Modular embeddings of Hilbert modular surfaces

May, 2003

1. Introduction

let K be a real quadratic number field of discriminant D. We denote by 𝔬_{K} the ring of integers and by SL(2, 𝔬_{K}) the Hilbert modular group. This group acts on the product of two upper half planes H = H^{+} and also on the product of an upper and a lower half plane H^{−}. The quotient of H × H^{±} by the Hilbert modular group is an algebraic surface. Hirzebruch constructed in both cases a certain non-singular projective model Y_{±}^{0}(D), which is conjectured to be a minimal model. We refer to the book [vdG] of van der Geer for a detailed description of these and even more general models*). Hirzebruch, Van der Geer, Van de Ven and Zagier (s. [HV], [HZ], [vGV]) proved the minimality conjecture in a finite number of cases, for the example in all cases where the surface is not of general type. Hermann [He1, He2] obtained an infinite series of surfaces Y_{−}^{0}(D), D ≡ 1 mod 8 (with further conditions), which are minimal. It is easy to see that the minimality conjecture is true if there exists a Hilbert cusp form of weight 2, whose zeros are located at so-called Hirzebruch-Zagier divisors. Hermann actually constructs in [He1] such modular forms using modular embeddings into the Siegel modular group of degree 2 and restricting classical theta constants.

It looks reasonable to look at other modular embeddings. In this context it is very useful to keep in mind that the Hilbert modular group is less more the orthogonal group of a certain lattice of signature (2, 2). The idea is to embed this lattice into lattices of signature (2, n). The theory of Borcherds products allows to construct modular forms with known zero divisors. There are cohomological obstructions for the existence of such modular forms. This is why we do not apply Borcherds constructions directly to the Hilbert modular group but to a bigger group O(2, n) which is cohomological trivial.

It is know that that weight of a non constant holomorphic modular from on O(2, n) is ≥ n/2 − 1 (the so-called singular weight). This means that we are restricted to the case n ≤ 6 if we want to get weight ≤ 2. In [FH] a distinguished (2, 6)-example has been studied. This example can be used to prove:

*) Van der Geer defines a Hilbert modular surface for every narrow ideal class. The two cases which we consider here correspond to the ideal classes of (1) and (\sqrt{D}).

1.1 Theorem. The models Y_{±}^{0}(D) are minimal if D ≢ 1 mod 8.

This result came up during detailed discussion with F. Hermann. He independently proved these results using slightly different techniques. He uses the fact that there is an exceptional isomorphism between the group O(2, 6) and the Hermitean symplectic group. The paper [FH] uses both realizations. My impression is that in a purely orthogonal point of view things get clearer and as I expect can be better generalized.

2. Orthogonal modular forms

Let V be a rational vector space equipped with a non degenerated symmetric bilinear form (·, ·)_{V} of index (2, n). We will use the same notation for the extension to a ℂ-bilinear form on V (ℂ) = V ⊗_{ℚ} ℂ. We consider the projective space of V (ℂ) with the natural projection

V (ℂ) − {0} ⟶ P (V (ℂ)), z ⟼ [z].

The set

K = K(V) = { [z] ∈ P (V (ℂ)); (z, z) = 0, (z, z̄) > 0 }

has two connected components. The orthogonal group O(V (ℝ)) acts on K and a certain subgroup of index two O^{+}(V (ℝ)) ⊂ O(V (ℝ)) preserves the two connected components.

We distinguish one of the two connected components of K and denote it by H. Then −H is the other component. We also denote by K̃ , H̃ the inverse images in V (ℂ) − {0}.

A subgroup Γ ⊂ O^{+}(V) is called a modular group, if there exists a lattice L ⊂ V such that Γ is commensurable with O(L). This condition is independent of the choice of L.

A modular form of weight k ∈ ℤ with respect to Γ and a given character v : Γ → ℂ^{•} is a holomorphic function f : H̃ → ℂ with the properties

1. f(tz) = t^{−k}f(z),

2. f(γ(z)) = v(γ)f(z).

3. f is regular at the cusps.

We explain condition 3 a little later. Up to some low dimensional cases it is automatically fulfilled.

§2. Orthogonal modular forms 3

Tube domain model

Let a ∈ V be an isotropic vector. We choose a second isotropic vector b ∈ V such that (a, b) = 1. Then H := ℚa + ℚb is a hyperbolic plane. There is an orthogonal decomposition V = H ⊕ V_{1}, where V_{1} now is of signature (1, n − 1). We write elements of V (ℂ) as

z = (z_{1}, z_{2}, Z) := z_{1}a + z_{2}b + Z, Z ∈ V_{1}(ℂ).

It is easy to verify that z_{1} ≠ 0 if z ∈ K̃ and we also have z_{1}z_{2} + q(Z) = 0. Hence we obtain an embedding

K ⟶ V_{1}(ℂ), z ⟼ z_{1}^{−1}Z.

The image consists of all Z ∈ V_{1}(ℂ) with the property q(Y) > 0. We denote the connected component which corresponds to H by T . This is the tube domain realization of H. It is of the form

T = V_{1}(ℝ) + iP,

where P is the so-called positive cone (one of the two connected components of the set defined by (Y, Y) > 0).

For a modular form f we consider

F = F_{a,b} : T ⟶ ℂ, F (Z) := f(1, −q(Z), Z).

This function determines f. For γ ∈ O^{+}(V (ℝ)) and Z ∈ T we define J(γ, Z) by

γ(1, −q(Z), Z) = (J(γ, Z), ∗, ∗).

This is an automorphy factor, J(γ_{1}γ_{2}, Z) = J(γ_{1}, γ_{2}Z)J(γ_{2}, Z). The transformation properties of a modular form are equivalent to

F (γZ) = v(γ)J(γ, Z)^{k}F (Z) (γ ∈ Γ).

There is a connection between J(γ, Z) and the Jacobian determinant of the transformation Z ↦ γ(Z). We denote this determinant by j(γ, Z). The connection is

j(γ, Z) = det(γ)J(γ, Z)^{−n}.

We consider the differential form dZ = dz_{1} ∧ … ∧ dz_{n+2} with respect to some basis of V (ℂ). It depends only up to a constant factor on the choice of this basis.

2.1 Lemma. Let f be a modular form of weight n with respect to the character v(γ) = det(γ). Then F (Z)dZ is an invariant differential form and conversely.

The tube domain model allows to introduce Fourier expansions. It is easy to show that for a modular form f the corresponding function F on the tube domain is periodic with respect to some lattice M ⊂ V_{1}. Hence it admits an expansion

F (Z) = ∑_{T ∈M′} α(T)e^{2πi(T,Z)}, α(T) = α_{a,b}(T),

where M′ denotes the dual lattice. Regularity at the “cusp” a means that α(T) only can be different from zero if T is in the closure of the positive cone. (This condition is independent on the choice of b.)

The modular form f is called a cusp form, if moreover

α(T) ≠ 0 ⟹ T ∈ P

for all choices of a.

We recall that the quotient H/Γ admits a natural “minimal” compactification, the Baily-Borel compactification, which carries a structure as projective variety X_{Γ} and which contains H/Γ as Zariski open subset. Moreover there exists a Zariski-open subset X_{Γ}^{0} ⊂ H/Γ such that Γ acts freely on its inverse image in H.

2.2 Proposition. Let ω = F (Z)dZ be a Γ-invariant holomorphic differential form on T which corresponds to a cusp form f. Then ω ∧ ω̄ is integrable on X_{Γ}^{0}. Corollary. The differential form ω is holomorphic (and not only meromorphic) on every non-singular model of X_{Γ}, for example a desingualization of X_{Γ}.

We omit the proof of this proposition and its corollary, which needs some insight into reduction theory. Instead of this we refer to [Fr1], III.2.1 and III.2.5 where the the analogous statement in the case of Siegel modular groups has been treated and to [Fr1], [FP], [Ba] for the most general cases of such extension theorems.

Heegner divisors

Let W ⊂ V be a subspace of signature (2, m). The orthogonal group O(W) is naturally embedded into O(V) and we can define the “projected group” Γ′ := Γ ∩ O(W). We assume that the choice of the connected components H′ of K(W) and H of K(V) have been made compatible. Then we obtain a natural map H′/Γ′ → H/Γ. This map extends to a finite algebraic map X_{Γ′} → X_{Γ}. The image is an algebraic subvariety.

§3. A special lattice 5

A very important special case is W = a^{⊥}, where a ∈ V is a vector of negative norm q(a). In this case the image of X_{Γ′} is of codimension one and is called a Heegner divisor. They generalize the so-called Hirzebruch-Zagier divisors. In this connection we mention, that the set of vectors a ∈ L of fixed norm q(a) ≠ 0 decomposes into finitely many Γ-orbits.

We introduced the notion of a cusp form. If f is a cusp form, then the constant Fourier coefficient α(0) vanishes for every choice of a. The converse is not true in general. We say that a modular form vanishes at the zero dimensional cusp belonging to a, if α_{a,b}(0) = 0. This condition is independent of the choice of b. Moreover it only depends on the Γ-orbit of the line ℚa.

2.3 Remark. Assume that f is a modular form such that for some isotropic a ∈ V there exists an element b ∈ V with the properties q(b) < 0, (a, b) = 0 and such that f vanishes along the Heegner divisor which corresponds to b. Then f vanishes at the zero dimensional cusp belonging to a.

The proof is trivial.  ⊔ ⊓

3. A special lattice

We consider the vector space V = ℚ^{8} with the quadratic form q(x) := x_{1}x_{2} + x_{3}x_{4} − x_{5}^{2} − x_{6}^{2} − x_{7}^{2} − x_{8}^{2}. The Gram matrix of the associated bilinear form with respect to the standard basis is

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

We have to consider several lattices in ℚ^{8}. First of all we take

L := ℤ^{8}.

We are interested in the orthogonal group O^{+}(L) of this lattice. We consider the dual lattice

L′ = ℤ^{4} × \frac{1}{2} ℤ^{4}.

The group O^{+}(L) acts also on on L′ and hence on 2L′/2L ≅ ℤ^{4}. The quadratic form induces a homomorphism

L′/L ⟶ ℤ/2ℤ, x̄ ⟼ 2x_{5} + ⋯ + 2x_{8} mod 2.

We see that the lattice

{ x ∈ 2L′; x_{5} + ⋯ + x_{8} ≡ 0 mod 2 }

is invariant under O(L).

We introduce the groups

\begin{aligned}𝔭 : & = \{ x ∈ ℤ^{4}; x_{1} + x_{2} + x_{3} + x_{4} ≡ 0 mod 2 \}, \\ 𝔬 : & = ℤ^{4} + ωℤ^{4}, ω = \frac{1}{2}(1, 1, 1, 1). \end{aligned}

We will use the isomorphism

(ℤ/2ℤ) × (ℤ/2ℤ) ⟶^{∼} 𝔬/𝔭, (1, 0) ⟼ ω, (1, 1) ⟼ (1, 0, 0, 0).

What we have seen is that

2ℤ^{4} × 𝔭

is invariant under O(L). The same is true for the dual of this lattice and hence for its double

ℤ^{4} × 𝔬.

This gives a remarkable inclusion

O(L) ⊂ O(ℤ^{4} × 𝔬).

The group O(ℤ^{4} × 𝔬) acts on

(ℤ^{4} × 𝔬)/(2ℤ^{4} × 𝔭) ≅ (ℤ/2ℤ)^{6}.

There is an induced “quadratic form”

q̄: (ℤ^{4} × 𝔬)/(2ℤ^{4} × 𝔭) ⟶ ℤ/2ℤ.

The subgroup of GL(6, ℤ/2ℤ), which preserves this form, will be denoted by O(6, ℤ/2ℤ). This is a copy of the Weyl group of the E_{6}-lattice. Its order is 51 840.

It is known that the homomorphism

O^{+}(ℤ^{4} × 𝔬) ⟶ O(6, ℤ/2ℤ)

is surjective. The group O^{+}(L) contains its kernel. Hence it corresponds to a certain subgroup of O(6, ℤ/2ℤ).

3.1 Remark. The subgroup G ⊂ O(6, ℤ/2ℤ) which fixes the vector

\overline{(0, 0, 0, 0, 1, 0, 0, 0)} ∈ (ℤ^{4} × 𝔬)/(2ℤ^{4} × 𝔭)

has index 36. Its order is 1440 = 2 · 6!. The group O^{+}(L) is the inverse image of G in O^{+}(ℤ^{4} × 𝔬) and hence has also index 36 in this group.

The structure of G is easy to determine. First of all there is an S_{6} in G. To see this we consider the subgroup ℤ^{4}/2ℤ^{4} of (ℤ^{4} × 𝔬)/(2ℤ^{4} × 𝔭), which is embedded by x ↦ (x, 0). The quadratic form of this group is x_{1}x_{2} + x_{3}x_{4}. The orthogonal group O(4, ℤ/2ℤ) of this ℤ^{4}/2ℤ^{4} is a subgroup of O(6, 𝔽_{2}) (trivial action on the second factor 𝔬/𝔭). The group O(4, ℤ/2ℤ) acts on the 6 anisotropic elements of (ℤ/2ℤ)^{4} (sometimes called “odd characteristics”). This defines an isomorphism

O(4, ℤ/2ℤ) ⟶^{∼} S_{6}.

In this way we obtain a subgroup of G which is isomorphic to S_{6}.

Now we consider the second factor 𝔬/𝔭. The transformation

(x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (x_{1}, −x_{2}, −x_{3}, −x_{4})

can be extended to O(6, ℤ/2ℤ) (trivial action on the first factor (ℤ/2ℤ)^{4}). We obtain an element of order two in O(6, ℤ/2ℤ) which generates a copy of ℤ/2ℤ. This group commutes with the embedded S_{6}. Hence we obtain a copy of S_{6} × ℤ/2ℤ in G.

3.2 Remark. The group G equals the constructed copy of S_{6} × ℤ/2ℤ.

We have to consider a fundamental character on G, namely the product of the trivial character on S_{6} and of the non trivial character on ℤ/2ℤ. We denote this character by χ and use the same letter for the induced character

χ : O^{+}(L) ⟶ {±1}.

We compute this character for special elements. If u ∈ V is an isotropic vector and v ∈ V is orthogonal on u then the Eichler transformation

x ⟼ E(u, v)(a) = a + (a, u)v − (a, v)u − q(v)(a, u)u

is an element of O^{+}(V). For u = (0, 1, 0) and v = (0, 0, A) (A ∈ V_{1}) we obtain the transformation

(x_{1}, x_{2}, X) ⟼ (x_{1}, x_{2} − (A, X) − q(A)x_{1}, X + x_{1}A).

Its effect on the tube domain is simply the translation Z ↦ Z + A.

3.3 Remark. Let A ∈ ℤ^{6}. Then the value of the character χ at the Eichler transformation is

χ(E((0, 1, 0), (0, 0, A))) = (−1)^{A_{1}+···+A_{6}}.

From [FH] the following result is taken. (One has to combine Lemma 11.4 with Theorem 11.5 and one has to understand the translation from the symplectic to the orthogonal world, which also is explained in this paper.)

3.4 Proposition. There exists a modular form θ of weight two on the group O^{+}(L) with respect to the character χ. This form vanishes at a point z ∈ H, if and only if z is orthogonal to a vector a ∈ L′ of norm q(a) = −1/4. The multiplicity is one, the zero-locus is irreducible in H/O^{+}(L).

We also need the behaviour of θ at the zero dimensional cups. Let ℚa ⊂ V be a non-zero isotropic line. We choose a ∈ L′ primitive, i.e. ℚa ∩ L′ = ℤa. It is known (see for example [FH]) that two such a are equivalent mod O′(L) if there images in L′/L agree. This gives:

3.5 Lemma. The set of isotropic lines in V decomposes into two O^{+}(L)- orbits. The first orbit consists of all ℚa, such that a ∈ L′ is a primitive element which is contained already in L. A representative of this is given by (1, 0, …, 0). A representative of the second orbit is (1, 1, 0, 0, 1/2, 1/2, 1/2, 1/2). The modular form θ vanishes at all zero-dimensional cusps from the first orbit.

4. The minimality property

Let Γ ⊂ O^{+}(V) be a modular group and X_{Γ} the Baily-Borel compactification of the quotient. In general X_{Γ} is singular, hence one considers non-singular projective models X instead of X_{Γ}. It is always possible to find a non-singular model X̃ above X_{Γ} and X. This means that there exist surjective holomorphic mappings

π : X̃ ⟶ X, p : X̃ ⟶ X_{Γ}.

4.1 Definition. An irreducible subvariety A ⊂ X_{Γ} of codimension 1 is potentially blown down, if there exists a desingularization p : X̃ → X_{Γ} and a surjective holomorphic map π : X̃ → X onto another non-singular model such that π(p^{−1}(A)) is of codimension > 1.

§5. Good embeddings 9

4.2 Minimality property. The group Γ has the minimality property if every irreducible subvariety A ⊂ X_{Γ} of codimension one, which is potentially blown down, is a Heegner divisor.

This formulation of the minimality property has the advantage that it can be generalized to arbitrary Hermitean quotients. Minimality results for the Siegel modular variety have been obtained in [Fr2] and for Hilbert modular varieties of arbitrary dimension but high level in [Fr3].

Let p : X → Y be a birational proper holomorphic map of non singular varieties. It is well-known that a holomorphic differential form ω on X of top degree vanishes on a subvariety A of codimension one, if p(A) is of codimension > 1. Hence we see:

4.3 Remark. The group Γ has the minimality property if there exists a cusp form of weight n with respect to the character det, whose zeros are located at Heegner divisors.

The construction of such a cusp form is a major problem. In the following we describe some examples.

5. Good embeddings

We consider the lattice L = ℤ^{8}. We are interested in subspaces

W ⊂ V = L(ℚ) = L ⊗_{ℤ} ℚ

of signature (2, 2). We assume that a modular group Γ ⊂ O^{+}(W) is given. We need certain properties.

5.1 Property. For every γ ∈ Γ there exists an element γ̃ ∈ O^{+}(L) which extends γ and which has the further property

det(γ) = χ(γ̃).

Then the restriction of the modular form θ defines a Γ-invariant holomorphic differential form. We want to have that this does not vanish identically.

5.2 Property. The orthogonal complement W^{⊥} of W in V contains no vector x ∈ L′ with q(x) = −1/4.

Finally we want to have that the restriction of θ is a cusp form. The cusps of Γ correspond to isotropic vectors of W. We are interested in the case where the quadratic space W has the property that isotropic subspaces have dimension ≤ 1. It follows that a modular form is a cusp form if it vanishes along the 0-dimensional boundary (all coefficients a(0) vanish). From 2.3 we are lead to consider:

5.3 Property. Every isotropic line in W belongs (considered as line in V) to the first orbit in the sense of 3.5. There are no isotropic planes in W.

5.4 Proposition. Let W be a quadratic space over ℚ of signature (2, 2) and let Γ ⊂ O^{+}(W) be a modular group. Assume that there exists an isometric embedding W ↪ V such that the properties 5.1, 5.2 and 5.3 hold. Then there exists a Γ-invariant cuspidal holomorphic differential form whose zeros are located along Heegner divisors.

6. The Hilbert modular group as orthogonal group

We consider a real quadratic field K ⊂ ℝ with discriminant D. We denote the non trivial Galois automorphism of K by a ↦ a′. Let S(a) = a + a′ be the trace and N(a) = aa′ the norm of an element a ∈ K. We consider on M_{2}(K) the involution

M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ⟼ M′ := ( \begin{matrix}a′ & c′ \\ b′ & d′ \end{matrix} ) .

(So M′ means not the transposed matrix). We consider the space of all “Hermitean matrices” H = H′,

Sym^{2}(K) := \{ H = ( \begin{matrix}h_{0} & h_{1} \\ h′_{1} & h_{2} \end{matrix} ); h_{0}, h_{2} ∈ ℚ, h_{1} ∈ K \}.

This is a 4-dimensional rational vector space. In the following ϵ is 1 or −1. We equip this space with the symmetric bilinear form (a, b), such that the associated quadratic form is

\frac{(H, H)}{2} = −ϵ det H = −ϵ(h_{0}h_{2} − N(h_{1})).

In both cases the signature of this quadratic space is (2, 2). Both cases ϵ = ±1 lead to isomorphic quadratic spaces. An isomorphism is obtained by

( \begin{matrix}h_{0} & h_{1} \\ h′_{1} & h_{2} \end{matrix} ) ⟼ ( \begin{matrix}h_{0} & εh_{1} \\ ε′h′_{1} & −h_{2} \end{matrix} ) ,

where ε denotes some element of K of norm −1. Usually ε is not integral. Therefore we prefer to separate both cases.

We consider the group

G(K) = { M ∈ GL(2, K); det M > 0, N(det M) = 1 }.

§6. The Hilbert modular group as orthogonal group 11

This group acts on Sym^{2}(K) by means of

G(K) × Sym^{2}(K) ⟶ Sym^{2}(K), (M, H) ⟼ MHM′

and preserves the quadratic forms. Hence we get a homomorphism of G(K) into the orthogonal group of Sym^{2}(K),

G(K) ⟶ O(Sym^{2}(K), ± det).

The substitution

σ : Sym^{2}(K) ⟶ Sym^{2}(K), ( \begin{matrix}h_{0} & h_{1} \\ h′_{1} & h_{2} \end{matrix} ) ⟼ ( \begin{matrix}h_{2} & −ϵh_{1} \\ −ϵh′_{1} & h_{0} \end{matrix} )

is not contained in the image of G(K). Hence we consider the semidirect product

Ĝ(K) := G(K) ∪ G(K)σ

with the group of order two generated by σ, which acts on G(K) by means of the automorphism

( \begin{matrix}a & b \\ c & d \end{matrix} ) ⟼ ( \begin{matrix}d′ & −ϵb′ \\ −ϵc′ & a′ \end{matrix} ) .

The homomorphism G(K) → O(Sym^{2}(K)) extends to a homomorphism

Ĝ(K) ⟶ O(Sym^{2}(K)), σ ⟼ σ.

The given embedding K ⊂ ℝ ⊂ ℂ induces an embedding Sym^{2}(K) ⟶ M_{2}(ℂ) and hence an identification Sym^{2}(K)⊗_{ℚ} ℂ = M_{2}(ℂ). The subset K(Sym^{2}(K)) consists of all matrices of the form

( \begin{matrix}τ_{1}τ_{2} & τ_{1} \\ τ_{2} & 1 \end{matrix} ); ϵ Im(τ_{1}) Im(τ_{2}) > 0.

Let H denote the usual upper half plane and H^{ϵ} = ϵH the upper or lower half plane corresponding to the two cases ϵ = ±1. So H^{ϵ} is the upper half plane for ϵ = −1.

H × H^{ϵ} ↪ K(Sym^{2}(K)), (τ_{1}, τ_{2}) ⟼ ( \begin{matrix}τ_{1}τ_{2} & τ_{1} \\ τ_{2} & 1 \end{matrix} )

identifies H × H^{ϵ} with one of the two connected components of K(Sym^{2}(K)). An easy calculation shows:

6.1 Remark. The image of Ĝ(K) is contained in O^{+}(Sym^{2}(K)). The induced action of Ĝ(K) on H × H^{ϵ} is given by the formulae

M(τ_{1}, τ_{2}) = (\frac{aτ_{1} + b}{cτ_{1} + d}, \frac{a′τ_{2} + b′}{c′τ_{2} + d′} ), σ(τ_{1}, τ_{2}) = −ϵ(\frac{1}{τ_{2}} , \frac{1}{τ_{1}} ).

We want to construct an embedding of Sym^{2}(K) into the quadratic space V = ℚ^{8} introduced in the previous section. For this purpose we decompose V = ℚ×ℚ×V_{1}, where V_{1} = ℚ^{6} carries the quadratic form q(y) = y_{1}y_{2}−y_{3}^{3}−⋯−y_{6}^{2}. We take only embeddings of special form

φ : Sym^{2}(K) ⟶ V, ( \begin{matrix}h_{0} & h_{1} \\ h′_{1} & h_{2} \end{matrix} ) ⟼ (h_{0}, −ϵh_{2}, φ(h_{1})),

where φ : K ↪ V_{1} is an isometric embedding with the property q(φ(a)) = ϵN(a). A ℚ-basis of K is given by 1 and \sqrt{D}. Hence we have to define two vectors A, B ∈ V_{1} = ℚ^{6} with the properties

\begin{aligned}q(A) & = A_{1}A_{2} − A_{2}^{2} − ⋯ − A_{6}^{2} = −ϵ, \\ q(B) & = B_{1}B_{2} − B_{2}^{2} − ⋯ − B_{6}^{2} = ϵD, \\ (A, B) & = A_{1}B_{2} + A_{2}B_{1} − 2A_{3}B_{3} − ⋯ − 2A_{6}B_{6} = 0. \end{aligned}

7. The standard Hilbert modular groups

The standard Hilbert modular group is Γ_{K} = SL(2, 𝔬_{K}), where 𝔬_{K} denotes the ring of integer of K. It is known that this group is generated by strict upper and lower triangular matrices. Hence it is generated by the elements

( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) and σ ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) σ.

As in the previous section explained, we choose two orthogonal vectors A, B ∈ V_{1} = ℚ^{6} of norm q(A) = −ϵ, q(B) = ϵD to obtain an embedding of Sym^{2}(K) into V_{1}. We check property 5.1 for Γ_{K}. The element σ is extended to the involution σ̃ of L = ℤ^{8},

σ̃(x_{1}, x_{2}, X) = −ϵ(x_{2}, x_{1}, X).

It is easy to check that this element is contained in O^{+}(L) and that χ(σ̃) = +1. The determinant of σ (considered as element of O(Sym^{2}(K)) is −1. Hence property 5.1 is not true for σ. (This means that we cannot prove the minimality conjecture for the symmetrized Hilbert modular group Γ̂_{K} := Γ_{K} ∪ σΓ_{K}).

We now treat the two cases ϵ = ±1 separately.

Case I. ϵ = −1: In this case we need two vectors A, B ∈ ℚ^{6} with the property q(A) = 1, q(B) = −D.

§7. The standard Hilbert modular groups 13

We take A = (1, 1, 0, …, 0) and check the remaining conditions for B. The orthogonality relations are

B_{1} = −B_{2} and B_{1}^{2} + B_{3}^{2} + ⋯ B_{6}^{2} = D (5 squares).

We check property 5.1 for translation matrices: They correspond to Eichler transformations which can be extended to Eichler transformations in O(V) in an obvious way. The condition is that A and (DA + B)/2 are integral and moreover the sum of the six entries have to be even (3.3). For A this is true and for B it means

B_{1} + D, B_{3}, …, B_{6} ∈ 2ℤ, 2D + B_{3} + ⋯ + B_{6} ≡ 0 mod 4.

The next property, which we have to check is 5.2. Let x = (x_{1}, x_{2}, X) ∈ L′ = ℤ^{4} × \frac{1}{2} ℤ^{4} be a vector of norm −1/4 in the orthogonal complement of W. Then x_{1} = x_{2} = 0 and X ∈ ℤ^{2} + \frac{1}{2} ℤ^{4} is orthogonal to A, which means X_{1} = −X_{2}. The norm condition means X_{1}^{2} + X_{3}^{2} + ⋯ + X_{6}^{2} = 1/4. The solutions are X = ±(0, … 0, 1/2, 0, …, 0) with at least two zeros at the beginning. Such an X can be orthogonal to B only if one of the B_{3}, …, B_{6} is zero.

Finally we have to check 5.4. We have to consider isotropic lines in the image of W. The last for components of an element in such a line is a multiple of (B_{3}, …, B_{6}). If this line is in the second orbit in the sense of 3.5 (what we don’t want to have) then there exists a rational scalar t such that tB_{3}, … tB_{6} are odd. This is impossible if we demand that

B_{3} ≡ 0 mod 4, B_{4} ≡ 2 mod 4.

Collecting all conditions we that we have to find a decomposition of D as a sum of 5 squares with some additional properties:

D = B_{1}^{2} + B_{3}^{2} + ⋯ + B_{6}^{2},
B_{1} + D, B_{3}, …, B_{6} ≡ 0 mod 2, 2D + B_{3} + ⋯ + B_{6} ≡ 0 mod 4,
B_{3} ≡ 0 mod 4, B_{4} ≡ 2 mod 4,
B_{3} ≠ 0, …, B_{6} ≠ 0.

Subcase I.1: D even: Then D = 4d with d ≡ 2 or 3 mod 4 All B_{i} have to be even, B_{i} = 2b_{i}. We take b_{1} =∈ {0, 2} if d ≡ 3 mod 4 and b_{1} ∈ {1, 3} if d ≡ 3 mod 4. Then d − b_{1}^{2} ≡ 2 mod 4. We have to solve d − b_{1}^{2} = b_{3}^{2} + ⋯ b_{6}^{2}. It follows that b_{3} + ⋯ + b_{6} is even. Because this expression is different from 0 mod 4 not all of the b_{i} can be even or odd. Hence we can assume that b_{3} is even and b_{4} odd. The only condition which remains is that b_{3}, …, b_{6} all are different form 0. A solution exists if d ≠ 2, 3, 6, 7, 15. For this one needs the following lemma, which follows from standard facts about the number of representations of natural number as a sum of m ≤ 4 squares.

7.1 Lemma. A natural number n which is not divisible by 8 and which is different from 1, 2, 3, 5, 6, 9, 11, 14, 17, 29, 41 can be written as sum of 4 squares all different from 0.

Subcase I.2: D ≡ 5 mod 8: Then B_{1} has to be odd, the others even, B_{i} = 2b_{i} for i ≥ 3. We take B_{1} = 1 or B_{1} = 3. Then (D − B_{1}^{2})/4 is odd. We need a solution of (D − B_{1}^{2})/4 = b_{3}^{2} + ⋯ + b_{6}^{2}. Automatically b_{3} + ⋯ + b_{6} has to be odd, which was one of the conditions and furthermore not all of the b_{i} can be even or odd. The only condition which remains is that none of the b_{i} is zero. Lemma 7.1 shows that a solution exists if D ≠ 5, 13, 21. Subcase I.3: D ≡ 1 mod 8. There exists no solution. (It can be checked that any other choice of A will not remedy the situation.)

Case II. ϵ = 1: In this case we need two vectors A, B ∈ ℚ^{6} with the property q(A) = −1, q(B) = D.

We take A = (1, −1, 0, …, 0) and check the remaining conditions for B. The orthogonality relations are

B_{1} = B_{2} and B_{3}^{2} + ⋯ B_{6}^{2} = B_{1}^{2} − D.

The integrality conditions give

B_{1} ≡ D mod 2, B_{3} = 2b_{3}, … B_{6} = 2b_{6} (even), b_{3} + ⋯ + b_{6} ≡ D mod 2.

We check 5.2: Let x = (x_{1}, x_{2}, X) ∈ L′ = ℤ^{4}× \frac{1}{2} ℤ^{4} be a vector of norm −1/4 in the orthogonal complement of W. Then x_{1} = x_{2} = 0 and X = (X_{1}, ⋯, X_{6}) ∈ ℤ^{2} × \frac{1}{2} ℤ^{4} satisfies

X_{1} = X_{2}, −X_{1}^{2} + X_{3}^{2} + ⋯ + X_{6}^{2} = \frac{1}{4}, X_{1}B_{1} − B_{3}X_{3} − ⋯ − B_{6}X_{6} = 0.

From the Cauchy-Schwarz inequality

(X_{3}B_{3} + ⋯ + X_{6}B_{6})^{2} ≤ (X_{3}^{2} + ⋯ + X_{6})^{2}(B_{3}^{2} + ⋯ + B_{6})^{2}

we obtain

X_{1}^{2}B_{1}^{2} ≤ (X_{1}^{2} + \frac{1}{4})(B_{1}^{2} − D) or X_{1}^{2} ≤ \frac{B_{1}^{2}}{4D} .

We will construct B such that B_{1}^{2} < 4D. Then the solutions for X are X = ±(0, … 0, 1/2, 0, …, 0) with at least two zeros at the beginning. Such an X can be orthogonal to B only if one of the B_{3}, …, B_{6} is zero.

Condition 5.3 is fulfilled if b_{3} is even and b_{4} is odd. Altogether we need B_{1}, b_{3}, …, b_{6} with the properties

B_{1} ≡ D mod 2, B_{1}^{2} < 4D,

b_{3}^{2} + ⋯ b_{6}^{2} = \frac{B_{1}^{2} − D}{4} ≡ D mod 2,

b_{3} ≠ 0, …, b_{6} ≠ 0, b_{3} ≡ 0 mod 2, b_{4} ≡ 1 mod 2.

The condition (B_{1}^{2} − D)/4 ≡ D mod 2 is not realizable for D ≡ 1 mod 8. the ansatz B_{1} = [\sqrt{D}] + x with \sqrt{D}/2 + 1 ≤ x < \sqrt{D} guarantees B_{1}^{2} < D and (B_{1}^{2} − D)/4 ≥ D/8. Now it is easy to see (using 7.1) that a solution exists up to some exceptional cases. All the exceptional cases belong to the finitely many cases treated by Hirzebruch and van de Ven (s. [vdG] for more details). Hence our main result 1.1 is completely proved.

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