j. differential geometry

65 (2003) 513-560

ANTI-HOLOMORPHIC MULTIPLICATION AND A REAL ALGEBRAIC MODULAR VARIETY

MARK GORESKY & YUNG SHENG TAI

Abstract

An anti-holomorphic multiplication by the integers O_{d} of a quadratic imaginary number field, on a principally polarized complex abelian variety A_{ℂ} is an action of O_{d} on A_{ℂ} such that the purely imaginary elements act in an anti-holomorphic manner. The coarse moduli space X_{ℝ} of such A (with appropriate level structure) is shown to consist of finitely many isomorphic connected components, each of which is an arithmetic quotient of the quaternionic Siegel space, that is, the symmetric space for the complex symplectic group. The moduli space X_{ℝ} is also identified as the fixed point set of a certain anti-holomorphic involution τ on the complex points X_{ℂ} of the Siegel moduli space of all principally polarized abelian varieties (with appropriate level structure). The Siegel moduli space X_{ℂ} admits a certain rational structure for which the involution τ is rationally defined. So the space X_{ℝ} admits the structure of a rationally defined, real algebraic variety.

1. Introduction

1.1

Let 𝔥_{n} = Sp(2n, ℝ)/U(n) be the Siegel upper half space of rank n and let Γ = Sp(2n, ℤ). The quotient Γ\𝔥_{n} has three remarkable properties:

(a) It has the structure of a quasi-projective complex algebraic variety.

(b) It is a coarse moduli space for principally polarized abelian varieties.

(c) It has a natural compactification (the Baily-Borel Satake compactification) which admits a model defined over the rational numbers.

The first author’s research was partially supported by NSF grant DMS-0139986. Received 05/30/2003.

Among the many missing ingredients in the theory of automorphic forms for groups of non-Hermitian type are analogues of these three facts. The associated locally symmetric spaces do not appear to have an algebraic structure; they do not appear to be associated with a natural class of elliptic curves or abelian varieties, and although they have many compactifications, there does not appear to be a canonical or “best” one. In the early 1970’s G. Shimura, J. Millson and M. Kuga [10] asked whether it might be possible to address these shortcomings by realizing a locally symmetric space W for a group of non-Hermitian type as a subspace of a locally symmetric space X for a group of Hermitian type; perhaps interpreting W as a moduli space of a class of real abelian varieties. These ideas were partially investigated by A. Adler [1], H. Jaffee [10], S. Kudla [12], K.-Y. Shih [22], and G. Shimura [24, 26, 27]. In [24], Shimura showed that results of this type cannot be expected in general. He found a moduli space X_{ℂ} (for a certain class of abelian varieties) which had a model defined over ℝ, such that the locus X_{ℝ} of real points did not represent a moduli space for the corresponding real abelian varieties.

1.2

We wish to revisit this question for quotients

W = Γ\Y_{n} (1.2.1)

of the symmetric space Y_{n} = Sp(2n, ℂ)/U(n, ℍ) (the “quaternionic Siegel space”, cf. §10) by the principal congruence subgroup

Γ = Sp(2n, O_{d})[M]

of Sp(2n, O_{d}) of level M. Here, d < 0 is a square-free integer and O_{d} is the ring of integers in the quadratic imaginary number field ℚ(\sqrt{d}). If M ≥ 3 then the space W is a smooth manifold of (real) dimension 2n^{2} + n. It does not have an (obvious) algebraic structure. In the case n = 1, W is an arithmetic quotient of the real hyperbolic 3-space, Y_{1}.

In this paper we show, for appropriate level M, that a certain disjoint union X_{ℝ} of finitely many copies of W admits analogs to all three of the above statements. That is:

(a) The smooth manifold X_{ℝ} is the set of fixed points of an antiholomorphic involution τ of a quasi-projective complex algebraic variety X_{ℂ}.

(b) The space X_{ℝ} may be naturally identified with the (coarse) moduli space of n-dimensional abelian varieties with level M structure and with anti-holomorphic multiplication (see below) by O_{d}.

(c) The complex variety X_{ℂ} and the involution τ admit a model that is defined over the rationals ℚ.

1.3

The algebraic variety X_{ℂ} is just the Siegel moduli space Γ(M)\𝔥_{2n} of principally polarized abelian varieties with level M structure. The involution τ extends to an anti-holomorphic involution of the Baily-Borel Satake compactification X of X_{ℂ} and hence defines a real structure on X. In §9 we make use of a result of Shimura [25] to prove an analogue of statement (c) above by showing that X admits a rational structure that is compatible with this real structure.

1.4

In this paper we introduce the concept of anti-holomorphic multiplication of the ring of integers O_{d} on a principally polarized abelian variety A: it is an action of O_{d} on A by real endomorphisms which are compatible with the polarization, such that the purely imaginary elements of O_{d} act in an anti-holomorphic manner; see §7.3 for the precise definition. If such an action exists then the (complex) dimension of A is even (so elliptic curves do not admit anti-holomorphic multiplication). The definition of anti-holomorphic multiplication extends in an obvious manner to more general CM fields, cf. §11.1. This appears to be a very interesting structure which merits further study.

1.5

The key technical tool in this paper, which appears to be a missing ingredient in the earlier work on this question, is Proposition 7.7, an analog of the lemma of Comessatti and Silhol ([29]). It describes “normal forms” for the period matrix of an abelian variety with anti-holomorphic multiplication. This in turn relies on a structure theorem (Proposition 6.4) for symplectic modules over a Dedekind ring.

1.6

The results in this paper, the parallel results for GL(n, ℝ) in [8], the paper [14], and recent results of [2] and [31] suggest that there are rich, largely unexplored phenomena involving real structures and moduli space interpretations of arithmetic quotients of non-Hermitian symmetric spaces. In this paper we have chosen particular arithmetic groups for which the results are (relatively) easy to state and prove, and for which the associated Shimura variety is defined over the rational numbers. Although it is possible to establish similar results for many other arithmetic groups, we do not know to what extent these results may be generalized to arbitrary arithmetic groups. (See also §11.)

1.7

We would like to thank J. van Hamel for useful conversations, and an anonymous referee for carefully reading the manuscript and making a number of useful suggestions. The first author is grateful to the Institute for Advanced Study for its hospitality and support during the period in which this research was carried out.

2. Statement of results

2.1

Throughout this paper we fix a square-free integer d < 0 and denote by O_{d} the ring of integers in the quadratic imaginary number field ℚ(\sqrt{d}). Let Q_{0} be the “standard” symplectic form, whose matrix is J = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) . For any ring R we use any of the standard notations Sp(2n, R), Sp(R^{2n}, Q_{0}), or Sp(R^{2n}, J) to denote the symplectic group consisting of all g ∈ GL(2n, R) such that ^{t}gJg = J, or equivalently, Q_{0}(gx, gy) = Q_{0}(x, y) for all x, y ∈ R^{2n}. It consists of matrices

g = (\begin{matrix}A & B \\ C & D \end{matrix})

such that

g^{−1} = (\begin{matrix}^{t}D & −^{t}C \\ −^{t}B & ^{t}A \end{matrix}) .

Let K_{s} ⊂ Sp(2n, ℂ) be the maximal compact subgroup that is fixed under the Cartan involution θ_{s}(g) = ^{t}(g)^{−1}. It is isomorphic to the unitary group U(n, ℍ) over the quaternions. Let Y_{n} = Sp(2n, ℂ)/K_{s} be the associated symmetric space. It is not compact and not Hermitian; for n = 1 it is the real hyperbolic 3-space. In §10 (which is not needed for the main results in this paper) we describe Y_{n} as a certain quaternionic Siegel space, on which Sp(2n, ℂ) acts by fractional linear transformations.

Let G = Sp(4n, ℝ) and let τ : G → G be the involution τ(g) = NgN^{−1} of (5.8.2). Then τ commutes with the Cartan involution θ : G → G of (5.8.2) so it passes to an (anti-holomorphic) involution (also denoted τ) on the (usual) Siegel space 𝔥_{2n} ≅ G/G^{θ}, which is given by (5.13.1), τ(Z) = 𝔟Z^{t}𝔟^{−1}. In §5.8 we describe an injective homomorphism

ϕ = Ψ ∘ ψ : Sp(2n, ℂ) → Sp(4n, ℝ)

such that ϕθ_{s} = θϕ and whose image is exactly the set of fixed points G^{τ} of τ. It has the property that

ϕ^{−1}(Sp(4n, ℤ)) = Sp(2n, O_{d}),

an arithmetic group that we will denote by Λ_{0}.

Lemma 2.2. The mapping ϕ passes to a closed embedding ϕ : Y_{n} → 𝔥_{2n} whose image is the set 𝔥_{2n}^{τ} of points fixed by the anti-holomorphic involution τ.

The proof appears in §5.15. Let Γ ⊂ Sp(4n, ℤ) be a torsion-free arithmetic subgroup which is preserved by the involution τ. If d ≡ 1 (mod 4) then assume also that Γ is contained in the principal congruence subgroup Γ(2) of level 2. Set X = Γ\𝔥_{2n} and let π : 𝔥_{2n} → X be the projection. Let Λ = ϕ^{−1}(Γ) ⊂ Sp(2n, ℂ) so that ϕ(Λ) = Γ^{τ} is the τ-invariants in Γ. Set W = Λ\Y_{n}. Then ϕ also passes to a closed embedding ϕ : W → X whose image is π(ϕ(Y_{n})). If h ∈ G we denote by hϕ(W) = π(hϕ(Y_{n})). Set

Γ̃ = {γ ∈ Sp(4n, ℤ) | τ(γ)γ^{−1} ∈ Γ} .

Let ⟨τ⟩ = {1, τ} be the group generated by τ and let H^{1}(⟨τ⟩, Γ) be the (nonabelian) cohomology of Γ.

Theorem 2.3. There is a canonical isomorphism

H^{1}(⟨τ⟩, Γ) ≅ Γ\Γ̃/ϕ(Λ_{0}). (2.3.1)

The involution τ passes to an anti-holomorphic involution τ : X → X and hence defines a real structure on X. The set of real points X_{ℝ} = X^{τ} is the disjoint union

X_{ℝ} = ∐_{h}^{h}ϕ(W) (2.3.2)

of finitely many disjoint translations of ϕ(W), indexed by

h ∈ Γ\Γ̃/ϕ(Λ_{0}) ≅ H^{1}(⟨τ⟩, Γ).

The proof is in §3.10. For the next two results we fix an integer M ≥ 3. If d ≡ 1 (mod 4) then we assume also that M is even. Denote by Γ(M) the principal congruence subgroup of level M in Sp(4n, ℤ). In the preceding theorem, take Γ = Γ_{M} = Γ(M) ∩ τ(Γ(M)). Then Equation (5.10.1) says that the arithmetic group Λ = ϕ^{−1}(Γ) is the principal congruence subgroup Sp(2n, O_{d})(M) of level M in the symplectic group over O_{d}.

Let X denote the Baily-Borel compactification of X = Γ_{M}\𝔥_{2n}. It carries the structure of a complex projective algebraic variety. In §9.9 we prove the following.

Theorem 2.4. The locus X_{ℝ} is the set of real points of a quasiprojective algebraic variety which has a model defined over the rational numbers. That is, there exists a holomorphic embedding in projective space X → ℙ^{m} such that the image of X is defined over the rational numbers ℚ, and such that the involution τ : X → X is the restriction of an anti-holomorphic involution τ : ℙ^{m} → ℙ^{m}, also defined over ℚ, which preserves X.

If A is an abelian variety with a principal polarization and a level M structure, an anti-holomorphic multiplication by O_{d} on A is a homomorphism O_{d} → End_{ℝ}(A) which is compatible with the polarization and level structures, such that \sqrt{d} acts as an anti-holomorphic mapping, cf. §7.3. In §8.6 we prove the following:

Theorem 2.5. The real algebraic variety X_{ℝ} may be canonically identified with the coarse moduli space of abelian varieties with principal polarization, level M structure, and anti-holomorphic multiplication by the ring O_{d}.

In summary, this coarse moduli space X_{ℝ} (of abelian varieties with anti-holomorphic multiplication) may be realized as the locus of real points of an algebraic variety defined over ℚ. It consists of finitely many isomorphic connected components, each of which is diffeomorphic to the arithmetic quotient (or locally symmetric space)

W = Sp(2n, O_{d})(M)\Sp(2n, ℂ)/U(n, ℍ).

3. Nonabelian cohomology

3.1

Let H be a group and let τ : H → H be an involution. Let ⟨τ⟩ be the group {1, τ} and let H^{1}(⟨τ⟩, H) be the first nonabelian cohomology set. For any γ ∈ H let f_{γ} : ⟨τ⟩ → H be the mapping f_{γ}(1) = 1 and f_{γ}(τ) = γ. Then f_{γ} is a 1-cocycle iff γτ(γ) = 1, in which case its cohomology class is denoted [f_{γ}]. Two cocycles f_{γ} and f_{γ′} are cohomologous iff there exists h ∈ H so that γ′ = τ(h)γh^{−1}.

Let G be a reductive algebraic group defined over ℝ, let θ be a Cartan involution with K = G^{θ} the maximal compact subgroup of θ-fixed points, and let E = G/K be the resulting symmetric space. Suppose τ : G → G is an involution which commutes with θ. Denote by G^{τ} , K^{τ} , and E^{τ} the corresponding fixed point sets in G, K, and E respectively. For notational simplicity we will often write ĝ for τ(g). The coboundary δ : E^{τ} → H^{1}(⟨τ⟩, K) may be defined as follows. If g ∈ G and if gK ∈ E^{τ} then τ(gK) = τ(g)K = gK so there exists k ∈ K so that

ĝ = gk. (3.1.1)

Applying τ to this equation gives g = gkk̂, hence k defines a 1-cocycle f_{k} = δ(gK).

Proposition 3.2. The cohomology sequence

1 ⟶ K^{τ} ⟶ G^{τ} ⟶ E^{τ} ⟶^{δ} H^{1}(⟨τ⟩, K) ⟶^{i} H^{1}(⟨τ⟩, G)

is exact. Moreover:

1. The mapping δ is trivial.

2. The mapping i is a bijection.

3. The inclusion G^{τ} ⊂ G induces a diffeomorphism

G^{τ} /K^{τ} ≅ E^{τ} .

3.3 Proof

Exactness of the cohomology sequence is standard ([21] §5.4). In the paragraph below we will show that δ is a locally constant mapping. Since τ acts by isometries, E^{τ} is connected and in fact the unique geodesic between any two points x, x′ ∈ E^{τ} is fixed under τ. It follows that δ takes E^{τ} to a single cohomology class which, taking g = 1 in Equation (3.1.1), is necessarily trivial. Hence ker(i) is trivial. It follows by “twisting” ([21] §5.3) that the mapping i is injective. Part (3) also follows: clearly the mapping G^{τ} /K^{τ} → E^{τ} is well-defined and injective; and part (1) guarantees that it is also surjective.

Now we will show that δ is locally constant. First observe that if k ∈ K is sufficiently close to the identity and if f_{k} is a 1-cocycle, then it is also a coboundary. For in this case we may write k = exp(k̇) where k̇ ∈ 𝔨 = Lie(K). Let τ′ : 𝔨 → 𝔨 be the differential of τ. From kτ(k) = 1 we obtain τ′(k̇) = −k̇. Then the element a = exp(−\frac{1}{2} k̇) satisfies τ(a)a^{−1} = a^{−2} = k which shows that the cohomology class defined by k is trivial.

Now suppose that gK, g_{0}K ∈ E^{τ} . Set τ(g) = gk and τ(g_{0}) = g_{0}k_{0} and let u = k_{0}^{−1}k. Since k̂ = k^{−1} and k̂_{0} = k_{0}^{−1} we find that uk_{0}^{−1}ûk_{0} = 1. This means that u ∈ K defines a 1-cocycle in H^{1}(⟨μ⟩, K) where μ : K → K is the involution μ(v) = k_{0}^{−1}v̂k_{0}. By the preceding paragraph, if g, g_{0} are sufficiently close then this cocycle gives the trivial cohomology class so there exists a ∈ K such that

u = μ(a)a^{−1} = k_{0}^{−1}âk_{0}a^{−1}

or k = âk_{0}a^{−1}. This says that the cocycles defined by k and by k_{0} are cohomologous, which completes the proof that δ is locally constant.

Finally it remains to be shown that i is surjective. In fact there is a splitting j : H^{1}(⟨τ⟩, G) → H^{1}(⟨τ⟩, K). Let G = KP be the Cartan decomposition of G that is determined by θ. Then P = exp(𝔭) where 𝔭 is the −1-eigenspace of θ on 𝔤 = Lie(G), so that θ(p) = p^{−1} ∈ P for all p ∈ P. Let g = kp ∈ G and suppose that f_{g} is a 1-cocycle. Then k̂p̂kp = 1 or

(k̂k)(k^{−1}p̂k) = p^{−1} ∈ P.

It follows from the Cartan decomposition that

k̂k = 1 and k^{−1}p̂kp = 1 (3.3.1) so we may define j(f_{g}) = f_{k}. We claim that f_{k} represents the same cohomology class as f_{g} in H^{1}(⟨τ⟩, G), from which it will follow that j is well-defined and that i is surjective. Equation (3.3.1) says that f_{p} is a 1-cocycle for the involution μ of G defined by μ(x) = k^{−1}x̂k. If p = exp ṗ, set a = exp(−ṗ/2). If μ′ : 𝔭 → 𝔭 denotes the differential of μ then μ′(ṗ) = −ṗ so μ(a) = a^{−1} and p = μ(a)a^{−1}. It follows that g = kp = kk^{−1}âka^{−1} = âka^{−1} which says that f_{g} and f_{k} are cohomologous.

q.e.d.

3.4

For the remainder of this section we assume G is a reductive algebraic group defined over ℚ, that θ is a Cartan involution of G, and that τ is an involution of G that commutes with θ. We often write ĝ for τ(g). Let G = G(ℝ) denote the group of real points, K = G^{θ} the corresponding maximal compact subgroup, and E = G/K the associated symmetric space. Fix an arithmetic subgroup Γ ⊂ G(ℚ) and let π : E → X = Γ\E be the projection.

To every 1-cocycle f_{γ} in H^{1}(⟨τ⟩, Γ) we associate the “γ-twisted” involutions τγ : E → E by x ↦ τ(γx) and τγ : Γ → Γ by γ′ ↦ τ(γγ′γ^{−1}). Let

E^{τγ} = {x ∈ E |τ(x) = γx} (3.4.1)

be the fixed point set in E of the involution τγ and let Γ^{τγ} be the fixed group in Γ of the involution τγ. Set X(τγ) = π(E^{τγ}). Recall the following theorem of Rohlfs ([17], [18], [19], [8]):

Theorem 3.5. Suppose Γ is torsion-free. Then the association f_{γ} ↦ X^{τγ} determines a one to one correspondence between the cohomology set H^{1}(⟨τ⟩, Γ) and the connected components of the fixed point set X^{τ} .

3.6 Proof

The twisted involution τγ : E → E acts by isometries so ([11] I §13.5) the fixed point set E^{τγ} is nonempty. If x, x′ ∈ E^{τγ} then the unique geodesic joining them is also fixed by τγ, so E^{τγ} is connected. Its image in X is a connected subset X(τγ) of X^{τ} which depends only on the cohomology class of f_{γ}. It is easy to check that f_{γ} and f_{γ′} are cohomologous iff X(τγ) ∩ X(τγ′) ≠ ϕ. q.e.d.

3.7

In general the cohomology set H^{1}(⟨τ⟩, Γ) may be difficult to compute, the connected component X(τγ) may be difficult to describe, and distinct connected components may fail to be isomorphic. We will introduce additional hypotheses which will allow us to address these three issues. Let Γ̃ ⊂ G(ℚ) be a τ stable arithmetic group that contains Γ. Let θ : G → G be the Cartan involution corresponding to K. Consider the following hypotheses:

(1) G is Zariski connected and the fixed subgroup G^{τ} = G^{τ} (ℝ) is Zariski connected.

(2) H^{1}(⟨τ⟩, K) is trivial.

(3) τ acts trivially on Γ\Γ̃.

(4) H^{1}(⟨τ⟩, Γ) → H^{1}(⟨τ⟩, Γ̃) is trivial.

(5) Γ is torsion-free.

Lemma 3.8. Suppose Γ ⊂ Γ̃ ⊂ G(ℚ) are τ-stable arithmetic subgroups. Suppose the Cartan involution θ commutes with τ. Then the following statements hold:

(a) Under hypothesis (1) above, G^{τ} is reductive, θ restricts to a Cartan involution of G^{τ} , and K^{τ} is a maximal compact subgroup of G^{τ} .

(b) Under hypothesis (2) above, the mapping G^{τ} /K^{τ} → E^{τ} is an isomorphism. If γ ∈ G and if f_{γ} is a 1-cocycle, then under hypothesis (2), its class in H^{1}(⟨τ⟩, G) is trivial if and only if E^{τγ} is nonempty.

(c) Under hypothesis (3) above, the association γ ↦ γ̂γ^{−1} defines a mapping Γ̃ → Γ which passes to an injection

Γ\Γ̃/Γ̃^{τ} → H^{1}(⟨τ⟩, Γ). (3.8.1)

Under hypotheses (3) and (4) this injection is a bijection.

(d) Under hypothesis (4), for each cohomology class

[f_{γ}] ∈ H^{1}(⟨τ⟩, Γ)

there exists h ∈ Γ̃ such that γ = τ(h)h^{−1}, in which case,

E^{τγ} = hE^{τ} and Γ^{τγ} = ^{h}Γ^{τ} = hΓ^{τ} h^{−1}.

(e) Under hypotheses (4), (3), and (5) the fixed point set X^{τ} is the disjoint union of isomorphic copies

X^{τ} = ∐_{h∈Γ\Γ̃/Γ̃^{τ}} ^{h}Γ^{τ} \hE^{τ}

of the quotient Γ^{τ} \E^{τ} .

In summary, if hypotheses (1) through (5) are satisfied, then the fixed point set X^{τ} consists of finitely many isomorphic copies of the arithmetic quotient Γ^{τ} \G^{τ} /K^{τ} indexed by H^{1}(⟨τ⟩, Γ) ≅ Γ\Γ̃/Γ̃^{τ} .

3.9 Proof

Part (a) is proven in [20] Chapt. 1 Thm. 4.2 and Cor. 4.5 (pages 15 and 17). Now consider part (b). Clearly G^{τ} /K^{τ} ⊂ E^{τ} so it suffices to show that G^{τ} acts transitively on E^{τ} . Let x = gK ∈ E^{τ} . Then τ(g)K = gK so the element k = g^{−1}τ(g) lies in K. Moreover, f_{k} is a cocycle, so by hypothesis (2) there exists u ∈ K with k = uτ(u)^{−1} = g^{−1}τ(g). Then gu ∈ G^{τ} and x = gK = guK. To prove the second statement in part (b), let f_{γ} be a 1-cocycle and suppose there exists a point gK ∈ E^{τγ}. Then γ̂ĝK = gK so there exists (a unique) k ∈ K with γg = ĝk, or γ = ĝkg^{−1}. Hence f_{γ} is cohomologous to f_{k}, which is trivial by (2). Part (c) follows from the long exact cohomology sequence for the groups Γ ⊂ Γ̃ and “twisting”, however it is also easy to verify directly. Let γ ∈ Γ̃. By (3) there exists a unique a ∈ Γ such that γ̂ = aγ. Moreover, f_{a} is a 1-cocycle, so we have defined a mapping ϕ : Γ̃ → H^{1}(⟨τ⟩, Γ). Suppose γ′ ∈ Γ̃ determines the same cohomology class ϕ(γ), that is, suppose a′ = γ′(γ′)^{−1} = b̂ab^{−1} for some b ∈ Γ. Let x = γ^{−1}b^{−1}γ′. Then x ∈ Γ̃^{τ} because x̂ = γ^{−1}b̂^{−1}γ′ = γ^{−1}b̂^{−1}b̂ab^{−1}γ′ = γ^{−1}a^{−1}ab^{−1}γ′ = x. Consequently γ′ ∈ ΓγΓ̃^{τ} , which verifies the injectivity statement. Hypothesis (4) immediately implies that ϕ is surjective, which proves (c). Part (d) is straightforward. Part (e) follows from Rohlfs’ theorem and parts (a)-(d). q.e.d.

3.10 Proof of Theorem 2.3

This follows from Lemma 3.8 provided we can verify hypotheses (1) through (5) of §3.7. Of these, (1), (3) and (5) are obvious. Hypothesis (2) will be proven in Proposition 5.11 and hypothesis (4) will be proven in Proposition 6.10. q.e.d.

4. Remarks on involutions

4.1

Let V be a real vector space with a symplectic form S and a nondegenerate symmetric bilinear form R and let Sp(V, S) be the group of linear automorphisms of V that preserve S. Let V^{∗} be the dual vector space. If g ∈ GL(V) define g^{∗} ∈ GL(V^{∗}) by g^{∗}(λ)(v) = λ(g^{−1}v) for any λ ∈ V^{∗}. Let N ∈ GL(V) and suppose that N^{2} = dI for some real number d. Define automorphisms τ and θ of GL(V) by

τ(g) = NgN^{−1} and R(gu, v) = R(u, θ(g)^{−1}v)

(for all u, v ∈ V). Then τ and θ are involutions, and in fact θ is a Cartan involution: its fixed point set is the orthogonal group O(V, R). Define S^{♯} : V → V^{∗} by S^{♯}(u)(v) = S(u, v). Let S^{♭} : V^{∗} → V be the mapping that is uniquely determined by the relation S(S^{♭}(λ), x) = λ(x) for any λ ∈ V^{∗} and x ∈ V. Then S^{♭} = (S^{♯})^{−1} . Define R^{♯}, R^{♭} similarly.

Lemma 4.2. The following statements hold:

1. If S(Nu, v) = S(u, Nv) (all u, v ∈ V) then τ preserves Sp(V, S).

2. If R(Nu, v) = −R(u, Nv) (all u, v ∈ V) then τθ = θτ.

3. If R^{♭}S^{♯}R^{♭}S^{♯} = cI is a multiple of the identity, then the involution θ preserves the symplectic group Sp(V, S) and its restriction to Sp(V, S) is a Cartan involution.

4.3 Proof

Part (1) is straightforward. For part (2), compute

R(Nθ(g)N^{−1}u, v) = R(u, Ng^{−1}N^{−1}v) = R(θ(NgN^{−1})u, v),

so τθ(g) = θτ(g). For part (3) consider the following diagram:

V ⟶^{S^{♯}} V^{∗} ⟶^{R^{♭}} V ⟶^{S^{♯}} V^{∗} ⟶^{R^{♭}} V
g↓  ^{g^{∗}}↓  ↓^{θ(g)}  ↓^{θ(g)^{∗}}  ↓^{g}
V ⟶_{S^{♯}} V^{∗} ⟶_{R^{♭}} V ⟶_{S^{♯}} V^{∗} ⟶_{R^{♭}} V.

The second and fourth square commute for every g ∈ GL(V). The first square commutes iff g ∈ Sp(V, S) (which we assume). The outside rectangle commutes by hypothesis. It follows that the third square also commutes, but this is equivalent to the statement that θ(g) ∈ Sp(V, S). Finally, it follows from [20] I Thm. 4.2 that the restriction of θ to Sp(V, S) is also a Cartan involution. q.e.d.

5. An involution on the symplectic group

5.1

In this section we construct an involution τ on Sp(4n, ℝ) which preserves a certain maximal compact subgroup K and which passes to an involution τ on the Siegel space. This involution is first constructed in a coordinate-free manner, but with respect to a non-standard symplectic form (S_{2}), and is denoted τ_{h}, see Lemma 5.6. Then we change coordinates so as to convert S_{2} to the usual symplectic form, and obtain the involution τ. The impatient reader may skip directly to the matrix descriptions (5.8.2) and (5.13.1), which could be used as an (unmotivated) definition of τ.

5.2 The number field

Throughout this paper we fix a square-free integer d < 0 and choose a square root, \sqrt{d}. Let O_{d} be the ring of integers in the quadratic imaginary number field ℚ(\sqrt{d}), that is, O_{d} = ℤ + ℤω where ω = \sqrt{d} if d ≢ 1 (mod 4) and ω = (1 + \sqrt{d})/2 if d ≡ 1 (mod 4). Let h : ℂ^{r} → ℝ^{2r} be the vector space isomorphism

h(x_{1} + ωy_{1}, x_{2} + ωy_{2}, …, x_{r} + ωy_{r}) = (x_{1}, y_{1}, x_{2}, y_{2}, …, x_{r}, y_{r}).

Then there is a unique homomorphism

ψ_{r} : M_{r×r}(ℂ) → M_{2r×2r}(ℝ). (5.2.1)

such that h(gz) = ψ_{r}(g)h(z) for all z ∈ ℂ^{r}. It takes the matrix

(a_{ij} + ωb_{ij})

(1 ≤ i, j ≤ r; a_{ij}, b_{ij} ∈ ℂ) to the matrix that consists of 2×2 blocks z_{ij} = ψ_{1}(a_{ij} + ωb_{ij}) where

ψ_{1}(a + ωb) = (\begin{matrix}a & db \\ b & a \end{matrix}) or (\begin{matrix}a & mb \\ b & a + b \end{matrix}) (5.2.2)

when d ≢ 1 (mod 4) or d = 4m + 1 respectively.

The complex linear mapping ℂ^{r} → ℂ^{r} given by multiplication by \sqrt{d} therefore corresponds to a real linear mapping N_{h} = ψ_{r}(\sqrt{d}I) : ℝ^{2r} → ℝ^{2r}. As in §4 define the involution τ_{h} : GL(2r, ℝ) → GL(2r, ℝ) by

τ_{h}(g) = N_{h}gN_{h}^{−1}. (5.2.3)

5.3

Take r = 2n. Fix a complex symplectic form s : ℂ^{2n} × ℂ^{2n} → ℂ. Using the isomorphism h we obtain a bilinear mapping S : ℝ^{4n} × ℝ^{4n} → ℝ^{2} whose components we denote by S_{1} and S_{2}, that is, s(h^{−1}u, h^{−1}v) = S_{1}(u, v) + ωS_{2}(u, v) for all u, v ∈ ℝ^{2n}. Then S_{1} and S_{2} are (real) symplectic forms on ℝ^{4n} and we denote by Sp(ℝ^{4n}, S_{i}) the corresponding symplectic groups. Since s(\sqrt{d}x, y) = s(x, \sqrt{d}y) = \sqrt{d}s(x, y) we have

(\begin{matrix}S_{1}(N_{h}x, y) \\ S_{2}(N_{h}x, y) \end{matrix}) = (\begin{matrix}S_{1}(x, N_{h}y) \\ S_{2}(x, N_{h}y) \end{matrix}) = β (\begin{matrix}S_{1}(x, y) \\ S_{2}(x, y) \end{matrix}) (5.3.1)

for all x, y ∈ ℝ^{4n}, where β is the matrix for N_{h} for r = 1; see §5.7. Using (5.3.1) it is easy to see that:

Lemma 5.4. The mapping ψ_{2n} restricts to an isomorphism

Sp(ℂ^{2n}, s) → Sp(ℝ^{4n}, S_{1}) ∩ Sp(ℝ^{4n}, S_{2}). (5.4.1)

For i = 1, 2 the involution τ_{h} preserves the group Sp(ℝ^{4n}, S_{i}). The subgroup of Sp(ℝ^{4n}, S_{i}) that is fixed by this involution is exactly the intersection (5.4.1).

5.5 Choice of s and θ

Let us take s to be the standard symplectic form Q_{0} on ℂ^{2n} whose matrix is J = ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ); we write Sp(2n, ℂ) = Sp(ℂ^{2n}, Q_{0}). We obtain symplectic forms S_{1} and S_{2} on ℝ^{4n} by §5.3. Take G = Sp(ℝ^{4n}, S_{2}).

Let H be the positive definite Hermitian form on ℂ^{2n} given by

H(z, w) = z · w.

Use the isomorphism h to convert H into a mapping ℝ^{4n}×ℝ^{4n} → ℂ and let R_{h} and S_{h} be the real and imaginary parts of this bilinear mapping, that is, H(h^{−1}x, h^{−1}y) = R_{h}(x, y)+iS_{h}(x, y). By §4 the positive definite form R_{h} determines a Cartan involution θ_{h} on GL(4n, ℝ).

Lemma 5.6. The involutions θ_{h} and τ_{h} commute. Both τ_{h} and θ_{h} preserve the symplectic group Sp(ℝ^{4n}, S_{i}) (for i = 1, 2).

Consequently θ_{h} restricts to a Cartan involution θ_{h} on

G_{h} = Sp(ℝ^{4n}, S_{2})

(resp. θ_{s} on G_{s} = Sp(2n, ℂ)) ([20] I, Thm. 4.2). The fixed point set K_{h} = G_{h}^{θ_{h}} (resp. K_{s} = G_{s}^{θ_{s}}) is a maximal compact subgroup of G_{h} (resp. of G_{s}). The involution τ_{h} passes to an involution τ_{h} of the symmetric space D_{h} = G_{h}/K_{h}.

The proof of Lemma 5.6 consists of verifying the conditions (1), (2), and (3) of Lemma 4.2 (for S = S_{i}), which amount to several calculations with matrices (cf. §5.7). q.e.d.

5.7 Matrix descriptions

If e is a k × k matrix let

Diag^{n}(e) = Diag(e, e, …, e)

be the nk × nk matrix with n identical diagonal blocks, each consisting of e. Let Sp(e) be the 2k × 2k matrix ( \begin{matrix}0 & e \\ −e & 0 \end{matrix} ) . We shall use the following 2 × 2 matrices.

symbol d ≢ 1 (mod 4) d = 4m + 1
β  ( \begin{matrix}0 & d \\ 1 & 0 \end{matrix} )  ( \begin{matrix}−1 & 2m \\ 2 & 1 \end{matrix} )
μ  ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix} )  ( \begin{matrix}1 & 0 \\ 0 & m \end{matrix})
ν  ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )  ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} )
r  ( \begin{matrix}1 & 0 \\ 0 & −d \end{matrix} )  ( \begin{matrix}1 & 1/2 \\ 1/2 & −m \end{matrix} )
r′  ( \begin{matrix}−d & 0 \\ 0 & 1 \end{matrix} )  ( \begin{matrix}−m & −1/2 \\ −1/2 & 1 \end{matrix} )

Set 𝔟 = Diag^{n}(β), 𝔲 = Diag^{n}(μ), 𝔳 = Diag^{n}(ν), 𝔯 = Diag^{n}(r), and 𝔯′ = Diag^{n}(r′). The bilinear forms R_{h}, S_{1}, and S_{2} have matrices R_{h} = Diag(𝔯, 𝔯), S_{1} = Sp(𝔲), and S_{2} = Sp(𝔳). The matrix for N_{h} is N_{h} = Diag(𝔟, 𝔟). The Cartan involution on GL(4n, ℝ) is θ_{h}(g) = R_{h}^{−1} ^{t}g^{−1}R_{h}. To prove Lemma 5.6 it is necessary to verify condition (3) of Lemma 4.2, which amounts to checking that R_{h}^{−1}S_{i}R_{h}^{−1}S_{i} = const · I_{4n} for i = 1, 2, a task which may be safely assigned to the undergraduate assistant.

5.8

The symplectic form S_{2} is integrally equivalent to the standard symplectic form Q_{0} whose matrix is

J = J_{2n} = (\begin{matrix}0 & I_{2n} \\ −I_{2n} & 0 \end{matrix}) .

which is easier to compute with. An isomorphism

Ψ : Sp(ℝ^{4n}, S_{2}) → Sp(4n, ℝ) = Sp(ℝ^{4n}, Q_{0})

is given by Ψ(g) = T gT^{−1} where

T = (\begin{matrix}I_{2n} & 0 \\ 0 & Diag^{n}(ν) \end{matrix})

so that

Q_{0}(T x, T y) = S_{2}(x, y) for all x, y ∈ ℝ^{4n}. (5.8.1)

Using the isomorphism Ψ the involutions τ_{h}, θ_{h} become the following involutions τ, θ on Sp(4n, ℝ) = Sp(ℝ^{4n}, Q_{0}):

τ(g) = NgN^{−1} and θ(g) = R^{−1} ^{t}g^{−1}R (5.8.2)

where

N = T^{−1}N_{h}T = (\begin{matrix}𝔟 & 0 \\ 0 & ^{t}𝔟 \end{matrix}) and R = T^{−1}R_{h}T = (\begin{matrix}𝔯 & 0 \\ 0 & 𝔯′ \end{matrix}) . (5.8.3)

In particular, it follows from (5.3.1) that

Q_{0}(Nx, y) = Q_{0}(x, Ny) for all x, y ∈ ℝ^{4n}. (5.8.4)

The involution τ preserves the maximal compact subgroup K that is fixed by the Cartan involution θ on Sp(4n, ℝ). The induced mappings on the symmetric space can also be explicitly described. The symmetric space for Sp(4n, ℝ) may be identified with the Siegel space

𝔥_{2n} = {Z ∈ M_{2n×2n}(ℂ) | ^{t}Z = Z, Im(Z) > 0} (5.8.5)

on which g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(4n, ℝ) acts by fractional linear transformations, g · Z = (AZ + B)(CZ + D)^{−1}. The maximal compact subgroup K is the stabilizer of the following basepoint

x_{1} = \frac{i}{\sqrt{−d}}Diag^{n} (\begin{matrix}−d & 0 \\ 0 & 1 \end{matrix}) or x_{1} = \frac{i}{\sqrt{−d}}Diag^{n} (\begin{matrix}−2m & −1 \\ −1 & 2 \end{matrix})

if d ≢ 1 (mod 4) or if d = 4m + 1 respectively. The symmetric space for Sp(ℝ^{4n}, S_{2}) may be identified with

𝔥_{2n}𝔳 = {Z𝔳 ∈ M_{2n×2n}(ℂ)| Z ∈ 𝔥_{2n}} (5.8.6)

on which g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(ℝ^{4n}, S_{2}) acts by fractional linear transformations, g · Z𝔳 = (AZ𝔳 + B)(CZ𝔳 + D)^{−1}. The mapping Ψ passes to a mapping which we also denote by Ψ : 𝔥_{2n}𝔳 → 𝔥_{2n} and which is given by Ψ(W) = W 𝔳^{−1}. Then Ψ(gW) = Ψ(g)Ψ(W) for all g ∈ Sp(ℝ^{4n}, S_{2}) and all W ∈ 𝔥_{2n}𝔳. The mapping ψ : Y_{n} → 𝔥_{2n}𝔳 will be described in §10. In summary we have a commutative diagram, the last line of which provides the names of the involutions associated with a given column:

K_{s}  ⟶  K_{h}  ⟶  K
↓  ↓  ↓
Sp(2n, ℂ) ⟶^{ψ} Sp(ℝ^{4n}, S_{2}) ⟶^{Ψ} Sp(ℝ^{4n}, Q_{0})
↓  ↓  ↓
Y_{n}  ⟶^{ψ}  𝔥_{2n}𝔳  ⟶^{Ψ}  𝔥_{2n}.
θ_{s}  θ_{h}, τ_{h}  θ, τ

Definition 5.9. Define ϕ : Sp(2n, ℂ) → Sp(4n, ℝ) and ϕ : Y_{n} → 𝔥_{2n} to be ϕ = Ψ ∘ ψ in the above diagram.

5.10 Remark

Let M be a nonzero integer and let GL(4n, ℤ)[M] be the principal congruence subgroup of level M. It follows from (5.2.2) that

ψ^{−1}(GL(4n, ℤ)[M]) = Sp(2n, O_{d})[M]

is the principal congruence subgroup of level M. Since Ψ ∈ GL(4n, ℤ) it also follows that

ϕ^{−1}(Sp(4n, ℤ)[M]) = Sp(2n, O_{d})[M]. (5.10.1)

Proposition 5.11. The nonabelian cohomology sets H^{1}(⟨τ⟩, K) and H^{1}(⟨τ⟩, Sp(4n, ℝ)) are both trivial.

5.12 Proof

By Proposition 3.2 it suffices to show that H^{1}(⟨τ⟩, K) is trivial. Let 𝔟_{0} = Diag^{n} ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) . Then 𝔟_{0}^{−1} = ^{t}𝔟_{0} = −𝔟_{0}. We claim there exists is an isomorphism Φ : (K, τ) ≅ (U(2n), τ′) where τ′(u) = 𝔟_{0}u𝔟_{0}^{−1} for all u ∈ U(2n). Assuming the claim for the moment, let us prove that H^{1}(⟨τ′⟩, U(2n)) is trivial. Let μ : GL(2n, ℂ) → GL(2n, ℂ) be the involution μ(A) = 𝔟_{0} ^{t}A^{−1}𝔟_{0}^{−1}. Its restriction to U(2n) coincides with τ′. To prove H^{1}(⟨τ′⟩, U(2n)) is trivial, by Proposition 3.2 it suffices to show that i : H^{1}(⟨μ⟩, U(2n)) → H^{1}(⟨μ⟩, GL(2n, ℂ)) is the trivial mapping. So let u ∈ U(2n) and assume that uμ(u) = 1. Then u𝔟_{0} ^{t}u^{−1}𝔟_{0}^{−1} = 1 so u𝔟_{0} is antisymmetric. Regarding u𝔟_{0} as a bilinear form, it is nondegenerate, so it can be converted into the symplectic form 𝔟_{0} by a change of basis. In other words, there exists A ∈ GL(2n, ℂ) such that Au𝔟_{0} ^{t}A = 𝔟_{0} or

u = A^{−1}𝔟_{0} ^{t}A^{−1}𝔟_{0}^{−1} = A^{−1}μ(A) (5.12.1)

This equation says that the cocycle defined by u becomes trivial in H^{1}(⟨μ⟩, GL(2n, ℂ)) as desired.

The isomorphism Φ : (K, τ) → (U(2n), τ′) is obtained by changing the basepoint x_{1} ∈ 𝔥_{2n} (whose isotropy group is K = G^{θ}) to the basepoint iI_{2n} ∈ 𝔥_{2n} (whose isotropy group we denote by K′ = G^{θ′} ≅ U(2n)). Let

𝔞 = (\begin{matrix}Diag^{n}(α) & 0 \\ 0 & Diag^{n}(α′) \end{matrix}) ∈ GSp(4n, ℝ) and N_{0} = (\begin{matrix}𝔟_{0} & 0 \\ 0 & ^{t}𝔟_{0} \end{matrix})

where α, α′ are defined as follows:

symbol d ≢ 1 (mod 4) d ≡ 1 (mod 4)
α  ( \begin{matrix}1 & 0 \\ 0 & \sqrt{−d} \end{matrix} )  ( \begin{matrix}2 & 1 \\ 0 & \sqrt{−d} \end{matrix} )
α′  ( \begin{matrix}\sqrt{−d} & 0 \\ 0 & 1 \end{matrix} )  ( \begin{matrix}\sqrt{−d} & 0 \\ −1 & 2 \end{matrix} )

Define Φ : Sp(4n, ℝ) → Sp(4n, ℝ) by Φ(g) = 𝔞g𝔞^{−1}. The mapping Φ converts the commuting involutions θ, τ into commuting involutions θ′(g) = ^{t}g^{−1} and τ′(g) = N_{0}gN_{0}^{−1}, so it takes (K, τ) to (K′, τ′), (however the mapping Φ does not preserve the integral structure). The identification U(2n) ≅ K′ ⊂ Sp(4n, ℝ) is given by A + iB ↦ ( \begin{matrix}A & B \\ −B & A \end{matrix} ) . Restricting the involution τ′ to U(2n) gives τ′(u) = 𝔟_{0}u𝔟_{0}^{−1}. q.e.d.

5.13 The involution on Siegel space

There is a unique involution τ : 𝔥_{2n} → 𝔥_{2n} so that τ(gZ) = τ(g)τ(Z) for all g ∈ Sp(4n, ℝ) and Z ∈ 𝔥_{2n}; it is given by

τ(Z) = Ẑ = 𝔟Z^{t}𝔟^{−1} (5.13.1)

where 𝔟 = Diag^{n}(β). If Z = (z_{ij}) is divided into 2×2 blocks z_{ij} then Z ∈ 𝔥_{2n}^{τ} is fixed under τ iff z_{ij} = β\overline{z}_{ij} ^{t}β (for 1 ≤ i, j ≤ n), or

(5.13.2)  z_{ij} = (\begin{matrix}dz_{ij} & w_{ij} \\ w_{ij} & z_{ij} \end{matrix})  if d ≢ 1 (mod 4)

(5.13.3)  z_{ij} = σ^{−1}(\begin{matrix}dz_{ij} & w_{ij} \\ w_{ij} & z_{ij} \end{matrix})^{t}σ^{−1}  if d ≡ 1 (mod 4)

for some z_{ij}, w_{ij} ∈ ℂ, and where σ = ( \begin{matrix}2 & 1 \\ 0 & 1 \end{matrix} ) . Conversely, if Z ∈ 𝔥_{2n} is divided into 2×2 blocks z_{ij} = ( \begin{matrix}x_{ij} & w_{ij} \\ y_{ij} & z_{ij} \end{matrix} ) then Z = τ(Z) iff x_{ij} = dz_{ij} and y_{ij} = w_{ij}. These are linear equations in the coordinates, so 𝔥_{2n}^{τ} is an open subset of a certain linear subspace of the space of symmetric 2n × 2n matrices.

Proposition 5.14. The embedding ϕ : Sp(2n, ℂ) → Sp(4n, ℝ) passes to an embedding ϕ : Y_{n} = Sp(2n, ℂ)/K_{s} → 𝔥_{2n} whose image is the fixed point set 𝔥_{2n}^{τ}. In particular, ϕ(Y_{n}) is a real algebraic submanifold of 𝔥_{2n}. If g ∈ Sp(4n, ℝ) and if g ≠ ±I then ϕ(Y_{n}) ∩ 𝔥_{2n}^{g} is a proper real algebraic subvariety of ϕ(Y_{n}), where 𝔥_{2n}^{g} denotes the points in 𝔥_{2n} that are fixed by g.

5.15 Proof

The first two statements follow from Proposition 5.11 and Lemma 3.8 part (b). If g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(4n, ℝ) then the points Z ∈ 𝔥_{2n} that are fixed by g satisfy

AZ + B = ZCZ + ZD (5.15.1)

which is a system of linear and quadratic equations in the matrix entries for Z, so this fixed point set is a real algebraic subvariety of 𝔥_{2n}, as is its intersection with 𝔥_{2n}^{τ}. We will now show that this intersection is a proper subvariety of 𝔥_{2n}^{τ} unless g = ±I. We consider only the case d ≢ 1 (mod 4); the case d ≡ 1 (mod 4) is similar.

Let Y be the 2n × 2n matrix consisting of 2×2 blocks along the diagonal y_{i} = ( \begin{matrix}−dy_{i} & 0 \\ 0 & y_{i} \end{matrix} ) (for 1 ≤ i ≤ n) where y_{i} > 0. It follows from (5.13.2) that itY ∈ 𝔥_{2n}^{τ} for all t > 0. If itY is fixed under g then (5.15.1) gives t^{2}Y CY = B and Y D = AY from which it follows that B = C = 0. Since g is symplectic we also obtain D = ^{t}A^{−1}. So we are reduced to considering those matrices A ∈ GL(2n, ℝ) such that Z = AZ^{t}A for all Z ∈ 𝔥_{2n}^{τ}.

We outline one of many possible ways to see this implies A = ±I. By taking Z = Diag(z_{1}, z_{2}, …, z_{n}) to consist of 2×2 blocks z_{i} = ( \begin{matrix}dz_{i} & r_{i} \\ r_{i} & z_{i} \end{matrix} ) along the diagonal, and by varying one block but fixing the others, we may conclude that A = Diag(a_{1}, a_{2}, …, a_{n}) also consists of 2×2 blocks, and that z_{i} = a_{i}z_{i} ^{t}a_{i}. Comparing real and imaginary parts of this equation gives a_{i} = ±I. It is then easy to see that the signs must all coincide. q.e.d.

5.16 γ-real points

If γ ∈ Sp(4n, ℤ) and γγ̂ = I (that is, if f_{γ} is a 1-cocycle), then a point Z ∈ 𝔥_{2n} is said to be a γ real point if τ(Z) = γZ, the set of which was denoted 𝔥_{2n}^{τγ} in (3.4.1). If Γ ⊂ Sp(4n, ℤ) is a torsion-free subgroup that is preserved by the involution τ then the set of τ fixed points in the quotient Γ\𝔥_{2n} is precisely the image of the set

𝔥_{2n}^{τΓ} = ⋃_{γ∈Z^{1}(⟨τ⟩,Γ)} 𝔥_{2n}^{τγ}.

6. Symplectic O_{d} modules

6.1

The main result in this section (Proposition 6.4), which classifies symplectic O_{d} modules, will be used in the proof of both main theorems (Theorem 2.3 and Theorem 8.5) of this paper. First, it is used to prove Proposition 6.10, which verifies hypothesis (4) (vanishing of nonabelian cohomology) of Lemma 3.8, which in turn is used to prove Theorem 2.3. Proposition 6.4 is also used in the proof of the Comessatti lemma (Proposition 7.7), which in turn is used to prove Theorem 8.5. Throughout this section we fix a square-free integer d < 0 and let O_{d} denote the ring of integers in the quadratic imaginary number field ℚ(\sqrt{d}).

6.2

Recall [3] (VII.10 Prop. 24) that a finitely generated module P over the Dedekind domain O_{d} is torsion-free iff it is projective. If such a module P has rank n, then there exist v_{1}, v_{2}, …, v_{n} ∈ P such that

P ≅ O_{d}v_{1} ⊕ O_{d}v_{2} ⊕ ⋯ ⊕ O_{d}v_{n−1} ⊕ Iv_{n} (6.2.1)

for some fractional ideal I.

Now suppose P_{0} ⊂ P is a submodule. Then there exist submodules P_{1}, P_{2} ⊂ P such that P = P_{1} ⊕ P_{2} and so that P_{1} ⊃ P_{0} and P_{1} ⊗ ℚ = P_{0} ⊗ ℚ. For, let M = P/P_{0} and consider its torsion-free quotient M/M^{tor} where M^{tor} denotes the torsion submodule of M. The preceding paragraph implies the composition

P → M → M/M^{tor} (6.2.2)

admits a splitting M/M^{tor} → P whose image we denote by P_{2}. Then P ≅ P_{1} ⊕ P_{2} where

P_{1} = {x ∈ P | rx ∈ P_{0} for some r ∈ O_{d}}

is the kernel of the composition (6.2.2).

6.3 Polarizations

Let Q : ℝ^{2r} × ℝ^{2r} → ℝ be a symplectic form and let L ⊂ ℝ^{2r} be a lattice. We say that Q is a principal polarization of L if Q takes integer values on L and if, for some basis of L (and hence for any basis of L), the matrix for Q has determinant 1. In this case there exists a symplectic basis of L, meaning an ordered basis such that the resulting matrix for Q is

J_{r} = (\begin{matrix}0 & I_{r} \\ −I_{r} & 0 \end{matrix}) .

Now suppose r = 2n and suppose that L is also an O_{d} module. Let us write b · v for the action of b ∈ O_{d} on a vector v ∈ L. We say the action of O_{d} is compatible with the polarization Q if

Q(b · u, v) = Q(u, b · v) (6.3.1)

for all u, v ∈ L and b ∈ O_{d}, cf. Equation (7.3.4). It follows that Q(b · u, u) = 0 for all b ∈ O_{d} and all u ∈ L.

Proposition 6.4. Suppose Q is a symplectic form on ℝ^{4n} that principally polarizes a lattice L ⊂ ℝ^{4n}. Suppose L has an O_{d} structure that is compatible with the polarization Q. Then there exists

u_{1}, …, u_{n}, v_{1}, …, v_{n} ∈ L

such that the following ordered collection is a symplectic basis for L:

{u_{1}, ω · u_{1}, …, u_{n}, ω · u_{n}, ω · v_{1}, v_{1}, …, ω · v_{n}, v_{n}}
if d ≢ 1 (mod 4), and

{u_{1}, ω · u_{1}, …, u_{n}, ω · u_{n}, (ω − 1) · v_{1}, v_{1}, …, (ω − 1) · v_{n}, v_{n}}

if d ≡ 1 (mod 4).

In particular, L ≅ L_{1} ⊕ L_{2} is the direct sum of the free Lagrangian submodules

L_{1} = O_{d}u_{1} ⊕ ⋯ ⊕ O_{d}u_{n} and L_{2} = O_{d}v_{1} ⊕ ⋯ ⊕ O_{d}v_{n}.

In either case, with respect to this basis, the matrix for the action of \sqrt{d} ∈ O_{d} is the matrix N of Equation (5.8.3) (cf. §7.3).

Proposition 6.4 will be proven by induction on n.

6.5 The case n = 1

By §6.2, L = O_{d}v ⊕ Iw for some v, w ∈ L and some fractional ideal I. So a ℤ basis of L is given by

{v, ω · v, aw, bω · w} (6.5.1)

for some a, b ∈ ℚ. The symplectic form Q vanishes on O_{d}v since Q(v, ω · v) = Q(ω · v, v) = −Q(v, ω · v), and it similarly vanishes on Iw. So with respect to this basis, the matrix for Q is ( \begin{matrix}0 & P \\ − tP & 0 \end{matrix} ) where P is some integer matrix. On the other hand, Q is a principal polarization of L, so det P = ±1. Apply P^{−1} to the basis {aw, bω · w} of Iw to obtain a new basis {x, y} of Iw. Then the matrix of Q with respect to the basis {v, ω · v, x, y} is J_{2}. Now let us determine the relationship between x, y, and ω · y. Set x = a′y + b′ω · y for some a′, b′ ∈ ℚ. Then

1 = Q(v, x) = a′Q(v, y) + b′Q(v, ω · y) = b′Q(ω · v, y) = b′
0 = Q(ω · v, x) = a′Q(ω · v, y) + b′Q(ω · v, ω · y) = a′ + Q(ω^{2} · v, y)

= \begin{cases}a′ + dQ(v, y) = a′ & if d ≢ 1 (mod 4) \\ a′ + Q((ω + m) · v, y) = a′ + 1 & if d ≡ 1 (mod 4). \end{cases}

Hence x = ω · y if d ≢ 1 (mod 4) and x = (ω − 1) · y if d ≡ 1 (mod 4), as desired. In either case, ω · y ∈ Iw so Iw = O_{d}y is free.

6.6 The case n > 1

We will prove in Lemma 6.7 below (by a somewhat roundabout argument) that there exist elements x, y ∈ L so that

Q(x, y) = 0 and Q(x, ω · y) = 1. (6.6.1)

It follows that Q(ω · x, y) = 1 and Q(ω · x, ω · y) = 0. Let P_{0} be the O_{d} span of {x, y}. It has a ℤ basis {x, ω · x, ω · y, y} with respect to which the matrix of Q|P_{0} is J_{2}.

We claim that L splits as a direct sum, L = P_{0}⊕L_{2} of O_{d} submodules (of ℤ rank 4 and rank 4n-4 respectively) such that the restriction Q|L_{2} is a principal polarization. By induction, the lattice L_{2} has a basis of the desired type, from which it follows that L does also.

The claim is proven as follows. Using §6.2 there exists a splitting L = P_{1} ⊕ P_{2} by submodules P_{1} and P_{2} such that P_{1} ⊃ P_{0} and P_{1} ⊗ ℚ = P_{0} ⊗ ℚ. Let {u, v, z, w} be a ℤ basis for P_{1} and let Q_{1} be the matrix of Q with respect to this basis. If A denotes the matrix that transforms this basis of P_{1} ⊗ ℚ into the basis {x, ω · x, ω · y, y}, then J_{2} = AQ_{1} ^{t}A. Since these are matrices of integers, it follows that det(A) = ±1 hence P_{0} = P_{1}. The next step is to modify the complement P_{2} to obtain a complement L_{2} which is principally polarized.

By §6.2 we may write P_{2} = O_{d}w_{1} ⊕⋯⊕O_{d}w_{2n−3} ⊕Iw_{2n−2} for some w_{i} ∈ P_{2}. For 1 ≤ i ≤ 2n − 2 set

w′_{i} = w_{i} − λ_{i}x − μ_{i}ω · x − γ_{i}ω · y − ν_{i}y.

Then there are unique choices of integers λ_{i}, μ_{i}, γ_{i}, ν_{i} ∈ ℤ so that each w′_{i} is Q orthogonal to P_{0}. Let L_{2} be the O_{d} span of the vectors w′_{i} (for 1 ≤ i ≤ 2n − 2). Then L = P_{0} ⊕ L_{2} and L_{2} is orthogonal to P_{0}. With respect to any choice of ℤ basis for L_{2} (and the above basis for P_{0}) the matrix for Q is

(\begin{matrix}J_{2} & 0 \\ 0 & Q_{2} \end{matrix}) (6.6.2)

where Q_{2} is some integer matrix. However Q is a principal polarization, so the determinant of the matrix (6.6.2) is 1, from which it follows that the determinant of Q_{2} is also 1. Therefore, the restriction of Q to L_{2} is a principal polarization, as desired. The rest of this section is dedicated to proving the existence of the elements x, y, which we now state precisely.

Lemma 6.7. Fix n ≥ 2. Suppose L ⊂ ℝ^{4n} is a lattice that is principally polarized by the symplectic form Q, and suppose L admits a compatible action of O_{d}. Then there exists x, y ∈ L so that (6.6.1) holds.

6.8 Proof

If {u_{1}, …, u_{r}} is a collection of vectors in L let

⟨u_{1}, …, u_{r}⟩

denote their vector space span in ℝ^{2n} and let

⟨u_{1}, …, u_{r}⟩^{⊥}

be the Q-annihilator of this span. Since Q is integral on L, the intersection L ∩ ⟨u_{1}, …, u_{r}⟩^{⊥} is a lattice in ⟨u_{1}, …, u_{r}, ⟩^{⊥}.

Step 1. There exists a Lagrangian O_{d} submodule L_{0} ⊂ L.

Suppose by induction that a ℚ linearly independent collection of vectors

{u_{1}, ω · u_{1}, u_{2}, ω · u_{2}, …, u_{r}, ω · u_{r}} ⊂ L (6.8.1)

has been found so that Q vanishes on their ℚ span U_{r} (with the case r = 0 being trivial). The Q-annihilator U_{r}^{⊥} has dimension 2n − r and the intersection L ∩ U_{r}^{⊥} is a lattice in U_{r}^{⊥}. If r < n then there exists a vector u_{r+1} ∈ U_{r}^{⊥} ∩ L which is not contained in U_{r}. We claim the collection {u_{1}, ω ·u_{1}, …, u_{r}, ω ·u_{r}, u_{r+1}, ω ·u_{r+1}} is linearly independent and that Q vanishes on its vector space span U_{r+1}.

Suppose that ω · u_{r+1} is a linear combination of the other vectors in this collection, say,

ω · u_{r+1} = ∑_{i=1}^{r}(a_{i}u_{i} + b_{i}ω · u_{i}) + cu_{r+1}

for some rational numbers a_{i}, b_{i}, and c. Multiplying by ω and collecting terms gives

(d − c^{2})u_{r+1} = ∑_{i=1}^{r}(a_{i} + cb_{i})ω · u_{i} + (b_{i}d + ca_{i})u_{i})

if d ≢ 1 (mod 4). But d − c^{2} < 0 so this contradicts the linear independence of (6.8.1). The case of d ≡ 1 (mod 4) is similar.

Step 2. There exists a Lagrangian O_{d} submodule L_{1} ⊂ L and a submodule L_{2} ⊂ L so that L = L_{1} ⊕ L_{2}.

This follows from §6.2 and in fact L_{1} ⊗ ℚ = L_{0} ⊗ ℚ.

Now set L_{1} = O_{d}y_{1} ⊕ O_{d}y_{2} ⊕ ⋯ ⊕ O_{d}y_{n−1} ⊕ Iy_{n} for some fractional ideal I. Then there exist rational numbers a, b ∈ ℚ so that the collection {y_{1}, ω · y_{1}, …, y_{n−1}, ω · y_{n−1}, y_{n}, ay_{n} + bω · y_{n}} forms a ℤ basis of L_{1}.

Step 3. Choose any ℤ basis for L_{2}. Together with the preceding basis for L_{1} this gives a basis for L = L_{1} ⊕ L_{2} with respect to which the matrix of Q is

(\begin{matrix}0 & T \\ −^{t}T & ∗ \end{matrix})

for some matrix T of integers. It follows that det(T) = ±1. Applying T^{−1} to this basis gives a new ℤ basis for L_{2} such that the matrix for Q is

(\begin{matrix}0 & I \\ −I & ∗ \end{matrix}) .

Denote this new basis by {z_{1}, x_{1}, z_{2}, x_{2}, …, z_{n}, x_{n}}. Then

Q(y_{1}, x_{1}) = 0 and Q(ω · y_{1}, x_{1}) = 1.

Therefore the elements y = y_{1} and x = −x_{1} satisfy (6.6.1). q.e.d.

6.9 Application to nonabelian cohomology

Proposition 6.4 may be used to construct an O_{d} module structure on certain lattices. Let τ be the involution of Sp(4n, ℤ) = Sp(ℤ^{4n}, Q_{0}) defined in §5.7. Let Γ ⊂ Sp(4n, ℤ) be a torsion-free (arithmetic) subgroup that is preserved under τ. If d ≡ 1 (mod 4) then suppose also that Γ is contained in the principal congruence subgroup Γ(2) of level 2. Set

Γ̃ = {h ∈ Sp(4n, ℤ) | ĥh^{−1} ∈ Γ} .

Proposition 6.10. The mapping

H^{1}(⟨τ⟩, Γ) → H^{1}(⟨τ⟩, Γ̃)

is trivial.

6.11 Proof

Let γ ∈ Γ and suppose f_{γ} is a 1-cocycle, that is, γγ̂ = γNγN^{−1} = I, cf. Equation (5.8.3). It follows that (Nγ)^{2} = NγNγ = N^{2} = dI. Using Equation (5.8.4) we obtain

Q_{0}(Nγu, Nγv) = dQ_{0}(u, v)

and hence

Q_{0}(Nγu, v) = Q_{0}(u, Nγv)

for all u, v ∈ ℝ^{4n}. We use this to define a Q_{0}-polarized O_{d} module structure on the standard lattice ℤ^{4n} by letting \sqrt{d} act through Nγ, that is, define

(a + b\sqrt{d}) · u = au + bNγu

whenever a, b ∈ ℤ If d ≡ 1 (mod 4) it is necessary to check that the action of ω = (1 + \sqrt{d})/2 also preserves the lattice ℤ^{4n}, however this follows from the fact that γ ≡ I (mod 2) when d ≡ 1 (mod 4).

So we may apply Proposition 6.4 to conclude that ℤ^{4n} admits a symplectic basis with respect to which the matrix of \sqrt{d} is N. In other words, there exists h ∈ Sp(4n, ℤ) such that Nγ = hNh^{−1}. Using the fact that N = N^{−1}dI we conclude that

γ = N^{−1}hNh^{−1} = NhN^{−1}h^{−1} = ĥh^{−1}

from which it follows that h ∈ Γ̃ and that the cocycle f_{γ} is a coboundary. q.e.d.

7. Anti-holomorphic multiplication

7.1

In this section we recall [13] some standard facts and notation concerning abelian varieties. Let L ⊂ ℂ^{r} be a lattice (that is, a free abelian subgroup of rank 2r so that L ⊗_{ℂ} ℝ → ℂ^{2r} is an isomorphism of real vector spaces). Then A = ℂ^{r}/L is a complex torus. If ω_{1}, ω_{2}, …, ω_{r} is a basis for the space of holomorphic 1-forms on A, and if v_{1}, v_{2}, …, v_{2r} is a basis for L then the corresponding period matrix Ω is the matrix with entries Ω_{ij} = ∫_{v_{j}} ω_{i}. If v′_{i} = ∑_{j} A_{ij}v_{j} and if ω′_{i} = ∑_{j} B_{ij}ω_{j} are new bases then the resulting period matrix is

Ω′ = BΩ^{t}A. (7.1.1)

A real symplectic form Q on ℂ^{r} is compatible with the complex structure if Q(iu, iv) = Q(u, v) for all u, v ∈ ℂ^{r}, (not to be confused with Equation (7.3.4) below). A compatible form Q is positive if the symmetric form R(u, v) = Q(iu, v) is positive definite. If Q is compatible and positive then it is the imaginary part of a unique positive definite Hermitian form H = R+iQ. Let L ⊂ ℂ^{r} be a lattice and let H = R+iQ be a positive definite Hermitian form on ℂ^{r}. Recall that Q is a principal polarization of L if L admits a basis such that the resulting matrix for Q is J_{r} (cf. §6.3). A principally polarized abelian variety is a pair (A = ℂ^{r}/L, H = R + iQ) where H is a positive definite Hermitian form on ℂ^{r} and where L ⊂ ℂ^{r} is a lattice that is principally polarized by Q.

Each Z ∈ 𝔥_{r} determines a principally polarized abelian variety

(A_{Z}, H_{Z})

as follows. Let Q_{0} be the standard symplectic form on ℝ^{2r} = ℝ^{r} ⊕ ℝ^{r} with matrix J = J_{r} = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) (with respect to the standard basis of ℝ^{r} ⊕ ℝ^{r}). Let F_{Z} : ℝ^{r} ⊕ ℝ^{r} → ℂ^{r} be the real linear mapping with matrix (Z, I), that is,

F_{Z} ( \begin{matrix}x \\ y \end{matrix}) = Zx + y.

Then

Q_{Z} = (F_{Z})_{∗}(Q_{0}) (7.1.2)

is a compatible, positive symplectic form that principally polarizes the lattice

L_{Z} = F_{Z}(ℤ^{r} ⊕ ℤ^{r}). (7.1.3)

(In fact F_{Z}(standard basis) is a symplectic basis for L_{Z}.) The Hermitian form corresponding to Q_{Z} is

H_{Z}(u, v) = Q_{Z}(iu, v) + iQ_{Z}(u, v) = ^{t}u(Im(Z))^{−1}v

for u, v ∈ ℂ^{r}. The pair (A_{Z} = ℂ^{r}/L_{Z}, H_{Z}) is the desired principally polarized abelian variety. If z_{1}, z_{2}, …, z_{r} are the standard coordinates on ℂ^{r} then, with respect to the above symplectic basis of L, the differential forms dz_{1}, dz_{2}, …, dz_{r} have period matrix Ω = (Z, I).

7.2

The principally polarized abelian varieties (A_{Z} = ℂ^{r}/L_{Z}, H_{Z}) and (A_{Ω} = ℂ^{r}/L_{Ω}, H_{Ω}) are isomorphic iff there exists a complex linear mapping ξ : ℂ^{r} → ℂ^{r} such that ξ(L_{Ω}) = L_{Z} and ξ_{∗}(H_{Ω}) = H_{Z}. Set

h = ^{t}(F_{Z}^{−1}ξF_{Ω}) = ( \begin{matrix}A & B \\ C & D \end{matrix} ) . Then: h ∈ Sp(2r, ℤ), Ω = h · Z, and

ξ(M) = ^{t}(CZ + D)M for all M ∈ ℂ^{r}, which is to say that the following diagram commutes:

\begin{matrix}( \begin{matrix}x \\ y \end{matrix}) & ℝ^{r} ⊕ ℝ^{r} & ⟶_{F_{h·Z}} & ℂ^{r} & M \\ ↓ & ↓ & & ↓^{ξ} & ↓ \\ ^{t}h ( \begin{matrix}x \\ y \end{matrix}) & ℝ^{r} ⊕ ℝ^{r} & ⟶_{F_{Z}} & ℂ^{r} & ^{t}(CZ + D)M \end{matrix} (7.2.1)

(since h · Z is symmetric). The relationship between the mapping F_{Z} and the involution τ is the following. Let N be as in (5.8.3). If Z ∈ 𝔥_{r} and Ẑ = 𝔟Z^{t}𝔟^{−1} then this diagram commutes:

\begin{matrix}( \begin{matrix}x \\ y \end{matrix}) & ℝ^{r} ⊕ ℝ^{r} & ⟶_{F_{Z}} & ℂ^{r} & M \\ ↓ & ↓ & & ↓ & ↓ \\ ^{t}N ( \begin{matrix}x \\ y \end{matrix}) & ℝ^{r} ⊕ ℝ^{r} & ⟶_{FẐ} & ℂ^{r} & 𝔟M. \end{matrix} (7.2.2)

7.3

A real endomorphism of a principally polarized abelian variety (A = ℂ^{r}/L, H = R + iQ) is an ℝ-linear mapping f : ℂ^{r} → ℂ^{r} such that f(L) ⊂ L; two such being considered equivalent if they induce the same mapping A → A. As in §5.2, fix a square-free integer d < 0 and let O_{d} denote the ring of integers in the number field ℚ(\sqrt{d}). Recall (for example, from [23] Equation (5.5.12) or [27] or [7] §3.1.1) that a complex multiplication by the ring O_{d} on A is a ring homomorphism Φ : O_{d} → End_{ℝ}(A) such that Φ(1) = I, and for all b ∈ O_{d} and u, v ∈ ℂ^{r},

(7.3.1) Φ(b) : ℂ^{r} → ℂ^{r} is complex linear,

Q(Φ(b)u, v) = Q(u, Φ(b)v). (7.3.2)

(If r = 1 then (7.3.2) follows from (7.3.1) and the relation Q(u, v) = Q(iu, iv).) In analogy with the above, let us say that an anti-holomor-phic multiplication by the ring O_{d} is a ring homomorphism Ψ : O_{d} → End_{ℝ}(A) such that Ψ(1) = I and so that the mapping κ = Ψ(\sqrt{d}) : ℂ^{2n} → ℂ^{2n} satisfies

(7.3.3)  κ(au) = aκ(u)
(7.3.4)  Q(κ(u), v) = Q(u, κ(v))

for all a ∈ ℂ and u, v ∈ ℂ^{r}. (Consequently, Ψ(1) is complex linear and κ = Ψ(\sqrt{d}) is anti-linear.) In this case, Ψ is determined by the mapping κ, and Equation (7.3.4) implies that

Q(Ψ(b)(u), v)) = Q(u, Ψ(b)(v)) (7.3.5)

for all b ∈ O_{d}. If such an anti-holomorphic multiplication exists then r is even (and in fact ⟨u, v⟩ = Q(κ(u), v) + \sqrt{d}Q(u, v) is a complex symplectic form on ℂ^{r} with respect to the complex structure defined by κ/\sqrt{−d}). Equivalently, a choice of anti-holomorphic multiplication by O_{d}, if one exists, is a choice of O_{d}-module structure on L that satisfies (7.3.3) and (7.3.5) for all u, v ∈ L and all b ∈ O_{d}.

7.4

For the remainder of this section take r = 2n and write Q_{0} for the (standard) symplectic form on ℝ^{4n} whose matrix is J = J_{2n} with respect to the standard basis of ℝ^{4n}. The following lemma states that certain points (the γ-real points, for appropriately chosen γ) in the Siegel space correspond to abelian varieties with anti-holomorphic multiplication. We use the involution τ defined in (5.8.2) and (5.13.1) and the corresponding matrix N of (5.8.3); see also §5.16.

Lemma 7.5. Let γ ∈ Sp(4n, ℤ) and suppose γγ̂ = 1. If d ≡ 1 (mod 4) then assume also that γ ≡ I (mod 2). Fix Z ∈ 𝔥_{2n}^{τγ}. Then the mapping

κ_{Z} = F_{Z} ∘ ^{t}(Nγ) ∘ F_{Z}^{−1} : ℂ^{2n} → ℂ^{2n} (7.5.1)

defines an anti-holomorphic multiplication by O_{d} on the principally polarized abelian variety (A_{Z}, H_{Z}).

7.6 Proof

Set η = ^{t}(Nγ). Then

η^{2} = ^{t}(NγNγ) = ^{t}(NγdN^{−1}γ) = d^{t}(γ̂γ) = dI

so the same is true of κ_{Z}. Also, η = ^{t}γ ( \begin{matrix}^{t}𝔟 & 0 \\ 0 & 𝔟^{−1} \end{matrix}) (\begin{matrix}I & 0 \\ 0 & dI \end{matrix}) . The first

two factors are in Sp(4n, ℝ) so Q_{0}(ηu, ηv) = dQ_{0}(u, v) for all u, v ∈ ℝ^{4n}. Hence Q_{Z}(κ_{Z}u, κ_{Z}v) = dQ_{Z}(u, v) for all u, v ∈ ℂ^{2n} which implies (7.3.4). The mapping κ_{Z} preserves the lattice L_{Z} = F_{Z}(ℤ^{2n} ⊕ ℤ^{2n}) since η preserves the integer lattice. If d ≡ 1 (mod 4) and γ ≡ I (mod 2) then ^{t}(Nγ) + I ≡ 0 (mod 2) (since β, 𝔟, and N are all ≡ I (mod 2), see §5.7). This shows that \frac{1}{2}(I + η) preserves the lattice ℤ^{2n} ⊕ ℤ^{2n}, hence O_{d} preserves L_{Z}.

Finally we check that κ_{Z} : ℂ^{2n} → ℂ^{2n} is anti-linear. Let γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) . By (7.2.1) and (7.2.2) the following diagram commutes:

\begin{matrix}ℝ^{2n} ⊕ ℝ^{2n} & ⟶_{F_{Z}} & ℂ^{2n} & M \\ _{tN}↓ & & ↓ & ↓ \\ ℝ^{2n} ⊕ ℝ^{2n} & ⟶_{FẐ} & ℂ^{2n} & 𝔟M \\ _{tγ}↓ & & ↓ & ↓^{ξ} \\ ℝ^{2n} ⊕ ℝ^{2n} & ⟶_{F_{γ^{−1}·Ẑ}} & ℂ^{2n} & ^{t}(CZ + D)𝔟M. \end{matrix} (7.6.1)

But Z = γ^{−1} · Ẑ so the bottom arrow is also F_{Z}. Then κ_{Z} is the composition along the right-hand vertical column and it is given by M ↦ ^{t}(CZ + D)𝔟M which is anti-linear. q.e.d.

The following proposition is an analog of the lemma ([29], [4]) of Comessatti and Silhol.

Proposition 7.7. Suppose A = (ℂ^{2n}/L, H = R + iQ) is a principally polarized abelian variety with antiholomorphic multiplication κ : ℂ^{2n} → ℂ^{2n} by O_{d}. Then there exists a basis for the holomorphic 1-forms on A and there exists a symplectic basis for L such that the resulting period matrix is Ω = (Z, I) for some Z ∈ 𝔥_{2n}^{τ} which is fixed under the involution τ.

7.8 Proof

Throughout this section, in order to simplify notation, but at the risk of some confusion with the usual multiplication, we will write b · v rather than Ψ(b)v, for any b ∈ O_{d} and v ∈ ℂ^{2n}. First consider the case d ≢ 1 (mod 4). By Proposition 6.4 and by interchanging the u’s and v’s, there exist u_{1}, …, u_{n}, v_{1}, …, v_{n} ∈ L so that the ordered collection

{ω · u_{1}, u_{1}, …, ω · u_{n}, u_{n}; v_{1}, ω · v_{1}, …, v_{n}, ω · v_{n}}

is a symplectic basis for L.

The space of holomorphic 1-forms on A is 2n-dimensional, so for each i (1 ≤ i ≤ n) there exist a unique holomorphic 1-form η_{i} such that the following holds for all j (with 1 ≤ j ≤ n):

∫_{v_{j}} η_{i} = 0 and ∫_{ω·v_{j}} η_{i} = δ_{ij}.

Set η′_{i} = κ^{∗}η_{i} = κ^{∗}η_{i}. Then the collection {η′_{1}, η_{1}, …, η′_{n}, η_{n}} is an ordered basis for the holomorphic 1-forms on A. Let us compute the period matrix with respect to these bases. Calculate that

∫_{ω·v_{j}} η′_{i} = ∫_{κv_{j}} κ^{∗}η_{i} = d ∫_{v_{j}} η_{i} = 0

∫_{v_{j}} η′_{i} = ∫_{κv_{j}} η_{i} = δ_{ij}.

It follows that the second “half” of the period matrix is the identity. Now let z_{ij} = ∫_{u_{j}} η_{i} and w_{ij} = ∫_{u_{j}} η′_{i} for 1 ≤ i, j ≤ n. Then by a similar calculation, the first half of the period matrix consists of 2×2 blocks,

z_{ij} = (\begin{matrix}∫_{ω·u_{j}} η′_{i} & ∫_{u_{j}} η′_{i} \\ ∫_{ω·u_{j}} η_{i} & ∫_{u_{j}} η_{i} \end{matrix}) = (\begin{matrix}dz_{ij} & w_{ij} \\ w_{ij} & z_{ij} \end{matrix})

which implies by (5.13.2) that Z ∈ 𝔥_{2n}^{τ}.

Now consider the case d = 4m + 1. In this case Proposition 6.4 guarantees the existence of vectors u_{1}, …, u_{n}, v_{1}, …, v_{n} ∈ L so that the ordered collection

{(ω − 1) · u_{1}, u_{1}, …, (ω − 1) · u_{n}, u_{n}; v_{1}, ω · v_{1}, …, v_{n}, ω · v_{n}} (7.8.1)

is a symplectic basis for L. For each i (1 ≤ i ≤ n) there exist a unique holomorphic 1-form η_{i} such that ∫_{v_{j}} η_{i} = 0 and ∫_{ω·v_{j}} η_{i} = δ_{ij}. Set η′_{i} = κ^{∗}(η_{i}). Then

∫_{v_{j}} η′_{i} = ∫_{κv_{j}} η_{i} = ∫_{(2ω−1)·v_{j}} η_{i} = 2

∫_{ω·v_{j}} η′_{i} = ∫_{κω·v_{j}} η_{i} = ∫_{ω·v_{j}} η_{i} = 1

since \sqrt{d}ω = 2m+ω. So the second half of the period matrix is Diag^{n}(σ) where σ = ( \begin{matrix}2 & 1 \\ 0 & 1 \end{matrix} ) . Set z_{ij} = ∫_{u_{j}} η_{i} and w_{ij} = ∫_{u_{j}} η′_{i}. Then a simple calculation gives

(\begin{matrix}∫_{(ω−1)·u_{j}} η′_{i} & ∫_{u_{j}} η′_{i} \\ ∫_{(ω−1)·u_{j}} η_{i} & ∫_{u_{j}} η_{i} \end{matrix}) = (\begin{matrix}dz_{ij} & w_{ij} \\ w_{ij} & z_{ij} \end{matrix}) ( \begin{matrix}\frac{1}{2} & 0 \\ −\frac{1}{2} & 1 \end{matrix} ) .

So the period matrix is (Z′Diag^{n}(^{t}σ^{−1}), Diag^{n}(σ)) where Z′ consists of

2×2 blocks z_{ij} = ( w_{ij} z_{ij}^{dz_{ij} w_{ij}} ) . By (7.1.1), changing the basis

{η′_{1}, η_{1}, …, η′_{n}, η_{n}}

by the action of Diag^{n}(σ^{−1}) will give a period matrix (Z, I) where

Z = Diag^{n}(σ^{−1})Z′Diag^{n}(^{t}σ^{−1}).

By (5.13.3) the point Z ∈ 𝔥_{2n}^{τ} is fixed under τ as claimed. q.e.d.

8. A coarse moduli space for abelian varieties with anti-holomorphic multiplication

8.1 Level structures

Let (A = ℂ^{2n}/L, H = R+iQ) be a principally polarized abelian variety. A level M structure on A is a choice of basis {U_{1}, …, U_{2n}, V_{1}, …, V_{2n}} for the M-torsion points of A that is symplectic, in the sense that there exists a symplectic basis

{u_{1}, …, u_{2n}, v_{1}, …, v_{2n}}

for L such that

U_{i} ≡ \frac{u_{i}}{M} and V_{i} ≡ \frac{v_{i}}{M} (mod L)

(for 1 ≤ i ≤ 2n). For a given level M structure, such a choice

{u_{1}, …, u_{2n}, v_{1}, …, v_{2n}}

determines a mapping

F : ℝ^{2n} ⊕ ℝ^{2n} → ℂ^{2n} (8.1.1) such that F (ℤ^{2n} ⊕ ℤ^{2n}) = L, by F (e_{i}) = u_{i} and F (f_{i}) = v_{i} where {e_{1}, …, e_{2n}, f_{1}, …, f_{2n}} is the standard basis of ℝ^{2n} ⊕ ℝ^{2n}. The choice {u_{1}, …, u_{2n}, v_{1}, …, v_{2n}} (or equivalently, the mapping F) will be referred to as a lift of the level M structure. It is well-defined modulo the principal congruence subgroup Γ(M), that is, if F′ : ℝ^{2n} ⊕ ℝ^{2n} → ℂ^{2n} is another lift of the level structure, then F′ ∘ F^{−1} ∈ Γ(M).

Suppose (A, H, κ) is a principally polarized abelian variety with anti-holomorphic multiplication by O_{d} as in §7.3. A level M structure {U_{1}, …, U_{2n}, V_{1}, …, V_{2n}} on A is compatible with κ if for some (and hence for any) lift F of the level structure, the following diagram commutes (mod L):

\begin{matrix}\frac{1}{M} (ℤ^{2n} ⊕ ℤ^{2n}) & ⟶_{F} & \frac{1}{M} L \\ ^{t}N↓ & & ↓κ \\ \frac{1}{M} (ℤ^{2n} ⊕ ℤ^{2n}) & ⟶_{F} & \frac{1}{M} L \end{matrix} (8.1.2)

where N is the matrix (5.8.3).

We will refer to the collection

A = (A = ℂ^{2n}/L, H = R + iQ, κ, {U_{i}, V_{j}})

as a principally polarized abelian variety with anti-holomorphic multiplication and level M structure. If

A′ = (A′ = ℂ^{2n}/L′, H′ = R + iQ, κ′, {U′_{i}, V′_{j}})

is another such, then an isomorphism A ≅ A′ is a complex linear mapping ψ : ℂ^{2n} → ℂ^{2n} such that ψ(L) = L′, ψ_{∗}(H) = H′, ψ_{∗}(κ) = κ′, and such that for some (and hence for any) lift

{u_{1}, …, u_{2n}, v_{1}, …, v_{2n}}

and

{u′_{1}, …, u′_{2n}, v′_{1}, …, v′_{2n}}

of the level structures,

ψ (\frac{u_{i}}{M} ) ≡ \frac{u′_{i}}{M} and ψ (\frac{v_{j}}{M} ) ≡ \frac{v′_{j}}{M} (mod L).

Define V (d, M) to be the set of isomorphism classes of principally polarized abelian varieties with anti-holomorphic multiplication by O_{d} and level M structure.

8.2

If Z ∈ 𝔥_{2n}, then for any M ≥ 1 we define the standard level M structure on the abelian variety (A_{Z}, H_{Z}) to be the basis

{F_{Z}(e_{i}/M), F_{Z}(f_{j}/M)} (mod L)

where

{e_{1}, …, e_{2n}, f_{1}, …, f_{2n}}

is the standard basis of ℝ^{2n} ⊕ ℝ^{2n}.

Lemma 8.3. Let γ ∈ Sp(4n, ℤ) and let Z ∈ 𝔥_{2n}^{τγ}, that is, Ẑ = γ · Z. Let M ≥ 3. Then the standard level M structure on the abelian variety (A_{Z}, H_{Z}) is compatible with the anti-holomorphic multiplication κ_{Z} iff γ ∈ Γ_{M} = Γ(M) ∩ Γ̂(M).

8.4 Proof

It follows immediately from diagram (7.6.1) that γ ∈ Γ(M) iff the standard level M structure on (A_{Z}, H_{Z}) is compatible with κ_{Z}. Since Γ(M) is torsion-free, γ̂γ = I which implies γ ∈ Γ̂(M); hence γ ∈ Γ_{M}. q.e.d.

By Lemma 8.3, each point Z ∈ 𝔥_{2n}^{Γ(M)} determines a principally polarized abelian variety

A_{Z} = (A_{Z}, H_{Z}, κ_{Z}, {F_{Z}(e_{i}/M), F_{Z}(f_{j}/M)})

with anti-holomorphic multiplication and (compatible) level M structure.

Theorem 8.5. Fix M ≥ 3. If d ≡ 1 (mod 4), assume also that M is even. Then the association Z ↦ A_{Z} determines a one to one correspondence between the real points (2.3.2) X_{ℝ} of X = Γ_{M}\𝔥_{2n} and the set V (d, M) of isomorphism classes of principally polarized abelian varieties with anti-holomorphic multiplication by O_{d} and (compatible) level M structure.

8.6 Proof

A point x ∈ X is real iff it is the image of a Γ_{M}-real point Z ∈ 𝔥_{2n}^{τΓ(M)}. If two Γ_{M}-real points Z, Ω determine isomorphic varieties, say ψ : A_{Ω} ≅ A_{Z} then by (7.2.1) there exists h ∈ Sp(4n, ℤ) such that Ω = h·Z. Since the isomorphism ψ preserves the level M structures, it follows also from (7.2.1) that h ∈ Γ(M). We claim that h ∈ Γ_{M}. Let Ẑ = γ_{Z} · Z and Ω̂ = γ_{Ω} · Ω, with γ_{Z}, γ_{Ω} ∈ Γ_{M}. Putting diagram (7.6.1) for Z together with the analogous diagram for Ω and diagram (7.2.1), and using the fact that ψ_{∗}(κ_{Ω}) = κ_{Z} gives a diagram

ℝ^{2n} ⊕ ℝ^{2n} ⟶_{F_{Ω}} ℂ^{2n}
t(γ_{Ω}^{−1}ĥγ_{Z})↓  ↓κ_{Z}ψκ_{Ω}^{−1} = ψ
ℝ^{2n} ⊕ ℝ^{2n} ⟶_{F_{Z}} ℂ^{2n}

from which it follows that ^{t}(γ_{Ω}ĥγ_{Z}) ∈ Γ(M), hence ĥ ∈ Γ(M), hence h ∈ Γ_{M}.

So it remains to show that every principally polarized abelian variety with anti-holomorphic multiplication and level M structure, A = (A, H, κ, {U_{i}, V_{j}}) is isomorphic to some A_{Z}. By the Comessatti lemma (Proposition 7.7) there exists Z′ ∈ 𝔥_{2n}, such that Ẑ′ = Z′, and there exists an isomorphism

ψ′ : (A_{Z′}, H_{Z′}, κ_{Z′}) ≅ (A, H, κ)

between the principally polarized abelian varieties with anti-holomor-phic multiplication. However the isomorphism ψ′ must be modified because it does not necessarily take the standard level M structure on (A_{Z′}, H_{Z′}, κ_{Z′}) to the given level M structure on (A, H, κ).

Choose a lift {u_{1}, …, u_{2n}, v_{1}, …, v_{2n}} of the level M structure and let F : ℝ^{2n} ⊕ ℝ^{2n} → ℂ^{2n} be the corresponding mapping (8.1.1). Define

(8.6.1)  ^{t}g^{−1} = F^{−1} ∘ ψ′ ∘ F_{Z′} ∈ Sp(4n, ℤ)
(8.6.2)  Z = g · Z′
(8.6.3)  γ = ĝg^{−1} = N^{−1}gNg^{−1}.

As in §7.2, if g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) define ξ : ℂ^{2n} → ℂ^{2n} by ξ(w) = ^{t}(CZ + D)w. Define ψ = ψ′ ∘ ξ. We will show that γ ∈ Γ_{M}, that Ẑ = γ · Z, and that ψ induces an isomorphism ψ : A_{Z} → A of principally polarized abelian varieties with anti-holomorphic multiplication and compatible level M structures.

In the following diagram, F is the mapping (8.1.1) associated to the lift of the level M structure. The bottom square commutes by the definition of g, while the top square commutes by (7.2.1):

\begin{matrix}ℝ^{2n} ⊕ ℝ^{2n} & ⟶^{F_{Z}} & ℂ^{2n} \\ _{tg}↓ & & ↓^{ξ} \\ ℝ^{2n} ⊕ ℝ^{2n} & ⟶_{F_{Z′}} & ℂ^{2n} \\ _{tg^{−1}}↓ & & ↓^{ψ′} \\ ℝ^{2n} ⊕ ℝ^{2n} & ⟶^{F} & ℂ^{2n}. \end{matrix}. (8.6.4)

First let us verify that ξ : (A_{Z}, H_{Z}, κ_{Z}) → (A_{Z′}, H_{Z′}, κ_{Z′}) is an isomorphism of principally polarized varieties with anti-holomorphic multiplication by O_{d}. It follows from (8.6.4) that ξ_{∗}(L_{Z}) = L_{Z′} and ξ_{∗}(H_{Z}) = H_{Z′}. We claim that ξ_{∗}(κ_{Z}) = κ_{Z′}, that is, κ_{Z′} = ξκ_{Z}ξ^{−1}. But this follows from direct calculation using ξ = F_{Z′} ^{t}gF_{Z}, κ_{Z} = F_{Z} ^{t}(Nγ)F_{Z}^{−1}, κ_{Z′} = F_{Z′} ^{t}NF_{Z′} and (8.6.3) (and it is equivalent to the statement that the pushforward by ^{t}g of the involution ^{t}(Nγ) on ℝ^{2n} ⊕ ℝ^{2n} is the involution ^{t}N). It follows that

ψ_{∗}(κ_{Z}) = κ. (8.6.5)

We claim that the standard level M structure on (A_{Z}, H_{Z}) is compatible with κ_{Z}. By construction, the mapping ψ takes the standard level M structure on (A_{Z}, H_{Z}) to the given level M structure on (A, H). By assumption, the diagram (8.1.2) commutes (mod L). By (8.6.4), F = ψ ∘ F_{Z}. Using (8.6.5) it follows that the diagram

\frac{1}{M} (ℤ^{2n} ⊕ ℤ^{2n}) ⟶_{F_{Z}} \frac{1}{M} L_{Z}
tN↓  ↓^{κ_{Z}}
\frac{1}{M} (ℤ^{2n} ⊕ ℤ^{2n}) ⟶_{F_{Z}} \frac{1}{M} L_{Z}

commutes (mod L_{Z}), which proves the claim. It also follows from Lemma 8.3 that γ ∈ Γ_{M}.

In summary, we have shown that

(A_{Z}, H_{Z}, κ_{Z}, {F_{Z}(e_{i}/M), F_{Z}(f_{j}/M)})

is a principally polarized abelian variety with anti-holomorphic multiplication and (compatible) level M structure, and that the isomorphism ψ preserves both the anti-holomorphic multiplication and the level structures. q.e.d.

9. Rational structure

9.1

Let g ↦ ĝ = NgN^{−1} be the involution on Sp(4n, ℝ) where N denotes the matrix (5.8.3), as in §5.7. The resulting anti-holomorphic involution on 𝔥_{2n} is given by Z ↦ Ẑ = 𝔟Z^{t}𝔟^{−1}. As in §8, fix a level M ≥ 1 and let Γ = Γ_{M} = Γ(M) ∩ Γ̂(M). It is well-known ([5] §V Thm. 2.5) that the arithmetic quotient Γ(M)\𝔥_{2n} admits a model that is defined over a certain cyclotomic field. For our purposes, however, we need a model that is defined over a subfield of the real numbers, and we need it for the slightly different arithmetic quotient X = Γ_{M}\𝔥_{2n}. For these facts we will use results of [25].

Theorem 9.2. There exists a projective embedding X → ℂℙ^{r} and there exists an anti-holomorphic involution τ on ℂℙ^{r} such that:

• The closure X is the Baily-Borel Satake compactification of X.

• As a projective algebraic variety, X is defined over ℚ.

• The involution τ is rationally defined and preserves X.

• The restriction τ|X coincides with the involution (induced by) τ of §5.13.

In summary, the set X_{ℝ} described in Theorem 2.3 forms the set of real points of a complex quasi-projective algebraic variety defined over ℚ. The proof will occupy the rest of this section.

Proposition 9.3. The complex vector space of (holomorphic) Γ_{M}- modular forms on 𝔥_{2n} is spanned by modular forms with rational Fourier coefficients.

9.4 Proof

Let G = GSp(4n), let 𝔸 be the adeles of ℚ and let S ⊂ G(𝔸)^{+} be an open subgroup containing ℚ^{×}G(ℝ)^{+} (where + denotes the identity component). Let Γ_{S} = S ∩ G(ℚ). Suppose that Γ ⊂ G(ℚ) is an arithmetic group which is contained in Γ_{S} and that:

(1) S/ℚ^{×}G(ℝ)^{+} is compact.

(2) Γ · ℚ^{×} = Γ_{S}.

(3) The set

Δ = \{(\begin{matrix}I_{2n} & 0 \\ 0 & tI_{2n} \end{matrix}) ∈ G(𝔸) | t ∈ Π_{p} ℤ_{p}^{×}\}

is contained in S.

Then [25] Thm. 3 (ii) states that the complex vector space of Γ-modular forms with weight k on 𝔥_{2n} is spanned by those forms whose Fourier coefficients are in the finite abelian extension k_{S} of ℚ that is determined by the set S. To apply this to our setting, let S_{1}(M) be the collection of elements x ∈ G(𝔸)^{+} such that each p-component x_{p} ∈ GL(2n, ℤ_{p}) and satisfies

x_{p} ≡ (\begin{matrix}I_{2n} & 0 \\ 0 & a_{p}I_{2n} \end{matrix}) (mod M · ℤ_{p})

for some a_{p} ∈ ℤ_{p}^{×}. Define S(M) = S_{1}(M)·ℚ^{×}, let S′(M) = NS(M)N^{−1}, and S = S(M) ∩ S′(M). Then hypothesis (1) is satisfied. It is easy to see that S(M), S′(M) both contain Δ, hence hypothesis (3) is satisfied for the set S. In this case, k_{S} = ℚ and

Γ_{S} = S ∩ G(ℚ) = (Γ(M) ∩ Γ̂(M)) · ℚ^{×} = Γ_{M} · ℚ^{×}

which verifies hypothesis (2). q.e.d.

9.5

Let I_{−} = ( \begin{matrix}I_{2n} & 0 \\ 0 & −I_{2n} \end{matrix} ) . Its action by fractional linear transformations

maps the Siegel lower half space 𝔥_{2n}^{−} to the upper half space 𝔥_{2n}, that is, I_{−} · Z = −Z. Hence, for any holomorphic mapping f : 𝔥_{2n} → ℂ we may define f′ : 𝔥_{2n} → ℂ by

f′(Z) = f(I_{−} · N · Z) = f(−𝔟Z^{t}𝔟^{−1}).

Proposition 9.6. If f : 𝔥_{2n} → ℂ is a holomorphic Γ-modular form of weight k, with rational Fourier coefficients, then f′ is also a holomorphic Γ-modular form of weight k, and

f(Ẑ) = \overline{f′(Z)} (9.6.1)

for all Z ∈ 𝔥_{2n}.

9.7 Proof

Suppose that f(γ · Z) = j(γ, Z)^{k}f(Z) for all γ ∈ Γ and all Z ∈ 𝔥_{2n} where

j (( \begin{matrix}A & B \\ C & D \end{matrix} ) , Z) = det(CZ + D)

is the standard automorphy factor. Then j(I_{−}N, Z) = det(−^{t}𝔟) is independent of Z. Let γ ∈ Γ and set

γ′ = I_{−}NγN^{−1}I_{−}^{−1} ∈ Γ.

Then

f′(γ · Z) = f(γ′ · I_{−}N · Z)
= j(γ′, I_{−}N · Z)^{k}f′(Z)
= det(−^{t}𝔟)^{k}j(γ, Z)^{k} det(−^{t}𝔟)^{−k}f′(Z)
= j(γ, Z)^{k}f′(Z).

which shows that f′ is Γ-modular of weight k. Next, with respect to the standard maximal parabolic subgroup P_{0} (which normalizes the standard 0-dimensional boundary component), the modular form f has a Fourier expansion,

f(Z) = ∑_{s} a_{s} exp (2πi⟨s, Z⟩)

which is a sum over lattice points s ∈ L^{∗} where L = Γ ∩ Z(U_{0}) is the intersection of Γ with the center of the unipotent radical U_{0} of P_{0} and where a_{s} ∈ ℚ. Then

f(Ẑ) = ∑_{s} a_{s} exp (2πi⟨s, 𝔟Z^{t}𝔟^{−1}⟩)

= \overline{∑_{s} a_{s} exp (2πi⟨s, −𝔟Z^{t}𝔟^{−1}⟩)}

= \overline{f(−𝔟Z^{t}𝔟^{−1})} = \overline{f′(Z)}.

q.e.d.

9.8

The Baily-Borel compactification X of X is the obtained by embedding X holomorphically into ℂℙ^{m} using m + 1 (Γ-)modular forms (say f_{0}, f_{1}, …, f_{m}) of some sufficiently high weight k, with rational Fourier coefficients, and then taking the closure of the image. Define an embedding Φ : X → ℂℙ^{2m+1} by

Φ(Z) = (f_{0}(Z) : f_{1}(Z) : ⋯: f_{m}(Z) : f′_{0}(Z) : f′_{1}(Z) : ⋯: f′_{m}(Z)) .

Denote these homogeneous coordinate functions by x_{j} = f_{j}(Z) and y_{j} = f′_{j}(Z). Define an involution σ : ℂℙ^{2m+1} → ℂℙ^{2m+1} by σ(x_{j}) = y_{j} and σ(y_{j}) = x_{j}. Then Equation (9.6.1) says that this involution is compatible with the embedding Φ, that is, for all Z ∈ X we have:

σΦ(Z) = Φ(Ẑ).

Define Ψ : ℂℙ^{2m+1} → ℂℙ^{2m+1} by setting ξ_{j} = x_{j} + y_{j} and η_{j} = i(x_{j} − y_{j}) for 0 ≤ j ≤ m. Let Y = ΨΦ(X) and let Y denote its closure.

Proposition 9.9. The composition ΨΦ : X → ℂℙ^{2m+1} is a holomorphic embedding which induces an isomorphism of complex algebraic varieties X → Y . The variety Y is defined over the rational numbers, and the real points of Y are precisely the image of those points Z ∈ X such that Ẑ = Z.

9.10 Proof

The image ΨΦ(X) is an algebraic subvariety of projective space that is preserved by complex conjugation, so it is defined over ℝ. The real points are obtained by setting ξ_{j} = ξ_{j} and η_{j} = η_{j} which gives x_{j} = y_{j} and y_{j} = x_{j} hence Φ(Z) = σΦ(Z), or Z = Ẑ. The Fourier coefficients of ξ_{j} and η_{j} are in ℚ[i] so the image ΨΦ(X) is defined over ℚ[i]. Since it is also invariant under Gal(ℂ/ℝ), it follows that ΨΦ(X) is defined over ℚ. q.e.d.

9.11 Remark

The embedding (f_{0} : f_{1} : ⋯: f_{m}) : X ↪ ℂℙ^{m} determines the usual rational structure on X, and the resulting complex conjugation is that induced by Z ↦ −Z for Z ∈ 𝔥_{2n}.

10. The symmetric space for Sp(2n, ℂ)

10.1

In this section we sketch a proof of the well-known (but difficult to reference) fact that the complex symplectic group acts transitively on the quaternionic Siegel space by fractional linear transformations, and that the stabilizer of each point is a maximal compact subgroup. These facts are not needed in the rest of the paper, however they may help to make this symmetric space look a little more familiar.

10.2 A quaternion algebra

Associated to the imaginary quadratic field ℚ(\sqrt{d}) we consider the quaternion algebra ℍ over ℝ that is generated by \underline{1}, \underline{i}, \underline{j}, \underline{k} with \underline{i}^{2} = \underline{k}^{2} = d\underline{1}, \underline{j}^{2} = −\underline{1} and \underline{i}\underline{j} = \underline{k}. If w = r\underline{1}+s\underline{i}+x\underline{j}+y\underline{k} set w = r\underline{1}−s\underline{i}−x\underline{j}−y\underline{k} and w^{∗} = r\underline{1} + s\underline{i} − x\underline{j} + y\underline{k}. If we embed θ : ℂ → ℍ by \sqrt{d} ↦ \underline{i} then θ(z) = θ(z) and we may write ℍ = θ(ℂ) ⊕ \underline{k}θ(ℂ). Define the purely quaternionic part of such an element w ∈ ℍ to be

Qu(w) = \underline{k}^{−1}(x\underline{j} + y\underline{k}) = y + \frac{x}{d}\underline{i}.

The mapping θ extends to an injective algebra homomorphism θ : M_{n×n}(ℂ) → M_{n×n}(ℍ) by applying θ to each matrix entry. For notational convenience we shall often omit the use of the symbol θ.

If A, B ∈ GL(n, ℍ) then ^{t}(AB)^{∗} = (^{t}B^{∗})(^{t}A^{∗}). If Qu(A) = 0 then A\underline{k} = \underline{k}A. An element A ∈ M_{n×n}(ℍ) is Hermitian if A = ^{t}A. In this case ⟨z, z⟩_{A} = ^{t}zAz is real, for all z ∈ ℍ^{n}. The element A is positive definite (written A > 0) if ⟨z, z⟩_{A} > 0 for all nonzero z. The unitary group U(n, ℍ) (sometimes denoted Sp(n)) consists of those A ∈ M_{n×n}(ℍ) such that A^{−1} = ^{t}A.

10.3

Define the quaternionic Siegel space

Y_{n} = {W ∈ M_{n×n}(ℍ) | W^{∗} = ^{t}W, Qu(W) > 0} .

Proposition 10.4. The symplectic group Sp(2n, ℂ) acts transitively on the quaternionic Siegel space Y_{n} by fractional linear transformations: if g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) then

g · W = (AW + B)(CW + D)^{−1}

where we have identified θ(A) with A, etc. Moreover,

Qu(g · W) = ^{t}(\overline{CW + D})^{−1}Qu(W)(CW + D)^{−1}. (10.4.1)

The stabilizer of the basepoint W_{0} = \frac{1}{\sqrt{−d}} \underline{k}I_{n} ∈ Y_{n} is the unitary group U(n, ℍ) over the quaternions, which is embedded in Sp(2n, ℂ) by

A + \frac{1}{\sqrt{−d}}B\underline{k} ↦ (\begin{matrix}A & B \\ −B & A \end{matrix}) .

10.5 Proof

Equation (10.4.1) may be verified by a (tedious) direct computation. It follows, for any g ∈ Sp(2n, ℂ), that W ∈ Y_{n} iff g · W ∈ Y_{n}. The remaining statements may be verified by direct computation. q.e.d.

10.6

Define the homomorphism μ : ℍ → M_{2×2}(ℂ) as follows:

d ≢ 1 (mod 4)  d = 4m + 1
μ(\underline{i})  ( \begin{matrix}0 & d \\ 1 & 0 \end{matrix} )  σ^{−1} ( \begin{matrix}0 & d \\ 1 & 0 \end{matrix} ) σ
μ(\underline{j}) ( \begin{matrix}\sqrt{−1} & 0 \\ 0 & −\sqrt{−1} \end{matrix} ) σ^{−1} ( \begin{matrix}\sqrt{−1} & 0 \\ 0 & −\sqrt{−1} \end{matrix} ) σ
μ(\underline{k}) ( \begin{matrix}0 & −d\sqrt{−1} \\ \sqrt{−1} & 0 \end{matrix} ) σ^{−1} ( \begin{matrix}0 & −d\sqrt{−1} \\ \sqrt{−1} & 0 \end{matrix} ) σ

where σ = ( \begin{matrix}2 & 1 \\ 0 & 1 \end{matrix} ) . This mapping extends to a homomorphism

μ : M_{r×r}(ℍ) → M_{2r×2r}(ℂ)

which replaces each matrix entry with the 2×2 block defined above. The following fact is immediate:

Lemma 10.7. The composition

M_{r×r}(ℂ) ⟶^{θ} M_{r×r}(ℍ) ⟶^{μ} M_{2r×2r}(ℂ)

takes values in M_{2r×2r}(ℝ) and it coincides with the mapping ψ_{r} of (5.2.1). In particular, it restricts to the injective homomorphism ψ_{n} : Sp(2n, ℂ) → Sp(ℝ^{4n}, S_{2}) of §5.7.

Proposition 10.8. The mapping μ takes the quaternionic Siegel space Y_{n} ⊂ M_{n×n}(ℍ) diffeomorphically to the symmetric space

𝔥_{2n}𝔳 ⊂ M_{2n×2n}(ℂ)

for Sp(ℝ^{4n}, S_{2}). Its image is the fixed point set of the involution τ_{h}. Moreover, for each g ∈ Sp(2n, ℂ) and W ∈ Y_{n} we have

μ(g · W) = ψ_{n}(g) · μ(W). (10.8.1)

10.9 Proof

Since μ and θ are algebra homomorphisms, for

g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(2n, ℂ)

we find,

μ(g · W) = μ ((θ(A)W + θ(B)) (θ(C)W + θ(D))^{−1})
= (μθ(A)μ(W) + μθ(B))(μθ(C)μ(W) + μθ(D))^{−1}
= ψ(g) · μ(W)

which verifies (10.8.1). A direct calculation shows that μ takes the base point W_{0} = \frac{1}{\sqrt{−d}} \underline{k}I_{n} ∈ Y_{n} to the following base point x_{2} ∈ 𝔥_{2n}𝔳,

x_{2} = \frac{i}{\sqrt{−d}}Diag^{n} (\begin{matrix}0 & −d \\ 1 & 0 \end{matrix}) or

x_{2} = \frac{i}{\sqrt{−d}}Diag^{n}σ^{−1} (\begin{matrix}0 & −d \\ 1 & 0 \end{matrix}) σ = \frac{i}{\sqrt{−d}}Diag^{n} (\begin{matrix}−1 & −2m \\ 2 & 1 \end{matrix})

depending on whether d ≢ 1 (mod 4) or d = 4m + 1 respectively. It follows from (5.8.6) that μ takes Y_{n} to 𝔥_{2n}𝔳 and it further follows from Proposition 5.14 that its image is precisely the fixed point set under τ_{h}.

q.e.d.

11. Concluding remarks

11.1

The definition of anti-holomorphic multiplication given in §7 extends in an obvious manner to more general CM fields. Let F be a totally real, degree m extension of ℚ and let E = F [\sqrt{d}] be a totally imaginary quadratic extension of F (with d ∈ O_{F}). Let (A = ℂ^{2n}/L, H = R + iQ) be a principally polarized abelian variety as in §7. Then an antiholomorphic multiplication by the ring of integers O_{E} is a homomorphism Ψ : O_{E} → End_{ℝ}(A) such that Ψ(O_{F}) ⊂ End_{ℂ}(A), such that κ = Ψ(\sqrt{d}) : ℂ^{2n} → ℂ^{2n} is anti-linear (κ(ax) = aκ(x) for all a ∈ ℂ and x ∈ ℂ^{r}), and such that Q(Ψ(b)x, y) = Q(x, Ψ(b)y) for all b ∈ O_{E} and y ∈ ℂ^{2n}. One might then expect (1) that the moduli space of principally polarized abelian varieties with anti-holomorphic multiplication by O_{E} and appropriate level structure may be identified with the locus of real points in a corresponding Hilbert-Siegel modular variety, and (2) that it consists of finitely many copies of Γ\D where Γ ⊂ Sp(2n, O_{e}) is an appropriate level subgroup and where D = Y_{n} × ⋯ × Y_{n} is a product of m copies of the symmetric space Y_{n} = Sp(2n, ℂ)/U(n, ℍ).

11.2

One might ask whether the closure of X_{ℝ} in the Baily-Borel Satake compactification X coincides with the locus of real points (X)_{ℝ} of the Baily-Borel compactification. Although we do not know the answer to this question, in [9] we were able to show, in the case n = 1 (that is, when X_{ℝ} is an arithmetic quotient of real hyperbolic 3-space), that the difference (X)_{ℝ} − X_{ℝ} consists at most of finitely many points.

11.3

In [8] we consider a different rational structure on the Siegel modular variety X = Γ\𝔥_{n} and a different anti-holomorphic involution τ′, such that the resulting locus of real points (let us call it X′_{ℝ}) may be naturally identified with the moduli space of real abelian varieties (with appropriate level structure); and we show that this moduli space consists of finitely many copies of the locally symmetric space Λ\GL(n, ℝ)/O(n) (for appropriate principal congruence subgroups Γ and Λ). The involution τ′ arises from an involution on Sp(2n, ℝ) whose fixed point set is GL(n, ℝ). In this paper also, the key technical tool is the lemma of Comessatti and Silhol. Although the outline of [8] is parallel to that of the present paper, the technical details are completely different and we do not yet know how to formulate or prove the most natural general statement along these lines. Interesting related results are described in [2].

References

[1] A. Adler, Antiholomorphic involutions of analytic families of abelian varieties, Trans. Amer. Math. Soc. 254 (1979) 69–94, MR 0539908 (81f:10035), Zbl 0423.14024.

[2] D. Allcock, J. Carlson & D. Toledo, Real cubic surfaces and real hyperbolic geometry, C. R. Math. Acad. Sci. Paris 337 (2003) 185–188, MR 2001132.

[3] N. Bourbaki, Commutative Algebra, Hermann/Addison Wesley, Reading MA, 1972, MR 0360549 (50 #12997), Zbl 0279.13001.

[4] H. Comessatti, Sulle varietà abeliane reali I, II, Ann. Mat. Pura. Appl. 2 (1924) 67-106 and 4 (1926) 27–72.

[5] G. Faltings & C.-L. Chai, Degeneration of Abelian Varieties, Ergeb. Math., 22, Springer Verlag, New York, 1990, MR 1083353 (92d:14036), Zbl 0744.14031.

[6] A. Fröhlich & M. Taylor, Algebraic Number Theory, Cambridge University Press, Cambridge, 1991, MR 1215934 (94d:11078), Zbl 0744.11001.

[7] B. Gordon, Canonical models of Picard modular surfaces, in ‘The Zeta Functions of Picard Modular Surfaces’, Publ. CRM, Montreal, 1988, 1–29, MR 1155224 (93a:14023), Zbl 0756.14011.

[8] M. Goresky & Y.-S. Tai, The moduli space of real abelian varieties with level structure, Comp. Math. 139 (2003) 1–27, MR 2024963.

[9] M. Goresky & Y.-S. Tai, Abelian surfaces with anti-holomorphic multiplication, preprint, I.A.S., 2001, math.AG/0108099.

[10] H. Jaffee, Real forms of Hermitian symmetric spaces, Bull. Amer. Math. Soc. 81 (1975) 456–458, MR 0412490 (54 #613), Zbl 0312.32018.

[11] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978, MR 0514561 (80k:53081), Zbl 0451.53038.

[12] S. Kudla, On the R-forms of certain algebraic varieties, Bull. Amer. Math. Soc. 81 (1975) 471–473, MR 0379505 (52 #410).

[13] H. Lange & C. Birkenhake, Complex Abelian Varieties, Grundlehren Math. Wiss., 302, Springer Verlag, Berlin, 1992, MR 1217487 (94j:14001), Zbl 0779.14012.

[14] N. Nygaard, Construction of some classes in the cohomology of Siegel modular threefolds, Comp. Math. 97 (1995) 173–186, MR 1355124 (96h:14037), Zbl 0838.11046.

[15] P. Raševskiǐ, The connection of the set of points of a Lie group that are fixed under one of its automorphisms, Funk. Anal. i Priložen 6 (1972) 97–98 (English translation: Funct. Anal. Appl 6 (1972) 341–342), MR 0311841 (47 #403).

[16] P. Raševskiǐ, A theorem on the connectedness of the subgroup of a simply connected Lie group that commutes with one of its automorphisms, Trudy Moskov. Mat. Obšč 30 (1974) 3–22 (English translation: Trans. Moscow Math. Soc. 30 (1974) 1–24), MR 0463350 (57 #3303).

[17] J. Rohlfs, Arithmetisch definierte Gruppen mit Galoisoperation, Invent. Math. 48 (1978) 185–205, MR 0507801 (80j:20043), Zbl 0391.14007.

[18] J. Rohlfs, The Lefschetz number of an involution on the space of classes of positive definite quadratic forms, Comment. Math. Helv. 56 (1981) 272–296, MR 0630954 (83a:10037), Zbl 0474.10019.

[19] J. Rohlfs & J. Schwermer, Intersection numbers of special cycles, J. Amer. Math. Soc. 6 (1993) 755–778, MR 1186963 (94a:11075), Zbl 0811.11039.

[20] I. Satake, Algebraic Structures Of Symmetric Domains, Princeton University Press, Princeton NJ, 1980, MR 0591460 (82i:32003), Zbl 0483.32017.

[21] J.P. Serre, Galois Cohomology, Springer Verlag, Berlin, 1997, MR 1466966 (98g:12007), Zbl 0902.12004.

[22] K.-Y. Shih, Anti-holomorphic automorphisms of arithmetic automorphic function fields, Ann. of Math. 103 (1976) 81–102, MR 0466027 (57 #5910), Zbl 0328.10024.

[23] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Math. Soc. Jap., 1971, reprinted Princeton University Press, Princeton NJ, 1994, MR 1291394 (95e:11048), Zbl 0872.11023.

[24] G. Shimura, On the field of rationality for an abelian variety, Nagoya Math. J. 45 (1972) 167–178, MR 0306215 (46 #5342), Zbl 0243.14012.

[25] G. Shimura, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen Math-Phys. 17 (1975) 261–268, MR 0485706 (58 #5528), Zbl 0332.32024.

[26] G. Shimura, On the real points of an arithmetic quotient of a bounded symmetric domain, Math. Ann. 215 (1975) 135–164, MR 0572971 (58 #27992), Zbl 0394.14007.

[27] G. Shimura, On abelian varieties with complex multiplication, Proc. London Math. Soc. 34 (1977) 65–86, MR 0572987 (58 #28008).

[28] R. Silhol, Compactifications of moduli spaces in real algebraic geometry, Inv. Math. 107 (1992) 151–202, MR 1135469 (93i:14018), Zbl 0777.14014.

[29] R. Silhol, Real abelian varieties and the theory of Comessatti, Math. Z. 181 (1982) 345–362, MR 0678890 (85j:14077), Zbl 0492.14015.

[30] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., 80, American Mathematical Society, Providence RI, 1968, MR 0230728 (37 #6288), Zbl 0164.02902.

[31] M. Yoshida, A hyperbolic structure on the real locus of the moduli space of marked cubic surfaces, Topology 40 (2001) 469–473, MR 1838991 (2002b:14016).

School of Mathematics Institute for Advanced Study

Princeton NJ

Dept. of Mathematics Haverford College

Haverford PA