Anti-holomorphic multiplication and a real algebraic ...goresky/pdf/newhyp.jour.pdf · j. differential geometry 65 (2003) 513-560 ANTI-HOLOMORPHIC MULTIPLICATION AND A REAL ALGEBRAIC

j. differential geometry

65 (2003) 513-560

ANTI-HOLOMORPHIC MULTIPLICATION AND AREAL ALGEBRAIC MODULAR VARIETY

MARK GORESKY & YUNG SHENG TAI

AbstractAn anti-holomorphic multiplication by the integers Od of a quadratic imag-inary number field, on a principally polarized complex abelian variety AC

is an action of Od on AC such that the purely imaginary elements act inan anti-holomorphic manner. The coarse moduli space XR of such A (withappropriate level structure) is shown to consist of finitely many isomor-phic connected components, each of which is an arithmetic quotient of thequaternionic Siegel space, that is, the symmetric space for the complex sym-plectic group. The moduli space XR is also identified as the fixed point setof a certain anti-holomorphic involution τ on the complex points XC of theSiegel moduli space of all principally polarized abelian varieties (with ap-propriate level structure). The Siegel moduli space XC admits a certainrational structure for which the involution τ is rationally defined. So thespace XR admits the structure of a rationally defined, real algebraic variety.

1. Introduction

1.1

Let hn = Sp(2n, R)/U(n) be the Siegel upper half space of rank n andlet Γ = Sp(2n, Z). The quotient Γ\hn 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 vari-eties.

(c) It has a natural compactification (the Baily-Borel Satake compact-ification) 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.

513

514 m. goresky & y.s. tai

Among the many missing ingredients in the theory of automorphicforms for groups of non-Hermitian type are analogues of these threefacts. The associated locally symmetric spaces do not appear to have analgebraic structure; they do not appear to be associated with a naturalclass of elliptic curves or abelian varieties, and although they have manycompactifications, there does not appear to be a canonical or “best”one. In the early 1970’s G. Shimura, J. Millson and M. Kuga [10] askedwhether it might be possible to address these shortcomings by realizinga locally symmetric space W for a group of non-Hermitian type as asubspace of a locally symmetric space X for a group of Hermitian type;perhaps interpreting W as a moduli space of a class of real abelianvarieties. 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 expectedin general. He found a moduli space XC (for a certain class of abelianvarieties) which had a model defined over R, such that the locus XR ofreal points did not represent a moduli space for the corresponding realabelian varieties.

1.2

We wish to revisit this question for quotients

W = Γ\Yn(1.2.1)

of the symmetric space Yn = Sp(2n, C)/U(n, H) (the “quaternionicSiegel space”, cf. §10) by the principal congruence subgroup

Γ = Sp(2n,Od)[M ]

of Sp(2n,Od) of level M. Here, d < 0 is a square-free integer and Od

is the ring of integers in the quadratic imaginary number field Q(√

d).If M ≥ 3 then the space W is a smooth manifold of (real) dimension2n2 + n. It does not have an (obvious) algebraic structure. In the casen = 1, W is an arithmetic quotient of the real hyperbolic 3-space, Y1.

In this paper we show, for appropriate level M, that a certain disjointunion XR of finitely many copies of W admits analogs to all three ofthe above statements. That is:

(a) The smooth manifold XR is the set of fixed points of an anti-holomorphic involution τ of a quasi-projective complex algebraicvariety XC.

anti-holomorphic multiplication 515

(b) The space XR may be naturally identified with the (coarse) modulispace of n-dimensional abelian varieties with level M structure andwith anti-holomorphic multiplication (see below) by Od.

(c) The complex variety XC and the involution τ admit a model thatis defined over the rationals Q.

1.3

The algebraic variety XC is just the Siegel moduli space Γ(M)\h2n ofprincipally polarized abelian varieties with level M structure. The in-volution τ extends to an anti-holomorphic involution of the Baily-BorelSatake compactification X of XC and hence defines a real structure onX. In §9 we make use of a result of Shimura [25] to prove an analogueof statement (c) above by showing that X admits a rational structurethat is compatible with this real structure.

1.4

In this paper we introduce the concept of anti-holomorphic multiplica-tion of the ring of integers Od on a principally polarized abelian varietyA: it is an action of Od on A by real endomorphisms which are com-patible with the polarization, such that the purely imaginary elementsof Od act in an anti-holomorphic manner; see §7.3 for the precise def-inition. If such an action exists then the (complex) dimension of A iseven (so elliptic curves do not admit anti-holomorphic multiplication).The definition of anti-holomorphic multiplication extends in an obviousmanner to more general CM fields, cf. §11.1. This appears to be a veryinteresting structure which merits further study.

1.5

The key technical tool in this paper, which appears to be a missing ingre-dient in the earlier work on this question, is Proposition 7.7, an analog ofthe lemma of Comessatti and Silhol ([29]). It describes “normal forms”for the period matrix of an abelian variety with anti-holomorphic mul-tiplication. 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], thepaper [14], and recent results of [2] and [31] suggest that there arerich, largely unexplored phenomena involving real structures and modulispace interpretations of arithmetic quotients of non-Hermitian symmet-ric spaces. In this paper we have chosen particular arithmetic groupsfor which the results are (relatively) easy to state and prove, and forwhich the associated Shimura variety is defined over the rational num-bers. Although it is possible to establish similar results for many otherarithmetic groups, we do not know to what extent these results may begeneralized to arbitrary arithmetic groups. (See also §11.)

1.7

We would like to thank J. van Hamel for useful conversations, and ananonymous referee for carefully reading the manuscript and making anumber of useful suggestions. The first author is grateful to the Institutefor Advanced Study for its hospitality and support during the period inwhich this research was carried out.

2. Statement of results

2.1

Throughout this paper we fix a square-free integer d < 0 and denoteby Od the ring of integers in the quadratic imaginary number fieldQ(

√d). Let Q0 be the “standard” symplectic form, whose matrix is

J =(

0 I−I 0

). For any ring R we use any of the standard notations

Sp(2n, R), Sp(R2n, Q0), or Sp(R2n, J) to denote the symplectic groupconsisting of all g ∈ GL(2n, R) such that tgJg = J, or equivalently,Q0(gx, gy) = Q0(x, y) for all x, y ∈ R2n. It consists of matrices

g =(

A BC D

)such that

g−1 =(

tD − tC− tB tA

).


Let Ks ⊂ Sp(2n, C) be the maximal compact subgroup that is fixedunder the Cartan involution θs(g) = t(g)−1. It is isomorphic to theunitary group U(n, H) over the quaternions. Let Yn = Sp(2n, C)/Ks

be the associated symmetric space. It is not compact and not Hermi-tian; for n = 1 it is the real hyperbolic 3-space. In §10 (which is notneeded for the main results in this paper) we describe Yn as a certainquaternionic Siegel space, on which Sp(2n, C) acts by fractional lineartransformations.

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 (alsodenoted τ) on the (usual) Siegel space h2n

∼= G/Gθ, 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 φθs = θφ and whose image is exactly the set of fixed pointsGτ of τ. It has the property that

φ−1(Sp(4n, Z)) = Sp(2n,Od),

an arithmetic group that we will denote by Λ0.

Lemma 2.2. The mapping φ passes to a closed embedding φ :Yn → h2n whose image is the set hτ

2n of points fixed by the anti-holomorphicinvolution τ.

The proof appears in §5.15. Let Γ ⊂ Sp(4n, Z) be a torsion-freearithmetic subgroup which is preserved by the involution τ. If d ≡1 (mod 4) then assume also that Γ is contained in the principal con-gruence subgroup Γ(2) of level 2. Set X = Γ\h2n and let π : h2n → Xbe the projection. Let Λ = φ−1(Γ) ⊂ Sp(2n, C) so that φ(Λ) = Γτ isthe τ -invariants in Γ. Set W = Λ\Yn. Then φ also passes to a closedembedding φ : W → X whose image is π(φ(Yn)). If h ∈ G we denote byhφ(W ) = π(hφ(Yn)). Set

Γ =γ ∈ Sp(4n, Z) | τ(γ)γ−1 ∈ Γ

.

Let 〈τ〉 = 1, τ be the group generated by τ and let H1(〈τ〉, Γ) be the(nonabelian) cohomology of Γ.

Theorem 2.3. There is a canonical isomorphism

H1(〈τ〉, Γ) ∼= Γ\Γ/φ(Λ0).(2.3.1)


The involution τ passes to an anti-holomorphic involution τ : X → Xand hence defines a real structure on X. The set of real points XR = Xτ

is the disjoint union

XR =∐h

hφ(W )(2.3.2)

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

h ∈ Γ\Γ/φ(Λ0) ∼= H1(〈τ〉, Γ).

The proof is in §3.10. For the next two results we fix an integerM ≥ 3. If d ≡ 1 (mod 4) then we assume also that M is even. Denote byΓ(M) the principal congruence subgroup of level M in Sp(4n, Z). In thepreceding theorem, take Γ = ΓM = Γ(M) ∩ τ(Γ(M)). Then Equation(5.10.1) says that the arithmetic group Λ = φ−1(Γ) is the principalcongruence subgroup Sp(2n,Od)(M) of level M in the symplectic groupover Od.

Let X denote the Baily-Borel compactification of X = ΓM\h2n. Itcarries the structure of a complex projective algebraic variety. In §9.9we prove the following.

Theorem 2.4. The locus XR is the set of real points of a quasi-projective algebraic variety which has a model defined over the rationalnumbers. That is, there exists a holomorphic embedding in projectivespace X → Pm such that the image of X is defined over the rationalnumbers Q, and such that the involution τ : X → X is the restrictionof an anti-holomorphic involution τ : Pm → Pm, also defined over Q,which preserves X.

If A is an abelian variety with a principal polarization and a levelM structure, an anti-holomorphic multiplication by Od on A is a ho-momorphism Od → EndR(A) which is compatible with the polarizationand level structures, such that

√d acts as an anti-holomorphic mapping,

cf. §7.3. In §8.6 we prove the following:

Theorem 2.5. The real algebraic variety XR may be canonicallyidentified with the coarse moduli space of abelian varieties with principalpolarization, level M structure, and anti-holomorphic multiplication bythe ring Od.

In summary, this coarse moduli space XR (of abelian varieties withanti-holomorphic multiplication) may be realized as the locus of realpoints of an algebraic variety defined over Q. It consists of finitely many


isomorphic connected components, each of which is diffeomorphic to thearithmetic quotient (or locally symmetric space)

W = Sp(2n,Od)(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 thegroup 1, τ and let H1(〈τ〉, H) be the first nonabelian cohomology set.For any γ ∈ H let fγ : 〈τ〉 → H be the mapping fγ(1) = 1 and fγ(τ) =γ. Then fγ is a 1-cocycle iff γτ(γ) = 1, in which case its cohomologyclass is denoted [fγ ]. Two cocycles fγ and fγ′ are cohomologous iff thereexists h ∈ H so that γ′ = τ(h)γh−1.

Let G be a reductive algebraic group defined over R, let θ be a Car-tan involution with K = Gθ the maximal compact subgroup of θ-fixedpoints, and let E = G/K be the resulting symmetric space. Supposeτ : G → G is an involution which commutes with θ. Denote by Gτ , Kτ ,and Eτ the corresponding fixed point sets in G, K, and E respectively.For notational simplicity we will often write g for τ(g). The cobound-ary δ : Eτ → H1(〈τ〉, K) may be defined as follows. If g ∈ G and ifgK ∈ Eτ then τ(gK) = τ(g)K = gK so there exists k ∈ K so that

g = gk.(3.1.1)

Applying τ to this equation gives g = gkk, hence k defines a 1-cocyclefk = δ(gK).

Proposition 3.2. The cohomology sequence

1 −→ Kτ −→ Gτ −→ Eτ δ−→ H1(〈τ〉, K) i−→ H1(〈τ〉, G)

is exact. Moreover:

1. The mapping δ is trivial.

2. The mapping i is a bijection.

3. The inclusion Gτ ⊂ G induces a diffeomorphism

Gτ/Kτ ∼= Eτ .


3.3 Proof

Exactness of the cohomology sequence is standard ([21] §5.4). In theparagraph below we will show that δ is a locally constant mapping. Sinceτ acts by isometries, Eτ is connected and in fact the unique geodesicbetween any two points x, x′ ∈ Eτ is fixed under τ. It follows that δ takesEτ 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: clearlythe mapping Gτ/Kτ → Eτ 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 ifk ∈ K is sufficiently close to the identity and if fk is a 1-cocycle, thenit is also a coboundary. For in this case we may write k = exp(k)where k ∈ k = Lie(K). Let τ ′ : k → k be the differential of τ. Fromkτ(k) = 1 we obtain τ ′(k) = −k. Then the element a = exp(−1

2 k)satisfies τ(a)a−1 = a−2 = k which shows that the cohomology classdefined by k is trivial.

Now suppose that gK, g0K ∈ Eτ . Set τ(g) = gk and τ(g0) = g0k0

and let u = k−10 k. Since k = k−1 and k0 = k−1

0 we find that uk−10 uk0 = 1.

This means that u ∈ K defines a 1-cocycle in H1(〈µ〉, K) where µ : K →K is the involution µ(v) = k−1

0 vk0. By the preceding paragraph, if g, g0

are sufficiently close then this cocycle gives the trivial cohomology classso there exists a ∈ K such that

u = µ(a)a−1 = k−10 ak0a

−1

or k = ak0a−1. This says that the cocycles defined by k and by k0 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 : H1(〈τ〉, G) → H1(〈τ〉, K). Let G = KP be the Cartandecomposition of G that is determined by θ. Then P = exp(p) wherep is the −1-eigenspace of θ on g = Lie(G), so that θ(p) = p−1 ∈ P forall p ∈ P. Let g = kp ∈ G and suppose that fg is a 1-cocycle. Thenkpkp = 1 or

(kk)(k−1pk) = p−1 ∈ P.

It follows from the Cartan decomposition that

kk = 1 and k−1pkp = 1(3.3.1)


so we may define j(fg) = fk. We claim that fk represents the samecohomology class as fg in H1(〈τ〉, G), from which it will follow that jis well-defined and that i is surjective. Equation (3.3.1) says that fp

is a 1-cocycle for the involution µ of G defined by µ(x) = k−1xk. Ifp = exp p, set a = exp(−p/2). If µ′ : p → p denotes the differential ofµ then µ′(p) = −p so µ(a) = a−1 and p = µ(a)a−1. It follows that g =kp = kk−1aka−1 = aka−1 which says that fg and fk are cohomologous.

q.e.d.

3.4

For the remainder of this section we assume G is a reductive algebraicgroup defined over Q, that θ is a Cartan involution of G, and that τ isan involution of G that commutes with θ. We often write g for τ(g). LetG = G(R) denote the group of real points, K = Gθ the correspondingmaximal compact subgroup, and E = G/K the associated symmetricspace. Fix an arithmetic subgroup Γ ⊂ G(Q) and let π : E → X = Γ\Ebe the projection.

To every 1-cocycle fγ in H1(〈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

Eτγ = x ∈ E |τ(x) = γx(3.4.1)

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

Theorem 3.5. Suppose Γ is torsion-free. Then the associationfγ → Xτγ determines a one to one correspondence between the coho-mology set H1(〈τ〉, Γ) and the connected components of the fixed pointset Xτ .

3.6 Proof

The twisted involution τγ : E → E acts by isometries so ([11] I §13.5)the fixed point set Eτγ is nonempty. If x, x′ ∈ Eτγ then the uniquegeodesic joining them is also fixed by τγ, so Eτγ is connected. Itsimage in X is a connected subset X(τγ) of Xτ which depends onlyon the cohomology class of fγ . It is easy to check that fγ and fγ′ arecohomologous iff X(τγ) ∩ X(τγ′) = φ. q.e.d.


3.7

In general the cohomology set H1(〈τ〉, Γ) may be difficult to compute,the connected component X(τγ) may be difficult to describe, and dis-tinct connected components may fail to be isomorphic. We will intro-duce additional hypotheses which will allow us to address these threeissues. Let Γ ⊂ G(Q) be a τ stable arithmetic group that contains Γ.Let θ : G → G be the Cartan involution corresponding to K. Considerthe following hypotheses:

(1) G is Zariski connected and the fixed subgroup Gτ = Gτ (R) isZariski connected.

(2) H1(〈τ〉, K) is trivial.

(3) τ acts trivially on Γ\Γ.

(4) H1(〈τ〉, Γ) → H1(〈τ〉, Γ) is trivial.

(5) Γ is torsion-free.

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

(a) Under hypothesis (1) above, Gτ is reductive, θ restricts to a Cartaninvolution of Gτ , and Kτ is a maximal compact subgroup of Gτ .

(b) Under hypothesis (2) above, the mapping Gτ/Kτ → Eτ is an iso-morphism. If γ ∈ G and if fγ is a 1-cocycle, then under hypothesis(2), its class in H1(〈τ〉, G) is trivial if and only if Eτγ is nonempty.

(c) Under hypothesis (3) above, the association γ → γγ−1 defines amapping Γ → Γ which passes to an injection

Γ\Γ/Γτ → H1(〈τ〉, Γ).(3.8.1)

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

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

[fγ ] ∈ H1(〈τ〉, Γ)

there exists h ∈ Γ such that γ = τ(h)h−1, in which case,

Eτγ = hEτ and Γτγ = hΓτ = hΓτh−1.


(e) Under hypotheses (4), (3), and (5) the fixed point set Xτ is thedisjoint union of isomorphic copies

Xτ =∐

h∈Γ\Γ/Γτ

hΓτ\hEτ

of the quotient Γτ\Eτ .

In summary, if hypotheses (1) through (5) are satisfied, then thefixed point set Xτ consists of finitely many isomorphic copies of thearithmetic quotient Γτ\Gτ/Kτ indexed by H1(〈τ〉, Γ) ∼= Γ\Γ/Γτ .

3.9 Proof

Part (a) is proven in [20] Chapt. 1 Thm. 4.2 and Cor. 4.5 (pages 15 and17). Now consider part (b). Clearly Gτ/Kτ ⊂ Eτ so it suffices to showthat Gτ acts transitively on Eτ . Let x = gK ∈ Eτ . Then τ(g)K = gKso the element k = g−1τ(g) lies in K. Moreover, fk is a cocycle, so byhypothesis (2) there exists u ∈ K with k = uτ(u)−1 = g−1τ(g). Thengu ∈ Gτ and x = gK = guK. To prove the second statement in part(b), let fγ be a 1-cocycle and suppose there exists a point gK ∈ Eτγ .Then γgK = gK so there exists (a unique) k ∈ K with γg = gk,or γ = gkg−1. Hence fγ is cohomologous to fk, which is trivial by (2).Part (c) follows from the long exact cohomology sequence for the groupsΓ ⊂ Γ and “twisting”, however it is also easy to verify directly. Letγ ∈ Γ. By (3) there exists a unique a ∈ Γ such that γ = aγ. Moreover,fa is a 1-cocycle, so we have defined a mapping φ : Γ → H1(〈τ〉, Γ).Suppose γ′ ∈ Γ determines the same cohomology class φ(γ), that is,suppose a′ = γ′(γ′)−1 = bab−1 for some b ∈ Γ. Let x = γ−1b−1γ′. Thenx ∈ Γτ because x = γ−1b−1γ′ = γ−1b−1bab−1γ′ = γ−1a−1ab−1γ′ =x. Consequently γ′ ∈ ΓγΓτ , 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’ theoremand 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 provenin 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 nonde-generate symmetric bilinear form R and let Sp(V, S) be the group oflinear automorphisms of V that preserve S. Let V ∗ be the dual vectorspace. If g ∈ GL(V ) define g∗ ∈ GL(V ∗) by g∗(λ)(v) = λ(g−1v) forany λ ∈ V ∗. Let N ∈ GL(V ) and suppose that N2 = dI for some realnumber d. Define automorphisms τ and θ of GL(V ) by

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

(for all u, v ∈ V ). Then τ and θ are involutions, and in fact θ is a Cartaninvolution: its fixed point set is the orthogonal group O(V, R). DefineS : V → V ∗ by S(u)(v) = S(u, v). Let S : V ∗ → V be the mappingthat is uniquely determined by the relation S(S(λ), x) = λ(x) for anyλ ∈ 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 RSRS = cI is a multiple of the identity, then the involutionθ preserves the symplectic group Sp(V, S) and its restriction toSp(V, S) is a Cartan involution.

4.3 Proof

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

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

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

VS

−−−→ V ∗ R

−−−→ VS

−−−→ 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 firstsquare commutes iff g ∈ Sp(V, S) (which we assume). The outside rect-angle commutes by hypothesis. It follows that the third square also com-mutes, but this is equivalent to the statement that θ(g) ∈ Sp(V, S). Fi-nally, 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 pre-serves a certain maximal compact subgroup K and which passes to aninvolution τ on the Siegel space. This involution is first constructed in acoordinate-free manner, but with respect to a non-standard symplecticform (S2), and is denoted τh, see Lemma 5.6. Then we change coordi-nates so as to convert S2 to the usual symplectic form, and obtain theinvolution τ. The impatient reader may skip directly to the matrix de-scriptions (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 asquare root,

√d. Let Od be the ring of integers in the quadratic imag-

inary number field Q(√

d), that is, Od = Z + Zω where ω =√

d ifd ≡ 1 (mod 4) and ω = (1 +

√d)/2 if d ≡ 1 (mod 4). Let h : Cr → R2r

be the vector space isomorphism

h(x1 + ωy1, x2 + ωy2, . . . , xr + ωyr) = (x1, y1, x2, y2, . . . , xr, yr).

Then there is a unique homomorphism

ψr : Mr×r(C) → M2r×2r(R).(5.2.1)

such that h(gz) = ψr(g)h(z) for all z ∈ Cr. It takes the matrix

(aij + ωbij)


(1 ≤ i, j ≤ r; aij , bij ∈ C) to the matrix that consists of 2×2 blockszij = ψ1(aij + ωbij) where

ψ1(a + ωb) =(

a dbb a

)or(

a mbb a + b

)(5.2.2)

when d ≡ 1 (mod 4) or d = 4m + 1 respectively.The complex linear mapping Cr → Cr given by multiplication by

√d

therefore corresponds to a real linear mapping Nh = ψr(√

dI) : R2r →R2r. As in §4 define the involution τh : GL(2r, R) → GL(2r, R) by

τh(g) = NhgN−1h .(5.2.3)

5.3

Take r = 2n. Fix a complex symplectic form s : C2n × C2n → C. Usingthe isomorphism h we obtain a bilinear mapping S : R4n × R4n → R2

whose components we denote by S1 and S2, that is, s(h−1u, h−1v) =S1(u, v) + ωS2(u, v) for all u, v ∈ R2n. Then S1 and S2 are (real) sym-plectic forms on R4n and we denote by Sp(R4n, Si) the correspondingsymplectic groups. Since s(

√dx, y) = s(x,

√dy) =

√ds(x, y) we have(

S1(Nhx, y)S2(Nhx, y)

)=(

S1(x, Nhy)S2(x, Nhy)

)= β

(S1(x, y)S2(x, y)

)(5.3.1)

for all x, y ∈ R4n, where β is the matrix for Nh for r = 1; see §5.7. Using(5.3.1) it is easy to see that:

Lemma 5.4. The mapping ψ2n restricts to an isomorphism

Sp(C2n, s) → Sp(R4n, S1) ∩ Sp(R4n, S2).(5.4.1)

For i = 1, 2 the involution τh preserves the group Sp(R4n, Si). Thesubgroup of Sp(R4n, Si) that is fixed by this involution is exactly theintersection (5.4.1).

5.5 Choice of s and θ

Let us take s to be the standard symplectic form Q0 on C2n whosematrix is J =

(0 In

−In 0

); we write Sp(2n, C) = Sp(C2n, Q0). We obtain

symplectic forms S1 and S2 on R4n by §5.3. Take G = Sp(R4n, S2).


Let H be the positive definite Hermitian form on C2n given by

H(z, w) = z · w.

Use the isomorphism h to convert H into a mapping R4n×R4n → C andlet Rh and Sh be the real and imaginary parts of this bilinear mapping,that is, H(h−1x, h−1y) = Rh(x, y)+iSh(x, y). By §4 the positive definiteform Rh determines a Cartan involution θh on GL(4n, R).

Lemma 5.6. The involutions θh and τh commute. Both τh and θh

preserve the symplectic group Sp(R4n, Si) (for i = 1, 2).

Consequently θh restricts to a Cartan involution θh on

Gh = Sp(R4n, S2)

(resp. θs on Gs = Sp(2n, C)) ([20] I, Thm. 4.2). The fixed point setKh = Gθh

h (resp. Ks = Gθss ) is a maximal compact subgroup of Gh (resp.

of Gs). The involution τh passes to an involution τh of the symmetricspace Dh = Gh/Kh.

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

5.7 Matrix descriptions

If e is a k × k matrix let

Diagn(e) = Diag(e, e, . . . , e)

be the nk × nk matrix with n identical diagonal blocks, each consistingof e. Let Sp(e) be the 2k×2k matrix

(0 e−e 0

). We shall use the following

2 × 2 matrices.

symbol d ≡ 1 (mod 4) d = 4m + 1

β(

0 d1 0

) (−1 2m2 1

)µ

(1 00 d

)( 1 0

0 m )

ν ( 0 11 0 ) ( 0 1

1 1 )

r(

1 00 −d

) (1 1/2

1/2 −m

)r′

(−d 00 1

) (−m −1/2−1/2 1

)


Set b = Diagn(β), u = Diagn(µ), v = Diagn(ν), r = Diagn(r), andr′ = Diagn(r′). The bilinear forms Rh, S1, and S2 have matrices Rh =Diag(r, r), S1 = Sp(u), and S2 = Sp(v). The matrix for Nh is Nh =Diag(b, b). The Cartan involution on GL(4n, R) is θh(g) = R−1

htg−1Rh.

To prove Lemma 5.6 it is necessary to verify condition (3) of Lemma 4.2,which amounts to checking that R−1

h SiR−1h Si = const · I4n for i = 1, 2,

a task which may be safely assigned to the undergraduate assistant.

5.8

The symplectic form S2 is integrally equivalent to the standard sym-plectic form Q0 whose matrix is

J = J2n =

(0 I2n

−I2n 0

).

which is easier to compute with. An isomorphism

Ψ : Sp(R4n, S2) → Sp(4n, R) = Sp(R4n, Q0)

is given by Ψ(g) = TgT−1 where

T =

(I2n 0

0 Diagn(ν)

)

so that

Q0(Tx, Ty) = S2(x, y) for all x, y ∈ R4n.(5.8.1)

Using the isomorphism Ψ the involutions τh, θh become the followinginvolutions τ, θ on Sp(4n, R) = Sp(R4n, Q0):

τ(g) = NgN−1 and θ(g) = R−1 tg−1R(5.8.2)

where

N = T−1NhT =

(b 0

0 tb

)and R = T−1RhT =

(r 0

0 r′

).(5.8.3)

In particular, it follows from (5.3.1) that

Q0(Nx, y) = Q0(x, Ny) for all x, y ∈ R4n.(5.8.4)


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

h2n =Z ∈ M2n×2n(C)

∣∣ tZ = Z, Im(Z) > 0

(5.8.5)

on which g =(

A BC D

)∈ Sp(4n, R) acts by fractional linear transforma-

tions, g · Z = (AZ + B)(CZ + D)−1. The maximal compact subgroupK is the stabilizer of the following basepoint

x1 =i√−d

Diagn

(−d 0

0 1

)or x1 =

i√−d

Diagn

(−2m −1

−1 2

)if d ≡ 1 (mod 4) or if d = 4m + 1 respectively. The symmetric space forSp(R4n, S2) may be identified with

h2nv = Zv ∈ M2n×2n(C)| Z ∈ h2n(5.8.6)

on which g =(

A BC D

)∈ Sp(R4n, S2) acts by fractional linear transfor-

mations, g · Zv = (AZv + B)(CZv + D)−1. The mapping Ψ passes toa mapping which we also denote by Ψ : h2nv → h2n and which is givenby Ψ(W ) = Wv−1. Then Ψ(gW ) = Ψ(g)Ψ(W ) for all g ∈ Sp(R4n, S2)and all W ∈ h2nv. The mapping ψ : Yn → h2nv will be described in §10.In summary we have a commutative diagram, the last line of whichprovides the names of the involutions associated with a given column:

Ks −−−→ Kh −−−→ K

↓ ↓ ↓

Sp(2n, C)ψ−−−→ Sp(R4n, S2)

Ψ−−−→ Sp(R4n, Q0)

↓ ↓ ↓

Ynψ−−−→ h2nv

Ψ−−−→ h2n.

θs θh, τh θ, τ

Definition 5.9. Define φ : Sp(2n, C) → Sp(4n, R) and φ : Yn →h2n 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,Od)[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,Od)[M ].(5.10.1)

Proposition 5.11. The nonabelian cohomology sets H1(〈τ〉, K)and H1(〈τ〉,Sp(4n, R)) are both trivial.

5.12 Proof

By Proposition 3.2 it suffices to show that H1(〈τ〉, K) is trivial. Letb0 = Diagn

(0 −11 0

). Then b

−10 = t

b0 = −b0. We claim there exists isan isomorphism Φ : (K, τ) ∼= (U(2n), τ ′) where τ ′(u) = b0ub

−10 for

all u ∈ U(2n). Assuming the claim for the moment, let us prove thatH1(〈τ ′〉,U(2n)) is trivial. Let µ : GL(2n, C) → GL(2n, C) be the invo-lution µ(A) = b0

tA−1b−10 . Its restriction to U(2n) coincides with τ ′. To

prove H1(〈τ ′〉,U(2n)) is trivial, by Proposition 3.2 it suffices to showthat i : H1(〈µ〉,U(2n)) → H1(〈µ〉,GL(2n, C)) is the trivial mapping.So let u ∈ U(2n) and assume that uµ(u) = 1. Then ub0

tu−1b−10 = 1

so ub0 is antisymmetric. Regarding ub0 as a bilinear form, it is non-degenerate, so it can be converted into the symplectic form b0 by achange of basis. In other words, there exists A ∈ GL(2n, C) such thatAub0

tA = b0 or

u = A−1b0

tA−1b−10 = A−1µ(A)(5.12.1)

This equation says that the cocycle defined by u becomes trivial inH1(〈µ〉,GL(2n, C)) as desired.

The isomorphism Φ : (K, τ) → (U(2n), τ ′) is obtained by chang-ing the basepoint x1 ∈ h2n (whose isotropy group is K = Gθ) to thebasepoint iI2n ∈ h2n (whose isotropy group we denote by K ′ = Gθ′ ∼=U(2n)). Let

a =

(Diagn(α) 0

0 Diagn(α′)

)∈ GSp(4n, R) and N0 =

(b0 0

0 tb0

)

where α, α′ are defined as follows:


symbol d ≡ 1 (mod 4) d ≡ 1 (mod 4)

α(

1 00√−d

) (2 10√−d

)α′

(√−d 00 1

) (√−d 0−1 2

)Define Φ : Sp(4n, R) → Sp(4n, R) by Φ(g) = aga−1. The mappingΦ converts the commuting involutions θ, τ into commuting involutionsθ′(g) = tg−1 and τ ′(g) = N0gN−1

0 , so it takes (K, τ) to (K ′, τ ′), (how-ever the mapping Φ does not preserve the integral structure). Theidentification U(2n) ∼= K ′ ⊂ Sp(4n, R) is given by A + iB →

(A B−B A

).

Restricting the involution τ ′ to U(2n) gives τ ′(u) = b0ub−10 . q.e.d.

5.13 The involution on Siegel space

There is a unique involution τ : h2n → h2n so that τ(gZ) = τ(g)τ(Z)for all g ∈ Sp(4n, R) and Z ∈ h2n; it is given by

τ(Z) = Z = bZ tb−1(5.13.1)

where b = Diagn(β). If Z = (zij) is divided into 2×2 blocks zij thenZ ∈ hτ

2n is fixed under τ iff zij = βzijtβ (for 1 ≤ i, j ≤ n), or

zij =

dzij wij

wij zij

if d ≡ 1 (mod 4)(5.13.2)

zij = σ−1

dzij wij

wij zij

tσ−1 if d ≡ 1 (mod 4)(5.13.3)

for some zij , wij ∈ C, and where σ = ( 2 10 1 ) . Conversely, if Z ∈ h2n is

divided into 2×2 blocks zij =( xij wij

yij zij

)then Z = τ(Z) iff xij = dzij

and yij = wij . These are linear equations in the coordinates, so hτ2n is

an open subset of a certain linear subspace of the space of symmetric2n × 2n matrices.

Proposition 5.14. The embedding φ : Sp(2n, C) → Sp(4n, R)passes to an embedding φ : Yn = Sp(2n, C)/Ks → h2n whose imageis the fixed point set hτ

2n. In particular, φ(Yn) is a real algebraic sub-manifold of h2n. If g ∈ Sp(4n, R) and if g = ±I then φ(Yn) ∩ h

g2n is a

proper real algebraic subvariety of φ(Yn), where hg2n denotes the points

in h2n that are fixed by g.


5.15 Proof

The first two statements follow from Proposition 5.11 and Lemma 3.8part (b). If g =

(A BC D

)∈ Sp(4n, R) then the points Z ∈ h2n 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 entriesfor Z, so this fixed point set is a real algebraic subvariety of h2n, as isits intersection with hτ

2n. We will now show that this intersection isa proper subvariety of hτ

2n unless g = ±I. We consider only the cased ≡ 1 (mod 4); the case d ≡ 1 (mod 4) is similar.

Let Y be the 2n × 2n matrix consisting of 2×2 blocks along thediagonal yi =

(−dyi 0

0 yi

)(for 1 ≤ i ≤ n) where yi > 0. It follows from

(5.13.2) that itY ∈ hτ2n for all t > 0. If itY is fixed under g then (5.15.1)

gives t2Y CY = B and Y D = AY from which it follows that B = C = 0.Since g is symplectic we also obtain D = tA−1. So we are reduced toconsidering those matrices A ∈ GL(2n, R) such that Z = AZ tA for allZ ∈ hτ

2n.

We outline one of many possible ways to see this implies A = ±I. Bytaking Z = Diag(z1, z2, . . . , zn) to consist of 2×2 blocks zi =

(dzi riri zi

)along the diagonal, and by varying one block but fixing the others,we may conclude that A = Diag(a1, a2, . . . , an) also consists of 2×2blocks, and that zi = aizi

tai. Comparing real and imaginary parts ofthis equation gives ai = ±I. It is then easy to see that the signs mustall coincide. q.e.d.

5.16 γ-real points

If γ ∈ Sp(4n, Z) and γγ = I (that is, if fγ is a 1-cocycle), then a pointZ ∈ h2n is said to be a γ real point if τ(Z) = γZ, the set of which wasdenoted h

τγ2n 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 thequotient Γ\h2n is precisely the image of the set

hτΓ2n =

⋃γ∈Z1(〈τ〉,Γ)

hτγ2n.


6. Symplectic Od modules

6.1

The main result in this section (Proposition 6.4), which classifies sym-plectic Od 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 toprove Proposition 6.10, which verifies hypothesis (4) (vanishing of non-abelian cohomology) of Lemma 3.8, which in turn is used to prove The-orem 2.3. Proposition 6.4 is also used in the proof of the Comessattilemma (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 Od

denote the ring of integers in the quadratic imaginary number fieldQ(

√d).

6.2

Recall [3] (VII.10 Prop. 24) that a finitely generated module P over theDedekind domain Od is torsion-free iff it is projective. If such a moduleP has rank n, then there exist v1, v2, . . . , vn ∈ P such that

P ∼= Odv1 ⊕Odv2 ⊕ · · · ⊕ Odvn−1 ⊕ Ivn(6.2.1)

for some fractional ideal I.

Now suppose P0 ⊂ P is a submodule. Then there exist submodulesP1, P2 ⊂ P such that P = P1 ⊕ P2 and so that P1 ⊃ P0 and P1 ⊗Q = P0 ⊗ Q. For, let M = P/P0 and consider its torsion-free quotientM/M tor where M tor denotes the torsion submodule of M. The precedingparagraph implies the composition

P → M → M/M tor(6.2.2)

admits a splitting M/M tor → P whose image we denote by P2. ThenP ∼= P1 ⊕ P2 where

P1 = x ∈ P | rx ∈ P0 for some r ∈ Od

is the kernel of the composition (6.2.2).


6.3 Polarizations

Let Q : R2r × R2r → R be a symplectic form and let L ⊂ R2r be alattice. We say that Q is a principal polarization of L if Q takes integervalues on L and if, for some basis of L (and hence for any basis of L), thematrix for Q has determinant 1. In this case there exists a symplecticbasis of L, meaning an ordered basis such that the resulting matrix forQ is

Jr =

0 Ir

−Ir 0

.

Now suppose r = 2n and suppose that L is also an Od module. Let uswrite b · v for the action of b ∈ Od on a vector v ∈ L. We say the actionof Od 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 ∈ Od, cf. Equation (7.3.4). It follows that Q(b ·u, u) = 0 for all b ∈ Od and all u ∈ L.

Proposition 6.4. Suppose Q is a symplectic form on R4n that prin-cipally polarizes a lattice L ⊂ R4n. Suppose L has an Od structure thatis compatible with the polarization Q. Then there exists

u1, . . . , un, v1, . . . , vn ∈ L

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

u1, ω · u1, . . . , un, ω · un, ω · v1, v1, . . . , ω · vn, vnif d ≡ 1 (mod 4), and

u1, ω · u1, . . . , un, ω · un, (ω − 1) · v1, v1, . . . , (ω − 1) · vn, vnif d ≡ 1 (mod 4).

In particular, L ∼= L1 ⊕ L2 is the direct sum of the free Lagrangiansubmodules

L1 = Odu1 ⊕ · · · ⊕ Odun and L2 = Odv1 ⊕ · · · ⊕ Odvn.

In either case, with respect to this basis, the matrix for the action of√d ∈ Od 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 = Odv ⊕ 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 Odv since Q(v, ω ·v) = Q(ω · v, v) = −Q(v, ω · v), and it similarly vanishes on Iw. Sowith respect to this basis, the matrix for Q is

(0 P

− tP 0

)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 anew basis x, y of Iw. Then the matrix of Q with respect to the basisv, ω · v, x, y is J2. 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(ω2 · v, y)

=

a′ + dQ(v, y) = a′ if d ≡ 1 (mod 4)a′ + Q((ω + m) · v, y) = a′ + 1 if d ≡ 1 (mod 4).

Hence x = ω · y if d ≡ 1 (mod 4) and x = (ω − 1) · y if d ≡ 1 (mod 4), asdesired. In either case, ω · y ∈ Iw so Iw = Ody is free.

6.6 The case n > 1

We will prove in Lemma 6.7 below (by a somewhat roundabout argu-ment) 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 P0 be the Od

span of x, y. It has a Z basis x, ω · x, ω · y, y with respect to whichthe matrix of Q|P0 is J2.

We claim that L splits as a direct sum, L = P0⊕L2 of Od submodules(of Z rank 4 and rank 4n-4 respectively) such that the restriction Q|L2

is a principal polarization. By induction, the lattice L2 has a basis ofthe desired type, from which it follows that L does also.

The claim is proven as follows. Using §6.2 there exists a splittingL = P1 ⊕P2 by submodules P1 and P2 such that P1 ⊃ P0 and P1 ⊗Q =P0 ⊗ Q. Let u, v, z, w be a Z basis for P1 and let Q1 be the matrix of


Q with respect to this basis. If A denotes the matrix that transformsthis basis of P1 ⊗ Q into the basis x, ω · x, ω · y, y, then J2 = AQ1

tA.Since these are matrices of integers, it follows that det(A) = ±1 henceP0 = P1. The next step is to modify the complement P2 to obtain acomplement L2 which is principally polarized.

By §6.2 we may write P2 = Odw1⊕· · ·⊕Odw2n−3⊕Iw2n−2 for somewi ∈ P2. For 1 ≤ i ≤ 2n − 2 set

w′i = wi − λix − µiω · x − γiω · y − νiy.

Then there are unique choices of integers λi, µi, γi, νi ∈ Z so that eachw′

i is Q orthogonal to P0. Let L2 be the Od span of the vectors w′i (for

1 ≤ i ≤ 2n − 2). Then L = P0 ⊕ L2 and L2 is orthogonal to P0. Withrespect to any choice of Z basis for L2 (and the above basis for P0) thematrix for Q is J2 0

0 Q2

(6.6.2)

where Q2 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 thatthe determinant of Q2 is also 1. Therefore, the restriction of Q to L2 is aprincipal polarization, as desired. The rest of this section is dedicated toproving the existence of the elements x, y, which we now state precisely.

Lemma 6.7. Fix n ≥ 2. Suppose L ⊂ R4n is a lattice that isprincipally polarized by the symplectic form Q, and suppose L admits acompatible action of Od. Then there exists x, y ∈ L so that (6.6.1) holds.

6.8 Proof

If u1, . . . , ur is a collection of vectors in L let

〈u1, . . . , ur〉

denote their vector space span in R2n and let

〈u1, . . . , ur〉⊥

be the Q-annihilator of this span. Since Q is integral on L, the inter-section L ∩ 〈u1, . . . , ur〉⊥ is a lattice in 〈u1, . . . , ur, 〉⊥.


Step 1. There exists a Lagrangian Od submodule L0 ⊂ L.

Suppose by induction that a Q linearly independent collection ofvectors

u1, ω · u1, u2, ω · u2, . . . , ur, ω · ur ⊂ L(6.8.1)

has been found so that Q vanishes on their Q span Ur (with the caser = 0 being trivial). The Q-annihilator U⊥

r has dimension 2n − r andthe intersection L ∩ U⊥

r is a lattice in U⊥r . If r < n then there exists

a vector ur+1 ∈ U⊥r ∩ L which is not contained in Ur. We claim the

collection u1, ω ·u1, . . . , ur, ω ·ur, ur+1, ω ·ur+1 is linearly independentand that Q vanishes on its vector space span Ur+1.

Suppose that ω · ur+1 is a linear combination of the other vectors inthis collection, say,

ω · ur+1 =r∑

i=1

(aiui + biω · ui) + cur+1

for some rational numbers ai, bi, and c. Multiplying by ω and collectingterms gives

(d − c2)ur+1 =r∑

i=1

(ai + cbi)ω · ui + (bid + cai)ui)

if d ≡ 1 (mod 4). But d − c2 < 0 so this contradicts the linear indepen-dence of (6.8.1). The case of d ≡ 1 (mod 4) is similar.

Step 2. There exists a Lagrangian Od submodule L1 ⊂ L and asubmodule L2 ⊂ L so that L = L1 ⊕ L2.

This follows from §6.2 and in fact L1 ⊗ Q = L0 ⊗ Q.

Now set L1 = Ody1 ⊕Ody2 ⊕ · · ·⊕Odyn−1 ⊕Iyn for some fractionalideal I. Then there exist rational numbers a, b ∈ Q so that the collectiony1, ω · y1, . . . , yn−1, ω · yn−1, yn, ayn + bω · yn forms a Z basis of L1.

Step 3. Choose any Z basis for L2. Together with the precedingbasis for L1 this gives a basis for L = L1 ⊕L2 with respect to which thematrix of Q is 0 T

− tT ∗


for some matrix T of integers. It follows that det(T ) = ±1. ApplyingT−1 to this basis gives a new Z basis for L2 such that the matrix for Qis 0 I

−I ∗

.

Denote this new basis by z1, x1, z2, x2, . . . , zn, xn. Then

Q(y1, x1) = 0 and Q(ω · y1, x1) = 1.

Therefore the elements y = y1 and x = −x1 satisfy (6.6.1). q.e.d.

6.9 Application to nonabelian cohomology

Proposition 6.4 may be used to construct an Od module structure oncertain lattices. Let τ be the involution of Sp(4n, Z) = Sp(Z4n, Q0) de-fined in §5.7. Let Γ ⊂ Sp(4n, Z) be a torsion-free (arithmetic) subgroupthat is preserved under τ. If d ≡ 1 (mod 4) then suppose also that Γ iscontained in the principal congruence subgroup Γ(2) of level 2. Set

Γ =

h ∈ Sp(4n, Z)∣∣∣ hh−1 ∈ Γ

.

Proposition 6.10. The mapping

H1(〈τ〉, Γ) → H1(〈τ〉, Γ)

is trivial.

6.11 Proof

Let γ ∈ Γ and suppose fγ is a 1-cocycle, that is, γγ = γNγN−1 = I,cf. Equation (5.8.3). It follows that (Nγ)2 = NγNγ = N2 = dI. UsingEquation (5.8.4) we obtain

Q0(Nγu, Nγv) = dQ0(u, v)

and hence

Q0(Nγu, v) = Q0(u, Nγv)


for all u, v ∈ R4n. We use this to define a Q0-polarized Od modulestructure on the standard lattice Z4n by letting

√d act through Nγ,

that is, define

(a + b√

d) · u = au + bNγu

whenever a, b ∈ Z If d ≡ 1 (mod 4) it is necessary to check that theaction of ω = (1 +

√d)/2 also preserves the lattice Z4n, 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 Z4n admits a

symplectic basis with respect to which the matrix of√

d is N. In otherwords, there exists h ∈ Sp(4n, Z) such that Nγ = hNh−1. Using thefact that N = N−1dI we conclude that

γ = N−1hNh−1 = NhN−1h−1 = hh−1

from which it follows that h ∈ Γ and that the cocycle fγ is a coboundary.q.e.d.

7. Anti-holomorphic multiplication

7.1

In this section we recall [13] some standard facts and notation concern-ing abelian varieties. Let L ⊂ Cr be a lattice (that is, a free abeliansubgroup of rank 2r so that L ⊗C R → C2r is an isomorphism of realvector spaces). Then A = Cr/L is a complex torus. If ω1, ω2, . . . , ωr isa basis for the space of holomorphic 1-forms on A, and if v1, v2, . . . , v2r

is a basis for L then the corresponding period matrix Ω is the matrixwith entries Ωij =

∫vj

ωi. If v′i =∑

j Aijvj and if ω′i =∑

j Bijωj are newbases then the resulting period matrix is

Ω′ = BΩ tA.(7.1.1)

A real symplectic form Q on Cr is compatible with the complex struc-ture if Q(iu, iv) = Q(u, v) for all u, v ∈ Cr, (not to be confused withEquation (7.3.4) below). A compatible form Q is positive if the sym-metric form R(u, v) = Q(iu, v) is positive definite. If Q is compatibleand positive then it is the imaginary part of a unique positive definiteHermitian form H = R+iQ. Let L ⊂ Cr be a lattice and let H = R+iQ


be a positive definite Hermitian form on Cr. Recall that Q is a princi-pal polarization of L if L admits a basis such that the resulting matrixfor Q is Jr (cf. §6.3). A principally polarized abelian variety is a pair(A = Cr/L, H = R + iQ) where H is a positive definite Hermitian formon Cr and where L ⊂ Cr is a lattice that is principally polarized by Q.

Each Z ∈ hr determines a principally polarized abelian variety

(AZ , HZ)

as follows. Let Q0 be the standard symplectic form on R2r = Rr ⊕ Rr

with matrix J = Jr =(

0 I−I 0

)(with respect to the standard basis of

Rr ⊕ Rr). Let FZ : Rr ⊕ Rr → Cr be the real linear mapping withmatrix (Z, I), that is,

FZ ( xy ) = Zx + y.

Then

QZ = (FZ)∗(Q0)(7.1.2)

is a compatible, positive symplectic form that principally polarizes thelattice

LZ = FZ(Zr ⊕ Zr).(7.1.3)

(In fact FZ(standard basis) is a symplectic basis for LZ .) The Hermitianform corresponding to QZ is

HZ(u, v) = QZ(iu, v) + iQZ(u, v) = tu(Im(Z))−1v

for u, v ∈ Cr. The pair (AZ = Cr/LZ , HZ) is the desired principally po-larized abelian variety. If z1, z2, . . . , zr are the standard coordinates onCr then, with respect to the above symplectic basis of L, the differentialforms dz1, dz2, . . . , dzr have period matrix Ω = (Z, I).

7.2

The principally polarized abelian varieties (AZ = Cr/LZ , HZ) and (AΩ

= Cr/LΩ, HΩ) are isomorphic iff there exists a complex linear map-ping ξ : Cr → Cr such that ξ(LΩ) = LZ and ξ∗(HΩ) = HZ . Seth = t(F−1

Z ξFΩ) =(

A BC D

). Then: h ∈ Sp(2r, Z), Ω = h · Z, and


ξ(M) = t(CZ + D)M for all M ∈ Cr, which is to say that the fol-lowing diagram commutes:

( xy ) Rr ⊕ Rr −−−→

Fh·ZCr M ξ

th ( x

y ) Rr ⊕ Rr −−−→FZ

Cr t(CZ + D)M

(7.2.1)

(since h · Z is symmetric). The relationship between the mapping FZ

and the involution τ is the following. Let N be as in (5.8.3). If Z ∈ hr

and Z = bZ tb−1 then this diagram commutes:

( xy ) Rr ⊕ Rr −−−→

FZ

Cr M tN ( x

y ) Rr ⊕ Rr −−−→F

Z

Cr bM .

(7.2.2)

7.3

A real endomorphism of a principally polarized abelian variety (A =Cr/L, H = R + iQ) is an R-linear mapping f : Cr → Cr such thatf(L) ⊂ L; two such being considered equivalent if they induce the samemapping A → A. As in §5.2, fix a square-free integer d < 0 and letOd denote the ring of integers in the number field Q(

√d). Recall (for

example, from [23] Equation (5.5.12) or [27] or [7] §3.1.1) that a complexmultiplication by the ring Od on A is a ring homomorphism Φ : Od →EndR(A) such that Φ(1) = I, and for all b ∈ Od and u, v ∈ Cr,

Φ(b) : Cr → Cr is complex linear,(7.3.1)

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 Od is a ring homomorphism Ψ : Od →EndR(A) such that Ψ(1) = I and so that the mapping κ = Ψ(

√d) :

C2n → C2n satisfies

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


for all a ∈ C and u, v ∈ Cr. (Consequently, Ψ(1) is complex linear andκ = Ψ(

√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 ∈ Od. If such an anti-holomorphic multiplication exists thenr is even (and in fact 〈u, v〉 = Q(κ(u), v) +

√dQ(u, v) is a complex

symplectic form on Cr with respect to the complex structure defined byκ/

√−d). Equivalently, a choice of anti-holomorphic multiplication by

Od, if one exists, is a choice of Od-module structure on L that satisfies(7.3.3) and (7.3.5) for all u, v ∈ L and all b ∈ Od.

7.4

For the remainder of this section take r = 2n and write Q0 for the(standard) symplectic form on R4n whose matrix is J = J2n with re-spect to the standard basis of R4n. The following lemma states thatcertain points (the γ-real points, for appropriately chosen γ) in theSiegel space correspond to abelian varieties with anti-holomorphic mul-tiplication. We use the involution τ defined in (5.8.2) and (5.13.1) andthe corresponding matrix N of (5.8.3); see also §5.16.

Lemma 7.5. Let γ ∈ Sp(4n, Z) and suppose γγ = 1. If d ≡1 (mod 4) then assume also that γ ≡ I (mod 2). Fix Z ∈ h

τγ2n. Then

the mapping

κZ = FZ t(Nγ) F−1Z : C2n → C2n(7.5.1)

defines an anti-holomorphic multiplication by Od on the principally po-larized abelian variety (AZ , HZ).

7.6 Proof

Set η = t(Nγ). Then

η2 = t(NγNγ) = t(NγdN−1γ) = d t(γγ) = dI

so the same is true of κZ . Also, η = tγ

tb 0

0 b−1

I 0

0 dI

. The first

two factors are in Sp(4n, R) so Q0(ηu, ηv) = dQ0(u, v) for all u, v ∈R4n. Hence QZ(κZu, κZv) = dQZ(u, v) for all u, v ∈ C2n which implies


(7.3.4). The mapping κZ preserves the lattice LZ = FZ(Z2n ⊕ Z2n)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 1

2(I + η) preserves the lattice Z2n ⊕ Z2n, henceOd preserves LZ .

Finally we check that κZ : C2n → C2n is anti-linear. Let γ =(

A BC D

).

By (7.2.1) and (7.2.2) the following diagram commutes:

R2n ⊕ R2n −−−→FZ

C2n M

tN

R2n ⊕ R2n −−−→

FZ

C2n bM

tγ

ξ

R2n ⊕ R2n −−−−→F

γ−1·Z

C2n t(CZ + D)bM .

(7.6.1)

But Z = γ−1 · Z so the bottom arrow is also FZ . Then κZ is thecomposition along the right-hand vertical column and it is given byM → t(CZ + D)bM which is anti-linear. q.e.d.

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

Proposition 7.7. Suppose A = (C2n/L, H = R + iQ) is a prin-cipally polarized abelian variety with antiholomorphic multiplication κ :C2n → C2n by Od. Then there exists a basis for the holomorphic 1-formson A and there exists a symplectic basis for L such that the resultingperiod matrix is Ω = (Z, I) for some Z ∈ hτ

2n which is fixed under theinvolution τ.

7.8 Proof

Throughout this section, in order to simplify notation, but at the risk ofsome confusion with the usual multiplication, we will write b · v ratherthan Ψ(b)v, for any b ∈ Od and v ∈ C2n. First consider the case d ≡1 (mod 4). By Proposition 6.4 and by interchanging the u’s and v’s,there exist u1, . . . , un, v1, . . . , vn ∈ L so that the ordered collection

ω · u1, u1, . . . , ω · un, un; v1, ω · v1, . . . , vn, ω · vn

is a symplectic basis for L.


The space of holomorphic 1-forms on A is 2n-dimensional, so foreach i (1 ≤ i ≤ n) there exist a unique holomorphic 1-form ηi such thatthe following holds for all j (with 1 ≤ j ≤ n):∫

vj

ηi = 0 and∫

ω·vj

ηi = δij .

Set η′i = κ∗ηi = κ∗ηi. Then the collection η′1, η1, . . . , η′n, ηn is an or-

dered basis for the holomorphic 1-forms on A. Let us compute the periodmatrix with respect to these bases. Calculate that∫

ω·vj

η′i =∫

κvj

κ∗ηi = d

∫vj

ηi = 0

∫vj

η′i =∫

κvj

ηi = δij .

It follows that the second “half” of the period matrix is the identity.Now let zij =

∫uj

ηi and wij =∫uj

η′i for 1 ≤ i, j ≤ n. Then by a similarcalculation, the first half of the period matrix consists of 2×2 blocks,

zij =

∫ω·ujη′i∫uj

η′i∫ω·uj

ηi

∫uj

ηi

=

dzij wij

wij zij

which implies by (5.13.2) that Z ∈ hτ

2n.Now consider the case d = 4m + 1. In this case Proposition 6.4

guarantees the existence of vectors u1, . . . , un, v1, . . . , vn ∈ L so that theordered collection

(ω − 1) · u1, u1, . . . , (ω − 1) · un, un; v1, ω · v1, . . . , vn, ω · vn(7.8.1)

is a symplectic basis for L. For each i (1 ≤ i ≤ n) there exist a uniqueholomorphic 1-form ηi such that

∫vj

ηi = 0 and∫ω·vj

ηi = δij . Set η′i =κ∗(ηi). Then ∫

vj

η′i =∫

κvj

ηi =∫

(2ω−1)·vj

ηi = 2

∫ω·vj

η′i =∫

κω·vj

ηi =∫

ω·vj

ηi = 1


since√

dω = 2m+ω. So the second half of the period matrix is Diagn(σ)where σ = ( 2 1

0 1 ) . Set zij =∫uj

ηi and wij =∫uj

η′i. Then a simplecalculation gives∫(ω−1)·uj

η′i∫uj

η′i∫(ω−1)·uj

ηi

∫uj

ηi

=

dzij wij

wij zij

12 0

−12 1

.

So the period matrix is(Z ′Diagn( tσ−1), Diagn(σ)

)where Z ′ consists of

2×2 blocks zij =(

dzij wij

wij zij

). By (7.1.1), changing the basisη′1, η1, . . . , η

′n, ηn

by the action of Diagn(σ−1) will give a period matrix (Z, I) where

Z = Diagn(σ−1)Z ′Diagn( tσ−1).

By (5.13.3) the point Z ∈ hτ2n is fixed under τ as claimed. q.e.d.

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

8.1 Level structures

Let (A = C2n/L, H = R+iQ) be a principally polarized abelian variety.A level M structure on A is a choice of basis U1, . . . , U2n, V1, . . . , V2nfor the M -torsion points of A that is symplectic, in the sense that thereexists a symplectic basis

u1, . . . , u2n, v1, . . . , v2n

for L such that

Ui ≡ui

Mand Vi ≡

vi

M(mod L)

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

u1, . . . , u2n, v1, . . . , v2n

determines a mapping

F : R2n ⊕ R2n → C2n(8.1.1)


such that F (Z2n ⊕ Z2n) = L, by F (ei) = ui and F (fi) = vi wheree1, . . . , e2n, f1, . . . , f2n is the standard basis of R2n ⊕ R2n. The choiceu1, . . . , u2n, v1, . . . , v2n (or equivalently, the mapping F ) will be re-ferred to as a lift of the level M structure. It is well-defined modulo theprincipal congruence subgroup Γ(M), that is, if F ′ : R2n ⊕ R2n → C2n

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 Od as in §7.3. A level M struc-ture U1, . . . , U2n, V1, . . . , V2n on A is compatible with κ if for some(and hence for any) lift F of the level structure, the following diagramcommutes (mod L):

1M

(Z2n ⊕ Z2n

)−−−→

F

1M L

tN

κ

1M

(Z2n ⊕ Z2n

)−−−→

F

1M L

(8.1.2)

where N is the matrix (5.8.3).We will refer to the collection

A =(A = C2n/L, H = R + iQ, κ, Ui, Vj

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

A′ =(A′ = C2n/L′, H ′ = R + iQ, κ′,

U ′

i , V′j

)is another such, then an isomorphism A ∼= A′ is a complex linear map-ping ψ : C2n → C2n such that ψ(L) = L′, ψ∗(H) = H ′, ψ∗(κ) = κ′, andsuch that for some (and hence for any) lift

u1, . . . , u2n, v1, . . . , v2n

and u′

1, . . . , u′2n, v′1, . . . , v

′2n

of the level structures,

ψ( ui

M

)≡ u′

i

Mand ψ

( vj

M

)≡

v′jM

(mod L).

Define V (d, M) to be the set of isomorphism classes of principally polar-ized abelian varieties with anti-holomorphic multiplication by Od andlevel M structure.


8.2

If Z ∈ h2n, then for any M ≥ 1 we define the standard level M structureon the abelian variety (AZ , HZ) to be the basis

FZ(ei/M), FZ(fj/M) (mod L)

where

e1, . . . , e2n, f1, . . . , f2n

is the standard basis of R2n ⊕ R2n.

Lemma 8.3. Let γ ∈ Sp(4n, Z) and let Z ∈ hτγ2n, that is, Z = γ ·Z.

Let M ≥ 3. Then the standard level M structure on the abelian variety(AZ , HZ) is compatible with the anti-holomorphic multiplication κZ iffγ ∈ ΓM = Γ(M) ∩ Γ(M).

8.4 Proof

It follows immediately from diagram (7.6.1) that γ ∈ Γ(M) iff the stan-dard level M structure on (AZ , HZ) is compatible with κZ . Since Γ(M)is torsion-free, γγ = I which implies γ ∈ Γ(M); hence γ ∈ ΓM . q.e.d.

By Lemma 8.3, each point Z ∈ hΓ(M)2n determines a principally po-

larized abelian variety

AZ = (AZ , HZ , κZ , FZ(ei/M), FZ(fj/M))

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

Theorem 8.5. Fix M ≥ 3. If d ≡ 1 (mod 4), assume also thatM is even. Then the association Z → AZ determines a one to onecorrespondence between the real points (2.3.2) XR of X = ΓM\h2n andthe set V (d, M) of isomorphism classes of principally polarized abelianvarieties with anti-holomorphic multiplication by Od and (compatible)level M structure.

8.6 Proof

A point x ∈ X is real iff it is the image of a ΓM -real point Z ∈ hτΓ(M)2n .

If two ΓM -real points Z,Ω determine isomorphic varieties, say ψ : AΩ∼=

AZ 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 ∈ ΓM . Let Z = γZ · Z andΩ = γΩ · Ω, with γZ , γΩ ∈ ΓM . Putting diagram (7.6.1) for Z togetherwith the analogous diagram for Ω and diagram (7.2.1), and using thefact that ψ∗(κΩ) = κZ gives a diagram

R2n ⊕ R2n −−−→FΩ

C2n

t(γ−1Ω hγZ)

κZψκ−1Ω = ψ

R2n ⊕ R2n −−−→FZ

C2n

from which it follows that t(γΩhγZ) ∈ Γ(M), hence h ∈ Γ(M), henceh ∈ ΓM .

So it remains to show that every principally polarized abelian va-riety with anti-holomorphic multiplication and level M structure, A =(A, H, κ, Ui, Vj) is isomorphic to some AZ . By the Comessatti lemma(Proposition 7.7) there exists Z ′ ∈ h2n, such that Z ′ = Z ′, and thereexists an isomorphism

ψ′ : (AZ′ , HZ′ , κZ′) ∼= (A, H, κ)

between the principally polarized abelian varieties with anti-holomor-phic multiplication. However the isomorphism ψ′ must be modifiedbecause it does not necessarily take the standard level M structure on(AZ′ , HZ′ , κZ′) to the given level M structure on (A, H, κ).

Choose a lift u1, . . . , u2n, v1, . . . , v2n of the level M structure andlet F : R2n ⊕ R2n → C2n be the corresponding mapping (8.1.1). Define

tg−1 = F−1 ψ′ FZ′ ∈ Sp(4n, Z)(8.6.1)Z = g · Z ′(8.6.2)

γ = gg−1 = N−1gNg−1.(8.6.3)

As in §7.2, if g =(

A BC D

)define ξ : C2n → C2n by ξ(w) = t(CZ + D)w.

Define ψ = ψ′ ξ. We will show that γ ∈ ΓM , that Z = γ · Z, and thatψ induces an isomorphism ψ : AZ → A of principally polarized abelianvarieties with anti-holomorphic multiplication and compatible level Mstructures.

In the following diagram, F is the mapping (8.1.1) associated tothe lift of the level M structure. The bottom square commutes by the


definition of g, while the top square commutes by (7.2.1):

R2n ⊕ R2n FZ−−−→ C2n

tg

ξ

R2n ⊕ R2n −−−→FZ′

C2n

tg−1

ψ′

R2n ⊕ R2n F−−−→ C2n.

.(8.6.4)

First let us verify that ξ : (AZ , HZ , κZ) → (AZ′ , HZ′ , κZ′) is an iso-morphism of principally polarized varieties with anti-holomorphic multi-plication by Od. It follows from (8.6.4) that ξ∗(LZ) = LZ′ and ξ∗(HZ) =HZ′ . We claim that ξ∗(κZ) = κZ′ , that is, κZ′ = ξκZξ−1. But this fol-lows from direct calculation using ξ = FZ′ tgFZ , κZ = FZ

t(Nγ)F−1Z ,

κZ′ = FZ′ tNFZ′ and (8.6.3) (and it is equivalent to the statement thatthe pushforward by tg of the involution t(Nγ) on R2n ⊕ R2n is the in-volution tN). It follows that

ψ∗(κZ) = κ.(8.6.5)

We claim that the standard level M structure on (AZ , HZ) is com-patible with κZ . By construction, the mapping ψ takes the standardlevel M structure on (AZ , HZ) 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 FZ . Using (8.6.5) it follows that the diagram

1M

(Z2n ⊕ Z2n

)−−−→

FZ

1M LZ

tN

κZ

1M

(Z2n ⊕ Z2n

)−−−→

FZ

1M LZ

commutes (mod LZ), which proves the claim. It also follows fromLemma 8.3 that γ ∈ ΓM .

In summary, we have shown that

(AZ , HZ , κZ , FZ(ei/M), FZ(fj/M))

is a principally polarized abelian variety with anti-holomorphic multipli-cation and (compatible) level M structure, and that the isomorphism ψpreserves both the anti-holomorphic multiplication and the level struc-tures. q.e.d.


9. Rational structure

9.1

Let g → g = NgN−1 be the involution on Sp(4n, R) where N denotesthe matrix (5.8.3), as in §5.7. The resulting anti-holomorphic involutionon h2n is given by Z → Z = bZ t

b−1. As in §8, fix a level M ≥ 1 andlet Γ = ΓM = Γ(M) ∩ Γ(M). It is well-known ([5] §V Thm. 2.5) thatthe arithmetic quotient Γ(M)\h2n admits a model that is defined overa certain cyclotomic field. For our purposes, however, we need a modelthat is defined over a subfield of the real numbers, and we need it forthe slightly different arithmetic quotient X = ΓM\h2n. For these factswe will use results of [25].

Theorem 9.2. There exists a projective embedding X → CPr andthere exists an anti-holomorphic involution τ on CPr 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) τof §5.13.

In summary, the set XR described in Theorem 2.3 forms the set ofreal points of a complex quasi-projective algebraic variety defined overQ. The proof will occupy the rest of this section.

Proposition 9.3. The complex vector space of (holomorphic) ΓM -modular forms on h2n is spanned by modular forms with rational Fouriercoefficients.

9.4 Proof

Let G = GSp(4n), let A be the adeles of Q and let S ⊂ G(A)+ bean open subgroup containing Q×G(R)+ (where + denotes the identitycomponent). Let ΓS = S ∩ G(Q). Suppose that Γ ⊂ G(Q) is an arith-metic group which is contained in ΓS and that:

(1) S/Q×G(R)+ is compact.

(2) Γ · Q× = ΓS .


(3) The set

∆ =

I2n 0

0 tI2n

∈ G(A)∣∣∣∣ t ∈ Π

pZ×

p

is contained in S.

Then [25] Thm. 3 (ii) states that the complex vector space of Γ-modularforms with weight k on h2n is spanned by those forms whose Fouriercoefficients are in the finite abelian extension kS of Q that is determinedby the set S. To apply this to our setting, let S1(M) be the collectionof elements x ∈ G(A)+ such that each p-component xp ∈ GL(2n, Zp)and satisfies

xp ≡

I2n 0

0 apI2n

(mod M · Zp)

for some ap ∈ Z×p . Define S(M) = S1(M)·Q×, let S′(M) = NS(M)N−1,

and S = S(M) ∩ S′(M). Then hypothesis (1) is satisfied. It is easy tosee that S(M), S′(M) both contain ∆, hence hypothesis (3) is satisfiedfor the set S. In this case, kS = Q and

ΓS = S ∩ G(Q) = (Γ(M) ∩ Γ(M)) · Q× = ΓM · Q×

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

9.5

Let I− =(

I2n 00 −I2n

). Its action by fractional linear transformations

maps the Siegel lower half space h−2n to the upper half space h2n, that

is, I− · Z = −Z. Hence, for any holomorphic mapping f : h2n → C wemay define f ′ : h2n → C by

f ′(Z) = f(I− · N · Z) = f(−bZ tb−1).

Proposition 9.6. If f : h2n → C is a holomorphic Γ-modularform of weight k, with rational Fourier coefficients, then f ′ is also aholomorphic Γ-modular form of weight k, and

f(Z) = f ′(Z)(9.6.1)

for all Z ∈ h2n.


9.7 Proof

Suppose that f(γ · Z) = j(γ, Z)kf(Z) for all γ ∈ Γ and all Z ∈ h2n

where

j((

A BC D

), Z)

= det(CZ + D)

is the standard automorphy factor. Then j(I−N, Z) = det(− tb) is in-

dependent of Z. Let γ ∈ Γ and set

γ′ = I−NγN−1I−1− ∈ Γ.

Then

f ′(γ · Z) = f(γ′ · I−N · Z)

= j(γ′, I−N · Z)kf ′(Z)

= det(− tb)kj(γ, Z)k det(− t

b)−kf ′(Z)

= j(γ, Z)kf ′(Z).

which shows that f ′ is Γ-modular of weight k. Next, with respect tothe standard maximal parabolic subgroup P0 (which normalizes thestandard 0-dimensional boundary component), the modular form f hasa Fourier expansion,

f(Z) =∑

s

as exp (2πi〈s, Z〉)

which is a sum over lattice points s ∈ L∗ where L = Γ ∩ Z(U0) is theintersection of Γ with the center of the unipotent radical U0 of P0 andwhere as ∈ Q. Then

f(Z) =∑

s

as exp(2πi〈s, bZ t

b−1〉)

=∑

s

as exp(2πi〈s,−bZ t

b−1〉)

= f(−bZ tb−1) = f ′(Z).

q.e.d.


9.8

The Baily-Borel compactification X of X is the obtained by embed-ding X holomorphically into CPm using m + 1 (Γ-)modular forms (sayf0, f1, . . . , fm) of some sufficiently high weight k, with rational Fouriercoefficients, and then taking the closure of the image. Define an embed-ding Φ : X → CP2m+1 by

Φ(Z) =(f0(Z) : f1(Z) : · · · : fm(Z) : f ′

0(Z) : f ′1(Z) : · · · : f ′

m(Z)).

Denote these homogeneous coordinate functions by xj = fj(Z) andyj = f ′

j(Z). Define an involution σ : CP2m+1 → CP2m+1 by σ(xj) =yj and σ(yj) = xj . Then Equation (9.6.1) says that this involution iscompatible with the embedding Φ, that is, for all Z ∈ X we have:

σΦ(Z) = Φ(Z).

Define Ψ : CP2m+1 → CP2m+1 by setting ξj = xj + yj and ηj =i(xj − yj) for 0 ≤ j ≤ m. Let Y = ΨΦ(X) and let Y denote its closure.

Proposition 9.9. The composition ΨΦ : X → CP2m+1 is a holo-morphic embedding which induces an isomorphism of complex algebraicvarieties 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 ∈ Xsuch that Z = Z.

9.10 Proof

The image ΨΦ(X) is an algebraic subvariety of projective space thatis preserved by complex conjugation, so it is defined over R. The realpoints are obtained by setting ξj = ξj and ηj = ηj which gives xj = yj

and yj = xj hence Φ(Z) = σΦ(Z), or Z = Z. The Fourier coefficients ofξj and ηj are in Q[i] so the image ΨΦ(X) is defined over Q[i]. Since itis also invariant under Gal(C/R), it follows that ΨΦ(X) is defined overQ. q.e.d.

9.11 Remark

The embedding (f0 : f1 : · · · : fm) : X → CPm determines the usualrational structure on X, and the resulting complex conjugation is thatinduced by Z → −Z for Z ∈ h2n.


10. The symmetric space for Sp(2n, C)

10.1

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

10.2 A quaternion algebra

Associated to the imaginary quadratic field Q(√

d) we consider thequaternion algebra H over R that is generated by 1, i, j,k with i2 = k2 =d1, j2 = −1 and ij = k. If w = r1+si+xj+yk set w = r1−si−xj−ykand w∗ = r1 + si − xj + yk. If we embed θ : C → H by

√d → i then

θ(z) = θ(z) and we may write H = θ(C) ⊕ kθ(C). Define the purelyquaternionic part of such an element w ∈ H to be

Qu(w) = k−1(xj + yk) = y +x

di.

The mapping θ extends to an injective algebra homomorphism θ :Mn×n(C) → Mn×n(H) by applying θ to each matrix entry. For no-tational convenience we shall often omit the use of the symbol θ.

If A, B ∈ GL(n, H) then t(AB)∗ = ( tB∗)( tA∗). If Qu(A) = 0 thenAk = kA. An element A ∈ Mn×n(H) is Hermitian if A = tA. In this case〈z, z〉A = tzAz is real, for all z ∈ Hn. The element A is positive definite(written A > 0) if 〈z, z〉A > 0 for all nonzero z. The unitary groupU(n, H) (sometimes denoted Sp(n)) consists of those A ∈ Mn×n(H)such that A−1 = tA.

10.3

Define the quaternionic Siegel space

Yn =W ∈ Mn×n(H) | W ∗ = tW, Qu(W ) > 0

.

Proposition 10.4. The symplectic group Sp(2n, C) acts transi-tively on the quaternionic Siegel space Yn by fractional linear transfor-mations: if g =

(A BC D

)then

g · W = (AW + B)(CW + D)−1


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

Qu(g · W ) = t(CW + D)−1Qu(W )(CW + D)−1.(10.4.1)

The stabilizer of the basepoint W0 = 1√−d

kIn ∈ Yn is the unitary groupU(n, H) over the quaternions, which is embedded in Sp(2n, C) by

A +1√−d

Bk →

A B

−B A

.

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 ∈ Yn iff g · W ∈ Yn. Theremaining statements may be verified by direct computation. q.e.d.

10.6

Define the homomorphism µ : H → M2×2(C) as follows:

d ≡ 1 (mod 4) d = 4m + 1

µ(i)(

0 d1 0

)σ−1

(0 d1 0

)σ

µ(j)(√

−1 00 −

√−1

)σ−1

(√−1 00 −

√−1

)σ

µ(k)(

0 −d√−1√

−1 0

)σ−1

(0 −d

√−1√

−1 0

)σ

where σ = ( 2 10 1 ) . This mapping extends to a homomorphism

µ : Mr×r(H) → M2r×2r(C)

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

Lemma 10.7. The composition

Mr×r(C) θ−−−→ Mr×r(H)µ−−−→ M2r×2r(C)

takes values in M2r×2r(R) and it coincides with the mapping ψr of(5.2.1). In particular, it restricts to the injective homomorphism ψn :Sp(2n, C) → Sp(R4n, S2) of §5.7.


Proposition 10.8. The mapping µ takes the quaternionic Siegelspace Yn ⊂ Mn×n(H) diffeomorphically to the symmetric space

h2nv ⊂ M2n×2n(C)

for Sp(R4n, S2). Its image is the fixed point set of the involution τh.Moreover, for each g ∈ Sp(2n, C) and W ∈ Yn we have

µ(g · W ) = ψn(g) · µ(W ).(10.8.1)

10.9 Proof

Since µ and θ are algebra homomorphisms, for

g =(

A BC D

)∈ 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 basepoint W0 = 1√

−dkIn ∈ Yn to the following base point x2 ∈ h2nv,

x2 =i√−d

Diagn

0 −d

1 0

or

x2 =i√−d

Diagnσ−1

0 −d

1 0

σ =i√−d

Diagn

−1 −2m

2 1

depending on whether d ≡ 1 (mod 4) or d = 4m + 1 respectively. Itfollows from (5.8.6) that µ takes Yn to h2nv and it further follows fromProposition 5.14 that its image is precisely the fixed point set under τh.

q.e.d.


11. Concluding remarks

11.1

The definition of anti-holomorphic multiplication given in §7 extendsin an obvious manner to more general CM fields. Let F be a totallyreal, degree m extension of Q and let E = F [

√d] be a totally imaginary

quadratic extension of F (with d ∈ OF ). Let (A = C2n/L, H = R+ iQ)be a principally polarized abelian variety as in §7. Then an anti-holomorphic multiplication by the ring of integers OE is a homomor-phism Ψ : OE → EndR(A) such that Ψ(OF ) ⊂ EndC(A), such thatκ = Ψ(

√d) : C2n → C2n is anti-linear (κ(ax) = aκ(x) for all a ∈ C and

x ∈ Cr), and such that Q(Ψ(b)x, y) = Q(x, Ψ(b)y) for all b ∈ OE andy ∈ C2n. One might then expect (1) that the moduli space of principallypolarized abelian varieties with anti-holomorphic multiplication by OE

and appropriate level structure may be identified with the locus of realpoints in a corresponding Hilbert-Siegel modular variety, and (2) thatit consists of finitely many copies of Γ\D where Γ ⊂ Sp(2n,Oe) is anappropriate level subgroup and where D = Yn × · · · × Yn is a productof m copies of the symmetric space Yn = Sp(2n, C)/U(n, H).

11.2

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

11.3

In [8] we consider a different rational structure on the Siegel modularvariety X = Γ\hn and a different anti-holomorphic involution τ ′, suchthat the resulting locus of real points (let us call it X ′

R) may be naturallyidentified with the moduli space of real abelian varieties (with appro-priate level structure); and we show that this moduli space consists offinitely many copies of the locally symmetric space Λ\GL(n, R)/O(n)(for appropriate principal congruence subgroups Γ and Λ). The invo-lution 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 ofComessatti and Silhol. Although the outline of [8] is parallel to thatof the present paper, the technical details are completely different andwe do not yet know how to formulate or prove the most natural generalstatement along these lines. Interesting related results are described in[2].

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http://www.ams.org/mathscinet-getitem?mr=0379505

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School of Mathematics

Institute for Advanced Study

Princeton NJ

Dept. of Mathematics

Haverford College

Haverford PA








Anti-holomorphic multiplication and a real algebraic ...goresky/pdf/newhyp.jour.pdf · j. differential geometry 65 (2003) 513-560 ANTI-HOLOMORPHIC MULTIPLICATION AND A REAL ALGEBRAIC

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