Modular forms for the even unimodular lattice of signature (2,10)

Eberhard Freitag Riccardo Salvati Manni Mathematisches Institut Dipartimento di Matematica, Im Neuenheimer Feld 288 Piazzale Aldo Moro, 2 D69120 Heidelberg I-00185 Roma, Italy.

freitag@mathi.uni-heidelberg.de   salvati@mat.uniroma1.it

2005

Introduction

More than forty years ago, Igusa proved in [Ig] a “Fundamental Lemma”, according to which, theta constants give an explicit generically injective map from some modular varieties related to the symplectic group Sp(n, ℝ) into the projective space. The embedded varieties satisfy some quartic relations, the so called Riemann’s relations. We expect that similar results hold for orthogonal instead of symplectic groups, where one should use typical “orthogonal constructions” instead of theta series.

Some years ago, Borcherds described in [Bo1] two methods for constructing modular forms on modular varieties related to the orthogonal group O(2, n). They are the so called Borcherds’ additive and multiplicative lifting. The multiplicative lifting has been used by Borcherds himself and other authors to construct modular forms with known vanishing locus and interesting properties. The additive lifting has been used to construct explicit maps from some modular varieties related to O(2, 4), O(2, 6), O(2, 8), O(2, 10) and also for some unitary groups as U(1, 4) and U(1, 5). cf. [FH], [AF], [Fr2], [Fr3], [FS], [Ko1], [Ko2].

In this paper we try a more systematic treatment in certain level 2 cases: We are mainly interested in even lattices L of signature (2, n), such that the discriminant L′/L is a vector space over the field of two elements and such the the induced quadratic form q : L′/L → 𝔽_{2} is of even type. This means that the dimension 2m of L′/L is even and that q is of the form

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

with respect to a suitable basis. This implies also that n ≡ 2 mod 8. To every such finite geometry (𝔽_{2}^{2m}, q) we attach a certain graded algebra R_{m}/I_{m}. We expect that this algebra is related to the structure of certain modular varieties. There are three cases where this can be verified. The first one has been investigated by Kondo’s in connection with his research about the moduli space of Enriques surfaces. The lattice here is U ⊕ \sqrt{2}U ⊕ (−\sqrt{2}E_{8}) and in this case m = 5. In section 11 we describe in some detail this case again, giving the connection with our results and correcting some error in Kondo’s paper.

Another case, related to the configuration space of 8 points in the projective line, appears in [Ko2] and [Koi]. It is related to the lattice U ⊕ \sqrt{2}U ⊕ (−D_{4}) ⊕ (−D_{4}), i.e. we have m = 3 in this case.

A third member is studied in this paper, namely the lattice \sqrt{2}(U ⊕ U ⊕ (−E_{8})). Here m = 6. In all three cases the ideal I_{m} ⊂ R_{m} occurs. The modular variety, which we investigate, is a covering of Kondo’s variety (modular variety of Enriques surfaces), which can be treated in a similar manner. But we have a bigger symmetry group O(𝔽_{2}^{12}) instead of O(𝔽_{2}^{10}) with the effect that some difficulties of Kondo’s approach disappear.

The ideal I_{m} is defined by quadratic relations in a polynomial ring R_{m}. Kondo considers instead of this only quartic relations. In [Ko2] and [Koi] has been explained in the case m = 3 that the quartic relations are consequences of quadratic relations. This is true for arbitrary m, hence also in Kondo’s O(2, 10)-case.

1. Lattices

A lattice L is a free abelian group together with a symmetric non degenerated bilinear quadratic form

q_{L} : L ⟶ ℚ.

The associated bilinear form is (x, y) = q_{L}(x, y) − q_{L}(x) − q_{L}(y). We extend (x, y) to a ℝ-bilinear from on V = L ⊗_{ℤ} ℝ and moreover to a ℂ-bilinear form on L ⊗_{ℤ} ℂ and we use the notation q_{L}(x) = (x, x)/2 also in this situation.

The norm of a vector a ∈ V is (a, a) = 2q_{L}(a). The lattice is called even if q_{L}(a) is integral for all a. The dual lattice L′ ⊂ L ⊗_{ℤ} ℝ consists of all elements a such that (a, x) is integral for all x ∈ L. The lattice is called unimodular if L = L′. We recall that an even unimodular lattice of signature (m, n) exists if and only if m−n is divisible by 8 and that it is determined up to isomorphism in this case. This lattice is denoted by II_{m,n}. We use the usual notation U = II_{1,1}.

In this paper we will treat mainly even lattices, such that 2L′ ⊂ L. Let us assume that V is of signature (2, n). We recall that there is a unique subgroup of index two O^{+}(V) ⊂ O(V), which contains all reflections

x ⟼ x − \frac{(a, x)}{q_{L}(a)}a

for vectors a ∈ V with negative q_{L}(a). We also consider the integral orthogonal group O(L) and

O^{+}(L) = O(L) ∩ O^{+}(V).

§2. Finite geometries and related projective varieties 3

Since L is even, q_{L} : L′ → ℚ factors through a map

q̄_{L} : L′/L ⟶ ℚ/ℤ,

which is called the discriminant of (L, q_{L}).

We denote by Aut(A) the group of automorphisms of an abelian group and by

O(L′/L) = O(L′/L, q̄_{L}) ⊂ Aut(L′/L)

the subgroup which preserves q̄_{L}. The kernel of the map O(L) ⟶ O(L′/L) is called the discriminant kernel. We define

Γ_{L} = kernel(O^{+}(L) ⟶ O(L′/L)).

Let H be the bounded symmetric domain of type IV associated to O(V), then we shall consider the modular variety H/Γ_{L}.

2. Finite geometries and related projective varieties

We consider 𝔽_{2}^{2m} equipped with the quadratic form

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

and the associated bilinear form

(x, y) = q(x + y) − q(x) − q(y).

The orthogonal group (group of all automorphims which preserve q) is denoted by O(𝔽_{2}^{2m}). Its order is

2^{m(m−1)+1}(2^{m} − 1) ∏_{i=1}^{m−1}(2^{2i} − 1).

The number of isotropic elements is 2^{m−1}(2^{m}+1) and the number of anisotropic elements is 2^{m−1}(2^{m} − 1). We will need the following refined information:

2.1 Lemma. 1) Let a ∈ 𝔽_{2}^{2m} be a non-zero isotropic element.

The number of all non-zero isotropic elements b with (a, b) = 0 is

2^{m−1}(2^{m−1} + 1) − 1

and the number of all anisotropic elements b with (a, b) = 0 is

2^{m−1}(2^{m−1} − 1).

The number of all isotropic elements with (a, b) ≠ 0 is

2^{2(m−1)}.

and the number anisotropic elements b with (a, b) ≠ 0 is also

2^{2(m−1)}.

2) Let a ∈ 𝔽_{2}^{2m} be an anisotropic element.

The number of all non-zero isotropic elements b with (a, b) = 0 is

2^{2(m−1)} − 1

and the number of all anisotropic elements b with (a, b) = 0 is

2^{2(m−1)}.

The number of all isotropic elements with (a, b) ≠ 0 is

2^{m−1}(2^{m−1} + 1).

and the number anisotropic elements b with (a, b) ≠ 0 is also

2^{m−1}(2^{m−1} − 1).

A sub-space A ⊂ 𝔽_{2}^{2m} is called totally isotropic if all elements are isotropic. The maximal totally isotropic spaces are of dimension m. They form one orbit under the orthogonal group.

2.2 Lemma. Let A ⊂ 𝔽_{2}^{2m} be a totally isotropic subspace of dimension m−2. There are precisely 6 maximal totally isotropic spaces (of dimension m) which contain A.

Proof. We decompose 𝔽_{2}^{2m} = 𝔽_{2}^{2m−4}⊕𝔽_{2}^{4}. Let A ⊂ 𝔽_{2}^{2m−4} be a maximal totally isotropic subspace. Then A can be consider as a (m − 2)-dimensional totally isotropic space in 𝔽_{2}^{2m}. The maximal totally isotropic subspaces which contain it are of the form A ⊕ B where B ⊂ 𝔽_{2}^{4} is a two dimensional totally isotropic subspace of 𝔽_{2}^{4}. It is easy to check that there are 6 such spaces. ⊔ ⊓

We list the non-zero elements of these subspaces B:

I_{1} (1,0,0,0) (0,0,1,0) (1,0,1,0)
I_{2} (1,0,0,0) (0,0,0,1) (1,0,0,1)
I_{3} (0,1,0,0) (0,0,0,1) (0,1,0,1)
I_{4} (0,1,0,0) (0,0,1,0) (0,1,1,0)
I_{5} (1,0,0,1) (0,1,1,0) (1,1,1,1)
I_{6} (1,0,1,0) (0,1,0,1) (1,1,1,1)

This table shows:

2.3 Lemma. Let A ⊂ 𝔽_{2}^{2m} be a totally isotropic subspace of dimension m−2. The space which is generated by the characteristic functions of the maximal (=m-dimensional) totally isotropic subspaces containing A has dimension 5. If I_{1}, …, I_{6} denotes these subspaces in a suitable ordering and χ_{1}, …, χ_{6} are their characteristic functions then one has the relation

χ_{1} + χ_{3} + χ_{5} = χ_{2} + χ_{4} + χ_{6}

and furthermore the nine spaces I_{i} ∩ I_{j} with odd i and even j are (m − 1)- dimensional.

Proof. This is a calculation inside 𝔽_{2}^{4} which can be done by means of the above

table.  ⊔ ⊓

Recall that the dimension of a maximal totally isotropic subspace of 𝔽_{2}^{2m} is m. A typical example is the subspace defined by x_{2} = x_{4} = ⋯ = x_{2m} = 0. We consider now (m − 1)-dimensional totally isotropic subspaces N ⊂ 𝔽_{2}^{2m}, for example that one which is defined by the additional condition x_{2m−1} = 0.

2.4 Lemma. Let N ⊂ 𝔽_{2}^{2m} be a (m − 1)-dimensional totally isotropic subspace. There exist exactly two cosets different from N, which contain only isotropic vectors and one coset which contains only anistropic vectors.

Proof. Because we have a transitive action of the orthogonal group on the set of (m − 1)-dimensional totally isotropic subspaces, it is sufficient to treat the above example. In this case the relevant orbits are represented vectors of the

form (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ∗, ∗).  ⊔ ⊓

2.5 Definition. A star is a set of 2^{m−1} anisotropic vectors of 𝔽_{2}^{2m} which form a coset of a (m − 1)- dimensional totally isotropic space.

This isotropic space is determined by the star. A simple calculation shows:

2.6 Lemma. Let M ⊂ 𝔽_{2}^{2m} be star with the underlying (m − 1)-dimensional totally isotropic subspace N ⊂ 𝔽_{2}^{2m}. There exist precisely two m-dimensional (i.e. maximal) totally isotropic subspaces which contain N. The difference of their characteristic functions is an element of ℂ[𝔽_{2}^{2m}] with the following property. It changes the sign if one applies the reflections x ↦ x + (a, x)a along the 2^{m−1} anisotropic vectors a of the associated star.

Next we consider all (m − 1)-dimensional totally isotropic subspaces which contain our (m − 2)-dimensional standard space A. They are of the form A + 𝔽_{2}α, where α is a non-zero isotropic. Hence there are nine such spaces. We list them and the corresponding star (=unique orbit of this space which contains only anisotropic elements): We only list the 𝔽_{2}^{4}-components. The first 2m − 4 components vary arbitrarily in A.

totally isotropic spaces corresponding star
I_{1} ∩ I_{2}  (1,0,0,0)  O_{1} (0,0,1,1) (1,0,1,1)
I_{3} ∩ I_{4}  (0,1,0,0)  O_{2} (0,0,1,1) (0,1,1,1)
I_{1} ∩ I_{4}  (0,0,1,0)  O_{3} (1,1,0,0) (1,1,1,0)
I_{3} ∩ I_{2}  (0,0,0,1)  O_{4} (1,1,0,0) (1,1,0,1)
I_{1} ∩ I_{6}  (1,0,1,0)  O_{5} (0,1,1,1) (1,1,0,1)
I_{5} ∩ I_{2}  (1,0,0,1)  O_{6} (0,1,1,1) (1,1,1,0)
I_{5} ∩ I_{4}  (0,1,1,0)  O_{7} (1,0,1,1) (1,1,0,1)
I_{3} ∩ I_{6}  (0,1,0,1)  O_{8} (1,0,1,1) (1,1,1,0)
I_{5} ∩ I_{6}  (1,1,1,1)  O_{9} (1,1,0,0) (0,0,1,1)

Recall that every (m − 1)-dimensional totally isotropic space B is associated to a star, which is the unique coset consisting of anistropic elements. There are two m-dimensional totally isotropic spaces which contain B.

2.7 Remark. Let ψ_{i} be the characteristic functions of the nine stars above. They satisfy the relations

\begin{matrix}ψ_{1} + ψ_{3} = ψ_{8} + ψ_{9} \\ ψ_{1} + ψ_{4} = ψ_{7} + ψ_{9} \\ ψ_{1} + ψ_{5} = ψ_{2} + ψ_{7} \end{matrix}  \begin{matrix}ψ_{1} + ψ_{6} = ψ_{2} + ψ_{8} \\ ψ_{2} + ψ_{3} = ψ_{6} + ψ_{9} \\ ψ_{3} + ψ_{4} = ψ_{5} + ψ_{9} \end{matrix}  \begin{matrix}ψ_{3} + ψ_{5} = ψ_{4} + ψ_{6} \\ ψ_{3} + ψ_{7} = ψ_{4} + ψ_{8} \\ ψ_{5} + ψ_{8} = ψ_{6} + ψ_{7} \end{matrix}

These relations are related to certain quadratic relations of characteristic functions.

2.8 Lemma. Let A ⊂ 𝔽_{2}^{2m} be an totally isotropic subspace of dimension m−2 and I_{1}, …, I_{6} the maximal totally isotropic subspaces in a suitable ordering. Denote by χ_{1}, …, χ_{6} : 𝔽_{2}^{m} → ℂ their characteristic functions. There are the quadratic relations

\begin{aligned}(χ_{1} − χ_{2})(χ_{1} − χ_{4}) & = (χ_{3} − χ_{6})(χ_{5} − χ_{6}) \\ (χ_{1} − χ_{2})(χ_{3} − χ_{2}) & = (χ_{5} − χ_{4})(χ_{5} − χ_{6}) \\ (χ_{1} − χ_{2})(χ_{1} − χ_{6}) & = (χ_{3} − χ_{4})(χ_{5} − χ_{4}) \\ (χ_{1} − χ_{2})(χ_{5} − χ_{2}) & = (χ_{3} − χ_{4})(χ_{3} − χ_{6}) \\ (χ_{3} − χ_{4})(χ_{1} − χ_{4}) & = (χ_{5} − χ_{2})(χ_{5} − χ_{6}) \\ (χ_{3} − χ_{4})(χ_{3} − χ_{2}) & = (χ_{1} − χ_{6})(χ_{5} − χ_{6}) \\ (χ_{1} − χ_{4})(χ_{1} − χ_{6}) & = (χ_{3} − χ_{2})(χ_{5} − χ_{2}) \\ (χ_{1} − χ_{4})(χ_{5} − χ_{4}) & = (χ_{3} − χ_{2})(χ_{3} − χ_{6}) \\ (χ_{1} − χ_{6})(χ_{3} − χ_{6}) & = (χ_{5} − χ_{2})(χ_{5} − χ_{4}) \end{aligned}

The proofs of 2.7 and 2.8 are trivial.  ⊔ ⊓

There are many other relations between characteristic functions. But the mentioned here will improve to be important, because in some cases they lead to relations between certain modular forms.

We remark that the number of (m − 2)-dimensional totally isotropic spaces in 𝔽_{2}^{2m} is

\frac{5}{2^{m} + 1} · ∏_{i=1}^{m−2} \frac{(2^{2m+1−i} − 1)}{(2^{m−1−i} − 1)}

Fortunately the 9 quadratic relations collaps to one if one takes the linear relation 2.3 into account. This gives us:

2.9 Theorem. Let A be a (m − 2)-dimensional totally isotropic space. Let P be the five dimensional space which is generated by the characteristic functions of maximal totally isotropic spaces containing A (see 2.3). There is a distinguished element ±R_{A} ∈ Sym^{2}(P) which for the standard A can be read off from 2.8 and is defined for general A in an obvious way (using decompositions of a union of two stars as union of two other stars). This element is non-zero and contained in the kernel of

Sym^{2}(P) ⟶ ℂ[𝔽_{2}^{2m}].

A homogenous ideal

For each maximal totally isotropic subspace V ⊂ 𝔽_{2}^{2m} we consider a variable X_{A} and we consider the polynomial ring ℂ[… X_{A} …] in all these variables. We are interested in the subring R_{m} := ℂ[… X_{A} − X_{B} …]. which is generated by differences X_{A} − X_{B}. It is sufficient to take a fixed A and arbitrary B. Hence R_{m} is a polynomial ring in one variable less. For the generation of R_{m} it is also sufficient to take those with dim(A ∩ B) = m − 1. Replacing the characteristic function by the corresponding variable, we obtain from 2.3 and 2.8 a finite set of linear and quadratic relations of the form X_{1} + X_{3} + X_{5} − X_{2} − X_{4} − X_{6} (s. 2.3) and from elements of the form (X_{1}−X_{2})(X_{1}−X_{4})−(X_{3}−X_{6})(X_{5}−X_{6}) as explained in 2.8. They are contained in R_{m} and we can consider the ideal

I_{m} ⊂ R_{m}

generated by them. We believe that this ideal is of interest and related to modular varieties. This paper contains a contribution in the case m = 6. There are related investigations in the cases m = 3 and m = 5, cf. [Ko1] and [Ko2]. One knows

dim(R_{m}/I_{m}) = \{ \begin{matrix}1 & if m = 1 \\ 3 & if m = 2 \\ 6 & if m = 3 \end{matrix}

Unfortunately not much more is known about this ideal. A result which has an application to modular varieties is that in the ideal certain quartic relations can be found: To explain this we consider an totally isotropic subspace A ⊂ 𝔽_{2}^{2m} of dimension m − 3, where we assume m ≥ 3. Let B be a coset which consists of anistropic elements only. There are precisely 15 stars containing B. These 15 stars belong to 30 maximal totally isotropic spaces. We describe them concretely. For this we can assume 𝔽^{2m} = 𝔽^{2(m−3)} ⊕ 𝔽_{2}^{6} , A is the standard totally isotropic space in 𝔽^{2(m−3)} defined by x_{2} = x_{4} = ⋯ = x_{2m−6} = 0 and B is the coset A+(0,…,0,1,1,0,0,0,0). Then the thirty spaces are of the form A ⊕ X, where X is a maximal totally isotropic subspace in 𝔽_{2}^{6}. We list the 30 spaces giving a basis in each case:

\begin{matrix}1 & (0, 0, 0, 0, 0, 1) & (0, 0, 0, 1, 0, 0) & (0, 1, 0, 0, 0, 0) \\ 2 & (0, 0, 0, 0, 0, 1) & (0, 0, 0, 1, 0, 0) & (1, 0, 0, 0, 0, 0) \\ 3 & (0, 0, 0, 0, 0, 1) & (0, 0, 1, 0, 0, 0) & (0, 1, 0, 0, 0, 0) \\ 4 & (0, 0, 0, 0, 0, 1) & (0, 0, 1, 0, 0, 0) & (1, 0, 0, 0, 0, 0) \\ 5 & (0, 0, 0, 0, 0, 1) & (0, 1, 0, 1, 0, 0) & (1, 0, 1, 0, 0, 0) \\ 6 & (0, 0, 0, 0, 0, 1) & (0, 1, 1, 0, 0, 0) & (1, 0, 0, 1, 0, 0) \\ 7 & (0, 0, 0, 0, 1, 0) & (0, 0, 0, 1, 0, 0) & (0, 1, 0, 0, 0, 0) \\ 8 & (0, 0, 0, 0, 1, 0) & (0, 0, 0, 1, 0, 0) & (1, 0, 0, 0, 0, 0) \\ 9 & (0, 0, 0, 0, 1, 0) & (0, 0, 1, 0, 0, 0) & (0, 1, 0, 0, 0, 0) \\ 10 & (0, 0, 0, 0, 1, 0) & (0, 0, 1, 0, 0, 0) & (1, 0, 0, 0, 0, 0) \\ 11 & (0, 0, 0, 0, 1, 0) & (0, 1, 0, 1, 0, 0) & (1, 0, 1, 0, 0, 0) \\ 12 & (0, 0, 0, 0, 1, 0) & (0, 1, 1, 0, 0, 0) & (1, 0, 0, 1, 0, 0) \\ 13 & (0, 0, 0, 1, 0, 0) & (0, 1, 0, 0, 0, 1) & (1, 0, 0, 0, 1, 0) \\ 14 & (0, 0, 0, 1, 0, 0) & (0, 1, 0, 0, 1, 0) & (1, 0, 0, 0, 0, 1) \\ 15 & (0, 0, 0, 1, 0, 1) & (0, 0, 1, 0, 1, 0) & (0, 1, 0, 0, 0, 0) \end{matrix} 16 (0,0,0,1,0,1)(0,0,1,0,1,0)(1,0,0,0,0,0) 17 (0,0,0,1,0,1)(0,1,0,0,0,1)(1,0,1,0,1,0) 18 (0,0,0,1,0,1)(0,1,1,0,1,0)(1,0,0,0,0,1) 19 (0,0,0,1,1,0)(0,0,1,0,0,1)(0,1,0,0,0,0) 20 (0,0,0,1,1,0)(0,0,1,0,0,1)(1,0,0,0,0,0) 21 (0,0,0,1,1,0)(0,1,0,0,1,0)(1,0,1,0,0,1) 22 (0,0,0,1,1,0)(0,1,1,0,0,1)(1,0,0,0,1,0) 23 (0,0,1,0,0,0)(0,1,0,0,0,1)(1,0,0,0,1,0) 24 (0,0,1,0,0,0)(0,1,0,0,1,0)(1,0,0,0,0,1) 25 (0,0,1,0,0,1)(0,1,0,0,0,1)(1,0,0,1,1,0) 26 (0,0,1,0,0,1)(0,1,0,1,1,0)(1,0,0,0,0,1) 27 (0,0,1,0,1,0)(0,1,0,0,1,0)(1,0,0,1,0,1) 28 (0,0,1,0,1,0)(0,1,0,1,0,1)(1,0,0,0,1,0) 29 (0,0,1,1,1,1)(0,1,0,1,0,1)(1,0,0,1,1,0) 30 (0,0,1,1,1,1)(0,1,0,1,1,0)(1,0,0,1,0,1)

2.10 Proposition. Let X_{i} be the variables corresponding to the thirty maximal totally isotropic spaces in 𝔽_{2}^{2m} in the table above. The quartic polynomial

\begin{matrix}(X_{29} − X_{30})(X_{17} − X_{18})(X_{27} − X_{28})(X_{7} − X_{8})+ \\ (X_{11} − X_{12})(X_{13} − X_{14})(X_{15} − X_{16})(X_{21} − X_{22}) \end{matrix}

is contained in the ideal I_{m}.

We proceed as in 2.2 , We decompose 𝔽_{2}^{2m} = 𝔽_{2}^{2m−6} ⊕ 𝔽_{2}^{6}. Let C ∈ 𝔽_{2}^{2m−6} be a maximal totally isotropic subspace. Then C can be considered as a m − 3 dimensional totally isotropic space in 𝔽_{2}^{2m}. The maximal totally isotropic subspaces which contain it are of the form C ⊕ B where B ⊂ 𝔽_{2}^{6} is a three dimensional totally isotropic subspace of 𝔽_{2}^{6}. So we reduced ourselves to the study of totally isotropic subspaces in 𝔽_{2}^{6} . In this case we obtain 420 quartic relations, cf [Ko2] , 36 quadratic relations and 36 linear relations between the 105 stars. Using the computer algebra system SINGULAR, we got that there are 14 linear independent quadratic relations and the generated ideal contains the quartics. A similar calculation seems to be performed in [Koi].

§3. Orthogonal modular forms 9

3. Orthogonal modular forms

In the rest of this text, L denotes an even lattice of signature (2, n). We consider the subset of L ⊗_{ℤ} ℂ defined by

(z, z) = 0, (z, z̄) > 0.

It is an analytic manifold, which consists of two connected components, which can be exchanged by the map z ↦ z̄. We denote by H̃_{n} one of the components and by H_{n} its image in the complex projective space P (L ⊗_{ℤ} ℂ). There is a subgroup of index two O^{+}(V) of the orthogonal group O(V) (V = L ⊗_{ℤ} ℝ) which preserves the two connected components. This group acts on H̃_{n} holomorphically. For any z = x + iy ⊂ H̃_{n} we consider the vector space

W = ℝx + ℝiy ∈ V := L ⊗_{ℤ} ℝ.

This is a positive definite subspace of V. Moreover the space W depends only on the image of z in H_{n}. This defines a bijection between H_{n} and the set of two dimensional positive definite subspaces of V.

The full group O(V) acts on the set of two-dimensional positive definite subspaces. Using the bijection above, on obtains an action of the whole O(V) (not only O^{+}(V)) on H_{n}. But this action is non-holomorphic for g ∈ O(V) − O^{+}(V). Tt is given by the real analytic map z ↦ \overline{g(z)}.

We recall the notion of an (holomorphic or meromorphic) orthogonal modular form of weight k ∈ ℤ with respect to a subgroup Γ ⊂ O^{+}(L) of finite index and with respect to a character v : Γ → ℂ^{•}. It is a (holomorphic or meromorphic) function f : H̃_{n} → ℂ with the properties

\begin{aligned}f(γz) & = v(γ)f(z), \\ f(tz) & = t^{−k}f(z). \end{aligned}

A condition at the cusps has to be added. This condition is in most cases automatically satisfied, for example for n ≥ 3. The vector space of holomorphic forms, which are regular at the cusps, is denoted by [Γ, k, v] and by [Γ, k] when v is trivial. These are finite dimensional spaces and moreover, the algebra

A(Γ) = ⨁_{k∈ℤ}[Γ, k]

is finitely generated.

Let Mp(2, ℤ) be the metaplectic cover of SL(2, ℤ). The elements of Mp(2, ℤ) are pairs (M, \sqrt{cτ + d}), where M = \binom{ab}{cd} ∈ SL(2, ℤ), and \sqrt{cτ + d}) denotes a holomorphic root on H of cτ + d. It is well known that Mp(2, ℤ) is generated by

\begin{aligned}T & = (\binom{1 1}{0 1}, 1) , \\ S & = (\binom{0 − 1}{1 0}, \sqrt{τ}) , Re \sqrt{τ} > 0. \end{aligned}

One has the relations S^{2} = (ST)^{3} = Z, where Z = (\binom{−1 0}{0−1}, i) is the standard generator of the center of Mp(2, ℤ).

Recall that there is a unitary representation ρ_{L} of Mp(2, ℤ) on the group algebra ℂ[L′/L].

ϱ_{L}(T) = (e^{2πiq_{L}(α)})_{α∈L′/L} (diagonal matrix)
ϱ_{L}(S) = \begin{matrix}\sqrt{i} & n−2 \\ √ & \overline{|L′/L|} \end{matrix}(e^{−2πi(α,β)})_{α,β∈L′/L}.

This representation is the Weil representation attached to the finite quadratic form (L′/L, q̄_{L}).

We recall the notion of an elliptic modular form with respect to a finite dimensional representation ϱ : Mp(2, ℤ) → GL(W). Let k ∈ (1/2)ℤ and f : H → W be a holomorphic function. Then f is called modular form of weight k with respect to ρ if

f(Mτ) = \sqrt{cτ + d}^{2k}ϱ(M, \sqrt{cτ + d})f(τ)

for all (M, \sqrt{cτ + d}) ∈ Mp(2, ℤ) and if f is holomorphic at i∞. We denote the space of all these modular forms by [SL(2, ℤ), k, ϱ]. A form is called a cusp form if it vanishes at ∞. The space [SL(2, ℤ), k, ϱ] decomposes into the direct sum of the subspace of cusp forms and into the space of Eisenstein series which can be defined as orthogonal complement with respect to the Petersson scalar product. An Eisenstein series is determined by its constant Fourier coefficient.

The additive lift

In [Bo1], Borcherds defined for integral k + n/2 a linear map

[SL(2, ℤ), k, ϱ_{L}] ⟶ [Γ_{L}, k + n/2 − 1],

which generalizes constructions of Saito-Kurokawa, Shimura, Maa”s, Gritsenko, Oda et.al. We are interested in this construction in the case k = 0. Modular forms of weight zero are constant, hence we get for even n a map

ℂ[L′/L]^{SL(2,ℤ)} ⟶ [Γ_{L}, n/2 − 1].

These orthogonal modular forms are the simplest examples of modular forms of several variables. The weight n/2 − 1 is the so-called singular weight. Every modular form of weight 0 < k < n/2 − 1 vanishes. Modular forms of singular weight which are cusp-forms vanish identically*).

There is a remarkable special case. Assume that L is even and selfdual. Then the Weil representation is the trivial one-dimensional representation and we get a linear map ℂ → [O^{+}(L), n/2 − 1]. The image of 1 is a very distinguished modular form. Borcherd’s formula for the Fourier coefficients show that it is different from 0 and that it agrees with a modular form which has already considered earlier by Gritsenko and in a special case also by Krieg. We call it here Gritsenko’s singular modular form.

We recall from [Fr2] some facts about the Baily Borel compactification of H_{n}/Γ for a subgroup of finite index Γ ⊂ O^{+}(L). There are one- and zerodimensional boundary components which correspond to the Γ-orbits of two and one-dimensional totally isotropic subspaces of L ⊗_{ℤ} ℚ. We recall that the value of an orthogonal modular form F ∈ [Γ, k] at a non-zero isotropic element α ∈ L ⊗_{ℤ} ℚ can be defined. One chooses β ∈ L ⊗_{ℤ} ℚ such that (α, β) = 1 and defines

F (α) : lim_{Im τ→∞} F (τα + 2i(α, β)β).

It is easy to check that this limit exists (using the Fourier expansion of F as described in [Bo1]) and that this limit is independent of the choice of β and that it depends only on the Γ-orbit of α. One has F (tα) = t^{−k}F (α).

Representing the zero-dimensional boundary components by primitive elements α ∈ L′ we obtain:

When the weight k is even, F (α) (for α ∈ L′ primitive isotropic) can be considered as function on the set of zero-dimensional boundary points of H_{n}/Γ.

Additive lifts of constants have a basic property:

3.1 Lemma. Let F be a singular modular form, which is the additive lift of a constant elliptic modular form. When all values F (α), α ∈ L′, primitive and isotropic, are zero, then F is identically zero.

This follows from Borcherds’ description of the Fourier expansion of F [Bo1], theorem 14.3. The formulae there show that the ”‘restriction”’ of F to the onedimensional boundary components are elliptic Eisenstein series, which hence are orthogonal to cusp-forms. Eisenstein series vanish identically if they vanish at all cusps. We don’t want to give more details. We only mention that Borcherds constructed examples of non-trivial singular modular forms which vanish at all zero dimensional cusps. So these cannot be additive lifts. A systematic treatment of them is a major unsolved problem.

*) The theory of singular modular forms is well established in the Siegel case. For the orthogonal case there seems to be no good reference at the moment. The proofs for the basic facts are the same.

Because the value at a cusp appears as constant Fourier coefficient, we can use Borcherds formula from from [Bo1], theorem 14.3 to compute the values of additive lifts at the cusps. (In the first line of the final formula in 5. one has to replace c_{δz}(0) by c_{δz/N}(0).) We only reproduce the formula in a very special case. Let B(n) be the n-th Bernoulli number.

3.2 Proposition. Assume that n ≡ 2 mod 4 and that 2L′ ⊂ L. Let F be the additive lift of C ∈ ℂ[L′/L]^{SL(2,ℤ)}. Then F (α) at a primitive isotropic element α ∈ L′ is

−\frac{B(n/2 − 1)}{n − 2}C(0) if α ∈ L

and otherwise

−\frac{B(n/2 − 1)}{n − 2}(C(0) + C(α)(1 − 2^{n/2−1}))

We immediately obtain:

3.3 Proposition. Assume that n ≡ 2 mod 4, n > 2 and that 2L′ ⊂ L. Assume furthermore that a linear subspace H ⊂ ℂ[L′/L]^{SL(2,ℤ)} with the following property is given: All elements of H vanish at the zero element of L′/L. Then there exists a non-zero constant γ = γ(n) such that for any C ∈ H with corresponding additive lift F = F_{C} and every primitive isotropic α ∈ L′ which is not contained in L the formula

F (α) = γ · C(α)

holds. Corollary. If in addition every non zero element ᾱ ∈ L′/L with q̄_{L}(ᾱ) = 0 can be represented by an isotropic element of L′, then the additive lift

H ⟶ [Γ, n/2 − 1]

is injective.

Let α ∈ L′ be an element of the dual lattice and n < 0 a negative integer. The Heegner divisor H(α, n) ⊂ H_{n} is the union of all

v^{⊥} ∩ H_{n} (v^{⊥} orthogonal complement of v in P (V ⊗_{ℝ} ℂ)),

where v runs through all elements in L′ with

v ≡ α mod L and q_{L}(v) = n.

§4. Level-two-cases 13

We consider H(α, n) as a divisor on H_{n} by attaching multiplicity 1 to all components. We have H(α, n) = H(−α, n), more precisely, this divisor depends only on the image of α in (L′/L)/ ± 1.

The existence of an orthogonal modular forms whose zero divisor is a given Heegner divisor is regulated by Borcherds’ space of obstructions

[SL(2, ℤ), n/2 + 1, ϱ̄_{L}].

3.4 Theorem. Assume n > 2. Then the space of obstructions contains an Eisenstein series ∑ b_{α}(n)e^{2πinτ} with constant Fourier coefficient

b_{α}(0) = \{ \begin{matrix}−1/2 & if α = 0, \\ 0 & else. \end{matrix}

A finite linear combination

∑_{α∈(L′/L)/±1, n<0} C(α, n)H(α, n) (C(α, n) ∈ ℤ)

is the divisor of a meromorphic modular form F of weight

k ∑_{n∈ℤ, α∈L′/L} b_{α}(n)C(α, −n)

if this number is integral and if for every cusp form f in the the space of obstructions,

f_{α}(τ) = ∑_{n∈ℚ} a_{α}(n) exp(2πinτ),

the relation

∑_{n<0, α∈L′/L} a_{α}(−n)C(α, n) = 0

holds.

(The assumption that k is integral can be omitted if one introduces modular forms of non-necessarily integral weight).

4. Level-two-cases

We want to consider level-two situations which means that we assume 2L′ ⊂ L. Then L′/L is a vector space over the field 𝔽_{2} of two elements. We use the embedding

𝔽_{2} ⟶ ℚ/ℤ, 0 ⟼ 0 + ℤ, 1 ⟼ 1/2 + ℤ.

The quadratic form q̄_{L} : L′/L → ℚ/ℤ then can be considered (as usual in the theory of quadratic forms over finite fileds) as function

q̄_{L} : L′/L ⟶ 𝔽_{2}.

For the rest of this paper we make the

4.1 Assumption. The lattice L is even and has signature (2, n). It has the further property 2L′ ⊂ L. The 𝔽_{2}-vector space L′/L admits a basis such that

q̄_{L}(x) := x_{1}x_{2} + ⋯ + x_{2m−1}x_{2m} (2m = dim(L′/L)).

(This means that the finite quadratic form is of “even type”.)

It is well-known that the signature n−2 mod 8 is determined by the associated finite quadratic form. In our case it follows

n ≡ 2 mod 8.

The Weil representation on ℂ[𝔽_{2}^{m}] factors through SL(2, ℤ/2ℤ). It is a real representation with the action of ϱ_{L}(S) and

ϱ_{L}(T) = ((−1)^{q(α)})_{α∈𝔽_{2}^{m}} (diagonal matrix)

We compute the character of this representation. Recall that SL(2, ℤ/2ℤ) is isomorphic to the symmetric group S_{3}. Hence it has three conjugacy classes, which are characterized by their orders 1,2,3. Representatives are the matrices E, S, ST (where E denotes the unit matrix). The traces are given by

trace(ϱ(E)) = 2^{2m}
trace(ϱ(T)) = 2^{m−1}(2^{m} + 1) − 2^{m−1}(2^{m} − 1) = 2^{m}
trace(ϱ(ST)) = \frac{1}{2^{m}} (2^{m−1}(2^{m} + 1) − 2^{m−1}(2^{m} − 1)) = 1

The character table is

\begin{matrix}E & T & ST \\ 1 & 1 & 1 \\ 1 & −1 & 1 \\ 2 & 0 & −1 \end{matrix}

We obtain 4.2 Lemma. We have

dim ℂ[𝔽_{2}^{2m}]^{SL(2,ℤ)} = 2^{m−1} + \frac{1}{3}2^{2m−1} + \frac{1}{3}.

In the case m = 6 this is 715.

On ℂ[𝔽_{2}^{2m}] also O(𝔽_{2}^{2m}) acts and this action commutes with the Weil representation. Hence the group SL(2, ℤ/2ℤ) also acts on the space of O(𝔽_{2}^{2m})- invariants

ℂ[𝔽_{2}^{2m}]^{O(𝔽_{2}^{2m})}.

This space is three dimensional and generated by three elements E_{0}, E_{+}, E_{−}. Here E_{0} means the characteristic function of the zero element, E_{+} is the characteristic function of the set of non-zero isotropic elements and E_{−} is the characteristic function of the non isotropic elements. Using this basis, we get a representation

ϱ̃: SL(2, ℤ/2ℤ) ⟶ GL(3, ℂ).

From 2.1 we obtain:

1) The matrix ϱ̃(T) is the diagonal matrix with the entries 1, 1, −1.

2) The diagonal entries of ϱ̃(S) are

\frac{1}{2^{m}} , \frac{1}{2} − \frac{1}{2^{m}} , \frac{1}{2}.

We obtain

traceϱ̃(E) = 3, traceϱ̃(T) = 1, traceϱ̃(ST) = 0.

A glance at the character table shows:

4.3 Lemma. The space

ℂ[𝔽_{2}^{2m}]^{O(𝔽_{2}^{2m})}

decomposes under the Weil representation into an irreducible two dimensional and a trivial one-dimensional representation. The space

ℂ[𝔽_{2}^{2m}]^{O(𝔽_{2}^{2m})×SL(2,ℤ)}

is one dimensional. It is generated by the function which assigns (2^{m−1} + 1) to the zero element, 1 to all non-zero isotropic and 0 to the anisotropic elements of 𝔽_{2}^{2m}.

4.4 Proposition. The additive lift

ℂ[𝔽_{2}^{2m}]^{SL(2,ℤ)} ⟶ [Γ_{L}, n/2 − 1]

is injective.

Proof. We decompose the space ℂ[𝔽_{2}^{2m}]^{SL(2,ℤ)}. We consider the subspace H of all functions C, which satisfy the linear equation

C(0) = 0.

The full invariant element is not contained in H. Hence ℂ[𝔽_{2}^{2m}]^{SL(2,ℤ)} is the direct sum of the one-dimensional invariant space and H . Since 3.3 applies, we obtain that the additive lift restricted to H is injective. The image contains no full invariant form. But the image of the invariant element is a full invariant form, which is non-zero. Hence we have injectivitiy on the whole. ⊔ ⊓

It is possible to construct explicit elements of ℂ[𝔽_{2}^{12}]^{SL(2,ℤ)}. The following lemma is well-known, [Ko1], [Sch].

4.5 Lemma. The characteristic function of a maximal totally isotropic subspace of 𝔽_{2}^{2m} is SL(2, ℤ)-invariant.

The sum of all these characteristic functions is invariant under the full modular group. Hence we obtain:

4.6 Lemma. Assume that ℂ[𝔽_{2}^{2m}]^{SL(2,ℤ)} decomposes under O(𝔽_{2}^{m}) into the sum of a one-dimensional and an irreducible subspace H. Then the space ℂ[𝔽_{2}^{2m}]^{SL(2,ℤ)} is spanned by the characteristic functions of maximal totally isotropic subspaces. Remark. The irreducibility of H can be checked for m ≤ 6.

There is another important invariant element, which belongs to an arbitrary non-zero isotropic element:

4.7 Remark. Let a be a non-zero isotropic element. Let A be the set of all maximal totally isotropic subspaces which contain a and B the complementary set. Denote by χ_{A} the characteristic function of a maximal totally isotropic subspace. Consider the element

∑_{A∈A} χ_{A} − 2^{m−2} ∑_{B∈B} χ_{B}.

It is a multiple of the function that assumes the value 2^{m−2} on the null vector, −2^{m−2} on the vector a, 1 on the isotropic vectors b with (a, b) = 1 and 0 on the remaining vectors.

§5. Some Borcherds’ products 17

We omit the easy proof. In the Special case a = (1, 0, …, 0) we get

2^{m−2} for x = (0, 0, …, 0)
−2^{m−2} for x = (1, 0, …, 0)
1  for isotropic x with x_{2} = 1
0  else

Now we consider the even and self dual lattice II_{2,n}, n ≡ 2 mod 8. The discriminant group is zero. To obtain a non-trivial discriminant group we rescale this lattice and consider

L = \sqrt{2} II_{2,n}, L′ = L/2.

Then L′/L is a vector space of dimension 2+n over the field 𝔽_{2} of two elements. We can choose an isomorphism 𝔽_{2}^{2+n} → L′/L such that the quadratic form is

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

Let us assume n = 10, the group O(𝔽^{12}) contains a simple subgroup of index two. The character table of this group is known. For example the command “CharTable(O12+(2).2)” of the computer algebra system GAP gives it. The representations of lowest dimensions have dimension 1, 651 and 714 and they are unique in these cases. Hence we obtain:

4.8 Proposition. The additive lift of ℂ[𝔽_{2}^{12}]^{SL(2,ℤ)} to [O^{+}(II_{2,10}[2], 4] is injective and decomposes into the one-dimension trivial and the unique 714dimensional irreducible representation.

5. Some Borcherds’ products

We first investigate Borcherds’ products for the full modular group O^{+}(II_{2,n}), n ≡ 2 mod 8. The space of obstructions is the space of usual elliptic modular forms of weight n/2 + 1. The cases n = 10, 18, 26 are of particular interest because there are no elliptic cusp forms of weight 6, 10 and 14. It is known that the group O^{+}(II_{2,n}) acts transitively on the set of all primitive vectors of a given norm. We study the two lowest cases, vectors of norm (x, x) = −2 and (x, x) = −4. Before we continue, we modify slightly our notation for the Heegner divisors. We denote by H(m) the set of all elements of H_{m}, which are orthogonal to some vector x ∈ II_{2,n} with (x, x) = m. For α ∈ 𝔽^{2+n} we denote by H_{α}(m) the set of all points of H_{n}, which are orthogonal to some x ∈ II_{2,n} with (x, x) = 2m and x ≡ α mod 2. Hence we have

H(m) = ⋃_{α∈𝔽^{2+n}} H_{α}(m).

With the notations of section 1 we have H(m) = H(0, m) where the underlying lattice for the definition of H(0, m) is II_{2,n}. If we identify α with an element of L′/L where L = \sqrt{2}II_{2,n} we get because of the rescaling factor

H_{α}(m) = H(α, m/2).

The Heegner divisors H(−1) and H(−2) are irreducible in H_{n}/O^{+}(II_{2,n}). This follows from the fact that two primitive vectors of II_{2,n} of the same norm are equivalent mod O^{+}(II_{2,n}) and that elements of norm −2 and −4 are automatically primitive. We compute the weights of the modular forms with this divisor. They are regulated by the Eisenstein series of weight 6, 10 and 14. Their Fourier expansion, normalized such the constant coefficient is −1/2, starts as follows:

weight 6 : − \frac{1}{2} + 252q + 8316q^{2} + ⋯
weight 10 : − \frac{1}{2} + 132q + 67716q^{2} + ⋯
weight 14 : − \frac{1}{2} + 12q + 98316q^{2} + ⋯

5.1 Lemma. In the cases n = 10, 18, 26 there exists a modular form on the full group O^{+}(II_{2,n}) whose zero divisor is the irreducible divisor H(−1). The weights are 252, 132, 12. There also exists a modular forms whose zero divisor is the irreducible divisor H(−2). The weights are 8316, 67716, 98316.

We now investigate the level two case.

5.2 Lemma. The divisor H_{α}(−1) is not empty if and only if α ∈ 𝔽_{2}^{m} is anisotropic. In this case it is irreducible in H_{n}/O^{+}(II_{2,n})[2]. Hence H(−1) considered in H_{n}/O^{+}(II_{2,n})[2] has 2^{m−1}(2^{m} − 1) irreducible components. The divisor H_{α}(−2) is not empty if and only if α ∈ 𝔽_{2}^{m} is a non zero isotropic. In this case it is irreducible in H_{n}/O^{+}(II_{2,n})[2]. Hence H(−2) considered in H_{n}/O^{+}(II_{2,n})[2] has 2^{m−1}(2^{m} + 1) − 1 irreducible components.

We are interested in the case that the modular form, which belongs to H(−1) or H(−2), splits into a product of forms with irreducible divisors H_{α}(−1) or H_{α}(−2). But then the number of components should divide the weight of this modular form. There is one distinguished case where this happens. In the case II_{2,10} and the divisor H(−2) the weight has been computed as 8316 (s. 5.1) and the number of components is 2079. One has 8316/2079 = 4. Thus we are lead to

5.3 Proposition. For every non-zero isotropic α ∈ 𝔽_{2}^{12} (their number is 2079) there exists a modular form on the congruence group of level two O^{+}(II_{2,10})[2] with divisor H_{α}(−2). The weight of this form is 4.

§6. Projection of groups 19

Proof. We want to apply Borcherd’s obstruction theory. For this purpose we have to compute the space of elliptic cusp forms of weight 6 with respect to the Weil representation on ℂ[𝔽_{2}^{12}]. The Weil representation is trivial on the principal congruence subgroup of level 2 of the elliptic modular group. The space of cusp forms of weight 6 of this group has dimension one, a generating element is η^{12}, where η denotes the Dedekind η-function. The expansion of η^{12} is of the form

η(τ)^{12} = ∑_{n>0} a_{n}e^{2πinτ} , a_{n} ≠ 0 ⟹ n ∈ \frac{1}{2} + ℤ.

Hence no a_{n} ≠ 0 with integral n occurs. We see that actually all H_{α}(m) with even negative m (and non zero isotropic α) are divisors of modular forms.

We shall consider particular additive lifts of the elements from ℂ[𝔽_{2}^{12}]^{SL(2,ℤ)} and identify them with Borcherds products.

5.4 Proposition. Let M ⊂ 𝔽_{2}^{12} be a star (2.5). The additive lift of the corresonding Weil invariant (2.6) is a modular form in [O^{+}(II_{2,10})[2], 4] whose zero divisor consists precisely of the 32 Heegner divisors H_{α}(−1) with α ∈ M.

Proof. Since the additive lift changes its sign under the 2^{5} reflections, there must be zeros along the 2^{5} Heegner divisors. We have to show that there are no other zeros. For this one takes the product with respect to all a and divides by a suitable power with the form of weight 252 from 5.1. The result is a form

of weight 0 which has to be constant.  ⊔ ⊓

For similar statements cf [Ko1] and [Ko2].

5.5 Definition. Let M be a star. The divisor consisting of the 32 Heegner divisors H_{α}(−1) with α ∈ M is called the associated star divisor.

We consider now the additive lift of the Weil invariant, which has been attached to an arbitrary non zero isotropic element in 4.7. We will show that this is also a Borcherds product.

5.6 Theorem. Let α ∈ 𝔽_{2}^{12} be a non-zero isotropic vector. The Borcherds product with divisor H_{α}(−2) as described in 5.3 is contained in the additive lift space. It is (up to a constant factor) the additive lift of the Weil invariant defined in 4.7.

All what we have to show is that the additive lift vanishes along H_{α}(−2). The proof of this is quite involved because we have now reflection with fixes this divisor. We postpone the proof to the next section.

6. Projection of groups

Let L be an even lattice of signature (2, n) and M ⊂ L a sublattice of signature (2, m). If the connected components H̃_{n}, H̃_{m} have been properly chosen then there is a natural inclusion H̃_{m} ⊂ H̃_{n}, which induces an inclusion of the corresponding half planes H_{m} ⊂ H_{n}. Let Γ ⊂ O^{+}(L) a subgroup of finite index. We consider the subgroup of all g ∈ Γ with g(M) = M. There is a natural homomorphism of this subgroup into O(M) and it is easy to see that this subgroup —let’s denote it by Γ′— is contained in O^{+}(M). We call this group the projected group. If v, v′ are compatible characters of Γ, Γ′ there is a natural restriction map

[Γ, k, v] ⟶ [Γ′, k, v′].

6.1 Theorem (Kneser). Assume that L is a lattice of index (2, m), m ≥ 4, which has Witt-rank two (which means that L×_{ℤ} ℚ contains a totally isotropic subspace of dimension two). Assume furthermore that L contains a vector of norm −2 and that the following two conditions hold:

a) There exists a sublattice L_{1} of rank ≥ 5 whose discriminant is not divisible by 3.

b) There exists a sublattice of rank ≥ 6 with odd discriminant. Then the discriminant kernel Γ_{L} is generated by reflections x ↦ x + (a, x)a along norm −2 vectors a.

The lattice D_{m} consists of all x ∈ ℚ^{m} such that x_{1} + ⋯ + x_{m} is even. The quadratic form is

q(x) = \frac{1}{2}(x_{1}^{2} + ⋯ + x_{m}^{2}).

We apply Knesers result 6.1 to the lattice ℤ^{4} × (−D_{m}) where m ≥ 3. The lattices A_{1} and A_{2} can be embedded into D(m), m ≥ 3. Hence we can take L_{1} = ℤ^{4} × (−A_{1}) (discriminant 2) and L_{2} = ℤ^{4} × (−A_{2}) (discriminant 3).

6.2 Lemma. The discriminant kernels Γ_{L} are generated by reflections along vectors of norm −2 in the cases

L = L_{m} = ℤ^{4} × (−D_{m}) with m ≥ 3.

We need another tool.

A vanishing result

It may happen that a Heegner divisor is the divisor of a Borcherds product. In this case the weight of this product is given by a certain Fourier coefficient of an elliptic Eisenstein series. This Fourier coefficient is defined even if the Heegner divisor is not a Borcherds product. We denote this number the virtual weight of the Heegner divisor. We refer to the paper [BK] of Bruinier and Kuss for the definition and computation of these numbers. We want to use the following non-trivial result of Bruinier:

6.3 Proposition (Bruinier). Let L be an even lattice of signature (2, m) with m ≥ 4 and f a non vanishing (entire) modular form of weight k on the discrimant kernel Γ_{L}. Assume that f vanishes along a certain Heegner divsor H (and maybe also on another other divisor). Then k is greater or equal than the virtual weight of H.

Now we have the tools for the Proof of 5.6. We take the realization II_{2,10} = ℤ^{4} × (−E_{8}) and we choose a concrete isomorphism II_{2,10}/2II_{2,10} ≅ 𝔽_{2}^{12}. We consider a vector a of norm -4. It can be taken inside −E_{8}. It defines a non-zero isotropic vector α in 𝔽_{2}^{12}. This vector defines a Weil invariant (s. 4.7). We denote by f the additive lift of it. This is modular form with trivial character under the stabilizer Γ of α inside O^{+}(II_{2,10}). From its concrete description in connection with 3.3 follows that it vanishes at all cusps which are defined by isotropic elements orthogonal to a.

All what we have to show is that f vanishes along the orthogonal complement of a. The orthogonal complement of a a norm 4 vector inside E_{8} is isomorphic to the root lattice D_{7}. Generally inside D_{m+1} sits D_{m} as orthogonal complement of a root. Hence we can consider the chain

ℤ^{4} × (−D_{2}) ⊂ ⋯ ℤ^{4} × (−D_{7}) ⊂ ℤ^{4} × (−E_{8}).

We denote the corresponding half planes by H_{4} ⊂ ⋯ ⊂ H_{10}. The claim is that f vanishes on the nine dimensional H_{9}. This will be done inductively starting form H_{4}.

The case D_{2} is exceptional, one can not apply Kneser’s theorem. This lattice is generated by two orthogonal vectors (1, ±1) of norm -2. Hence it is isomorphic to A_{1} × A_{1}. From the explicit description of the automorphism groups [CS] follows that every automorphism of D_{2} extends to an automorphism of D_{7} and this extends to an automorphism of E_{8} which fixes a. The orthogonal group O^{+}(ℤ^{4} × −(A_{1} × A_{1})) is isomorphic to the extended Hermitian modular group of degree two of the Gauss number field [FH]. This group is generated by Eichler transformations and by the automorphisms of A_{1} × A_{1}. Eichler transformations of course extend to O^{+}(ℤ^{4} × (−E_{8}). Hence we obtain that the projected group of Γ is the full O^{+}(ℤ^{4} × (−D_{2})) which can be identified with the extended Hermitian modular group. From the structure theorem [Fr3] one can see every symmetric cusp form of weight 4 vanishes.

The next step is to prove that f vanishes on H_{5}. It is a modular form with respect to a certain projected group Γ_{5}. Now we can apply Kneser’s result to prove that this group contains the discrimant kernel of ℤ^{4} ×(−D_{5}). The image of H_{4} is contained in the Heegner divisor which is defined by a root. But this Heegner divisor is irreducible. This follows from the following well-known fact:

6.4 Lemma. Let L be an even lattice of which contains a copy of II_{2,2}. Then any two primitive vectors a, b ∈ L′ with the same image in L′/L are equivalent mod Γ_{L}.

We have seen that the image of H_{4} inside H_{5}/Γ_{5} is a Heegner divisor and that f vanishes along it. Now we can apply Brunier’s result 6.3. A concrete calculation with coefficients of Eisenstein series gives the virtual weight 9. Because this exceeds the actual weight 4, the form f must vanish on H_{5}. In the same way we prove the vanishing inductively on H_{6}, …, H_{9}. The corresponding virtual

weights are 8, 7, 6, 5. This completes the proof of 5.6.  ⊔ ⊓

7. Relations

In this section we shall consider algebraic relations between modular forms that are additive liftings.

Since the additive lifting is a linear map, the linear relations between the characteristic functions described in 2.3 of course imply the same relations between the corresponding modular forms. But for the algebraic relations this is not clear. We only can see that an algebraic relation between modular forms implies the same relation between their values at the cusps. If we are inside the space H (defined by C(0) = 0 then 3.3 shows that the input functions C ∈ (𝔽_{2}^{2m})^{SL(2,ℤ)} (considered as functions on 𝔽_{2}^{m}) satisfy the same relation. Hence there is a chance that the quadratic relations in (𝔽_{2}^{2m})^{SL(2,ℤ)}, which have been described in 2.8 also hold for the corresponding modular forms. We don’t know, whether this is always true, but it is at least true when there are modular forms whose zero set are star divisors. This is the case when n = 10 and m = 3, 5, 6.

7.1 Lemma. Let A ⊂ 𝔽_{2}^{12} be a 4-dimensional totally isotropic subspace above and I_{1}, …, I_{6} the maximal totally isotropic subspaces containing A and in suitable ordering (as described in 2.8.) Denote by f_{1}, …, f_{6} the additive lifts

§8. Embedded lattices 23

of their characteristic functions. There are the quadratic relations

\begin{aligned}(f_{1} − f_{2})(f_{1} − f_{4}) & = (f_{3} − f_{6})(f_{5} − f_{6}) \\ (f_{1} − f_{2})(f_{3} − f_{2}) & = (f_{5} − f_{4})(f_{5} − f_{6}) \\ (f_{1} − f_{2})(f_{1} − f_{6}) & = (f_{3} − f_{4})(f_{5} − f_{4}) \\ (f_{1} − f_{2})(f_{5} − f_{2}) & = (f_{3} − f_{4})(f_{3} − f_{6}) \\ (f_{3} − f_{4})(f_{1} − f_{4}) & = (f_{5} − f_{2})(f_{5} − f_{6}) \\ (f_{3} − f_{4})(f_{3} − f_{2}) & = (f_{1} − f_{6})(f_{5} − f_{6}) \\ (f_{1} − f_{4})(f_{1} − f_{6}) & = (f_{3} − f_{2})(f_{5} − f_{2}) \\ (f_{1} − f_{4})(f_{5} − f_{4}) & = (f_{3} − f_{2})(f_{3} − f_{6}) \\ (f_{1} − f_{6})(f_{3} − f_{6}) & = (f_{5} − f_{2})(f_{5} − f_{4}) \end{aligned}

These relations are induced by decompositions of a union of two disjoint stars as union of two other stars as for example O_{1} ∪ O_{3} = O_{8} ∪ O_{9}.

Proof. The point is that the divisors of both side agree by 5.4, because they are the unions of the corresponding star divisors. Hence both sides are equal up to a constant factor. The normalizing factor follows from 2.8. ⊔ ⊓

We recall that the 9 quadratic relations collaps to one if one takes the linear relation 2.3 into account.

In section 2 we introduced a certain ideal I_{6} generated by linear and quadratic relations in the polynomial ring ℂ[… X_{A} − X_{B} …], where X_{A} are formal variables attached to maximal totally isotropic subspaces. We can summarize

7.2 Proposition. There is a natural homomorphism

R_{6}/I_{6} ⟶ A(O^{+}(II_{2,10})[2]),

8. Embedded lattices

Let us assume that the L is a sublattice of finite index of another even lattice M. Then we have

L ⊂ M ⊂ M′ ⊂ L′.

The orthogonal groups O^{+}(M) and O^{+}(L) are usually not contained in each other but for the discrimant groups one has

Γ_{L} ⊂ Γ_{M}.

To prove this one considers some element g ∈ Γ_{L} and an element x ∈ M′. Because x is contained also in L′ we have g(x) − x ∈ L ⊂ M. This shows g(x) ∈ M′. Hence g defines an automorphism of M′ and then also for M = M′′. From g(x) − x ∈ M we see g ∈ Γ_{M}.

Following Scheithauer we define an injective map

Ψ : ℂ[M′/M]^{SL(2,ℤ)} → ℂ[L′/L]^{SL(2,ℤ)}.

as follows: We extend a function M′/M → ℂ by zero to a function L′/M (i.e. the values are zero on the complement L′/M −M′/M) and then compose the result with the natural projection L′/L → L′/M. Scheithauer [Sch] proved that Weil invariants go to Weil invariants.

8.1 Lemma. The diagram

ℂ[M′/M]^{SL(2,ℤ)} ⟶ ℂ[L′/L]^{SL(2,ℤ)}
↓  ↓
[Γ_{M}, n/2 − 1]  ⊂ [Γ_{L}, n/2 − 1]

commutes.

Proof. The proof follows from the definition of the theta lift [Bo1]. We explain it briefly for the lattice M. We consider a point τ form the usual upper half plane and a z ∈ H_{n}. Let A be the two dimensional positive definit subspace of V = M ⊗_{ℤ} ℝ and B its orthogonal complement. We decompose an arbitrary v ∈ V as v = v_{A} + v_{B} with v_{A} ∈ A and v_{B} ∈ B. The Siegel theta function corresponding to a coset M + α form M′/M is defined as

θ_{M+α}(τ, z) = ∑_{λ∈M+α} eπi(τ(λ_{A}, λ_{A}) + τ̄(λ_{B}, λ_{B})).

This can be considered as a function with values in ℂ[M′/M]. Using a standard pairing on ℂ[M′/M] we can pair the Siegel theta function with an element C ∈ ℂ[M′/M]^{SL(2,ℤ)} and integrate then along τ (with respect to an invariant volume element). The result essentially is the additive lift of C. Now we compare the Siegel theta series with respect to the two lattices L ⊂ M. For α ∈ M′ one obviously has

θ_{M+α} = ∑_{β∈M/L} θ_{L+α+β}.

From this follows 8.1 immediatly.  ⊔ ⊓

8.2 Remark. Let M be an even lattice and L ⊂ M a sub-lattice of finite index. The image of ℂ[M′/M]^{SL(2,ℤ)} under the map Ψ consists of all elements C ∈ ℂ[L′/L]^{SL(2,ℤ)}, which are periodic under M/L.

§9. Distinguished points 25

Proof. Let C ∈ ℂ[L′/L]^{SL(2,ℤ)}. Then

C(α) = \begin{matrix}\sqrt{i} & n−2 \\ √ & \overline{|L′/L|} \end{matrix} ∑_{β∈L′/L} C(β)e^{−2πi(α,β)}.

If C is periodic we can write

C(α) = \begin{matrix}\sqrt{i} & n−2 \\ √ & \overline{|L′/L|} \end{matrix} ∑_{β∈L′/M} C(β) ∑_{γ∈M/L} e^{−2πi(α,β+γ)}.

The inner sum is zero if α is not contained in M′. Hence we can consider C as a function on M′/M and it is clear that this function is invariant under

SL(2, ℤ).  ⊔ ⊓

9. Distinguished points

The aim of this and next section is to prove the generic injectivity of the map from the modular variety H_{10}/O^{+}(II_{2,10})[2] to the projective space of dimension 714. We recall some basic facts from [Ni] about primitive lattices.

Let (L, (·, ·)) be an even lattice. There is a natural isomorphism

L′ ≅ Hom_{ℤ}(L, ℤ).

Let M ⊂ L be a sublattice. Then there is a natural map L′ → M′. The lattice M is called primitive inside L if L/M is free. Then the sequence

0 ⟶ M ⟶ L ⟶ L/M ⟶ 0

splits and L′ → M′ is surjective. We mention that M is primitive inside L if and only if

ℚM ∩ L = M.

A lattice M is called maximal if there is no even over-lattice M ⊂ M̃, M ≠ M̃, of the same rank. We see that maximal lattices M are primitive in every even over-lattice M ⊂ L. Lattices with square free determinant are maximal.

Recall that for a quadratic space of signature (2, n), n > 0, the group O^{+}(V) is generated by reflections along vectors of negative norm. Let V = V_{1} ⊕ V_{2} be an orthogonal decomposition into a positive definite V_{1} and a negative definite V_{2}. An automorphism of V which fixes this decomposition is contained in O^{+}(V) if and only of the determinant on V_{1} is positive.

Again we consider an even unimodular lattice L and a primitive sublattice M ⊂ L. We ask whether every primitive sublattice N ⊂ L which is isometric to M can be obtained from M by applying an automorphism of L. For this we have to consider the orthogonal complements M^{⊥} and N^{⊥} inside L. We know that there discrimants agree. Hence they define the same genus. We now make the rather strong assumption that M^{⊥} is unique in its genus. Then M^{⊥} and N^{⊥} are isometric lattices. We choose isometries

M ⟶^{∼} N and M^{⊥} ⟶^{∼} N^{⊥}.

They define an isometric isomorphism

M + M^{⊥} ⟶^{∼} N + N^{⊥}.

This isomorphism extends to L if and only if the glue groups agree. Recall that the glue group of M + M^{⊥} in L is of the form

A = { (a, σa), a ∈ M′/M },

where σ : M′/M → (M′)^{⊥}/M^{⊥} is an isomorphism (changing the sign of the quadratic form). We assume that every isometric automorphism of (M′)^{⊥}/M^{⊥} extends to an element of O(M). Then we can obtain by a suitable choice of the isomorphism M^{⊥} → N^{⊥} that the glue groups match. We see:

9.1 Remark (Nikulin). Let L be a unimodular lattice and M ⊂ L a primitive sublattice such that the lattice M^{⊥} ⊂ L is unique in its genus. Assume that every isometry of the discrimant group of M^{⊥} extends to an isometry of M^{⊥}. Then the group O(L) acts transitively on the set of isometric embeddings M ↪ L.

This applies to the following situation. Let L = II_{2,10} and M = A_{2}. We can take the realization L = II_{2,2} × (−E_{8}) and realize A_{2} inside II_{2,2}, for example as the lattice generated by (1, 1, 0, 0; 0, …, 0) and (1, 0, 1, 1; 0, …, 0). We see that a realization of A_{2} in L exists such that the orthogonal complement is −(A_{2} × E_{8}). From Kneser’s classification of definite lattices of determinant ≤ 3 up to dimension 17 follows that this lattice is unique in its genus. The discriminant is isomorphic to ℤ/3ℤ. The non trivial automorphism x ↦ −x extends. Hence 9.1 applies. We can even more replace O(L) by O^{+}(L) because A_{2} admits an automorphism of determinant -1.

9.2 Lemma. There is precisely one O^{+}(L)-orbit of sublattices M ⊂ L = II_{2,10} isometric to A_{2}.

We want to determine the subgroup of O^{+}(L) which fixes a given A_{2}. We start with the full group O(L). This group acts on the set of two dimensional positive definite subspaces in a natural way. The automorphism group of A_{2} has order 12. By a result of Eichler one knows

O(A_{2} × E_{8}) ≅ O(A_{2}) × O(E_{8}).

The order of O(E_{8}) is 696 729 600. We get

#O(A_{2} × (−(A_{2} × E_{8}))) = 12 · 12 · 696 729 600 = 2^{18} · 3^{7} · 5^{2} · 7.

Only half of this group survives in O(L) because of the matching condition.

9.3 Lemma. The subgroup of O(II_{2,10}), which stabilizes a given A_{2}, has order 2^{17} · 3^{6} · 5^{2} · 7.

Next we consider the congruence group

O(II_{2,10})[2] := kernel(O(II_{2,10}) ⟶ O(𝔽_{2}^{12})).

The number of all O(II_{2,10})[2]-orbits of A_{2}-sublattices is the index [O(𝔽_{2}^{12}) : H], where H denotes the image of the stabilizer described in 9.3. To get the order of H we have to know the order of the kernel of the homomorphism

O(A_{2} × (−(A_{2} × E_{8}))) → O(𝔽_{2}^{12}).

When M is a primitive sublattice of an even lattice N, one has M ∩ 2N = 2M. This shows that in the kernel there are only signs changes of the occurring A_{2}, −A_{2} and −E_{8}. Hence the kernel has order 8. Because of the gluing condition only order 4 survives. From 9.3 we obtain #H = 2^{15} · 3^{6} · 5^{2} · 7. Hence the number of O(II_{2,10})[2]-orbits of A_{2}-sublattices is 2^{16} · 3 · 7 · 11 · 17 · 31. Let M be a lattice which is isomorphic to A_{2} × E_{8}, then there is a unique sublattice of M which is ismorphic to A_{2} and whose orthogonal complement is isomorphic to E_{8}. This follows from the mentioned result of Eichler. Hence an A_{2}-lattice inside II_{2,10} defines a unique sublattice isomorphic to −A_{2}. There images in 𝔽_{2}^{12} define an ordered pair (A, B) of orthogonal hyperbolic planes. Every pair occurs because the orthogonal group acts transitively on them. The number of these pairs is easy to compute, one obtains 2^{16} · 3 · 7 · 11 · 17 · 31, the same number as above. Hence we obtain:

9.4 Lemma. The O(II_{2,10})[2]-orbits of sublattices of II_{2,10} isomorphic to A_{2} are in one-to-one correspondence to the ordered pairs (A, B) of orthogonal hyperbolic planes in 𝔽_{2}^{12}. Their number is 2^{16} · 3 · 7 · 11 · 17 · 31.

The situation for O^{+}(II_{2,10})[2] is slightly different. The reason is that an automorphism which stabilizes an embedded A_{2} is in O^{+} if and only if it has positive determinant on A_{2}. Hence the number of O^{+}(II_{2,10})[2]-orbits is the double of the number given in 9.4. This can be understood as follows:

9.5 Definition. Two O^{+}(II_{2,10})[2]-classes of sublattices of II_{2,10} of type A_{2} are called companions, if there exists an element g ∈ O(II_{2,10})[2], which is not contained in O^{+}(II_{2,10})[2] such that g maps one class to the other.

As an example we realize II_{2,10} as II_{2,2} × (−E_{8}). We consider the lattice generated by (1, 1, 0, 0; 0, …, 0) and (1, 0, 1, 1; 0, …, 0). The automorphism which changes the signs of the first two components is not in O^{+}. Hence the companion class is represented by the lattice with basis (−1, −1, 0, 0; 0, …, 0) and (−1, 0, 1, 1; 0, …, 0).

9.6 Remark. Companions are equivalent mod O(II_{2,10})[2] and also equivalent mod O^{+}(II_{2,10}) but not equivalent mod O^{+}(II_{2,10})[2].

Our method also shows that the stabilizer of a lattice of type A_{2} (embedded in II_{2,10}) inside O^{+}(II_{2,10})[2] only consists of the identity and its negative. We also can restate our results in terms of modular varieties:

9.7 Proposition. The set of points of type A_{2} in

H_{10}/Γ[2] (Γ[2] = O^{+}(II_{2,10})[2])

consists of 2^{16} · 3 · 7 · 11 · 17 · 31 pairs of companions. The map H_{10} → H_{10}/Γ[2] is locally biholomorphic at the A_{2}-points. Especially they define smooth points in H_{10}/Γ[2].

10. Mapping to the projective space

We have to consider the Baily-Borel compactification \overline{H_{10}/Γ[2]}, which is an algebraic variety [BB]. We want to use a basis F_{0}, …, F_{713} of the 714-dimensional irreducible part of the additive lift space to define a regular map from it to P^{713}(ℂ). For this we need

10.1 Lemma. The modular forms of the 714-dimensional space of additive lifts don’t have common zeros in \overline{H_{10}/Γ[2]}.

Proof. The more involved part is the interior. Here the argument is the same as in Kondo’s case and we omit it. (Actually it can be obtained as a consequence of Kondo’s case using 11.1.) For the boundary one can argue as follows. The irreducible components of the boundary are modular curves corresponding to SL(2, ℤ)[2]. The restriction of the 714-dimensional space contains a form of weight 4, which is not invariant under the full modular group. The space of all elliptic modular forms is three-dimensional and generated by ϑ_{1}^{8}, ϑ_{2}^{8}, ϑ_{3}^{8}, where ϑ_{1}, ϑ_{2}, ϑ_{3} denote the Jacobi theta constants with respect to the characteristics (0, 0), (1, 0), (0, 1). Recall that ϑ_{1}^{4} = ϑ_{2}^{4} + ϑ_{3}^{4}. The space generated by the the three eighth powers splits under SL(2, ℤ) into a one dimensional space generated by f = ϑ_{1}^{8} + ϑ_{2}^{8} + ϑ_{3}^{8} and an irreducible two dimensional space generated by ϑ_{1}^{8} − ϑ_{2}^{8}, ϑ_{1}^{8} − ϑ_{3}^{8} and ϑ_{2}^{8} − ϑ_{3}^{8}. This space must be in the restriction of the 714-dimensional space. But this two dimensional space has no joint zero. This follows from the fact that f^{2} is in the second symmetric power of this two dimensional space (consider the sum of the (ϑ_{i}^{8} − ϑ_{j}^{8})^{2}). ⊔ ⊓

Due to a result of Hilbert we obtain from 10.1: The map

\overline{H_{10}/Γ[2]} ⟶ P^{713}(ℂ)

§10. Mapping to the projective space 29

defined by a basis of the additive lift space is a finite regular map. We denote by

A(Γ[2]) = ∑_{r=0}^{∞}[Γ[2], r]

the graded algebra of modular forms (similarly for other groups) and by

B(Γ[2]) = ℂ[F_{0}, …, F_{714}] ⊂ A(Γ[2])

the subring which is generated by the 715-dimensional additive lift space. Be aware that we include the invariant form and not only the 714-dimensional part.

10.2 Lemma. The finite map

\overline{H_{10}/Γ[2]} = proj(A(Γ[2])) ⟶ proj(B(Γ[2])) (↪ P^{714}(ℂ))

is locally biholomorphic at the A_{2}-points.

Proof. Let A_{2} × (−A_{2} × E_{8}) ↪ II_{2,10} an embedding which describes an A_{2}- point. In −(A_{2} × E_{8}) exist 10 linearly independent vectors of norm -4. We know that there exist ten modular forms in the additive lift space whose zero divisor consists of the corresponding Heegner divisors. These define locally at the point a coordinate frame. ⊔ ⊓

We want to proof that the map in 10.2 is generically injective. We follow closely Kondo ([Ko1], section 6). First of all we consider for a two-dimensional positive definite subspace W ⊂ II_{2,10} ⊗_{ℤ} ℝ the set of all norm -2 vectors in II_{2,10}, which are orthogonal to W. The image of this set in 𝔽_{2}^{12} is denoted by

Δ_{W} := { α ∈ 𝔽_{2}^{12}; α image of a norm -2 vector a ∈ II_{2,10} orthogonal to W }.

If W is an A_{2}-point (i.e. the vector space generated by a A_{2}-sublattice of II_{2,10}), the set Δ_{W} contains 123 elements. This can be seen if one takes the realization

II_{2,10} = II_{2,2} × (−E_{8})

and for W the space generated by (1, 1, 0, 0; 0, …, 0) and (1, 0, 1, 1; 0, …, 0). The orthogonal norm -2 vectors are Vectors of norm -2 orthogonal to them are ±(0, 0, −1, 1; 0), ±(1, −1, −1, 0; 0), ±(1, −1, 0, −1; 0) and (0, 0, 0, 0, x) with x a root of −E_{8}. This description also shows:

10.3 Remark. Let ϱ be an A_{2}-point and A, B ⊂ 𝔽_{2}^{12} the corresponding pair (9.4). The set Δ_{ϱ} consists of two types: First type: Three points inside A ⊕ B. Second type: 120 points orthogonal to A ⊕ B.

The set Δ_{W} only depends on the Γ[2]-equivalence class of W. Hence we can define the set Δ_{ϱ} also for points ϱ ∈ H_{10}/Γ[2]:

10.4 Lemma. Let ϱ, ϱ′ ∈ H_{10}/Γ[2] be two points. Assume that one of them is an A_{2}-points and furthermore that Δ_{ϱ} = Δ_{ϱ′}. Then the two points are equal or companions.

The following two lemmas are similar to Kondo’s 6.4 and 6.5 in [Ko1]:

10.5 Lemma. Let Δ = Δ_{ϱ} for a A_{2}-point ϱ. Assume that α ∈ 𝔽_{2}^{12} is a non-isotropic element, which is not contained in Δ. Then there exists a star, which contains α and has empty intersection with Δ.

This is a finite problem which can be checked by calculation. One can assume that ϱ is our standard point. There is a group isomorphic to S_{3} × O(E_{8}/2E_{8}) which stabilizes Δ and also its complement. The complement consists of 5 orbits, which can be represented by

(0, 0, 1, 1; 0, 1, 0, 0, 0, 0, 0, 0) (1, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0)
(1, 1, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0) (0, 0, 0, 1; 1, 1, 0, 0, 0, 0, 0, 0)

(0, 1, 0, 0; 1, 1, 0, 0, 0, 0, 0, 0) In the first four cases we take the following star: The underlying totally isotropic space is spanned by

(1, 1, 1, 1; 0, 1, 0, 0, 0, 0, 0, 0) (1, 1, 0, 1; 1, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) The star is the coset which contains (1, 1, 0, …, 0). In the last case, we consider the even totally isotropic space spanned by

(1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0) (0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)

The star is the coset which contains (0, 0, 1, 1; 0, 0, …, 0).  ⊔ ⊓

10.6 Lemma. Let Δ = Δ_{ϱ} for a A_{2}-point ϱ. Assume that α ∈ Δ is a non-isotropic element.

a) Assume that α is of the first type (see 10.3). There exists a star S such that S ∩ Δ = {α}.

b) Assume that α, β ∈ Δ are two orthogonal elements of the second type. There exists a star S such that S ∩ Δ = {α, β}.

Proof. Again we assume that ϱ is the standard point. Let α be of first type. Up to the action of the group S_{3} we can take α = (0, 0, 1, 1; 0, …, 0). We can take the star

(∗, 0, 1, 1, ∗, 0, ∗, 0, ∗, 0, ∗, 0).

Let α, β ∈ Δ be orthogonal elements of second type. Using the action of O(E_{8}/2E_{8}), we can assume

α = (0, 0, 0, 0; 1, 1, 0, …, 0), β = (0, 0, 0, 0; 1, 1, 0, 0, 0, 0, 1, 0).

In this case we can take the star α + M where M is the totally isotropic space spanned by the vectors

(1, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0), (0, 1, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0),
(0, 0, 1, 0; 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 1; 0, 0, 0, 0, 0, 1, 0, 0,),
(0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1). ⊔ ⊓

Before we continue we recall that a non-isotropic element α ∈ 𝔽_{2}^{12} defines a certain Heegner divisor H_{α}(−1) in H_{10}/Γ[2]. It is the fixed point set of the reflection corresponding to a norm -2 representative a ∈ II_{2,10}. Hence we call this divisor a mirror. We also notice that a point ϱ is contained in this mirror if and only if α ∈ Δ_{ϱ}. If S ⊂ 𝔽_{2}^{12} is a star, we can consider the star divisor which consists of the corresponding 32 mirrors. The star divisor of a star S ⊂ 𝔽_{2}^{12} contains ϱ if and only of S ∩ Δ_{ϱ} is not empty.

10.7 Lemma. Let ϱ ∈ H_{10}/Γ[2] be an A_{2}-point and let ϱ′ be an arbitrary point which is different from ϱ and its companion. Then there exists a star which contains one of the two points but not both.

Proof. We assume that the two points cannot be separated. If there exists an α ∈ Δ_{ϱ′} − Δ_{ϱ} we find by 10.5 a star which contains ϱ′ but not ϱ. This shows Δ_{ϱ′} ⊂ Δ_{ϱ}. Let be M = Δ_{ϱ} − Δ_{ϱ′}. Every α ∈ M must be of second kind, because otherwise we could use 10.6, a) to separate the two points. In M two elements cannot be orthogonal, because otherwise we could separate the two points using 10.6, b). The same argument shows that every element of second type which is orthogonal to an element of M is contained in Δ_{ϱ′}. Before we continue we fix a notation:

An E_{8}-root system in E_{8}/2E_{8} is a system of anisotropic vectors α_{1}, …, α_{8} with the property that

(α_{1}, α_{2}) = (α_{2}, α_{3}) = ⋯ (α_{7}, α_{8}) = 1 and (α_{5}, α_{8}) = 1.

Claim. Let M ⊂ E_{8}/2E_{8} be a subset containing only anisotropic elements, such that any two elements of M are not orthogonal. Then the complement of M contains an E_{8}-root system. Proof of the claim. The cases where M is empty or contains only one element is trivial. Hence M contains at least two elements α, β. We use the standard realization E_{8}/2E_{8} = 𝔽_{2}^{8} with q(x) = x_{1}x_{2} + ⋯ + x_{7}x_{8} and can then assume

α = (1, 1, 0, 0, 0, 0, 0, 0,), β = (0, 1, 1, 0, 1, 0, 1, 1).

The system

(1, 1, 1, 0, 0, 0, 0, 0) (0, 0, 1, 1, 1, 0, 0, 0) (0, 0, 0, 0, 1, 1, 1, 0) (0, 0, 0, 0, 0, 0, 1, 1)
(1, 1, 0, 0, 0, 0, 1, 0) (0, 1, 0, 1, 1, 1, 0, 0) (0, 1, 0, 1, 0, 1, 1, 1 (1, 0, 1, 1, 0, 0, 0, 0)

is an E_{8}-root system. The first five are orthogonal to α the rest to β. Hence

they are all in the complement.  ⊔ ⊓

Proof of 10.7 continued. We want to apply this to the following situation. As above a representative of ϱ belongs to a decomposition II_{2,10} = II_{2,2} × (−E_{8}) The set M = Δ_{ϱ} − Δ_{ϱ′} belongs to the part E_{8}/2E_{8} in this decomposition (second type). From the above follows that Δ_{ϱ′} contains an E_{8}-root system. Together with the three elements of the first type we obtain: The set Δ_{ϱ′} contains an A_{2} × E_{8}-root system in an obvious sense. Now we have to recall the definition of Δ_{ϱ′}. It is the image of all elements of norm -2 contained in a ten-dimensional negative definite sublattice of II_{2,10}. From the classification of root lattices by their Dynkin diagram we see that the lattice generated by these roots is a copy of −(A_{2} × E_{8}). This means that ϱ′ is an A_{2}-point. Hence Δ_{ϱ′} contains 123 elements and we get Δ_{ϱ} = Δ_{ϱ′}. Now 10.4 completes the proof of 10.7. ⊔ ⊓

We also have to take boundary points into account. Recall that there are one-dimensional and zero dimensional boundary components. Each onedimensional boundary component corresponds to a two dimensional totally isotropic space B. It belongs to the group SL(2, ℤ)[2] and the three cusp classes of this group correspond to the three non-zero isotropics of B. Let α ∈ 𝔽_{2}^{12} be an anisotropic element, which is orthogonal to B. Then the three zero-dimensional boundary points are contained in the closure of the Heegner divisor H_{α}(−1). The number of these α is 480. Let ϱ now be any A_{2}-point. As we know it is contained in 123 Heegner divisors H_{β}(−1). Hence there exists an α such that the closure of H_{α}(−1) contains the three cusps but not ϱ. Now we use 10.5 and obtain a star modular form f_{S} with the following property: It doesn’t vanish at ϱ but it vanishes in the three cusps in consideration. Now we use that every cusp from of weight 4 for the group SL(2, ℤ)[2] vanishes. Hence we obtain:

10.8 Lemma. Let ϱ be an A_{2}-point. For every one-dimensional boundary component there exist a star modular form which vanishes along this boundary component but does not vanish at ϱ.

From 10.7, 10.8 and 10.2 we know that the map \overline{H_{10}/Γ[2]} → proj(B(Γ[2])) has covering degree at most two. We want to show that it is one. Because we cannot separate companions directly by our methods, we have to take a small detour. We take invariants under the group O(𝔽_{2}^{12}) to go down to the full modular group Γ = O^{+}(II_{2,10}). We claim the this finite group acts faithfully on B(Γ[2]). This follows from the fact that O(𝔽_{2}^{12}) has a simple subgroup of index 2 (the kernel of the so-called Dixon invariant) and that this subgroup cannot act trivially on B(Γ[2]). Set

B(Γ) := B(Γ[2])^{O(𝔽_{2}^{12})}.

We consider the diagram

\overline{H_{10}/Γ[2]} ⟶ proj(B(Γ[2])
↓  ↓
\overline{H_{10}/Γ} ⟶ proj(B(Γ)

§11. Enriques surfaces, Kondo’s approach 33

Since both vertical arrows have the same covering degree, it is sufficient to show that the second row has covering degree one. But this follows from

10.9 Lemma. The finite map

\overline{H_{10}/Γ} = proj(A(Γ)) ⟶ proj(B(Γ))

is locally biholomorphic at the A_{2}-point. The A_{2}-point is unique in its fibre.

This follows from our considerations about the level two case and from the fact that companions have the same image in level 1.

10.10 Theorem. The map

\overline{H_{10}/Γ[2]} → proj(B(Γ[2]))

is everywhere regular finite and birational. The ring of modular forms of weight divisible by 4

A^{(4)}(Γ[2]) := ⨁_{r=0}^{∞}[Γ[2], 4r]

is the normalization of the ring B(Γ[2]) (the ring generated by the 715 dimensional additive lift space). Using 7.2 we get a finite map

H_{10}/O^{+}(II_{2,10})[2] ⟶ proj(R_{6}/I_{6}).

11. Enriques surfaces, Kondo’s approach

Denote by U the unimodular lattice ℤ × ℤ with quadratic form (x, x) = 2x_{1}x_{2}. Kondo investigated in [Ko1] the case of the lattice

M = U ⊕ \sqrt{2}U ⊕ (−\sqrt{2}E_{8}).

We recall that thus case is related to the moduli space of marked Enriques surfaces, i.e. Enriques surfaces with a choice of level 2 structure of the Picard lattice.

Our lattice

L = \sqrt{2}II_{2,10} ≅ \sqrt{2}U ⊕ \sqrt{2}U + ⊕(−\sqrt{2}E_{8})

can be embedded into Kondo’s lattice by means of

\sqrt{2}U ⟶ U, \sqrt{2}(x_{1}, x_{2}) ⟼ (x_{1}, 2x_{2}).

The image consists of all integral pairs (x_{1}, x_{2}) with even x_{2}. The dual of this lattice is (1/2)ℤ × ℤ. Hence M′ ⊂ L′ corresponds to

ℤ ⊕ ℤ ⊕ \frac{1}{\sqrt{2}} ℤ ⊕ \frac{1}{\sqrt{2}} ℤ ⊕ \frac{−1}{\sqrt{2}}E_{8} ⊂ \frac{1}{2} ℤ ⊕ ℤ ⊕ \frac{1}{\sqrt{2}} ℤ ⊕ \frac{1}{\sqrt{2}} ℤ ⊕ \frac{−1}{\sqrt{2}}E_{8}

Assume that x = (x_{1}, x_{2}, x_{3}/\sqrt{2}, x_{4}/\sqrt{2}; 𝔵/\sqrt{2}) is a primitive element of M′, which is not primitive in L′. Then x_{3} = 2y_{3}, x_{4} = 2y_{4} must be even and 𝔵 = 2𝔶 with 𝔶 ∈ E_{8}. This gives x = (x_{1}, x_{2}, y_{3}\sqrt{2}, y_{4}\sqrt{2}; 𝔶\sqrt{2}). Hence x is contained in M. This shows that we can apply 8.1. Kondo proved that ℂ[M′/M]^{SL(2,ℤ)} is the direct sum of a full invariant one-dimensional space and a 186-dimensional space H which is irreducible under O(M′/M) = 𝔽_{2}^{8}. The elements C ∈ H satisfy C(0) = 0. Hence the additive lift space of H —which is of dimension 186 by Kondo— appears as a subspace of our 714-dimensional space:

11.1 Proposition. The embedding of L = \sqrt{2}U ⊕ \sqrt{2}U ⊕ (−\sqrt{2}E_{8}) ≅ \sqrt{2}II_{2,10} into M = U ⊕ \sqrt{2}U ⊕ (−\sqrt{2}E_{8}) defines an embedding of Kondo’s 186dimensional space of modular forms of weight four into our 714-dimensional space.

The main result Kondo’s paper is that the 186-dimensional space defines birational map ψ from this moduli space, i.e. H_{10}/Γ_{M} onto its image in P^{185}(ℂ).

Unfortunately the proof is not correct, in fact he proves that the map ψ is holomorphic onto H_{10}/Γ_{M} and is locally biholomorphic at a special point. But his map does not extend to \overline{H_{10}/Γ_{M}}.

However Kondo’s argument works with a small modification. In fact we know that the additive lifting from ℂ[𝔽_{2}^{10}]^{SL(2,ℤ)} to [Γ_{M}, 4] in injective. Thus we get another modular form that is relative to O^{+}(M), so also in this case we add the full invariant form to the 186 dimensional space of modular forms.

Using the results of proposition 3.2 , it can be easily checked that this form does not vanishes along those 0-dimensional points that are the base locus for the map ψ . In fact these are points parametrized by primitive isotropic elements of M′ that are also in M.

We denote by

A(Γ_{M}) = ∑_{r=0}^{∞}[Γ_{M}, r]

the graded algebra of modular forms and by

B(Γ_{M}) = ℂ[G_{0}, …, G_{186}] ⊂ A(Γ_{M})

the subring which is generated by the 187-dimensional additive lift space. The correct modification of Kondo’s statement is:

§12. Relation to a result of Koike 35

11.2 Theorem. The map

\overline{H_{10}/Γ_{M}} → proj(B(Γ_{M}))) (↪ P^{186}(ℂ))

is everywhere regular finite and birational. The ring of modular forms of weight divisible by 4

A^{(4)}(Γ_{M}) := ⨁_{r=0}^{∞}[Γ_{M}, 4r]

is the normalization of the ring B(Γ_{M}) (the ring generated by the 187 dimensional additive lift space).

We consider now [Γ_{M}, r] as subspace of [Γ_{L}, r] using this concrete imbedding. The image of M in L′/L ≅ 𝔽_{2}^{12} is a one dimensional subspace. We consider two six-dimensional (maximal) totally isotropic subspaces of L′/L which contain the image of M. The difference of their characteristic functions defines a modular form in [Γ_{L}, 4]. From 8.1 and 8.2 follows that this modular form is contained in Kondo’s space [Γ_{L}, 4]. Let now A be a four dimensional totally isotropic subspace of L′/L which contains the image of M. From 2.9 we obtain quadratic relations between modular forms inside Kondo’s space [Γ_{M}, 4].

We recall that in its papers Kondo’s defined some quartic relations. We don’t want to go into the details of Kondo’s paper and mention just that Kondo’s quartic relations are those we defined in 2.10.

11.3 Proposition. Kondo’s quartic relations are a consequence of our quadratic relations.

Proof. This follows from 2.10. More precisely, the model which Kondo describes

in [Ko1] is nothing else but proj(R_{5}/I_{5}).  ⊔ ⊓

12. Relation to a result of Koike

There is another interesting case, which belongs to the lattice N = U ⊕ \sqrt{2}U ⊕ (−D_{4}) ⊕ (−D_{4}). In this case we have N′/N ≅ 𝔽_{2}^{6}. Obviously Kondo’s lattice M can be embedded into N. Hence we obtain enbeddings Γ_{L} ⊂ Γ_{M} ⊂ Γ_{N}. The space ℂ[N′/N]^{SL(2,ℤ)} splits into a full invariant one-dimensional and a 14-dimensional irreducible space under S_{8} ≅ O(𝔽_{2}^{6}). The additive lift of the 14-dimensional space appears as 14-dimensional subspace of Kondo’s 185dimensional space and hence of our 714-dimensional space. We get a homomorphism

R_{3}/I_{3} ⟶ A(Γ_{N}).

Computer algebra shows that the dimension of R_{3}/I_{3} is 6 (the dimension of the associated projective variety is 5). The lattice N admits a structure as Hermitian lattice of signature (1, 5) over the ring of Gau”s integers (s. [Ko2]). This defines a certain 5-ball B_{5} inside H_{10}. The corresponding ball quotient is the configuration space X(2, 8) of 8 points in the projective line [MY]. We obtain a rational map

X(2, 8) ⟶ proj(R_{3}/I_{3}).

Koike [Koi] proved that this is a biholomorphic map. So we recover Koike’s observation that certain quartic relations are consequences of quadratic ones and even more that all these relation live already in the ten-dimensional space H_{10}/Γ_{L}.

13. Final Remark

in 10.10 we used the full 715-dimensional additive lift space. But the relations which we described play in the 714-dimensional irreducible part. It may be true that the invariant form is not necessary, because it may be possible that the second symmetric power of the 714- and 715-dimensional space is the same. We don’t know whether this is true or not.

The invariant form in the 715-dimensional space possible being superfluously does not detain it to be in some sense very basically. To explain this we consider a rational orthogonal transformation g ∈ O^{+}(II_{2,10} ⊗_{ℤ} ℚ) with the property

gO^{+}(II_{2,10}[2])g^{−1} ⊂ O^{+}(II_{2,10}).

Then one can transform f with g to obtain a form on O^{+}(II_{2,10})[2]. To give a simple example we realize II_{2,10} as

II_{2,10} = II_{1,1} × II_{1,9}.

We take for g the orthogonal transformation

(a, b, C) ⟼ (a/2, 2b, C).

To understand it better we consider the standard embedding of H_{10} into II_{1,9} ⊗_{ℤ} ℂ: The modular form f is determined by the function

F (Z) := f(1, ∗, Z),

where the star is taken such that the entry is in the zero-quadric. Now the effect on F is given by the transformation F (Z) ↦ F (2Z). It is possible to compute the values at the cusps and to prove in this way that the transformed form is contained in the 715-dimensional additive lift space. This gives:

§13. Final Remark 37

13.1 Proposition. The 715-dimensional additive lift space (4.8) in the space of modular forms [O^{+}(II_{2,10})[2], 4] is generated as O(𝔽_{2}^{10})-module by the single form F (2Z), where F (Z) denotes the full invariant form, expressed in standard coordinates.

References

[AF] Allcock, D. and Freitag, E.: Cubic surfaces and Borcherds products, Commentarii Math. Helv. Vol. 77, Issue 2, 270–296 (2002)

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

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

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

[BK] Brunier, J. Kuss, M.: Eisensetin series attached to lattices and modular forms on orthogonal groups, Manuscr. Math. 106, 443–450 (2001)

[CS] J. H. Conway, N. J. A. Sloane. Sphere packings, lattices and groups. Springer-Verlag New York, Grundlehren der mathematischen Wissenschaften 290, (1988)

[Fa] Fay, J.: On the Riemann-Jacobi formula, Nachr. Akad. Wiss. Gtt., II. Math.-Phys. Kl. 1979 61-73 (1979)

[FH] Freitag, E. Hermann, C.F.: Some modular varieties in low dimension, Advances in Math. 152, 203-287 (2000)

[Fr1] Freitag, E.: Modulformen zweiten Grades zum rationalen und Gau”sschen Zahlk”orper, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 1. Abh. (1967)

[Fr2] Freitag, E.: Some modular forms related to cubic surfaces Kyungpook Math. J. 43, No.3, 433-462 (2003)

[Fr3] Freitag, E.: Comparison of different models of the moduli space of marked cubic surfaces, Proceedings of Japanese-German Seminar, Ryushi-do, edited by T. Ibukyama and W. Kohnen, 74-79 (2002)

[FS] Freitag, E. Salvati-Manni, R.: Some modular varieties of low dimension II, preprint (2004)

[Ig] Igusa, J.: On the graded ring of theta-constants Am. J. Math. 86,, 219-246 (1964).

[Kn] Kneser, M.: Erzeugung ganzzahliger orthogonaler Gruppen durch Spiegelungen, Math. Ann. 255, 453–462 (1981)

[Ko1] Kondo, S.: The moduli space of Enriques surfaces and Borcherds products, J. Algebraic Geometry 11, 601-627(2002)

[Ko2] Kondo, S.: The moduli space of 8 points on P^{1}(ℂ) and automorphic forms, preprint (2005)

[Koi] Koike, K.: The projective embedding of the configuration space X(2, 8), preprint (2005)

[MY] Matsumoto, K. Yoshida, M.: Configuration space of 8 points on the projective line and a 5-dimensional Picard modular group, Compositio Mtah. 86, 265–280 (1993)

[Ni] Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications, Math. USSR Izvestija 14, No 1 (1980)

[Sch] Scheithauer, N.: Moonshine for Conway’s group, Habilitationsschrift, Heidelberg (2004)