Some modular varieties of low dimension

May 1998

Eberhard Freitag Mathematisches Institut Im Neuenheimer Feld 288 D69120 Heidelberg

FREITAG@ MATHI.UNI-HEIDELBERG.DE

Carl Friedrich Hermann Mathematisches Institut Universität Mannheim D68131 Mannheim

HERMANN@ FERMAT.MATH.UNI-MANNHEIM.DE

Introduction

A classical result of Igusa states that the Siegel modular variety of degree and level two is isomorphic to a quartic in a four dimensional projective space the isomorphism being given by theta series. One very remarkable application of this result states that the degree d(Δ) of the Humbert surface of discrimant Δ occurs as Fourier-coefficient of a certain elliptic modular form [vdG1].

The modular quartic is well-known from invariant theory. It is the moduli space of six points in P^{2}(ℂ) that lie on a quadric and it is the dual of the Segre cubic which is the moduli space of six points in P^{1}(ℂ). There are some other examples where a projective model of a modular variety is given by theta series this model being also related to invariant theory and such that the degrees of modular divisors are Fourier coefficients of an elliptic modular form.

For example in the paper [He] a four-dimensional analogue has been treated. The Siegel modular group of degree 2 has been replaced by the Hermitean modular group (which is a unitary group) with respect to the ring of Gauss integers. This case is related to the moduli space of six points in P^{2}(ℂ).

One of our motivations was to look for more examples also in higher dimensions. An interesting infinite series of modular varieties which looks promising belongs to the orthogonal group O(2, n). Because of some well-known exceptional isomorphisms for algebraic groups of low dimension the mentioned two examples belong to this series.

The algebraic geometry of these varieties seems to be very rich. One reason for this is that there are many divisors —generalized Humbert surfaces— which come from embeddings O(2, n−1) → O(2, n). We call them quadratic divisors. The intersection theory of those divisors seems to by arithmetically interesting.

Generally spoken it is hard to find projective models for modular varieties. So it seems to be reasonable to modify the question as follows: Find arithmetic groups Γ ⊂ SO(2, n) such that the Baily Borel compactification X(Γ) admits a finite covering

X(Γ) → P^{n}(ℂ)

of known degree and such that the ramification divisor can explicitly be described as linear combination of quadratic divisors. In this paper we propose some examples for further study. One example is five dimensional and belongs to the group O(2, 5). We consider a subgroup Γ_{5} ⊂ O(2, 5) which leaves a certain lattice invariant. This is our “full modular group”. We have to consider a certain normal subgroup Γ_{5}[𝔭] ⊂ Γ_{5}. The quotient group is isomorphic with the Weyl group of the E_{6}-root system,

Γ_{5}/Γ_{5}[𝔭] ⟶ W (E_{6}).

The Weyl group W (E_{6}) is a reflection group which contains a simple subgroup of index 2 and of order 25 920. It has a unique 6-dimensional irreducible linear representation. Hence we have actions of W (E_{6}) on X(Γ_{5}) and on P^{5}ℂ. We construct a covering map

X(Γ_{5})[𝔭] ⟶ P^{5}ℂ,

which is equivariant with respect to these actions. The ramification divisor is irreducible and belongs to an embedded SO(2, 4)-variety. There is an exceptional isogeny between SO(2, 4) and the Hermitean symplectic group in the sense of Hel Braun. The modular groups we have to consider belong to the field ℚ(\sqrt{−3}). The group Γ_{4} is more or less the Hermitean modular group with respect to this field and the level subgroup Γ_{4}[𝔭] which we have to consider is the usual congruence group of level 2. The factor group is the unitary group U(4, 𝔽_{4}), where 𝔽_{4} denotes the field of four elements equipped with the non-trivial automorphism. This field occurs because 2 is a prime in ℚ(\sqrt{−3}). It is a classical result that SU(4, 𝔽_{4}) is isomorphic to the subgroup of index 2 of W (E_{6}). We will recover this ismorphism from our geometric configurations. Similar observations are described in papers of van Geemen [vG] and Hunt [H].

The modular variety X(Γ_{4}[𝔭]) is a two-fold covering of a hypersurface I 32 ⊂ P^{5}ℂ. With help of a calculator the equation of this hypersurface could be determined. The degree is 32. It is invariant under the action of W (E_{6}) and hence very distinguished.

Hunt discusses in [H] a certain quintic (=hypersurface of degree 5) I 5 ⊂ P^{5}ℂ. This quintic is the unique quintic which is invariant under W (E_{6}). Hunt conjectured that it is related with a four dimensional modular variety. One argument is that it contains several well-known modular verieties of dimension 3. One of them is the Segre cubic whose dual belongs to the Siegel modular group of level 2. Another one is the Nieto quintic which belongs to the level-2 subgroup of the paramodular group with respect to polarization type (1, 3). We refer to the papers [H] and [BN] for more details. It is very remarkable that both groups occur in our situation:

We take the intersection of I 32 with a certain hyperplane (which is related to a certain quadratic divisor on the modular side). This intersection is a hypersurface in P^{4}ℂ. It has two irreducible components. One of them is the Igusa quartic with multiplicity 3. The other can be defined by a polynomial of degree 20 (which we give explicitly). This hypersurface of degree 20 belongs to the paramodular group of level 2 and polarization type (1, 3).

The following picture opens: There is a close connection between Hunts I 5 and our I 32 and also between the Nieto quintic and our hypersurface of degree 20.

What could this connection be? The dual hypersurface X̂ of a hypersurface X ⊂ P^{n}(ℂ) is a hypersurface in the dual projective space. It consists of all tangent hyperplanes of X. One has to explain carefully what this means in the singular points of X. The singularities of X can be responsible for the fact that it is difficult to predict the degree of the dual variety. Hunt conjectures that I 32 is the dual of I 5 and converse*). Both should be birational equivalent. So Hunts conjecture that I 5 is modular would have been proved in a weak sense. By the way there are many hints that I 5 and I 32 are related with the moduli space of cubic surfaces. We want to come back to this point at another occasion.

Our 5-dimensional variety appears in this paper as a subvariety of a 6dimensional modular variety which belongs to O(2, 6). This group is isogenous with the quaternary symplectic group of degree two which has been studied by Krieg [K]. The group Γ_{6} which we consider is Kriegs quarternary modular group, more precisely an extension of index 4. The group Γ_{6}[𝔭] appears in this description more or less as principal congruence subgroup of level 𝔭, where 𝔭 is the two-sided ideal of the ring of Hurwitz integers generated by an element of norm 2. We construct 36 theta series which are modular forms on Γ_{6}[𝔭]. The group Γ_{6} permutes these functions upto constant factors. They play a similar role as the three Jacobi theta series in case of the elliptic modular group or the ten even thetas in the case of Siegel modular forms of genus two and similarly in the case of the Hermitean modular group with respect to the Gauß number field. The vector space generated by the 36 thetas is 6. So in contrast to the classical cases the 36 thetas must have common zeros. One of our basic observations is that the zero divisor of each of the 36 series is an irreducible “quadratic divisor” in X(Γ_{6}[𝔭]). These 36 quadratic divisors intersect in exactly one point ϱ. The existence of such a distinguished point in a modular variety is a very remarkable fact. It is connected with the fact that the homorphism

Γ_{6} ⟶ Γ_{6}/Γ_{6}[𝔭]

has a section. The factor group is isomorphic to W (E_{6}). Hence we obtain a realization as semidirect product

Γ_{6} = Γ_{6}[𝔭] · W (E_{6}).

*) This has been proved meanwhile and will be published elsewhere.

In this way W (E_{6}) appears precisely as the stabilizer of a representative of ϱ. The determination of the zeros of the 36 thetas is of course basic for our investigation. We propose two proofs of this result. The first one rests on the known fact that every divisor in a modular variety X(Γ), Γ ⊂ O(2, n), has to intersect the Baily Borel boundary if one excludes some special low dimensional cases (as for examples the Hilbert modular surfaces). This follows from results of Harris and Li [HL] and also from the theory of the Borcherds products. The second method rests directly on the theory of Borcherds products. Borcherds constructed in [Bo1,Bo2] a marvellous machine which produces from certain vector valued elliptic modular forms meromorphic modular forms on O(2, n) with known poles and zeros all lying on quadrics. The weight, poles and zeros of such modular forms depend on the Fourier coefficients of the the starting elliptic modular forms. We refer to [GN] for some more examples of Borcherds products.

As a consequence of the existence of the base point ϱ we do not obtain X(Γ_{6}[𝔭]) as covering of a 6-dimensional projective space. But we succeed to get it as a covering of a twisted projective space by constructing a further modular form which does not vanish in ϱ.

This should be a basic result for a further investigation of the quaternary modular variety. Further modular forms have to been constructed and relations have to be determined. There are some results of Krieg in this direction.

This paper makes use of the link between orthogonal groups and symplectic groups. The orthogonal side is used to investigate the quadratic divisors, the symplectic side is used in different contexts, for example the theta series are constructed in the symplectic context, the group U(4, 𝔽_{4}) comes from this side. The connection between the SO(2, n)-world and the symplectic groups comes from modular embeddings of SO(2, n) into symplectic groups Sp(N, ℝ). These embeddings have been described by Satake [Sa] in a very general context. It uses the spin-covering of the orthogonal group. In the first sections we describe briefly a short and elementary approach to the symplectic embeddings and obtain in this way the well known exceptional isogenies between of O(2, n) for n = 3, 4, 6 with the symplectic group, the Hermitean symplectic group and the quaternary symplectic group (all of degree 2).

In this paper we jump between the two worlds as we find it convenient. It seems to us that the symplectic world is important in this context. We cannot avoid it completely bacause we have to derive theta relations and it is not clear for us how to do this in the orthogonal context. The symplectic modular forms have been investigated by many authors and we could use several results. The theory of modular forms on the orthogonal group is less developped. The orthogonal context is unavoidable for our purpose. The group O(2, 5) for example is not isogenous to a symplectic group. So if someone wants to rewrite this paper in a pure world he should use the orthogonal side and not the symplectic one.

There are more interesting configurations in X(Γ_{6}[𝔭]) as we mentioned in this introduction. Some of them will be described in the text. Many of them are still undiscovered and cases of dimension n > 6 are not treated here. We consider this paper only as a first step into a new world of geometric configurations with high symmetry and beauty.

For helpful discussions we have to thank S. Böge, Borcherds, Gritsenko, Krieg, Runge and van Geemen. O. Herrmann supported us with computer calculations. Our special thanks are due to Bruce Hunt who helped us during an intensive period of detailed discussions and who also supported us with computer calculations. For example his conjecture on the duality between I 5 and I 32 has been supported through such calculations.

I. The orthogonal group of signature (2,n)

1. The real orthogonal group and the spin-group

Let V be a finite dimensional real vector space which is equipped with a quadratic form

q : V ⟶ ℝ.

The associated bilinear form is

⟨a, b⟩ = q(a + b) − q(a) − q(b).

We always assume that this bilinear form is nondegenerate. In suitable coordinates it has the form

q(x) = x_{1}^{2} + ⋯ + x_{m}^{2} − x_{m+1}^{2} − ⋯ − x_{m+n}^{2},

where (m, n) is the so-called signature. We are mainly interested in the case m = 2.

The orthogonal group O(V, q) consists of all g ∈ GL(V) which preserve the quadratic form q. It is determined up to isomorphism by the signature. Hence we can use the notation

O(m, n) = O(V, q).

For our purpose it is convenient to introduce the spin-covering of the orthogonal group. We recall briefly its definition:

Let C(V) denote the Clifford algebra of the quadratic space (V, q). This is a algebra which contains V as a subvector space and which is generated by this space. The defining relations are

ab + ba = ⟨a, b⟩ (a, b ∈ V).

We denote the even part of the Clifford algebra by C^{+}(V). By definition it is generated by the two-products ab, a, b ∈ V. The so-called main involution of C(V) is denoted by a ↦ a′. This is an involutive anti-automorphism which acts on V as the negative of the identity,

(a + b)′ = a′ + b′, (ab)′ = b′a′, a′ = −a for a ∈ V.

The even part C^{+}(V) is invariant under this involution.

The spin-group Spin(V) consists of all g ∈ C^{+}(V) with the properties

g′g = 1 and gV g^{−1} = V.

The resulting transformations of V

x ↦ gxg^{−1}

are orthogonal. This defines a homomorphism

Spin(V) ⟶ O(V).

The image is the so-called spinor kernel. This spinor kernel is the connected component of the orthogonal group. The spin group is a two-fold covering of the spinor kernel O(V). We use the notations

Spin(m, n) = Spin(V), O(m, n) = O(V).

We now assume m = 2, n ≥ 2.

We replace our quadratic form by the equivalent one

q(x) := x_{1}x_{2} + x_{3}x_{4} − x_{5}^{2} − ⋯ − x_{n+2}^{2}.

The Gram matrix of the associate bilinear form is

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

§1. The real orthogonal group and the spin-group 7

We denote the standard basis of V by

f_{1}, …, f_{4}, e_{1}, …, e_{n−2}.

By means of the subspaces

H_{1}(ℝ) := ℝf_{1} + ℝf_{2}, H_{2}(ℝ) := ℝf_{3} + ℝf_{4}, V_{0} := ℝe_{1} + ⋯ + ℝe_{n−2}

we obtain an orthogonal decomposition

V = H_{1}(ℝ) ⊕ H_{2}(ℝ) ⊕ V_{0}.

The spaces H_{1}(ℝ), H_{2}(ℝ) are hyperbolic planes,

⟨f_{1}, f_{1}⟩ = ⟨f_{2}, f_{2}⟩ = ⟨f_{3}, f_{3}⟩ = ⟨f_{4}, f_{4}⟩ = 0, ⟨f_{1}, f_{2}⟩ = ⟨f_{3}, f_{4}⟩ = 1.

The restricion of q to V_{0} is negative definit

⟨e_{i}, e_{j}⟩ = −2δ_{ij}, (1 ≤ i, j ≤ n − 2).

The following description of the structure of the Clifford algebra in this special case will be very useful for a link between the spin group and the symplectic group.

Let 𝔄 by an involutive algebra, i.e. an unital associative ℝ-algabra which is equipped with an involution a ↦ a′. We extend this involution to matrices with entries from 𝔄 by the formula

(m_{ij})^{∗} := (m′_{ji}).

1.1 Lemma. There exists a commutative diagram of algebras

C(V) ⟶^{∼} M_{4}(C(V_{0}))
∪  ∪
C^{+}(V) ⟶^{∼} M_{4}(C^{+}(V_{0})).

The main involution of C(V) corresponds to the involution

( \begin{matrix}a & b \\ c & d \end{matrix} ) ⟼ ( \begin{matrix}d^{∗} & −b^{∗} \\ −c^{∗} & a^{∗} \end{matrix} ) .

(The blocs a, b, c, d are 2 × 2-matrices). The diagram is defined by the assignments

f_{1} ⟼ e_{1} ( \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ) ,

f_{2} ⟼ e_{1} ( \begin{matrix}0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 \end{matrix} ) ,

f_{3} ⟼ e_{1} ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) ,

f_{4} ⟼ e_{1} ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ) ,

e_{1} ⟼ e_{1} ( \begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) ,

e_{i} ⟼ e_{i} ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , 2 ≤ i ≤ n − 2.

There is a similar representation of the spin group Spin(1, n) by 2 × 2-matrices [EGM]. Proof of lemma 1.1. The images of the basis elements satisfy the defining relations of the Clifford algebra. By the universal property of the Clifford algebra this extends to a homomorphism C(V) → M_{4}(C(V_{0})). One verifies that this homomorphism is surjective. A dimension argument shows that it is an isomorphism. The rest is clear. ⊔ ⊓

The (Hermitean) symplectic group (of degree two) Sp(2, 𝔄) of an involutive algebra 𝔄 consists of all 4 × 4-matrices M with entries from 𝔄, such that

M^{∗}IM = I,  I = ( \begin{matrix}0 & e \\ −e & 0 \end{matrix} )  (e = unit matrix).

From the above lemma we obtain

1.2 Lemma. The restricion of the homomorphism 1.1 defines an embedding

Spin(V) ⟶ Sp(2, 𝔄),

where 𝔄 is the algebra C^{+}(V_{0}) equipped with the main involution.

The structure of the algebra 𝔄 can be described.

§2. The action of the orthogonal group of signature (2, n) on a tube domain 9

1.3 Lemma. Let be n − 3 = 8q + r, 0 ≤ r < 7. The algebra C^{+}(V_{0}) of the negative definite space V_{0} is isomorphic

\{ \begin{matrix}M_{2^{4q}}(ℝ) & r = 0, \\ M_{2^{4q}}(ℂ) & r = 1, \\ M_{2^{4q}}(ℍ) & r = 2, \\ M_{2^{4q}}(ℍ) × M_{2^{4q}}(ℍ) & r = 3, \\ M_{2^{4q+1}}(ℍ) & r = 4, \\ M_{2^{4q+2}}(ℂ) & r = 5, \\ M_{2^{4q+3}}(ℝ) & r = 6, \\ M_{2^{4q+3}}(ℝ) × M_{2^{4q+3}}(ℝ) & r = 7. \end{matrix}

The main involution corresponds in all cases to the standard involution A ↦ Ā′ resp. (A_{1}, A_{2}) ↦ (Ā′_{1}, Ā′_{2}).

We will not prove this lemma in full generality, because we don’t need it. The special case n = 6 will be treated in section 3.

Using the standard representations of ℂ resp. ℍ by real 2 × 2- resp. 4 × 4 matrices we obtain in all cases an embedding of Spin(2, n) into a real symplectic group Sp(N, ℝ). These embeddings are compatible with the complex structures of the associated Hermitean symmetric domains. Those embeddings have been described by Satke in a more general context using different methods [Sa].

2. The action of the orthogonal group of signature (2,n) on a tube domain.

We give two different descriptions of the Hermitean symmetric domain associated to O(2, n), the first one in the orthogonal, the second one in the symplectic context.

Again we assume that the signature of (V, q) is (2, n). We look at V as an orthogonal sum of two hyperbolic planes and a vector space V_{0} equipped with a negative definite quadratic form

V = H_{1}(ℝ) ⊥ H_{2}(ℝ) ⊥ V_{0}.

We consider the complexification

V (ℂ) := V ⊗_{ℝ} ℂ.

The bilinear form ⟨ , ⟩ is extended to a ℂ-bilinear form on V (ℂ). We have to consider the associated complex projective space P (V (ℂ)), which consists of all one-dimensional subspaces of V. We have a natural projection

V (ℂ) − {0} ⟶ P (V (ℂ)), z ⟼ [z].

The zero-quadric consists of all elements [z] with ⟨z, z⟩ = 0. This is a complex manifold. We have to consider the subset

K = K(V_{0}) := { [z] ∈ P (V (ℂ)); ⟨z, z⟩ = 0, ⟨z, z̄⟩ > 0 }.

The defining conditions are equivalent with

⟨x, y⟩ = 0, ⟨x, x⟩ = ⟨y, y⟩, ⟨y, y⟩ > 0 (z = x + iy).

It is easy to see and well-known that this set has two connected components. The group O(V) acts on K(V_{0}). A distinguished point of K(V_{0}) is

[i, i, 1, 1; 0, …, 0].

The connected component of K(V_{0}) which contains this point is denoted by K^{+} = K(V_{0})^{+}. The subgroup of O(V) which preserves K(V_{0})^{+} is denoted by O^{+}(V). It is an open subgroup of index two. The negative of the identity − id is contained in O^{+}(V). It acts as identity on K. Therefore the group O^{+}(V)/{± id} will be more important for us than O^{+}(V) itselve.

Sometimes it is useful to have the following modified description of K.

2.1 Remark. The assignment

z ↦ W := ℝx + ℝy

defines a bijection between K^{+}(V_{0}) and the set of all two-dimensional subspaces W ⊂ V such that the restriction of q to W is positive definite.

This is clear because every positive definite subspace admits an orthogonal basis of vectors of equal length. ⊔ ⊓

It is useful to have a realization of K^{+} as a tube domain. For this purpose we notice that for [z] ∈ K none of the 4 hyperbolic components (the first four) vanishes. This is clear because H_{1} ⊕ V_{0} contains no two dimensional positive definite subspace. This means that each element [z] ∈ K contains a unique representant z with the property z_{4} = 1. From the condition ⟨z, z⟩ = 0 we obtain that z is of the form

z = (z_{0}, z_{1}, −z_{0}z_{1} − q(𝔷), 1, 𝔷).

We obtain:

2.2 Remark. Consider the domain of all

(z_{0}, z_{2}, 𝔷) ∈ H_{1}(ℂ) × V_{0}(ℂ); y_{0}y_{2} + q(𝔶) > 0.

The assignment

(z_{0}, z_{2}, 𝔷) ⟼ [z_{0}, z_{2}, −z_{0}z_{2} − q(𝔷), 1, 𝔷)]

is a biholomorphic mapping from this domain onto K. The connected component K^{+} is defined by y_{0} > 0.

We want to describe the action of the spin group in this model. For this purpose we need an embedding of V_{0} into the even part of the Clifford algebra,

V_{0} ⟶ C^{+}(V_{0}), w ⟼ e_{1}w.

We denote the image of V_{0} by

W := e_{1}V_{0} ⊂ C^{+}(V_{0}).

2.3 Remark. The vector space

W := e_{1}V_{0}

is invariant under the main involution x ↦ x′ of C^{+}(V_{0}). A basis of W is given by

e_{1}e_{2}, …, e_{1}e_{n−2} together with − 1 = e_{1}e_{1}.

The involution x ↦ x′ fixes the last vector and changes the sign of the other ones. This means that the involution x ↦ −x′ is a reflection.

This involution x ↦ −x′ is a reflection in the Euclidean sense if one equippes W with the following positive definit scalar product:

(e_{1}x, e_{1}y) = −\frac{1}{2}⟨x, y⟩ (x, y ∈ V_{0}).

A simple calculation shows:

2.4 Remark. We have

x′x = (x, x) (x ∈ W).

We denote by X the set of all ”Hermitean“ 2 × 2-matrices

x = ( \begin{matrix}x_{0} & x_{1} \\ x′_{1} & x_{2} \end{matrix} ) where x_{0}, x_{2} ∈ ℝ, x_{1} ∈ W.

By definition the determinant of a matrix x ∈ X is the real number

det x := x_{0}x_{2} − x′_{1}x_{1} = x_{0}x_{2} − (x_{1}, x_{1}).

If the matrix x ∈ X is an invertible element of the algebra M_{2}(C^{+}(V_{0})), the inverse is contained in ℝ[x]. This implies det x ≠ 0. Conversely each matrix with this property is invertible and one has

x^{−1} = \frac{1}{det x} ( \begin{matrix}x_{2} & −x_{1} \\ −x′_{1} & x_{0} \end{matrix} ) .

A matrix x ∈ X is called positive — x > 0 —, if

x_{0} > 0 and det x > 0.

Then x_{2} is positive too. The positive cone P is defined by

P := {y ∈ X; y > 0}.

One easily verifies that the positive cone consists of the elements of squares of invertible x.

The tube domain associated with this cone is

H := X + iP ⊂ Z := X ⊗_{ℝ} ℂ.

This domain is the same as the one considered in 2.2, if we identify V_{0} with W by means of the map z_{1} ↦ e_{1}z_{1}. The advantage of this modification is that we obtain now a very simple formula for the action of the spin group.

2.5 Proposition. The spin group Spin(V) acts via the representation 1.2 on the tube domain H as group of biholomorphic transformations by

Mz := (az + b)(cz + d)^{−1}, M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Spin(V), z ∈ H.

The assignment

z = ( \begin{matrix}z_{0} & z_{1} \\ z′_{1} & z_{2} \end{matrix} ) ⟼ [z_{0}, z_{2}, − det z, 1, e_{1}z_{1}]

defines a biholomorphic map from H onto K^{+}. This map is equivariant in the sense that the diagram

Spin(2, n) × H ⟶ H
↓  ↓
O(2, n) × K ⟶ K

commutes.

Proof. Special elements of the group Sp(2, 𝔄) are

( \begin{matrix}e & h \\ 0 & e \end{matrix} ) , h ∈ X; ( \begin{matrix}u′ & 0 \\ 0 & u^{−1} \end{matrix} ) where u = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) , t ∈ W.

Explicit conjugation of the matrices in lemma 1.1 shows that they and their conjugates belong to the spin group. We give one example:

( \begin{matrix}e & h \\ 0 & e \end{matrix} ) , h = ( \begin{matrix}0 & e_{1}e_{2} \\ −e_{1}e_{2} & 0 \end{matrix} ) .

The vector x_{1}f_{1} + ⋯ + x_{4}f_{4} + y_{1}e_{1} + ⋯ + y_{n−2}e_{n−2} corresponds to the matrix

\begin{matrix}e_{1} ( \begin{matrix}A & B \\ C & A′ \end{matrix} ) + yE, where A = ( \begin{matrix}−y_{1} & x_{1} \\ −x_{2} & y_{1} \end{matrix} ) , B = ( \begin{matrix}0 & x_{3} \\ −x_{3} & 0 \end{matrix} ) , \\ C = ( \begin{matrix}0 & x_{4} \\ −x_{4} & 0 \end{matrix} ) , y = y_{2}e_{2} + ⋯ y_{n−2}e_{n−2}. \end{matrix}

A straightforward calculation gives

( \begin{matrix}e & h \\ 0 & e \end{matrix} ) (e_{1} ( \begin{matrix}A & B \\ C & A′ \end{matrix} ) + yE) ( \begin{matrix}e & −h \\ 0 & e \end{matrix} ) = e_{1} ( \begin{matrix}A & B̃ \\ C & A′ \end{matrix} ) + ỹE

with

x̃_{3} = x_{3} + x_{4} − 2y_{2}, ỹ = y_{1}e_{1} + (y_{2} − x_{4})e_{2} + y_{3}e_{3} + ⋯ y_{n−2}e_{n−2}.

We obtain a commutative diagramm

( \begin{matrix}z_{0} & z_{1} \\ ∗ & z_{2} \end{matrix} )  ⟼  [z_{0}, z_{2}, ∗, 1, e_{1}z_{1}]
↓^{−}  ↓^{−}
( \begin{matrix}z_{0} & z_{1} + e_{1}e_{2} \\ ∗ & z_{2} \end{matrix} ) ⟼ [z_{0}, z_{2}, ∗, 1, e_{1}(z_{1} + e_{1}e_{2})]

as stated in proposition 2.5. The complete proof of proposition 2.5 will follow from

2.6 Proposition. In the following list the left side contains elements of the spin group and the right side the corresponding orthogonal transformations. These orthogonal transformations are described by the image of vectors (x_{1}, x_{2}, x_{3}, x_{4}; y) ∈ V, y ∈ V_{0}. (We identify V_{0} and W by means of y ↦ e_{1}y.)

( \begin{matrix}1 & h & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & −h′ & 1 \end{matrix} )  (x_{1} + ⟨h, y⟩ − q(h)x_{2}, x_{2}, x_{3}, x_{4}; y − x_{2}h)

( \begin{matrix}1 & 0 & 0 & 0 \\ h′ & 1 & 0 & 0 \\ 0 & 0 & 1 & −h \\ 0 & 0 & 0 & 1 \end{matrix} )  (x_{1}, x_{2} + ⟨h, y⟩ − q(h)x_{1}, x_{3}, x_{4}; y − x_{1}h)

( \begin{matrix}1 & 0 & 0 & h \\ 0 & 1 & h′ & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )  (x_{1}, x_{2}, x_{3} + ⟨h, y⟩ − q(h)x_{4}, x_{4}; y − x_{4}h)

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & h & 1 & 0 \\ h′ & 0 & 0 & 1 \end{matrix} )  (x_{1}, x_{2}, x_{3}, x_{4} + ⟨h, y⟩ − q(h)x_{3}; y − x_{3}h)

( \begin{matrix}1 & 0 & t & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )  (x_{1} + tx_{4}, x_{2}, x_{3} − tx_{2}, x_{4}; y)

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )  (x_{1}, x_{2} + tx_{4}, x_{3} − tx_{1}, x_{4}; y)

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ t & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )  (x_{1}, x_{2} − tx_{3}, x_{4} + tx_{1}; y)

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & t & 0 & 1 \end{matrix} )  (x_{1} − tx_{3}, x_{2}, x_{3}, x_{4} + tx_{2}; y)

The proof is given by straightforward calculations.  ⊔ ⊓

The corresponding orthogonal transformations in 2.5 are so-called Eichler transformations which we will introduce a little later (4.2, 5.1). It is well known and follows from the Eichler calculus in section 4, that these special elements and their transposed generate the spin group. The commutativity of the diagram in 2.5 has to be proved only for the generators. This is easy.

⊔ ⊓

§3. The quaternary symplectic group as orthogonal group 15

3. The quaternary symplectic group as orthogonal group

We consider the special case n = 6. Let’s recall the basic properties of the field ℍ of Hamilton quaternions: We denote the standard basis of this field by 1, i_{1}, i_{2}, i_{3},

ℍ = ℝ + ℝi_{1} + ℝi_{2} + ℝi_{3}.

The defining relations are

i_{1}^{2} = i_{2}^{2} = i_{3}^{2} = −1,
i_{1}i_{2} = −i_{2}i_{1} = i_{3}, i_{2}i_{3} = −i_{3}i_{2} = i_{1}, i_{3}i_{1} = −i_{1}i_{3} = i_{2}.

The conjugate x̄ of a quaternion x = x_{0} + x_{1}i_{1} + x_{2}i_{2} + x_{3}i_{3} is

x̄ := x_{0} − x_{1}i_{1} − x_{2}i_{2} − x_{3}i_{3}.

The norm of a quaternion is the nonnegative real number

|x|^{2} = x̄x = x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2},

its real part is

Re x := x_{0} = \frac{x + x̄}{2}.

The following lemma implies a special case (n = 6) of Lemma 1.3.

3.1 Lemma. Assume n = 6. There exists an isomorphism of algebras

C^{+}(ℍ) ⟶^{∼} ℍ × ℍ,

given by

e_{1}e_{2} ↦ (i_{1}, i_{1}), e_{1}e_{3} ↦ (i_{2}, i_{2}), e_{1}e_{4} ↦ (i_{3}, −i_{3}).

The main involution of C^{+}(V_{0}) corresponds to the standard involution (a, b) ↦ (ā, b̄) on ℍ × ℍ.

The proof is easy and omitted.

In the following we take as model for V_{0} the field of quaternions V_{0} := ℍ with the basis

e_{1} := 1, e_{2} := i_{1}, e_{3} := i_{2}, e_{4} := i_{3}

(and with the quadratic form q(x) = −x̄x). We obtain a homomomorphism

Spin(V) ⟶ Sp(2, ℍ) × Sp(2, ℍ).

The projection to one of the factors gives a homomorphism

Spin(V) ⟶ Sp(2, ℍ).

To be concrete we take the projection to the first factor*). We will see that this homomorphism is an isogeny, which means that it has a finite kernel and that the image is a subgroup of finite index. We introduced a vector space W = e_{1}V_{0}. Projecting it we get an isomorphism

W ⟶^{∼} ℍ.

The composition of the maps

ℍ = V_{0} ⟶ W ⟶ ℍ

is the identity. In this way we may identify

V_{0} = W = ℍ.

The main-involution x ↦ x′ on W corresponds to the standard conjugation x ↦ x̄ on ℍ. Now X (ℍ) is the set of all Hermitean matrices

x = ( \begin{matrix}x_{0} & x_{1} \\ x̄_{1} & x_{2} \end{matrix} ) , x_{0}, x_{2} ∈ ℝ, x_{1} ∈ ℍ

and

P(ℍ) = {y ∈ X (ℍ); y_{0} > 0, det y > 0} (det y := y_{0}y_{2} − |y|^{2}).

The associated tube-domain is

H(ℍ) = X (ℍ) + iP(ℍ) ⊂ Z(ℍ).

The group Sp(2, ℍ) acts on H_{6} by means of the usual formula

z ↦ (az + b)(cz + d)^{−1}.

The ”determinant“

det : X (ℍ) ⟶ ℝ, det(x) = x_{0}x_{2} − |x_{1}|^{2},

extends to a holomorphic function on the complexification

det : Z(ℍ) ⟶ ℂ,

*) The occurance of the two projections is connected with the existence of two spin representations. As soon as one developes a theory of vector-valued modular forms both spin representations will occur. In this paper only scalar valued modular forms will be treated. Therefore it is sufficient to restrict to one of the spin representations which means to consider one of the two projections.

which can be described explicitly as follows: We extend the conjugation x ↦ x̄ on ℍ as ℂ-linear map to ℍ ⊗_{ℝ} ℂ. We denote this extension by z ↦ z′. (This involution changes the signs of i_{1}, i_{2}, i_{3} ∈ ℍ but fixes i ∈ ℂ. Hence we avoid to use the notation z ↦ z̄.) We have

(x + iy)′ = x̄ + iȳ.

The elements of Z(ℍ) are of the form

z = ( \begin{matrix}z_{0} & z_{1} \\ z′_{1} & z_{2} \end{matrix} ) , z_{0}, z_{2} ∈ ℂ, z_{1} ∈ ℍ ⊗_{ℝ} ℂ.

We obviously have

det z = z_{0}z_{2} − z_{1}z′_{1}.

The product z_{1}z′_{1} (which is taken in the algebra ℍ ⊗_{ℝ} ℂ) is contained in the ground field ℂ.

The embedding of H(ℍ) into the zero-quadric in P (V (ℂ)) is given by the formula

z ⟼ [z_{0}, z_{2}, − det z, 1, −z_{1}].

If we denote by Bihol(H(ℍ)) the group of all biholomorphic transformations we obtain embeddings

O^{+}(V)/{± id} ⊂ Bihol(H(ℍ)), Sp(2, ℍ)/{±E} ⊂ Bihol(H(ℍ)).

It is well known that the second group is a subgroup of index 2 in Bihol(H(ℍ)) [Kr]. The whole group is generated by the additional involution z ↦ z′. This substitution is contained in the image of O^{+}(V) but not in the image of the special orthogonal group. We obtain identifications

Sp(2, ℍ)/{±E} = SO^{+}(V)/{± id} ⊂ O^{+}(V)/{± id} = Bihol(H(ℍ)).

Explicit formulae for this isomorphism on the level of standard generators result from proposition 2.6. But it is not possible to find simple closed formulae in general. The reason is that the isomorphism Sp(2, ℍ)/{±E} → SO^{+}(V)/{± id} does not lift to a homomorphism Sp(2, ℍ) → SO^{+}(V). If one replaces the skew field ℍ by one of the commutative subfields ℝ, ℂ, the homomorphism lifts [He].

In the following we consider O^{+}(V)/{± id} and Sp(2, ℍ)/{±E} as subgroups of

Ω(ℍ) := Bihol(H(ℍ)).

The compact group O(4, ℝ) = O(ℍ) also acts faithfully on H(ℍ) because it is contained in O^{+}(V). An element σ ∈ O(4, ℝ) acts as

( \begin{matrix}z_{0} & z_{1} \\ ∗ & z_{2} \end{matrix} ) ⟼ ( \begin{matrix}z_{0} & σ(z_{1}) \\ ∗ & z_{2} \end{matrix} ) .

18 I. The orthogonal group of signature (2,n)

We obtain an embedding

O(4, ℝ) ⟶ Ω(ℍ).

Every orthogonal transformation of determinant 1 can be written in the form

z ⟼ azb̄

with quaternions a, b of absolute value 1. The pair (a, b) is determined upto sign. This means that we have an isomorphism

SO(4, ℝ) ≅ (ℍ^{∗} × ℍ^{∗})/{±(1, 1)}.

These transformations belong to diagonal matrices in the symplectic group: 3.2 Remark. The matrix

( \begin{matrix}U & 0 \\ 0 & Ū^{−1} \end{matrix} ) , U = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) ,

corresponds modulo ± id to the orthogonal transformation

(x_{1}, x_{2}, x_{3}, x_{4}; y) ⟼ (\frac{|b|}{|a|}x_{1}, \frac{|a|}{|b|}x_{2}, |a||b|x_{3}, \frac{x_{4}}{|a||b|}; \frac{ayb̄}{|a||b|}).

The proof is trivial.  ⊔ ⊓

But the orthogonal transformation

τ(x) = x̄

(more precisely its image in Ω(ℍ)) is not contained in the connected component. As we already mentioned

Ω(ℍ) = Ω(ℍ)^{0} ∪ τΩ(ℍ)^{0}.

II. Modular groups and quadratic divisors

4. Integral orthogonal groups

The integral orthogonal group with respect to a lattice L ⊂ V is defined as

O(L) := { g ∈ O(V); g(L) = L }.

The group O(L) acts on the set of vetors a ∈ L of a given ”norm“ q(a). We need information about the orbits.

Eichler calculus

We assume that the lattice L is even*), i.e. q(a) ∈ ℤ for all a ∈ L. This implies that L is contained in its dual lattice

L^{∗} := { a ∈ V; ⟨a, x⟩ ∈ ℤ for all x ∈ L }.

The discriminant group is the finite abelian group

Dis(L) := L^{∗}/L.

The group O(L) acts on Dis(L). The discriminant kernel of O(L) is

O^{dis}(L) := kernel (O(L) ⟶ Aut(Dis(L))).

The quadratic form q usually is not integral on L^{∗}. It is sometimes useful to replace it by an integral multiple.

4.1 Definition. Let L ⊂ L^{∗} be an integral lattice, d the smallest natural number such that dq is integral on L^{∗}. The discriminant of a vector a ∈ L^{∗} is defined as

Δ(a) := −dq(a).

The induced function on the discrimant group is

Δ̄ : Dis(L) = L^{∗}/L ⟶ ℤ/dℤ.

For a detailed study of integral orthogonal groups the Eichler transformations E(u, v) play a fundamental role:

*) Because of ⟨x, x⟩ = 2q(x) this means that ⟨x, x⟩ is even for all x ∈ L.

4.2 Remark. Let u be an isotropic element, i.e. q(u) = 0, and v an element which is orthogonal to u, i.e. ⟨u, v⟩ = 0. The element 1 + uv is contained in the spin group. The image in the orthogonal group is

E(u, v)(a) = a − ⟨a, u⟩v + ⟨a, v⟩u − q(v)⟨a, u⟩u (⟨v, v⟩ = 2q(v)).

In the case u, v ∈ L this element is contained in the discriminant kernel.

We are interested in the very special situation where L contains two hyperbolic planes. This means by definition that L admits an orthogonal decomposition

L = H_{1} ⊕ H_{2} ⊕ L_{0},

H_{1}, H_{2} are two integral hyperbolic planes, i.e.

\begin{aligned}H_{1} & = ℤf_{1} + ℤf_{2}, \\ H_{2} & = ℤf_{3} + ℤf_{4}, \end{aligned}  \begin{aligned}q(x_{1}f_{1} + x_{2}f_{2}) & = x_{1}x_{2}, \\ q(x_{3}f_{3} + x_{4}f_{4}) & = x_{3}x_{4}. \end{aligned}

4.3 Definition. Let be

L = H_{1} ⊕ H_{2} ⊕ L_{0}.

The group

EO(L_{0}) ⊂ O^{dis}(L)

is generated by all Eichler transformations of the form E(f_{i}, v), 1 ≤ i ≤ 4, where v ∈ L is orthogonal to f_{i}.

The group O(L) acts also on the dual lattice L^{∗}. We want to determine the orbits of the subgroup EO(L_{0}) with respect to this action. A vector a ∈ L^{∗} is called primitive (in L^{∗}) if

ℚa ∩ L^{∗} = ℤa.

Every nonzero vector of L^{∗} is an integral multiple of a primitive one. The group O(L) acts on the set of primitive vectors. Therefore it is sufficient to determine the orbits of the primitive vectors.

4.4 Lemma. Let L be a lattice which contains two hyperbolic planes. So EO(L_{0}) is defined. Two primitive vectors a, b ∈ L^{∗} of the same norm q(a) = q(b) are contained in the same orbit of EO(L_{0}) iff they have the same image in the discrimant group Dis(L).

This result was communicated to one of the authors by V. Gritsenko and apostrophed as well-known. Because the result was new for us we include a proof.

Proof of 4.4. There exists an isomorphism

H_{1} × H_{2} ⟶^{∼} M_{2}(ℤ),

§4. Integral orthogonal groups 21

such that q corresponds to the determinant on M_{2}(ℤ). The group SL(2, ℤ) acts on M_{2}(ℤ) by multiplication from both sides. This gives a homomorphism

SL(2, ℤ) × SL(2, ℤ) ⟶ O(H_{1} ⊕ H_{2}).

The image is a subgroup of index two of O^{+}(H_{1} ⊕H_{2}). The theorem of elementary divisors for SL(2, ℤ) shows, that every element a ∈ L^{∗} can be transformed into L_{1}^{∗}, where L_{1} := {0} ⊕ H_{2} ⊕ L_{0}.

Consider now two vectors a, b as in 4.4. We may assume that both are contained in L_{1}^{∗}. Because both vectors are primitive there exist a′, b′ ∈ L_{1} such that

⟨a, a′⟩ = ⟨b, b′⟩ = 1.

A simple calculation shows

E(f_{1}, a′)(a) = a − f_{1}, E(f_{1}, b′)(b) = b − f_{1}.

Another straightforward computation gives

E(f_{2}, b − a)(a − f_{1}) = b − f_{1}.  ⊔ ⊓

The dual lattice of L is H_{1} + H_{2} + L_{0}^{∗}. The discriminant groups of L and L_{0} are the same,

Dis(L) = Dis(L_{0}).

The orthogonal group O(L_{0}) is embedded in O(L). An element of O(L_{0}) acts trivially on H_{1} + H_{2}. In our applications the restriction of q to L_{0} will be definite. Then O(L_{0}) is a finite group.

4.5 Lemma. We use the notations of 4.3. Let Γ ⊂ O(L) be a subgroup which contains EO(L). Assume that Γ acts transitively on the sets of elements of Dis(L) with the same value of Δ̄. Then Γ acts transitively on the set of primitive vectors of L^{∗} of a given norm.

We consider some examples. Our starting vector space is

V = ℝ^{4} ⊕ ℍ

equipped with the quadratic form

q(x_{1}, x_{2}, x_{3}, x_{4}, h) := x_{1}x_{2} + x_{3}x_{4} − |h|^{2}.

The first lattice which we consider is L = ℤ^{4} ⊕ L_{0} where L_{0} = 𝔬 is the ring of Hurwitz quaternions. It contains

𝔬_{0} := ℤ + ℤi_{1} + ℤi_{2} + ℤi_{3}

as subgroup of index two which is extended by the element

ω := \frac{1 + i_{1} + i_{2} + i_{3}}{2} .

The quadratic form q is integral on 𝔬. The dual lattice is

𝔬^{∗} = 𝔭/2, 𝔭 := {a ∈ 𝔬_{0}; a_{1} + ⋯ + a_{4} ≡ 0 mod 2}.

The set 𝔭 is a two-sided ideal,

𝔭 = (1 + i_{1})𝔬 = 𝔬(1 + i_{1}).

We have

L^{∗}/L ≅ (𝔭/2)/𝔬 ≅ 𝔬/𝔭.

The discriminant of an element a ∈ L^{∗} is

Δ(a) = −2q(a).

The quotient

𝔽_{4} := 𝔬/𝔭

is a field of 4 elements which are represented by 0, 1, ω, ω̄. The orthogonal group Σ of 𝔬 contains all multiplications by units of 𝔬. The units are the 24 elements

±1, ±i_{1}, ±i_{2} ± i_{3}, \frac{±1 ± i_{1} ± i_{2} ± i_{3}}{2} .

From this it is clear that the transformation x ↦ ωx generates a subgroup of the orthogonal group of 𝔬 which acts transitively on the three nonzero elements of 𝔽_{4}. The zero-element and the nonzero-elements are seperated by Δ̄.

Recall that the orthogonal group of 𝔬 is considered as a subgroup of O(L). Now 4.5 applies and shows that the group

⟨EO(𝔬), x ↦ ωx⟩

acts transitively on the set of primitive vectors a ∈ L^{∗} of a fixed norm. In the next example we replace 𝔬 by the three-dimensional sublattice

L_{0} := {a ∈ 𝔬; a_{4} = 0}.

We have

L_{0} = ℤ + ℤi_{1} + ℤi_{2}.

The dual lattice is

L_{0}^{∗} = \frac{1}{2}L_{0}.

The discriminant is

Δ(a) = −4q(a).

For an element a ∈ L^{∗} the number of odd entries in 2a is determined by Δ̄(a). This implies that two elements of Dis(L_{1}) which are images of primitive elements of the same norm are equivalent mod O(3, ℤ) = O(L_{0}). This group consists of permutations combined with possible changes of signs. We obtain that the group

⟨EO(L_{0}), O(3, ℤ)⟩ (L_{0} = {x ∈ 𝔬; x_{4} = 0})

acts transitively on the set of primitive vectors of a fixed norm. It is sufficient to take one of the subgroups SO(3, ℤ), S_{3} instead of O(3, ℤ).

Another sublattice of 𝔬 which will play a role is

L_{0} := {x ∈ 𝔬; x_{1} = x_{2}}.

This lattice is generated by ω, i_{2}, i_{3}. Its dual lattice is

L_{0}^{∗} = {x ∈ ℍ; x_{1} = x_{2}, x_{1} + x_{2} + x_{3} + x_{4} ∈ ℤ, 2x_{3}, 2x_{4} ∈ ℤ}.

The index of L_{0} in L_{0}^{∗} is 4. The quotient group is represented by

0, \frac{1 + i_{1}}{2} , \frac{1 + i_{1}}{4} + \frac{1}{2}i_{2}, \frac{1 + i_{1}}{4} + \frac{1}{2}i_{3}.

The discriminant is

Δ(a) = −8q(a).

Only the two cosets of \frac{1+i_{1}}{4} + \frac{1}{2}i_{2} and \frac{1+i_{1}}{4} + \frac{1}{2}i_{3} cannot be seperated by Δ̄ = Δ mod 8. They are intertwined by the orthogonal transformation

x ↦ i_{1}x̄,

which leaves L_{0} invariant. We obtain that the group

⟨EO(L_{0}), x ↦ i_{1}x̄⟩

acts transitively on the set of primitive vectors of a fixed norm.

We collect these examples and some others, which can be treated similarly in the following list.

4.6 Lemma. The following list contains lattices Γ = L_{0} ⊂ 𝔬 such that the groups ⟨EO(L_{0}, O(L_{0})⟩ (L = H_{1} ⊕ H_{2} ⊕ L_{0}) act transitively on the set of primitive vectors in L^{∗} of a given norm. All these lattices are of the form L_{0} = V_{0} ∩ 𝔬 where V_{0} ⊂ ℍ is a subspace. The list contains the lattices, their duals and the discriminant Δ as multiple of the quadratic form q (4.1).

1. L_{0} = 𝔬, L_{0}^{∗} = 𝔭/2, Δ = −2q.

2. L_{0} = { x ∈ 𝔬; x_{1} = x_{2} }, L_{0}^{∗} = { x ∈ ℍ; x_{1} = x_{2}, x_{1}+x_{2}+x_{3}+x_{4} ∈ ℤ, 2x_{3}, 2x_{4} ∈ ℤ }, Δ = −8q.

3. L_{0} = { x ∈ 𝔬; x_{4} = 0 }, L_{0}^{∗} = L_{0}/2, Δ = −4q.

4. L_{0} = { x ∈ 𝔬; x_{1} = x_{2} = 0 (or x_{3} = x_{4} = 0) } L_{0}^{∗} = L_{0}/2, Δ = −4q.

5. L_{0} = { x ∈ 𝔬; x_{1} = x_{2}, x_{4} = 0 }, L_{0}^{∗} = \{ (\frac{x_{1}}{4} , \frac{x_{1}}{4} , \frac{x_{3}}{2} , 0); x_{1}, x_{2} ∈ ℤ \}, Δ = −8q.

6. L_{0} = { x ∈ 𝔬; x_{1} = x_{2} = x_{3} }, L_{0}^{∗} = \{ (\frac{x_{1}}{6} , \frac{x_{1}}{6} , \frac{x_{1}}{6} , \frac{x_{1}}{2} + x_{4}); x_{1}, x_{2} ∈ ℤ \}, Δ = −3q.

7. L_{0} = { x ∈ 𝔬; x_{1} = x_{2} = x_{3} = 0 }, L_{0}^{∗} = L_{0}/2, Δ = −4q.

8. L_{0} = { x ∈ 𝔬; x_{1} = x_{2}, x_{3} = x_{4} = 0 } L_{0}^{∗} = L_{0}/4, Δ = −8q.

9. L_{0} = { x ∈ 𝔬; x_{1} = x_{2} = x_{3}, x_{4} = 0 } L_{0}^{∗} = L_{0}/6, Δ = −12q. All these groups are contained in O^{+}(L).

We need good generators for several groups.

4.7 Definition. A subgroup Γ ⊂ O(V) is nicely generated if it is generated by Eichler transformations E(f_{i}, v) and by transformations σ ∈ O(L_{0}).

In the case L_{0} = 0 the group O^{+}(L) is generated by the image of SL(2, ℤ) × SL(2, ℤ) and by a further transformation σ of order two, namely the transformation which interchanges the two components of f_{1}, f_{2} in the first hyperbolic plane H_{1}. We can use this as first step of an induction to prove the following result:

4.8 Proposition. Let L_{0} be one of the sublattices of 𝔬 which occur in 4.6. The group O^{+}(L) = O(L) ∩ O^{+}(V) is nicely generated, i.e.

O^{+}(L) = ⟨EO(L_{0}), O(L_{0})⟩.

Proof. Let L_{0} be one of the lattices in the list. One first has to know that the transformation σ of order two above can be expressed by Eichler transformations. This is more or less well known. We omit the proof because it would be no harm to add σ to the generators. We choose a line ℚa ⊂ L_{0} ⊗_{ℤ} ℚ. Let

L_{1} = { x ∈ L_{0}; x ⊥ a }

§5. Congruence subgroups of level 2 25

be the maximal sublattice of L_{0} which is orthogonal to a. The stabilizer of the line ℚa in O(L_{0}) is denoted by O(L_{0})_{ℚa}.

Claim. The line ℚa can be choosen such that the following two conditions hold:

a) The lattice L_{1} is zero or also a lattice of the list.

b) The natural map O(L_{0})_{ℚa} → O(L_{1}) is surjective.

It is easy to exhibit a vector a in each case. For example in the case L_{0} = 𝔬 one can take a = 1. In general one can apply lemma 4.9 below.

By means of 4.6 the claim for the lattice L_{0} reduces to the claim for the

lattice L_{1}.  ⊔ ⊓

4.9 Lemma. Let L_{0} ⊂ L_{1} be two lattices of the list in 4.6. Every element of the finite group O(L_{0}) is the restriction of an element of O(L_{1}) (which automatically stabilizes the orthogonal complement of L_{0} in L_{1}).

Corollary. Every element of O(ℤ^{4} ⊕ L_{0}) is the restriction of an element of O(ℤ^{4} ⊕ L_{1}).

Proof. In most cases the automorphism groups are small and contain only permutations and sign changes of the components. The exceptions are the lattices 1. 2. and 6. The lattice 1. (more precisely its negative) is isomorphic to the root lattice D_{4} which consist of all x ∈ ℤ^{4} with even x_{1} + ⋯ x_{4}. The quadratic form is (x_{1}^{2} + ⋯ + x_{4}^{2})/2. A concrete isomorphism is given by

ω ↦ (0, 1, −1, 0),  i_{1} ↦ (−1, −1, −0, −0),
i_{2} ↦ (1, −1, −0, −0),  i_{3} ↦ (−0, 0, 1, −1).

Automorphisms are given by permutations and sign changes*). The second lattice in 4.6 corresponds in this realization to the lattice A_{3} and the sixth one to a lattice A_{2}. The lattice A_{n} consists of all vectors x ∈ ℤ^{n+1} with x_{1}+⋯+x_{n} = 0. The only automorphisms are permutations possibly combined

with x ↦ −x. This proves lemma 4.9.  ⊔ ⊓

5. Congruence subgroups of level 2

We want to introduce also congruence subgroups of integral orthogonal groups. For this purpose we consider sub-lattices of maximal rank

L_{2} ⊂ L_{1} ⊂ L ⊗_{ℤ} ℚ

*) This is not the full group of automorphisms which will be determinated later.

such that both are invariant under O(L). Then O(L) acts on L_{1}/L_{2} and one can consider the kernel of the homomorphism

O(L) ⟶ Aut(L_{1}/L_{2}).

We already considered this kernel in the case L_{1} = L^{∗} and L_{2} = L. In this section we consider the cases L_{1} = L and L_{2} = M, where M is of the form

M = 2H_{1} ⊕ 2H_{2} ⊕ M_{0}, M_{0} ⊂ L_{0}.

There are several interesting choices for M_{0}. In the case 2L^{∗} ⊂ L we can take M = 2L^{∗} which means M_{0}^{∗} = 2L_{0}^{∗} but there will occur also other interesting choices. In any case we will assume that M is invariant under O(L). We then denote the above kernel by

O(L)[M] = kernel(O(L) ⟶ Aut(L/M)).

We want to derive a transitivity result similar to lemma 4.4. This means that we have to determine the orbit of a primitive element a ∈ L^{∗} under the congruence subgroup. Again we want to apply Eichler calculus. We list some special Eichler elements.

5.1 Remark. Using the coordinates

V = H_{1}(ℝ) ⊕ H_{2}(ℝ) ⊕ V_{0} = ℝf_{1} + ℝf_{2} + ℝf_{3} + ℝf_{4} + V_{0}

we have the following special Eichler elements:

\begin{aligned}E(f_{1}, f_{3})(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1} + x_{4}, x_{2}, x_{3} − x_{2}, x_{4}; y), \\ E(f_{1}, f_{4})(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1} + x_{3}, x_{2}, x_{3}, x_{4} − x_{2}; y), \\ E(f_{2}, f_{3})(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1}, x_{2} + x_{4}, x_{3} − x_{1}, x_{4}; y), \\ E(f_{2}, f_{4})(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1}, x_{2} + x_{3}, x_{3}, x_{4} − x_{1}; y), \\ E(f_{1}, h)(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1} + ⟨h, y⟩ − q(h)x_{2}, x_{2}, x_{3}, x_{4}; y − x_{2}h), \\ E(f_{2}, h)(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1}, x_{2} + ⟨h, y⟩ − q(h)x_{1}, x_{3}, x_{4}; y − x_{1}h), \\ E(f_{3}, h)(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1}, x_{2}, x_{3} + ⟨h, y⟩ − q(h)x_{4}, x_{4}; y − x_{4}h), \\ E(f_{4}, h)(x_{1}, x_{2}, x_{3}, x_{4}; y) & = (x_{1}, x_{2}, x_{3}, x_{4} + ⟨h, y⟩ − q(h)x_{3}; y − x_{3}h). \end{aligned}

Here h denotes an element of V_{0}.

The congruence subgroup is normal in O(L). If we want to determine the orbit of an element a under O(L)[M] we can replace a by g(a) for a g ∈ O(L). We want to choose such a g such that one of the four hyperbolic coordinates is odd. This is possible in all the cases we are interested in because of 5.2 Remark. Let a = (x_{1}, x_{2}, x_{3}, x_{4}; y) ∈ L^{∗} be a primitive element. The element (1, x_{1}x_{2} + x_{2}x_{3}, 0, 0; y) is also a primitive element which has the same norm as a. In all the cases we considered in lemma 4.6 the two elements are equivalent mod O(L).

The proof is trivial.  ⊔ ⊓

In the following we need an assumption on M_{0} which means that we only consider a level 2 case. We have to assume that L_{0}^{∗} can be multplied into M_{0} by a power of 2,

2^{m}L_{0}^{∗} ⊂ M_{0} (m > 0).

5.3 Lemma. Assume that 2^{m}L_{0}^{∗} ⊂ M_{0} for a suitable natural number m. Let (x, y) ∈ L^{∗} be a primitive vector such that at least one of the hyperbolic coordinates x_{1}, …, x_{4} is odd. Then there exists a mod O(L)[M] equivalent vector (x′_{1}, …, x′_{4}; y′) such that

a) (x, y) ≡ (x′, y′) mod M.

b) x′ is primitive in ℤ^{4}.

Proof. We can assume that x_{1} is odd. The vector a = (x, y) is primitive in L^{∗}. Therefore there exists a vector b ∈ L such that ⟨a, b⟩ = 1. From the assumption on M we obtain a vector c ∈ M such that ⟨a, c⟩ = 2^{m}. We decompose c = (ξ, η). Now we apply an Eichler transformation

(x, y) ⟼ (x′, y′) = (x_{1}, x_{2}, x_{3}, x_{4} + ⟨y, tη⟩ − q(tη)x_{4}, y − x_{4}tη).

Here t is an arbitrary even integer. The condition a) in 5.3 is satisfied. We want to find an even integer t such that x′ is coprime. This means that the greatest common divisor of (x_{1}, x_{2}, x_{3}, x_{4} −⟨y, tη⟩) for some t is 1. Let d denote the greatest common divisor of x_{4} and ⟨y, 2η⟩. The arithmetic progression x_{4} − ⟨y, tη⟩ contains elements dP, where P is an arbitrary large prime. We choose P coprime to the greatest common divisor of (x_{1}, x_{2}, x_{3}). Then the greatest common divisor of (x_{1}, x_{2}, x_{3}, dP) is the same as that of (x_{1}, x_{2}, x_{3}, x_{4}, 2⟨y, η⟩). From the choice of η we know that ⟨y, η⟩ is the sum of a linear combination of x_{1}, …, x_{4} and a pure power of 2. This means that the only prime which divides (x_{1}, …, x_{4}, ⟨y, η⟩) can be two. But x_{1} is odd. This proves lemma 5.3. ⊔ ⊓

We need a variant on the theorem of elementary divisors for SL(2, ℤ)[2] instead of SL(2, ℤ). Here SL(2, ℤ)[2] denotes the principal congruence subgroup of level 2 of the elliptic modular group.

5.4 Lemma. let M be an integral 2 × 2 matrix with coprime entries and with the property

M ≡ ( \begin{matrix}1 & 0 \\ 0 & ∗ \end{matrix} ) mod 2.

Then the diagonal matrix with diagonal (1, det M) is contained in the double coset SL(2, ℤ)[2]M SL(2, ℤ)[2].

Proof. The only problem is to get a 1 into the diagonal. From the theorem about the elementary divisors one knows that the above diagonal matrix — we call it D— is contained in SL(2, ℤ)M SL(2, ℤ). Consider the usual system of representatives of SL(2, ℤ)/ SL(2, ℤ)[2] and denote it by M_{1}, … M_{6}. The coefficients of the M_{i} are contained in { 0, ±1 }. We obtain that one of the matrices M_{i}DM_{j} is contained in the double coset SL(2, ℤ)[2]M SL(2, ℤ)[2]. Inspection of the several cases shows that the first or the second row of the possible M_{i}DM_{j} is coprime. Then we can use the following well known fact: Let (a, b) be a coprime pair of integers such that a is odd and b even. There exists a matrix M ∈ SL(2, ℤ)[2] such that (a, b)M = (1, 0). This gives the proof of lemma 5.4. ⊔ ⊓

Another simple observation is

5.5 Remark. let a = (x_{1}, x_{2}, x_{3}, x_{4}; y) ∈ L^{∗} be an element with the property x_{1} = 1, x_{3} = x_{4} = 0. Then there exists a substitution g ∈ O(L)[M] such that g(a) = (x′_{1}, x′_{2}, x′_{3}, x′_{4}; y′) has the following properties

a) x′_{1} = 1, x′_{3} = x′_{4} = 0.

b) The component y′ is contained in a given system of representatives of L_{0}^{∗}/M_{0}.

The proof is easy. One uses Eichler elements.  ⊔ ⊓

The component x′_{2} which does not occur in 5.5 is determined by the other ones and by the discriminant.

Now we have to distinguish two cases. A primitive element a of L^{∗} can be contained in L but this must not be so. The difference comes from the fact that O(L)[M] acts trivial on L/M (by definition) but not necessarily trivial on L^{∗}/M. In the first case the reduced representant constructed in 5.5 is uniquely determined.

5.6 Proposition. Assume that O(L) acts transitively on the set of primitive vectors of L^{∗} of a given norm. Assume furthermore that L_{0}^{∗} can be multiplied into M_{0} by a power of 2. Let a, b ∈ L^{∗} be two primitive vectors whose images in L^{∗}/M agree. Then the two vectors are equivalent mod O(L)[M]. Remark. It is sufficient to replace O(L)[M] by the smallest normal subgroup, which contains all Eichler elements E(f_{i}, v), v ∈ M, v ⊥ f_{i}. Corollary. Let G be the image of O(L)[M] in Aut(L^{∗}/M). (This is a finite group). Two primitive elements a, b ∈ L^{∗} of the same norm are equivalent mod O(L)[M] if and only if their cosets mod M are equivalent mod G.

Proof. Let be

a = (x, y), b = (x′, y′), a ≡ b mod M.

We mentioned already that we can assume x_{1} = 1, x_{3} = x_{4} = 0. We then have that x′_{1} is odd and x′_{3} and x′_{4} are even. From 5.3 we can assume that x′ is primitive and from 5.4 that x′_{1} = 1, x′_{3} = x′_{4} = 0. Now 5.5 gives the desired reduction. ⊔ ⊓

Proposition 5.6 gives a complete answer for the action of O(L)[M] on vectors which are primitive in L^{∗} but contained already in L. In this case two primitive elements of L^{∗} are equivalent if and only if their cosets mod M agree. In the other cases a problem concerning the action of O(L)[M] on L^{∗}/M remains. In each given case this action can be described. The next proposition is a result of general nature in this connection. We have to consider the finite group O(L_{0}) and denote by O(L_{0})[M_{0}] its intersection with O(L)[M]. This group acts on L^{∗}. The group O(L_{0}) acts trivial on the hyperbolic components x_{1}, …, x_{4}.

5.7 Proposition. Let L_{0} ⊂ K_{0} ⊂ L_{0}^{∗} be an intermediate lattice such that L + K_{0} is invariant under O(L)*). The group O(L)[M] acts transitively on the set of vectors a of a given norm, which are contained in L^{∗} − K and primitive in L^{∗} if the following conditions are satisfied:

1. The group L^{∗} can be multiplied into M by a power of 2.

2. The group O(L) acts transitively on the set of primitive vectors of L^{∗} of a given norm.

3. The images ā, b̄ ∈ L_{0}^{∗}/M_{0} of two primitive vectors a, b ∈ L_{0}^{∗} − K_{0} of the same norm are equivalent mod O(L_{0})[M_{0}].

4. For a primitive vector a in L^{∗} which is not contained in K there exists a vector b ∈ M such that ⟨a, b⟩ is odd.

5. The values of the quadratic form q are even on M. Remark. One can replace O(L)[M] by the subgroup generated by the Eichler elements E(f_{i}, v) and the finite group O(L_{0})[M_{0}].

Proof. Let a = (x, y) ∈ L^{∗} − L be a vector which is primitive in L^{∗}. Using 4. and 5. we find an Eichler transformation which changes the hyperbolic components of a mod 2 arbitrarily. Using 3. we may change the class of y mod M_{0}. The rest of the proof is similar to that of proposition 5.6. ⊔ ⊓

We want to apply these results to the lattice L_{0} = 𝔬 and to the various sublattices of lower dimension of 𝔬 which have been introduced already in 4.6. They are of the form

L_{0} = 𝔬 ∩ V_{0}, V_{0} ⊂ ℍ subspace.

We equip them with the sublattice

M_{0} = L_{0} ∩ 𝔭.

We consider the congruence groups O(L)[M], M = 2H_{1} ⊕ 2H_{2} ⊕ M_{0} and their finite part O(L_{0})[M_{0}].

*) Examples are K_{0} = L_{0} and K_{0} = L_{0} + 2L_{0}^{∗} (K = L + 2L^{∗}).

5.8 Lemma. In the case L_{0} = 𝔬, M_{0} = 𝔭 the finite group O(L_{0})[M_{0}] is generated by arbitrary permutations of the components and sign changes of an even number of components. This means that O(L_{0})[M_{0}] is a subgroup of index two of O(4, ℤ). If L_{0} is one of the lattices in 4.6 and M_{0} = L_{0} ∩ 𝔬 then O(L_{0})[M_{0}] is the group of all orthogonal transformations which are restrictions of an element of O(𝔬)[𝔭].

Proof. The first part of the lemma follows from the explicit description of the group O(𝔬) ≅ O(D_{4}). The lattice D_{4} is discussed in detail in [CS]. The group of automorphisms has order 1152. (The Weyl group W (D_{4}) ≅ (ℤ/2ℤ)^{3} · S_{4} is a subgroup of index 6 of Aut(D_{4})). We will describe this group a little later using quaternions. The second part of the lemma is a consequence of the first part and the description of the automorphism groups of the lattices L_{0} as given in the proof of lemma 4.9. ⊔ ⊓

For induction arguments we need an anologue of 4.9.

5.9 Lemma. Let L_{0} ⊂ L_{1} be two lattices from the list in 4.6. With one exception every element of O(L_{0})[M_{0}] is the restriction of an element O(L_{1})[M_{1}] (M_{1} = L_{1} ∩ 𝔬). The exception is given by the pair

L_{1} = { x; x_{1} = x_{2} }, L_{0} = { x; x_{1} = x_{2} = 0 }.

In this exceptional case only a subgroup of index two Õ(L_{0})[M_{0}] ⊂ O(L_{0})[M_{0}] extends to O(L_{1})[M_{1}]. Corollary. In all cases but the exceptional case we have corresponding finite maps of the Baily Borel compactifications

X(O(L_{0})[M_{0}]) → X(O(L_{1})[M_{1}]),

which are birational onto its image. In the exceptional case the same is true for

X(Õ(L_{0})[M_{0}]) → X(O(L_{1})[M_{1}]).

This can be checked case by case. The corollary follows from [Fr3].  ⊔ ⊓

In the exceptional case the transformation x_{4} ↦ −x_{4} does not extend to an element of O(L_{1})[M_{1}]. (It extends to an element of O(𝔬)[𝔭] namely (x_{1}, x_{2}, x_{3}, x_{4}) ↦ (−x_{1}, x_{2}, x_{3}, −x_{4}) but this transformation does not preserve L_{1}).

5.10 Lemma. The assumptions of proposition 5.6 are satisfied for all L_{0} listed in 4.6 (with M_{0} = L_{0} ∩ 𝔭).

The proof is trivial.  ⊔ ⊓

We use the lemma to obtain a result about generators of the congruence subgroups.

§6. Quadratic divisors 31

5.11 Proposition. For all lattices L_{0} listed in 4.6 (and M_{0} = L_{0} ∩ 𝔭) the group O(L)[M] is the smallest normal subgroup of O^{+}(L) which contains all Eichler transformations E(f_{i}, v) and the finite group O(L_{0})[M_{0}].

1. Corollary. The group O(L)[M] acts trivial on the finite group K/M, where K = L + 2L^{∗}.

2. Corollary. The group O(L)[M] is contained in O^{+}(L).

The proof is given by induction. The induction starts with the lattice L_{0} = 0 and uses the fact that the group SL(2, ℤ)[2] is generated by three matrices

( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 2 & 1 \end{matrix} ) , ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) .

The image of the homomorphism SL(2, ℤ)[2]×SL(2, ℤ)[2] → O(H_{1}⊕H_{2})[2H_{1}⊕ H_{2}] is surjective. (This is different from the level one case where an involution σ occurs.) For the induction step one chooses a lattice L_{1} ⊂ L_{0} of the list of corank 1. We can avoid the exceptional case in 5.9. There exists a primitive vector a ∈ L_{0}^{∗} such that L_{1} is the orthogonal complement of a in L_{0}. Let now be given an arbitrary element g ∈ O(L)[M]. We consider the vector g(a). We use lemma 5.10 to find an element g_{1} in the normal subgroup with the property g_{1}(a) = g(a). The transformation g_{1}g^{−1} acts as identity on H_{1} ⊕ H_{2} ⊕ L_{1} and we can imply induction. This completes the proof of the proposition. The corollary follows because the Eichler transformations act trivial on K/M.

⊔ ⊓

5.12 Lemma. The assumptions of proposition 5.7 are satisfied in the following two cases:

1. L_{0} = 𝔬, K_{0} = L_{0}, M_{0} = 𝔭.

2. L_{0} = { x ∈ 𝔬; x_{1} = x_{2} }, K_{0} = L_{0} + 2L_{0}^{∗}, M_{0} = L_{0} ∩ 𝔭.

Proof. We only have to verify 3. and 4. in 5.7.

Case 1 (L_{0} = 𝔬): The set (L_{0}^{∗} − L_{0})/M can be represented by 12 elements of norm −1/2. A typical one is (1 + i_{1})/2. They are equivalent under O(L_{0})[M_{0}). To prove 4. we observe ⟨(1 + i_{1})/2, 1 + i_{2}⟩ = 1. This proves lemma 5.10.

Case 2: The set (L_{0}^{∗} − K)/M consists of 8 elements. They can be represented by elements of norm −3/8. A typical on is \frac{1+i_{1}}{4} + \frac{i_{2}}{2} . They are obviously

equivalent under O(L_{0})[M_{0}]. The scalar product with 1 + i_{1} is 1.  ⊔ ⊓

It is natural to ask whether O(L)[M] is generated by Eichler elements E(f_{i}, v) and by the finite group O(L_{0})[M_{0}], that is whether it is nicely generated in the sense of 4.7. One can reduce this question from the starting six dimensional to a five dimensional case (x_{1} = x_{2}) using the second part of 5.12. In a similar way one can reduce it to four dimensional case (x_{1} = x_{2} = x_{3}). But then the obvious method of reduction ends.

6. Quadratic divisors

Let v ∈ V, v ≠ 0 be a fixed vector of our quadratic space. We consider its orthogonal complement with respect to our ℂ-bilinear pairing ⟨ , ⟩ in P (V (ℂ)) and the intersection of this orthogonal complement with the ”halfplane“ K^{+}(V_{0}). We denote this set by Q(v) and use the same notation for the corresponding set in H(V_{0}). If we write v in the form

v = (α_{1}, α_{2}, α_{3}, α_{4}, β) (β ∈ ℍ),

the equations for Q(v) in the model H(V_{0}) are

α_{1}z_{2} + α_{2}z_{0} − α_{3} det z + α_{4} + ⟨β, z_{1}⟩ = 0.

This is a quadratic equation and Q(v) is a quadric. The group O^{+}(V) acts on the set of vectors v ∈ V and also on the set of quadrics. One verifies by direct calculation:

6.1 Lemma. Let g ∈ O^{+}(V) and v ∈ V (v ≠ 0). Then

M(Q(v)) = Q(M(v)).

An easy consequence from remark 2.1 is

6.2 Lemma. The quadric Q(v) is not empty if and only if q(v) < 0. In this case v is determined by Q(v) up to a real factor.

Now we assume that v is a rational vector of negative norm, i.e. v ∈ L ⊗_{ℤ} ℚ for our given lattice L ⊂ V. We multiply v with a constant factor such that v is contained in the dual lattice L^{∗} and is primitive (ℚv ∩ L^{∗} = ℤv). After this normalization v is determined by Q(v) upto the sign. We introduced the discriminant of a primitive vector in L^{∗} (s. 4.1). We call this also the discriminant of the quadric Q(v),

Δ(Q(v)) := Δ(v).

From 4.6 we obtain

6.3 Lemma. The groups Γ introduced in 4.6 act transitively on the set of quadrics of a given discriminant.

In the previous section we determined the orbits of the quadrics for a wide class of congruence subgroups of level 2. We exhibit some special cases which will be investigated in the following.

§7. The quaternary modular group 33

6.4 Proposition. Let L = H_{1} ⊕ H_{1} ⊕ L_{0} where L_{0} = 𝔬 is the Hurwitz ring of integers with the quadratic form q(x) = −|x^{2}|. Let be

M := 2L^{∗} = 2H_{1} ⊕ 2H_{2} ⊕ 𝔭.

Let Δ be a discriminant of L, i.e. a natural number of the form −2q(a), a ∈ L^{∗}. If Δ is odd there is precisley one equivalence class mod O(L)[M] of quadrics of discriminant Δ. The number of equivalence classes is 36 if Δ ≡ 2 mod 4 and 27 if Δ ≡ 0 mod 4.

Proof. We can use the propositions 5.6 and 5.7. The two cases are not mixed because q(a) is not integral for a ∈ L^{∗} − L.

To prove 6.4 we have to decide which elements ā ∈ L/2L^{∗} can be represented by an element a of L of a given discriminant and which is primitive in L^{∗}. Of course ā has to be different from 0. The set of non zero elements of L/2L^{∗} decomposes into two orbits under the action of O(L). The orbits are distinguished by the discriminant mod 4 which is well defined on L/2L^{∗}. The two orbits are represented by the two elements (1, 0, 0, 0; 1) and (1, −1, 0, 0; 1). These elements are equivalent mod 2L^{∗} to (1, Δ/2−1, 0, 0; 1) where Δ ≡ 2 mod 4 in the first case and ≡ 0 mod 4 in the second case. Thus we have proved: Let Δ be an even natural number and ā ∈ L/2L^{∗} a non-zero element whose discriminant is Δ mod 4. Then ā can be represented by an element a ∈ L of discrimant Δ and which is primitive in L^{∗}. To prove proposition 6.4 we only have to verify that L/2L^{∗} contains 36 elements of discriminant 2 mod 4 and 27 = 64−36−1 non zero elements of discriminant 0 mod 4. ⊔ ⊓

We treat a five dimensional case:

6.5 Proposition. Let L = H_{1} ⊕ H_{2} ⊕ L_{0}, where L_{0} ⊂ 𝔬 is the sublattice of the Hurwitz integers defined by x_{1} = x_{2} (equipped with q(x) = −|x|^{2}). Let be

M = L ∩ 2H_{1} ⊕ 2H_{2} ⊕ 𝔭.

All quadrics of discriminant 3 are equivalent mod O(L)[M]. There are 36 equivalence classes for each of the discriminants 4 and 8.

Proof. We start with the discriminant 3. The vectors of this type are contained in the complement of K = L + 2L^{∗}. We apply 5.7 with this K. There are 8 elements in L^{∗}/K whose discriminant is three mod 8. A typical one is represented by (1 + i_{1})/2 + i_{2}/2. The other ones ore obtained by replacing j by k and changing 4 signs. So 3. in 5.7is fulfilled. Condition 4. is satisfied because the scalar product with 1 + i_{1} is -1.

The vectors of discriminant 4 and 8 are contained in K. The group O(L)[M] acts trivial on K/M (1. Corollary of proposition 5.11). The statement follows

from 5.6.  ⊔ ⊓

7. The quaternary modular group

In this section we start to investigate the special case O(2, 6). We switch into the symplectic world because we want to introduce theta series whose transformation formalism and relations can be described very easy in the symplectic context. To compare the two worlds it is better to divide the groups by the center, i.e. we consider all groups as subgroups of the group of biholomorphic transformation of the tube domain:

7.1 Notation. For any negative definite lattice L_{0} we denote by Γ(L_{0}) the image of O^{+}(L) in the group of biholomorphic mappings Ω(V_{0}) of the corrsponding tube domain.

We are now interested in the case L_{0} = 𝔬 (with quadratic form q(x) = −x̄x). The quaternary modular group is

Sp(2, 𝔬),

where 𝔬 is the ring of Hurwitz integers. By a result of Krieg [Kr] the group Sp(2, 𝔬) is generated by all matrices \binom{AB}{CD}, where B or C is zero and where

A (and hence D = Ā^{−1}) is a triangular matrix. This result in connection with proposition 2.6 shows that Sp(2, 𝔬)/{±E} is contained in Γ(𝔬).

7.2 Remark. The group Sp(2, 𝔬)/{±E} is a normal subgroup of index 4 in Γ(𝔬). The quotient is isomorphic to (ℤ/2ℤ) × (ℤ/2ℤ). It is generated by the cosets of the orthogonal transformations

x ↦ x̄, x ↦ αxᾱ where α := \frac{1 + i_{1}}{\sqrt{2}} .

We need frequently the formulae

\begin{aligned}ωxω̄ & = x_{1} + x_{3}i_{1} + x_{4}i_{2} + x_{2}i_{3}, \\ αxᾱ & = x_{1} + x_{2}i_{1} + x_{4}i_{2} − x_{3}i_{3}. \end{aligned}

Proof. of 7.2. From 4.8 follows that Γ(𝔬) is generated by the subgroups Sp(2, 𝔬)/{±E} and the orthogonal group of the definite lattice 𝔬,

Σ := O(𝔬).

The orthogonal group Σ contains the subgroup*)

Σ_{0} = (𝔬^{•} × 𝔬^{•})/{±(1, 1)},

*) The multiplicative group of 𝔬 is denoted by 𝔬^{•}. Recall that 𝔬^{∗} = 𝔭/2 is the dual lattice in our notation.

which acts by x ↦ axb̄. This subgroup is contained in Sp(2, 𝔬)/{±E}. It corresponds to the group of diagonal matrices. We know the order of Σ(≅ Aut(D_{4}). It follows that Σ_{0} is a subgroup of index 4 in Σ. Explicit computation shows that this subgroup is normal. The factor group is generated by the two orthogonal transformations x ↦ x̄, αxᾱ. They generate a subgroup which has only the unit element common with Sp(2, 𝔬)/{±E}. Now 7.2 is clear. ⊔ ⊓

let 𝔞 be a two-sided ideal in 𝔬. The principal congruence subgroup of level 𝔞 in the quaternary modular group is

Sp(2, 𝔬)[𝔞] := kernel(Sp(2, 𝔬) ⟶ Sp(2, 𝔬/𝔞)).

We are mainly interested in the ideal 𝔞 = 𝔭. The quotient ring

𝔽_{4} := 𝔬/𝔭

is a field of 4 elements. Conjugation on 𝔬 induces the unique non-trivial automorphism of 𝔽_{4}. We denote it again by a ↦ ā. In accordance to our notations Sp(2, 𝔽_{4}) denotes here the Hermitean(!) symplectic group, which is defined by

M̄′IM = I where I = ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ) .

The field 𝔽_{4} has characteristic 2. Therefore we can replace −E by E and hence I by a symmetric matrix. This means that Sp(2, 𝔽_{4}) is a unitary group. We use the notation

U(4, 𝔽_{4}) := Sp(2, 𝔽_{4}).

It contains SU(4, 𝔽_{4}) as subgroup of index 3. This group can be found in Conways atlas. It is a simple group of order 25 920.

7.3 Remark. The natural homomorphism

Sp(2, 𝔬) ⟶ Sp(2, 𝔽_{4})

is surjective and induces an isomorphism

Sp(2, 𝔬)/ Sp(2, 𝔬)[𝔭] ⟶^{∼} Sp(2, 𝔽_{4}).

Proof. The group Sp(2, 𝔽_{4}) is generated by diagonal matrices and the standard unipotent matrices (B = 0 or C = 0, A and D strict triangular). Therefore it is sufficient to verify:

a) The natural map 𝔬^{•} ↦ 𝔽_{4}^{•} is surjective.

b) Every invariant element (ā = a) of 𝔽_{4} is the image of an invariant (real) element of 𝔬.

Both statements are trivial. ⊔ ⊓

We need a certain symmetrization of Sp(2, 𝔬)[𝔭]. We want to add a certain subgroup

O′(4, ℤ) ⊂ O(4, ℤ) ⊂ Σ = O(𝔬).

7.4 Definition. The group O′(4, ℤ) conists of all transformations

(x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (±x_{σ(1)}, ±x_{σ(2)}, ±x_{σ(3)}, ±x_{σ(4)}),

where σ is an arbitrary permutation and where the number of minus-signs is even.

This is a subgroup of index 2 in O(4, ℤ).

7.5 Definition. The group Γ(𝔬)[𝔭] is defined as

Γ(𝔬)[𝔭] = ⟨Sp(2, 𝔬)[𝔭]/{±E}, O′(4, ℤ)⟩.

This group plays a central role in this paper because it is the precise invariancegroup of a system of 36 theta functions which we will introduce in the next section.

7.6 Lemma. The group Γ(𝔬)[𝔭] contains Sp(2, 𝔬)[𝔭]/{±E} as normal subgroup of index 6. The quotient group is isomorphic to S_{3}. It can be generated by the images of

x ⟼ ωxω̄, x ⟼ αx̄ᾱ.

Proof. The transfomations

x ↦ axb̄ where a, b ∈ {±1 ± i_{1}, ±i_{2}, ±i_{3}}

are contained in O′(4, ℤ) as well as in Sp(2, 𝔬)[𝔭]/{±E}.

7.7 Lemma. The group Γ(𝔬)[𝔭] is a normal subgroup of Γ(𝔬). The quotient group is naturally isomorphic with an extension of index 2 from SU(4, 𝔽_{4}).

Proof. There are three things to show:

1. The group O′(4, ℤ) is normal in Σ.

2. The group Σ normalizes Sp(2, 𝔬)[𝔭]/{±E}.

3. Let be σ ∈ O′(4, ℤ) and g ∈ Sp(2, 𝔬)/{±E}. Then gσg^{−1} ∈ Sp(2, 𝔬)/{±E}. Ad 1.: The group O′(4, ℤ) contains 2^{6} · 3 elements. The index in Σ is 6. We obtain that Σ is generated by O′(4, ℤ) and the two substitutions x ↦ x̄ und x ↦ ωx. Obviously the first one normalizes O′(4, ℤ). We have to show that this is true for the substitution h(x) := ωx too. The subgroup (𝔬_{0}^{•} ×𝔬_{0}^{•})/{±(1, 1)} is normalized because 𝔬_{0}^{•} is a normal subgroup of 𝔬^{•}. The index in O′(4, ℤ) is 6.

This group is generated by the subgroup and the two substitions g_{1}(x) := ωxω̄, g_{2}(x) := αx̄ᾱ. It is sufficient to show that hg_{1}h^{−1} and hg_{2}h^{−1} are contained in O′(4, ℤ). ⊔ ⊓ Ad 2.: This is obvious. Ad 3.: The group Γ(𝔬) is generated by Σ and by the unipotent standard matrices. We already know that O′(4, ℤ) is normal in Σ. Hence we may assume that g is one of the unipotent generators. One has to show

a ∈ 𝔬 ⟹ a − σ(a) ∈ 𝔭  (σ ∈ O(4, ℤ)).  ⊔ ⊓

We now want to compare the “symplectic” congruence subgroup Γ(𝔬)[𝔭] with its orthogonal version:

7.8 Lemma. Let O(L)[M] be the kernel of

O^{+}(L) ⟶ Aut(L/2L^{∗}), L = H_{1} + H_{2} + 𝔬.

The natural homomorphism O^{+}(L) → Γ(𝔬) = O^{+}(L)/{± id} induces an isomorphism

O(L)[M]/{± id} ⟶ Γ(𝔬)[𝔭].

We especially have

O^{+}(L)/O(L)[M] ≅ Γ(𝔬)/Γ(𝔬)[𝔭].

Proof. It follows from the propositions 5.11 and 2.6 that O(L)[M]/{± id} is contained in Γ(𝔬)[𝔭]. We have to show equality. This means that the natural homomorphism

O^{+}(L)/O(L)[M] ⟶ Γ(𝔬)/Γ(𝔬)[𝔭]

is an isomorphism. We know already that this homomorphism is surjective. Therefore it suffices to show the inequality

#O^{+}(L)/O(L)[M] ≤ #Γ(𝔬)/Γ(𝔬)[𝔭] = 2 · 25 920.

The group L/2L^{∗} is a vector space of dimension 6 over 𝔽_{2}. The quadratic form q induces a mod 2 quadratic form q̄ on this vector space. It is easy to check (and well-known) that the order of the finite orthogonal group O(L/2L^{∗}, q̄) is 51 840. This completes the proof of lemma 7.8.

7.9 Lemma. The groups Γ(𝔬)/Γ(𝔬)[𝔭] and O(L^{∗}/L, q̄) are naturally isomorphic.

We will see a little later that Γ(𝔬)/Γ(𝔬)[𝔭] is isomorphic to the Weyl group W (E_{6}) of the E_{6}-root system. In this way one recovers the well-known fact that the W (E_{6}) is isomorphic to an orthogonal group over the field of two elements and that it contains SU(4, 𝔽_{4}) as simple subgroup of index 2.

8. A distinguished point

We use the following notation. Let L be an arbitrary lattice with quadratic form q. Then L(d) denotes the same abelian group, L = L(d) but with the new quadratic form a ↦ dq(a). Here d can be an arbitrary non-zero number.

8.1 Proposition. There exists a sub-lattice of the lattice

L^{∗}(−1) (L^{∗} = H_{1} ⊕ H_{2} ⊕ 𝔭/2)

which is isomorphic to the lattice E_{6}. The group O^{+}(L) acts transitively on the set of all these sub-lattices.

Proof. The lattice E_{6} is a certain even positive definite 6-dimensional lattice of determinant 3. We refer to [CS] for a detailed description. This lattice can be characterized as follows: It is the only positive definite integral lattice of rank 6 which is generated by vectors of norm 2 and which contains at least 72 vectors of norm 2. (It contains precisely 72.) We consider the following 6 elements of norm −1 of L^{∗}:

A_{1} = (0, 0, 0, 0; i_{3}), A_{2} = (0, 0, 0, 0; ω),  A_{3} = (1, 0, 0, 0; −ω̄),
A_{4} = (0, 1, 0, 0; −ω̄), A_{5} = (1, 1, 1, −1; −ω̄), A_{6} = (0, 0, 1, 0; ^{−}i_{2}).

One verifies that the sublattice of L, which is generated by the vectors A_{1} …, A_{6}, is a copy of E_{6}(−1). The orthogonal complement in L is spanned by the two vectors

B_{1} = (2, 2, 2, 0; −1 + i_{1}), B_{2} = (0, 0, 2, 2; i_{1} − i_{2}).

Let now E ⊂ L^{∗}(−1) be an arbitrary lattice which is isomorphic to E_{6}. Every element a ∈ L^{∗} of integral norm q(a) is contained in L. Therefore E is contained already in L. We obtain a natural map

E/2E ⟶ L/2L^{∗}.

Claim 1. This map is an isomorphism. Proof. Both sides are ℤ/2ℤ-vector spaces of dimension 6. Therefore it is sufficient to proof that the map is injective. For this we consider the group of all g ∈ O^{+}(L) which stabilize E. This group contains the Weyl group of E. It is known that E/2E is irreducible under the Weyl group. ⊔ ⊓

We consider the orthogonal complement W ⊂ L of E in L. This is a positive definite sublattice of rank two. From the first claim we know that W ⊂ 2L^{∗}. This implies ⟨x, x⟩ ∈ 4ℤ for all x ∈ W. Claim 2. W is isomorphic with A_{2}(2), i.e. a Gram matrix is given by ({42 \atop 24}).

§8. A distinguished point 39

Proof. A_{2} is upto isomorphism the unique even integral lattice of determinant 3. We know already that W (1/2) is an even integral lattice. Therefore it is sufficient to show that det W = 12:

Let be E(ℚ) = E ⊗_{ℤ} ℚ. We have L ∩ E(ℚ) = E because E is a maximal lattice. This implies that E is primitive in L, i.e. L/E is free. The orthogonal complement W is of course also primitive in L. From the primitivity follows that the two natural maps

L^{∗} → E^{∗} and L^{∗} → W^{∗}

are surjective. We consider the diagram

E(ℚ) ⊕ W (ℚ) ⊂ L(ℚ)
∪ ∪ ∪
E^{∗} ⊕ W^{∗} ⊃ L^{∗}
∪ ∪ ∪
E ⊕ W ⊂ L .

The two maps

L^{∗} ⟶ E^{∗}/L^{∗} ∩ E^{∗}, L^{∗} ⟶ W^{∗}/L^{∗} ∩ W^{∗}

(induced by the inclusion L^{∗} ⊂ E^{∗} ⊕ W^{∗} and the projections from E^{∗} ⊕ W^{∗} to E^{∗} and W^{∗}) have the same kernel: An element a = b + c with a ∈ L^{∗}, b ∈ E^{∗}, c ∈ W^{∗} is in the kernel if c, d both are contained in L^{∗}. We obtain an isomorphism

W^{∗}/(W (ℚ) ∩ L^{∗}) ⟶^{∼} E^{∗}/(E(ℚ) ∩ L^{∗}).

Using det E = [E^{∗} : E] (and similarly for W) we obtain the formula

det W · [E^{∗} ∩ L^{∗} : E] = det E · [W^{∗} ∩ L^{∗} : W].

In our special situation the determinants det L = 4 and det E = 3 are coprime. From this we derive Claim. a) E = E(ℚ) ∩ L^{∗}. b) L^{∗}/L ≅ (W (ℚ) ∩ L^{∗})/W. Proof. a) From a ∈ L^{∗} ∩ E^{∗} follows det(L)a ∈ L and det(E)a ∈ E, especially a ∈ L ∩ E^{∗} = E.

b) We have to show L^{∗} ⊂ L + W^{∗}. For this we decompose an element a ∈ L^{∗} as a = b + c, b ∈ E^{∗}, c ∈ W^{∗}. From det(L)a ∈ L and det(E)b ∈ E follows that

det(L)a and det(E)a both are contained in L + W^{∗}.  ⊔ ⊓
A consequence of the claim is det W = 12.  ⊔ ⊓

We obtain now that the index of E ⊕ W in L is three, which means

L/E ⊕ W ≅ ℤ/3ℤ.

The proof of 8.1 has been reduced to a kind of Witt theorem:

Two embeddings of E_{6} ⊕ A_{2}(2) into L(−1) are equivalent mod O(L).

Problems of this type have been investigated by Nikulin [Ni]. The question depends heavily on L/(A_{2}(2) + E_{6}) which is ℤ/3ℤ in our case. It is easy to show that one can apply [Ni], Corollary 1.5.2 to prove proposition 8.1. ⊔ ⊓

The Weyl group W (E_{6}) is generated by the reflections associated to the vectors of norm 2. It is a subgroup of index 2 of Aut(E_{6}), which is generated by the Weyl group and the transformation x ↦ −x. We fix an embedding E_{6} → E ⊂ L(−1). We want to determine the subgroup G ⊂ O^{+}(L) which leaves E invariant. Each reflection with respect to one of the 72 minimal vectors of E extends naturally to a reflection of the whole lattice L(−1). These reflections are contained in O^{+}(L) *) Hence the restriction homomorphism G → Aut(E_{6}) is surjective. We want to show that the kernel of this homomorphism is trivial. For this we use the above model for E. The vector

A := \frac{A_{1} + A_{2} + A_{3} + A_{4} + A_{5} − B_{1} − B_{2}}{3}

is contained in L. If g ∈ G is an element of the kernel then g leaves the orthogonal complement W invariant and it has the further property that (B_{1} + B_{2} − g(B_{1}) − g(B_{2}))/3 is contained in L. There is only one automorphism of W which has this property, namly B_{1} ↔ B_{2}. But this automorphism is not contained in the spinor kernel. This means that we found the subgroup G ⊂ O^{+}(L) is naturally isomorphic to Aut(E_{6}). The group W (E_{6}) does not contain − id. It therefore defines a subgroup of Γ(𝔬) = O^{+}(L)/{± id}. We consider the homomorphism

W (E_{6}) → Γ(𝔬) ⟶ Γ(𝔬)/Γ(𝔬)[𝔭].

The restriction of this homomorphism to the simple subgroup (the intersection with SO(L)) is injectiv. One checks easily that a reflection is not contained in Γ(𝔬)[𝔭]. Hence the homomorphism is injective on the whole Weyl group. Comparing the orders of the two finite groups we obtain:

8.2 Proposition. The homomorphism

Γ(𝔬) ⟶ Γ(𝔬)/Γ(𝔬)[𝔭]

admits a section. We have a decomposition as semidirect product Γ(𝔬) = Γ(𝔬)[𝔭] · W (E_{6}).

For a fixed embedding E_{6} → E ⊂ L(−1) we can consider the orthogonal complement W ⊂ V. The restriction of q to W is positive definite. By 2.1 W defines a point ϱ in the tube-domain H(ℍ). This is a fixed point for the image of W (E_{6}). The orbit of this point under Γ(𝔬) is uniquely determined. This means that the modular variety H(ℍ)/Γ(𝔬) contains a distinguished point. The stabilizer of ϱ in O^{+}(L) is given by the group G. We obtain:

*) A reflection with respect to a vector a is contained in the spinor kernel if and only if its norm is negative.

8.3 Proposition. The stabilizer of ϱ in Γ(𝔬) is isomorphic with W (E_{6}). The homomorphism

Γ(𝔬)_{ϱ} ⟶ Γ(𝔬)/Γ(𝔬)[𝔭] ≅ W (E_{6})

is an isomorphism. Corollary. The distinguished point in H/Γ(𝔬) has precisely one inverse image in H/Γ(𝔬)[𝔭].

We denote by X(Γ) the Baily-Borel compactification of H/Γ, where H is one of our introduced half planes and Γ one of our arithmetic groups. We will investigate the Galois covering

X(Γ(𝔬)[𝔭]) ⟶ X(Γ(𝔬)).

We want to study the geometry of quadratic divisors. For every natural number Δ we have an irreducible divisor in X(Γ(𝔬)) which is the image of all the quadrics of discrimant Δ. We denote this divisor by

Q(Δ) ⊂ X(Γ(𝔬)).

We reformulate 6.4 as follows:

8.4 Proposition. The inverse image of Q(Δ) in X(Γ(𝔬)[𝔭]) remains irreducible if and only if Δ ≡ 1 mod 2. It decomposes into 36 components if Δ ≡ 2 mod 4 and into 27 components if Δ ≡ 0 mod 4.

Combining this with 5.6 we obtain:

8.5 Remark. Two quadrics Q(a) and Q(b) of the same even discriminant are in the same orbit under Γ(𝔬)[𝔭] iff the images of a and b in L/2L^{∗} agree.

In general it is difficult to compute the intersection of quadratic divisors. A very remarkable result in this direction is:

8.6 Proposition. The intersection of the 36 quadratic divisors in X(Γ(𝔬)[𝔭]) of discriminant 2 consists exactly of one point, namely the distinguished point (8.3). This point is smooth. Even its image in X(Γ)[𝔬] is smooth.

Proof. In a first step we show that the 36 divisors have no common zero at the Baily-Borel boundary. For this one needs some information about the compactification. We don’t want to go into details and mention only that the typical boundary values are given by ”z_{2} → i∞“.

Let z ∈ H be a point which represents an intersection point of the 36 quadratic divisors. The set of all vectors v ∈ L^{∗} such that z ∈ Q(v) (including 0) is a six dimensional lattice which contains 36 pairs ±v of vectors of norm −1. This means that this lattice is a copy of E_{6}(−1). The point z therefore has to be a representative of the distinguished point 8.3.

By a well known result of Chevalley the smoothness of the point even in X(Γ(𝔬)) comes from the fact that W (E_{6}) is a reflection group. ⊔ ⊓

It is easy to find an explixcit representative z of the point ϱ on the tube domain.

8.7 Remark. The point ϱ can be represented by

\frac{1}{4} ( \begin{matrix}2 & 1 − i_{1} + 2i_{2} \\ 1 + i_{1} − 2i_{2} & 2 \end{matrix} ) + i \frac{\sqrt{3}}{4} ( \begin{matrix}2 & 1 + i_{1} \\ 1 − i_{1} & 2 \end{matrix} ) ∈ H

For the proof one has to use the translation from a two dimensional positive

definite space to a point in the tube domain (2.1, 2.2).  ⊔ ⊓

9. Moduli Interpretation

Most of the results of this paper admit a moduli interpretation. For our results and proofs this interpretation is not really necessary, it is behind the scenes. So the reader who is not interested in it, can skip this section.

Lattices L of signature (2, n), n ≤ 19 arise in the theory of moduli of algebraic surfaces. The symmetric space associated to L is then interpreted as a period domain. We recall briefly some basic facts. Let X be a K3-surface. The group H^{2}(X, ℤ) equiped with the intersection form is the unique unimodular lattice of signature (3, 19) which is called the K3-lattice. A possible realization is

K = H ⊥ H ⊥ H ⊥ E_{8}(−1) ⊥ E_{8}(−1).

By the Chern class map the Picard group Pic(X) is embedded as a sublattice of the K3-lattice. Its signature is (1, t), t ≤ 19. The number 1 + t is the socalled Picard number of X. Any lattice M of signature (1, t), t ≤ 19 that is embeddable into the K3-lattice occurs as a Picard lattice of some K3-surface. The group

{ γ ∈ O^{+}(K); γ(m) = m for all m ∈ M }

may be identified with a subgroup Γ of O^{+}(M^{⊥}). This is a congruence subgroup in the sence we introduced in section 5. The quotient H/Γ may be interpreted as moduli space of pseudo-ample-M-polarized K3-surfaces. From the uniqueness of the point ϱ in the previous section one can derive:

9.1 Theorem. There is a K3-surface X with maximal Picard number (20) and with Picard lattice of type

H ⊥ E_{8} ⊥ D_{4} ⊥ E_{6}.

The lattice of transcendental cycles is isomorphic to A_{2}(2). This surface is unique up to isomorphism.

§10. Quaternary theta series 43

It is an interesting problem to find an explicit description of this distinguished K3-surface.

III. Theta series on the quaternary modular variety

10. Quaternary theta series

We want to introduce modular forms on H(ℍ). We will define theta series and prefer for this purpose the symplectic model

Ω(ℍ) = Bihol(H(ℍ)) = Sp(2, ℍ)/{±E} ∪ τ Sp(2, ℍ)/{±E} (τ(z) = z′).

The embedding

ℍ ⟶ M_{2}(ℂ), a ⟼ ǎ := ( \begin{matrix}a_{1} + ia_{2} & a_{3} + ia_{4} \\ −a_{3} + ia_{4} & a_{1} − ia_{2} \end{matrix} ) ,

extends to an embedding

M_{n}(ℍ) ⟶ M_{2n}(ℂ), A ⟼ Ǎ.

This gives

H(ℍ) ⟶ M_{4}(ℂ), X + iY ⟼ X̌ + iY̌ .

By

j(M, Z) := det(ČŽ + Ď)

we obtain an automorphy factor of Sp(2, ℍ). It is invariant under M ↦ −M and defines an automorphy factor

j : Sp(2, ℍ)/{±E} × H(ℍ) → H(ℍ).

10.1 Lemma. One has

j(g^{τ} , τ(z)) = j(g, z) (g^{τ} := τ ∘ g ∘ τ).

Proof. This formula is true for g = g_{1}g_{2} if it is true for g_{1}, g_{2} instead of g. Hence it is sufficient to verify it on generators. ⊔ ⊓

As a corollary we obtain:

10.2 Lemma. The automorphy factor j extends by

j(τ, z) = 1

to an automorphy factor

j : Ω_{2}(ℍ) × H_{2}(ℍ) ⟶ H_{2}(ℍ).

One has j(σ, z) = 1 for all σ ∈ O(4, ℝ).

Modular forms with respect to a subgroup Γ ⊂ Γ(𝔬) of weight r ∈ ℤ and character v : Γ → ℂ^{•} are holomorphic functions on H(ℍ) satisfying the functional equation

f(g(z)) = v(g)j(g, z)^{r}f(z) for g ∈ Γ.

We denote by [Γ, r, v] the vector space of all such modular forms. When v is trivial we write simply [Γ, r]. We should mention that our normalization of the weight differs from that which Krieg [Kr] or Borcherds [Bo1] use. Their weight is the double one which we use. We prefer this normalization because we want that the basic functions which we will use have weight 1.

We are especially interested in the group Γ = Γ(𝔬)[𝔭].

We introduce quaternary theta series. They can be considered as restriction of standard theta constants with certain half integral characteristics on the Siegel half plane of genus 16: We use the subring

𝔬_{0} := ℤ + ℤi_{1} + ℤi_{2} + ℤi_{3}

of index 2 of the ring of all Hurwitz integers 𝔬.

ϑ \left[{a \atop b}\right] (z) := ∑_{g∈𝔬_{0}^{2}} exp πi\{z[g + a/2] + Re b̄′g\}.

Here a and b can be arbitrary elements from ℍ (or from ℍ ⊗_{ℝ} ℂ if one wants). The pair m = (a, b)′ is called the characteristic of the theta series. We are interested in the following 16 characteristics:

a = (1 + i_{1} + i_{2} + i_{3})α, b = (1 + i_{1} + i_{2} + i_{3})β,

where α and β are columns of two elements from {0, 1}.

10.3 Definition. We denote the 16 theta series with characteristics

( \begin{matrix}α \\ β \end{matrix} ) ∈ \{ \begin{matrix}0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 \end{matrix} \}

by the letters X_{1}, … X_{10} and U_{1}, …, U_{6}. The vector space generated by these 16 theta series is denoted by Θ.

These series have been considered already by Krieg [Kr].

10.4 Proposition. The 16 theta series introduced above are modular forms of weight 1 with respect to the group Γ(𝔬)[𝔭] and with respect to the trivial character. The full modular group Γ(𝔬) acts on the space Θ by the formula

ϑ(z) ⟼ j(g, z)^{−1}ϑ(g(z)).

The dimension of the space Θ is six. Corollary. The group W (E_{6}) acts on the 6-dimensional space Θ. The restriction to the subgroup of index 2 (which is isomorphic to SU(4, 𝔽_{4})) is up to isomorphims the unique irreducible representation of dimension 6 of this group.

Proof. One can use the modular embedding which comes from the standard representation of ℍ in M_{4}(ℝ). Under this standard representation 𝔬_{0} (but not the whole 𝔬) is contained in M_{4}(ℤ). We obtain a modular embedding of Sp(2, 𝔬_{0}) into Sp(8, ℤ). The inverse image of the principal congruence subgroup of level 2 contains Sp(2, 𝔬)[𝔭]. Our 16 theta series are obtained by restricting standard theta constants of degree 8 (s. [Fr1], I.3). From this one obtains easily that the 16 theta series are modular forms on Sp(2, 𝔬)[𝔭] and even more on Γ(𝔬)[𝔭]. One still has to prove that the multiplier systems are trivial. For this purpose it is convenient to use proposition 5.11. By direct calculation one verifies that the multipliers are trivial on the “generators” E(f_{i}, v). We omit details.

A similar argument shows also that the group Sp(2, 𝔬_{0}) acts on the 16 thetas by permutation upto constant factors. But it is not clear from this consideration that the whole group Sp(2, 𝔬) and even Γ(𝔬) acts on this space. To prove this we need

10.5 Lemma. The group Sp(2, 𝔬) is generated by Sp(2, 𝔬_{0}) and by the diagonal matrix with entries (1, ω, 1, ω).

Proof. The congruence group of level 𝔭 is contained in Sp(2, 𝔬_{0}). Therefore it is sufficient to show that Sp(2, 𝔽_{4}) is generated by Sp(2, 𝔽_{2}) and the image of the diagonal matrix. It is known that Sp(2, 𝔽_{4}) is generated by the standard involution, translation matrices and diagonal matrices. The image of ω generates the multiplicative group of 𝔽_{4}. The two elements 1 and ω define a basis of the 𝔽_{2}-vector space 𝔽_{4}. (One can show more: Sp(2, 𝔽_{2}) is a maximal subgroup of Sp(2, 𝔽_{4}).) ⊔ ⊓ Proof of 10.4 continued. We need the transformation behaviour of the theta functions under z_{1} ↦ z_{1}ω. The problem is that ω does not act on 𝔬_{0}. Therefore we decompose 𝔬_{0} into cosets mod 𝔭. This gives us a decomposition of each theta series into four series of the type

ϑ_{1}[m] = ∑_{g∈𝔭^{2}} exp πi{z[g + a/2] + Re(b̄′g)}.

46 III. Theta series on the quaternary modular variety

One has

ϑ \left[{a \atop b}\right] = ϑ_{1} \left[{c \atop b}\right] + e^{πi Re b_{1}}ϑ_{1} \left[{d \atop b}\right] + e^{πi Re b_{2}}ϑ_{1} \left[{e \atop b}\right] + e^{πi Re(b_{1}+b_{2})}ϑ_{1} \left[{f \atop b}\right]

where

c = a; d = a − ( \begin{matrix}2 \\ 0 \end{matrix} ) , e = a − ( \begin{matrix}0 \\ 2 \end{matrix} ) , f = a − ( \begin{matrix}2 \\ 2 \end{matrix} ) .

We specialize this formula to

b = 2βω with β ∈ \{0, 1\}^{2} and ω := \frac{1 + i_{1} + i_{2} + i_{3}}{2}

and obtain

ϑ_{1} \left[{a \atop b}\right] (z) = ϑ_{1} \left[{a \atop 0}\right] (z).

Using the notation

ϑ_{1}(a)(z) := ϑ_{1} \left[{a \atop 0}\right] (z),

we obtain

ϑ \left[{a \atop b}\right] = ϑ_{1}(c) + e^{πiβ_{1}}ϑ_{1}(d) + e^{πiβ_{2}}ϑ_{1}(e) + e^{πi(β_{1}+β_{2})}ϑ_{1}(f).

We derive some relations: From the simple formula

z[gε] = z[g] for ε̄ε = 1, ε ∈ ℍ

follows

\begin{matrix}ϑ \left[{a \atop b}\right] = ϑ \left[{aε \atop bε}\right] & for ε̄ε = 1, ε ∈ 𝔬_{0}, \\ ϑ_{1} \left[{a \atop b}\right] = ϑ_{1} \left[{aε \atop bε}\right] & for ε̄ε = 1, ε ∈ 𝔬. \end{matrix}

We are lead to consider the following 6 functions: 10.6 Definition. The theta series

ϑ_{1}(a), a ∈ \{ \begin{matrix}0 \\ 0 \end{matrix} {0 \atop 2} {2 \atop 0} {2 \atop 2} \begin{matrix}2 \\ 2ω \end{matrix} \begin{matrix}2 \\ 2ω̄ \end{matrix} \}

§11. A formula of Kuß and zero divisors of theta functions 47

are denoted by Y_{1}, …, Y_{6}. From the above formulae we derive 10.7 Remark. We have

X_{1} =Y_{1} + Y_{2} + Y_{3} + Y_{4}  X_{2} =Y_{1} + Y_{2} − Y_{3} − Y_{4}
X_{3} =Y_{1} − Y_{2} + Y_{3} − Y_{4}  X_{4} =Y_{1} − Y_{2} − Y_{3} + Y_{4}
X_{5} =2Y_{3} + Y_{5} + Y_{6}  X_{6} =2Y_{3} − Y_{5} − Y_{6}
X_{7} =2Y_{2} + Y_{5} + Y_{6}  X_{8} =2Y_{2} − Y_{5} − Y_{6}
X_{9} =2Y_{4} + Y_{5} + Y_{6}  X_{10} =2Y_{4} − Y_{5} − Y_{6}
U_{1} = Y_{6} − Y_{5} = −U_{2} = −U_{3} = U_{4} = −U_{5} = U_{6}.

it is easy to see that this system of linear equations is solvable, i.e. the 6 thetas Y_{j} can be reconstructed from the 16 original ones. The six Y_{j} are linearly independent. This can be proved by looking at some Fourier coefficients. We shall see a little later that they are even algebraically independent and hence skip the proof of the linear independency.

The advantage of the series ϑ_{1}(z) is that the summation is taken over 𝔭 which is invariant under multiplication with units from 𝔬. This makes it easy to show that that all elements from O(𝔬) act on Θ. The group Γ(𝔬) is generated by Sp(2, 𝔬_{0})/{±E} and O(𝔬). This follows from 4.6 (or from Kriegs result and 7.2.)

It remains to show that the sixdimensional representation of SU(4, 𝔽_{4}) is irreducible. For this one considers the restriction of this representation to the subgroup Sp(2, 𝔽_{2}), which is isomorphic to the symmetric group S_{6}. Under this group the representation splits into two irreducible representations, the subspaces of Θ which is generated by the 10 even thetas α′β ≡ 0 mod 2 and by the 6 odd ones. The first one is five dimensional the second one dimensional. One easily verifies that both representations are irreducible. They are not invariant under SU(4, 𝔽_{4}).

This completes the proof of proposition 10.4.

11. A formula of Kuß and zero divisors of theta functions

The six “odd” theta series U_{i} (10.3) agree upto sign. Let’s take one of them, U := U_{1}:

U(z) := ϑ \left[{a \atop b}\right] (z) where a_{1} = b_{1} = 2ω, a_{2} = b_{2} = 0.

We want to determine the zeros of U. We may express this also as difference

U = Y_{6} − Y_{5}.

The following formula has been proved by Kuß who gave us the kindly permission to use it here.

11.1 Propositition. We have

U(z) = ∑_{T} a(T)e^{πi(t_{0}z_{0}+t_{2}z_{2}+t_{11}z_{11}+t_{12}z_{12}+t_{13}z_{13}+t_{14}z_{14})},

where t_{0}, t_{2}, t_{11}, t_{12}, t_{13}, t_{14} run through all sixtuples of odd integers with the property

t_{0} > 0, t_{2} > 0, 4t_{0}t_{2} = t_{11}^{2} + t_{12}^{2} + t_{13}^{2} + t_{14}^{2}.

The Fourier coefficients of U are

a(T) = (−1)^{\frac{t_{11}+t_{12}+t_{13}+t_{14}}{2}} ∑_{d|gcd(t_{0},t_{2},t_{11},t_{12},t_{13},t_{14})} d.

This formula shows that U is a so-called Kurokawa-lift (arithmetical lift in the sense of Gritsenko [Gr]).

Proof of proposition 11.1. The Fourier coefficients are

a(t_{0}, t_{2}, t_{1}) = ∑_{\begin{matrix}|g_{1}+ω|^{2}= & t_{0} \\ |g_{2}|^{2}= & t_{2} (g_{1},g_{2}∈𝔬_{0}) \\ \overline{g_{1}+ω}·g_{2}= & t_{1} \end{matrix}} (−1)^{g_{11}+g_{12}+g_{13}+g_{14}}.

The formula shows that t_{0} is odd if a(T) ≠ 0. Taking U_{3} instead of U_{1} we see that t_{2} has to be odd too.

The key of the proof is the following recursion formula:

a(t_{0}, t_{1}, t_{2}) = a(\frac{t_{0}}{p} , t_{1}, pt_{2}) − p · a(\frac{t_{0}}{p^{2}} , \frac{t_{1}}{p} , t_{2}) + p · a(\frac{t_{0}}{p} , \frac{t_{1}}{p} , \frac{t_{2}}{p} ),

which is valid for odd primes. This formula is a consequence of

11.2 Lemma. Let n and m be two coprime numbers. Every element c ∈ 𝔬_{0} of norm |c|^{2} = mn can be written in exactly 8 different manners as product c = ab where a, b are elements of 𝔬_{0} with norms |a|^{2} = m and |b|^{2} = n.

This is a result of Hurwitz. It is usually expressed in the ring 𝔬 of all Hurwitz integers instead of 𝔬_{0}. But there is no difference because every element of 𝔬 is the product of an element of 𝔬_{0} and a unit in 𝔬.

The proof of proposition 11.1 is an easy consequence of these considerations.

We skip the details.  ⊔ ⊓

We determine the precise invariance group for U.

11.3 Remark. The group

Γ(𝔬_{0}) = ⟨Sp(2, 𝔬_{0})/{±E}, O(4, ℤ)⟩,

which is generated by Sp(2, 𝔬_{0})/{±E} and by O(4, ℤ) contains Sp(2, 𝔬_{0})/{±E} as normal subgroup of index 12. The factor group is generated by the images of

x ⟼ ωxω̄, x ⟼ αxᾱ, x ⟼ x̄.

Corollary. The index of Γ(𝔬_{0}) in Γ(𝔬) is 36.

Proof. The unit group 𝔬_{0}^{•} of 𝔬_{0} consists of the 8 elements ±1, ±i_{1}, ±i_{2}, ±i_{3}. The group

(𝔬_{0}^{•} × 𝔬_{0}^{•})/{±(1, 1)} ⊂ O(4, ℤ)

is a subgroup of index 12. The three substitions in 11.3 normalize the subgroup (𝔬_{0}^{•} × 𝔬_{0}^{•})/{±(1, 1)}. It is clear that they generate O(4, ℤ) together with the

subgroup.  ⊔ ⊓

11.4 Lemma. We have

U ∈ [Γ(𝔬_{0}), 1, v].

The character v is determined by the following property:

1. The restriction of v to Sp(2, 𝔬_{0})/{±E} is the nontrivial character

Sp(2, 𝔬_{0})/{±E} ⟶ Sp(2, 𝔽_{2}) ≅ S_{6} ⟶ S_{6}/A_{6} ≅ {±1}.

2. For σ ∈ O(4, ℤ) one has

v(σ) = (−1)^{a(σ)},

where a(σ) denotes the number of minus-signs in

(x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (±x_{σ(1)}, ±x_{σ(2)}, ±x_{σ(3)}, ±x_{σ(4)})

The transformation property under O(4, ℤ) is clear from 11.1. We have

Sp(2, 𝔬_{0})/ Sp(2, 𝔬)[𝔭] ≅ Sp(2, ℤ/2ℤ).

Therefore Sp(2, 𝔬_{0}) is generated by Sp(2, 𝔬)[𝔭] and by Sp(2, ℤ). The transformation law for the standard generators of Sp(2, ℤ) is obvious. It is convenient to use the fact that Sp(2, gz) has only one non trivial character. From 11.1 follows that U changes its sign under z_{0} ↦ z_{0} + 1. ⊔ ⊓

The formula of Kuß shows that U vanishes identically on the subset

z ∈ H(ℍ), z_{1} = i_{1}z_{12} + i_{2}z_{13} + i_{3}z_{14},

which means that one of the four components of z (namely the first one) is zero. This is a quadric of discrimant 2. Hence U vanishes in X(Γ(𝔬)[𝔭]) on one of the 36 irreducible divisors defined by the quadrics of discrimant 2. A fundamental result for this paper states:

11.5 Theorem. The zero divisor of U in X(Γ(𝔬)[𝔭]) is irreducible and consists of one of the 36 irreducible components of the divisor definied by quadrics of discrimant 2. It is represented by the quadric z_{11} = 0.

Corollary. Up to constant factors the modular form U has 36 transformed forms*) under the action of Γ(𝔬). The zero divisors of these 36 modular forms on X(Γ(𝔬)[𝔭]) are precisely the 36 irreducible components of the discrimant-2-divisor.

Corollary. The common set of zeros of all modular forms in the space Θ consists of precisely one point in X(Γ(𝔬)[𝔭]).

Remark: That there are common zeros is clear: It is known that the ring of modular forms is integral over a subring if this subring has no common zero. This means that a space of modular forms on a six dimensional modular variety without common zero has at least dimension seven.

Proof of 11.5. The formula of Kuß shows that U vanishes on the divisor z_{11} = 0. Let g_{1}, …, g_{36} denote a system of representatives of Γ(𝔬_{0})\Γ(𝔬). Then

Δ := ∏_{j=1}^{36} U^{(j)} (U^{(j)} = U|g_{j})

is a modular form with respect to the full modular group Γ(𝔬). This means that theorem 11.5 follows from

11.6 Theorem. There exists a modular form Δ of weight 36 on Γ(𝔬) whose zero divisor on X(Γ(𝔬)) is the quadratic divisor of discrimant 2, which is irreducible. The multiplier system of Δ is given by the non-trivial character

Γ(𝔬) ⟶ W (E_{6}) ⟶ W (E_{6})/ SU(4, 𝔽_{4}) = {±1}.

Proof. We have two proofs for this result. The first one uses Borcherds products [Bo1], [Bo2]. Borcherds constructed a marvellous machine which produces modular forms on orthogonal groups. The following results are contained in [Bo1], theorem 13.3. Let (L, q) be a lattice with an integral quadratic form of signature (2, n). The input is a vector valued elliptic modular form f, the output a modular form B(f) on an orthogonal group. This map is multiplicative, B(f + h) = B(f)B(h). The form f = (f_{g}) is a vector whose components are holomorphic functions on the upper usual half plane parametrized by the discriminant group g ∈ L^{∗}/L. The assumptions on f are: f_{g}(τ + 1) = e^{2πiq(g)}f_{g}(τ) 1.

*) We will give the explicit formulae for these 36 functions during the proof of 15.1.

f_{g}(−\frac{1}{τ} ) = \sqrt{\frac{τ}{i}}^{2−n} \frac{1}{\sqrt{#L^{∗}/L}} ∑_{h∈L^{∗}/L} e^{−2πi⟨g,h⟩}f_{h}(τ). 2.

3. The components are meromorphic at the cusp infinity.

This means that f_{g} admits a Forier expansion

f_{g}(τ) = ∑_{k>>−∞} a_{g}(k)e^{2πikτ} .

The indices k are rational numbers with bounded denominators. The associated Borcherds product B(f) is a meromorphic function on the tube domain associated with L. It transforms like a modular form of weight a_{0}(0)/4 (in our normalization of the weight) under a certain subgroup of finite index of O^{+}(L). It is not necessary to give the definition of this group, because we can derive it from the zeros and poles. The zeros and poles lie on quadrics. If v ∈ L^{∗} is a primitive vector then the order of B(f) at the corresponding quadric Q(v) is

∑_{g∈F} a_{g}(q(g)) where F := (ℚv ∩ L^{∗})/ ± 1.

This sum is finite.

We apply this theorem to our lattice L = H_{1} ⊕ H_{2} ⊕ 𝔬. Representatives of L^{∗}/L are 0, \frac{1+i_{1}}{2} , \frac{1+i_{2}}{2} , \frac{1+i_{3}}{2} . So the modular form has 4 components f_{1}, …, f_{4} with the transformation law

f_{1}(τ + 1) = f_{1}(τ), f_{j}(τ + 1) = −f_{j}(τ) for 2 ≤ j ≤ 4, ; \begin{matrix}2\sqrt{\frac{τ}{i}} ^{4}f_{1}(−\frac{1}{τ} ) = & f_{1}(τ) & +f_{2}(τ) & +f_{3}(τ) & +f_{4}(τ) & , \\ 2\sqrt{\frac{τ}{i}} ^{4}f_{2}(−\frac{1}{τ} ) = & f_{1}(τ) & +f_{2}(τ) & −f_{3}(τ) & −f_{4}(τ) & , \\ 2\sqrt{\frac{τ}{i}} ^{4}f_{3}(−\frac{1}{τ} ) = & f_{1}(τ) & −f_{2}(τ) & +f_{3}(τ) & −f_{4}(τ) & , \\ 2\sqrt{\frac{τ}{i}} ^{4}f_{4}(−\frac{1}{τ} ) = & f_{1}(τ) & −f_{2}(τ) & +f_{3}(τ) & −f_{4}(τ) & . \end{matrix}

The following solution (besides others) has been found by M. Bundschuh (Diplomarbeit, Heidelberg). We are grateful for his permission to use it here.

For the construction one uses three classical theta constants:

ϑ(τ) : = ∑_{n=−∞}^{∞} exp πin^{2}τ,

ϑ̃(τ) : = ∑_{n=−∞}^{∞}(−1)^{n} exp πin^{2}τ,

˜ϑ̃(τ) : = ∑_{n=−∞}^{∞} exp πi(n + 1/2)^{2}τ.

Using their well-known transformation formulae one can show that the following functions

\begin{aligned}f_{0} & = 128\frac{ϑ^{20} + ϑ̃^{20}}{(ϑϑ̃˜ϑ̃)^{8}} − \frac{640(ϑ^{4} + ϑ̃^{4})˜ϑ̃^{4} + 320(ϑ^{8} − ϑ̃^{8})}{3(ϑϑ̃˜ϑ̃)^{4}}, \\ f_{1} & = f_{2} = f_{3} = \frac{128}{3} \frac{ϑ^{20} − ϑ̃^{20} + 2˜ϑ̃^{20}}{(ϑϑ̃˜ϑ̃)^{8}} + \frac{640˜ϑ̃^{8} − 320(ϑ^{8} + ϑ̃^{8})}{3(ϑϑ̃˜ϑ̃)^{4}} \end{aligned}

give a solution of the problem. The Fourier expansions are

f_{0}(τ) = q^{−2} + 144 + 70524q^{2} + …, f_{1}(τ) = 5120q + … (q = exp πiτ).

The weight of the corresponding Borcherds product is 144/4 = 36. The zero divisor comes from the term q^{−2}. There are no poles.

The existence of the modular form f can be obtained in a different (easier) way also from [Bo2]. In this paper Borcherds proved that the space of obstructions for a modular form f with prescribed principal part a_{g}(k), k ≤ 0 is the space of modular forms h of the type

F_{g}(τ + 1) = e^{−2πiq(g)}F_{g}(τ) 1.

F_{g}(−\frac{1}{τ} ) = −\sqrt{\frac{τ}{i}}^{2+n} \frac{1}{\sqrt{#L^{∗}/L}} ∑_{h∈L^{∗}/L} e^{2πi⟨g,h⟩}F_{h}(τ). 2.

3. F is holomorphic at the cusp infinity.

For every solution f the expression ∑_{g} f_{g}F_{g} is a modular from of weight two, whose zero coefficient has to vanish by the residue theorem. This means

∑_{g,k} a_{g}(−k)A_{g}(k) = 0,

if we denote by A_{g}(k) the Fourier coefficients of F_{g}. Applying Serre duality Borcherds showed [Bo2] that this condition is also satisfactory for the existence of f with prescribed a_{g}(−k), k ≥ 0. Bundschuh showed that the space of obstructions has dimension one in our case and that is generated by the following form with the property F_{1} = F_{2} = F_{3}:

\begin{aligned}F_{0} = & 2˜ϑ̃^{8} − ϑ^{8} − ϑ̃^{8} & = −1 + 144q + 912q^{2} + ⋯ \\ F_{1} = & ϑ^{8} − ϑ̃^{8} & = 16q^{1/2} + 448q^{3/2} + ⋯ \end{aligned}

This gives a second proof for 11.6.

There is also a direct method to determine zeros of a given modular form. This method rests on 11.7 Proposition. let L be a lattice of signature (2, n), n ≥ 2 such that L ⊗_{ℤ} ℚ contains a direct sum of two hyperbolic planes. Let Γ ⊂ O^{+}(2, n) be a subgroup which is commensurable with O(L). Then every irreducible subvariety of codimension one of X(Γ) meets the Baily Borel boundary.

This result has been proved by Harris and Li. It follows also from the existence of Borcherds products by induction on n. ⊔ ⊓

From proposition 11.7 follows:

11.8 Corollary. Let f be a (holomorphic) modular form on Γ which vanishes on a certain subvariety Y ⊂ X(Γ) of codimension one. Assume that for every modular F on Γ which vanishes on Y and for every point a in the Baily Borel boundary of X(Γ) the germ f_{a} divides a suitable power of F_{a}. Then Y is the precise zero set of f.

We apply this method in a case which will not be investigated in this paper further, namely in case of the lattice

L = H_{1} ⊕ H_{2} ⊕ L_{0}, L_{0} = { x ∈ 𝔬; x_{4} = 0 } .

This means that L_{0} is the lattice ℤ^{3} with the standard form x_{1}^{2} + x_{2}^{2} + x_{3}^{2}. For f we take the restriction of the ten theta series

ϑ \left[{a \atop b}\right] (z) := ∑_{g∈𝔬_{0}^{2}} exp πi\{z[g + a/2] + Re b̄′g\}.

with “even” characteristics

a = (1 + i_{1} + i_{2} + i_{3})α, b = (1 + i_{1} + i_{2} + i_{3})β, α′β ≡ 0 mod 0.

11.9 Proposition. The product f of the ten even theta series restricted to the five dimensional subspace z_{14} = 0 of the quaternionic half plane is a modular form with respect to the group Γ(L_{0}). It vanishes on the manifolds

z_{11} = ±z_{12} ± z_{13}

in first order and every zero is mod Γ(L_{0}) equivalent to one of them.

Proof. The lattice L_{0} occurs in 4.6. From 4.8 follows that it is nicely generated. The transfomation law under the generators is easy to describe.

It is not quite obvious that the product vanishes on the described manifolds. To prove this we observe that the transformation z_{1} ↦ ωz_{1}ω̄ transforms U_{1} into −(X_{10} + U)/2. (We use the notations of 10.3.) From the Kuß formula we know that U vanishes on z_{14} = 0. This gives the zeros of X_{10} on this manifold. (It would be easy now to derive 11.9 from 11.5.)

We now want to apply 11.8 to show that f has no other zeros than the described ones. We restrict ourselves to the essential case of a zero dimensional boundary component. (The other cases are easier.) Let’s take the standard cusp at infinity. It’s local ring O_{∞} consists of Fourier series of the type

∑_{t≥0} a(t) exp πiσ(tz).

Here t runs over a ceratin lattice of positive semidefinite matrices t. The coefficients a(t) have the invariance property a(t[u]) = a(t) under the group 𝔘 of all transformations of the type z ↦ z[u] which are contained in Γ(L_{0}). Due to a result of Knöller [Kn] (compare [FK]) one obaines the formal completion Ô_{∞} of O_{∞} if one takes all formal series of this type. This formal ring is a subring of a bigger integral domain: We use the notations

X_{0} = exp 2πiσ(z_{0}), X_{10} = exp πiσ(z_{11})), ….

The ring of formal power series in X_{0}, X_{2} over the ring of Laurent polynomials in X_{11}, X_{12}, X_{13}

Õ_{∞} = ℂ{X_{11}, X_{12}, X_{13}}[[X_{0}, X_{2}]]

is an integral domain. The element f in 11.9 is divisible in this ring by the polynomial

16X_{10}^{−4}X_{11}^{−4}X_{12}^{−4}·
(X_{10}X_{11} + X_{12})(X_{10}X_{11}X_{12} − 1)(X_{10} − X_{11}X_{12})(X_{10}X_{12} − X_{11})
(X_{10}X_{12} + X_{11})(X_{10} + X_{11}X_{12})(X_{10}X_{11}X_{12} + 1)(X_{10}X_{11} − X_{12}).

The quotient of f by this polynomial contains a constant coefficient and is hence a unit in Õ_{∞}. From this consideration we obtain: Let F be an element of O_{∞} which vanishes on the manifolds z_{11} = ±z_{12} ± z_{13}. Then F is divisible by f in the ring Õ_{∞},

F (z) = f(z)h(z), h(z) = ∑_{t} c(t) exp πiσ(tz).

This implies a formal equation F (z) = f(z)h(z[u]). But now a problem arises. It is not clear that c(t) is invariant under the group 𝔘. As soon as we have proved this invariance the proof is complete because then F/f is contained in Ô_{∞} and hence in O_{∞} by a flatness argument. We prove the invariance for a typical element of a system of generators, namely

u = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) hence c(t[u]) = c ( \begin{matrix}t_{0} & t_{0} + t_{1} \\ ∗ & t_{0} + 2t_{11} + t_{2} \end{matrix} ) .

The idea is to construct an integral domain of formal series which contains Õ_{∞} and the series ∑_{t} c(t[u]) exp πiσ(tz). It is easy to see that the ring

((ℂ{X_{11}, X_{12}}){{X_{10}}})[[X_{0}, X_{2}]]

has this property*).

*) For a commutative ring R with unity R{{X}} denotes the ring of formal Laurent series ∑_{n} a_{n}X^{n} such that a_{n} is zero for almost all negative n.

§12. The quaternary modular variety as covering of a twisted projective space 55

12. The quaternary modular variety as covering of a twisted projective space

We need a modular form on Γ(𝔬) which does not vanish in the distinguished point ϱ.

12.1 Lemma. There exists a quaternary modular form f of weight 12 with respect to the full modular group Γ(𝔬) whose zero set in X(Γ(𝔬)) is precisely the quadratic divisor Q(1). The character of f is the unique (quadratic) nontrivial character on Γ(𝔬) which is trivial on the subgroup of index 2 which is generated by Sp(2, 𝔬)/{±E} and by the orthogonal transformation z_{1} ↦ z′_{1}. This character is non trivial on Γ(𝔬)[𝔭]. The modular form f does not vanish in the distinguished point ϱ.

Proof. The quickiest way to prove the existence of f is to look at the space of obstructions for Borcherds products (see proof of 11.6). For further use the following explicite construction as theta series may be useful.

For the construction of f we use a kind of multiplicative Hecke operator. By means of the matrix

P := ( 1 + i_{1} 1 \frac{1+i_{1}}{2} 1 )

we define

V_{1} := U|P^{−1}  V_{1}(Z) = \frac{1}{2}U ( ∗  z_{2}^{\frac{z_{0}}{2} \frac{z_{1}(1+i_{1})}{2}} ) .

Using the matrices

M_{2} : = ( \begin{matrix}E & H \\ 0 & E \end{matrix} ) with H = ( \begin{matrix}0 & 1 + i_{2} \\ 1 − i_{2} & 0 \end{matrix} ) and
M_{1} : = ( \begin{matrix}U & 0 \\ 0 & Ū^{′−1} \end{matrix} ) with U = ( \begin{matrix}1 & 0 \\ 1 + i_{2} & 1 \end{matrix} ) .

we construct

V_{1}, V_{2} := V_{1}|M_{1}, V_{3} := V_{1}|M_{2}, V_{4} := V_{1}|M_{1}M_{2}.

If we apply to each of the V_{i} the three even permutations of the variables z_{12}, z_{13}, z_{14}, we obtain 12 theta series. Their product f is a modular form of weight 12 with respect to some congruence subgroup. We determine the zeros of f. We know already that U vanishes at all quadrics Q(v), where v = (x; y) ∈ L^{∗} is a primitive vector of norm q(v) = −1 and such that the hyperbolic coordinates x are even and such that y ≡ 1 mod 𝔭 and that U has no other zeros. The orders are one. If we apply P^{−1} to v we get (3.2) upto a constant factor the vector

(\frac{x_{1}}{2} , x_{2}, \frac{x_{3}}{2} , x_{4}; y \frac{1 + i_{1}}{2} ).

This is a vector of norm −1/2 in L^{∗} and automatically primitive. It is congruent (∗, 0, ∗, 0; (1 + i_{1})/2 mod 2L^{∗} and every vector of this type occurs. It is now easy to show that the set of zeros of f is the union of all quadratic divisors of discriminant 1 (norm −1/2). It is worth while to mention that f can also be constructed as a Borcherds product.

The zero divisor of f is invariant under Γ(𝔬). Therefore f is a modular from with respect to the full group Γ(𝔬). The multiplier is trivial on z_{1} ↦ −z′_{1} because f does not vanish on z_{11} = 0. It is non trivial on z_{13} ↔ −z_{14} because f vanishes at the fixed point set of this transformation in first order. The multiplier system now can be determined by an indirect method. We use a result of Krieg and Walcher [KW], which states that Sp(2, 𝔬)[𝔭] is contained in the commutator subgroup of Sp(2, 𝔬). Hence the restriction of the character to Γ(𝔬)[𝔭] factors through the quotient group S_{3}. It must be the unique non trivial character of this group because Γ(𝔬)[𝔭] contains the transformation z_{1} ↦ −αz′_{1}ᾱ. This character of Γ(𝔬)[𝔭] admits two extensions to Γ(𝔬) because the quotient group W (E_{6}) contains a simple subgroup of index two. We must take the extension which is trivial on z_{1} ↦ z′_{1}. The rest of 12.1 is clear. ⊔ ⊓

The modular form f has a non-trivial multiplier even on Γ(𝔬)[𝔭]). Therefore we have to consider f^{2} instead of f which has a trivial multiplier system (even on Γ(𝔬)).

The basis Y_{1}, … Y_{6} of Θ together with f^{2} gives a system of seven modular forms without common zero in X(Γ(𝔬)[𝔭]). They are algebraically independent and generate a weighted polynomial ring ℂ[ϑ_{1}, …, ϑ_{6}, f]. The weights of the variables are w = (1, 1, 1, 1, 1, 1, 24). The projective algebraic variety which is related to this graded algebra is the weighted projective space P^{6}(ℂ)^{w}. It can be described as the quotient of the usual projective space P^{6}(ℂ) by the cyclic group generated by

[z_{0}, z_{1}, …, z_{6}] ↦ [ζz_{0}, z_{1}, …, z_{6}], ζ = exp (\frac{2πi}{24}) .

A concrete realization as projective variety can be given in the projective space of dimension \binom{24}{6}. The mapping

\begin{matrix}P^{6}(ℂ) & ⟶ P^{\binom{12}{6}}(ℂ), \\ z & ⟼ (z_{0}^{ν_{1}} ⋯ z_{5}^{ν_{6}}, z_{6}) (ν_{1} + ⋯ + ν_{6} = 24), \end{matrix}

factors through the quotient.

12.2 Theorem. Let Y_{1}, …, Y_{6} be the basis of Θ and let f be the form of weight 12 which does not vanish in the distinguished point ϱ. The assignment

z ⟼ [Y_{1}(z)^{ν_{1}} ⋯ Y_{6}(z)^{ν_{6}}, f^{2}] (ν_{1} + ⋯ + ν_{6} = 24)

defines a finite holomomorphic (ramified) covering

X(Γ(𝔬)[𝔭]) ⟶ P^{6}(ℂ)^{w} (w = (1, 1, 1, 1, 1, 1, 24)).

The covering degree is 24.

Proof. We consider the covering in the distinguished point. This point is unique in its fibre. This means that we can compute the covering degree in a small neigbourhood of this point. In a small neighbourhood exists a holomorphic 12-th root h of f. The covering degree in question is 24 times the covering degree of

[z] ⟼ (\frac{Y_{1}(z)}{h(z)}, …, \frac{Y_{6}(z)}{h(z)})

locally at ϱ. But the space generated by the Y_{i} admits a basis whose zero divisors form a coordinate frame. Hence the latter covering degree is one.

⊔ ⊓

It is a major problem to determine the ring of all modular forms for Γ_{6}(𝔬). From 12.2 we obtain 6 algebraically independent modular forms of weights 2, 4, 5, 6, 9, 12, 24. The ring of all modular forms is an integral extension of the polynomial ring generated by them. It seems to be very difficult to get all generators and the relations between them. Further examples of modular forms can be constructed using variants of the construction of f. Let be f_{1} = V_{1}V_{2}V_{3}V_{4} be the product of the 4 translates of U constructed above and f_{1}, f_{2}, f_{3} the three modular forms which one obtains applying the even permutations of z_{12}, z_{13}, z_{14}. We have f = f_{1}f_{2}f_{3} and f_{1}+f_{2}+f_{3} = 0. The expression g := f_{1}f_{2} + f_{1}f_{3} + f_{2}f_{3} is a modular form of weight 8 with respect to the full modular group Γ_{6}(𝔬) but a multiplier system of order 3. This modular form has been found also by Krieg [Kr2] using completely different methods. He constructed a Kurokawa lift of weight 4 (Theorem 1, Corollary 1 in [Kr2]). We can prove that our form g is up to a constant factor the square of this Kurokawa lift.

IV. Some more modular varieties

13. A five dimensional case

In §4 we introduced several sub-lattices L_{0} ⊂ 𝔬. All those lattices have been of the form

L_{0} = 𝔬 ∩ V_{0},

where V_{0} is a subvector space of ℍ. For each of those sublattices the group O^{+}(L) is nicely generated.

13.1 Definition. For an arbitrary sublattice L_{0} ⊂ 𝔬 the group Γ(L_{0}) is the image of O^{+}(L), L = H_{1} ⊕ H_{2} ⊕ L_{0}, in the group of biholomorphic automorphisms of H(V_{0}).

This means

Γ(L_{0}) = O^{+}(L)/{± id}.

13.2 Remark. Let L_{0} be one of the lattices in the list 4.6. The group Γ(L_{0}) consists of all biholomorphic automorphisms which can be extended to an element of Γ(𝔬).

Proof. This is an immediate consequence of 4.8 and 4.9.  ⊔ ⊓

13.3 Definition. The subgroup

Γ(L_{0})[𝔭] ⊂ Ω(L_{0}) = Bihol(H(V_{0}))

consists of all transformations which extend to an element from Γ(𝔬)[𝔭].

The group Γ(L_{0})/Γ(L_{0})[𝔭] is in general not contained as a subgroup in Γ(𝔬)/Γ(𝔬)[𝔭]. It is only a subquotient. If N denots the stabilizer of H(V_{0}) in Γ(𝔬) and N[𝔭] the intersection with Γ(𝔬)[𝔭] then we have a natural surjective homomorphism

N/N[𝔭] ⟶ Γ(L_{0})/Γ(L_{0})[𝔭].

§13. A five dimensional case 59

To determine the kernel we need 13.4 Remark. Let g be a biholomorphic automorphism of H(ℍ) which leaves the diagonal z_{1} = 0 pointwise invariant. Then g is contained in O(ℍ), which means that g is given as an orthogonal transformation z_{1} ↦ σ(z_{1}).

We recall (7.4 and 7.5) that

Γ(𝔬)[𝔭] ∩ O(ℍ) = O′(4, ℤ).

An immediate consequence of this and remark 13.4 is

13.5 Remark. The natural homomorphism

N/N[𝔭] ⟶ Γ(L_{0})/Γ(L_{0})[𝔭]

is an isomorphism if and only if every element of the finite group O(𝔬) which acts as identity on L_{0} is contained in O′(4, ℤ). In this case we can consider Γ(L_{0})/Γ(L_{0})[𝔭] as a subgroup of Γ(𝔬)/Γ(𝔬)[𝔭].

We are interested in modular subvarieties of X(Γ(𝔬)[𝔭]) which do not contain the distinguished point ϱ. Since the lattice E_{6} is even all quadratic divisors with odd discriminant have this property. It follows that the distinguished point ϱ is not contained in a quadratic divisor of discriminant 1. We obtain:

13.6 Proposition. The inverse image of the divisor Q(1) ⊂ X(Γ(𝔬)) in X(Γ(𝔬)[𝔭]) remains irreducible. The distinguished point is not contained in this divisor.

A concrete representant of Q(1) is given by H(ℍ′) ⊂ H(ℍ), where ℍ′ is defined by

ℍ′ := { x = x_{1} + x_{2}i_{1} + x_{3}i_{2} + x_{4}i_{3}; x_{1} = x_{2} }.

In the following we use the notations 𝔬′ = 𝔬 ∩ ℍ′ and

Γ_{6} = Γ(𝔬)  Γ_{6}[𝔭] = Γ(𝔬)[𝔭]  H_{6} = H(ℍ)
Γ_{5} = Γ(𝔬′)  Γ_{5}[𝔭] = Γ(𝔬′)[𝔭]  H_{5} = H(ℍ′)

The group Γ_{5} is a homomorphic image of a subgroup N ⊂ Γ_{6}, namely the normalizer of the divisor z_{11} = z_{12}. The kernel of N → Γ_{5} in this case is a group of two elements (the permutations of z_{11}, z_{12}). This group is contained in Γ_{6}[𝔭]. We can conclude from remark 13.5 that Γ_{5}/Γ_{5}[𝔭] is naturally contained in Γ_{6}/Γ_{6}[𝔭] and from proposition 8.4:

13.7 Proposition. We have natural isomorphisms

Γ_{5}/Γ_{5}[𝔭] ≅ Γ_{6}/Γ_{6}[𝔭] ≅ W (E_{6}).

From the determination of the zeros of the theta functions and from 13.6 now follows:

13.8 Proposition. The functions from Θ have no common zero in H_{5}. A basis ϑ_{0}, …, ϑ_{5} defines a finite covering

ϑ : X(Γ_{5}[𝔭]) ⟶ P^{5}ℂ.

We want to determine the degree of this covering as well as the ramification divisor. For this we have to consider quadrics in H_{5}. We determined the dual lattice of 𝔬′ already in section 4. The discriminant of a vector a ∈ (H_{1}⊕H_{2}⊕𝔬′) is −8|a|^{2}.

Furthermore we need some information about the ramification of the map

X(Γ_{5}[𝔭]) ⟶ X(Γ_{5}).

We determine the inverse image of the irreducible divisors Q(Δ), where Δ is 3, 4 or 8. Representatives of the discriminant group are the vectors

\frac{1 + i_{1}}{4} + i_{2}, \frac{1 + i_{1}}{2} , i_{2}.

The equations of the corresponding divisors (inside z_{11} = z_{12}) are

z_{11} + z_{13} = 0, z_{11} = z_{12} = 0, z_{13} = 0.

We reformulate proposition 6.5 in a geometric form.

13.9 Proposition. Let Q(Δ) ⊂ X(Γ_{5}) denote the irreducible divisor which is defined by the quadrics of discriminant Δ. The inverse image with respect to the covering

X(Γ_{5}[𝔭]) ⟶ X(Γ_{5})

has one components in the case Δ = 3. In the cases Δ = 4 and Δ = 8 it has 36 components.

The natural map

X(Γ_{5}[𝔭]) ⟶ X(Γ_{6}[𝔭])

is birational onto its image. We can consider the intersection (=inverse image) of a quadratic divisor*) in X(Γ_{6})[𝔭] with X(Γ_{5})[𝔭]. This is a union of finitely many quadratic divisors. It is not always easy to determine them explicitely.

We want to determine the zeros of the 36 theta series on X(Γ_{5})[𝔭]. It is sufficient to do this for one of them, for example for U.

*) A quadratic divisor in X(Γ) for one of our groups Γ is by definition an irreducible subvariety of codimension one whose inverse image in the tube domain is an Γ-orbit of quadrics Q(v).

13.10 Proposition. The zero divisor of U on X(Γ_{5})[𝔭] consists of two irreducible components. One of them has discriminant 4 the other 8. They are represented by the vectors (0, 0, 0, 0; (1 + i_{1})/2 resp. (0, 0, 0, 0; i_{2}). Their equations (inside x_{1} = x_{2}) are x_{1} = x_{2} = 0 resp. x_{3} = 0.

Proof. The zeros of U are given by the quadrics Q(v) where

v = (α_{1}, α_{2}, α_{3}, α_{4}; β) ∈ L^{∗}

is a primitive vector with the properties

α_{1}α_{2} + α_{3}α_{4} − |β|^{2} = −1, α_{i} even, β ≡ 1 mod 𝔭 (β ∈ 𝔬).

We want to intersect Q(v) with z_{11} = z_{12}. What we obtain is a quadric Q(w) on the five dimensional domain which is defined by the projection

w := (α_{1}, α_{2}, α_{3}, α_{4}; (β_{1} + β_{2})\frac{1 + i_{1}}{2} + β_{3}i_{2} + β_{4}i_{3}).

If the norm of w is not negative this intersection is empty. We assume that q(β) < 0. Because of |w|^{2} ≥ |v|^{2} the only possibilities are q(w) = −1 and q(w) = −1/2. This implies |β_{1} − 1 − β_{2}| ≤ 1. It follows that w is primitive and congruent mod M to one of the two vectors described in 13.10. This proves

proposition 13.10.  ⊔ ⊓

There are many similar results as proposition 13.10. One starts with a modular form on H_{6} and intersects its zero divisor with H_{5}. Another interesting case is the modular form V_{2} = U|P^{−1}M_{1} which occured in the proof of lemma 12.1. Its zeros are translates of the zeros of U. If one intersects the zero divisor one obtains a union of quadrics of discriminant 4. We omit the proof because it is similar to that of proposition 13.10. Taking a multiplicative symmetrization of V_{2}|H_{5} we obtain a modular form on the whole of Γ_{5}. Its zero divisor must be a multiple of the union of all quadrics of discriminant 4. it is not difficult to compute the order of the zero and the weight. The result is:

13.11 Proposition. There exists a modular form on the full group Γ_{5} of a weight 9k/2, k ∈ ℕ, such that the zero divisor on X(Γ_{5}) is the quadratic divisor of discriminant 4 with multiplicity k.

Remark. By Cousins theorem this modular form is the k-th power of a modular form of weight 9/2*).

*) We did not yet introduce modular forms of half integral weight. In this connection it is sufficient to say that W^{2} is a modular form of weight 9.

Again the powerful method of Borcherds products applies. The discriminant group L^{∗}/L consists of four elements represented by

0, \frac{1 + i_{1}}{2} , \frac{1 + i_{1}}{4} + \frac{1}{2}i_{2}, \frac{1 + i_{1}}{4} + \frac{1}{2}i_{3}.

By a calculation of Bundschuh the space of obstructions (s. §11) for the existence of a Borcherds product (see section 11) again is one- dimensional and generated by the following form with form F_{2} = F_{3}:

\begin{aligned}F_{0} = & ϑ^{7} + ϑ̃^{7} − 2ϑ^{4}ϑ̃^{3} − 2ϑ^{3}ϑ̃^{4} & = 1 − 108q − 450q^{2} + ⋯ \\ F_{1} = & ϑ^{7} − ϑ̃^{7} + 2ϑ^{4}ϑ̃^{3} − 2ϑ^{3}ϑ̃^{4} & = −18q^{1/2} − 232q^{3/2} + ⋯ \\ F_{2} = & ϑ^{4}˜ϑ̃^{3} + ϑ̃^{4}˜ϑ̃^{3} & = −8q^{3/8} − 216q^{11/8} + ⋯ \end{aligned}

The coefficient −18 in F_{2} produces the form in 13.11.  ⊔ ⊓

More results of the type 13.11 can be obtained if one replaces U by the other satellites under Γ_{6} or by suitable Borcherds products.

13.12 Proposition. There exists a modular form of weight 4 on Γ_{5}, such that the zero divisor on X(Γ_{5}) is the quadratic divisor of discriminant 3 (with multiplicity one).

This follows immediately from the computation of the space of obstructions above. The weight four comes from the coefficients −8 in F_{2} = F_{3}. ⊔ ⊓

We want to obtain information about the ramification divisor of the map X(Γ_{5})[𝔭] → P^{5}ℂ. For this purpose we need the Wronski determinant: Let f_{j}(z_{1}, …, z_{n}), 1 ≤ j ≤ n be a system of n + 1 functions of n variables (varying in some open domain of ℂ^{n}. The Wronski determinant is defined as

W (f_{0}, …, f_{n}) := det ( \begin{matrix}f_{0} & … & f_{n} \\ ∂f_{0}/∂z_{1} & … & ∂f_{n}/∂z_{1} \\ ⋮ & & ⋮ \\ ∂f_{0}/∂z_{n} & … & ∂f_{n}/∂z_{n} \end{matrix} )

It is well-known that the Wronsky determinant is a modular form. In our case its weight is 17/2. The ramification of the map

ϑ : H_{5} ⟶ P^{5}ℂ

is the zero divisor of the Wronski determinant W of the θ_{1}, …, θ_{6}. We claim that this map ramifies along the quadrics of discriminant 3 and 4. We take the two representatives

\frac{1+i_{1}}{2} − \frac{i_{2}}{2}  x_{11} = x_{12} = x_{13}  (discriminant 3)
\frac{1+i_{1}}{2}  x_{11} = x_{12} = 0  (discrminant 4)

The above quadric of discriminant 4 is the fixed point set of the involution x_{11} ↦ −x_{11}, x_{12} ↦ −x_{12}. The theta map is invariant under this involution. Hence ϑ ramifies along the quadratic divisors of discriminant 4.

The situation of discriminant 3 is a little bit more complicated because the quadric of discriminant 3 is not the fixed point set of an involution. One has to use that the theta functions are defined on H_{6}. Here we can use the involution which interchanges x_{11} and x_{13}. The theta map on H_{6} ramifies along x_{11} = x_{13} because the theta functions are invariant under this involution. It is easy to prove that the restriction of the theta map to H_{5} ramifies along the intersection of the set x_{11} = x_{13} with H_{5}. This intersection is x_{11} = x_{12} = x_{13}.

We have proved that W is a modular form of weight 17/2 which vanishes on the quadrics of discrimant 3 and 4. From propositions 13.11 and 13.12 follows that there exists a modular form of weight 9/2 + 4 = 17/2 with precisely these zeros. The quotient of W and this form has to be constant.

13.13 Proposition. The Wronski determinant W of the six thetas is a modular form of weight 17/2 with respect to the full modular group Γ_{5}. The zero divisor of this modular form is precisely the union of the quadrics of discriminant 3 and 4, both with multiplicity 1

We obtain the following important application:

13.14 Theorem. The ramification divisor of the theta map

X(Γ_{5}[𝔭]) ⟶ P^{5}ℂ

is irreducible and defined by the quadrics of discriminant 3. The covering degree of the theta map is eight.

The computation of the covering degree is a bit lengthy and we will not give all details of the proof. The main part of the proof is contained in the following section.

To compute covering degrees we use the following well known formula. Let f : X → Y be a finite map of irreducible normal algebraic varieties and B ⊂ Y an irreducible subvariety of codimension one. Let f^{−1}(B) = A_{1} ∪ … ∪ A_{m} be the decomposition of its inverse image into irreducible components. Let d_{i} be the covering degree of A_{i} → B and e_{i} the ramification degree of f along A_{i}. Then the total degree of f is ∑ e_{i}d_{i}.

We apply this to the map ϑ : X(Γ_{5}[𝔭]) → P^{5}ℂ and B = P^{4}ℂ where the P^{4}ℂ is embedded in P^{5}ℂ by the linear equation w_{0} = 0 (w = [w_{0}, …, w_{5}]). The inverse image of B is the zero divisor of U. By 13.10 it consists of two irreducible subvarieties A_{1}, A_{2} say. The map ϑ is unramified along both. The variety A_{1} resp. A_{2} is represented by the equation z_{11} = z_{12} resp. z_{13} = 0. In other terms: A_{1} is the image of the map

φ_{1} : X_{1} := X(Õ(L_{0})[M_{0}]) ⟶ X(Γ_{5}[𝔭]),

L_{0} being the exceptional case in 5.9. And A_{2} is the image of

φ_{2} : X_{2} := X(O(L)[𝔭]) ⟶ X(Γ_{5}[𝔭]),

where L corresponds to the lattice 5 in the list 4.6. For the proof of 13.14 it is therefore enough to show that the degree of ϑ ∘ φ_{1} is 2 and the degree of ϑ ∘ φ_{2} is 6. To do this we relate the X_{i} with certain Hermitean modular varieties.

14. Some Hermitean modular groups

We are lead to consider especially the quadrics of discriminant 3, 4 and 8 in H_{4} with respect to the lattice 𝔬′ which has been defined by x_{3} = x_{4}. They are related to three two dimensional sublattices

\begin{aligned}𝔬_{3} & := \{ x ∈ 𝔬; x_{2} = x_{3} = x_{4} \}, \\ 𝔬_{4} & = \{ x ∈ 𝔬; x_{3} = x_{4} = 0 \}, \\ 𝔬_{8} & := \{ x ∈ 𝔬; x_{2} = 0, x_{3} = x_{4} \}. \end{aligned}

14.1 Remark. The lattices 𝔬_{a}, a ∈ {3, 4, 8} generate a commutative subfield of ℍ which is isomorphic to the imaginary quadratic number field ℚ(\sqrt{−a}). The ring of integers of this field is precisely 𝔬_{a}.

We have to consider the modular groups with respect to these lattices. We denote them and the corresponding subgroups by

\begin{aligned}Γ_{4}(\sqrt{−a}) & := Γ(𝔬_{a}), \\ Γ_{4}(\sqrt{−a}) [𝔭] & := Γ(𝔬_{a})[𝔭]. \end{aligned}

It is easy to see that the groups Γ_{4}(𝔬_{d}) are more or less the Hermitean modular groups in the sense of Hel Braun, more precisely: Let σ denote the conjugation a ↦ ā in O(𝔬_{a}). We have a natural identification

Γ_{4}(\sqrt{−a}) = ⟨Sp(2, 𝔬_{a})/\{±E\}, σ⟩.

The action of Γ_{4}(\sqrt{−a}) is given by the formulae in 2.5. The group Γ_{4}(\sqrt{−a}) [𝔭] is the usual principal congruence subgroup of level 𝔭, where 𝔭 now is the unique ideal in 𝔬_{a} of norm 2 for a = 4, 8 and 𝔭 = 2𝔬_{a} for a = 3. The Galois group of the covering X(Γ_{4}(\sqrt{−a}) [𝔭]) → X(Γ_{4}(\sqrt{−a})) is the symmetric group S_{6} for a = 4, 8 and is the Weyl group W (E_{6}) in case a = 3. The case a = 4 has been investigated in the literature [He],[Ma] and it has been shown that there is an isomorphism

Θ : X(Γ_{4}(\sqrt{−4}) [𝔭]) ⟶ P^{4}(ℂ).

§14. Some Hermitean modular groups 65

Comparing Θ with our ϑ we get ϑ ∘ φ_{1} = j ∘ Θ ∘ η, where η denotes a two-fold covering

η : X_{1} ⟶ X(Γ_{4}(\sqrt{−4} [𝔭])

and j : P^{4}(ℂ) → P^{5}(ℂ) a suitable embedding. From this we get deg(ϑ ∘ φ_{1}) = 2. The covering η can be described more precisely as follows: There is a homomorphism

det : Sp(2, 𝔬_{4}) ⟶ {±1}

given by the determinant. Let G denote its kernel. Then G/{±E} is a normal subgroup of Γ_{4}(\sqrt{−4}) the factor group being isomorphic to ℤ/2ℤ × ℤ/2ℤ and generated by σ and an element γ of determinant −1. We have

Õ(L)[𝔭] ≅ (G ∪ σ ∘ γG)/{±E}.

Using results from [He] we obtain

14.2 Proposition. The theta map defines a two fold covering

X(Γ_{4}(\sqrt{−4})[𝔭]) ⟶ P^{4}(ℂ) ⊂ P^{5}(ℂ),

where P^{4}(ℂ) denotes a certain linear subspace of P^{5}(ℂ). This covering is branched over the Igusa quartic.

The prove of theorem 13.14 follows now from

14.3 Proposition. The theta map

ϑ : X(Γ_{4}(\sqrt{−8}) [𝔭]) ⟶ P^{4}(ℂ)

is of degree six.

We recall that our basic space of theta functions is spanned by 36 thetas which are obtained if one applies Γ_{6} to one of the ϑ[𝔪] (see 10.3). We have to restrict them to H_{4}, where H_{4} denotes the half plane which corresponds to 𝔬_{8}. It is defined inside H_{6} by z_{12} = 0 and z_{13} = z_{14}. One of the thetas (namely U) vanishes identically on H_{4}. The others consist of two Γ_{4}(\sqrt{−8}) orbits with 10 resp. 15 elements. Their zero divisors can be determined by means of 11.5. In any case the zero divisor has two components. We choose the orbit which is represented by ϑ[𝔪] with 𝔪′ = (1, 1, 1, 1). For the convenience of the reader we describe one case more explicitely. To do so, we leave the orthogonal setting and formulate everything in terms that are more common from the literature.

14.4 Lemma. Define

A_{2} := ( \begin{matrix}\sqrt{−2} & 0 \\ 0 & 1 \end{matrix} ) , A_{6} := ( \begin{matrix}2 + \sqrt{−2} & 0 \\ 0 & 1 \end{matrix} ) .

Any zero of ϑ[𝔪] with 𝔪′ = (1, 1, 1, 1) on H_{4} is equivalent mod Γ_{4}(\sqrt{−8} [𝔭]) to a point in

B_{k} := A_{k}H_{3}Ā_{k}^{−1},

where H_{3} denotes the usual Siegel upper halfplane of degree two*) (embedded in H_{6} as z_{12} = z_{13} = z_{14} = 0) and k = 2 or k = 6. The vanishing order of ϑ[𝔪] on B_{2} is two and one on B_{6}. The group of all transformations of A_{k}H_{3}Ā_{k}^{−1} which extend to Γ_{4}(\sqrt{−8}) corresponds on H_{3} to an extension Γ̂_{1,k} of the paramodular group Γ_{1,k}. There is an involution in Γ_{4}(\sqrt{−8})[𝔭]) fixing B_{2} pointwise.

Here the paramodular group Γ_{1,k} is defined as the subgroup of Sp(2, ℚ)/{±E} consisting of all ±({AB \atop CD} ) such that all entries but b_{11} are integral and

b_{11} ∈ \frac{1}{k} ℤ, a_{21}, c_{11}, c_{12}, c_{21}, d_{12} ∈ kℤ.

This group can be extended by I_{k},

I_{k} := ± ( \begin{matrix}R_{k} & 0 \\ 0 & R_{k}^{′−1} \end{matrix} ) , R_{k} = \frac{1}{\sqrt{k}} ( \begin{matrix}0 & 1 \\ −k & 0 \end{matrix} ) .

The extension is denoted by

Γ̂_{1,k} = Γ_{1,k} ∪ I_{k}Γ_{1,k}.

In case k = 6 there is a further involution

J_{3} = ± ( \begin{matrix}S_{3} & 0 \\ 0 & S_{3}^{′−1} \end{matrix} ) , S_{3} = \frac{1}{\sqrt{3}} ( \begin{matrix}3 & −1 \\ −6 & 3 \end{matrix} ) .

Define

Γ̃_{1,6} = Γ̂_{1,6} ∪ J_{3}Γ̂_{1,6} and Γ̃_{1,2} = Γ̂_{1,2}.

The congruence subgroup Γ_{1,k}(2) of level two is defined inside Γ_{1,k} by

a_{11} ≡ a_{22} ≡ d_{11} ≡ d_{22} ≡ 1 mod 2 and kb_{11} ≡ b_{22} ≡ \frac{1}{k} c_{11} ≡ c_{22} ≡ 0 mod 2.

Finally

Γ̃_{1,6}(2) = Γ_{1,6}(2) ∪ J_{3}Γ_{1,6}(2).

*) This group belongs to O(2, 3). Therefore we prefer the notation H_{3} instead of the frequently used notation H_{2}.

14.5 Lemma. Let be

Λ_{k} = ( \begin{matrix}A_{k} & 0 \\ 0 & Ā_{k}^{′−1} \end{matrix} ) .

The normalizor of A_{k}H_{3}Ā′_{k} in Γ_{4}(\sqrt{−8}) [𝔭] is the group Λ_{2}Γ_{1,2}Λ_{2}^{−1} in the case k = 2 and Λ_{6}Γ̃_{1,6}Λ_{6}^{−1} in the case k = 6.

From lemma 14.4, 14.5 we get that the inverse image of the hyperplane corresponding to ϑ[𝔪] is the union of two unramified divisors and that we obtain two generically injective mappings

\begin{matrix}ψ_{2} : X(Γ_{1,2}(2)) & ⟶ X(Γ_{4}(\sqrt{−8})[𝔭]), \\ ψ_{6} : X(Γ̃_{1,6}(2)) & ⟶ X(Γ_{4}(\sqrt{−8})[𝔭]) \end{matrix}

whose images are these divisors. Proposition 14.3 states deg(ϑ ∘ ψ_{2}) + deg(ϑ ∘ ψ_{6}) = 6. In [He] it has been proved that deg(ϑ ∘ ψ_{2}) = 1. We therefore need

14.6 Proposition. The map

ϑ ∘ ψ_{6} : X(Γ̃_{1,6}(2)) ⟶ P^{3}(ℂ)

is a 5-fold covering.

One proceeds along the same lines. There is a ϑ such that the zero divisor of its restriction to X(Γ̃_{1,6}(2)) is 2S_{12}(2) + S_{24}(2), where S_{d}(2) is a Hilbert modular surface associated to ℚ(\sqrt{d}). That S_{12}(2) occurs with multiplicity two corresponds to the fact that there is no involution in Γ̃_{1,6}(2) fixing pointwise a given quadric that represents S_{12}(2) on H_{3}. Proposition 14.6 follows from

deg(ϑ : S_{12}(2) ⟶ P^{2}(ℂ)) = 1 and deg(ϑ : S_{24}(2) ⟶ P^{2}(ℂ)) = 3.

The first statement can be extracted from [He]. We give some information concerning the second. Hilbert modular surfaces associated to ℚ(\sqrt{6}) have been investigated in the paper [vdG2] of van der Geer. Let o be the ring of integers in ℚ(\sqrt{6}) and Γ(24, p) be the principal congruence subgroup of SL(2, o) associated with the unique ideal p of norm 2. Let ρ denote the involution (z_{1}, z_{2}) ↦ (z_{2}, z_{1}) and τ the Hurwitz-Maass involution. Define G (resp. G(2)) to be the extensions of SL(2, o) (resp. Γ(24, p)) by both ρ and τ. We have G/G(2) ≅ S_{3}. The surface in question is

S_{24}(2) = \overline{H_{2}/G(2)},

where H_{2} is the product of two usual upper half planes. This is a rational surface (the rational surface Y/τ from [vdG2], p. 168 is a double cover of S_{24}(2)). The curve F_{N}, N = 4, 16 has 3 components F_{N}^{i}, i = 1, 2, 3 on S_{24}(2). Using the same formula as in [He], p. 116 but with A =(1/\sqrt{12})\binom{1 1}{−\sqrt{6}\sqrt{6}} we get a generically injective map S_{24}(2) → X(Γ̃_{1,6}(2)) and we get theta functions. The space of theta functions is 3-dimensional and there is a basis ϑ_{i} such that the zero divisor of ϑ_{i} is F_{4}^{i} + F_{16}^{i}. (Note that F_{4}^{i} is pointwise fixed by τ.) There are three cusps s_{1}, s_{2}, s_{3} and we have s_{j} ∈ F_{4}^{i} + F_{16}^{i} iff i = j. For i ≠ j the intersection F_{4}^{i} ∩ F_{4}^{j} is empty and the intersections F_{4}^{i} ∩ F_{16}^{j} resp. F_{16}^{i} ∩ F_{16}^{j} consist precisely of one point each. These two points constitute the set of common zeros of ϑ_{i} and ϑ_{j}. From these data we get

deg(ϑ : F_{4}^{i} ⟶ P^{1}(ℂ)) = 1 and deg(ϑ : F_{16}^{i} ⟶ P^{1}(ℂ)) = 2.

This finishes the proof of proposition 14.6.  ⊔ ⊓

15. A four dimensional case

We restrict the theta map to the irreducible divisor which is defined by the quadrics of discriminant 3.

We have seen that they are related to the subring

𝔬_{3} = ℤ + ℤω ⊂ 𝔬.

The quotient field is isomorphic to ℚ(\sqrt{−3}), which is the same as the cyclotomic field ℚ(ζ_{3}), where ζ_{3} denotes a third root of unity. The isomorphism is given by identifying ω and ζ_{3}. We also notice that 𝔭 ∩ ℚ(\sqrt{−3}) is the ideal generated by 2 hence we write

Γ_{4}(\sqrt{−3})[2] := Γ(𝔬_{3})[𝔭].

The image of X(Γ_{4}(\sqrt{−3}) [2]) under the theta map is a hypersurface in P^{5}ℂ. This hypersurface is invariant under the action of W (E_{6}). We want to give the explicit equation of this hypersurface and therefore recall some known facts about the invariants of the group W (E_{6}).

We consider the natural action of this group on the algebra of polynomials on E_{6} ⊗_{ℤ} ℚ, which is the symmetric tensor algebra of E_{6}^{∗} ⊗_{ℤ} ℚ. The minimal vectors of the dual lattice E_{6}^{∗} are called fundamental weights. The dual lattice E_{6}^{∗} contains 27 pairs ±v of fundamental weights. The set of fundamental weights decomposes into 2 orbits under the action of W (E_{6}). Let u_{0}, …, u_{26} be one of the two orbits. Changing all signs one obtains the other one. Let us denote

j_{k} := ∑_{i=0}^{26} u_{i}^{k}.

§15. A four dimensional case 69

It is well known [Hu] that the ring of invariant polynomials is generated by the J_{k} for k = 2, 5, 6, 8, 9, 12. Next we identify our space Θ of theta functions with E_{6}^{∗} ⊗_{ℤ} ℂ. This is possible because there is upto equivalence only one sixdimensional representation of W (E_{6}). We denote the elementes of Θ which correspond to 4u_{i} by U_{i}. The system U_{0}, …, U_{26} is determined up to a (common) sign.

We need some information about the action of W (E_{6}) on Θ to determine the fundamental weights. This group contains the subgroup Sp(2, 𝔽_{2}) which is isomorphic to S_{6}. We already know that S_{6} acts on U = Y_{6} − Y_{5} by means of its non trivial character. Following [He] the action of S_{6} on the space generated by X_{1}, … X_{10} can be described as follows: We introduce

W_{1} = −X_{9}, W_{2} = X_{4}, W_{3} = X_{5}, W_{4} = −X_{1}, W_{5} = X_{10}, W_{6} = X_{7}.

Then we introduce

6·F_{1}=7·W_{1}+5·W_{2}+9·W_{3}+7·W_{4}+5·W_{5}+3·W_{6},
6·F_{2}=7·W_{2}+5·W_{3}+9·W_{4}+7·W_{5}+5·W_{6}+3·W_{1},
6·F_{3}=7·W_{3}+5·W_{4}+9·W_{5}+7·W_{6}+5·W_{1}+3·W_{2},
6·F_{4}=7·W_{4}+5·W_{5}+9·W_{6}+7·W_{1}+5·W_{2}+3·W_{3},
6·F_{5}=7·W_{5}+5·W_{6}+9·W_{1}+7·W_{2}+5·W_{3}+3·W_{4},
6·F_{6}=7·W_{6}+5·W_{1}+9·W_{2}+7·W_{3}+5·W_{4}+3·W_{5}.

We have F_{1} +⋯+F_{6} = 0. This gives an isomorphism from the space generated by X_{1} … X_{10} to the hyperplane in ℂ^{6} defined by the equation F_{1}+⋯+F_{6} = 0. The group S_{6} acts on ℂ^{6} by permutation of the coordinates. The hyperplane is an invariant subspace. This gives the action of S_{6} on the space generated by X_{1} …, X_{10}. These ten functions are transitively permuted under S_{6}.

To get information about the action of the whole W (E_{6}) we consider two transformations which are not contained in the subgroup S_{6}. The first one is the orthogonal transformation z_{1} ↦ ωz_{1}ω̄. It transforms U = Y_{6} − Y_{5} into Y_{5} − Y_{4} = −(X_{10} + U). Appling the subgroup S_{6} we obtain that in the orbit of U the 20 functions (±X_{j} + U) are contained.

To get the missing 16 we apply the translation z_{1} ↦ z_{1} + ω. It fixes Y_{i} for i = 1, 2, 3, 5 and changes the sign of the two remaining ones. Now it is easy to get the whole orbit of U. It consists of the following 36 functions plus their negatives:

1. The twenty functions Y_{i} ± Y_{j}, 1 < j < i ≤ 6.

2. Six linear combinations of Y_{i} where the coefficients consist of five 1 ond one -1. A typical is −Y_{1} + Y_{2} + ⋯ + Y_{6}.

3. Ten linear combinations of Y_{i} where three coefficients are 1 and the other three coefficients are −1. The coefficient of Y_{1} has to be 1:

Next we consider the scalar product ⟨ , ⟩ of the real vector space generated by the Y_{i} which is invariant under W (E_{6}). This is determined upto a constant factor and we normalize it such that ⟨U, U⟩ = 2. The 36 functions must have the same norm. Now it is easy to derive:

15.1 Lemma. The ℤ-module generated by the six functions

\begin{matrix}− \frac{1}{2}(−Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6}), Y_{2} − Y_{3}, \\ − \frac{1}{2}(Y_{1} + Y_{2} + Y_{3} − Y_{4} − Y_{5} − Y_{6}), Y_{3} − Y_{4}, Y_{4} − Y_{5}, Y_{5} − Y_{6} \end{matrix}

is invariant under the action of W (E_{6}). The Gram matrix of this basis is the following matrix representing the E_{6}-lattice:

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

Using this explicit description it is no problem to compute the dual lattice and its minimal vectors. Here is the result:

15.2 Proposition. Let Y_{1}, … Y_{6} denote the basis of Θ. The following 27 elements are permuted under W (E_{6}). Dividing them by 4 and choosing a W (E_{6})-linear isomorphism E_{6}^{∗} ⊗_{ℤ} ℂ ≅ Θ one obtains 27 fundamental weights.

U_{0} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}
U_{1} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}
U_{2} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}
U_{3} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{4} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{5} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}
U_{6} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{7} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{8} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{9} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{10} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{11} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{12} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{13} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{14} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{15} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{16} = 4 · Y_{1}
U_{17} = −2 · Y_{1} + 6 · Y_{2}
U_{18} = −2 · Y_{1} − 6 · Y_{2}
U_{19} = −2 · Y_{1} + 6 · Y_{3}
U_{20} = −2 · Y_{1} − 6 · Y_{3}

U_{21} = −2 · Y_{1} + 6 · Y_{4}
U_{22} = −2 · Y_{1} − 6 · Y_{4}
U_{23} = −2 · Y_{1} + 6 · Y_{5}
U_{24} = −2 · Y_{1} − 6 · Y_{5}
U_{25} = −2 · Y_{1} + 6 · Y_{6}
U_{26} = −2 · Y_{1} − 6 · Y_{6}

The image of X(Γ_{4}(\sqrt{−3}) [2]) under the theta map is a hypersurface in the five dimensional projective space. We want to determine the equation of this hypersurface. Here is it:

15.3 Theorem. The image of X(Γ_{4}(\sqrt{−3}) [2]) under the theta map is a hypersurface I = I 32 ⊂ P^{5}ℂ of degree 32 which is invariant under the action of the Weyl group W (E_{6}). The equation of this hypersurface is given by the following linear combination of 53 monomials. Let be

J_{k} := ∑_{i=0}^{26} U_{i}^{k},

where U_{i} is defined in 15.2.

2^{32} ·3^{16} ·5·7^{2} ·J_{8}^{4} −2^{34} ·3^{14} ·5·7^{2} ·J_{5}^{2}J_{6}J_{8}^{2} +2^{38} ·3^{13} ·5·7·J_{5}^{3}J_{8}J_{9} −2^{39} ·3^{12} ·7^{2} ·J_{5}^{4}J_{12} +2^{34} · 3^{10} · 7^{2} · 61 · J_{5}^{4}J_{6}^{2} −2^{35} · 3^{14} · 5 · 7^{2} · J_{2}J_{6}J_{8}^{3} −2^{35} · 3^{14} · 5^{2} · 7 · J_{2}J_{5}J_{8}^{2}J_{9} +2^{38} · 3^{12} · 5 · 7^{2} · J_{2}J_{5}^{2}J_{8}J_{12} +2^{36} · 3^{10} · 5 · 7^{3} · J_{2}J_{5}^{2}J_{6}^{2}J_{8} −2^{36} · 3^{11} · 5 · 7 · 13 · J_{2}J_{5}^{3}J_{6}J_{9} +2^{35} · 3^{10} · 7^{2} · 13 · J_{2}J_{5}^{6} −2^{34} · 3^{12} · 5^{2} · 7^{2} · J_{2}^{2}J_{8}^{2}J_{12} +2^{33} · 3^{10} · 5 · 7^{2} · 113 · J_{2}^{2}J_{6}^{2}J_{8}^{2} +2^{35} · 3^{11} · 5^{2} · 7 · 17 · J_{2}^{2}J_{5}J_{6}J_{8}J_{9} −2^{35} · 3^{11} · 5 · 7^{3} · J_{2}^{2}J_{5}^{2}J_{6}J_{12} −2^{34} · 3^{9} · 5^{2} · 7^{2} · J_{2}^{2}J_{5}^{2}J_{6}^{3} −2^{35} · 3^{10} · 7^{2} · 29 · J_{2}^{2}J_{5}^{4}J_{8} +2^{35} · 3^{10} · 5^{4} · J_{2}^{3}J_{8}J_{9}^{2} +2^{36} · 3^{10} · 5^{2} · 7^{2} · J_{2}^{3}J_{6}J_{8}J_{12} −2^{35} · 3^{8} · 5 · 7^{2} · 41 · J_{2}^{3}J_{6}^{3}J_{8} −2^{36} · 3^{9} · 5^{3} · 7 · J_{2}^{3}J_{5}J_{9}J_{12} −2^{35} · 3^{7} · 5^{2} · 7 · 193 · J_{2}^{3}J_{5}J_{6}^{2}J_{9} −2^{31} · 3^{12} · 5 · 7^{2} · J_{2}^{3}J_{5}^{2}J_{8}^{2} +2^{33} · 3^{8} · 7^{2} · 367 · J_{2}^{3}J_{5}^{4}J_{6} +2^{34} · 3^{8} · 5^{3} · 7^{2} · J_{2}^{4}J_{12}^{2} +2^{26} · 3^{10} · 5^{3} · 7^{4} · J_{2}^{4}J_{8}^{3} −2^{33} · 3^{8} · 5^{4} · 13 · J_{2}^{4}J_{6}J_{9}^{2} −2^{34} · 3^{6} · 5^{2} · 7^{2} · 41 · J_{2}^{4}J_{6}^{2}J_{12} +2^{32} · 3^{4} · 5 · 7^{2} · 41^{2} · J_{2}^{4}J_{6}^{4} −2^{27} · 3^{14} · 5 · 7^{2} · J_{2}^{4}J_{5}^{2}J_{6}J_{8} +2^{30} · 3^{7} · 5^{2} · 7 · 11 · 31 · J_{2}^{4}J_{5}^{3}J_{9} −2^{27} · 3^{8} · 5^{3} · 7^{2} · 139 · J_{2}^{5}J_{6}J_{8}^{2} −2^{28} · 3^{7} · 5^{4} · 7 · 59 · J_{2}^{5}J_{5}J_{8}J_{9} +2^{30} · 3^{6} · 5^{3} · 7^{2} · J_{2}^{5}J_{5}^{2}J_{12} +2^{28} ·3^{4} ·5·7^{2} ·10903·J_{2}^{5}J_{5}^{2}J_{6}^{2} −2^{27} ·3^{6} ·5^{4} ·7^{4} ·J_{2}^{6}J_{8}J_{12} +2^{26} ·3^{4} ·5^{3} ·7^{2} ·11^{2} ·41·J_{2}^{6}J_{6}^{2}J_{8} +2^{27} · 3^{5} · 5^{4} · 7 · 457 · J_{2}^{6}J_{5}J_{6}J_{9} −2^{24} · 3^{4} · 7^{2} · 179 · 593 · J_{2}^{6}J_{5}^{4} +2^{27} · 3^{8} · 5^{5} · J_{2}^{7}J_{9}^{2} +2^{28} ·3^{4} ·5^{4} ·7^{2} ·41·J_{2}^{7}J_{6}J_{12} −2^{27} ·3^{2} ·5^{3} ·7^{2} ·41^{2} ·J_{2}^{7}J_{6}^{3} +2^{23} ·3^{4} ·5·7^{2} ·64937·J_{2}^{7}J_{5}^{2}J_{8} +2^{17} · 3^{4} · 5^{2} · 7^{2} · 659947 · J_{2}^{8}J_{8}^{2} −2^{18} · 3^{2} · 5 · 7^{2} · 239 · 23747 · J_{2}^{8}J_{5}^{2}J_{6} −2^{19} · 3^{2} · 5^{2} · 7^{2} · 561947 · J_{2}^{9}J_{6}J_{8} −2^{19} · 3^{5} · 5^{3} · 7 · 17 · 179 · J_{2}^{9}J_{5}J_{9} −2^{18} · 3^{4} · 5^{3} · 7^{2} · 11 · 67 · J_{2}^{10}J_{12} +2^{17}·5^{2}·7^{2}·19·23·41·109·J_{2}^{10}J_{6}^{2} +2^{16}·3^{2}·5^{3}·7^{2}·4373·J_{2}^{11}J_{5}^{2} +2^{10}·3^{2}·5^{4}·7^{4}·11·67·J_{2}^{12}J_{8} −2^{11} · 5^{4} · 7^{2} · 11 · 41 · 67 · J_{2}^{13}J_{6} +5^{3} · 7^{2} · 11^{2} · 67^{2} · J_{2}^{16}

The covering degree of

X(Γ_{4}(\sqrt{−3}) [2]) ⟶ I 32

is two.

Proof. Let d denote the degree of the hypersurface and b the covering degree of X(Γ_{4}(\sqrt{−3}) [2]) → I. In a first step of the proof one has to show

db = 64.

Do do this we consider two subvarieties A, B ⊂ X(Γ_{5}[𝔭]). The first one A is the ramification divisor, hence irreducible. The variety B consists of the 36 quadratic divisors of discriminant 4. We know that A and B both are zero divisors of modular forms. We want to avoid multiplier systems. Therefore we consider the cycles kA and kB instead of A and B. For suitable even k we have (13.11, 13.12): kA is the zero divisor of a modular form of weight 9k/2 and kB is the zero divisor of a modular form of weight 8k/2. This means that 8kA and 9kB are linearly equivalent divisors. Next we consider a generic one dimensional linear subspace G ⊂ P^{5}ℂ such that the inverse image ϑ^{∗}(G) intersects A and B properly. Because of the linear equivalence we have

8ϑ^{∗}(G) · A = 9ϑ^{∗}(G) · B.

The right hand side can be computed. A standard formula of intersection theory states that it equals ϑ^{∗}(G)·B = G·ϑ_{∗}(B). This is the degree of the cycle f_{∗}(B). Each irreducible component of B gives the same contribution. Each one has to counted with multiplicity two, because each irreducible component of B is a two-fold covering of its image. This image is a linear subspace and meets therefore G in 1 point. We obtain

G · ϑ_{∗}(B) = 36 · 2 = 72.

Applying intersection theory again we obtain ϑ^{∗}(G)·A = G·ϑ_{∗}(A). But ϑ_{∗}(A) is the hypersurface counted with multiplicity b. This gives us

bd = G · ϑ_{∗}(A) = 64.

As we have a bound for the degree of the hypersurface it is possible to find its equation by means of a computer calculation. The idea of the computation is to use the Fourier expansion of the thetas. The computation Fourier coefficients for the 6 basis functions Y_{i} is very easy. But it is not so easy to multiply them. We need enough Fourier coefficients of the J_{k}. There are 704 monomials of degree 64, for which coefficients have to be computed.

In our case it turned out to be convenient to specialize the modular form to certain ”ergodic surfaces“. These surfaces are obtained by specialising the two variables z_{11}, z_{12} = z_{13} = z_{14}. We took specializations of the form

exp πz_{11} = a, exp πz_{12} = b

with suitable natural numbers a, b. After this specialisation it is very easy to multiply the modular forms and one keeps enough information to prove the following two facts:

1. The dimension of the space of linear relations of the 66 monomials of degree 32 is less or equal 1.

§16. Three dimensional cases 73

2. The dimension of the space of linear relations of the 704 monomials*) of degree 64 is less or equal 66.

This computation can be done modulo a prime. In the second case one has to compute the rank of a 704 × 704-matrix.

From 1. follows that there is at most one relation (up to constant factor) in degree 32. This polynomial has been determined by means of a computer. It is printed in theorem 15.3. We still do not know whether this polynomial gives really a relation. The first relation still could be in degree 64. With help of 2. one shows that the only possible relations in degree 64 are products of the polynomial 15.3 with an arbitrary polynomial of degree 32. The dimension of the space of these polynomials is 66. But the relation is irreducible! Hence degree 64 is excluded and the relation must be given by the distinguished polynomial of degree 32. ⊔ ⊓

Not all of the 66 momomials occur in the relation, 13 coefficients vanish.

16. Three dimensional cases

If one considers one dimensional sublattices of 𝔬 one obtains Siegel threefolds. The best known belongs to the sublattice

ℤ = { x ∈ 𝔬; x_{2} = x_{3} = x_{4} }.

The corresponding group —in our notation Γ(ℤ) is nothing else but the usual Siegel modular group Sp(2, ℤ) (modulo center) of degree two. The subgroup Γ(ℤ)[𝔭] corresponds to the principal congruence subgroup of level two. The restriction of our theta map defines an embedding of the corresponding modular variety of level two into P^{5}ℂ. It is contained in the same four dimensional subspace P^{4}ℂ as described in 14.2. It is a hypersurface which is known as Igusa quartic. We give the equation in our coordinates.

16.1 Proposition. The restriction of the theta map with basis functions Y_{j}, 1 ≤ j ≤ 5, defines an embedding

X(Γ(ℤ)[𝔭]) = X(Sp(2, ℤ)[2]) ⟶ P^{4}ℂ.

The image is the quartic defined by the polynomial

Y_{1}^{2}Y_{5}^{2} − 2Y_{1}Y_{2}Y_{3}Y_{4} + Y_{2}^{2}Y_{3}^{2} + Y_{2}^{2}Y_{4}^{2} − Y_{2}^{2}Y_{5}^{2} + Y_{3}^{2}Y_{4}^{2} − Y_{3}^{2}Y_{5}^{2} − Y_{4}^{2}Y_{5}^{2} + Y_{5}^{4}.

*) To be precise: We mean relations which can be read of finitely many Fourier coefficients of our specialized theta series.

Another example which is known comes from the one dimensional lattice

{ x ∈ 𝔬; x_{1} = x_{2}, x_{3} = x_{4} = 0 }.

The corresponding group is the (extended) paramodular group of polarization type [1, 2] and the congruence subgroup of level 𝔭 is just the congruence subgroup of level two. In this case the theta map defines an biholomorphic map to a P^{3}(ℂ) (s. [He], Fr]).

16.2 Proposition. Let be

L_{0} := { x ∈ 𝔬; x_{1} = x_{2}, x_{3} = x_{4} = 0 }.

The restriction of the theta map defines a biholomorphic map

X(Γ(L_{0})[𝔭]) ⟶ P^{3}(ℂ) ⊂ P^{5}ℂ,

where the three dimensional linear subspace is defined by

Y_{5} = Y_{6}, Y_{4} = 2Y_{6}.

A further interesting one dimensional lattice is

{ x ∈ 𝔬; x_{1} = 0, x_{2} = x_{3} = x_{4} }.

The corresonding modular group is the extended paramodular group of polarization type [1, 3]. This group and the corresponding congruence subgroup has been investigated in [BN].

16.3 Theorem. Let P be the polynomial of degree 32 in the variables Y_{1}, …, Y_{6}. We specialize this polynomial by setting Y_{5} = Y_{6}. The resulting polynomial is called P_{0}. This polynomial is divisible by the third power of the polynomial Q of degree 4 described in 16.1. The quotient S := P_{0}/Q^{3} is a polynomial of degree 20 which is irreducible over the field of complex numbers.

If one expands the polynomial P_{0} as polynomial in Y_{1}, …, Y_{5} one gets a big data set (several MB). It is better to use a different representation which uses the fact that S still has an invariance under the group S_{6} ⊂ W (E_{6}). Following [He] the S_{6}-invariance can be described by means of the functions F_{1}, …, F_{6} which we introduced for the proof of lemma 15.1. The expressions

G_{k} = ∑_{j=1}^{6} F_{j}^{k} (2 ≤ k ≤ 6)

are the basic S_{6}-invarints.

16.4 Proposition. The polynomial S in 16.3 can be written as polynomial in the functions G_{2}, …, G_{6}. Up to a constant factor is

+2^{16} · 3^{4} · 5^{2} · G_{4}G_{5}^{2}G_{6} −2^{18} · 3^{2} · 5^{2} · 19 · G_{2}^{2}G_{3}^{2}G_{5}^{2} +2^{18} · 3^{4} · 5 · G_{2}G_{3}G_{5}^{3} −2^{19} · 3^{4} · G_{5}^{4} −2^{13} · 5^{4} · 71 · G_{2}^{4}G_{3}^{4} +2^{15} · 3 · 5^{3} · 11 · 13 · G_{2}^{3}G_{3}^{3}G_{5} +2 · 3 · 5^{6} · 17 · 19 · 797 · 2843 · G_{2}^{10} +2^{13} · 3^{3} · 5^{4} · 47 · G_{2}^{4}G_{6}^{2} +2^{5} · 3 · 5^{5} · 37 · 389 · G_{2}^{4}G_{4}^{3} +2^{8} · 3^{2} · 5^{4}137 · 613 · G_{2}^{6}G_{4}^{2} +2^{13} ·3^{3} ·5^{4} ·G_{2}^{2}G_{4}G_{6}^{2} +2^{14} ·3·5^{5} ·7·G_{2}^{4}G_{3}^{2}G_{6} −2^{11} ·3^{3} ·5^{4} ·G_{2}G_{4}^{3}G_{6} −2^{10} ·3^{2} ·5^{4} ·313· G_{2}^{3}G_{4}^{2}G_{6} −2^{7} · 3^{2} · 5^{4} · 13 · 6389 · G_{2}^{5}G_{4}G_{6} −2^{12} · 3 · 5^{4} · G_{2}^{2}G_{3}^{4}G_{4} +2^{12} · 3^{2} · 5^{4} · G_{2}G_{3}^{2}G_{4}^{3} +2^{12} · 3^{2} · 5^{4} · G_{2}G_{3}^{2}G_{4}^{3} +2^{10} · 3 · 5^{4} · 7 · 79 · G_{2}^{3}G_{3}^{2}G_{4}^{2} +2^{8} · 3 · 5^{4} · 7 · 23 · 431 · G_{2}^{5}G_{3}^{2}G_{4} +2^{12} · 3^{2} · 5^{4} · G_{2}^{2}G_{3}^{2}G_{4}G_{6} +2^{12} · 3^{2} · 5^{4} · G_{2}^{2}G_{3}^{2}G_{4}G_{6} −2^{16} · 3^{3} · 5^{3} · G_{2}G_{4}G_{3}G_{5}G_{6} +2^{15} ·3^{2} ·5^{3} ·G_{2}G_{3}^{3}G_{4}G_{5} −2^{11} ·3^{2} ·5^{3} ·13·17·G_{2}^{2}G_{3}G_{4}^{2}G_{5} +2^{7} ·3^{3} ·5^{4} ·G_{4}^{5} −2^{12} ·3^{3} ·5^{3} · G_{3}G_{4}^{3}G_{5} −2^{14} · 3^{3} · 5^{2} · 7 · G_{2}G_{4}^{2}G_{5}^{2} −2^{15} · 3^{3} · 5^{2} · G_{3}^{2}G_{4}G_{5}^{2} −2^{8} · 3^{6} · 5^{3} · 14503 · G_{3}G_{2}^{6}G_{5} +2^{18} · 3^{3} · 5^{3} · 7 · G_{2}^{2}G_{5}^{2}G_{6} −2^{14} · 3^{3} · 5^{4} · 13 · G_{2}^{3}G_{4}G_{5}^{2} −2^{7} · 3^{2} · 5^{4} · 2260403 · G_{2}^{7}G_{6} +2^{7} · 3 · 5^{4} · 11 · 311 · 1069 · G_{2}^{7}G_{3}^{2} +3^{2} · 5^{5} · 106521161 · G_{2}^{8}G_{4} −2^{11} · 3^{4} · 5^{2} · 27281 · G_{2}^{5}G_{5}^{2} −2^{15} · 3^{2} · 5^{3} · 281 · G_{2}^{3}G_{3}G_{5}G_{6} −2^{8} · 3^{2} · 5^{3} · 48673 · G_{2}^{4}G_{3}G_{4}G_{5}.

We still have to show that S is irrreducible over the field of complex numbers. We know that it contains only one prime factor over ℂ. Therefore it is sufficient to prove irreducibility over ℚ. This can be done with help of a computer.

16.5 Theorem. Let be

L_{0} := { x ∈ 𝔬; x_{1} = 0, x_{2} = x_{3} = x_{4} }.

The restriction of the theta map

X(Γ(L_{0})[𝔭]) ⟶ P^{4}ℂ

is a covering of a certain hypersurface of degree 20. The equation of the hypersurface is given by the polynomial S (s. 16.4). The covering degree is one or two.

We believe that the degree is two but the method which we used to determine the covering degree in 12.2 does not apply directly because the hypersurface needs not to be normal. Our conviction comes from the paper [BN]. Our Γ(L_{0})[𝔭] is nothing else but the (extended) paramodular group of level (1, 3) considered there (compare lemma 14.4 and the following definition of Γ̂_{1,3}). The authors there constructed a two-sheeted rational map from X(Γ(L_{0})[𝔭]) to a certain quintic which we call the ”Nieto quintic“. It can be proved that the Nieto quintic is dual to our hypersurface of degree 20 (as Hunts I 5 is dual to our I 32). The Gauss map yields a birational map from the Nieto quintic to our hypersurface of degree twenty. We are confident that this map intertwines our map with that considered in [BN] but we did not prove it.

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