On p-adic Uniformization of Fake Projective Planes ^{1}

Fumiharu Kato

Abstract: In this paper we prove that fake projective planes (Mumford’s one and those considered by Ishida-Kato) are (connected) Shimura varieties. We also describe the level structures and minimal fields of definition explicitly.

1. Introduction

The aim of this paper is to exhibit some explicit examples of p-adic uniformization of Shimura surfaces of PEL-type. Each of our examples gives, on the complex analytic side, a surface of general type having complex unit-ball as the topological universal covering; more specifically, they are smooth surfaces having the same topological Betti numbers as ℂℙ^{2}, so-called “fake projective planes”, which are among the most interesting classes of ball-quotient surfaces in algebraic geometry.

One of the fake projective planes discussed in this paper was discovered by Mumford [Mu79]; p-adic uniformization was already the crucial strategy of construction there. Later in part of [IK98] Masanori Ishida and the author observed other possible fake projective planes according to Mumford’s idea, and proved there exist (at least) two more different fake projective planes, collaborated with the group-theorists’ classification of discrete groups acting on Bruhat-Tits buildings in [CMSZ93a] and [CMSZ93b]. As far as the author knows, these three are the only known fake projective planes.

However, these surfaces are, a priori, constructed as surfaces over the field of 2-adic numbers ℚ_{2}, not over ℂ. (A fake projective plane over an arbitrary field may be defined to be a smooth surface enjoying: c_{1}^{2} = 3c_{2} = 9, p_{g} = q = 0, and the canonical class is ample; if one considers this condition in the complex analytic category, it is equivalent to that the surface in question is a fake projective plane in the origial sense.) In this sense, although those constructions as above prove the “existence” of such surfaces, they are not really concrete as complex surfaces. For instance, topological or even some algebrogeometric properties of them (e.g. fundamental groups, field of definition, etc.) are quite obscure; even the arithmeticity of the fundamental groups seems unknown.

The theory of p-adic uniformization, on the other hand, involves surprizingly rich algebro-geometric and arithmetic aspects concerning with Shimura varieties. This was first observed by Čerednik [Če76] and Drinfeld [Dr76] for Shimura curves. The generalizations to higher dimensions have been developed over the past two decades; recently, Boutot-Zink [BZ95] and Rapoport-Zink [RZ96] gave powerfully extended generalities. These theories give a nice symmetric picture involving p-adic and complex uniformizations of certain Shimura varieties of PEL-type; viz. such a Shimura variety has both p-adic (for very special p) and complex analytic uniformizations, and they can be viewed symmetrically by means of “changing-invariants” trick.

Shimura varieties of this kind are, therefore, much expected to possess rich structures involving the interplay between p-adic and complex analysis. However, as far as the author knows, very few such examples are known in dimension two or high, in contract with numerous explicit examples of Shimura curves of this type (see, e.g. [Ku79][Ku94][An98]).

11991 Mathematics Subject Classification. Primary 14G35; Secondary 14J29, 11G15, 11Q25.

Sketch of Results. In this paper, we will see that the p-adic constructions of the three fake projective planes as above actually fit in with the picture of the theory by Boutot-Zink [BZ95] and Rapoport-Zink [RZ96], and give very explicit examples in dimension two. This implies, in particular, the following:

(1) These fake projective planes are realized as connected components of certain Shimura varieties, and hence, are solutions of certain moduli problems parametrizing suitable abelian varieties with additional structures.

(2) In particular, they are, as complex surfaces, arithmetic quotient of the complex unit-ball; moreover, we can describe the fundamental groups fairly explicitly in terms of congruence conditions.

(3) Besides, describing connected components of the Shimura varieties by means of the theory of complex multiplication, we can find out the minimal fields of definition for those surfaces. In fact, we will prove that they are defined over the absolute class fields of suitable imaginary quadratic fields.

The structure of this paper is as follows: In the next section, we will survey very briefly the theory of p-adic uniformizations of Shimura varieties, basically after Boutot-Zink [BZ95] and Rapoport-Zink [RZ96], in order to show how it works to come up with the results as above. The subsequent sections are devoted to present examples; in §3 we will discuss Mumford’s fake projective plane, and the other two fake projective planes in §4.

Acknowledgements. The author’s viewpoint of this work owes much to suggestions by Professor Yves André. The author expresses gratitude to him for valuable discussions. The author thanks Professor Yoichi Miyaoka for his interest in this work and several valuable suggestions on the manuscript. The author thanks Max-Planck-Institut für Mathematik Bonn for the nice hospitality.

Notation. The notation A ⊗ B (absence of the base ring) only occurs when A and B are ℚ-algebra, and tensors are taken over ℚ. For a field extension F/E, we denote by Res_{F/E} the Weil restriction. If E is a p-adic field, E^{ur} denotes the maximal unramified extension in an algebraic closure.

2. p-adic uniformization of Shimura varieties of PEL-type

This section surveys the generalities of p-adic uniformization of Shimura varieties of PEL-type. Our references on this subject are Rapoport [Ra90], Boutot-Zink [BZ95], and Rapoport-Zink [RZ82, §4][RZ96], through which the reader can always justify the sometimes sketchy statements in this section. Since the purpose of this section is to provide language to describe the examples in the subsequent sections, we do not really need the full generality, and will be fairly specific. For instance, knowledge of the theory over general CM fields is not needed, and so we will limit ourselves to the case only over imaginary quadratic fields, which will make the situation simpler and explicit.

A. The fundamental data. Throughout this section we fix an integer d > 2 and a prime number p. Let K be an imaginary quadratic extension of ℚ such that the prime p decomposes to the product 𝔭\overline{𝔭} of two distinct prime ideals in K. Let D be a central division algebra over K of dimension d^{2}. For any place ν of K we write D_{ν}: = D ⊗_{K} K_{ν}. We choose once for all an infinite place ε: K ↪ ℂ, i.e. a CM-type, of K. We assume: zw (a) D_{ε} is isomorphic to the matrix algebra M_{d}(ℂ).

(b) The Hasse invariant of D_{𝔭} (resp. D\overline{𝔭}) is \frac{1}{d} (resp. −\frac{1}{d}).

A positive involution on D is a ℚ-linear endomorphism x ↦ x^{∗} of D such that (xy)^{∗} = y^{∗}x^{∗}, x^{∗∗} = x for all x, y ∈ D, and that tr_{D/ℚ}(xx^{∗}) > 0 for all non-zero x ∈ D, where tr_{D/ℚ} denotes the reduced trace. Fix a positive involution ∗ on D such that its restriction to K coincides with the complex conjugation. The assumption (a) yields an isomorphism

Φ: D ⊗ ℝ ⟶^{∼} M_{d}(ℂ)

of ℝ-algebras. We may assmue that Φ gives an isomorphism between ℝ-algebras with involution, where M_{d}(ℂ) is equipped with the standard involution A ↦ ^{t}\overline{A}. Indeed, any involution of M_{d}(ℂ) is a standard one up to inner automorphisms. We fix a ∗-stable order O_{D} of D such that O_{D_{𝔭}}: = O_{D} ⊗_{O_{K}} O_{K_{𝔭}} (resp. O_{D\overline{𝔭}}: = O_{D} ⊗_{O_{K}} O_{K\overline{𝔭}}) is a maximal order of D_{𝔭} (resp. D\overline{𝔭}). For any finite place ν, O_{D_{ν}} denotes O_{D} ⊗_{O_{K}} O_{K_{ν}}.

We shall further introduce the following data: Let V be D viewed as a left Dmodule. Let b ∈ D be a non-zero element which is anti-symmetric, i.e. b^{∗} = −b. We define the anti-symmetric ℚ-bilinear form ψ = ψ_{b}: V × V → ℚ by ψ(x, y): = tr_{D/ℚ}(ybx^{∗}) for x, y ∈ V. The form ψ clealy satisfies ψ(αx, y) = ψ(x, α^{∗}y) for x, y ∈ V and α ∈ D. It is also easily verified that ψ is non-degenerate if b is invertible. By Γ we denote O_{D} viewed as a left O_{D}-module, and define Γ_{p}: = Γ ⊗_{ℤ} ℤ_{p}. Then we require: zw

(c) The form ψ induces a perfect paring ψ_{p}: Γ_{p} ⊗ Γ_{p} → ℤ_{p}.

Via the isomorphism Φ chosen as above, the element b can be represented by an anti-Hermitian matrix, written simply as Φ(b). We assume: zw

(d) There exists P ∈ M_{d}(ℂ) such that ^{t}\overline{P} Φ(b)P = diag(−\sqrt{−1}, …, −\sqrt{−1}, \sqrt{−1}).

B. The Shimura variety. Define an algebraic group G over ℚ as follows: Given a ℚ-algebra R, we set

G(R): = {g ∈ ((D ⊗ R)^{op})^{×} | ψ(xg, yg) = c(g)ψ(x, y), c(g) ∈ R^{×}, x, y ∈ V ⊗ R}.

Note that we have an equality Nm g ·\overline{Nm g} = c(g)^{d}, where Nm denotes the reduced norm of (D ⊗ R)^{op}/K ⊗ R.

To see that the algebraic group G_{ℝ} is isomorphic to a group of unitary similitudes, we consider, as in [RZ96, 6.37], a decomposition

V ⊗ ℝ ⟶^{∼} ℂ^{d} ⊗_{ℂ} W

so that D ⊗ ℝ ≅ M_{d}(ℂ) acts from the left on the right hand side by the usual matrix multiplication on the first factor. Since D ⊗ ℝ is simple, we have End_{D⊗ℝ}(V ⊗ ℝ) = (D ⊗ ℝ)^{op}, and the right action of (D ⊗ ℝ)^{op} ≅ M_{d}(ℂ)^{op} on V ⊗ ℝ can be seen as the usual matrix multiplications on W from the right. Then G_{ℝ} is the group of unitary similitudes with respect to the anti-Hermitian form H on W, where H is induced by ψ by the formula

ψ(z_{1} ⊗ w_{1}, z_{2} ⊗ w_{2}) = tr_{ℂ/ℝ}(^{t}\overline{z}_{1}z_{2}H(w_{1}, w_{2}))

for w_{1}, w_{2} ∈ W and z_{1}, z_{2} ∈ ℂ^{d}. By a suitable choice of basis, the form H is represented by the matrix J: = diag(−\sqrt{−1}, …, −\sqrt{−1}, \sqrt{−1}), and then we deduce that G_{ℝ} is isomorphic to GU(d − 1, 1), the group of unitary similitudes of (W, J).

We define a homomorphism h: Res_{ℂ/ℝ}𝔾_{m} → G_{ℝ} so that h(r), r ∈ ℝ^{×}, acts on W as the multiplication by r and h(\sqrt{−1}) acts as the multiplication by −J. By [Ko92, 4.1] the pair (G_{ℝ}, h) satisfies the conditions (1.5.1), (1.5.2), and (1.5.3) in [De71], so that we get the associated Shimura variety; for a sufficiently small level structure C, i.e. an open compact subgroup of G(𝔸_{f}), the corresponding Shimura variety will be denoted by S_{C}:

S_{C}: = G(ℚ) \ X_{∞} × G(𝔸_{f})/C,

where X_{∞} is the set of all conjugates of h under the action of G(ℝ), that is, in our case, the open unit-ball in ℂ^{d−1}. The Shimura variety S_{C} is a finite disjoint union of quasi-projective manifolds, which are arithmetic quotients of the complex unit-ball. The Shimura field will be denoted by E, which is ε(K) in our case. Given S_{C}, Sh_{C} stands for its canonical model, a quasi-projective variety over E.

A small remark on the structure of G: Consider a subgroup G_{1} of G defined by

G_{1}(R): = {u ∈ G(R) | c(u) = 1}

for a ℚ-algebra R. Then, in case d is odd, there exists an epimorphism

G_{1} × Res_{K/ℚ}𝔾_{m,K} ⟶ G

by (u, k) ↦ uk. Indeed, for any g ∈ G(R), we can take k = Nm g · c(g)^{m} so that gk^{−1} ∈ G_{1}(R), where d = 2m + 1.

C. The moduli problem over E. The Shimura variety Sh_{C} with respect to a level structure C ⊂ G(𝔸_{f}) represents the moduli functor \underline{Sh}_{C}: Sch_{E} → Set defined as follows: The Shimura data (G, h) gives rise to a Hodge decomposition V_{ℝ} = V_{0,ℂ}⊕V_{1,ℂ}, where V_{0,ℂ} (resp. V_{1,ℂ}) is the subspace of V_{ℝ} on which h(z) acts by z (resp. by \overline{z}). As the conjugacy class of h is defined over E, this decomposition comes from a decomposition V = V_{0} ⊕V_{1} defined over E. Then, for an E-scheme T, \underline{Sh}_{C}(T) is the set of all isomorphism classes of quadruples (A, ι, \overline{Λ}, \overline{η}) consisting of the following data: zw

(i) An abelian scheme A over T considered up to isogeny. (ii) A homomorphism ι: D → End^{∘}(A) such that, for any δ ∈ D,

tr(ι(δ)|Lie(A)) = tr(δ|V_{0}),

where End^{∘}(A): = End(A) ⊗ ℚ. (iii) A ℚ-homogeneous polarization \overline{Λ} of A such that ι gives a homomorphism of ℚ-

algebras respecting the involutions ∗ on D and the Rosatti involution on End^{∘}(A). (iv) A modulo C-class \overline{η} of D ⊗ 𝔸_{f}-linear symplectic similitudes

η: (V (A), Riemann form) ⟶^{∼} (V ⊗ 𝔸_{f}, ψ ⊗ 𝔸_{f}),

where V (A) denotes the rational Tate module of A.

D. Local model. ¿From now on, the level structure C will be always assumed to be of form C = C_{p}C^{p}, where C^{p} ⊂ G(𝔸_{f}^{p}) and: zw

(e) C_{p} is the maximal open compact subgroup of G(ℚ_{p}) consisting of those elements of G(ℚ_{p}) which map Γ_{p} onto itself.

Localizing the above moduli problem to the infinite place E ↪ ℂ, one gets the moduli problem of which the solution is given by the complex Shimura manifold S_{C}, which has the complex unit-ball uniformization. The theory of p-adic uniformization of Shimura varieties, which is the main subject of this section, asserts that, localizing Sh_{C} to the special place

ν: E ⟶ \overline{ℚ}_{p}

defined by the prime ε(𝔭), again one gets an interesting analytic variety which has a uniformization of a certain kind. In order to see this, it will be more convenient to extend the localized moduli problem over E_{ν} to that over the integer ring O_{E_{ν}}, which amounts to consider a model of Sh_{C} ⊗_{E} E_{ν}.

Let \underline{A}_{C}: Sch_{O_{Eν}} → Set be the moduli functor defined as follows: For an O_{E_{ν}}-scheme T the set \underline{A}_{C}(T) consists of the isomorphism classes of the quadruple (A, ι, \overline{Λ}, \overline{η}^{p}) of the following data: zw

(i)′ An abelian scheme A over T considered up to isogeny of degree prime to p. (ii)′ An injective homomorphism ι: O_{D} ↪ End^{(p)}(A) such that, for δ ∈ O_{D},

det_{O_{T}} (ι(δ)|Lie(A)) = det_{\overline{ℚ}_{p}}(δ|V_{0}),
where End^{(p)}(A): = End(A) ⊗ ℤ_{(p)}.

(iii)′ A ℚ-homogeneous polarization \overline{Λ} of A, which contains a polarization of degree prime to p, such that ι gives a homomorphism between ℤ_{(p)}-algebras with involution with O_{D} (resp. End^{(p)}(A)) considered together with ∗ (resp. the Rosatti involution).

(iv)′ A modulo C^{p}-class \overline{η}^{p} of D ⊗ 𝔸_{f}^{p}-linear symplectic similitudes

η: (V^{p}(A), Riemann form) ⟶^{∼} (V ⊗ 𝔸_{f}^{p}, ψ ⊗ 𝔸_{f}^{p}),

where V^{p}(A) denotes the prime-to-p part of the rational Tate module of A.

The condition in (ii)′ calls for an explanation. This condition has been introduced in [Ko92, §5], and, as explained there, the equality should be read as an identity of polynomial functions on δ ∈ O_{D}. Over schemes of characteristic zero, it can be replaced by the usual “trace” condition as in (ii), while it is crucial over positive characteristics.

By the standard argument as in [Ko92, §5], one sees that the étale sheafification of the functor \underline{A}_{C} can be represented by a quasi-projective scheme A_{C} over O_{E_{ν}} (even projective due to [BZ95, 1.1]).

We need to relate A_{C} to the Shimura variety Sh_{C}; to do this, one has to take a bit care of the difference between the data (iv) and (iv)′. Let (A, ι, \overline{Λ}, \overline{η}^{p}) be as above. By [RZ96, 6.39] the Tate module of A together with the pairing induced by the polarization is isomorphic to (Γ_{p}, ψ_{p}) as a symplectic O_{D_{p}}-module (cf. [RZ96, 6.38]). This gives a similitude between (V_{p}(A), Riemann form) and (V ⊗ ℚ_{p}, ψ ⊗ ℚ_{p}). So, if our group G satisfies the Hasse principle, then, combining with the datum (iv)′, we get a similitude as in (iv), and thus we get an isomorphism

A_{C} ⊗_{O_{Eν}} E_{ν} ⟶^{∼} Sh_{C} ⊗_{E} E_{ν},

i.e. the scheme A_{C} is giving a local model of Sh_{C}.

For verifying the Hasse principle for G, we refer the argument by Kottwitz [Ko92, §7]; our situation is of the case (A) in his sense, and we find that G satisfies the Hasse principle, for the center of G is an imaginary quadratic extension of ℚ.

E. p-adic uniformization. Now let κ be the residue field of \overline{E}_{ν}(≅ \overline{ℚ}_{p}), and take a point [A_{t}, ι_{t}, \overline{Λ}_{t}, \overline{η}_{t}^{p}] ∈ \underline{A}_{C}(κ). Let us denote by X_{t} the p-divisible group corresponding to the abelian variety A_{s}. The p-divisible group X_{t} has an O_{D_{p}}-action so that it is a formal O_{D_{p}}-module, where O_{D_{p}}: = O_{D} ⊗_{ℤ} ℤ_{p} ≅ O_{D_{𝔭}} × O_{D\overline{𝔭}}. By this action one has the decomposition

X_{t} = X_{1} × X_{2},

where X_{1} (resp. X_{2}) is a formal O_{D_{𝔭}} (resp. O_{D\overline{𝔭}})-module. The “determinant” condition in (ii)′ is equivalent with that X_{1} is a special formal O_{D_{𝔭}}-module in the sense of Drinfeld [Dr76]. By [Dr76] it is known that there is only one isogeny class of special formal O_{D_{p}}-module over κ.

Due to [RZ96, 6.39], the Tate module of X_{t} together with the pairing induced by the polarization is isomorphic to (Γ_{p}, ψ_{p}) as a symplectic O_{D_{p}}-module (cf. [RZ96, 6.38]). Since the involution ∗ gives an isomorphism O_{D\overline{𝔭}} ≅ O_{D_{𝔭}}^{op} and O_{D_{𝔭}}^{op} ≅ End_{O_{D𝔭}}(Γ_{𝔭}), there exists an isomorphism Γ_{p} ≅ Γ_{𝔭} × Γ_{𝔭}^{∨}, where Γ_{𝔭}: = Γ ⊗_{O_{K}} K_{𝔭}. Hence we have X_{2} ≅ X_{1}^{∨}, and the polarization on X_{t} ≅ X_{1} × X_{1}^{∨} has to be the tautological one (the Poincaré bundle), for it has been required to be compatible with the O_{D_{p}}-action. The significance of this fact is that, when we take the p-divisible group associated to the data as above, we can throw away the polarization!

Summarizingly, once we consider the associated p-divisible groups, we have only to concern with a single isogeny class. This enables us to deal with the moduli data on an equal footing, and leads us to the context of p-adic periods. So, fixing one point [A_{s}, ι_{s}, \overline{Λ}_{s}, \overline{η}_{s}^{p}] ∈ \underline{A}_{C}(κ), we shall be interested in classifying quasi-isogenies X_{s} → X_{t}; because of the rigidity property of such quasi-isogenies (cf. [RZ96, 3.62]), they form a padic formal scheme over ℤ_{p}^{ur}. In fact, Drinfeld [Dr76] proved that they are parametrized by a formal scheme Ω̂^{ur}, which is base extension of the formal scheme Ω over ℤ_{p} of which the Raynaud generic fiber is known to be the Drinfeld symmetric space.

Then, pushing out the fixed data [A_{s}, ι_{s}, \overline{Λ}_{s}, \overline{η}_{s}^{p}] by quasi-isogenies, and applying the theory of Serre-Tate local moduli, we get an epimorphism of formal schemes (the uniformization morphism)

Ω̂^{ur} × G(𝔸_{f}^{p})/C^{p} ⟶ Â_{C} ⊗ O_{E_{ν}}^{ur},

where Â_{C} denotes the ν-adic completion of A_{C}; the second factor on the left hand side serves for the level structure. In order to understand the fibers of this morphism, we introduce an algebraic group I over ℚ by setting, for a ℚ-algebra R,

I(R): = {φ ∈ (End_{D}^{∘} A_{s}) ⊗ R | φφ′ ∈ R^{×}},

where φ ↦ φ′ denotes the involution induced by the Rosatti involution. This is the algebraic group of quasi-isogenies of (A_{s}, ι_{s}), and so we can consider the natural action of I(ℚ) on Ω̂^{ur} × G(𝔸_{f})/C (cf. [BZ95, §1]). The algebraic group I is, in fact, an inner form of G such that I(𝔸_{f}^{p}) ≅ G(𝔸_{f}^{p}) and I(ℚ_{p}) is isomorphic to the group, denoted by J(ℚ_{p}), of all quasi-isogenies of the formal O_{D_{p}}-module X_{s}. Now the main theorem of the theory of p-adic uniformization of Shimura varieties asserts:

Theorem 2.1 ([BZ95, 1.12], [RZ96, 6.51]). We have an isomorphism of formal schemes

I(ℚ) \ Ω̂^{ur} × G(𝔸_{f})/C ⟶^{∼} Â_{C} ⊗ O_{E_{ν}}^{ur}.

Furthermore, this isomorphism is equivariant with respect to the Galois action of Gal(ℚ_{p}^{ur}/ℚ_{p}) ≅ Gal(E_{ν}^{ur}/E_{ν}) in a suitable sense (see the references cited above for the precise meaning). The left hand side is a finite disjoint union of quotients of Ω̂^{ur} by the groups

Γ_{g} = I(ℚ) ∩ (J(ℚ_{p}) × g^{−1}C^{p}g) ⊂ J(ℚ_{p})

with g ∈ G(𝔸_{f}^{p}), where I(ℚ) is now considered as a subgroup of I(𝔸_{f}) ≅ J(ℚ_{p}) × G(𝔸_{f}^{p}).

F. Calculation of the algebraic group I. It is important to describe the algebraic group I more explicitly. We should do this in the case that the following two conditions are satisfied: zw

(f) For any finite place ℓ which does not divide p, D_{ℓ} is isomorphic to the matrix algebra M_{d}(K_{ℓ}).

(g) For any finite prime ℓ which ramifies or is prime in K, there exists c ∈ Kℓ̃ with \overline{c} = −c such that Nm_{D⊗_{K}Kℓ̃/ℚ_{ℓ}}(cb) is in the image of Nm_{Kℓ̃/ℚ_{ℓ}}, where ℓ̃ is the prime of K lying above ℓ.

Note that the condition (g) is always satisfied if d is odd; indeed, setting f = Nm_{D⊗_{K}Kℓ̃/ℚ_{ℓ}}(cb) for some c, the norm of f · cb becomes a square number.

Let A_{s} be the abelian variety over κ fixed above. First we need to describe End_{D}^{∘} A_{s} with the Rosatti involution φ ↦ φ′. We claim:

Proposition 2.2. There exists an isomorphism

(End_{D}^{∘} A_{s}, φ ↦ φ′) ⟶^{∼} (M_{d}(K), τ: M ↦ ^{t}\overline{M}).

Proof. Let R be a commutative algebra over ℚ, and suppose we have an isomorphism (End_{D}^{∘} A_{s}) ⊗ R ≅ M_{d}(K) ⊗ R of algebras over R. Let us denote by ρ the involution on M_{d}(K)⊗R induced by the Rosatti involution on End_{D}^{∘} A_{s}. By Skolem-Noether theorem, there exists U ∈ M_{d}(K)⊗R such that M^{τ} = UM^{ρ}U^{−1} for any M ∈ M_{d}(K)⊗R. If we require U^{τ} = U, then U is uniquely determined up to constants in R^{×}. Then (M_{d}(K)⊗R, ρ) and (M_{d}(K)⊗R, τ) are isomorphic if and only if there exist f ∈ R^{×} and W ∈ M_{d}(K)⊗R such that fU = W W^{τ}. This observation shows that Isom((End_{D}^{∘} A_{s},′), (M_{d}(K), τ)) is a torsor over the ℚ-group scheme T defined by

T (R): = {W ∈ M_{d}(K) ⊗ R | W W^{τ} ∈ R^{×}}

for ℚ-algebra R. The group scheme T satisfies the Hasse principle, as is shown by an arguement similar to that in the end of D. So it suffices to find the desired isomorphism only locally at each place of ℚ.

At a finite place ℓ ≠ p, by Tate’s theorem and the level structure η_{s}, we have

(End_{D}^{∘} A_{s}) ⊗ ℚ_{ℓ} ≅ End_{D⊗ℚ_{ℓ}}(V ⊗ ℚ_{ℓ}) ≅ (D ⊗ ℚ_{ℓ})^{op} ≅ M_{d}(K) ⊗ ℚ_{ℓ}.

At the infinite place, we similarly have (End_{D}^{∘} A_{s}) ⊗ ℝ ≅ M_{d}(K) ⊗ ℝ by the assumption (a). So we treat these two cases at the same time. First, let ℓ be a place of ℚ which decomposes in K as ℓ = ν\overline{ν}. We look at the isomorphism (End_{D}^{∘} A_{s}) ⊗ ℚ_{ℓ} ≅ M_{d}(ℚ_{ν}) × M_{d}(ℚ_{ν}); the involution τ on the right hand side is now the switch of factors. By looking at the action of Rosatti involution on the rational Tate module, which is isomorphic to V ⊗ ℚ_{ℓ} by the level structure, we find

((End_{D}^{∘} A_{s}) ⊗ ℚ_{ℓ}, Rosatti) ≅ (D_{ℓ}^{op}, δ ↦ bδ^{∗}b^{−1}).

Hence, the action of ρ on M_{d}(ℚ_{ν}) × M_{d}(ℚ_{ν}) is described by γ ↦ (b, b)γ^{∗}(b^{−1}, b^{−1}), where ∗ is now the switch. So in this case one can choose U = (b^{−1}, b^{−1}), and then, W = (b^{−1}, 1) gives a solution. Also in the case that ℓ ramifies or remains prime in K, we can find f and W as above due to the assumption (g), for isomorphism classes of Hermitian forms of fixed rank over a local field are characterized by their determinant modulo the image of the norm mapping.

At p, we shall consider the Dieudonné isocrystal (N, 𝔽) associated to A_{s}; we have (End_{D}^{∘} A_{s}) ⊗ ℚ_{p} ≅ End_{D⊗ℚ_{p}^{ur}}(N, 𝔽) (cf. [RZ96, 6.29]). Since the associated p-divisible group splits like X_{s} ≅ X_{1} × X_{1}^{∨}, N must also split as

N ≅ V ⊗ ℚ_{p}^{ur} ≅ (V ⊗_{K} K_{𝔭}^{ur}) ⊕ (V ⊗_{K} Kur ; \overline{𝔭}).

Due to [RZ96, 3.59, 3.60], the isocrystal V ⊗_{K} K_{𝔭}^{ur}, together with the action of D_{𝔭} ⊗_{K_{𝔭}} K_{𝔭}^{ur}, is characterized by an isocrystal which is isotypic of slope zero. So the endomorphism algebra of it is isomorphic to M_{d}(K_{𝔭}). We can do the same at \overline{𝔭}, and thus we have

(End_{D}^{∘} A_{s}) ⊗ ℚ_{p} ≅ End_{D⊗ℚ_{p}^{ur}}(N, 𝔽) ≅ M_{d}(K) ⊗ ℚ_{p} ≅ M_{d}(ℚ_{p}) × M_{d}(ℚ_{p}).

In this case, the Rosatti involution is just the switch of factors, for the polarization on X_{s} is tautological, as we have seen before. So we find that in this case the solution W trivially exists, and hence the proposition has been proved. □

Corollary 2.3. The algebraic group I is isomorphic to a group of unitary similitudes with respect to the positive definite Hermitian form; more specifically, we have a functorial isomorphism

I(R) ≅ \{g ∈ (M_{d}(K) ⊗ R)^{×} | ^{t}\overline{g}g ∈ R^{×}\}

for any ℚ-algebra R.

Note that, since the prime p splits on K, I(ℚ_{p}) is isomorphic to GL_{d}(ℚ_{p}).

G. Connected components. Let us consider the torus T over ℚ defined by

T: = \{(k, f) ∈ Res_{K/ℚ}𝔾_{m,K} × 𝔾_{m,ℚ} | k\overline{k} = f^{d}\},

and define an epimorphism ϑ: G → T by sending g ↦ (Nm_{D^{op}/K} g, c(g)), where Nm_{D^{op}/K} is the reduced norm on D^{op}. The kernel of ϑ is the derived group G^{der} of G. By [De71, §2] we know that the morphism ϑ induces the canonical morphism of Shimura varieties

S_{C} = G(ℚ) \ X_{∞} × G(𝔸_{f})/C ⟶ T (ℚ)_{+} \ T (𝔸_{f})/ϑ(C)

which induces the Galois-equivariant bijection between the set of all connected components of both sides.

The description of the torus differs according to the parity of d; viz., those when d is even or odd (cf. [Ko92, §7]). When d is even, say d = 2m, then (k, f) ↦ (kf^{−m}, f) gives an isomorphism between T and U_{1} ×𝔾_{m,ℚ}, where U_{1} is the kernel of the norm map from Res_{K/ℚ}𝔾_{m,K} to 𝔾_{m,ℚ}. When d is odd, say d = 2m + 1, then (k, f) ↦ kf^{−m} gives an isomorphism between T and Res_{K/ℚ}𝔾_{m,K}.

The Shimura variety T (ℚ)_{+} \ T (𝔸_{f})/ϑ(C) in case d is odd can be described fairly explicitly in terms of the theory of complex multiplication. More precisely, the Shimura variety T (ℚ)_{+} \ T (𝔸_{f})/ϑ(C) has the canonical model defined over the Shimura field E = ε(K), and the Galois group Gal(F/E) acts simply transitively, where F is the ray class field of E corresponding to the level structure ϑ(C). Since the canonical morphism as above induces the morphism between the canonical models which is equivariant with the Galois action, we know that the number of connected components of S_{C} is equal to the order of Gal(F/E), and that each connected component of S_{C} has F as the field of definition.

3. The first example: Mumford’s fake projective plane

In this and following sections we will present explicit examples of the p-adic uniformization of Shimura varieties discussed in the previous section with d = 3, p = 2. Each Shimura variety in our examples has connected components isomorphic to algebraic surfaces of a special kind, so-called, fake projective planes, i.e. compact complex surfaces of general type which have the same topological Betti numbers as ℂℙ^{2}. The first example is related to the Mumford’s fake projective plane, which was discovered by Mumford [Mu79].

Let us first fix the fundamental data as in A of §2: Let ζ = ζ_{7} be a primitive 7throot of unity, and set L: = ℚ(ζ); L is a cyclic extension of ℚ of degree 6 having an intermediate quadratic extension K: = ℚ(λ) (≅ ℚ(\sqrt{−7})), where λ = ζ + ζ^{2} + ζ^{4}. The Galois group of L/K is generated by the Frobenius map σ: ζ ↦ ζ^{2}, and that of K/ℚ is generated by the complex conjugation z ↦ \overline{z}. Note that the prime 2 decomposes on K, more explicitly, 2 = λ\overline{λ} gives the prime factorization, and the prime 7 ramifies. We fix an infinite place (CM-type) ε: K ↪ ℂ by λ ↦ \frac{−1+\sqrt{−7}}{2}. Note that the class number of K is 1.

Construction of D. We set μ: = λ/\overline{λ}, and define the central division algebra D over K of dimension 9 by

D: = ⨁_{i=0}^{2} LΠ^{i}

with Π^{3} = μ and Πz = z^{σ}Π for z ∈ L; D satisfies the conditions (a), (b), and (f) in §2. The involution ∗ on D is defined by z^{∗} = \overline{z} for z ∈ L and Π^{∗} = \overline{μ}Π^{2}. It is easily verified that ∗ is positive. As for the order, we set

O_{D}: = O_{L} ⊕ O_{L}\overline{λ}Π ⊕ O_{L}\overline{λ}Π^{2},

which is clealy stable under ∗. Moreover, one can easily verify that O_{D_{λ}} and O_{D\overline{λ}} are maximal.

Finally, we introduce a non-degenerate anti-symmetric ℚ-bilinear form ψ = ψ_{b} on V = D by putting b = (λ − \overline{λ}) − \overline{λ}Π + \overline{λ}Π^{2}. In order to check the conditions (c) and (d) it is convenient to have an explicit description of the isomorphism Φ: D_{ε} →^{∼} M_{3}(ℂ) of algebras with involutions. Let θ: L ↪ ℂ be given by ζ ↦ exp\frac{2iπ}{7} , and θ_{i}: = θ ∘ σ^{i} for i = 0, 1, 2. Then we have

V ⊗_{K,ε} ℂ ⟶^{∼} ⨁_{i=0}^{2} V ⊗_{L,θ_{i}} ℂ.

We look at the action of D on the factor V ⊗_{L,θ_{0}}ℂ; subject to the base {1⊗1, Π⊗1, Π^{2}⊗1} one obtains a matrix presentation of elements in D determined by

z ↦ [ \begin{matrix}z \\ & z^{σ^{2}} \\ & & z^{σ} \end{matrix}] and Π ↦ [ \begin{matrix}& & μ \\ 1 \\ & 1 \end{matrix} ] ,

which induces the desired isomorphism Φ. (Here we dropped θ_{0} and ε for simplicity.) Then we have

Φ(b) = [ \begin{matrix}λ − \overline{λ} & λ & −λ \\ −\overline{λ} & λ − \overline{λ} & λ \\ \overline{λ} & −\overline{λ} & λ − \overline{λ} \end{matrix} ] .

(Here ε(λ) is abbreviated simply to λ.) The characteristic polynomial is t^{3} − 3\sqrt{−7}t^{2} − 15t − \sqrt{−7}. In particular, the condition (d) is satisfied; as Nm_{D/K} b = 2λ + 1 is both λ and \overline{λ}-adic unit, we also verify (c).

Level structure. Let us first notice that, in the isomorphy in Corollary 2.3, the involution g ↦ ^{t}\overline{g} on the right hand side can be replaced by the one defined by g ↦ g′: = H^{−1} ^{t}\overline{g}H, where H: = Φ(h) with h = 3 + \overline{λ}Π + \overline{λ}Π^{2} ∈ D; viz.

H = [ \begin{matrix}3 & \overline{λ} & \overline{λ} \\ λ & 3 & \overline{λ} \\ λ & λ & 3 \end{matrix} ];

this is allowed since there exists an decomposition H = ^{t}\overline{W} W over ℚ according as 3 + \overline{λ}Π + \overline{λ}Π^{2} = (λ + Π)^{∗}(λ + Π).

To define the level structure, it suffices to define the prime-2-part C^{2}. But we need to have a non-trivial component only at 7; viz. we define C^{2} in the form C^{2} = C^{2,7}C_{7} with C_{7} ∈ G(ℚ_{7}) and C^{2,7} being the maximal one:

C^{2,7}: = {g ∈ G(𝔸_{f}^{2,7}) | Γ^{2,7}g ⊆ Γ^{2,7}},

with Γ^{2,7} = Γ ⊗_{ℤ} ℤ̂^{2,7}, where ℤ̂^{2,7} = lim ℤ/mℤ with the projective limit taken over m such that (2, m) = (7, m) = 1.

Now the component C_{7} is defined as follows: By Corollary 2.3, the fact stated below Theorem 2.1, and what we have just done above, we have an isomorphism

G(ℚ_{7}) ⟶^{∼} {g ∈ GL_{3}(ℚ̃_{7}) | g^{∗}Hg = c(g)H, c(g) ∈ ℚ_{7}^{×}},

where ℚ̃_{7} is the ramified quadratic extension of ℚ_{7}, and ∗ is the matrix transposition followed by the Galois action. Let C̃_{7} be the maximal open compact subgroup of G(ℚ_{7}) consisiting of g which maps Γ ⊗_{ℤ}ℤ_{7} to itself. We consider modulo \sqrt{−7} reduction of C̃_{7}, which is a subgroup G_{0} in GL_{3}(𝔽_{7}). The matrix H mod \sqrt{−7} is of rank 1, and has 2 dimensional null space N_{0}. Restricting elements in G_{0} to N_{0}, we obtain a homomorphism

π: C̃_{7} ⟶ GL_{2}(𝔽_{7}).

The last group has 2^{6}3^{2}7 elements. Let S be a 2-Sylow subgroup of GL_{2}(𝔽_{7})∩{g | det g = ±1}. Now we set C_{7}: = π^{−1}(S), and thus, the definition of the level structure is finished.

Theorem 3.1. The canonical model Sh_{C} of the Shimura variety obtained by the data as above has three geometrically connected non-singular projective surface over ℚ(ζ_{7}) such that the base change over ℚ_{2}(ζ_{7}) are isomorphic to the Mumford’s fake projective plane [Mu79].

Corollary 3.2. On the complex analytic side, the Mumford’s fake projective plane is an arithmetic quotient of the 2-dimensional complex unit-ball having the minimal field of definition E = ℚ(ζ_{7}).

More explicitly, indeed, it is the quotient of the 2-dimensional unit-ball by the arithmetic group Γ_{∞}: = G(ℚ) ∩ C ⊂ G(ℝ) ≅ GU(2, 1).

Proof of Theorem 3.1. First, we notice that we constructed the level structure C = C_{2}C^{2} precisely so that the associated discrete group Γ_{1} = I(ℚ) ∩ (J(ℚ_{2}) × C^{2}) modulo the center of J(ℚ_{2}) ≅ I(ℚ_{2}) ≅ GL_{3}(ℚ_{2}) coincides with the one constructed by Mumford in [Mu79]. Indeed, Mumford’s Γ was a subgroup of the unitary group with respect to H characterized by the same congruence condition as above. Also by [RZ96, 6.51] the isomorphism in Theorem 2.1 descends to ℚ_{2} ≅ E_{ν}. So it suffice to see that every element g in I(ℚ) can be written as g = uk with u ∈ I(ℚ) and k ∈ ℚ^{×} such that ^{t}\overline{u}Hu = H. But this follows from an argument similar to that we have remarked at the end of B of §2. Hence we deduce that one, and therefore, every connected component of the associated Shimura variety S_{C} is isomorphic to Mumford’s fake projective plane.

Next we look at the morphism ϑ: G → T as in G of §2 in order to observe connected components. To calculate ϑ(C) we have only to concern with the component at 7. We claim the following: Under the canonical identification T (ℚ_{7}) ≅ ℚ̃_{7}^{×}, where ℚ̃_{7}^{×} is the ramified quadratic extension of ℚ_{7}, we have ϑ(C_{7}) ≅ ±1 + \sqrt{−7} ℤ̃_{7}^{×}, where ℤ̃_{7}^{×} is the integer ring of ℚ̃_{7}^{×}. First we note that the map ϑ is described by

ϑ: C_{7} ∋ γ ↦ (^{t}\overline{γ}HγH^{−1})^{−1} det γ ∈ ℤ̃_{7}^{×}.

Since ±1 + 7 ℤ_{7}^{×} ⊂ C_{7} we immediately deduce that ϑ(C_{7}) contains ±1 + 7 ℤ_{7}^{×}. Moreover, since ±1+\sqrt{−7} ℤ̃_{7}^{×} ⊂ C_{7}, ϑ(C_{7}) contains the element k^{2}\overline{k}^{−1} for any k ∈ ±1+\sqrt{−7} ℤ̃_{7}^{×} ⊂ C_{7}; but, since k\overline{k} ∈ ϑ(C_{7}), we deduce that

(±1 + \sqrt{−7} ℤ̃_{7}^{×})^{3} = ±1 + \sqrt{−7} ℤ̃_{7}^{×} ⊆ ϑ(C_{7}).

Hence the claim follows if we check that (ϑ(C_{7}) mod \sqrt{−7}) = \{±1\}. As we have seen above any γ ∈ C_{7} has a decomposition γ = kθ in C_{7} with ^{t}\overline{θ}HθH^{−1} = 1. Since det ^{t}\overline{θ} det HθH^{−1} ≡ (det θ)^{2} ≡ 1 mod \sqrt{−7} we have ϑ(γ) ≡ ±k. But, since k = γθ^{−1} mod \sqrt{−7} belongs to P, the order of k is either one or two; hence k ≡ ±1 mod \sqrt{−7}. We therefore get ϑ(γ) ≡ ±1 mod \sqrt{−7}, which proves the claim.

Now in view of the theory of complex multiplication we find that the Shimura variety T (ℚ)_{+}\T (𝔸_{f})/ϑ(C) has three connected components having field of definition L = ℚ(ζ) which are permuted by Gal(L/K), since the ray class field corresponding to C is easily checked to be L. We therefore proved the all statements in the theorem. □

4. The second and third examples

The second and third examples contain fake projective planes as connected components, which are the ones constructued in [IK98].

In this section we work over the 45th-cyclotomic field ℚ(ζ_{45}); inside it are the subfields ℚ(ζ_{9}) and ℚ(ζ_{15}). We set λ: = ζ_{15} + ζ_{15}^{2} + ζ_{15}^{4} + ζ_{15}^{8}, of which the minimal polynomial is given by x^{2} − x + 4. We set K: = ℚ(λ) (≅ ℚ(\sqrt{−15})); the class number of K is 2. The prime 2 decomposes on K; more explicitly, 2O_{K} = Λ\overline{Λ} gives the prime factoization, where Λ = 2ℤ + (λ + 1)ℤ. The primes 3 and 5 ramify on K; viz. 3O_{K} = 𝔓_{3}^{2} with 𝔓_{3} = 3ℤ + (λ + 1)ℤ, and 5O_{K} = 𝔓_{5}^{2} with 𝔓_{5} = 5ℤ + (λ + 2)ℤ. A CM-type embedding ε: K ↪ ℂ is fixed by λ ↦ \frac{1+\sqrt{−15}}{2}. We define L to be the composite field of K and the maximal real subfield of ℚ(ζ_{9}); L is generated over K by ζ_{9} + ζ_{9}^{−1} of which the minimal polynomial over K is x^{3}−3x+1. The Galois group of L/K is generated by the Frobenius σ: ζ_{9} + ζ_{9}^{−1} ↦ ζ_{9}^{2} + ζ_{9}^{−2}, while that of K/ℚ is generated by the complex conjugation.

Putting μ: = \overline{λ}/λ, we can define D and ∗ by the same recipe as in §3; the conditions (a), (b), and (f) can be easily verified. The order O_{D} should be defined by

O_{D}: = O_{L} ⊕ O_{L}λΠ ⊕ O_{L}λΠ^{2}.

The ℚ-bilinear form ψ = ψ_{b} on V = D is given by the element b = (λ − \overline{λ}) − λΠ + λΠ^{2}, which guarantees the conditions (c) and (d). Indeed, one expresses elements in D by matrices in a similar way as in §3.

Level structures. In order to define level structures, we replace the involution g ↦ ^{t}\overline{g} on the right hand side of the isomorphy in Corollary 2.3 by g ↦ g′: = H^{−1} ^{t}\overline{g}H by

H = [ \begin{matrix}10 & 2λ − 6 & −λ + 3 \\ 2\overline{λ} − 6 & 10 & 2λ − 6 \\ −\overline{λ} + 3 & 2\overline{λ} − 6 & 10 \end{matrix} ] .

This replacement is possible since H = ^{t}\overline{A}A/3 with

A = [ \begin{matrix}0 & 0 & 2λ − 1 \\ λ − 2 & −2λ − 2 & 3 \\ −2λ + 4 & λ − 2 & −λ + 2 \end{matrix} ] .

We define two distinct level structures ^{I}C and ^{II}C which define two Shimura varieties. The both are expressed in the form C_{2}C^{2,3}C_{3}, with C_{2} and C^{2,3} being maximal, given by a formula similar to those in §3. the components at 3 are defined in terms of congruence conditions modulo 3O_{K} = 𝔓_{3}^{2}. Put t: = 2λ − 1 mod 3. Then the modulo 3 reduction of the maximal open subgroup C̃_{3} of I(ℚ_{3}) ≅ G(ℚ_{3}) is a subgroup of GL_{3}(𝔽_{3}[t]/(t^{2})). We define its subgroups ^{I}U and ^{II}U: ^{I}U is generated by the invertible scalar matrices and the following seven matrices:

Ih_{0} = [ \begin{matrix}t + 2 & 2t + 1 & 2t \\ t + 2 & 0 & 2t \\ 0 & 0 & t + 1 \end{matrix} ] , ^{I}h_{1} = [ \begin{matrix}2 & 0 & 1 \\ 2t + 2 & t + 1 & 2t + 1 \\ 0 & 0 & 2t + 2 \end{matrix} ] ,

Ih_{2} = [ \begin{matrix}0 & t + 1 & 2 \\ 1 & t & 2t \\ 0 & 0 & 2t + 2 \end{matrix} ] , ^{I}h_{3} = [ \begin{matrix}0 & 0 & 2t + 1 \\ 1 & 0 & 1 \\ 0 & 1 & t + 2 \end{matrix} ] ,

Ih_{4} = [ \begin{matrix}t + 1 & 2 & 0 \\ 0 & 2 & 2t + 2 \\ 0 & 0 & 2t + 2 \end{matrix} ] , ^{I}h_{5} = [ \begin{matrix}2 & 0 & 0 \\ 2t + 1 & t + 2 & 0 \\ 1 & 2 & 1 \end{matrix} ] ,

Ih_{6} = [ \begin{matrix}t & 2 & 0 \\ t + 2 & 2 & 2t + 1 \\ 2 & 0 & 0 \end{matrix} ] .

IIU is generated by the invertible scalar matrices and the following seven matrices:

IIh_{0} = [ \begin{matrix}t + 1 & t + 1 & 2t \\ 0 & 1 & t \\ 0 & 0 & t + 1 \end{matrix} ] , ^{II}h_{1} = [ \begin{matrix}t + 2 & t + 2 & 2t + 1 \\ t & 2 & 2 \\ t & 0 & 1 \end{matrix} ] ,

IIh_{2} = [ \begin{matrix}2 & 2t + 2 & t + 2 \\ t & t + 2 & 2 \\ t & t & 1 \end{matrix} ] , ^{II}h_{3} = [ \begin{matrix}0 & 2t + 2 & t + 1 \\ 2t + 1 & 1 & 2 \\ 0 & 0 & 2t + 1 \end{matrix} ] ,

IIh_{4} = [ \begin{matrix}2t + 2 & t + 2 & 0 \\ t & 2t + 2 & 2 \\ t & 2t & 1 \end{matrix} ] , ^{II}h_{5} = [ \begin{matrix}t + 2 & 0 & 0 \\ 1 & 2t + 2 & 0 \\ t & t & 2t + 1 \end{matrix} ] ,

IIh_{6} = [ \begin{matrix}1 & t + 2 & 0 \\ t + 1 & t & t + 1 \\ 2t & t & t + 1 \end{matrix} ] .

It is easily verified that the groups ^{I}U and ^{II}U are contained in the image of C̃_{3} under the reduction map. Now we define ^{I}C_{3} and ^{II}C_{3} to be the inverse image of ^{I}U and ^{II}U under the reduction map, respectively. The level structures ^{I}C and ^{II}C are defined to be C_{2}C^{2,3} ^{I}C_{3} and C_{2}C^{2,3} ^{II}C_{3}, respectively.

We denote the associated Shimura varieties by the above data with the level structure IC (resp. ^{II}C) by S_{^{I}C} (resp. S_{^{II}C}). We claim:

Theorem 4.1. The canonical model Sh_{^{I}C} (resp. Sh_{^{II}C}) of the Shimura variety S_{^{I}C} (resp. S_{^{II}C}) consists of two connected components, each of which is a geometrically connected non-singular projective surface over E′ = ℚ(\sqrt{−3}, \sqrt{5}), such that the base change of each connected component at the prime Λ is isomorphic over E′_{Λ} ≅ ℚ_{2^{2}} to the base change of the surface X_{Γ_{B_{1}},η} (resp. X_{Γ_{B_{2}},η}) obtained in [IK98].

Corollary 4.2. The two complex fake projective planes considered in [IK98], other than the Mumford’s one, are arithmetic quotients of the 2-dimensional complex unit-ball having the minimal field of definition E′ = ℚ(\sqrt{−3}, \sqrt{5}).

In order to prove the theorem, first we show that the discrete group

IΓ = I(ℚ) ∩ (J(ℚ_{2}) × ^{I}C^{2}) and ^{II}Γ = I(ℚ) ∩ (J(ℚ_{2}) × ^{II}C^{2})

coincide modulo center with those discovered by Cartwright, Mantero, Steger, and Zappa in [CMSZ93a] and [CMSZ93b]. More precisely, Γ_{B.1} and Γ_{B.2}, respectively, in their notation (see [CMSZ93b, §5]). These groups act on vertices of Bruhat-Tits building related to the Drinfeld Symmetric space Ω simply transitively, and, as was shown in [IK98], they are torsion-free. So, by Mumford’s criterion [Mu79], one deduces that the quotients of Ω by these groups are fake projective planes. Moreover, it was proved in [IK98] that these fake projective planes, including the Mumford’s one, are pairwise non-isomorphic.

Proposition 4.3. We have equalities ^{I}Γ = Γ_{B.1} and ^{II}Γ = Γ_{B.2} modulo scalar matrices.

Proof. The inclusions Γ_{B.1} ⊆ ^{I}Γ and Γ_{B.2} ⊆ ^{II}Γ can be seen without so much pain; actually, the matrices ^{I}h_{i} and ^{II}h_{i} are precisely the modulo 3 reductions of those g_{i} of the corresponding groups described in [CMSZ93b, pp. 182]. In order to show the other inclusions, we should calculate the stabilizer of one vertex of the Bruhat-Tits building. Let [M_{0}] be a vertex, i.e. the similarity class of a lattice, where M_{0} is the standard lattice ℤ_{2}^{3}. Elements in ^{I}Γ or ^{II}Γ has entries belonging to ℤ[λ, \frac{1}{2}]. Let γ be such a matrix in I(ℚ), and assume it fixes [M_{0}]. We can assume that γ is H-unitary due to the similar argument as in the end of B of §2; taking inverse if necessary, we may moreover assume γM_{0} ⊆ M_{0}. Then γ has entries in ℤ[λ, \frac{1}{2}] ∩ ℤ_{2} = ℤ[λ, \frac{1}{\overline{λ}}]. Since γ is H-unitary, we moreover deduce that they are actually belonging to ℤ[λ, \frac{1}{\overline{λ}}] ∩ \frac{1}{2·3·5}ℤ[\frac{1}{λ}] ⊆ ℤ[\frac{λ}{2}], where 2, 3, and 5 are the primes which appear in denominators of the entries of H^{−1}.

Lemma 4.4. The matrix γ must be of form ±S^{i} (i = 0, 1, 2), where

S = [ \begin{matrix}0 & −1 & 1 − λ \\ 1 & −1 & 3 − λ \\ 0 & 0 & 1 \end{matrix} ] .

(Note that S is of order 3.)

Proof of Lemma 4.4. Since γ is H-unitary, each column of γ has H-norm 10. There exist only finitely many such vectors having entries in ℤ[\frac{λ}{2}]. So if we have a complete list of such vectors, then all the possible γ can be determined. This procedure is elementary, but it would be too painful to perform by hand. The author carried it out by computer search collaborated with MAPLE V. Actually there are precisely 42 solutions for the vectors of norm 10. By checking the H-unitarity of all combinations of such vectors, the author got precisely 6 matrices as in the lemma. Note that the matrix S is precisely the one which appeared in [CMSZ93b, pp. 182] (denoted by s there). □

We continue the proof of Proposition 4.3. By the above observations, we only need to show that the similarity class of S does not belong to each of ^{I}Γ and ^{II}Γ modulo scalar matrices. Here we prove the proposition for ^{II}Γ, but exactly the same argument runs for the other. In the following we drop the superscript ^{II}–.

Since the modulo 𝔓_{3} reduction of the Hermitian form H is of rank 1, modulo 𝔓_{3} reduction of every element of Γ leaves invariant the 2-dimensional null space of H mod 𝔓_{3}. More explicitly, looking at

H mod 𝔓_{3} = [ \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} ] ,

one considers the upper left 2 × 2-minor of each P^{−1}γP mod 𝔓_{3} for γ ∈ Γ, where

P = [\begin{matrix}1 & 0 & 0 \\ −1 & 1 & 0 \\ 0 & −1 & 1 \end{matrix} ] .

This gives a mapping

ρ: Γ ⟶ GL_{2}(𝔽_{3}).

By definition of Γ, the image of ρ is generated by the upper left 2 × 2-minor of P^{−1}h_{i}P for i = 0, …, 6. Let us denote the resulting matrices by \overline{h}_{i} for i = 0, …, 6. Then one calculates

\overline{h}_{0} = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] , \overline{h}_{1} = \overline{h}_{2} = \overline{h}_{4} = [\begin{matrix}−1 & −1 \\ 0 & −1 \end{matrix} ] , \overline{h}_{3} = [ \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ] ,

\overline{h}_{5} = [\begin{matrix}−1 & 0 \\ 1 & −1 \end{matrix} ] , and \overline{h}_{6} = [ \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ] .

By this, it is easy to see that the image of ρ is exactly GL_{2}(𝔽_{3}). Let \overline{ρ}: U → GL_{2}(𝔽_{3}) be the reduction map which factors ρ, i.e. \overline{ρ}(h_{i}) = \overline{h}_{i}. By observing the fundamental relations among \overline{h}_{i}’s, one easily deduce that Ker \overline{ρ} is generated by scalar matrices with determinant 1 and some matrices of form h_{i_{1}} ⋯ h_{i_{N}} with N being multiples of 3. In particular, one sees that every element in Ker \overline{ρ} has determinant 1, for every h_{i} has determinant 2t + 1.

Now assume that a scalar times S were belonging to Γ; then U contains the matrix

s = [ \begin{matrix}2t + 1 & 1 & t + 1 \\ 0 & t + 1 & t \\ 0 & 2t & 1 \end{matrix} ] ,

the modulo 3 reduction of S. Since \overline{ρ}(s) = \overline{ρ}(h_{0}), we know s ∈ h_{0} ·Ker \overline{ρ}; but this violates with the fact det s = 1. Thus we conclude the proof of the proposition. □

Proof of Theorem 4.1. It remains only to prove the statements on connected components and the fields of definition. As in the first example, we shall look at the morphism \overline{ϑ}: ^{I}C_{3}, ^{II}C_{3} → (𝔽_{3}[t]/(t^{2}))^{×} given by uk ↦ (det u)k\overline{k}, where the conjugation \overline{·} is now given by t ↦ −t. The \overline{ϑ} factors through ^{I}U or ^{II}U, and in both cases, one easily sees that \overline{ϑ} is surjective. So, similarly as in the first example, our the level structures ϑ(^{I}C) and ϑ(^{II}C) of the torus T are the maximal one. So each of the Shimura varieties S_{^{I}C} and S_{^{II}C} has exactly two connected components defined over the absolute class field E′ ≅ ℚ(\sqrt{−3}, \sqrt{5}) of K. Since E′_{Λ} ≅ ℚ_{2^{2}} (the unramified quadratic extension of ℚ_{2}), and the 2-adic uniformizations by Γ_{B.1} and Γ_{B.2} were carried out over ℚ_{2}, the isomorphism in Theorem 2.1 descends onto ℚ_{2^{2}}. □

References

[An98] André, Y.: p-adic orbifolds and p-adic triangle groups, preprint, Institut de Math. Jussieu (1998).

[Bo97] Boutot, J-F.: Uniformisation p-adique des variétés de Shimura, Sém. Bourbaki, Exposé 831 (1997).

[BC91] Boutot, J-F., Carayol, H.: Uniformisation p-adique des courbes de Shimura: Les Théorèmes de Čerednik et de Drinfeld, Astérisque 196-197, Soc. Math. France (1991).

[BZ95] Boutot, J-F., Zink, Th.: The p-adic Uniformization of Shimura Curves, preprint 95–107, Universität Bielefeld (1995).

[CMSZ93a] Cartwright, D.I., Mantero, A.M., Steger, T., Zappa A.: Group acting simply transitively on the vertices of a building of type Ã_{2}, I, Geometriae Dedicata 47 (1993), 143–166.

[CMSZ93b] Cartwright, D.I., Mantero, A.M., Steger, T., Zappa A.: Group acting simply transitively on the vertices of a building of type Ã_{2}, II — the cases q = 2 and q = 3, Geometriae Dedicata 47 (1993), 167–223.

[Če76] Čerednik, I.V.: Uniformization of algebraic curves by discrete arithmetic subgroups of PGL_{2}(k_{w}) with compact quotient spaces, Mat. Sd., 100 (1976), 59–88.

[De71] Deligne, P.: Travaux de Shimura, Sém. Bourbaki Exposé 389, Lect. Notes in Math. 244 (1971), Springer, Heidelberg.

[Dr76] Drinfeld, V.G.: Coverings of p-adic symmetric regions, Funct. Anal. and Appl. 10 (1976), 29–40.

[IK98] Ishida, M., Kato, F.: The strong rigidity theorem for non-archimedean uniformization, Tôhoku Math. J., 50 (1998), 537–555.

[Ko92] Kottwitz, R.E.: Points on some Shimura varieties over finite fields, Journal Amer. Math. Soc. 5 (1992), 373–444.

[Ku79] Kurihara, A.: On some examples of equations defining Shimura curves and the Mumford uniformization, J. Fac. Sci. Uni. Tokyo, Sec. IA 25 (1979), 277–300.

[Ku94] Kurihara, A.: On p-adic Poincaré series and Shimura curves, Internat. J. Math. 5 (1994), 747–763.

[Mi90] Milne, J.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in Automorphic forms, Shimura varieties and L-functions, Clozel, L., Milne, J.S. ed. vol. I, Persp. in Math. 10, Acad. Press Boston (1990), 283–414.

[Mu79] Mumford, D.: An algebraic surface with K ample, (K^{2}) = 9, pg = q = 0, in Contribution to Algebraic Geometry, Johns Hopkins Univ. Press (1979), 233–244.

[Ra90] Rapoport, M.: On bad reduction of Shimura varieties, in Automorphic forms, Shimura varieties and L-functions, Clozel, L., Milne, J.S. ed. vol. II, Persp. in Math. 11, Acad. Press Boston (1990), 283–414.

[RZ82] Rapoport, M., Zink, Th.: Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), 21–101.

[RZ96] Rapoport, M., Zink, Th.: Period Spaces for p-divisible Groups, Annals of Math. Studies 141 Princeton University Press (1997).

Graduate School of Mathematics, Kyushu University, 6-10-1 Hakozaki Higashiku, Fukuoka 812, Japan.

Current Address: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26 53225 Bonn, Germany.