ON THE EQUATIONS DEFINING FAMILIES OF KUMMER SURFACES OF QUATERNIONIC MULTIPLICATION TYPE

AMNON BESSER

1. Introduction

Let B be a rational quaternion algebra which is unramified at infinity. Let M be an order in B. Denote by Γ the group of units with positive norm in M. We may fix an isomorphism B ⊗ ℝ ≅ M_{2}(ℝ) and this identifies Γ as a discrete subgroup of SL(2, ℝ). There is a well known action, through linear fractional transformations, of SL(2, ℝ) on the Poincaré upper half plane ℌ and we may therefore consider the quotient space Γ\ℌ.

The simplest case is when B = M_{2}(ℚ). Then Γ may be identified as a group of finite index in SL(2, ℤ). The resulting curves are called modular curves and have played a major role in modern arithmetic geometry. One key ingredient in the understanding of the arithmetic of modular curves is their interpretation as parameter varieties (moduli schemes) for families of elliptic curves with some extra structures. In another direction, modular forms, which are sections of certain line bundles above modular curves, have been studied extensively. The existence of parabolic elements in Γ allows an expansion of modular forms in Fourier series and one can use this to write explicit formulae for these forms. This, together with the simplicity of equations describing elliptic curves, makes it relatively easy to write equations of modular curves and the families of elliptic curves parameterized by them.

In contrast, if B ≠ M_{2}(ℚ), Poincaré already observed that Γ\ℌ is compact. In this case, there are no Parabolic elements in Γ and therefore no Fourier expansion is available. Shimura noticed around 1960 that a moduli interpretation is still available for these curves. They classify Abelian surfaces with a quaternionic multiplication i.e. an embedding of some order in B in their endomorphism ring. In the case M maximal the moduli problem gives a canonical model of Γ\ℌ defined over ℚ. These curves are therefore called Shimura curves.

Because there is no Fourier expansion in this case and because equations of Abelian surfaces are considerably more complicated then those of elliptic curves, It is much harder to write equations of Shimura curves and of the universal families of Abelian surfaces above them. Equations for Shimura curves were computed in certain cases by Kurihara [K] and by Jordan in his thesis [Jor]. The method is slightly artificial and uses the fact that the quotient of some Shimura curved by certain canonical automorphisms has genus 0. The aim of this work is to write equations for universal families above Shimura curves in a few particular cases. In fact, we write equations not for the Abelian surfaces themselves but rather for the associated Kummer surfaces obtained by dividing by −1.

We use two different methods. The first uses results of Nikulin on discriminant forms and the theory of elliptic surfaces to write elliptic fibrations of Kummer surfaces coming from Abelian surfaces with quaternionic multiplication. The second method uses embeddings into a Siegel threefold and the results of van der Geer describing the families of Kummer surfaces arising there.

After this work was completed we learned that Hashimoto and Murabayashi [Ha-Mu] have obtained some of these results independently using a method similar to our second method.

Here is an outline of the contents. In §2 we describe the main results of [N] about discriminant forms. In §3 we recall the description of the integral second cohomology of a QM Abelian surface given in [Be] and use it to compute the discriminant form of the Néron - Severi lattice of the associated Kummer surface. §4 recalls the theory of elliptic surfaces. In §5 we explain how the knowledge of the discriminant form of a K3 surface allows us to write nice elliptic fibrations of this surface. As an application we give elliptic fibrations for Kummer surfaces coming from Abelian surfaces with quaternionic multiplication by some orders with low discriminant. §6 and §7 describe the results of [G] about a Siegel threefold and Humbert surfaces and CM curves inside it respectively and give a few more details. §8 then uses the results in the previous 2 sections to write equations for the images of Shimura curves under the map to the Siegel threefold induced by the functor forgetting the QM. we comment on how this allows, due to a result of van der Geer, to write equations of universal families of Kummer surfaces above Shimura curves in these cases.

acknowledgment: I would like to thank Ron Livne for suggesting the problem and for his patient explanations and help and to Bruce Jordan for suggesting further work.

2. The work of Nikulin on Integral quadratic forms

This section reviews the work of Nikulin [N] on integral, symmetric, even bilinear forms and their associated discriminant forms. This work deals with the following problems:

• Classification of lattices.

• Existence and uniqueness of embeddings of one lattice into another.

By a lattice we mean the following:

Definition 2.1. A lattice is a free ℤ module of finite rank together with a nondegenerate symmetric bilinear form with values in ℤ.

The value of the form on x and y will be denoted x · y.

Definition 2.2. The parity character of a lattice S, par : S ⟶ ℤ/2ℤ is defined by par(x) = x · x + 2ℤ.

Our lattice is assumed even, i.e. that the parity character is trivial.

2.3. The signature of a lattice S is the signature of the induced bilinear form on S ⊗_{ℤ} ℝ. It is (+t_{+}, −t_{−}) if S ⊗ ℝ is isomorphic to ℝ^{t_{+}+t_{−}} with the quadratic form

∑_{i=1}^{t_{+}} x_{i}^{2} − ∑_{i=1}^{t_{−}} x_{i+t_{+}}^{2}

2.4. Given two lattices S_{1}, S_{2} we denote by S_{1} ⊕ S_{2} their orthogonal direct sum. S ≅ L means that S and L are isomorphic which means that there is an isometry between them. The symbol S(k) for a lattice S and an integer k stands for the lattice which as a group is S while the form is the one of S multiplied by k.

2.5. If S ⊂ L we say that S is primitive in L if L/S is torsion free. Let S^{⊥} denote the orthogonal complement of S in L. Clearly, S is primitive in L iff S = (S^{⊥})^{⊥}.

Definition 2.6. The hyperbolic plane U is the lattice generated by e and e′ satisfying e^{2} = (e′)^{2} = 0 and e · e′ = 1.

If we set s = e′ − e then s^{2} = −2 and s · e = 1.

Definition 2.7. A finite quadratic form (A, q) is a finite group A together with a map q : A ⟶ ℚ/2ℤ s.t.

• q(na) = n^{2}q(a) for all n ∈ ℤ and a ∈ A.

• The function b defined by b(a, a′) = \frac{1}{2}(q(a + a′) − q(a) − q(a′)) (mod 1) is a bilinear form on A with values in ℚ/ℤ.

We call the function b the bilinear form of q. (A, q) is called nondegenerate if b is.

We will often denote the form just by q and refer to A as A_{q}.

2.8. Given a lattice S, let S^{∗} denote the dual of S, S^{∗} := Hom(S, ℤ). There is a canonical embedding, induced by the form on S, S ↪ S^{∗}. Let A_{S} := S^{∗}/S. Putting

q_{S}(t + S) = t^{2} + 2ℤ, b_{S}(t_{1} + S, t_{2} + S) = t_{1} · t_{2} + ℤ

one gets a quadratic form q_{S} on A_{S} with the corresponding bilinear form b_{S} which is nondegenerate.

Definition 2.9. (A_{S}, q_{S}) is called the discriminant form of S.

One has the obvious relation q_{S_{1}⊕S_{2}} = q_{S_{1}} ⊕ q_{S_{2}}.

2.10. To compute q_{S}, one finds easily bases {e_{i}}_{i=1}^{n} and {e′_{j}}_{j=1}^{n} of S s.t. e_{i} · e′_{j} = δ_{ij}d_{i} with d_{i} ∈ ℤ − 0 (Smith normal form). It is then clear that S^{∗} is generated by {e_{i}/d_{i}}_{i=1}^{n} and therefore that S^{∗}/S is generated by all e_{i}/d_{i} with |d_{i}| > 1. These generators are independent and have order |d_{i}| respectively. Therefore, the order of A_{S} is | ∏_{i=1}^{n} d_{i}| = | det(e_{i} · e′_{j})|. This last quantity is independent of the choice of bases.

2.11. Clearly, the matrix E = (e_{i} · e_{j}) determines uniquely the isomorphism class of the lattice and two matrices E, E′ determine the same lattice iff there exists A ∈ GL(2, ℤ) s.t E′ = AEA^{t}.

Definition 2.12. det(E) is called the discriminant of the lattice.

A lattice is called unimodular if A_{S} = {0}. In view of the above, this is equivalent to the property that its discriminant is invertible.

2.13. there is an analogues notion of a p-adic lattice (free finite rank ℤ_{p} module with a symmetric nondegenerate form into ℤ_{p}) and corresponding notions to the ones introduced above. One has

Proposition 2.14. ([N, 1.7.1]) For a ℤ lattice K,(q_{K})_{p} = q_{K⊗ℤ_{p}} where (q_{K})_{p} is the p part of A_{K} with the induced form. In addition q_{K} = ⊕_{p}(q_{K})_{p}.

Proposition 2.15. ([N, 1.8.1 and 1.8.2])

a) Each p-adic lattice is isomorphic to a direct sum of the following lattices:

• K_{θ}^{(p)}(p^{k}) – 1 dimensional lattice defined by the matrix (see (2.11) ) (θp^{k})

for each θ ∈ ℤ_{p}^{×}/(ℤ_{p}^{×})^{2} and k ≥ 0. In addition, if p = 2

• U^{(2)}(2^{k}), V^{(2)}(2^{k}) – 2 dimensional defined by

(\begin{matrix}0 & 2^{k} \\ 2^{k} & 0 \end{matrix}) , (\begin{matrix}2^{k+1} & 2^{k} \\ 2^{k} & 2^{k+1} \end{matrix})

respectively with k ≥ 0.

b) Each finite quadratic form is a direct sum of forms of type: q_{θ}^{(p)}(p^{k}),u_{+}^{(2)}(2^{k}) and v_{+}^{(2)}(2^{k}) – the discriminant forms of K_{θ}^{(p)}(p^{k}), U^{(2)}(2^{k}) and V^{(2)}(2^{k}) respectively for k ≥ 1.

c) The following relations exist:

(1) (K_{θ}^{(p)}(p^{k}))^{2} ≅ (K_{θ′}^{(p)}(p^{k}))^{2}, if p ≠ 2;

(2) (U^{(2)}(2^{k}))^{2} ≅ (V^{(2)}(2^{k}))^{2};

(3) K_{θ}^{(2)}(2^{k}) ⊕ K_{θ′}^{(2)}(2^{k}) ≅ K_{5θ}^{(2)}(2^{k}) ⊕ K_{5θ′}^{(2)}(2^{k});

(4) (K_{θ}^{(2)}(2^{k}))^{2} ⊕ K_{θ′}^{(2)}(2^{k}) ≅ \begin{cases}V^{(2)}(2^{k}) ⊕ K_{−5θ′}^{(2)}(2^{k}), & if θ′ ≡ θ (mod 4), \\ U^{(2)}(2^{k}) ⊕ K_{−θ′}^{(2)}(2^{k}), & if θ′ ≡ −θ (mod 4); \end{cases}

(5) V^{(2)}(2^{k}) ⊕ K_{θ}^{(2)}(2^{k+1}) ≅ U^{(2)}(2^{k}) ⊕ K_{5θ}^{(2)}(2^{k+1});

(6) V^{(2)}(2^{k+1}) ⊕ K_{θ}^{(2)}(2^{k}) ≅ U^{(2)}(2^{k+1}) ⊕ K_{5θ}^{(2)}(2^{k});

(7) K_{θ}^{(2)}(2^{k}) ⊕ K_{θ′}^{(2)}(2^{k+1}) ≅ K_{θ+2θ′}^{(2)}(2^{k}) ⊕ K_{5(θ′−2θ)}^{(2)}(2^{k+1});

(8) K_{θ}^{(2)}(2^{k}) ⊕ K_{θ′}^{(2)}(2^{k+2}) ≅ K_{5θ}^{(2)}(2^{k}) ⊕ K_{5θ′}^{(2)}(2^{k+2});

(9) All relations on discriminant forms arising from the above relations. In addition

(10) q_{θ}^{(2)}(2) ≅ q_{5θ}^{(2)}(2).

We notice that the discriminant of a p-adic lattice is defined only (mod (ℤ_{p}^{×})^{2}).

Lemma 2.16. A 2 ≠ p-adic lattice with an invertible discriminant is determined by it. In particular, it is K_{1}^{(p)}(1) ⊕ K_{−1}^{(p)}(1) if the discriminant is −1.

Proof. Immediate from (2.15.c.1).

Lemma 2.17. an even 2-adic lattice of rank 2 with an invertible discriminant θ is determined to be V^{(2)}(1) if θ ≡ 3 (mod 8) and U^{(2)}(1) if θ ≡ 7 (mod 8).

Proposition 2.18. ([N, 1.7.1]) If q is a finite quadratic form, The number t_{+} − t_{−} for a lattice K with q ≅ q_{K} is independent of K modulo 8ℤ.

Definition 2.19. For a finite quadratic form q define the signature of q, sign(q) ∈ ℤ/8ℤ to be the value defined in (2.18).

sign(q) is clearly additive in q. It is thus sufficient to know its value on a set of generators for the finite quadratic forms. This is done in

Proposition 2.20. ([N, 1.11.2],[Wal])

(1) sign(q_{θ}^{(p)}(p^{k})) ≡ k^{2}(1 − p) + 4kη (mod 8) if p ≠ 2 where η is defined by (\frac{θ}{p}) = (−1)^{η} and (\frac{·}{·}) is the Legendre symbol.

(2) for p = 2 we have

sign(q_{θ}^{(2)}(2^{k})) ≡ θ + k \frac{θ^{2} − 1}{2}, sign(v_{+}^{(2)}(2^{k})) ≡ 4k, sign(u_{+}^{(2)}(2^{k})) ≡ 0

(mod 8).

We need one more invariant of q, denoted ℓ(q), which is the minimal number of generators for A_{q}. Note that ℓ(q) = max_{p} ℓ((q)_{p}).

2.21. Let S, S′ be lattices. An embedding S ↪ S′ is called an overlattice of S if S′/S is finite. For an overlattice S ⊂ S′ one has S ⊂ S′ ⊂ S′^{∗} ⊂ S^{∗}. We define H_{S′} := S′/S^{∗} ⊂ A_{S}.

Proposition 2.22. ([N, 1.4.1]) There is a 1−1 correspondence S′ ⟶ H_{S′} between overlattices of S and subgroups H ⊂ A_{S} which are isotropic (i.e. q|_{H} = 0). We have q_{S′} ≅ q_{S}|_{H_{S′}^{⊥} /H_{S′}}.

From this, one deduces the following description of primitive embeddings: If S ↪ L is a primitive embedding and K = S^{⊥} then L is an overlattice of S ⊕ K.

Proposition 2.23. ([N, 1.5.1]) With the above notation H_{L} ⊂ A_{S}⊕A_{K} is the graph Γ_{γ} of a monomorphism γ : H ⟶ A_{K} for some H ⊂ A_{S} s.t. q_{K} ∘ γ = −q_{S}|_{H}. One then has q_{L} ≅ q_{S} ⊕ q_{K}|_{Γ_{γ}^{⊥}/Γ_{γ}}. Conversely, given S and K, the data (H, γ) determines an embedding of S into some lattice s.t. S^{⊥} ≅ K.

Corollary 2.24. ([N, 1.6.2]) if U is a unimodular lattice, S ↪ U a primitive embedding and K = S^{⊥} then q_{K} ≅ q_{S(−1)}.

We now state the main results.

Theorem 2.25. ([N, 1.10.2 and 1.13.2])

(1) A lattice with signature (t_{+}, t_{−}) and discriminant form q exists if t_{+} +t_{−} > ℓ(q) and t_{+} − t_{−} ≡ sign(q) (mod 8).

(2) If, in addition, t_{+}, t_{−} > 0, t_{+} + t_{−} ≥ 3 and, for each prime p s.t. t_{+} + t_{−} <

2 + ℓ(q_{p}), one has • q_{p} ≅ q_{θ_{1}}^{(p)}(p^{k}) ⊕ q_{θ_{2}}^{(p)}(p^{k}) ⊕ q′_{p} for some k and some q′_{p} if p ≠ 2,

• q_{2} ≅ u_{+}^{(2)}(2^{k}) ⊕ q′_{2} or v_{+}^{(2)}(2^{k}) ⊕ q′_{2} or q_{θ_{1}}^{(2)}(2^{k}) ⊕ q_{θ_{2}}^{(2)}(2^{k+1}) ⊕ q′_{2} if

p = 2,

then there is a unique lattice up to isomorphism with these invariants.

(3) In particular, the conclusion of 2 holds if t_{+}, t_{−} > 0 and t_{+} + t_{−} ≥ 2 + ℓ(q) (since in this case t_{+} + t_{−} ≥ 3 unless the lattice is unimodular in which case the uniqueness is well known).

Two embeddings i, i′ : S ↪ L are said to be isomorphic if there exist an isometry α : L ⟶ L s.t. the diagram

S  S
i↓  ↓_{i′}
L ⟶^{α} L

commutes.

Theorem 2.26. ([N, 1.12.3 and 1.14.4])

(1) A lattice S with signature (t_{+}, t_{−}) and discriminant form q has a primitive embedding into some unimodular lattice of signature (l_{+}, l_{−}) if l_{+} ≥ t_{+}, l_{−} ≥ t_{−} and l_{+} + l_{−} − t_{+} − t_{−} > ℓ(q).

(2) Such an embedding is unique up to isomorphism if:

• l_{+} > t_{+}, l_{−} > t_{−}

• l_{+} + l_{−} − t_{+} − t_{−} ≥ ℓ(q_{p}) + 2 for all p ≠ 2.

• if l_{+} + l_{−} − t_{+} − t_{−} = ℓ(q_{2}) then q = u_{+}^{(2)}(2) ⊕ q′ or v_{+}^{(2)}(2) ⊕ q′.

Theorem 2.27. ([N, 1.9.4]) If the lattices S and S′ have the same signature and the same discriminant form then, for any prime p, S ⊗ ℤ_{p} ≅ S′ ⊗ ℤ_{p} as p-adic lattices.

3. The discriminant form of the Néron – Severi lattice of a QM Abelian surface and its associated Kummer surface

Let A be an Abelian surface with QM by an order M in a rational quaternion algebra (r.q.a.) B unramified at ∞. Then [Be, §3] A is complex analytically the Torus ℂ^{2}/I with I some M ideal embedded into ℂ^{2} via

B ↪ B ⊗ ℝ ≅ M_{2}(ℝ)⟶^{g_{τ}}ℂ^{2} , g_{τ} (Y) = Y \binom{τ}{1} (1)

for some τ in the upper half plane ℌ. This identifies I to H_{1}(A, ℤ). We refer the reader to §2 of [Be] for the necessary facts about rational quaternion algebras that will be used in this §.

Assume I is an ideal of discriminant D. Recall [Be, 2.14] that this means (in the notation of §2) that D^{2} is the discriminant of the form on I defined by tr(xy′) where y ⟶ y′ is the main involution of B.

Let B_{0} = Ker(tr : B ⟶ ℚ) an element b ∈ B_{0} defines a simplectic form on I with values in ℚ by

⟨x, y⟩ = tr(bxy′)

and using (1) we get an element of H^{2}(A, ℚ) which we denote by ι_{τ} (b).

We now quote a part of Theorem 3.10 in [Be] which tells us that if A is a generic Abelian surface of the required type (excluding a countable number of Abelian surfaces which have CM in the sense of [Be]) then the elements defined above give all the algebraic classes in H^{2}. More precisely, let NS(A) = H^{2}(A, ℤ) ∩ H^{1,1}(A, ℂ). Then, for a generic A

Theorem 3.1.

(1) ι_{τ} : B_{0} ⟶ NS(A) · ℚ is an isomorphism.

(2) For all x, y ∈ B_{0} ι(x) · ι(y) = D tr(xy′) where · is the pairing obtained by composing the trace from H^{4}(A, ℚ) to ℚ with the cup product.

NS(A) together with the cup product forms some lattice in the sense of §2 (The form is even since NS(A) ⊂ H^{2}(A, ℤ) and the form here is even). The main objective of this section is to compute the corresponding discriminant form.

Lemma 3.2. Under ι_{τ}, NS(A) corresponds to the group of elements b ∈ B_{0} for which tr(bxy′) ∈ ℤ for all x, y ∈ I.

We will compute the discriminant form only for Abelian surfaces of the type defined below.

Definition 3.3. An Abelian surface is called of type (D, N) if its ring of endomorphisms, M, is an Eichler order of level N in an r.q.a. B of discriminant D. (see [Be, 2.15] and [Be, 2.17]). An Abelian surface is called generic of type (D, N) if it is of type (D, N) and its Néron – Severi group has rank 3.

Since any left ideal of the Eichler order M is principal [Vig, III.5.7], We may assume I = M. In this case NS(A) corresponds simply to the ℤ module of elements b ∈ B_{0} s.t. tr(bx) ∈ M for all x ∈ M. We denote this module by M_{0}^{∗}. According to (3.1) and [Be, 2.18] the cup product corresponds on M_{0}^{∗} to

a · b = DN tr(ab′). (2)

3.4. To compute the discriminant form of M_{0}^{∗}, it is enough by (2.14) to compute the p-adic lattice M_{0}^{∗} ⊗ ℤ_{p} for all prime p. Let M_{p} := M ⊗ ℤ_{p} considered as an order in B_{p} = B ⊗ ℚ_{p}. Clearly M_{0}^{∗} ⊗ ℤ_{p} = (M_{p})_{0}^{∗}.

Proposition 3.5. Let M be an Eichler order of level N in an r.q.a. of discriminant D. For each prime p let n = n_{p} = ord_{p}(N) be the exponent of the largest power of p dividing N. For each prime p ≠ 2 let v = v_{p} denote a non residue mod p. The local lattice (M_{p})_{0}^{∗} with the bilinear form (2) is isomorphic to:

(1) If 2 ≠ p|D, K_{−2vDN/p}^{(p)}(p) ⊕ K_{−2DN/p}^{(p)}(1) ⊕ K_{2vDN/p}^{(p)}(1).

(2) If 2 ≠ p |̸D, K_{−2DN/p^{n}}^{(p)}(p^{n}) ⊕ K_{1}^{(p)}(1) ⊕ K_{−1}^{(p)}(1).

(3) If p = 2|D, K_{−5DN/2}^{(2)}(4) ⊕ V^{(2)}(1)

(4) If p = 2 |̸D, K_{−DN/2^{n}}^{(2)}(2^{n+1}) ⊕ U^{(2)}(1)

Proof. We can assume our local orders M_{p} are given in the following standard form [Be, 2.16].

• If 2 ≠ p|D

M_{p} = \{(\begin{matrix}a & b \\ pb^{σ} & a^{σ} \end{matrix}) , a, b ∈ O_{L_{p}}\} ⊂ M_{2}(O_{L_{p}})

where L_{p}/ℚ_{p} is the unique unramified quadratic extension, σ is the nontrivial automorphism of L_{p} and O_{L_{p}} = ℤ_{p} ⊕ ℤ_{p}\sqrt{v} is the ring of integers of L_{p} with v a nonresidue mod p.

• If p |̸D

M_{p} = \{(\begin{matrix}a & b \\ cp^{n} & d \end{matrix}) , a, b, c, d ∈ ℤ_{p}\}

inside M_{2}(ℚ_{p}) for some nonnegative integer n.

• If p = 2|D

M_{2} = \{(\begin{matrix}a & b \\ 2b^{σ} & a^{σ} \end{matrix}) , a, b ∈ O_{L_{2}}\} ⊂ M_{2}(O_{L_{2}})

where L_{2} may be taken to be ℚ_{2}(\sqrt{5}) and O_{L_{2}} = ℤ_{2} ⊕ ℤ_{2}(1 + \sqrt{5})/2

Lemma 3.6. A ℤ_{p} basis for (M_{p})_{0}^{∗} is given by:

\{(\begin{matrix}\sqrt{v} & 0 \\ 0 & −\sqrt{v} \end{matrix}) , (\begin{matrix}0 & p^{−1} \\ 1 & 0 \end{matrix}) , (\begin{matrix}0 & p^{−1}\sqrt{v} \\ −\sqrt{v} & 0 \end{matrix})\} if 2 ≠ p|D

\{(\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , (\begin{matrix}0 & p^{−n} \\ 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})\} if p |̸D and ord_{p}(N) = n

\{(\begin{matrix}\sqrt{5} & 0 \\ 0 & −\sqrt{5} \end{matrix}) , (\begin{matrix}0 & \frac{1}{2} \\ 1 & 0 \end{matrix}) , (\begin{matrix}0 & \frac{1+\sqrt{5}}{4} \\ \frac{1−\sqrt{5}}{2} & 0 \end{matrix})\} if p = 2|D

Proof. For a ramified prime p|D

M_{p}^{∗} = \{A ∈ B_{p} , tr [A (\begin{matrix}a & 0 \\ 0 & a^{σ} \end{matrix})] , tr [A (\begin{matrix}0 & b \\ pb^{σ} & 0 \end{matrix})] ∈ ℤ_{p}, all a, b ∈ O_{L_{p}}\}

= \{A = (\begin{matrix}c & d \\ pd^{σ} & c^{σ} \end{matrix}) , tr(ca), p tr(db) ∈ ℤ_{p}, all a, b ∈ O_{L_{p}}\}

= \{A = (\begin{matrix}c & d \\ pd^{σ} & c^{σ} \end{matrix}) , c, pd ∈ O_{L_{p}}^{∗}\}

= \{A = (\begin{matrix}c & d \\ pd^{σ} & c^{σ} \end{matrix}) , c, pd ∈ O_{L_{p}}\}

where the last equality follows since L_{p}/ℚ_{p} is unramified. This proves the ramified cases and the unramified one is easy.

We now compute the matrix representing the form (2) in our basis. The result is (recall that the main involution is the adjoint matrix):

\frac{DN}{p} (\begin{matrix}−2pv & 0 & 0 \\ 0 & −2 & 0 \\ 0 & 0 & 2v \end{matrix}) when 2 ≠ p|D

\frac{DN}{p^{n}} (\begin{matrix}−2p^{n} & 0 & 0 \\ 0 & 0 & −1 \\ 0 & −1 & 0 \end{matrix}) when p |̸D and ord_{p}(N) = n

\frac{DN}{2} (\begin{matrix}−20 & 0 & 0 \\ 0 & −2 & −1 \\ 0 & −1 & 2 \end{matrix}) when p = 2|D.

This proves the proposition since \frac{DN}{2} ( \begin{matrix}−2 & −1 \\ −1 & 2 \end{matrix} ) is a matrix representing the lattice V^{(2)}(1) by (2.17) and \frac{DN}{p^{n}} ( \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} ) represents K_{1}^{(p)}(1) ⊕ K_{−1}^{(p)}(1) by (2.16)).

Corollary 3.7. The discriminant form of the Néron – Severi lattice of a generic Abelian surface of type (D, N) is

⨁_{2≠p|D} q_{−2v_{p}DN/p}^{(p)}(p) ⊕ ⨁_{2≠p|N} q_{−2DN/p^{np}}^{(p)} (p^{n_{p}}) ⊕ \begin{cases}q_{−5DN/2}^{(2)}(4) & if2—D; \\ q_{−DN/2^{n_{2}}}^{(2)} (2^{n_{2}+1}) & otherwise. \end{cases}

3.8. Associated to an Abelian surface A is a K3 surface X = Kummer(A) called the Kummer surface associated to A. It is obtained from A by first dividing A by the action of the involution x ⟶ −x and then blowing up once each of the 16 double points coming from the points of A of order 2. For the theory of Kummer surfaces see [BPV] and [PS-S]. We will need the following result:

Definition 3.9. Given a projective surface Y, the orthogonal complement to NS- (Y) inside H^{2}(Y, ℤ) is called the transcendental lattice of Y and is denoted by T (Y).

Theorem 3.10. ([N1, Remark 2]) Let A be an Abelian surface and let X =- Kummer(A). Then

T (X) ≅ T (A)(2)

(recall from (2.4) that S(k) is the lattice S with the form multiplied by k)

Definition 3.11. Call a Kummer surface X = Kummer(A) generic of type (D, N) (or just of type (D, N)) if A is of type (D, N) in the sense of (3.3).

Proposition 3.12. Let X be a Kummer surface of type (D, N). The discriminant form of the Néron – Severi lattice of X is

⨁_{2≠p|D} q_{−v_{p}DN/p}^{(p)}(p) ⊕ ⨁_{2≠p|N} q_{−DN/p^{np}}^{(p)} (p^{n_{p}}) ⊕ \begin{cases}q_{−5DN/2}^{(2)}(8) ⊕ v_{+}^{(2)}(2) & if 2|D; \\ q_{−DN/2^{n_{2}}}^{(2)} (2^{n_{2}+2}) ⊕ u_{+}^{(2)}(2) & if 2 |̸D. \end{cases}

Proof. Let X = Kummer(A). We can easily see from (3.10.1) that T (A) ≅ NS(A)(−1). It thus follows from (3.10) that T (X) ≅ NS(A)(−2). Since NS(X) = T (X)^{⊥} in the unimodular lattice H^{2}(X, ℤ) it follows from (2.24) that

q_{NS(X)} = q_{T (X)(−1)} = q_{NS(A)(2)}.

The result now follows directly from (3.5).

Remark 3.13. One may avoid the use of (3.10.1) in the above argument as follows: From the fact that NS(A) and T (A) are complement to each other in the unimodular lattice H^{2}(A, ℤ) it follows by (2.24) that q_{T (A)} = q_{NS(A)(−1)}. By (2.27) we see that all localizations of T (A) and NS(A)(−1) are isomorphic. Arguing as above we find that the same holds for T (X) and NS(A)(−2) hence they have the same discriminant form. Then we continue as before.

4. Elliptic surfaces

Definition 4.1. An elliptic surface is a compact complex surface E together with a proper projection π : E ⟶ S, called an elliptic fibration, onto some compact curve S s.t. the generic fiber π^{−1}(s), s ∈ S is a nonsingular curve of genus 1.

Elliptic surfaces were studied systematically by Kodaira [Ko,Ko1] (see also [He], [BPV], [Pe], [Mi], [Mi-Pe]). Most of what we describe here is due to him.

4.2. Let E⟶^{π}S be an elliptic surface. One usually assumes there is no exceptional curve contained in the fibers. We also assume for the time being that π admits a section and in fact we choose one and name it 0 : S ⟶ E. This assumption simplifies things a bit.

The following data is associated with an elliptic surface π : E ⟶ S:

4.3. Denote by Σ the finite set of points Σ = {s_{1}, …, s_{n}} ⊂ S s.t. for s ∉ Σ π^{−1}(s) is nonsingular of genus 1. This set is called the exceptional set. For s ∈ Σ π^{−1}(s) is a singular curve which is either singular irreducible or decomposes into rational −2 curves. The possible types of singular curves were classified by Kodaira. They will be given in table 1.

Let E_{0} = π^{−1}(S − Σ) and j : S − Σ ↪ S the inclusion.

Definition 4.4. The homological invariant of π is the sheaf j_{∗}R_{1}π_{∗}^{0}(ℤ) where π^{0} is π restricted to E_{0}. Here, and elsewhere, R_{i}π_{∗}(ℤ) is the local system of homology groups.

The restriction of the homological invariant to S − Σ is locally free of rank 2. This sheaf is equivalent to a monodromy representation χ : π_{1}(S − Σ, ∗) ⟶ Aut(H_{1}(π^{−1}(∗), ℤ)) (choosing an arbitrary base point ∗ ∈ S − Σ). π_{1}(S − Σ, ∗) has generators a_{i}, b_{i}, i = 1, …, g =genus of S, which are generators of π_{1}(S, ∗), and c_{j}, j = 1, …, n which are loops around the elements s_{j} of Σ. These generators satisfy the unique relation

∏_{i=1}^{g}[a_{i}, b_{i}] ∏_{j=1}^{n} c_{j} = 1

with [a, b] = ab(ba)^{−1}. Therefore, the homological invariant is equivalent, after fixing the generators and an isomorphism ℤ^{2} ≅ H_{1}(π^{−1}(∗), ℤ), to giving matrices A_{i}, B_{i}, i = 1, …, g and C_{j}, j = 1, …, n in SL(2, ℤ) s.t

∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{n} C_{j} = I.

The GL(2, ℤ) conjugacy class of C_{j} depends only on the type of the fiber at s_{j}. In table 1 we indicate it by a choice of one representative.

Definition 4.5. The functional invariant J of E⟶^{π}S is the continuation to S of the function J(s) = j(π^{−1}(s)) from S − Σ to ℂ where j is the j invariant of an elliptic curve. J is a meromorphic function.

4.6. There is a simple relation between the homological and functional invariants of an elliptic surface. Choose some τ_{0} ∈ ℌ, the upper half plane, s.t. j(τ_{0}) := j(ℂ/⟨1, τ_{0}⟩) = J(∗). Then π^{−1}(∗) ≅ ℂ/⟨1, τ_{0}⟩ so one can pick the basis {1, τ_{0}} to H_{1}(π^{−1}(∗), ℤ). There is a multivalued analytic function τ : S ⟶ ℌ s.t. j(τ(s)) = J(s) for all s ∈ S − Σ. Analytic continuation around a loop x ∈ π_{1}(S − Σ, ∗) sends τ_{0} to some τ_{x} s.t. j(τ_{x}) = j(τ_{0}). Assuming that π^{−1}(∗) has no nontrivial automorphisms we see that

τ_{x} = (\begin{matrix}a & b \\ c & d \end{matrix}) τ_{0}

for the Möbius action of an element in SL(2, ℤ) determined up to sign. Since {1, τ(s)} is a basis for H_{1}(π^{−1}(s), ℤ) one finds easily

(\begin{matrix}a & b \\ c & d \end{matrix}) = ±χ(x).

Thus, the composition of χ with the projection from SL(2, ℤ) to PSL(2, ℤ) is determined by J.

Theorem 4.7. ([BPV, V.11.1],[Ko]) Given a functional and homological invariants with a compatible monodromy representation there exists a unique elliptic surface with these invariants.

Table 1: Singular fibers

\begin{matrix}type & m & m^{(1)} & ε & f & J & monodromy & q \\ I_{b} & b & b & b & 1 & b · ∞ & (\begin{matrix}1 & b \\ 0 & 1 \end{matrix}) & {A_{q} = ℤ/bℤ \atop 1^{2} = b^{−1} − 1} \\ II & 1 & 1 & 2 & 2 & {h · 0, \atop h ≡ 1(3)} & (\begin{matrix}1 & 1 \\ −1 & 0 \end{matrix}) & \{0\} \\ III & 2 & 2 & 3 & 2 & {h · 1, \atop h ≡ 1(2)} & (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) & q_{−1}^{(2)}(2) \\ IV & 3 & 3 & 4 & 2 & {h · 0, \atop h ≡ 2(3)} & (\begin{matrix}0 & 1 \\ −1 & −1 \end{matrix}) & q_{1}^{(3)}(3) \\ I_{b}^{∗} & b + 5 & 4 & b + 6 & 2 & b · ∞ & − (\begin{matrix}1 & b \\ 0 & 1 \end{matrix}) & q_{4−b}^{(2)}(4), 2 |̸b; ; (q_{2−b/2}^{(2)}(2))^{2}, ; b ≡ ±2(8); ; u_{+}^{(2)}(2), b ≡ 4(8); ; v_{+}^{(2)}(2), b ≡ 0(8) \\ II^{∗} & 9 & 1 & 10 & 2 & {h · 0, \atop h ≡ 2(3)} & − (\begin{matrix}0 & 1 \\ −1 & −1 \end{matrix}) & \{0\} \\ III^{∗} & 8 & 2 & 9 & 2 & {h · 1, \atop h ≡ 1(2)} & − (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) & q_{1}^{(2)}(2) \\ IV^{∗} & 7 & 3 & 8 & 2 & {h · 0, \atop h ≡ 1(3)} & − (\begin{matrix}1 & 1 \\ −1 & 0 \end{matrix}) & q_{−1}^{(3)}(3) \end{matrix}

4.8. Explanation of the table: The fibers of type I_{b},I_{b}^{∗},II^{∗},III^{∗} and IV^{∗} are described by their dual graphs, i.e. a vertex stands for a rational component, the number above the vertex counts multiplicity in the fiber and an edge between two vertices indicates a transversal intersection. The other types are described by writing their components (all of multiplicity 1) since they have either non proper intersection or triple points. I_{0} is just a nonsingular elliptic curve, I_{1} is a rational curve with a node and I_{2} consists of two components intersecting transversally at two distinct points. m (resp. m^{(1)}) is the number of components (resp. components of multiplicity 1). The ε column indicates the Euler characteristics of the fibers f = ε − (m − 1) is the ”exponent in the conductor” in [Ta,§7] (we will not explain this here). J denotes the value of the functional invariant at the singular fiber. This part of the table is copied from [Ta,§6]. The 0 section cuts each singular fiber at one of the components with multiplicity 1. Let ∗ denote a fiber. We look at M_{∗}, The lattice spanned by the components of ∗ not intersecting the 0 section with the cup pairing. Then q_{∗} denotes the discriminant form of this lattice.

Remark 4.9. We notice that M (hence q) is the same for I_{1} or II, I_{2} or III and I_{3} or IV.

Remark 4.10. M_{∗}(−1)⊗ℝ is an Euclidean space and since the dual graph is always a Coxeter graph it follows that the components not meeting 0 form a basis for a root system in M_{∗}(−1) ⊗ ℝ in the sense of Lie-algebra theory [H, 10.].

Lemma 4.11. The discriminant forms corresponding to the singular fibers are as listed in table 1.

Proof. For the I_{b} type this is given in [Mi-Pe,4.3]. For the other fibers one sees this as follows. It is a result of Raynaud [Shi,1.3] that the order of A_{q_{∗}} equals m_{∗}^{(1)}. Moreover, for the I_{b}^{∗} fibers this group is cyclic of order 4 iff b is odd. Since M_{∗} is negative definite we know its signature hence that of q_{∗} (2.19). This information, together with (2.20), is sufficient to determine the discriminant form.

4.12. The following pairs of singular fibers are related by the fact that their Euler characteristics differ by 6 and the associated monodromy changes sign: I_{b} and I_{b}^{∗}, II and IV^{∗},III and III^{∗},IV and II^{∗}.

Theorem 4.13. ([Shi]) For an elliptic fibration over a ℙ^{1} base and with a section, the invariants of the singular fibers satisfy the following relations:

(1) The sum of the Euler characteristics equals the Euler characteristics of the surface and is therefore divisible by 12 (equal to 24 for a K3 surface).

(2) ∑_{v}(m_{v} − 1) + 2+ rank of the Mordell – Weil group = Picard number of the surface (always ≤ 20 for a K3 surface).

4.14. An elliptic surface may be described by a Weierstrass model

y^{2} = 4x^{3} − g_{2}(t)x − g_{3}(t)

where the projection is on the t coordinate. The discriminant and the functional invariant are given by

Δ = g_{2}^{3} − 27g_{3}^{2}, J = \frac{g_{2}^{3}}{Δ} .

To get from this the description of singular fibers, write g_{2}, g_{3} in terms of the local coordinate s = t − t_{0} in the neighborhood of t_{0}. The change of variables

y ⟶ c^{3}(s)y, x ⟶ c^{2}(s)x

transforms the equation to

y^{2} = 4x^{3} − c^{4}(s)g_{2}x − c^{6}(s)g_{3}.

One may use these transformations to find a model where g_{2}, g_{3} have no poles in t_{0} and the order of Δ is minimal there. The fiber at t_{0} is singular iff Δ vanishes there and the order of vanishing equals the Euler characteristics of the fiber. Using this and the behavior of J one can deduce the type of the fiber from table 1.

4.15. Let c(t) = (t − a)/(t − b) be some rational function of t of degree 1. The quadratic twist by c(t) has equation

c(t)y^{2} = 4x^{3} − g_{2}x − g_{3}

or equivalently

y^{2} = 4x^{3} − c^{2}(t)g_{2}x − c^{3}(t)g_{3}.

It is easily seen that this transformation switches the pairs of singular fibers in (4.12). It is an involutive operation.

Proposition 4.16. Let J be a non constant rational function on ℙ^{1}(ℂ). Let Σ ⊂ ℙ^{1}(ℂ) be a finite set containing all points s s.t. J(s) ∈ {0, 1, ∞}. Suppose we are given, for each s ∈ Σ, a type of a singular fiber which is compatible with the behavior of J at s and such that the sum of the Euler characteristics of the types is divisible by 12. Then there exists a unique elliptic fibration above ℙ^{1}(ℂ) having J as a functional invariant and having singular fibers of the given type at s ∈ Σ and no other singular fibers.

Proof. (For a possible different approach see [Mi]) J gives a monodromy representation into PSL(2, ℤ) given by a set of matrices A_{s}, s ∈ Σ determined up to sign, satisfying the relation ∏ A_{s} = ±I and either ±A_{s} is compatible with the type of fiber we want at s. We need only show that if we choose A_{s} to to compatible with the type we get a representation in SL(2, ℤ), i.e. ∏ A_{s} = I, since the result will then follow from (4.7). But if it is not we may change one A_{s} to −A_{s} and get an elliptic surface. The sum of the Euler characteristics differ by 6 from the previous one (4.12) hence not divisible by 12 which contradicts (4.13).

Proposition 4.17. Let there be given a list of types of singular fibers with exponents of conductors f_{i}, i = 1, …, n and sum E of Euler characteristics. If there exist an elliptic surface above a ℙ^{1} base with these singular fibers than the component of highest dimension in the moduli space of these surfaces is

∑ f_{i} − 2 − \frac{E}{6} − ( number of fibers of types II, III, IV)

Proof. By the above proposition we need only determine the dimension of the space of possible J maps plus a correction if there are I_{0}^{∗} fibers, a case which we will consider in the end. Let d be the degree of J. By the discussion before (2.3) in [Mi-Pe] the J map (up to J ∼ J ∘ α with α an automorphism of ℙ^{1}) is equivalent to

• The set of multiple values (branch points) {v_{1}, …, v_{k}} of J together with the multiplicities e_{j}.

• Monodromy permutation σ_{i} ∈ S_{d} of the loop around v_{i} on the the fiber above a generic point.

These should satisfy the obvious formula arising from the Hurwitz formula

∑(e_{j} − 1) = 2d − 2

and also some compatibility conditions on the permutations. These are never damaged if the number of ramification points increases. Thus, since the list of fibers give us information about the ramification at {0, 1, ∞}, it is clear that the dimension of the space of J maps equals the number of ramification points outside {0, 1, ∞}. Let d_{i} be the multiplicities in which J gets ∞ (coming from I_{d_{i}} and I_{d_{i}}^{∗} fibers). Let n_{v} (resp. n_{0,i}) be the number of fibers where J has value v (resp. has value 0 with multiplicity i (mod 3)). We note that if J gets the value 1 (resp. 0) at a nonsingular fiber it must do so with even multiplicity (resp. divisible by 3). It is now clear that the maximal possible number of ramification points outside {0, 1, ∞} is reached when all the multiplicities are minimal and equals

2d − 2 − ∑(d_{i} − 1) − n_{0,2} − \frac{d − 2n_{0,2} − n_{0,1}}{3} · 2 − \frac{d − n_{1}}{2} =

− \frac{d}{6} + n_{∞} + \frac{1}{3}n_{0,2} + \frac{2}{3}n_{0,1} + \frac{1}{2}n_{1} − 2 =

\frac{1}{6} (6n_{∞} + 2n_{0,2} + 4n_{0,1} + 3n_{1} − d) − 2 =

(n_{∞} + n_{0,1} + n_{0,2} + n_{1}) − 2 − \frac{1}{6} (4n_{0,2} + 2n_{0,1} + 3n_{1} − ∑ d_{i}) .

A look at table 1 now shows that in the final brackets each singular fiber contributes precisely its Euler characteristic or its Euler characteristic−6 if it is a ∗ fiber. Thus we may write the expression above as

|\{ non * fibers \}| + 2 · |\{ * fibers \}| − 2 − \frac{E}{6}

We now notice that the I_{0}^{∗} fiber fits this formula precisely. The final expression is a result of the values of f in table 1.

Remark 4.18. In the K3 case assume there are no singular fibers of type II, III or IV. Then, an easy computation using (4.13) shows the above dimension is equal to 20−Picard number of an elliptic surface with the prescribed singular fibers and torsion Mordell - Weil group. This is in accordance with the fact that the dimension of the space of K3 surfaces with Picard number ρ is 20 − ρ.

5. Elliptic fibrations of K3 surfaces and discriminant forms

The purpose of this section is to show how the knowledge of the invariants of the Néron – Severi lattice of a K3 surface allows one to prove the existence of nice elliptic fibrations of these surfaces and determine precisely the singular fibers. We are especially interested in fibrations where the Mordell – Weil group of sections has as small as possible rank. In the ideal case where this rank is generically 0, We can in fact characterize precisely the K3 surfaces with the corresponding invariants of the NS lattice as the set of elliptic surfaces with prescribed singular fibers (see the remark at the end of this section).

5.1. Let X be a K3 surface and let D be a fixed ample divisor on X. Denote by L the Néron – Severi lattice of X. Recall that this means that as a ℤ module L is just the Néron – Severi group of X and the bilinear form is induced from the cup product which is known to be even. Let Aut(L) denote the group of isometries of L. We will be careless and often write D both for a divisor and for its class in the Néron – Severi lattice.

Proposition 5.2. Let e ∈ L satisfy the following properties:

(1) e · e = 0.

(2) e · D > 0.

(3) e is primitive, i.e. e can’t be written as e = me′ for ±1 ≠ m ∈ ℤ and e′ ∈ L.

(4) e · D = min{|x · D|, x ∈ Aut(L) · e}. Then e is represented by the fiber of an elliptic fibration π : X ⟶ ℙ^{1}(ℂ).

Proof. This is a reformulation of some results in [PS-S]. 1) and Riemann - Roch implies (see [PS-S, §1 Lemma 2]) that either e or −e is represented by an effective divisor. By 2) this is e. We now show that e · C ≥ 0 for any effective divisor C. It is enough to assume C is an irreducible curve. Assume e · C is negative. Then clearly C · C < 0 and C is contained in any effective divisor representing e, say e = nC + E′ with n ∈ ℕ and E′ effective not containing C. Assume the genus of C is g. Since on a K3 surface the canonical divisor is trivial it follows from the adjunction formula that C ·C = 2g −2. Thus C ·C = −2. In this case the reflection s(x) = x + (C · x)C is in Aut(L). We compute

s(e) = e + (C · e)C = nC + E′ + (C · (nC + E′))C

= nC + E′ + (C · E′ − 2n)C = E′ − nC + (C · E′)C.

By assumption, C is not contained in E′ so C · E′ ≥ 0. In addition, D has positive intersection with every effective divisor since it is ample. Therefore

s(e) · D ≥ (E′ − nC) · D > (−E′ − nC) · D = −e · D.

On the other hand, since by assumption C · e < 0 we have

s(e) · D = (e + (C · e)C) · D < e · D.

We thus get |s(e) · D| < e · D which contradicts 4).

Now, by [PS-S §3 Theorem 1], e is represented by some mE with E the fiber of an elliptic fibration and m = 1 by 3).

Corollary 5.3. If U ⊂ L is an embedding of the hyperbolic plane (2.6) into an even lattice L, e and s ∈ U satisfy e^{2} = 0, e · s = 1 and s^{2} = −2 and D ∈ L is an element with D^{2} > 0, take an isometry γ of L s.t. |γ(e) · D| is minimal among all elements of Aut(L). Let e′ = ±γ(e), s′ = ±γ(s) s.t. e′ · D, e′ · s′ > 0, Then, any isomorphism L ≅ NS(X) between L and the Néron – Severi lattice of a K3 surface which sends D to the class of an ample divisor sends e′ to the fiber of an elliptic fibration of X, π : X ⟶ ℙ^{1}(ℂ), and s′ to a section of this fibration.

In the next result, we do not distinguish between the types of fibers, listed in (4.9), which have the same discriminant form.

Proposition 5.4. Let there be given a list of singular fibers. Let S be a direct sum of the lattices corresponding to the singular fibers in the list. Let S ↪ S′ be an embedding s.t. S′ is negative definite of rank n and any −2 class in S′ is contained in S. Let L be a lattice of signature (+1, 1 − n) having the same discriminant form as S′ and D ∈ L s.t. D^{2} > 0. Then there exist elements e, s ∈ L s.t.: If α : L ⟶ NS(X) is an isomorphism with α(D) the class of an ample divisor, then α(e) and α(s) are respectively the classes of a fiber and a section of an elliptic fibration having the above list of singular fibers and possibly I_{1} fibers.

Proof. The lattice S′ ⊕ U has the same discriminant form and signature as L. Since L is not definite it follows from (2.25.2) that S′ ≅ L and we therefore find an embedding of U into L. We now choose γ as in (5.3). Composing γ with the above isomorphism we get a new isomorphism S′ ⊕ U ≅ L hence a new embedding U ⊂ L and elements e, s ∈ U satisfying the required properties except that we haven’t checked the type of fibers. Since U is unimodular U^{⊥} ≅ S′. Let M be one direct summand in S corresponding to some singular fiber. M(−1) contains a root system R as remarked in (4.10). We have a standard basis for R where the matrix representing the form is precisely the intersection matrix of the rational curves in the corresponding singular fibers. We first show that we can find in M classes of irreducible rational curves satisfying the same intersection relations. Let ϕ be the functional on M(−1) obtained by intersection with the ample class D. We may slightly deform ϕ to ϕ′ in such a way that ϕ′ doesn’t vanish on R but has the same sign as ϕ on all elements where ϕ ≠ 0. By [H, 10.1] there is a basis of M(−1) corresponding to ϕ′ which is the set of elements on which ϕ′ is positive which cannot be written as the sum of two other elements of this form. By [H, 10.3] all bases of a root system can be transformed one to the other by an isometry. It follows that the elements of our base have square −2, hence are either effective or their minus is effective. It follows that ϕ doesn’t vanish on these classes hence they have a positive intersection with the ample class. They are therefore effective and by their definition the effective representatives are irreducible hence are rational nonsingular curves. Finally, these curves have the right intersection relations. These are necessarily all the rational components of singular fibers not meeting the 0 section since there are no other −2 classes orthogonal to U.

Remark 5.5. The number of I_{1} fibers can be determined using (4.13). Since the Euler characteristic of I_{1} is 1 and that of II is 2 we see that 1 II fiber might occur instead of 2 I_{1} fibers but (4.17) shows that in a family of fibrations the generic member has only I_{1} fibers. A similar remark applies to replacing a III (resp. IV) by I_{1} and a I_{2} (resp. I_{3}) fiber.

We recall (3.12) and give a more clear description in a simple case:

Corollary 5.6. if D = 2p or D = 3p with p a prime, The discriminant form of a generic Kummer surface of type (D, 1) is:

• q_{−2v}^{(p)}(p) ⊕ q_{3p}^{(2)}(8) ⊕ v_{+}^{(2)}(2) If D = 2p
• q_{−3v}^{(p)}(p) ⊕ q_{p}^{(3)}(3) ⊕ q_{−3p}^{(2)}(4) ⊕ u_{+}^{(2)}(2) If D = 3p

where, as in (3.5), v = v_{p} is a nonresidue modulo p

5.7. We now describe the equations that can be found using these results. We write for short X_{D} do denote a generic Kummer surface of type (D, 1)

D=6 The discriminant form q = q_{−1}^{(3)}(3) ⊕ q_{1}^{(2)}(8) ⊕ v_{+}^{(2)}(2) By table 1 this is the sum of the forms associated to the singular fibers I_{8}, I_{0}^{∗}, IV^{∗} . Thus, by (5.4), X_{6} has an elliptic fibration with singular fibers I_{8}, I_{0}^{∗}, IV^{∗}, 2I_{1}. Eliminating the 2 stars using (4.15) we get a fibration with singular fibers I_{8}, II, 2I_{1}. An explicit equation for this fibration, found by Herfurtner [He, p. 336], is

y^{2} = 4x^{3} − (12t(t^{3} − 6t^{2} + 15t − 12))x − 4t(2t^{5} − 18t^{4} + 72t^{3} − 144t^{2} + 135t − 27).

with the II fiber at t = 0. Thus, for a generic Kummer surface of type X_{6} there exists some a ∈ ℂ s.t. X_{6} has an elliptic fibration with equation

\frac{t + a}{t}y^{2} = 4x^{3} −(12t(t^{3} −6t^{2} +15t−12))x−4t(2t^{5} −18t^{4} +72t^{3} −144t^{2} +135t−27).

D=14 The discriminant form is q_{1}^{(7)}(7) ⊕ q_{5}^{(2)}(8) ⊕ v_{+}^{(2)}(2). We can get this lattice from the following configuration of singular fibers: I_{8}, I_{7}, 4I_{2}, I_{1}. Indeed, the discriminant form may be computed using 2.15.c.4& 8

q_{1}^{(7)}(7) ⊕ q_{1}^{(2)}(8) ⊕ (q_{−1}^{(2)}(2))^{4} = q_{1}^{(7)}(7) ⊕ q_{1}^{(2)}(8) ⊕ q_{−1}^{(2)}(2) ⊕ q_{5}^{(2)}(2) ⊕ v_{+}^{(2)}(2)
= q_{1}^{(7)}(7) ⊕ q_{5}^{(2)}(8) ⊕ q_{−1}^{(2)}(2) ⊕ q_{1}^{(2)}(2) ⊕ v_{+}^{(2)}(2)

and, using (2.22), we see that we can cancel the factor q_{−1}^{(2)}(2) ⊕ q_{1}^{(2)}(2) and find an overlattice with the required discriminant form. We find: X_{14} has an elliptic fibration with singular fibers I_{8}, I_{7}, 4I_{2}, I_{1}.

D=15 The discriminant form is

q_{−1}^{(3)}(3) ⊕ q_{1}^{(5)}(5) ⊕ q_{1}^{(2)}(4) ⊕ u_{+}^{(2)}(2) ≅ q_{−1}^{(3)}(3) ⊕ q_{1}^{(5)}(5) ⊕ q_{5}^{(2)}(4) ⊕ v_{+}^{(2)}(2)

using 2.15.c.5. From table 1 we see that this corresponds to the singular fibers IV^{∗}, I_{5}, I_{4}, I_{0}^{∗}, I_{1}. We drop the stars to get fibers II, I_{5}, I_{4}, I_{1}. This surface appears on Herfurtner’s list on page 337 and plugging his result we get: X_{15} has an equation \frac{t+a}{t−1/8}y^{2} = 4x^{3} − g_{2}(t)x − g_{3}(t) with

g_{2}(t) = 3(8t − 1)(8t^{3} + 87t^{2} + 96t − 64),
g_{3}(t) = (8t − 1)(64t^{5} + 2^{4} · 5 · 13t^{4} + 5^{2} · 157t^{3} + 100t^{2} + 2^{7} · 5^{2}t − 2^{9}).

In all three cases we obtained a connected 1-parameter family of elliptic surfaces It therefore follows that, conversely, each member in the family is isomorphic to a Kummer surface of the appropriate type.

Remark 5.8. That the family is 1-dimensional could have been predicted since by (4.13) we can see that the Mordell - Weil group of each fibration has rank 0 and then by using (4.18).

Remark 5.9. The Néron - Severi lattice of Kummer(A) is determined up to a unique isomorphism once we identify the Néron - Severi lattice A (which we have universally done in section 3) and fix a numbering of the exceptional curves hence a numbering of the 2-torsion points of A i.e. a level 2 structure. We therefore expect a map from the moduli curve of such objects to the ℙ^{1} base of our family of fibrations. The determination of this map seems nontrivial and we do not discuss it here.

6. Siegel modular threefold with full level 2 structure

This section reviews, with some additional remarks, The explicit projective embedding of the Siegel modular threefold , Γ_{2}(2)\S_{2} , by theta functions, found in [G].

6.1. Let

S_{2} = \{τ = (\begin{matrix}τ_{1} & τ_{2} \\ τ_{2} & τ_{3} \end{matrix}) ∈ M_{2}(ℂ)| Im(τ) > 0\}

be the Siegel upper half space of dimension three. The group

Sp(4, ℝ) = \{g ∈ GL(4, ℝ)|gJg^{t} = J\}, J = (\begin{matrix}0 & I_{2} \\ −I_{2} & 0 \end{matrix})

acts on S_{2} according to the rule

g(τ) = (Aτ + B)(Cτ + D)^{−1}, g = (\begin{matrix}A & B \\ C & D \end{matrix}) .

6.2. Let Γ_{2} = Sp(4, ℤ) and Γ_{2}(N) = Ker(Γ_{2} ⟶ Sp(4, ℤ/Nℤ)). The quotient space S_{2}^{∗}(N) = Γ_{2}(N)\S_{2} classifies principally polarized Abelian surfaces with a full level N structure. It is nonsingular if N > 1. One constructs a projective embedding of S_{2}^{∗}(N) called the Satake compactification which is defined by \overline{S}_{2}(N) = Proj(⊕_{k≥0}M_{k}(Γ_{2}(N))) where M_{k}(Γ_{2}(N)) is the space of Siegel modular forms of weight k on Γ_{2}(N). In the case N = 2, van der Geer constructed an explicit projective embedding using theta functions as follows:

6.3. Let m = (m_{1}, …, m_{4}) be a fourtuple whose entries are taken from the set {0, 1} which we identify with the field with 2 elements F_{2}. We call such a fourtuple a characteristic. We call the characteristic m even or odd if m_{1}m_{3} + m_{2}m_{4} is even or odd respectively.

Definition 6.4. The theta function associated to the characteristic m is the holomorphic function on S_{2} × ℂ^{2} defined by

θ_{m}(τ, ζ) = ∑_{ξ∈ℤ^{2}} e (\frac{1}{2} (ξ + \frac{m′}{2} ) τ (ξ + \frac{m′}{2} )^{t} + (ξ + \frac{m′}{2} ) (ζ + \frac{m′′}{2} )^{t})

with m′ = (m_{1}, m_{2}), m′′ = (m_{3}, m_{4}) and e(x) = e^{2πix}. The theta constant associated to m is θ_{m}(τ) = θ_{m}(τ, 0)

The parity of θ w.r.t. ζ is the same as that of m. In particular, the odd theta constants are 0. The 10 even theta constants, raised to fourth power, are Siegel forms of weight 2 for Γ_{2}(2). This means that if we let g ∈ Sp(4, ℝ) act on functions according to the rule

f^{γ}(τ) = det(Cτ + D)^{−2}f(γ(τ))

then for γ ∈ Γ_{2}(2) and any m we have (θ_{m}^{4})^{γ} = θ_{m}^{4}.

6.5. It follows from the above that there is an action of Γ_{2}/Γ_{2}(2) ≅ Sp(4, F_{2}) on the theta’s. We now describe this action. Let m = (m_{1}, …, m_{4}) be a characteristic. Associate to it the quadratic form on F_{2}^{4} defined by Q_{m}(x_{1}, …, x_{4}) = ∑_{i=1}^{4} m_{i}x_{i} + ∑_{i=1}^{2} x_{i}x_{i+2}. This association defines a bijection between the set of characteristics and the set of quadratic forms Q whose associated bilinear form b(x, y) = Q(x + y) + Q(x) + Q(y) is the simplectic form x_{1}x_{3} + x_{2}x_{4} on F_{2}^{4}. By embedding F_{2}^{+} into ℚ/2ℤ we get from Q a finite quadratic form in the sense of (2.7). F_{2}^{4} splits as a direct orthogonal sum of two 2-dimensional spaces and a simple computation, using (2.15), gives:

Lemma 6.6.

Q_{m} ≅ \begin{cases}(u_{+}^{(2)}(2))^{2}, & ifm is even; \\ u_{+}^{(2)}(2) ⊕ v_{+}^{(2)}(2), & if mis odd. \end{cases}

Since Sp(4, F_{2}) acts on the set of quadratic forms according to Q^{γ}(x) = Q(γ(x)) and preserves the simplectic form we find an action on the set of the characteristics which we write m ⟶ m^{γ}.

Theorem 6.7. ([Ig])

(θ_{m}^{4})^{γ} = ±θ_{m^{γ}}^{4} .

6.8. The parity of the characteristic is preserved by the above action because of (6.6). Since there are 6 odd characteristics and the order of Sp(4, F_{2}) is 720 we see that the action on the odd characteristics define an isomorphism of Sp(4, F_{2}) with S_{6}, the symmetric group on 6 letters. with this isomorphism, Igusa computed that the representation of S_{6} on the space generated by the 10 even theta function is irreducible of dimension 5 and corresponds to the partition 6 = 2 + 2 + 2.

6.9. There exists an outer automorphism for the group S_{6} (in fact, it’s the only symmetric group with this property). It can be constructed as follows: There are 15 partitions of the set {1, …, 6} into 3 disjoint pairs (ij)(kl)(mn). One checks that there are precisely 6 sets of partitions each being maximal w.r.t. the property: no two partitions in the set contain a common pair. Each set is of size 5. The action of S_{6} permutes these sets and labeling these 1′, …, 6′ gives the outer automorphism. A partition (ij)(kl)(mn) belongs to precisely two sets r′ and s′. This defines a bijection between the set of partitions into pairs of {1, …, 6} and the set of pairs in {1′, …, 6′}. It follows easily that the square of the outer automorphism we found is inner. In addition, one finds that it interchanges the conjugacy classes in S_{6} corresponding to the partitions 2+2+2 and 2+1+1+1+1,

6.10. A look at character tables [Ig, p. 400] now shows that twisting by the outer automorphism gives a representation corresponding to 5 + 1 (not 3 + 3 as is misprinted in [G, p. 337]). This representation has the following model: The space is

{(x_{1}, …, x_{6}) ∈ ℂ^{6}, x_{1} + ⋯ + x_{6} = 0}

and S_{6} acts by permuting the coordinates.

6.11. According to [G, p. 323] the boundary components in the Satake compactification \overline{S}_{2}(2) may be described as follows: let T be a rank 4 free ℤ module together with a unimodular simplectic form (i.e. T is isomorphic to ℤ^{4} and the simplectic form is x_{1} ∧ x_{3} + x_{2} ∧ x_{4} with x_{1}, …, x_{4} the coordinate functions). Denote this form by (·, ·). S_{2} may be described as the set of all complex structures J on T_{ℝ} s.t. (Jx, Jy) = (x, y) and the form (Jx, x) is positive definite. The action of Sp(4, ℝ) on S_{2} is induced from its action on T_{ℝ}. To compactify Γ\S_{2}, simply add the Γ equivalence classes of J s.t. (Jx, x) is positive semidefinite and vanishes on a rationally defined subspace U of T (ℝ). It then easily follows that U is isotropic. Thus, the 1 (resp. 0)-dimensional boundary components correspond bijectively to the Γ equivalence classes of 1 (resp. 2)-dimensional isotropic subspaces U of T_{ℚ}. The component corresponding to U is the set of equivalence classes of J s.t. (Jx, x) is positive semidefinite and vanishes on U.

6.12. Sp(4, ℤ) acts transitively on the set of 1 or 2-dimensional isotropic subspaces of T_{ℚ}. It is then clear that the orbits under Γ_{2}(2) correspond bijectively to 1 or 2-dimensional isotropic subspaces of T_{ℤ}/2T_{ℤ} ≅ F_{2}^{4}. We note that there are 15 1 (resp. 2)-dimensional such subspaces and that each 2-dimensional isotropic subspace contains 3 1-dimensional isotropic subspaces whereas each 1-dimensional isotropic subspace is contained in 3 2-dimensional ones.

There are 6 maximal sets, which we number 1 to 6, each with 5 elements, of 1-dimensional isotropic subspaces of F_{2}^{4} s.t. no two subspaces in the same set are contained in the same 2-dimensional isotropic subspace. The action of Sp(4, F_{2}) permutes these maximal sets and defines an isomorphism with S_{6}. In fact, if m is an odd characteristic and Q_{m} the corresponding quadratic form (6.5), one can check that the set

{span{x}, x ∈ F_{2}^{4} − {0} and Q_{m}(x) = 0}

is such a maximal set (it is of course enough to check one m) hence the isomorphism with S_{6} is identical to that found in (6.8) by the action on the odd characteristics. Each 1-dimensional isotropic subspace is contained in precisely 2 maximal sets i, and j, and we can index it by the pair (ij). A 2-dimensional isotropic subspace is indexed by a partition of {1, …, 6} into 3 pairs. It is now clear that twisting the isomorphism between Sp(4, F_{2}) and S_{6} by the outer automorphism of S_{6} described in (6.9) gives the isomorphism obtained by the action of Sp(4, F_{2}) on the set {1′, …, 6′} of maximal sets of 2-dimensional isotropic subspaces no two contain a common 1-dimensional subspace. With this description a 0-dimensional boundary component is indexed by a pair (i′j′) and a 1-dimensional component by a partition (i′j′)(k′l′)(m′n′).

Theorem 6.13. ([G, Theorem 5.2]) The ten even theta constants, θ_{m}^{4}, embed \overline{S}_{2}(2) as a threefold in ℙ^{5} defined by the equations:

s_{1} = 0, s_{2}^{2} = 4s_{4}

where

s_{k} = ∑_{i=1}^{6} x_{i}^{k}

is the k-th power sum of the homogeneous coordinates x_{1}, …, x_{6}.

6.14. The boundary components form the singular locus of this threefold. These are easily found [G, Theorem 4.1]:

• The 1-dimensional boundary components are in 1 − 1 correspondence with partitions of {1, …, 6} into 3 disjoint pairs (ij)(kl)(mn). The corresponding component has equations

x_{i} = x_{j} , x_{k} = x_{l} , x_{m} = x_{n} , x_{i} + x_{k} + x_{m} = 0

• The 0-dimensional boundary components are in 1 − 1 correspondence with pairs (ij). The corresponding point is the one having in homogeneous coordinates x_{i} = x_{j} = 2 and x_{k} = −1 for k other then i or j.

It is clear that the indexation of the boundary components here is identical to the one described at the end of (6.12).

7. Humbert surfaces and CM curves on some Siegel threefolds

In this section we continue considering a moduli space of principally polarized Abelian surfaces with some level structure and are interested in the subspaces cut out by conditions on the Néron – Severi lattice of these surfaces. If the condition implies this lattice is at least 2-dimensional we get a surface called a Humbert surface. If it implies the lattice is at least 3-dimensional we get a curve which we call a CM curve. We will see that, from a different point of view, these objects are cut by conditions on the ring of endomorphisms. In particular, CM curves are images of Shimura or modular curves which is precisely why we are interested in them. The main objective is to provide some necessary tools for computing equations of a few Humbert surfaces and CM curves in the next section.

7.1. Let (A, ζ) be an Abelian surface together with a principal polarization ζ ∈ H^{2}(A, ℤ). We can always choose H_{1}(A, ℤ)⟶^{∼}ℤ^{4} s.t. the induced α : H^{2}(A, ℤ)- ⟶^{∼} ⋀^{2}(ℤ^{4})^{∗} sends ζ to the simplectic form x_{1} ∧ x_{3} + x_{2} ∧ x_{4} with x_{1}, …, x_{4} the coordinates on ℤ^{4}. As in (6.11), we may fix a lattice T_{ℤ} ≅ ℤ^{4} with a simplectic form ζ and view S_{2} as the space of complex structures J on T_{ℤ} ≅ ℤ^{4} s.t. ζ is a (1, 1) form plus a positivity constraint: ζ(Jx, x) positive definite, and S_{2}^{∗} is obtained from it by dividing out by automorphisms of T_{ℤ} fixing ζ (i.e. Sp(4, ℤ)).

7.2. Let ζ ∈ M ⊂ ⋀^{2}(T_{ℤ})^{∗} be a sublattice containing the polarization. Let S_{M} ⊂ S_{2} be the space of complex structures for which all elements in M are of type (1, 1). Let S_{M}^{∗}(k) be the image of S_{M} in S_{2}^{∗}(k).

7.3. Let M be an even lattice containing a class ζ of square 2. The free Abelian group N = M/ℤζ carries a ℤ valued form (not necessarily even) derived from the cup product on ⋀^{2} T_{ℤ}^{∗}:

⟨x, y⟩ = −2 (x − (\frac{1}{2}x · ζ) ζ) · (y − (\frac{1}{2}y · ζ) ζ) .

Alternatively we may describe it by saying that N(−2) is isomorphic to the orthogonal complement of ζ in 2M + ℤζ. The map x ⟶ x · ζ (mod 2) is a well defined character of N to ℤ/2ℤ. The following lemma is easy:

Lemma 7.4. There is a 1−1 correspondence between even lattices M of rank n+1 with a class ζ of square 2 and lattices N of rank n representing only integers which are congruent to 0 or 1 modulo 4. The correspondence is given by:

• N = M/ℤζ with the form defined above.

• M is the overlattice of N(−2) ⊕ ℤζ containing all elements of the form \frac{1}{2}(x + par(x)ζ) with par the character defined in (2.2).

Definition 7.5. Let N be a 1 (resp. 2)-dimensional lattice. The Humbert surface (of level k) H_{N} (resp. CM curve C_{N}) is the union of all S_{M}^{∗}(k) with M ⊂ ⋀^{2} T_{ℤ}^{∗} a primitive sublattice corresponding to N under (7.4). Alternatively it is the subspace of S_{2}^{∗}(k) of moduli points of Abelian surfaces s.t. the lattice corresponding to their Néron - Severi lattices contain N primitively. The Humbert surface G_{N} (resp. CM curve C′_{N}) is the union of all H_{N′} (resp. C_{N′}) over all overlattices N′ of N. They may be given the same definition as H_{N}, C_{N} dropping the condition of primitivity.

Remark 7.6. A 1-dimensional N is characterized by the square of a generator which we therefore denote by Δ and will refer to H_{Δ} instead of H_{N} and similarly for the other objects.

Lemma 7.7. ([G1, IX 2.2]) There is a 1-1 correspondence between points in S_{2} and 3-dimensional subspaces of ⋀^{2} T_{ℤ}^{∗} ⊗ ℝ which are orthogonal to ζ and on which the form Q induced by the cup product is negative definite.

Proof. Points in S_{2} correspond to polarized Hodge structures of type (1, 0), (0, 1) on T_{ℤ}^{∗}. The induced Hodge structure on ⋀^{2} T_{ℤ}^{∗} is of type (2, 0), (1, 1), (0, 2) and we associate to it the subspace H^{1,1} ∩ ⋀^{2} T_{ℤ}^{∗} ⊗ ℝ ∩ ζ^{⊥}. That this space is of the right type is the Hodge index theorem. Conversely, given a subspace W with the required properties, define W′ = (W ⊕ ℝζ)^{⊥}. Q restricted to this 2-dimensional subspace is negative definite and it follows that Q ⊗ ℂ vanishes on precisely two different rays ℂv, ℂv̄ in W′ ⊗ ℂ. Frobenius lemma implies v = v′ ∧ v′′ and thus defines a 2-dimensional space U = span{v′, v′′} in T_{ℤ}^{∗} ⊗ ℂ. We have U ∩ Ū = {0} because we clearly have v · v̄ ≠ 0. Thus, U and Ū define Hodge structures of type (1, 0), (0, 1) We need to show that ζ defines a polarization for one of them. This means that the form i(u, ū) is positive definite on U or Ū. Let u ∈ U. i(u, ū) = ζ · iu ∧ ū. w = iu ∧ ū ∈ ⋀^{2} T_{ℤ}^{∗} ⊗ ℝ is orthogonal to W′ and therefore w ∈ W ⊕ ℝζ. Also Q(w) = 0. Thus w cannot be orthogonal to ζ. This means i(u, ū) is definite on U and since on Ū it has inverse sign we obtain the result.

Corollary 7.8. H_{N} (C_{N}) is nonempty iff N is positive definite.

Proof. Given N, we look at the corresponding even lattice M defined in (7.4). From (2.26) we see that M embeds in ⋀^{2} T_{ℤ}^{∗} and we may further assume that ζ maps to x_{1} ∧ x_{3} + x_{2} ∧ x_{4}. Now apply (7.7).

7.9. Let A be an Abelian surface whose moduli point is in some H_{N} or C_{N}. We want to compute End(A). Let ( , ) denote the simplectic form x_{1} ∧ x_{3} + x_{2} ∧ x_{4} corresponding to the polarization ζ. Let x ∈ NS(A) correspond to some simplectic form ( , )_{x} on T_{ℤ}. Then there exist a unique endomorphism T_{x} of T_{ℤ} s.t. (a, b)_{x} = (T_{x}a, b). Since x is of type (1, 1) it follows that T_{x} commutes with the complex structure and therefore defines an element T_{x} in End(A).

Proposition 7.10. If rank NS(A) ≤ 3 then End(A) is the subring of End(T_{ℤ}) generated by {T_{x}, x ∈ NS(A)}.

Proof. We represent everything using the simplectic basis of T_{ℤ}. ( , ) is represented by the matrix J = ( \begin{matrix}0 & I_{2} \\ −I_{2} & 0 \end{matrix} ) . T_{x} then corresponds to a matrix satisfying T_{x}^{t}J = JT_{x}. This condition makes ( , )_{x} simplectic. In block form we discover

T_{x} = (\begin{matrix}A & bw \\ cw & A^{t} \end{matrix}) with w = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , A = (\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}) .

In this case we have (·, ·)_{x} = a_{11}x_{1} ∧ x_{3} + a_{22}x_{2} ∧ x_{4} + ⋯ so x · ζ = tr(A). It is easily seen that tr(A) = 0 iff Aw = −wA^{t}. One can then easily compute that if x · ζ = 0 then 2T_{x}^{2} = x^{2} I. It follows that the algebra generated by the T_{x} over ℚ is precisely the Clifford algebra of the orthogonal complement to ζ in NS(A) ⊗ ℚ. Since one knows the types of algebras that can appear as endomorphism algebras of Abelian varieties and the corresponding Néron - Severi groups (over ℚ)([Mum],[G1, IX 1.2]) we can see by dimension considerations that the proposition holds after tensoring with ℚ. We are left with showing that the ring generated by the T_{x} sits primitively in M_{2}(ℤ). In the case that dim NS(A) = 2, End(A) is of dimension 2 as well and there is nothing to show. In the remaining case, since, for a vector space V of dimension 2 equipped with a quadratic form, the Clifford algebra is generated over ℚ by the unit, V and v_{1} · v_{2} for any two basis elements, it will be enough to show the following: If there is some relation of the form T_{x_{1}}T_{x_{2}} = T_{x} modulo a prime p then there is some relation modulo p between x_{1}, x_{2}, ζ. To see this, reduce everything mod p and write the relation as

(\begin{matrix}A & bw \\ cw & A^{t} \end{matrix}) = (\begin{matrix}A_{1} & b_{1}w \\ c_{1}w & A_{1}^{t} \end{matrix})·(\begin{matrix}A_{2} & b_{2}w \\ c_{2}w & A_{2}^{t} \end{matrix}) = (\begin{matrix}A_{1}A_{2} − b_{1}c_{2} & b_{2}A_{1}w + b_{1}wA_{2}^{t} \\ c_{1}wA_{2} + c_{2}A_{1}^{t}w & A_{1}^{t}A_{2}^{t} − c_{1}b_{2} \end{matrix}) .

Assume first p ≠ 2. then we can modify A_{1}, A_{2} by multiples of I_{2} to make them traceless. We then find:

(b_{2}A_{1} − b_{1}A_{2})w = bw , w(c_{1}A_{2} − c_{2}A_{1}) = cw

hence b_{2}A_{1} − b_{1}A_{2} = bI_{2} and c_{1}A_{2} − c_{2}A_{1} = cI_{2}. If not all b_{1}, b_{2}, c_{1}, c_{2} are 0 we find a relation between A_{1}, A_{2}, I_{2} and it is easily seen that the same relation holds between x_{1}, x_{2}, ζ. If all are 0 we easily find that A_{1}, A_{2} commute hence there is again a relation between A_{1}, A_{2}, I_{2}. In case p = 2 one can simply check all cases and arrive at the same conclusion.

Corollary 7.11.

(1) If A corresponds to a point in H_{Δ} then

End(A) ≅ O_{Δ} = ℤ ⊕ ℤ\frac{Δ + \sqrt{Δ}}{2}

the unique order of discriminant Δ in a quadratic extension.

(2) If A corresponds to a point in H_{N} then End(A) is an order in a rational quaternion algebra B unramified at ∞ whose discriminant is disc(N)/4. A prime p ramifies in B iff

N ⊗ ℤ_{p} ≅ K_{θ_{1}}^{(p)}(p^{k_{1}}) ⊕ K_{θ_{2}}^{(p)}(p^{k_{2}}) and (θ_{1}p^{k_{1}}, θ_{2}p^{k_{2}})_{p} = −1,

where ( , )_{p} is the local Hilbert symbol at p, or p = 2 and N ⊗ℤ_{2} ≅ V^{(2)}(2^{k}) with k even.

Proof. 1. According to (7.4) NS(A) is generated by ζ and \frac{1}{2}(x + x^{2}ζ) with x satisfying x · ζ = 0 and x^{2} = −2Δ. In the proof of (7.10) we saw T_{x}^{2} = Δ and End(A) is generated as a module over ℤ by 1 and \frac{1}{2}(T_{x} + Δ).

2. The statement is local. For p ≠ 2 we have N ⊗ ℤ_{p} ≅ K_{θ_{1}}^{(p)}(p^{k_{1}}) ⊕ K_{θ_{2}}^{(p)}(p^{k_{2}}). The corresponding order is generated by anticommuting elements T_{x} and T_{y} with T_{x}^{2} = θ_{1}p^{k_{1}}, T_{y}^{2} = θ_{2}p^{k_{2}}. The discriminant is then easily computed to have order k_{1} +k_{2} and [Vig,p. 32] shows the relation with the Hilbert symbol. In particular, at ∞ it shows B is unramified. The case p = 2 is the same with respect to ramification. The discriminant however has to be divided by 4. To see this, assume N ⊗ ℤ_{2} has generators x, y and is represented by the matrix ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}). The matrix of the form tr(zw′) on the basis given by 1, T_{x}, T_{y}, T_{x}T_{y} is diagonal with entries 2, −2a, −2b, 2ab so the discriminant of the order spanned by this basis is 4ab. However, End(A)⊗ℤ_{2} is generated by (T_{x} + a)/2, (T_{y} + b)/2 and therefore contains the above order with index 16. Finally, the ramification for the case N ⊗ ℤ_{2} = V^{(2)}(2^{k}) or U^{(2)}(2^{k}) can be easily checked by noting that V^{(2)}(1) (resp. U^{(2)}(1)) contains with index 2 K_{1}^{(2)}(2) ⊕ K_{3}^{(2)}(2) (resp. K_{1}^{(2)}(2) ⊕ K_{−1}^{(2)}(2).

We denote the order corresponding to N in 2. by M_{N}.

7.12. It follows from a theorem of Humbert [G, Lemma 2.1] that there is just 1 component in \overline{S}_{2}. Let J be a complex structure on T s.t. the corresponding moduli point is in H_{Δ}. If A is the corresponding Abelian surface we know that End(A) ≅ O_{Δ}, The order of discriminant Δ, and this order acts on H_{1}(A, ℤ) hence on T_{ℤ}. This gives T_{ℤ} the structure of a polarized O_{Δ} module i.e. a module with a simplectic form ( , ) s.t. (ax, y) = (x, ay) for a ∈ O_{Δ}. Humbert’s theorem implies that there is a unique polarized O_{Δ} module structure on T (this is easy if the order is maximal). It is thus isomorphic to the following module: T_{ℤ} ≅ O_{Δ} ⊕ O_{Δ}^{∗} and the form is

((a, b), (c, d)) = tr det (\begin{matrix}a & b \\ c & d \end{matrix}) .

Here ^{∗} is the dual w.r.t. the form tr(xy) and the trace is from ℚ(\sqrt{Δ}) to ℚ. H_{Δ} is thus the image in \overline{S}_{2} of the set of all complex structures on T (with some positivity constraint) which commute with the action of O_{Δ}. Denote this set by ℌ_{T}

Suppose now that J and J′ give rise to isomorphic polarized Abelian surfaces A and A′. The isomorphism f : A⟶^{∼}A′ induces an automorphism f_{∗} of T_{ℤ}. Since f sends End(A) to End(A′) we see that, for a generic A, f_{∗} is either O_{Δ} linear or antilinear (w.r.t. the nontrivial automorphism σ of ℚ(\sqrt{Δ})). Since σ induces an antilinear automorphism of T preserving the form we see that f_{∗} = γ · σ^{i}, i = 0 or 1 with γ O_{Δ} linear preserving the form. One easily sees that

γ ∈ SL(O_{Δ} ⊕ O_{Δ}^{∗}) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) , a, d ∈ O_{Δ}, c, b^{−1} ∈ O_{Δ}^{∗}\} .

Thus, the unique component of H_{Δ} in S_{2}^{∗} is isomorphic to (SL(O_{Δ}⊕O_{Δ}^{∗})∪SL(O_{Δ}⊕ O_{Δ}^{∗})σ)\ℌ_{T}.

7.13. We now describe the number of components of H_{Δ} in \overline{S}_{2}(2). points in \overline{S}_{2}(2) correspond to principally polarized Abelian surfaces with a full level 2 structure. This level 2 structure on an Abelian variety A is just an isomorphism between the group of 2-torsion points of A with the simplectic form induced from the polarization and F_{2}^{4} with the standard simplectic form. If A is obtained from T_{ℤ} and a complex structure J a polarization is just a level structure on T i.e. an isomorphism of F_{2}^{4} with T_{ℤ}/2T_{ℤ} preserving the simplectic form. These form a Sp(4, F_{2}) orbit hence there are precisely 720. Let F_{Δ} = O_{Δ}/2O_{Δ}. Then T_{ℤ}/2T_{ℤ} is a F_{Δ} module isomorphic to (F_{Δ})^{2}. For each level structure on T we obtain a component of H_{Δ} in \overline{S}_{2}(2) in the same way as before. Two components will be identical if there is an element in SL(O_{Δ} ⊕ O_{Δ}^{∗}) ∪ SL(O_{Δ} ⊕ O_{Δ}^{∗})σ relating the corresponding level structure. Reducing mod 2 we see that this will be iff the two level structures are in the same SL(2, F_{Δ}) ∪ SL(2, F_{Δ})σ orbit.

Since O_{Δ} is generated over ℤ by the element α = (Δ + \sqrt{Δ})/2 which satisfies the equation α^{2} − Δα + (Δ^{2} − Δ)/4 = 0 we have the following cases:

• 2|Δ - ramified case, F_{Δ} ≅ F_{2}[ε] with ε^{2} = 0. In this case σ acts trivially on F_{Δ} and the order of SL(2, F_{Δ}) is 48. There are therefore 720/48 = 15 components.

• Δ ≡ 1 (mod 8) - split case, F_{Δ} ≅ F_{2} ⊕ F_{2}. the order of SL(2, F_{Δ}) is 36 and thus there are 720/(2 · 36) = 10 components.

• Δ ≡ 5 (mod 8) - inert case, F_{Δ} ≅ F_{4}. the order of SL(2, F_{Δ}) is 60 and thus there are 720/(2 · 60) = 6 components.

7.14. For the CM curves the situation is similar. Assume for simplicity M = M_{N} is an Eichler order in a r.q.a. B. Then T has the structure of a M module and since any ideal of an Eichler order is principal it may be identified with M. The polarization is given, according to (3.1), by (x, y) = tr(vxy′) with v ∈ B_{0} s.t. N(x) = D^{−1} (D the discriminant of M). Let M_{0}^{∗} = B_{0} ∩ M^{∗}. N is then identified with M_{0}^{∗}/ℤv (again from (3.1)). Let Γ be the group of norm 1 elements in M. ℌ may be identified with the set of complex structures on M commuting with the left M action and positive w.r.t. the polarization. Γ\ℌ is identified with the quotient of this space by automorphisms of M module M preserving the polarization. Forgetting the M action we obtain a map into S_{2}^{∗}. The image is a component of C_{N}. We may put a level structure on M as before by fixing an isomorphism preserving the form between F_{2}^{4} and M/2M. The subgroup of elements of Γ preserving the level is Γ[2] = Γ ∩ (1 + 2M). We therefore get maps (corresponding to the choice of level structure) of Γ[2]\ℌ onto components of C_{N} in S_{2}^{∗}(2). It follows that C_{N} touches the boundary iff B is a matrix algebra.

We now compute the number of components. Repeating the considerations in (7.13) we first compute the group Aut(T) of automorphisms of T which are twisted M linear and fix the polarization. f ∈ Aut(T) may be written f(x) = α(x) u with α a ring automorphisms of M (viewed as a twisted M automorphism of T = M) and u a unit in M. Since we know how u acts on the polarization (3.10) we have the following immediate lemma

Lemma 7.15. f^{∗}(ζ) = ζ iff either α^{∗}(ζ) = ζ and N(u) = 1 or α^{∗}(ζ) = −ζ and N(u) = −1.

α^{∗} is an isometry of H^{2} and as we saw it will preserve the orthogonal complement of ζ. It therefore induces an isometry α′ of N. Conversely, each isometry α′ of N lifts to an automorphism of its Clifford algebra and induces the automorphism α on M_{N}. It can be computed that α^{∗}(ζ) = det(α′)ζ. We obtain an exact sequence

1 ⟶ M_{+}^{×} ⟶ Aut(T) ⟶ Aut_{0}(N) ⟶ 1

where Aut_{0}(N) is the group of isometries of N if M_{N} has elements of norm −1 and the group of isometries with determinant 1 otherwise. The projection is given by f ⟶ f^{∗} det(f^{∗}). The next step is to consider the reduction map Aut(T)⟶^{r} Aut(M/2M). The map M_{+}^{×} ⟶ (M/2M)^{×} is onto. Thus, on the image, we get a split exact sequence

1 ⟶ (M/2M)^{×} ⟶ r(Aut(T)) ⟶ (Aut_{0}(N)/ Aut_{0}(N)[2]) ⟶ 1.

Aut_{0}(N)[2] is the subgroup of isometries inducing the identity on M_{N}/2M_{N}.

Lemma 7.16. α′ ∈ Aut_{0}(N) belongs to Aut_{0}(N)[2] iff

α′(x) − x = 2y , y ∈ N and 2|y^{2} , all x ∈ N.

The condition need only be checked on a basis of N and the condition about y^{2} is automatic if 2|x^{2}.

Proof. Immediate from the relation between N and M_{N} in (7.4) and (7.10).

Once we have computed these objects, The following proposition is immediate from the above discussion:

Proposition 7.17. Assume M_{N} is an Eichler order. Then, in S_{2}^{∗}, C_{N} has only 1 component. The number of components in S_{2}^{∗}(2) is

720/(|(M/2M)^{×}|| Aut_{0}(N)/ Aut_{0}(N)[2]|)

and the map from the Shimura curve with level structure 2 to the component is generically | Aut_{0}(N)[2]| to 1.

Remark 7.18. |(M/2M)^{×}| is 12 if M_{N} ⊗ ℚ is ramified at 2, and 4 (resp. 6) if it is not and 2 divides (resp. doesn’t divide) the level of M_{N}.

7.19. The intersections of the components of H_{Δ} with the boundary components may be computed as follows: We assume for simplicity that Δ is not a square. Recall (6.11) that a complex structure J on T is in the boundary if the form (Jx, x) is positive semidefinite and vanishes on a rationally defined subspace U of T. If T has an O_{Δ} module structure and J commutes with the module structure then U is clearly a ℚ(\sqrt{Δ}) vector space. It is therefore of ℚ dimension 2. We recover in particular the result of [G, 2.3] that H_{Δ} does not intersect the 1-dimensional boundary components if Δ is not a square.

Fix now a level structure on T. From the above and (6.12) we see that the corresponding component of H_{Δ} contains the 0-dimensional boundary components corresponding to the 2-dimensional isotropic subspaces of T_{ℤ}/2T_{ℤ} ≅ (F_{Δ})^{2} which are reductions of 1-dimensional O_{Δ} modules (note that such a module is automatically isotropic). We describe 2 cases:

• In the ramified case there are 7 F_{Δ} ≅ F_{2}[ε] invariant isotropic 2-dimensional subspaces of (F_{2}[ε])^{2}. These are {(0, 0), (ε, 0), (0, ε), (ε, ε)} and the 6 subspaces generated over F_{2}[ε] by (1, 0),(0, 1),(1, 1),(1, ε),(ε, 1) and (1, 1 + ε). Of these, only the first one cannot be lifted to a 1-dimensional O_{Δ} modules because it is not generated by a single element over F_{Δ}. One checks that this subspace has a nontrivial intersection with each of the other subspaces. If this subspace is labeled (ij), it follows from (6.12) that the boundary components correspond to the 6 pairs (kl) disjoint from (ij).

• In the inert case, F_{Δ} is a field with 4 elements hence (F_{Δ})^{2} contains 5 1dimensional F_{Δ} vector spaces and the intersection of any two is {0}. This is thus a maximal set of 2-dimensional isotropic spaces as in (6.12). If it is labeled i then the elements in the set are labeled (ij) with j ≠ i.

We have proved:

Proposition 7.20. Assume Δ is not a square. Then:

(1) If 2|Δ then H_{Δ} ⊂ \overline{S}_{2}(2) has 15 components labeled (H_{Δ})_{(ij)}, 1 ≤ i < j ≤ 6. The component (H_{Δ})_{(ij)} contains the 6 0-dimensional boundary components labeled by (kl) with {i, j} ∩ {k, l} = ϕ.

(2) If Δ ≡ 5 (mod 8) then H_{Δ} ⊂ \overline{S}_{2}(2) has 6 components labeled (H_{Δ})_{i}, i = 1, …, 6. The component (H_{Δ})_{i} contains the 5 0-dimensional boundary components labeled by (ij) with j ≠ i

We make a final remark concerning the relation between Humbert surfaces and CM curves:

Lemma 7.21. If N is a 2-dimensional lattice and x, y ∈ N have either x^{2} ≠ y^{2} or x ± y ∉ 2N then each component of C_{N} in \overline{S}_{2}(2) is contained in the intersection of two distinct component of H_{x^{2}} and H_{y^{2}}.

Proof. Under the correspondence N ⟶ M (7.4) x, y map to x′ = (x + x^{2}ζ)/2 and y′ = (y + y^{2}ζ)/2. We have either (x′)^{2} ≠ (y′)^{2} or x′ − y′ ∉ 2M. We now embed M in ⋀^{2} T_{ℤ}^{∗} s.t. ζ goes to the polarization. The lattices spanned by ζ, x′ and ζ, y′ respectively define components of H_{x^{2}} and H_{y^{2}} which are clearly distinct because of the conditions on x′, y′ and their intersection contains the component of C_{N} corresponding to M.

Remark 7.22. Clearly, x and −x give the same component. As a result, using also (7.8), we obtain

Corollary 7.23.

(1) If Δ_{1} ≠ Δ_{2} then H_{Δ_{1}} ∩ H_{Δ_{2}} is composed of all C_{N} with N of one of the

following two types

(a) N has a basis \{e, e′\} in which it is represented by a matrix ( \begin{matrix}Δ_{1} & x \\ x & Δ_{2} \end{matrix} ) whose determinant is positive and divisible by 4.

(b) N is an overlattice of a lattice of type (a). in which e, e′ are primitive.

(2) If Δ_{1} = Δ_{2} = Δ the intersection of all non identical components of H_{Δ} consists of all C_{N} of type (a). and some of type (b).

The relation with universal families of Kummer surfaces comes from

Theorem 7.24. Let X be the model of (6.13) for \overline{S}_{2}(2). If p is a point on X − H_{1} then the tangent space at p to X intersects X in a (singular) Kummer surface K_{p}. The singular points are p and the intersection with the 15 singular lines. This induces a level structure on K_{p} and the moduli point is precisely p.

8. Computation of the equations

In this §we compute a few equations for Humbert surfaces and CM curves based on the computations in [G]. We work in the model (6.13). According to (7.14) the CM curves are in fact images of Shimura curves and (7.24) lets us write universal families of Kummer surfaces above these Shimura curves by pulling back the family above S_{2}^{∗}(2).

Theorem 8.1. [G, p. 348] In the above model we have the following equations of Humbert surfaces:

(1) H_{1} has 10 components given by equations x_{i} + x_{j} + x_{k} = 0 for triples {i, j, k} ⊂ {1, …, 6} (clearly the complement triple {1, …, 6} − {i, j, k} gives the same equation).

(2) H_{4} consists of 15 components (H_{4})_{(ij)} for 1 ≤ i < j ≤ 6. (H_{4})_{(ij)} has equation x_{i} = x_{j}. The 15 components are permuted by S_{6}.

(3) H_{5} consists of 6 components (H_{5})_{j}, j = 1, …, 6. Let ρ_{k,j} denote the kth elementary symmetric function in the 5 coordinates different from x_{j}. Then, (H_{5})_{j} has the equation 2ρ_{2,j} + ρ_{1,j}^{2} = 0.

An easy computation, using the equations in (6.13), gives the following alternative description of H_{5}:

Lemma 8.2. (H_{5})_{j} can be given by the equation 3x_{j}^{2} = s_{2}.

We now list some relations between 2 variable quadratic forms. We denote the forms by a 2 × 2 matrix, write the equivalence relation as ∼ and use ⊂ to indicate an overlattice.

Lemma 8.3.

(\begin{matrix}5 & 3 \\ 3 & 5 \end{matrix}) ∼ (\begin{matrix}5 & 2 \\ 2 & 4 \end{matrix}) ⊂ (\begin{matrix}5 & 1 \\ 1 & 1 \end{matrix}) ∼ (\begin{matrix}8 & 6 \\ 6 & 5 \end{matrix}) ∼ (\begin{matrix}4 & 0 \\ 0 & 1 \end{matrix})

∼ (\begin{matrix}8 & 2 \\ 2 & 1 \end{matrix}) ⊃ (\begin{matrix}8 & 4 \\ 4 & 4 \end{matrix}) (\begin{matrix}8 & 4 \\ 4 & 5 \end{matrix}) ∼ (\begin{matrix}5 & 1 \\ 1 & 5 \end{matrix}) (3)

8.4. By (7.23) H_{1} ∩ H_{4} consists of 1 CM curve corresponding to the matrix ( \begin{matrix}4 & 0 \\ 0 & 1 \end{matrix} ). Since the orbit of (say) x_{1} + x_{2} + x_{3} = 0, x_{1} = x_{4} is contained in the boundary, this must be the orbit of x_{1} + x_{2} + x_{3} = 0, x_{1} = x_{2}.

8.5. The intersection of all pairs of components of H_{5} consists of the 2 CM curves corresponding to the matrices ( \begin{matrix}5 & 3 \\ 3 & 5 \end{matrix} ) and ( \begin{matrix}5 & 1 \\ 1 & 5 \end{matrix} ). The first of these should occur in the intersection of H_{1} and H_{4} according to (8.3). Intersecting (H_{5})_{1} and (H_{5})_{2} we find from (8.2) the equations x_{1}^{2} = x_{2}^{2} = s_{2}/3. Thus we find two components corresponding to x_{1} = x_{2} or x_{1} = −x_{2}. The first one is contained in H_{4} hence cannot correspond to ( \begin{matrix}5 & 1 \\ 1 & 5 \end{matrix} ) which doesn’t represent 4. It therefore corresponds to

( \begin{matrix}5 & 3 \\ 3 & 5 \end{matrix} ). we find:

Theorem 8.6.

C( \begin{matrix}5 & 1 \\ 1 & 5 \end{matrix} ) is given by 3x_{i}^{2} = s_{2}, x_{i} = −x_{j}, 1 ≤ i < j ≤ 6.

(this curve is the image of a Shimura curve of discriminant 6 and level structure 2 by ( 7.11.2))

8.7. We compute the equation of H_{8}. By (8.1.3), it consists of 15 components (H_{8})_{(ij)} for 1 ≤ i < j ≤ 6. By [G p. 347] the total degree is 30 so each component has degree 2. The component (H_{8})_{(12)} is fixed by permutations fixing the set {1, 2} according to (7.20) and since s_{1} = 0 we can write the equation of this component as

a(x_{1} + x_{2})^{2} + bx_{1}x_{2} + c(x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2}) = 0 (4)

for some a, b, c. From (7.20) we know that this component contains all boundary 0 components corresponding to subsets {i, j} of {3, 4, 5, 6} and in particular the point with homogeneous coordinates {−1, −1, 2, −1, −1, 2}. Substituting this point we find the equation 4a + b + 10c = 0. To find another equation we proceed as follows: From (8.3) we see that the intersection of H_{8} and H_{5} contains the CM curve corresponding to ( \begin{matrix}5 & 1 \\ 1 & 5 \end{matrix} ). Applying permutations fixing \{1, 2\}, we see that a component of this curve inside (H_{8})_{(12)} has one of the following equations:i. x_{1} =

−x_{2}, 3x_{1}^{2} = s_{2}, ii. x_{3} = −x_{4}, 3x_{3}^{2} = s_{2}, iii. x_{1} = −x_{3}, 3x_{1}^{2} = s_{2}. Since ( \begin{matrix}5 & 1 \\ 1 & 5 \end{matrix} )

represents 8 only once, it follows from (7.21) that this curve should not be on any other component of H_{8}. The only curve satisfying this is i. This implies x_{1}^{2} = x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2} and substituting x_{1} = −x_{2} in (4) we find

0 = a(x_{1}+x_{2})^{2}+bx_{1}x_{2}+c(x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}) = −bx_{1}^{2}+c(x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}) = −bx_{1}^{2}+cx_{1}^{2} and therefore b = c. Combining this with the previous equation we see that we may set a = 11, b = c = −4. We find

Proposition 8.8. (H_{8})_{(12)} is given by the equation

11(x_{1} + x_{2})^{2} − 4x_{1}x_{2} − 4(x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2}) = 0.

Theorem 8.9. The images of the Shimura curve of discriminant 10 and full level structure 2 in \overline{S}_{2}(2) are given by the orbit of the curve cut by equations

x_{1} + 5x_{2} = 0, 3x_{1}^{2} = s_{2}

Proof. We are looking for the curve C( \begin{matrix}5 & 0 \\ 0 & 8 \end{matrix} ) since, according to (7.11.2), it is the required image. The intersection of H_{8} and H_{5} should contain this along with curves corresponding to ( \begin{matrix}5 & 4 \\ 4 & 8 \end{matrix} ),( \begin{matrix}5 & 6 \\ 6 & 8 \end{matrix} ) which we have already identified using (8.3), and ( \begin{matrix}5 & 2 \\ 2 & 8 \end{matrix} ) which corresponds to an order of discriminant 9 and hence must intersect the boundary. We get the above equation as a component of the intersection of (8.8) with 3x_{1}^{2} = s_{2} and one checks that all other components either intersect the boundary or were found before.

Theorem 8.10. The images of the Shimura curve of discriminant 15 and full level structure 2 in \overline{S}_{2}(2) are given by the orbit of the curve cut by equations

11(x_{1} + x_{2})^{2} − 4x_{1}x_{2} − 4(x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2}) = 0, 6x_{1} + 5x_{2} + 5x_{3} = 0 (5)

Proof. This curve is obtained as the CM curve C( \begin{matrix}8 & 2 \\ 2 & 8 \end{matrix} ). When we intersect the different components of H_{8} we get this together with the curves C( \begin{matrix}8 & 0 \\ 0 & 8 \end{matrix} ), C( \begin{matrix}8 & 4 \\ 4 & 8 \end{matrix} ) and C( \begin{matrix}8 & 6 \\ 6 & 8 \end{matrix} ). There are two different ways for intersecting the components corresponding to the orbits of (H_{8})_{(12)} ∩ (H_{8})_{(13)} and (H_{8})_{(12)} ∩ (H_{8})_{(34)}. Taking the difference of the equations defining (H_{8})_{(12)} and (H_{8})_{(13)} we find (x_{2} −x_{3})(6x_{1} +5x_{2} +5x_{3}) = 0. Thus, the first type of intersections contain 2 components and the one corresponding to x_{2} = x_{3} is contained in H_{4}. This must correspond to ( \begin{matrix}8 & 6 \\ 6 & 8 \end{matrix} ) since this is the only form representing 4. To finish we have to prove that C( \begin{matrix}8 & 0 \\ 0 & 8 \end{matrix} ) and C( \begin{matrix}8 & 4 \\ 4 & 8 \end{matrix} ) cannot be contained in this intersection. It is not hard to see that the curve defined by (5) doesn’t intersect the boundary. (7.11.2) implies ( \begin{matrix}8 & 0 \\ 0 & 8 \end{matrix} ) correspond to an order in a matrix algebra hence the corresponding curve must intersect the boundary. Finally, ( \begin{matrix}8 & 4 \\ 4 & 8 \end{matrix} ) represents 4 in precisely 3 ways and these are not congruent 1 to the other mod 2. This implies by (7.21) that the corresponding curve should be in precisely 3 different components of H_{8}. On the other hand, It corresponds to an order in a quaternion algebra hence doesn’t touch the boundary. One can check directly that (5) is not in any other component of H_{8}. It can also be argued as follows: It is certainly not in (H_{8})_{(23)}. But if it is in any other component an application of a permutation preserving {1, 2, 3} would give two more component of H_{8} in which it is contained.

Using the above theorems and (7.24) one can write the equations of the universal Kummer surfaces above the Shimura curves corresponding to discriminants 6, 10, 15 and full level structure 2. We also remark that a computation using (7.15-7.18) shows that we got the right number of components hence that the curves we found are irreducible. In addition it gives that for discriminant 6 the map from the Shimura curve to it’s image is generically 1 − 1.

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University of California, Department of Mathematics, 405 Hilgard Avenue, Los Angeles, California 90024-1555

E-mail address: besser@math.ucla.edu