation

A = ℂA_{1} + ⋯ + ℂA_{5}, B = ℂB_{1} + ⋯ + ℂB_{5}, C = ℂC_{1} + ⋯ + ℂC_{5}.

The Burkhardt groups acts on them. We describe the action in some detail. For this we have to know the decomposition of some tensor products of theese representations into irreducible ones. The calculation can be performed using the known character table of G. We will not reproduce any character calculations here. It is very convenient but not necessary to use a computer algebra system like GAP.

An calculation with the characters shows: The 15 dimensional representation Sym^{2}(B) is irreducible. It is up to isomorphism the unique 15-dimensional representation of G. Each of the spaces A ⊗ C and Sym^{4}(A) contains precisely one 15-dimensional irreducible representation. This gives us:

Proposition 6. There exist unique G-invariant embeddings

σ : Sym^{2}(B) ⟶ A ⊗ C, τ : Sym^{2}(B) ⟶ Sym^{4}(A)

which are normalized by

σ(2^{3}3^{5} B_{1}^{2}) = A_{1}C_{1}, τ(2^{3}3^{5} B_{1}^{2}) = −A_{1}^{4} + 4A_{1}(∑_{i=2}^{5} A_{i}^{3}) − 24A_{2}A_{3}A_{4}A_{5}.

The expressions for B_{i}B_{j} as polynomials in the A-s and C-s can be written formally as

P ≡ σ(P) + τ(P) (P ∈ Sym^{2}(B)).

This is a 15-dimensional space of relations. The explicit expressions for the relations are:

2^{3}3^{5} B_{1}^{2} − A_{1}C_{1} + 24A_{2}A_{3}A_{4}A_{5} − 4A_{1}A_{5}^{3} − 4A_{1}A_{4}^{3} − 4A_{1}A_{3}^{3} − 4A_{1}A_{2}^{3} + A_{1}^{4} = 0 2^{3}3^{5} B_{1}B_{2} − 2A_{2}C_{2} + 24A_{1}A_{3}A_{4}A_{5} − 6A_{1}^{2}A_{2}^{2} = 0 2^{3}3^{5} B_{1}B_{3} − 2A_{3}C_{3} + 24A_{1}A_{2}A_{4}A_{5} − 6A_{1}^{2}A_{3}^{2} = 0 2^{3}3^{5} B_{1}B_{4} − 2A_{4}C_{4} + 24A_{1}A_{2}A_{3}A_{5} − 6A_{1}^{2}A_{4}^{2} = 0 2^{3}3^{5} B_{1}B_{5} − 2A_{5}C_{5} + 24A_{1}A_{2}A_{3}A_{4} − 6A_{1}^{2}A_{5}^{2} = 0 2^{3}3^{5} B_{2}^{2} − 2A_{1}C_{2} − 2A_{2}C_{1} + 16A_{2}A_{5}^{3} + 16A_{2}A_{4}^{3} + 16A_{2}A_{3}^{3} − 8A_{2}^{4} − 4A_{1}^{3}A_{2} = 0 2^{3}3^{5} B_{2}B_{3} − 2A_{4}C_{5} − 2A_{5}C_{4} + 24A_{2}^{2}A_{3}^{2} + 12A_{1}^{2}A_{4}A_{5} = 0 2^{3}3^{5} B_{2}B_{4} − 2A_{3}C_{5} − 2A_{5}C_{3} + 24A_{2}^{2}A_{4}^{2} + 12A_{1}^{2}A_{3}A_{5} = 0 2^{3}3^{5} B_{2}B_{5} − 2A_{3}C_{4} − 2A_{4}C_{3} + 24A_{2}^{2}A_{5}^{2} + 12A_{1}^{2}A_{3}A_{4} = 0 2^{3}3^{5} B_{3}^{2} − 2A_{1}C_{3} − 2A_{3}C_{1} + 16A_{3}A_{5}^{3} + 16A_{3}A_{4}^{3} − 8A_{3}^{4} + 16A_{2}^{3}A_{3} − 4A_{1}^{3}A_{3} = 0 2^{3}3^{5} B_{3}B_{4} − 2A_{2}C_{5} − 2A_{5}C_{2} + 24A_{3}^{2}A_{4}^{2} + 12A_{1}^{2}A_{2}A_{5} = 0 2^{3}3^{5} B_{3}B_{5} − 2A_{2}C_{4} − 2A_{4}C_{2} + 24A_{3}^{2}A_{5}^{2} + 12A_{1}^{2}A_{2}A_{4} = 0 2^{3}3^{5} B_{4}^{2} − 2A_{1}C_{4} − 2A_{4}C_{1} + 16A_{4}A_{5}^{3} − 8A_{4}^{4} + 16A_{3}^{3}A_{4} + 16A_{2}^{3}A_{4} − 4A_{1}^{3}A_{4} = 0 2^{3}3^{5} B_{4}B_{5} − 2A_{2}C_{3} − 2A_{3}C_{2} + 24A_{4}^{2}A_{5}^{2} + 12A_{1}^{2}A_{2}A_{3} = 0 2^{3}3^{5} B_{5}^{2} − 2A_{1}C_{5} − 2A_{5}C_{1} − 8A_{5}^{4} + 16A_{4}^{3}A_{5} + 16A_{3}^{3}A_{5} + 16A_{2}^{3}A_{5} − 4A_{1}^{3}A_{5} = 0

The space B ⊗ C splits into a one-dimensional and a 24-dimensional irreducible representation.

Lemma 1. The one-dimensional part of B ⊗ C is generated by the G-invariant element

H = B_{1} ⊗ C_{1} + ⋯ + B_{5} ⊗ C_{5}.

The associated modular form is non-zero. It is 40X_{5}.

There is another way to get this form which has been described in Proposition 1. This gives us further relations: A calculation with characters shows, that B ⊗ Sym^{3}(A) contains a unique 24 dimensional representation of G. Hence there exists a Gequivariant embedding

B ⊗ C ⟶ ℂH ⊕ (B ⊗ Sym^{3}(A)).

This map is determined only up to two parameters but the image space is unique. From Lemma 1 and Proposition 1 and the informations about the representations we obtain:

Proposition 7. There exists an G-invariant embedding

η : B ⊗ C ⟶ ℂH ⊕ (B ⊗ Sym^{3}(A))

such that on the level of modular forms one has the relations

P − η(P) ≡ 0 (P ∈ B ⊗ C).

This is a 24-dimensional space of relations. The explicit expressions for the relations are:

B_{1}C_{1} − 1/5H − 12A_{5}^{3}B_{1} − 12A_{4}^{3}B_{1} − 12A_{3}^{3}B_{1} − 12A_{2}^{3}B_{1} + 18A_{1}^{3}B_{1} − 3A_{1}^{2}A_{2}B_{2} −

3A_{1}^{2}A_{3}B_{3} − 3A_{1}^{2}A_{4}B_{4} − 3A_{1}^{2}A_{5}B_{5} = 0 B_{1}C_{2} − 21A_{1}^{2}A_{2}B_{1} + 36A_{3}A_{4}A_{5}B_{2} − 30A_{1}A_{2}^{2}B_{2} + 6A_{1}A_{2}A_{3}B_{3} + 6A_{1}A_{2}A_{4}B_{4} +

6A_{1}A_{2}A_{5}B_{5} = 0 B_{1}C_{3} − 21A_{1}^{2}A_{3}B_{1} + 36A_{2}A_{4}A_{5}B_{3} − 30A_{1}A_{3}^{2}B_{3} + 6A_{1}A_{3}A_{2}B_{2} + 6A_{1}A_{3}A_{4}B_{4} +

6A_{1}A_{3}A_{5}B_{5} = 0 B_{1}C_{4} − 21A_{1}^{2}A_{4}B_{1} + 36A_{2}A_{3}A_{5}B_{4} − 30A_{1}A_{4}^{2}B_{4} + 6A_{1}A_{4}A_{2}B_{2} + 6A_{1}A_{4}A_{3}B_{3} +

6A_{1}A_{4}A_{5}B_{5} = 0 B_{1}C_{5} + 3A_{1}^{2}A_{5}B_{1} + 30A_{1}A_{2}A_{5}B_{2} + 30A_{1}A_{3}A_{5}B_{3} + 30A_{1}A_{4}A_{5}B_{4} + 36A_{2}A_{3}A_{4}B_{5} −

6A_{1}A_{5}^{2}B_{5} = 0 B_{2}C_{1}+72A_{3}A_{4}A_{5}B_{1}−60A_{1}A_{2}^{2}B_{1}+12A_{5}^{3}B_{2}+12A_{4}^{3}B_{2}+12A_{3}^{3}B_{2}−48A_{2}^{3}B_{2}−3A_{1}^{3}B_{2}+

12A_{2}^{2}A_{3}B_{3} + 12A_{2}^{2}A_{4}B_{4} + 12A_{2}^{2}A_{5}B_{5} = 0 B_{2}C_{2} − 1/5H + 12A_{5}^{3}B_{1} + 12A_{4}^{3}B_{1} + 12A_{3}^{3}B_{1} − 24A_{2}^{3}B_{1} − 9A_{1}^{3}B_{1} − 24A_{1}^{2}A_{2}B_{2} +

3A_{1}^{2}A_{3}B_{3} + 3A_{1}^{2}A_{4}B_{4} + 3A_{1}^{2}A_{5}B_{5} = 0 B_{2}C_{3} + 36A_{2}^{2}A_{3}B_{1} + 9A_{1}^{2}A_{3}B_{2} + 36A_{1}A_{4}A_{5}B_{3} + 36A_{2}A_{5}^{2}B_{4} + 36A_{2}A_{4}^{2}B_{5} = 0 B_{2}C_{4} + 36A_{2}^{2}A_{4}B_{1} + 9A_{1}^{2}A_{4}B_{2} + 36A_{2}A_{5}^{2}B_{3} + 36A_{1}A_{3}A_{5}B_{4} + 36A_{2}A_{3}^{2}B_{5} = 0 B_{2}C_{5} + 36A_{2}^{2}A_{5}B_{1} + 9A_{1}^{2}A_{5}B_{2} + 36A_{2}A_{4}^{2}B_{3} + 36A_{2}A_{3}^{2}B_{4} + 36A_{1}A_{3}A_{4}B_{5} = 0 B_{3}C_{1}+72A_{2}A_{4}A_{5}B_{1}−60A_{1}A_{3}^{2}B_{1}+12A_{5}^{3}B_{3}+12A_{4}^{3}B_{3}−48A_{3}^{3}B_{3}+12A_{2}^{3}B_{3}−3A_{1}^{3}B_{3}+

12A_{3}^{2}A_{2}B_{2} + 12A_{3}^{2}A_{4}B_{4} + 12A_{3}^{2}A_{5}B_{5} = 0 B_{3}C_{2} + 36A_{2}A_{3}^{2}B_{1} + 36A_{1}A_{4}A_{5}B_{2} + 9A_{1}^{2}A_{2}B_{3} + 36A_{3}A_{5}^{2}B_{4} + 36A_{3}A_{4}^{2}B_{5} = 0 B_{3}C_{3} − 1/5H + 12A_{5}^{3}B_{1} + 12A_{4}^{3}B_{1} − 24A_{3}^{3}B_{1} + 12A_{2}^{3}B_{1} − 9A_{1}^{3}B_{1} − 24A_{1}^{2}A_{3}B_{3} +

3A_{1}^{2}A_{2}B_{2} + 3A_{1}^{2}A_{4}B_{4} + 3A_{1}^{2}A_{5}B_{5} = 0 B_{3}C_{4} + 36A_{3}^{2}A_{4}B_{1} + 36A_{3}A_{5}^{2}B_{2} + 9A_{1}^{2}A_{4}B_{3} + 36A_{1}A_{2}A_{5}B_{4} + 36A_{2}^{2}A_{3}B_{5} = 0 B_{3}C_{5} + 36A_{3}^{2}A_{5}B_{1} + 36A_{3}A_{4}^{2}B_{2} + 9A_{1}^{2}A_{5}B_{3} + 36A_{2}^{2}A_{3}B_{4} + 36A_{1}A_{2}A_{4}B_{5} = 0 B_{4}C_{1}+72A_{2}A_{3}A_{5}B_{1}−60A_{1}A_{4}^{2}B_{1}+12A_{5}^{3}B_{4}−48A_{4}^{3}B_{4}+12A_{3}^{3}B_{4}+12A_{2}^{3}B_{4}−3A_{1}^{3}B_{4}+

12A_{4}^{2}A_{2}B_{2} + 12A_{4}^{2}A_{3}B_{3} + 12A_{4}^{2}A_{5}B_{5} = 0 B_{4}C_{2} + 36A_{2}A_{4}^{2}B_{1} + 36A_{1}A_{3}A_{5}B_{2} + 36A_{4}A_{5}^{2}B_{3} + 9A_{1}^{2}A_{2}B_{4} + 36A_{3}^{2}A_{4}B_{5} = 0 B_{4}C_{3} + 36A_{3}A_{4}^{2}B_{1} + 36A_{4}A_{5}^{2}B_{2} + 36A_{1}A_{2}A_{5}B_{3} + 9A_{1}^{2}A_{3}B_{4} + 36A_{2}^{2}A_{4}B_{5} = 0 B_{4}C_{4} − 1/5H + 12A_{5}^{3}B_{1} − 24A_{4}^{3}B_{1} + 12A_{3}^{3}B_{1} + 12A_{2}^{3}B_{1} − 9A_{1}^{3}B_{1} − 24A_{1}^{2}A_{4}B_{4} + 3A_{1}^{2}A_{2}B_{2} +

3A_{1}^{2}A_{3}B_{3} + 3A_{1}^{2}A_{5}B_{5} = 0 B_{4}C_{5} + 36A_{4}^{2}A_{5}B_{1} + 36A_{3}^{2}A_{4}B_{2} + 36A_{2}^{2}A_{4}B_{3} + 9A_{1}^{2}A_{5}B_{4} + 36A_{1}A_{2}A_{3}B_{5} = 0 B_{5}C_{1}+72A_{2}A_{3}A_{4}B_{1}−20A_{1}A_{5}^{2}B_{1}+52A_{2}A_{5}^{2}B_{2}+52A_{3}A_{5}^{2}B_{3}+52A_{4}A_{5}^{2}B_{4}+12A_{4}^{3}B_{5}+

12A_{3}^{3}B_{5} + 12A_{2}^{3}B_{5} − 3A_{1}^{3}B_{5} − 8A_{5}^{3}B_{5} = 0 B_{5}C_{2} + 36A_{2}A_{5}^{2}B_{1} + 36A_{1}A_{3}A_{4}B_{2} + 36A_{4}^{2}A_{5}B_{3} + 36A_{3}^{2}A_{5}B_{4} + 9A_{1}^{2}A_{2}B_{5} = 0 B_{5}C_{3} + 36A_{3}A_{5}^{2}B_{1} + 36A_{4}^{2}A_{5}B_{2} + 36A_{1}A_{2}A_{4}B_{3} + 36A_{2}^{2}A_{5}B_{4} + 9A_{1}^{2}A_{3}B_{5} = 0 B_{5}C_{4} + 36A_{4}A_{5}^{2}B_{1} + 36A_{3}^{2}A_{5}B_{2} + 36A_{2}^{2}A_{5}B_{3} + 36A_{1}A_{2}A_{3}B_{4} + 9A_{1}^{2}A_{4}B_{5} = 0 B_{5}C_{5} − 1/5H − 24A_{5}^{3}B_{1} + 12A_{4}^{3}B_{1} + 12A_{3}^{3}B_{1} + 12A_{2}^{3}B_{1} + 9A_{1}^{3}B_{1} + 21A_{1}^{2}A_{2}B_{2} +

21A_{1}^{2}A_{3}B_{3} + 21A_{1}^{2}A_{4}B_{4} − 6A_{1}^{2}A_{5}B_{5} = 0

5. Ring of modular forms

We consider the obvious homomorphisms

\begin{matrix}ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}] & ⟶ A(Γ_{2}[3]), \\ ℂ[A_{1}, …, A_{5}, B_{1}, …, B_{5}, C_{1}, …, C_{5}] & ⟶ A(Γ′_{2}[3]). \end{matrix}

The main result of [F SM] states that the first homomorphism is surjective and that its kernel ℑ is generated by 20 relations, 5 in weight 5 and 15 in weight 6. We want to prove a similar result for the second homomorphism. For this we introduce the ideal 𝔍 generated by ℑ and the 40 relations of the previous section, one in weight three (proposition 3), 15 ind weight 4 (proposition 5) and 24 in weight 5 (proposition 6). We have to consider the rings

\begin{aligned}R & = ℂ[A_{1}, …, A_{5}, C_{1}, …, C_{5}]/I ≅ A(Γ_{2}[3]), \\ S & = ℂ[A_{1}, …, A_{5}, B_{1}, …, B_{5}, C_{1}, …, C_{5}]/𝔍 \end{aligned}

We have a commutative diagram, where all arrows define finite module extensions

R  ⟶  S
↓  ↓
A(Γ_{2}[3]) ⟶ A(Γ′_{2}[3])

and a corresponding diagram of finite maps

proj(R) ⟵ proj(S)
↑  ↑
X(Γ_{2}[3]) ⟵ X(Γ′_{2}[3])

Lemma 2. The map X(Γ′_{2}[3]) → proj(S) is bijective. Proof. It is easy to work out the fibres of the map X(Γ′_{2}[3]) → X(Γ_{2}[3]). Every point in the interior H_{2}/Γ_{2}[3] has precisely two inverse images, whereas the boundary points only have one inverse image. In other words, the map is a two fold covering, the ramification locus is precisely the boundary. For the proof of the remark we have to show that proj(S) → proj(R) has the same ramification behavior. It is clear that this map is surjective. Recall that the B-s vanish along the boundary but have no zero in common in the interior. This shows that every boundary point has only one inverse image in proj(S). In contrast we get two inverse images of an point in the interior, because we can change the sign of the B-coordinates.

Our aim is to prove that S → A(Γ′_{2}[3]) is an isomorphism. For this it is enough to show that S is a normal ring. The idea is to use Serre’s normality criterion, cf. [Se]. There is a graded version of Serre’s criterion which could be used, but we prefer to use the more standard affine version. Hence we consider Spec(S) insted of proj(S). There are natural projections

Spec(S) − {0} ⟶ proj(S), Spec(R) − {0} ⟶ proj(R).

We define the boundary part of Spec(R) to be the inverse image of the boundary part of proj(R) including {0} and define the boundary part of Spec(S) to be the inverse image of the boundary part of Spec(R).

Lemma 3. The complement of the boundary part of Spec(S) is in the regular locus of Spec(S).

Corollary. The codimensions of the singular locus of Spec(S) is greater or equal than two. Proof. We know that proj(R) is regular outside the boundary part. It is easy to derive from this that Spec(R) is regular outside the boundary. Let now

P := (a_{1}, …, a_{5}, b_{1}, …, b_{5}, c_{1}, … c_{5})

be the coordinates of a point of Spec(S) outside the boundary part. Then the point (a_{1}, …, a_{5}, c_{1}, … c_{5}) is regular in Spec(R). This means that the Jacobian of the 20 relations in the A-s and C-s has rank 6. We have to show that the Jacobian of all the 36 relations has rank 11. It is easy to get 5 additional columns in the Jacobian by using the relations in proposition 7 and their derivatives by the variables B_{i}. Again one has to use that the B-s have no common zero.

To get the normality of S we have to verify Serre’s criterium S2. This is fulfilled if S is an Cohen-Macaulay ring or which means the same, that S is an Cohen-Macauley module over the polynomial ring

R = ℂ[A_{1}, …, A_{5}].

Actually we will construct a free resolution of R-modules

0 ⟶ R^{12} ⟶ R^{12} ⟶ S ⟶ 0.

More precisely we consider the free R-module

F := R + RB_{1} + ⋯ + RB_{5} + RC_{1} + ⋯ + RC_{5} + RH ≅ R^{12}.

There is a natural map F → S, which is surjective because of the relations in S. We have to determine the kernel of this map, more precisely we need generators for the kernel as R-module.

It is convenient to equip F with a (non-associative) multiplication:

Lemma 4. There is a unique R-bilinear map

F × F ⟶ F, (X, Y) ⟼ X ∗ Y,

with the following properties:

1. P ∗ X = P X for P ∈ R.

2. The expressions C_{i} ∗ C_{j} are defined as in the first part of the paper.

3. The expressions B_{i} ∗ B_{j} and B_{i} ∗ C_{j} are defined using propositions 6 and 7.

4. H = B_{1} ∗ C_{1} + ⋯ B_{5} ∗ C_{5}.

5. B_{i} ∗ H = (B_{i} ∗ B_{1}) ∗ C_{1} + ⋯ (B_{i} ∗ B_{5}) ∗ C_{5}.

6. C_{i} ∗ H = (C_{i} ∗ C_{1}) ∗ B_{1} + ⋯ (C_{i} ∗ C_{5}) ∗ B_{5}.

7. H ∗ H = 40^{2}P_{10}.

Proof. We consider the sub-module

F′ = R + RB_{1} + ⋯ + RB_{5} + RC_{1} + ⋯ RC_{5}.

Using the prescriptions 1)-3), we get a bilinear map F′ × F′ → F. One has to check that 4) is already a consequence. Next one has to observe that B_{i} ∗ B_{j} and C_{i} ∗ C_{j} are contained in F′. Hence the definitions 5) and 6) are possible.

Lemma 5. The pairing on F has the following properties:

a) It is commutative.

b) It satisfies

(X ∗ Y) ∗ Z − X ∗ (Y ∗ Z) ∈ R_{0} ∗ F where R_{0} := A_{1}B_{1} + ⋯ + A_{5}C_{5}.

c) The natural projection p : F → S is homomorphism in the sense

p(X ∗ Y) = p(X)p(Y).

Proof. These properties have to be checked on the generators case by case. We omit details.

Next we consider

G := R_{0} ∗ F.

This is the sub-module genrated by the 12 elements:

\begin{aligned}R_{0} & = A_{1}B_{1} + ⋯ A_{5}B_{5} \\ R_{1} & = B_{1} ∗ R_{0} \\ R_{2} & = B_{2} ∗ R_{0} \\ R_{3} & = B_{3} ∗ R_{0} \\ R_{4} & = B_{4} ∗ R_{0} \\ R_{5} & = B_{5} ∗ R_{0} \end{aligned} \begin{aligned}R_{6} & = H ∗ R_{0} \\ R_{7} & = C_{1} ∗ R_{0} \\ R_{8} & = C_{2} ∗ R_{0} \\ R_{9} & = C_{3} ∗ R_{0} \\ R_{10} & = C_{4} ∗ R_{0} \\ R_{11} & = C_{5} ∗ R_{0} \end{aligned}

Hence

G = RR_{0} + RR_{1} + ⋯ + RR_{11} ⊂ F.

Lemma 4 shows that the star-product defines on F/G a structure as associative and commutative ring with unity and that the natural map

F/G ⟶ S

is a ring homomorphism. We claim that it is an isomorphism. Looking at the defining ideal 𝔍 of S on sees that the kernel of this homomorphism is generated by a five dimensional space of relations of weight 5, which has been introduced in the first part of the paper. This five dimensional space is spanned also by the five R_{0} ∗ B_{i} as one verifies by a direct computation. Now we know that we have an exact sequence

0 ⟶ G ⟶ F ⟶ S ⟶ 0.

Theorem 1. The R-module G is free of rank 12. We have the exact sequence of R-modules

0 ⟶ G ⟶ F ⟶ S ⟶ 0
∥ ∥ ∥
0 ⟶ R^{12} ⟶ R^{12} ⟶ S ⟶ 0

Proof. It remains to show that the 12 generators of G are linearly independent over R: We consider the action of ℤ/2ℤ on F which is trivial on the A-s and C-s but changes signs of the B-s. The space G splits into an even and an odd part:

G = G_{+} ⊕ G_{−}, G_{+} = RR_{1} + ⋯ + RR_{6}, G_{−} = RR_{0} + RR_{7} + ⋯ RR_{11}.

It is easy to check that G_{+} coincides with the space of relations which we introduced in the first part of our paper (and which we denoted by G) there. Hence we concentrate to the space G_{−}. The proof of theorem 1 follows from

Lemma 6. The elements R_{0}, R_{0} ∗ C_{1}, …, R_{0} ∗ C_{5} of F are linearly independent over ℂ[A_{1}, …, A_{5}].

The proof of the lemma (and hence of theorem 1) is done by a direct calculation which uses the explicit expressions for R_{0} ∗ C_{i}.

This theorem shows that S has a free resolution of lenght one, thus the “Auslander-Buchsbaum formula” implies that is a Cohen-Macaulay module over the ring ℂ[A_{1}, …, A_{5}], cf. [BH]. Now we know that S is a Cohen-Macaulay ring. Serre’s criterion shows that S is normal. Hence we obtain our main result.

Theorem 2. The natural homomorphism

S ⟶ A(Γ′_{2}[3])

is an isomorphism. This means that A(Γ′_{2}[3]) is generated by the 15 modular forms A_{1}, …, C_{5} and that the described relations are defining ones. The Hilbert function is

∑_{r=0}^{∞} dim[Γ′_{2}[3], r] X^{r} = \frac{(1 + X + X^{2})(1 + 5X^{2} + 5X^{3} + X^{5})}{(1 − X)^{4}}.

The ring A(Γ′_{2}[3]) is a Gorenstein ring.

6. Relation to older approach

Let us recall some basic facts about Thetanullwerte with characteristics. A characteristic m is a vector of ℚ^{2n}. We shall denote by a and b its first respectively second n-coordinates. The group Γ_{n} acts on the set of characteristic according to the following rule

M ( \begin{matrix}a \\ b \end{matrix} ) = ( \begin{matrix}D & −C \\ −B & A \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) + \frac{1}{2} ( \begin{matrix}diag(C D′) \\ diag(A B′) \end{matrix} ) .

Here with diag(S) we have denoted the vector which coordinates are the diagonal entries of the square matrix S.

For a half integral characteristic m we set e(m) = (−1)^{4a′b} and we say that m is even or odd according as e(m) = 1 or −1 We recall that the action of Γ_{n} on the set of half integral characteristics is not transitive, in fact the parity e(m) of the characteristics is an invariant, so we get two orbits.

The Thetanullwert of characteristic m is the holomorphic function

θ [ \begin{matrix}a \\ b \end{matrix} ] (Z) = ∑_{p∈ℤ^{n}} exp(πi(Z[p + a] + 2 (p + a)′b)).

For every s, u ∈ ℤ, we have

θ [ \begin{matrix}a + s \\ b + u \end{matrix} ] (Z) = exp(2πia′u)θ [ \begin{matrix}a \\ b \end{matrix} ] (Z).

Thus we shall consider Thetanullwerte with characteristics in ℚ^{2n}/ℤ^{2n}. The action of Γ_{n} induces a transformation formula on the Thetanullwerte. In fact for any M ∈ Γ_{n} we have

θ [ M ( \begin{matrix}a \\ b \end{matrix} ) ] (M · Z) = κ(σ)exp(πiϕ (( \begin{matrix}a \\ b \end{matrix} ) , M) det(CZ + D)^{\frac{1}{2}} θ [ \begin{matrix}a \\ b \end{matrix} ] (Z).

Here κ(M) is an eighth root of unity depending on M and on the choice of the sign of the root square of det(CZ + D)^{\frac{1}{2}} and

ϕ (( \begin{matrix}a \\ b \end{matrix} ) , M) = − ( B′D[a] + A′C[b] − 2 a′ B′Cb − diag(A)′ B′)(Da − Cb))

For more details we refer to [Ig].

The following Thetanullwerte will be useful for our purposes

θ [ \begin{matrix}σ + a \\ b \end{matrix} ] (qZ)

with qσ, 2a, 2b integral.

Relatively to Γ_{n,0}(q), the transformation formula for θ [ \begin{matrix}σ + a \\ b \end{matrix} ] (qZ) can be reduced to the above, since

θ [ \begin{matrix}σ + a \\ b \end{matrix} ] (qM · Z) = θ [ \begin{matrix}σ + a \\ b \end{matrix} ] (M̃ · qZ) =

χ(M̃)exp(πiϕ (M̃^{−1} ( \begin{matrix}σ + a \\ b \end{matrix} ) , M̃) det(CZ + D)^{\frac{1}{2}} θ [ M̃^{−1} ( \begin{matrix}σ + a \\ b \end{matrix} ) ] (qZ) .

Here

M̃ = ( \begin{matrix}A & qB \\ \frac{1}{q}C & D \end{matrix} ) and M̃^{−1} ( \begin{matrix}σ + a \\ b \end{matrix} ) = ( \begin{matrix}A′ & \frac{1}{q}C′ \\ qB′ & D′ \end{matrix} ) ( \begin{matrix}σ \\ 0 \end{matrix} ) + M̃^{−1} ( \begin{matrix}a \\ b \end{matrix} ) (1)

We observe that the set of these characteristics is stable under the action of Γ_{n,0}(q). We restrict ourselves to consider in details the situation related to the group Γ_{2}[3]. We introduce 5 characteristics

σ_{1} = \frac{1}{3} ( \begin{matrix}0 \\ 0 \end{matrix} ) σ_{2} = \frac{1}{3} ( \begin{matrix}1 \\ 0 \end{matrix} ) σ_{3} = \frac{1}{3} ( \begin{matrix}0 \\ 1 \end{matrix} ) σ_{4} = \frac{1}{3} ( \begin{matrix}1 \\ 1 \end{matrix} ) σ_{5} = \frac{1}{3} ( \begin{matrix}1 \\ −1 \end{matrix} )

We consider the thetanullwerte ϑ_{i} := ϑ [ \begin{matrix}σ_{i} \\ 0 \end{matrix} ] (qZ, 0). From [An] we deduce the fol-

lowing

Proposition 8. The theta series ϑ_{1}, …, ϑ_{5} are contained in [Γ_{2}[3, 6], 1/2, κ]. They span a 5-dimensional space V which is invariant under the Burkhardt group and with irreducible action. The action, up to the ambiguity given by the multiplier, is given by means of the following 4 matrices which generate Γ_{2}[1, 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 & 2 & 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 & 2 \\ 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 & 0 \\ 0 & 0 & 0 & 0 & ω^{2} \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). These modular forms induce an embedding, cf [BL] and [SM],

ϑ^{+} : H_{2}/Γ_{2}[3, 6] ⟶ ℙ^{4}.

This map is obviosly G-equivariant, but differently from what was wrongly stated in [SM], does not extend to all the boundary (we need to blow it up!). In any case it defines a birational map onto a subvariety of ℙ^{4}. This is the Hessian of Burkhardt’s quartic, cf [Bu] and [Cob]. The Hessian is defined by an equation that is invariant for the action of the Burkhardt ’s group, more precisely , we have

P_{10}(ϑ_{1}, …, ϑ_{5}) = det ( \begin{matrix}ϑ_{1}^{2} & 2ϑ_{2}^{2} & 2ϑ_{3}^{2} & 2ϑ_{4}^{2} & 2ϑ_{5}^{2} \\ ϑ_{2}^{2} & 2ϑ_{1}ϑ_{2} & 2ϑ_{4}ϑ_{5} & 2ϑ_{3}ϑ_{5} & 2ϑ_{3}ϑ_{4} \\ ϑ_{3}^{2} & 2ϑ_{4}ϑ_{5} & 2ϑ_{1}ϑ_{3} & 2ϑ_{2}ϑ_{5} & 2ϑ_{2}ϑ_{4} \\ ϑ_{4}^{2} & 2ϑ_{3}ϑ_{5} & 2ϑ_{2}ϑ_{5} & 2ϑ_{1}ϑ_{4} & 2ϑ_{2}ϑ_{3} \\ ϑ_{5}^{2} & 2ϑ_{3}ϑ_{4} & 2ϑ_{2}ϑ_{4} & 2ϑ_{2}ϑ_{3} & 2ϑ_{1}ϑ_{5} \end{matrix} ) ≡ 0.

We recall from [Hu] and [JSV] that the solutions of the system

( \begin{matrix}ϑ_{1}^{2} & 2ϑ_{2}^{2} & 2ϑ_{3}^{2} & 2ϑ_{4}^{2} & 2ϑ_{5}^{2} \\ ϑ_{2}^{2} & 2ϑ_{1}ϑ_{2} & 2ϑ_{4}ϑ_{5} & 2ϑ_{3}ϑ_{5} & 2ϑ_{3}ϑ_{4} \\ ϑ_{3}^{2} & 2ϑ_{4}ϑ_{5} & 2ϑ_{1}ϑ_{3} & 2ϑ_{2}ϑ_{5} & 2ϑ_{2}ϑ_{4} \\ ϑ_{4}^{2} & 2ϑ_{3}ϑ_{5} & 2ϑ_{2}ϑ_{5} & 2ϑ_{1}ϑ_{4} & 2ϑ_{2}ϑ_{3} \\ ϑ_{5}^{2} & 2ϑ_{3}ϑ_{4} & 2ϑ_{2}ϑ_{4} & 2ϑ_{2}ϑ_{3} & 2ϑ_{1}ϑ_{5} \end{matrix} ) ( \begin{matrix}B′_{1} \\ B′_{2} \\ B′_{3} \\ B′_{4} \\ B′_{5} \end{matrix} ) = ( \begin{matrix}0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} ) (2)

are modular forms. By construction, they span a five dimensional G-invariant space. This induces a map α into the Burkhardt’s quartic. As explained in [JSV] and in [Ge], this map factorizes throught H_{2}/Γ_{2}[3]. We will check it again, using modular forms.

First of all, we give explicitely B′_{i}. We fix an odd characteristic m and we consider the thetanullwerte

θ_{i} := e(σ′_{i}2b) ϑ [ \begin{matrix}σ_{i} + a \\ b \end{matrix} ] (3Z, 0), i = 2, …, 5.

We set

\begin{aligned}B′_{1} & = 6θ_{1}θ_{2}θ_{3}θ_{4} \\ B′_{2} & = θ_{1}(θ_{2}^{3} + θ_{3}^{3} − θ_{4}^{3}) \\ B′_{3} & = −θ_{2}(θ_{1}^{3} + θ_{3}^{3} + θ_{4}^{3}) \\ B′_{4} & = θ_{3}(−θ_{1}^{3} + θ_{2}^{3} + θ_{4}^{3}) \\ B′_{5} & = θ_{4}(θ_{1}^{3} + θ_{2}^{3} − θ_{3}^{3}). \end{aligned}

Proposition 9. The modular forms B′_{i} are in [Γ_{2}[3], 2, χ] and span the Bspace.

Proof. It is enough to prove that up to a multiplicative constant B_{1} and B′_{1} coincide. We analize the set of zeros of B′_{1}. We observe that on the diagonal B′_{1} vanishes, since the thetanullwerte along the diagonal split in the product of two Thetanullwerte of genus 1 and at least one of the terms of all product is an odd Thetanullwerte that is zero. [Γ_{2,0}[3] acts on B′_{1} changing the odd characteristic and also σ, cf (*) , but in any case the form of the product does not change, thus B′_{1} still vanishes along all [Γ_{2,0}[3] conjugate of the diagonal. Now let us consider the modular function

B′_{1}/B_{1}.

This has no poles , hence is a constant.

This proof implies that, up to a third root of unity, the definition of the modular forms B′_{i} does not depends from the choice of the odd characteristic. Moreover, using this presentation, is easy to check that they are cusp forms. In fact a simple computation shows that all B′_{i} vanish on the standard 1-dimensional boundary component which is the one containing

lim_{y→∞} \binom{τ 0}{0 iy}.

Thus they vanish at all 1-dimensional boundary components, hence they are cusp forms.

We conclude this section, recalling that (2) induces a 10 : 1 map

ψ^{+} : Hess(B) ⟶ B

that makes the following diagram commutative

H_{2}/Γ_{2}[3, 6] ⟶^{ϑ^{+}} Hess(B) ⊂ ℙ^{4}
π^{+}↓  ^{ψ^{+}}↓
H_{2}/Γ_{2}[3] ⟶^{α}  B

7. The modular forms B as lifts of elliptic modular forms: Results of Klaus Merz

Let V = ℝ^{5} be the quadratic space with the quadratic form

q(x_{1}, …, x_{5}) = x_{1}x_{2} + x_{3}x_{4} − x_{5}^{3}

and the associated bilinear form

⟨(x_{1}, …, x_{5}), (y_{1}, …, y_{5})⟩ = q(x+y)−q(x)−q(y) = x_{1}y_{2}+x_{2}y_{1}+x_{3}y_{3}+x_{4}y_{3}−2x_{5}y_{5}.

The orthogonal group O(V) contains a subgroup O^{+}(V) of index 2, which contains the reflections along vectors of negative norm. We consider the lattice

L = ℤ^{5} ⊂ V

and the integral orthogonal group

Γ_{L} := O^{+}(L) = O(L) ∩ O^{+}(V).

The dual lattice of L is

L′ = ℤ^{4} × \frac{1}{2}ℤ.

To introduce congruence subgroups we rescale the lattice and consider

\sqrt{l}L,  (\sqrt{l}L)′ = \frac{1}{\sqrt{l}} L′  (l ∈ ℕ).

The so-called discrimant kernel is

(\sqrt{l}L)′/(\sqrt{l}L) ≅ L′/lL.

We define the congruence sub-groups

Γ_{L}[l] := Kernel(O^{+}(L) ⟶ Aut(L′/lL)).

For l ≥ 2 they are contained in SO^{+}(V), which is the connected component of O(V). We want to introduce orthogonal modular forms on Γ_{L}[l]. For our purpose it is convenient to consider them as functions on the Siegel upper half plane ℍ_{2}. We embed it into the projective space of the complexification V_{ℂ} = ℂ^{5} by

ℍ_{2} ⟶ H_{2} ⊂ P (V_{ℂ}), Z = ( \begin{matrix}z_{0} & z_{1} \\ z_{1} & z_{2} \end{matrix} ) ⟼ [−detZ, 1, z_{0}, z_{1}, z_{2}].

The image H_{2} is an open subset, which sometimes is called the orthogonal half plane. We identify V with a certain space of matrices.

(x_{1}, …, x_{5}) ⟼ X = ( \begin{matrix}x_{5} & −x_{3} & 0 & −x_{1} \\ x_{4} & −x_{5} & x_{1} & 0 \\ 0 & −x_{2} & x_{5} & x_{4} \\ x_{2} & 0 & −x_{3} & −x_{5} \end{matrix} ) .

Let M ∈ Sp(2, ℝ) be a symplectic matrix. It is easy to check that X^{2} = q(x)E. Hence X ↦ MXM^{−1} defines an orthogonal transformation and we obtain a natural surjective homomorphism

Sp(2, ℝ) ⟶ SO^{+}(V),

which is compatible with the standard action of Sp(2, ℝ) on ℍ_{2} and of SO^{+}(V) on H_{2}. The kernel is ±E. Moreover the inverse image of SO^{+}(L) is precisely Sp(2, ℤ). Just for convenience we use this homomorphism to define the space [Γ, k, χ] of modular forms for a subgroup Γ ⊂ Γ_{L} of finite index and a character χ on Γ. By definition it equals [Γ̃, k, χ̃], where Γ̃ ⊂ Sp(2, ℤ) is the inverse image and χ̃ the pulled back character. Recall that we denoted by Γ_{2}[l] the principal congruence subgroup of the Siegel modular group. The groups Γ_{L}[l] and Γ_{2}[l] are related in a not quite trivial way. In the case l = 3 one can check:

Lemma 7. The inverse image of Γ_{L}[3] is Γ_{2}[3] ∪ −Γ_{2}[3].

We recall Borcherd’s additive lift ([Bo1, theorem 14.4): Let M = \sqrt{l}L. The input space for the additive lift is a certain space of vector valued elliptic modular forms. The values are in the group ring ℂ[M′/M]. More precisely the inputs are tuples of holomorphic functions f_{α}, α ∈ M′/M, on the upper half plane with the transformation property

\begin{aligned}f_{α}(τ + 1) & = e^{2πiq(α)}f_{α}(τ), \\ f_{α}(−\frac{1}{τ} ) & = \sqrt{#M′/M}^{\sqrt{i}−1} τ^{k} ∑_{β∈M′/M} e^{−2πi(α,β)}f_{β}(τ) \end{aligned}

and which are regular at the cusps. The additive lift is a linear map from this space to [Γ_{L}[k], k +1/2]. This lift is compatible with the action of the full group Γ_{L}, which acts on the inputs by permutation of the indices an α and on [Γ_{L}[k], k + 1/2] in the standard way. We want to use this construction in level 6. We have to consider

L′/6L = (ℤ/6Z)^{4} × ℤ/12ℤ.

There is a canonical splitting

L′/6L = (ℤ/3ℤ)^{5} × (ℤ/2ℤ)^{4} × (ℤ/4ℤ).

We have to understand the action of SO^{+}(L) ≅ Sp(2, ℤ)/± on the group ring

ℂ[L′/6L] = ℂ[(ℤ/3ℤ)^{5}] ⊗_{ℂ} ℂ[(ℤ/2ℤ)^{4} × (ℤ/4ℤ)].

The action of Sp(2, ℤ)/± is the tensor product of two actions. The first one factors through the Burkhardt group Sp(2, ℤ/3ℤ)/± and the second through Sp(2, ℤ/2ℤ).

Lemma 8. The space ℂ[(ℤ/3ℤ)^{5}] contains of each of the two irreducible five dimensional representations of the Burkhardt group precisely one copy. The space ℂ[(ℤ/2ℤ)^{4} × (ℤ/4ℤ)] contains a unique one dimensional sub-space on which Sp(2, ℤ/2ℤ) acts by its non-trivial character. The tensor product of the sum of the two five dimensional representations and the one-dimensional representation is a ten dimensional subspace W ⊂ ℂ[L′/6L].

Let C = (C_{α}) ∈ W. One checks that

η(τ)^{3}C

is an input modular form of weight 3/2. Its additive lift defines a linear map

W ⟶ [Γ_{L}[6], 2].

Using Borcherd’s explicit description of the Fourier coefficients of an additive lift it can be checked that this map is not 0. We know that this map is compatible with the action of Sp(2, ℤ). Hence the image are modular forms of level 3 and we obtain:

Proposition 10. The additive lift defines a linear map

W ⟶ [Γ_{2}[3], 2, χ],

where χ is the restriction of the non-trivial character of Sp(2, ℤ). This map is surjective, more precisely it is zero on one of the two five-dimensional summands and an isomorphism on the other.

We recall that [Γ_{2}[3], 2, χ] is spanned by the modular forms B_{1}, …, B_{5}. Hence these are actually additive lifts of η(τ)^{3}, which probably is the easiest description of them.

Following similar lines one can check (using [Bo2]) that B_{1} and its satellites are infinity products in the sense of Borcherds.

References

[An] Andrianov, A.N.: Quadratic forms and Hecke operators, Grundlehren math. Wiss. 286, Springer Verlag (1987)

[BB] Baily, W.L., and Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains, Annals of Math. 84, No 3, 442–528 (1966)

[Bo1] Borcherds, R.: Automorphic forms with singularities on Grassmannians, Invent. math. 132, 491–562 (1998)

[Bo2] Borcherds, R.: The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97, No 219–233 (1999)

[BH] Bruns, W., and Herzog J.: Cohen-Macaulay rings Cambridge Stud. Adv. Math. 39, Cambridge, University Press (1993)

[BL] Birkenhake, Ch.; Lange, H.: Cubic theta relations, J. Reine Angew. Math. 407, 167–177 (1990)

[Bu] Burkhardt, H.: Untersuchungen aus dem Gebiet der hyperelliptischen Modulfunktionen, II, Math. Ann. 38, 161–224 (1890), III, Math. Ann. 40, 313–343 (1892)

[Cob] Coble, A. B.: Point sets and allied cremona groups (Part III), Trans. AMS 18, 331–372, (1917).

[F SM] Freitag, E. Salvati-Manni, R.: The Burkhardt group and modular forms, Manuskripte der Forschergruppe Arithmatik, Universität Mannheim, Heidelberg, No 6 (2002)

[Ge] van der Geer, G.: Note on abelian schemes of level three, Math. Ann. 278 (1987), no. 1-4, 401–408.

[Hu] Hunt, B.: The geometry of some special arithmetic quotients, Lecture Notes in Math. 1637 Springer Verlag Berlin-Heidelberg (1996)

[Ig] Igusa, J.I. :On Siegel modular forms of genus two (II) Amer. J. Math. 86 , 392-412 (1964).

[JSV] de Jong, A. J.; Shepherd-Barron, N. I.; Van de Ven, A.: On the Burkhardt quartic, Math. Ann. Vol. 286 no. 1, 309-328 (1990)

[Se] Serre, J.P.: Algébre locale. Multiplicités, Lecture Notes in Mathematics 11, Berlin-New York: Springer-Verlag (1965)

[SM] Salvati Manni R.: On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure, Math. Z. Vol 221 no. 2, 231-241 (1996)