CM Points and Quaternion Algebras

C. Cornut and V. Vatsal

Received: December 23, 2004

Communicated by Peter Schneider

Abstract. This paper provides a proof of a technical result (Corollary 2.10 of Theorem 2.9) which is an essential ingredient in our proof of Mazur’s conjecture over totally real number fields [3].

2000 Mathematics Subject Classification: 14G35, 11G18, 11G15 Keywords and Phrases: Shimura curves, CM points

Contents

1 Introduction 264

2 CM points on quaternion algebras 266

2.1 CM points, special points and reduction maps . . . . . . . . . . 266 2.1.1 Quaternion algebras. . . . . . . . . . . . . . . . . . . . . 266 2.1.2 Algebraic groups . . . . . . . . . . . . . . . . . . . . . . 266 2.1.3 Adelic groups . . . . . . . . . . . . . . . . . . . . . . . . 266 2.1.4 Main objects . . . . . . . . . . . . . . . . . . . . . . . . 268 2.1.5 Galois actions . . . . . . . . . . . . . . . . . . . . . . . . 269 2.1.6 Further objects . . . . . . . . . . . . . . . . . . . . . . . 269 2.1.7 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 269 2.1.8 Level structures . . . . . . . . . . . . . . . . . . . . . . . 270

2.2 Main theorems: the statements . . . . . . . . . . . . . . . . . . 270 2.2.1 Simultaneous reduction maps . . . . . . . . . . . . . . . 270 2.2.2 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . 271 2.2.3 Surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . 272 2.2.4 Equidistribution . . . . . . . . . . . . . . . . . . . . . . 273

2.3 Proof of the main theorems: first reductions . . . . . . . . . . . 274 2.4 Further reductions . . . . . . . . . . . . . . . . . . . . . . . . . 277

2.4.1 Existence of a measure and proof of Proposition 2.5 . . 277 2.4.2 A computation. . . . . . . . . . . . . . . . . . . . . . . . 278

2.4.3 P -adic uniformization. . . . . . . . . . . . . . . . . . . . 281 2.5 Reduction of Proposition 2.13 to Ratner’s theorem . . . . . . . 282

2.5.1 Reduction of Proposition 2.21 . . . . . . . . . . . . . . . 282 2.5.2 Reduction of Proposition 2.22 . . . . . . . . . . . . . . . 283

2.6 Proof of Proposition 2.12. . . . . . . . . . . . . . . . . . . . . . 286 2.7 An application of Ratner’s Theorem . . . . . . . . . . . . . . . 288

3 The case of Shimura curves 291

3.1 Shimura Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 291 3.1.2 CM points . . . . . . . . . . . . . . . . . . . . . . . . . . 292 3.1.3 Integral models and supersingular points . . . . . . . . . 292 3.1.4 Reduction maps . . . . . . . . . . . . . . . . . . . . . . 293 3.1.5 Connected components . . . . . . . . . . . . . . . . . . 293 3.1.6 Simultaneous reduction maps . . . . . . . . . . . . . . . 294 3.1.7 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . 295

3.2 Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 3.2.1 CM points . . . . . . . . . . . . . . . . . . . . . . . . . . 296 3.2.2 Connected components . . . . . . . . . . . . . . . . . . 297 3.2.3 Supersingular points . . . . . . . . . . . . . . . . . . . . 297 3.2.4 Reciprocity laws . . . . . . . . . . . . . . . . . . . . . . 298 3.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 300

3.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 3.3.1 On the parameter ϵ = ±1 . . . . . . . . . . . . . . . . . 300 3.3.2 P -rational elements of Gal_{K}^{ab}. . . . . . . . . . . . . . . . 303

1 Introduction

Let F be a totally real number field of degree d, and let B denote a quaternion algebra over F. For the purposes of this introduction, we assume that either:

• B is definite, meaning that B_{v} = B ⊗ F_{v} is non-split for all real places v of F, or

• B is indefinite, meaning that B_{v} is split for precisely one real v.

We shall write G to denote the algebraic group over Q whose points over a Q-algebra A are the set (B ⊗ A)^{×}. Now let K be an imaginary quadratic extension of F. We suppose that there is given an embedding K → B. Then associated to the data of B and K, one can define a collection of points, the so-called CM points. The natural habitat for these points depends on whether B is definite or indefinite: in the former case, the CM points are just an infinite discrete set, whereas in the latter, they inhabit certain canonical algebraic curves, the Shimura curves, associated to the indefinite algebra B. Our goal in this paper is to study the distribution of these CM points in certain auxiliary spaces. The main result proven here is the key ingredient in our proof in [3] of certain non-vanishing theorems for certain automorphic L-functions over F and their derivatives. The theorems of [3] may be regarded as generalizations of Mazur’s conjectures in [12] when F = Q.

Our original intention was simply to write a single paper proving the nonvanishing theorems for the L-functions, using the connection between Lfunctions and CM points, and proving a basic nontriviality theorem for the latter. However, in the course of doing this, we realized that although the CM points in the definite and indefinite cases are a priori very different, the proof of the main nontriviality result on CM points runs along parallel lines. In light of this, it seemed somewhat artificial to give essentially the same arguments twice, once in each of the two cases. The present paper therefore presents a rather general result about CM points on quaternion algebras, which allows us to obtain information about CM points in both the definite and indefinite cases. The former case follows trivially, but the latter requires us to develop a certain amount of foundational material on Shimura curves, their various models, and the associated CM points.

Since this paper is neccessarily rather technical, we want to give an overview of the contents. The first part deals with the abstract results. The main theorems are given in Theorem 2.9 and Corollary 2.10. Although the statements are somewhat complicated, they are not hard to prove, in view of our earlier results [2], [18], [19], where all the main ideas are already present. As before, the basic ingredient is Ratner’s theorem on unipotent flows on p-adic Lie groups.

The second part is concerned with the applications of the abstract result to CM points on Shimura curves. We start with basic theory of Shimura curves, especially their integral models and reduction. In Section 3.1.1, we define the CM points and supersingular points, and establish the basic fact that the reduction of a CM point at an inert prime is a supersingular point. The basic result on CM points on Shimura curves is stated in Theorem 3.5. Section 3.2 gives a series of group theoretic descriptions of the various sets and maps which appear in Theorem 3.5, thus reducing its proof to a purely group theoretical statement, which may be deduced from the results in the first part of this paper.

The final two sections in the paper are meant to shed some light on related topics: section 3.3.1 investigates the dependence of Shimura curves on a certain parameter ϵ = ±1, while section 3.3.2 provides some insight on a certain subgroup of Gal(K^{ab}/K) which plays a prominent role in the statements of Theorem 3.5 and also appears in the André-Oort conjecture.

In conclusion, we mention that a fuller discussion of the circle of ideas and theorems that are the excuse for this paper may be found in the introduction of [3], where the main arithmetical applications are also spelled out. We would also like to thank Hee Oh for a number of useful conversations, and Nimish Shah for providing us with the proof of the crucial Lemma 2.30.

2 CM points on quaternion algebras

2.1 CM points, special points and reduction maps

We keep the following notations: F is a totally real number field, K is a totally imaginary quadratic extension of F and B is any quaternion algebra over F which is split by K. At this point we make no assumption on B at infinity. We fix once and for all, an F -embedding ι : K ↪ B and a prime P of F where B is split. We denote by ϖ_{P} ∈ F_{P}^{×} a local uniformizer at P. For any quaternion algebra B′ over F, we denote by Ram(B′), Ram_{f}(B′) and Ram_{∞}(B′) the set of places (resp. finite places, resp. archimedean places) of F where B′ ramifies.

2.1.1 Quaternion algebras.

Let S be a finite set of finite places of F such that

S1 ∀v ∈ S, B is unramified at v.

S2 |S| + |Ram_{f}(B)| + [F : Q] is even.

S3 ∀v ∈ S, v is inert or ramifies in K.

The first two assumptions imply that there exists a totally definite quaternion algebra B_{S} over F such that Ram_{f}(B_{S}) = Ram_{f}(B)∪S. The third assumption implies that there exists an F -embedding ι_{S} : K → B_{S}. We choose such a pair (B_{S}, ι_{S}).

2.1.2 Algebraic groups

We put

G = Res_{F/Q}(B^{×}), G_{S} = Res_{F/Q}(B_{S}^{×}),
T = Res_{F/Q}(K^{×}) and Z = Res_{F/Q}(F^{×}).

These are algebraic groups over Q. We identify Z with the center of G and G_{S}. We use ι and ι_{S} to embed T as a maximal subtorus in G and G_{S}. We denote by nr : G → Z and nr_{S} : G_{S} → Z the algebraic group homomorphisms induced by the reduced norms nr : B^{×} → F^{×} and nr_{S} : B_{S}^{×} → F^{×}.

2.1.3 Adelic groups

Let A_{f} denote the finite adeles of Q. We shall consider the following locally compact, totally discontinuous groups:

• G(A_{f}) = (B ⊗_{Q} A_{f})^{×}, G_{S}(A_{f}) = (B_{S} ⊗_{Q} A_{f})^{×}, T (A_{f}) = (K ⊗_{Q} A_{f})^{×} and Z(A_{f}) = (F ⊗_{Q} A_{f})^{×} with their usual topology.

• G(S) = ∏_{v∉S} B_{S,v}^{×} ×∏_{v∈S} F_{v}^{×} where ∏_{v∉S} B_{S,v}^{×} is the restricted product of the B_{S,v}^{×}’s over all finite places of F not in S, with respect to the compact subgroups R_{v}^{×} ⊂ B_{S,v}^{×}, where R_{v} is the closure in B_{S,v} of some fixed O_{F} -order R in B_{S}.

These groups are related by a commutative diagram of continuous morphisms:

\xymatrix{ & G(A_{f}) \ar[dr]_{ϕ_{S}} \ar[drr]^{nr} \\ T (A_{f}) \ar@{-->}[ur] \ar[dr] &  & G(S) \ar[r]^{_{nr{′ \atop S}}} & Z(A_{f}) \\  & G_{S}(A_{f}) \ar[ur]^{π_{S}} \ar[urr]_{_{nr_{S}}}}

In this diagram,

• T (A_{f}) → G(A_{f}) and T (A_{f}) → G_{S}(A_{f}) are the closed embeddings induced by ι and ι_{S}.

• nr : G(A_{f}) → Z(A_{f}) and nr_{S} : G_{S}(A_{f}) → Z(A_{f}) are the continuous, open and surjective group homomorphisms induced by nr and nr_{S}.

• nr′_{S} : G(S) → Z(A_{f}) is the continuous, open and surjective group homomorphism induced by nr_{S,v} : B_{S,v}^{×} → F_{v}^{×} for v ∉ S and by the identity on the remaining factors.

• π_{S} : G_{S}(A_{f}) = ∏_{v∉S} B_{S,v}^{×}×∏_{v∈S} B_{S,v}^{×} → G(S) = ∏_{v∉S} B_{S,v}^{×}×∏_{v∈S} F_{v}^{×} is the continuous, open and surjective group homomorphism induced by the identity on ∏_{v∉S} B_{S,v}^{×} and by the reduced norms nr_{S,v} : B_{S,v}^{×} → F_{v}^{×} on the remaining factors. It induces an isomorphism of topological groups between G_{S}(A_{f})/ker(π_{S}) and G(S). Since ker(π_{S}) ≃ ∏_{v∈S} B_{S,v}^{1} is compact, π_{S} is also a closed map.

The definition of

ϕ_{S} : G(A_{f}) = ∏_{v∉S} B_{v}^{×} × ∏_{v∈S} B_{v}^{×} → G(S) = ∏_{v∉S} B_{S,v}^{×} × ∏_{v∈S} F_{v}^{×}

is more involved. By construction, B_{v} and B_{S,v} are isomorphic for v ∉ S. We shall construct a collection of isomorphisms (ϕ_{v} : B_{v} → B_{S,v})_{v∉S} such that (1) ∀v ∉ S, ϕ_{v} ∘ ι = ι_{S} on K_{v}, and (2) the product of the ϕ_{v}’s yields a continuous isomorphism between ∏_{v∉S} B_{v}^{×} and ∏_{v∉S} B_{S,v}^{×}. Note that any two such families are conjugated by an element of ∏_{v∉S} K_{v}^{×}. Once such a family has been chosen, we may define the morphism ϕ_{S} by taking ∏_{v∉S} ϕ_{v} on ∏_{v∉S} B_{v}^{×} and nr_{v} : B_{v}^{×} → F_{v}^{×} on the remaining factors. It is then a continuous, open and surjective group homomorphism which makes the above diagram commute.

We first fix a maximal O_{F} -order R in B (respectively R_{S} in B_{S}). For all but finitely many v’s, (a) R_{v} ≃ M_{2}(O_{F_{v}}) ≃ R_{S,v} and (b) ι^{−1}(R_{v}) and ι_{S}^{−1}(R_{S,v}) are the maximal order of K_{v}. For such v’s we may choose the isomorphism ϕ_{v} : R_{v} ⟶^{≃} R_{S,v} in such a way that ϕ_{v} ∘ ι = ι_{S} on K_{v}. Indeed, starting with any isomorphism ϕ_{v}^{?} : R_{v} → R_{S,v}, we obtain two optimal embeddings ϕ_{v}^{?} ∘ ι and ι_{S} of O_{K_{v}} in R_{S,v}. By [20, Théorème 3.2 p. 44], any two such embeddings are conjugated by an element of R_{S,v}^{×}: the corresponding conjugate of ϕ_{v}^{?} has the required property. For those v’s that satisfy (a) and (b), we thus obtain an isomorphism ϕ_{v} : B_{v} → B_{S,v} such that ϕ_{v}(R_{v}) = R_{S,v} and ϕ_{v} ∘ ι = ι_{S} on K_{v}. For the remaining v’s not in S, we only require the second condition: ϕ_{v} ∘ι = ι_{S} on K_{v}. Such ϕ_{v}’s do exists by the Skolem-Noether theorem [20, Théorème 2.1 p. 6]. The resulting collection (ϕ_{v})_{v∉S} satisfies (1) and (2).

2.1.4 Main objects

Definition 2.1 We define the space CM of CM points, the space X (S) of special points at S and the space Z of connected components by

CM = \overline{T (Q)}\G(A_{f})
X (S) = \overline{G(S, Q)}\G(S)
Z = \overline{Z(Q)^{+}}\Z(A_{f})

where \overline{T (Q)} is the closure of T (Q) in T (A_{f}), \overline{G(S, Q)} is the closure of G(S, Q) = π_{S}(G_{S}(Q)) in G(S) and \overline{Z(Q)^{+}} is the closure of Z(Q)^{+} = F^{>0} in Z(A_{f}).

These are locally compact totally discontinuous Hausdorff spaces equipped with a right, continuous and transitive action of G(A_{f}) (with G(A_{f}) acting on X (S) through ϕ_{S} and on Z through nr). By [20, Théorème 1.4 pp. 61–64], X (S) and Z are compact spaces.

Definition 2.2 The reduction map RED_{S} at S, the connected component map c_{S} and their composite

\xymatrix{c : CM \ar[r]^{^{RED^{S}}} & X (S) \ar[r]^{c_{S}} & Z.}

are respectively induced by

\xymatrix{nr : G(A_{f}) \ar[r]^{ϕ_{S}} & G(S) \ar[r]^{^{nr{′ \atop S}}} & Z(A_{f}).}

Remark 2.3 Since ϕ_{S}(T (Q)) = π_{S}(T (Q)) ⊂ π_{S}(G_{S}(Q)) = G(S, Q), ϕ_{S} maps \overline{T (Q)} to \overline{G(S, Q)} and indeed induces a map CM → X (S). Similarly, c_{S} is welldefined since nr′_{S}(G(S, Q)) = nr_{S}(G_{S}(Q)) = Z(Q)^{+} (by the norm theorem [20, Théorème 4.1 p. 80]).

It follows from the relevant properties of nr, ϕ_{S} and nr′_{S} that c, RED_{S} and c_{S} are continuous, open and surjective G(A_{f})-equivariant maps. Since X (S) is compact, c_{S} is also a closed map.

Remark 2.4 The terminology CM points, special points and connected components is motivated by the example of Shimura curves: see the second part of this paper, especially section 3.2.

2.1.5 Galois actions

The profinite commutative group \overline{T (Q)}\T (A_{f}) acts continuously on CM, by multiplication on the left. This action is faithful and commutes with the right action of G(A_{f}). Using the inverse of Artin’s reciprocity map rec_{K} : \overline{T (Q)}\T (A_{f}) ⟶^{≃} Gal_{K}^{ab}, we obtain a continuous, G(A_{f})-equivariant and faithful action of Gal_{K}^{ab} on CM.^{1} Similarly, Artin’s reciprocity map rec_{F} : \overline{Z(Q)^{+}}\Z(A_{f}) ⟶^{≃} Gal_{F}^{ab} allows one to view Z as a principal homogeneous Gal_{F}^{ab}-space. From this point of view, c : CM → Z is a Gal_{K}^{ab}-equivariant map in the sense that for x ∈ CM and σ ∈ Gal_{K}^{ab},

c(σ · x) = σ |_{F^{ab}} ·c(x).

2.1.6 Further objects

For technical purposes, we will also need to consider the following objects:

• X_{S} = \overline{G_{S}(Q)}\G_{S}(A_{f}), where \overline{G_{S}(Q)} is the closure of G_{S}(Q) in G_{S}(A_{f}).

• q_{S} : X_{S} → X (S) is induced by π_{S} : G_{S}(A_{f}) → G(S).

The composite map c_{S} ∘ q_{S} : X_{S} → Z is induced by nr_{S} : G_{S}(A_{f}) → Z(A_{f}). By [20, Théorème 1.4 p. 61], X_{S} is compact. Note that q_{S} is indeed well defined since π_{S}(G_{S}(Q)) = G(S, Q). In fact, π_{S}(\overline{G_{S}(Q)}) = \overline{G(S, Q)} since π_{S} is a closed map: the fibers of q_{S} are the ker(π_{S})-orbits in X_{S}. In particular, q_{S} yields a G(S)-equivariant homeomorphism between X_{S}/ker(π_{S}) and X (S).

2.1.7 Measures

The group G^{1}(S) = ker(nr′_{S}) (resp. G_{S}^{1}(A_{f}) = ker(nr_{S})) acts on the fibers of c_{S} (resp. c_{S} ∘ q_{S}). In section 2.4.1 below, we shall prove the following proposition. Recall that a Borel probability measure on a topological space is a measure defined on its Borel subsets which assigns voume 1 to the total space.

1This action extends to a continuous, G(A_{f})-equivariant action of Gal(K^{ab}/F) as follows.

By the Skolem-Noether theorem, there exists an element b ∈ B^{×} such that x ↦ x^{b} = b^{−1}xb induces the non-trivial F -automorphism of K. In particular, b^{2} belongs to T (Q).

Multiplication on the left by b induces an involution ι on CM such that for all x ∈ CM and σ ∈ Gal_{K}^{ab}, ι(σx) = σ^{ι}ιx where σ ↦ σ^{ι} is the involution on Gal_{K}^{ab} which is induced by the nontrivial element of Gal(K/F).

Documenta Mathematica 10 (2005) 263–309

Proposition 2.5 The above actions are transitive and for each z ∈ Z,

(1) there exists a unique G_{S}^{1}(A_{f})-invariant Borel probability measure μ_{z} on (c_{S} ∘ q_{S})^{−1}(z), and (2) there exists a unique G^{1}(S)-invariant Borel probability measure μ_{z} on c_{S}^{−1}(z).

The uniqueness implies that these two measures are compatible, in the sense that the (proper) map q_{S} : (c_{S} ∘ q_{S})^{−1}(z) → c_{S}^{−1}(z) maps one to the other: this is why we use the same notation μ_{z} for both measures. Similarly, for any g ∈ G_{S}^{1}(A_{f}) (resp. G(S)), the measure μ_{z·g}(⋆g) equals μ_{z} on (c_{S} ∘ q_{S})^{−1}(z) (resp. on c_{S}^{−1}(z)).

2.1.8 Level structures

For a compact open subgroup H of G(A_{f}), we denote by CM_{H}, X_{H}(S) and Z_{H} the quotients of CM, X (S) and Z by the right action of H. We still denote by c, RED_{S} and c_{S} the induced maps on these quotient spaces:

\xymatrix{c : CM_{H} \ar[r]^{^{RED^{S}}} & X_{H}(S) \ar[r]^{c_{S}} & Z_{H}.}

Note that X_{H}(S) and Z_{H} are finite spaces, being discrete and compact. We have

Z_{H} = Z/nr(H) and X_{H}(S) = X (S)/H(S) ≃ X_{S}/H_{S}

where H(S) = ϕ_{S}(H) ⊂ G(S) and H_{S} = π_{S}^{−1}(H(S)) ⊂ G_{S}. The Galois group Gal_{K}^{ab} still acts continuously on the (now discrete) spaces CM_{H} and Z_{H}, and c is a Gal_{K}^{ab}-equivariant map.

2.2 Main theorems: the statements

2.2.1 Simultaneous reduction maps

Let 𝔖 be a nonempty finite collection of finite sets of non-archimedean places of F not containing P and satisfying conditions S1 to S3 of section 2.1.1. That is: each element of 𝔖 is a finite set S of finite places of F such that ∀v ∈ S, v is not equal to P, K_{v} is a field, and B_{v} is split, and |S| + |Ram_{f}(B)| + [F : Q] is even. For each S in 𝔖, we choose a totally definite quaternion algebra B_{S} over F with Ram_{f}(B_{S}) = Ram_{f}(B) ∪ S, an embedding ι_{S} : K → B_{S} and a collection of isomorphisms (ϕ_{v} : B_{v} → B_{S,v})_{v∉S} as in section 2.1.3. For each S in 𝔖, we thus obtain (among other things) an algebraic group G_{S} over Q, two locally compact and totally discontinuous adelic groups G_{S}(A_{f}) and G(S), a commutative diagram of continuous homomorphisms as in Section 2.1.3, a special set X (S) = \overline{G(S, Q)}\G(S), a reduction map RED_{S} : CM → X (S) and a connected component map c_{S} : X (S) → Z with the property that each fiber c_{S}^{−1}(z) of c_{S} has a unique Borel probability measure μ_{z} which is right invariant under G^{1}(S) = ker(nr′_{S}) (we refer the reader to section 2.1 for all notations).

Let ℜ be a nonempty finite subset of Gal_{K}^{ab} and consider the sequence

\xymatrix{CM \ar[r]^{Red} & X (𝔖, ℜ) \ar[r]^{C} & Z(𝔖, ℜ)}

where

• X (𝔖) = ∏_{S∈𝔖} X (S) and X (𝔖, ℜ) = ∏_{σ∈ℜ} X (𝔖) = ∏_{S,σ} X (S);

• Z(𝔖) = ∏_{S∈𝔖} Z and Z(𝔖, ℜ) = ∏_{σ∈ℜ} Z(𝔖) = ∏_{S,σ} Z;

• C : X (𝔖, ℜ) → Z(𝔖, ℜ) maps x = (x_{S,σ}) to C(x) = (c_{S}(x_{S,σ}));

• Red : CM → X (𝔖, ℜ) is the simultaneous reduction map which sends x to Red(x) = (RED_{S}(σ · x)).

We also put G(𝔖, ℜ) = ∏_{S,σ} G(S) and G^{1}(𝔖, ℜ) = ∏_{S,σ} G^{1}(S), so that G(𝔖, ℜ) acts on X (𝔖, ℜ) and Z(𝔖, ℜ), C is equivariant for these actions and its fibers are the G^{1}(𝔖, ℜ)-orbits in X (𝔖, ℜ). For z = (z_{S,σ}) in Z(𝔖, ℜ), the measure μ_{z} = ∏_{S,σ} μ_{z_{S,σ}} is a G^{1}(𝔖, ℜ)-invariant Borel probability measure on C^{−1}(z) = ∏_{S,σ} c_{S}^{−1}(z_{S,σ}). If g ∈ G(𝔖, ℜ) and z ∈ Z(𝔖, ℜ), μ_{z·g}(⋆g) = μ_{z} on C^{−1}(z). The Galois group Gal_{K}^{ab} acts diagonally on Z(𝔖, ℜ) = ∏_{S,σ} Z (through its quotient Gal_{F}^{ab}) and the composite map C ∘ Red : CM → Z(𝔖, ℜ) is Gal_{K}^{ab} - equivariant. For x ∈ CM, we shall frequently write x̄ = C ∘Red(x). Explicitly:

x̄ = C ∘ Red(x) = (σ · c(x))_{S,σ} ∈ Z(𝔖, ℜ) = ∏_{S,σ} Z.

2.2.2 Main theorem

In this section, we state the main results, without proofs. The proofs are long, and will be given later.

Definition 2.6 A P -isogeny class of CM points is a B_{P}^{×}-orbit in CM. If H ⊂ CM is a P -isogeny class and f is a C-valued function on CM, we say that f(x) goes to a ∈ C as x goes to infinity in H if the following holds: for any ϵ > 0, there exists a compact subset C(ϵ) of CM such that |f(x) − a| ≤ ϵ for all x ∈ H \ C(ϵ).

Remark 2.7 This definition can be somewhat clarified if we introduce the Alexandroff “one point” compactification ĈM = CM ∪ {∞} of the locally compact space CM. It is easy to see that the point ∞ ∈ ĈM lies in the closure of any P -isogeny class H (simply because P -isogeny classes are not relatively compact in CM). Our definition of “f(x) goes to a ∈ C as x goes to infinity in H” is then equivalent to the assertion that the limit of f |_{H} at ∞ exists and equals a.

Definition 2.8 An element σ ∈ Gal_{K}^{ab} is P -rational if σ = rec_{K}(λ) for some λ ∈ K̂^{×} whose P -component λ_{P} belongs to the subgroup K^{×} · F_{P}^{×} of K_{P}^{×}. We denote by Gal_{K}^{P −rat} ⊂ Gal_{K}^{ab} the subgroup of all P -rational elements.

In the above definition, rec_{K} : K̂^{×} ↠ Gal_{K}^{ab} is Artin’s reciprocity map. We normalize the latter by specifying that it sends local uniformizers to geometric Frobeniuses.

Theorem 2.9 Suppose that the finite subset ℜ of Gal_{K}^{ab} consists of elements which are pairwise distinct modulo Gal_{K}^{P −rat}. Let H ⊂ CM be a P -isogeny class and let G be a compact open subgroup of Gal_{K}^{ab} with Haar measure dg. Then for every continuous function f : X (𝔖, ℜ) → C,

x ↦ ∫_{G} f ∘ Red(g · x)dg − ∫_{G} dg ∫_{C^{−1}(g·x̄)} fdμ_{g·x̄}

goes to 0 as x goes to infinity in H.

2.2.3 Surjectivity

Let H be a compact open subgroup of G(A_{f}). Replacing CM, X and Z by CM_{H}, X_{H} and Z_{H} in the constructions of section 2.2.1, we obtain a sequence

\xymatrix{CM_{H} \ar[r]^{Red} & X_{H}(𝔖, ℜ) \ar[r]^{C} & Z_{H}(𝔖, ℜ)}

where

• X_{H}(𝔖, ℜ) = ∏_{S,σ} X_{H}(S) = X (𝔖, ℜ)/H(𝔖, ℜ) and

• Z_{H}(𝔖, ℜ) = ∏_{S,σ} Z_{H} = Z(𝔖, ℜ)/H(𝔖, ℜ) with

• H(𝔖, ℜ) = ∏_{S,σ} H(S), a compact open subgroup of G(𝔖, ℜ).

Applying the main theorem to the characteristic functions of the (finitely many) elements of X_{H}(𝔖, ℜ), we obtain the following surjectivity result. Let \overline{H} be the image of H in CM_{H}.

Corollary 2.10 For all but finitely many x in \overline{H},

Red(G · x) = C^{−1}(G · x̄) in X_{H}(𝔖, ℜ)

where x̄ = C ∘ Red(x) ∈ Z_{H}(𝔖, ℜ).

2.2.4 Equidistribution

When H = R̂^{×} for some Eichler order R in B, we can furthermore specify the asymptotic behavior (as x varies inside \overline{H}) of

Prob \{Red(G · x) = s\} =^{def} \frac{1}{|G · x|} |\{g · x; Red(g · x) = s, g ∈ G\}|

where s is a fixed point in X_{H}(𝔖, ℜ). To state our result, we first need to define a few constants. Let N = ∏_{Q} Q^{n_{Q}} be the level of R. By construction, the compact open subgroup H_{S} = π_{S}^{−1}ϕ_{S}(H) of G_{S}(A_{f}) equals R̂_{S}^{×} for some Eichler order R_{S} ⊂ B_{S} whose level N_{S} is the “prime-to-S” part of N: N_{S} = ∏_{Q∉S} Q^{n_{Q}}. For g ∈ G_{S}(A_{f}) and x = G_{S}(Q)gH_{S} in

X_{H}(S) ≃ X_{S}/H_{S} = \overline{G_{S}(Q)}\G_{S}(A_{f})/H_{S} = G_{S}(Q)\G_{S}(A_{f})/H_{S}

put O(g) = gR̂_{S}g^{−1} ∩ B_{S}. This is an O_{F} -order in B_{S} whose B_{S}^{×}-conjugacy class depends only upon x. The isomorphism class of the group O(g)^{×}/O_{F}^{×} also depends only upon x and since B_{S} is totally definite, this group is finite [20, p. 139]. The weight ω(x) of x is the order of this group: ω(x) = [O(g)^{×} : O_{F}^{×}]. The weight of an element s = (x_{S,σ}) in X_{H}(𝔖, ℜ) is then given by ω(s) = ∏_{S,σ} ω(x_{S,σ}). Finally, we put

Ω = \frac{1}{Ω(G)} · ( ∏_{S∈𝔖} \frac{Ω(F)}{Ω(B_{S}) · Ω(N_{S})})^{|ℜ|}

where

• Ω(F) = 2^{2[F:Q]−1}[O_{F}^{×} : O_{F}^{>0}]^{−1} |ζ_{F} (−1)|^{−1},

• Ω(B_{S}) = ∏_{Q∈Ram_{f} (B_{S})}(∥Q∥ − 1),

• Ω(N_{S}) = ∥N_{S}∥ · ∏_{Q|N_{S}}(∥Q∥^{−1} + 1) and

• Ω(G) is the order of the image of G in the Galois group Gal(F_{1}^{+}/F) of the narrow Hilbert class field F_{1}^{+} of F.

Here ∥·∥ denotes the absolute norm.

Corollary 2.11 For all ϵ > 0, there exists a finite set \overline{C}(ϵ) ⊂ \overline{H} such that for all s ∈ X_{H}(𝔖, ℜ) and x ∈ \overline{H} \ \overline{C}(ϵ),

|Prob \{Red(G · x) = s\} − \frac{Ω}{ω(s)}| ≤ ϵ

if s belongs to C^{−1}(G · x̄) and Prob {Red(G · x) = s} = 0 otherwise.

The remainder of this first part of the paper is devoted to the proofs of Proposition 2.5, Theorem 2.9, Corollary 2.10, Corollary 2.11.

2.3 Proof of the main theorems: first reductions

Notations

For a continuous function f : X (𝔖, ℜ) → C and x ∈ CM, we put

A(f, x) = ∫_{G} f ∘ Red(g · x)dg and B(f, x) = B(f, x̄) = ∫_{G} I(f, g · x̄)dg

where x̄ = C ∘ Red(x), with I(f, z) = ∫_{C^{−1}(z)} fdμ_{z} for z ∈ Z(𝔖, ℜ). Then the theorem says that for all ϵ > 0, there exists a compact subset C(ϵ) ⊂ CM such that,

∀x ∈ H, x ∉ C(ϵ) : |A(f, x) − B(f, x)| ≤ ϵ.

We claim that the functions x ↦ A(f, x) and x ↦ B(f, x) are well-defined. This is clear for A(f, x), as g ↦ f ∘ Red(g · x) is continuous on G. For B(f, x), we claim that g ↦ I(f, g · x̄) is also continuous. Since g ↦ g · x̄ is continuous, it is sufficient to show that z ↦ I(f, z) is continuous on Z(𝔖, ℜ). Note that for u ∈ G(𝔖, ℜ),

I(f, z ·u)−I(f, z) = ∫_{C^{−1}(z·u)} fdμ_{z·u} −∫_{C^{−1}(z)} fdμ_{z} = ∫_{C^{−1}(z)} (f(⋆u) − f) dμ_{z}.

Since f is continuous and X (𝔖, ℜ) is compact, f is uniformly continuous. It follows that I(f, z · u) − I(f, z) is small when u is small and z ↦ I(f, z) is indeed continuous. To prove the theorem, we may assume that f is locally constant. Indeed, there exists a locally constant function f′ : X (𝔖, ℜ) → C such that ∥f − f′∥ ≤ ϵ/3. If the theorem were known for f′, we could find a compact subset C(ϵ) ⊂ CM such that |A(f′, x) − B(f′, x)| ≤ ϵ/3 for all x ∈ H with x ∉ C(ϵ), thus obtaining

|A(f, x) − B(f, x)|
≤ |A(f, x) − A(f′, x)| + |A(f′, x) − B(f′, x)| + |B(f′, x) − B(f, x)|
≤ ϵ/3 + ϵ/3 + ϵ/3 = ϵ.

A decomposition of G · H · H

From now on, we shall thus assume that f is locally constant. Let H be a compact open subgroup of G(A_{f}) such that f factors through X (𝔖, ℜ)/H(𝔖, ℜ), where H(𝔖, ℜ) = ∏_{S,σ} H(S) with H(S) = ϕ_{S}(H). Then

• x ↦ A(x) = ∫_{G} f ∘ Red(g · x)dg factors through G\CM/H,

• z ↦ I(z) = ∫_{C^{−1}(z)} fdμ_{z} factors through Z(𝔖, ℜ)/H(𝔖, ℜ), hence

• x ↦ B(x) = B(x̄) = ∫_{G} I(g · x̄)dg factors through G\CM/H (where x̄ = C ∘ Red(x) ∈ Z(𝔖, ℜ) as usual). For a nonzero nilpotent element N ∈ B_{P}, the formula u(t) = 1 + tN defines a group isomorphism u : F_{P} → U = u(F_{P}) ⊂ B_{P}^{×}. We say that U = {u(t)} is a one parameter unipotent subgroup of B_{P}^{×}.

Proposition 2.12 There exists: (1) a finite set I, (2) for each i ∈ I, a point x_{i} ∈ H and a one parameter unipotent subgroup U_{i} = {u_{i}(t)} of B_{P}^{×}, and (3) a compact open subgroup κ of F_{P}^{×} such that

1. G · H · H = ⋃_{i∈I} ⋃_{n≥0} G · x_{i} · u_{i}(κ_{n}) · H, and

2. ∀i ∈ I and ∀n ≥ 0, G · x_{i} · u_{i}(κ_{n}) · H = G · x_{i}u_{i,n} · H,

where κ_{n} = ϖ_{P}^{−n}κ ⊂ F_{P}^{×} and u_{i,n} = u_{i}(ϖ_{P}^{−n}) ∈ u_{i}(κ_{n}).

Proof. Section 2.6.

Unipotent orbits: reduction of Theorem 2.9

This decomposition allows us to switch from Galois (=toric) orbits to unipotent orbits of CM points. To deal with the latter, we have the following proposition. We fix a CM point x ∈ H and a one parameter unipotent subgroup U = {u(t)} in B_{P}^{×}. We also choose a Haar measure λ = dt on F_{P}. Then Theorem 2.9 follows from Proposition 2.12 and

Proposition 2.13 Under the assumptions of Theorem 2.9, for almost all g ∈ Gal_{K}^{ab},

lim_{n→∞} \frac{1}{λ(κ_{n})} ∫_{κ_{n}} f ∘ Red(g · x · u(t))dt = ∫_{C^{−1}(g·x̄)} fdμ_{g·x̄}.

Proof. Section 2.5.

To deduce Theorem 2.9, we may argue as follows. By taking the integral over g ∈ G and using (a) Lebesgue’s dominated convergence theorem to exchange ∫_{G} and lim_{n}, and (b) Fubini’s theorem to exchange ∫_{G} and ∫_{κ_{n}}, we obtain:

lim_{n→∞} \frac{1}{λ(κ_{n})} ∫_{κ_{n}} A(x · u(t))dt = B(x).

This holds for all x and u. Then for x = x_{i} and u = u_{i}, we also know from part (2) of Proposition 2.12 that t ↦ A(x_{i} · u_{i}(t)) is constant on κ_{n}, equal to A(x_{i}u_{i,n}). In particular,

∀i ∈ I : lim_{n→∞} A(x_{i}u_{i,n}) = B(x_{i}).

Fix ϵ > 0 and choose N ≥ 0 such that

∀n > N, ∀i ∈ I : |A(x_{i}u_{i,n}) − B(x_{i})| ≤ ϵ.

Put C(ϵ) = ⋃_{i∈I} ⋃_{n=0}^{N} G · x_{i}u_{i}(κ_{n}) · H, a compact subset of CM. For any x ∈ H, there exists i ∈ I and n ≥ 0 such that x belongs to G · x_{i}u_{i,n}H, so that A(x) = A(x_{i}u_{i,n}) and B(x) = B(x_{i}u_{i,n}) = B(x_{i}). If x ∉ C(ϵ), n > N and |A(x) − B(x)| ≤ ϵ, QED.

Reduction of Corollaries 2.10 and 2.11

Let H be a compact open subgroup of G(A_{f}) and let f : X (𝔖, ℜ) → {0, 1} be the characteristic function of some s ∈ X_{H}(𝔖, ℜ), say s = s̃ · H(𝔖, ℜ) with s̃ ∈ X (𝔖, ℜ). The function z ↦ I(f, z) = ∫_{C^{−1}(z)} fdμ_{z} factors through Z_{H}(𝔖, ℜ) and equals 0 outside C(s) = C(s̃) · H(𝔖, ℜ). Let I(s) be its value on C(s). For x̃ ∈ CM, we easily obtain:

• A(f, x̃) = Prob {Red(G · x) = s} where x is the image of x̃ in CM_{H},

• B(f, x̃) = 0 if x̄ = C ∘ Red(x) does not belong to G · C(s), and

• B(f, x̃) = I(s)/Ω(G, H) otherwise, where Ω(G, H) is the common size of all G-orbits in Z(𝔖, ℜ)/H(𝔖, ℜ) ≃ ∏_{S,σ} Z/nr(H), which is also the size of the G-orbits in Z/nr(H).

If nr(H) is the maximal compact subgroup Ô_{F}^{×} of Z(A_{f}) (which occurs when H = R̂^{×} for some Eichler order R ⊂ B), Z/nr(H) ≃ Gal(F_{1}^{+}/F) and Ω(G, H) is the order of the image of G in Gal(F_{1}^{+}/F): Ω(G, H) = Ω(G). The main theorem asserts that for all ϵ > 0, there exists a compact subset C(ϵ) of CM such that for all x ∈ \overline{H} \ \overline{C}(ϵ) (where \overline{H} and \overline{C}(ϵ) are the images of H and C(ϵ) in CM_{H}),

|Prob {Red(G · x) = s} − I(s)/Ω(G, H)| ≤ ϵ

if s ∈ C^{−1}(G · x̄) and Prob \{Red(G · x) = s\} = 0 otherwise. Note that \overline{C}(ϵ) is finite, being compact and discrete. To prove the corollaries, it remains to (1) show that I(s) is nonzero and (2) compute I(s) exactly when H arises from an Eichler order in B. Write s = (x_{S,σ}) with x_{S,σ} = x̃_{S,σ}H_{S} in X (S)/H(S) ≃ X_{S}/H_{S} (S ∈ 𝔖, σ ∈ ℜ and x̃_{S,σ} ∈ X_{S}). Then I(s) = ∏_{S,σ} I(s)_{S,σ} with

I(s)_{S,σ} = ∫_{(c_{S}∘q_{S})^{−1}(z_{S,σ})} f_{S,σ}dμ_{z_{S,σ}}

where z_{S,σ} = c_{S} ∘ q_{S}(x̃_{S,σ}) ∈ Z and f_{S,σ} : X_{S} → {0, 1} is the characteristic function of x_{S,σ}.

Proposition 2.14 (1) For all S ∈ 𝔖 and σ ∈ ℜ, I(s)_{S,σ} > 0.

(2) If H = R̂^{×} for some Eichler order R ⊂ B of level N,

I(s)_{S,σ} = \frac{1}{ω(x_{S,σ})} · \frac{Ω(F)}{Ω(B_{S}) · Ω(N_{S})}

with ω(⋆), Ω(F), Ω(B_{S}) and Ω(N_{S}) as in section 2.2.4.

Proof. See section 2.4.2, especially Proposition 2.18.

In particular, I(s) > 0 and if H = R̂^{×} with R as above,

I(s) = \frac{1}{ω(s)} · ( ∏_{S∈𝔖} \frac{Ω(F)}{Ω(B_{S}) · Ω(N_{S})})^{|ℜ|} .

Thus we obtain Corollaries 2.10 and 2.11.

2.4 Further reductions

The arguments of the last section have reduced our task to proving Propositions 2.5, 2.12, 2.13, and 2.14. In this section, we make some further steps in this direction. Section 2.4.1 gives the proof of Proposition 2.5. Section 2.4.2 gives the proof of Proposition 2.14. Finally, Section 2.4.3 is a step towards Ratner’s theorem and the proof of Proposition 2.14. Throughout this section, S is a finite set of finite places of F subject to the condition S1 to S3 of section 2.1.1.

2.4.1 Existence of a measure and proof of Proposition 2.5

We shall repeatedly apply the following principle:

Lemma 2.15 [20, Lemme 1.2, p. 105] Suppose that L and C are topological groups with L locally compact and C compact. If Λ is a discrete and cocompact subgroup of L×C, the projection of Λ to L is a discrete and cocompact subgroup of L.

By [20, Théorème 1.4, p. 61], G_{S}^{1}(Q) diagonally embedded in G_{S}^{1}(A_{f}) × G_{S}^{1}(A_{∞}) is a discrete and cocompact subgroup. Since G_{S}^{1}(A_{∞}) is compact, G_{S}^{1}(Q) is also discrete and cocompact in G_{S}^{1}(A_{f}). Since the sequence

1 → ker(π_{S}) → G_{S}^{1}(A_{f}) → G^{1}(S) → 1

is split exact with ker(π_{S}) compact, G^{1}(S, Q) = π_{S}(G_{S}^{1}(Q)) is again a discrete and cocompact subgroup of G^{1}(S).

Lemma 2.16 The fibers of c_{S} ∘ q_{S} are the G_{S}^{1}(A_{f})-orbits in X_{S}. For g ∈ G_{S}(A_{f}) and x = \overline{G_{S}(Q)}g in X_{S}, the stabilizer of x in G_{S}^{1}(A_{f}) is a discrete and cocompact subgroup of G_{S}^{1}(A_{f}) given by Stab_{G_{S}^{1}(A_{f})}(x) = g^{−1}G_{S}^{1}(Q)g.

Proof. Fix x = \overline{G_{S}(Q)}g in X_{S} and put z = c_{S} ∘ q_{S}(x) = \overline{Z(Q)^{+}}nr_{S}(g) ∈ Z. The fiber of c_{S} ∘ q_{S} above z is the image of L = nr_{S}^{−1} (\overline{Z(Q)^{+}}nr_{S}(g)) in X_{S}

and the stabilizer of x in G_{S}^{1}(A_{f}) equals M = G_{S}^{1}(A_{f}) ∩ g^{−1}\overline{G_{S}(Q)}g. We have to show that L = \overline{G_{S}(Q)}gG_{S}^{1}(A_{f}) and M = g^{−1}G_{S}^{1}(Q)g. We break this up into a series of steps. Step 1: \overline{G_{S}(Q)}gG_{S}^{1}(A_{f}) is closed in G_{S}(A_{f}). This is equivalent to saying that the G_{S}^{1}(A_{f})-orbit of x is closed in X_{S}. Since M contains g^{−1}G_{S}^{1}(Q)g which is cocompact in G_{S}^{1}(A_{f}), M itself is cocompact in G_{S}^{1}(A_{f}). It follows that x · G_{S}^{1}(A_{f}) is compact, hence closed in X_{S}.

Step 2: L = \overline{G_{S}(Q)}gG_{S}^{1}(A_{f}). Since nr_{S} : G_{S}(A_{f}) → Z(A_{F}) is open, L is the closure of nr_{S}^{−1} (Z(Q)^{+}nr_{S}(g)) in G_{S}(A_{f}). The norm theorem [20, Théorème 4.1 p. 80] implies that nr_{S}^{−1} (Z(Q)^{+}nr_{S}(g)) = G_{S}(Q)gG_{S}^{1}(A_{f}) and then L = \overline{G_{S}(Q)}gG_{S}^{1}(A_{f}) by (1).

Step 3: \overline{G_{S}(Q)} = \overline{Z(Q)}G_{S}(Q) . This is easy. See for instance the proof of Corollary 3.10. Step 4: M = g^{−1}G_{S}^{1}(Q)g. Suppose that γ belongs to M = G_{S}^{1}(A_{f}) ∩ g^{−1}\overline{G_{S}(Q)}g. By (3), γ = g^{−1}λg_{Q}g for some λ ∈ \overline{Z(Q)} and g_{Q} ∈ G_{S}(Q) with nr_{S}(γ) = 1. Then α = λ^{2} = nr(g_{Q}^{−1}) belongs to \overline{Z(Q)}^{2} ∩ Z(Q)^{+} ⊂ Z(A_{f})^{2} ∩ Z(Q)^{+}. Since α belongs to Z(Q)^{+} ⊂ F^{×}, we may form the abelian extension F (\sqrt{α}) of F. Since α also belongs to Z(A_{f})^{2}, this extension splits everywhere and is therefore trivial: α = λ_{0}^{2} for some λ_{0} ∈ F^{×}. Then λ/λ_{0} is an element of order 2 in \overline{Z(Q)} ∩ Ô_{F}^{×}. Since \overline{Z(Q)} ∩ Ô_{F}^{×} = \overline{O_{F}^{×}} is isomorphic to the profinite completion of O_{F}^{×} (a finite type Z-module), λ/λ_{0} actually belongs to {±1}, the torsion subgroup of O_{F}^{×}. We have shown that λ belongs to Z(Q), hence γ = g^{−1}λg_{Q}g belongs to g^{−1}G_{S}(Q)g ∩ G_{S}^{1}(A_{f}) = g^{−1}G_{S}^{1}(Q)g.

Since (1) q_{S} identifies X_{S}/ker(π_{S}) with X (S) and (2) G_{S}^{1}(A_{f}) ≃ G^{1}(S) × ker(π_{S}) with ker(π_{S}) compact, we obtain:

Lemma 2.17 The fibers of c_{S} are the G^{1}(S)-orbits in X (S). For g ∈ G(S) and x = \overline{G(S, Q)}g in X (S), the stabilizer of x in G^{1}(S) is a discrete and cocompact subgroup of G^{1}(S) given by Stab_{G^{1}(S)}(x) = g^{−1}G^{1}(S, Q)g.

For z ∈ Z and x ∈ (c_{S} ∘ q_{S})^{−1}(z), the map g ↦ x · g induces a G_{S}^{1}(A_{f})- equivariant homeomorphism between Stab(x)\G_{S}^{1}(A_{f}) and (c_{S}∘q_{S})^{−1}(z). Similarly, any x ∈ c_{S}^{−1}(z) defines a G^{1}(S)-equivariant homeomorphism between Stab(x)\G^{1}(S) and c_{S}^{−1}(z). Proposition 2.5 easily follows.

2.4.2 A computation.

Any Haar measure μ^{1} on G_{S}^{1}(A_{f}) induces a collection of G_{S}^{1}(A_{f})-invariant Borel measures μ_{z}^{1} on the fibers (c_{S} ∘ q_{S})^{−1}(z) of c_{S} ∘ q_{S} : X_{S} → Z. These measures are characterized by the fact that for any compact open subgroup H_{S}^{1} of G_{S}^{1}(A_{f}) and any x ∈ (c_{S} ∘ q_{S})^{−1}(z),

μ_{z}^{1}(x · H_{S}^{1}) = \frac{μ^{1}(H_{S}^{1})}{|Stab_{H_{S}^{1}}(x)|}

(Stab_{H_{S}^{1}}(x) = Stab_{G_{S}^{1}(A_{f})}(x) ∩ H_{S}^{1} is indeed finite since Stab_{G_{S}^{1}(A_{f})}(x) is discrete while H_{S}^{1} is compact). One easily checks that μ_{z·g}^{1}(⋆g) equals μ_{z}^{1} on (c_{S} ∘ q_{S})^{−1}(z) for any g ∈ G_{S}(A_{f}). It follows that these measures assign the same volume λ to each fiber of c_{S} ∘ q_{S}, and μ_{z}^{1} = λμ_{z} on (c_{S} ∘ q_{S})^{−1}(z). We shall now simultaneously determine λ (or find out which normalization of μ^{1} yields λ = 1) and compute a formula for

φ_{z}(x) = μ_{z} (xH_{S} ∩ (c_{S} ∘ q_{S})^{−1}(z)) (x ∈ X_{S}, z ∈ Z)

where H_{S} is a compact open subgroup of G_{S}(A_{f}). The map z ↦ φ_{z}(x) factors through Z/nr_{S}(H_{S}) and equals 0 outside c_{S} ∘ q_{S}(xH_{S}) = c_{S} ∘ q_{S}(x) · nr_{S}(H_{S}). Let z_{1}, ⋯, z_{n} be a set of representatives for Z/nr_{S}(H_{S}) and for 1 ≤ i ≤ n, let x_{i,1}, ⋯, x_{i,n_{i}} be a set of representatives in (c_{S} ∘ q_{S})^{−1}(z_{i}) of

(c_{S} ∘ q_{S})^{−1}(z_{i}nr_{S}(H_{S}))/H_{S} = (c_{S} ∘ q_{S})^{−1}(z_{i}) · H_{S}/H_{S}.

The x_{i,j}’s then form a set of representatives for X_{S}/H_{S} and

∑i,jφ_{z_{i}}(x_{i,j}) = _{∑i}μ_{z_{i}} (∪_{j=1}^{n_{i}}x_{i,j}H_{S} ∩ (c_{S} ∘ q_{S})^{−1}(z_{i})) = _{∑i}1 = n (1)

since (x_{i,j}H_{S})_{j=1}^{n_{i}} covers (c_{S} ∘ q_{S})^{−1}(z_{i}). To compute φ_{z}(x), we may assume that z = c_{S} ∘ q_{S}(x). Choose g ∈ G_{S}(A_{f}) such that x = \overline{G_{S}(Q)}g and put H_{S}^{1} = H_{S} ∩ G_{S}^{1}(A_{f}). By Lemma 2.16, the map b ↦ x · b yields a bijection

g^{−1}G_{S}^{1}(Q)g\(g^{−1}\overline{G_{S}(Q)}g · H_{S}) ∩ G_{S}^{1}(A_{f})/H_{S}^{1} ⟶^{≃} xH_{S} ∩ (c_{S} ∘ q_{S})^{−1}(z)/H_{S}^{1}.

(2)

Note that g^{−1}\overline{G_{S}(Q)}g · H_{S} = g^{−1}G_{S}(Q)g · H_{S}. Let (a_{k}b_{k})_{k=1}^{m} be a set of representatives for the left hand side of (2), with a_{k} in g^{−1}G_{S}(Q)g, b_{k} in H_{S} and nr_{S}(a_{k}b_{k}) = 1. Since x · a_{k} = x and b_{k} normalizes H_{S}^{1},

φ_{z}(x) = ∑_{k=1}^{m} μ_{z} (x · a_{k}b_{k}H_{S}^{1}) = \frac{m}{|H_{S}^{1} ∩ g^{−1}G_{S}^{1}(Q)g|} × \frac{μ^{1}(H_{S}^{1})}{λ} · .

On the other hand, the map a_{k}b_{k} ↦ nr_{S}(a_{k}) = nr_{S}(b_{k})^{−1} yields a bijection between the left hand side of (2) and

nr_{S} (H_{S} ∩ g^{−1}G_{S}(Q)g) \nr_{S} (H_{S}) ∩ nr_{S}(G_{S}(Q)).

Since nr_{S}(G_{S}(Q)) = Z(Q)^{+}, we obtain

φ_{z}(x) = \frac{|q(g, H_{S})|}{|k(g, H_{S})|} × \frac{μ^{1}(H_{S}^{1})}{λ}  (3)

where k(g, H_{S}) and q(g, H_{S}) are respectively the kernel and cokernel of

gH_{S}g^{−1} ∩ G_{S}(Q) ⟶^{nr_{S}} nr_{S}(H_{S}) ∩ Z(Q)^{+}.

When H_{S} = R̂_{S}^{×} for some Eichler order R_{S} in B_{S}, the following simplifications occur:

• nr_{S}(H_{S}) = Ô_{F}^{×}, so that n = |Z/nr_{S}(H_{S})| = |F̂^{×}/F^{>0}Ô_{F}^{×}| = h_{F}^{+} is the

order of the narrow class group of F. Note that h_{F}^{+} = h_{F} · [O_{F}^{>0} : (O_{F}^{×})^{2}], where h_{F} is the class number of F and (O_{F}^{×})^{2} = {x^{2} | x ∈ O_{F}^{×}}.

• The map g ↦ L(g) = g · R̂_{S} ∩ B_{S} yields a bijection between X_{S}/H_{S} = G_{S}(Q)\G_{S}(A_{f})/H_{S} and the set of isomorphism classes of nonzero right R-ideals in B_{S}. Moreover, the left order O(g) of L(g) equals gR̂_{S}g^{−1}∩B_{S}, so that O(g)^{×} = gH_{S}g^{−1} ∩ G_{S}(Q).

• The following commutative diagram with exact rows

1 → O_{F}^{×} → O(g)^{×} → O(g)^{×}/O_{F}^{×} → 1
2 ↓ nr_{S} ↓ ↓
1 → O_{F}^{>0} → O_{F}^{>0} →  1

yields an exact sequence

1 → {±1} → k(g, H_{S}) → O(g)^{×}/O_{F}^{×} → O_{F}^{>0}/(O_{F}^{×})^{2} → q(g, H_{S}) → 1.

In particular,

\frac{|q(g, H_{S})|}{|k(g, H_{S})|} = \frac{[O_{F}^{>0} : (O_{F}^{×})^{2}]}{2 · [O(g)^{×} : O_{F}^{×}]}.

Combining this, (1), (3) and [20, Corollaire 2.3 p. 142], we obtain:

\frac{μ^{1}(H_{S}^{1})}{λ} = \frac{2^{[F:Q]} |ζ_{F} (−1)|^{−1}}{∥N_{S}∥ ·_{∏Q∈Ram_{f} (B_{S})}(∥Q∥ − 1) ·_{∏Q|N_{S}}(∥Q∥^{−1} + 1)}

where N_{S} is the level of R_{S}. This tells us how to normalize μ^{1} in order to have λ = 1. We have proven:

Proposition 2.18 Let H_{S} be a compact open subgroup of G_{S}(A_{f}). For x ∈ X_{S} and z ∈ Z,

μ_{z} (xH_{S} ∩ (c_{S} ∘ q_{S})^{−1}(z)) = \{ \begin{matrix}\frac{|q(g,H_{S})|}{|k(g,H_{S})|}μ^{1}(H_{S}^{1}) & if z ∈ c_{S} ∘ q_{S}(xH_{S}) \\ 0 & otherwise \end{matrix}

where x = \overline{G_{S}(Q)}g, k(g, H_{S}) and q(g, H_{S}) are as above, H_{S}^{1} = H_{S} ∩ G_{S}^{1}(A_{f}) and μ^{1} is the unique Haar measure on G_{S}^{1}(A_{f}) such that

μ^{1}(H_{S}^{1}) = \frac{2^{[F:Q]} |ζ_{F} (−1)|^{−1}}{_{∏Q∈Ram_{f} (B_{S})}(∥Q∥ − 1)}

Documenta Mathematica 10 (2005) 263–309

when H_{S} = R̂_{S}^{×} for some maximal order R_{S} ⊂ B_{S}. Moreover, if H_{S} = R̂_{S}^{×} for some Eichler order R_{S} ⊂ B_{S} of level N_{S},

\frac{|q(g, H_{S})|}{|k(g, H_{S})|}μ^{1}(H_{S}^{1}) = \frac{1}{[O(g)^{×} : O_{F}^{×}]} × \frac{Ω(F)}{Ω(B_{S}) · Ω(N_{S})}

with Ω(F), Ω(B_{S}) and Ω(N_{S}) as in section 2.2.4.

2.4.3 P -adic uniformization.

Suppose moreover that P does not belong to S (this is the case for all S ∈ 𝔖). Since B splits at P, so does B_{S}. Let H be a compact open subgroup of G_{S}^{1}(A_{f})^{P} = {x ∈ G_{S}^{1}(A_{f}) | x_{P} = 1}. For z ∈ Z, the right action of G_{S}^{1}(A_{f}) on c_{S} ∘ q_{S}^{−1}(z) induces a right action of B_{S,P}^{1} = {b ∈ B_{S,P}^{×} | nr_{S}(b) = 1} on c_{S} ∘ q_{S}^{−1}(z)/H.

Lemma 2.19 This action is transitive and the stabilizer of x ∈ c_{S} ∘ q_{S}^{−1}(z)/H is a discrete and cocompact subgroup Γ_{S}(x) of B_{S,P}^{1} . For x = \overline{G_{S}(Q)}gH (with g ∈ G_{S}(A_{f})), Γ_{S}(x) = g_{P}^{−1}Γ_{S}g_{P} where g_{P} ∈ B_{S,P}^{×} is the P -component of g and Γ_{S} = Γ_{S}(gHg^{−1}) is the projection to B_{S,P}^{1} of G_{S}^{1}(Q) ∩ {gHg^{−1} · B_{S,P}^{1} } ⊂ G_{S}^{1}(A_{f}). The commensurator of Γ_{S} in B_{S,P}^{×} equals F_{P}^{×}B_{S}^{×}.

Proof. The stabilizer of x̃ = \overline{G_{S}(Q)}g in G_{S}^{1}(A_{f}) equals Stab(x̃) = g^{−1}G_{S}^{1}(Q)g (by Lemma 2.16). The strong approximation theorem [20, Théorème 4.3, p. 81] implies that Stab(x̃)B_{S,P}^{1} H = G_{S}^{1}(A_{f}). Using Lemma 2.16 again, we obtain

(c_{S} ∘ q_{S})^{−1}(z) = x̃ · G_{S}^{1}(A_{f}) = x̃ · B_{S,P}^{1} H = x̃ · HB_{S,P}^{1} = x · B_{S,P}^{1} .

In particular, B_{S,P}^{1} acts transitively on (c_{S} ∘q_{S})^{−1}(z)/H. An easy computation shows that Γ_{S}(x) = g_{P}^{−1}Γ_{S}g_{P} with Γ_{S} as above. Put U = gHg^{−1} ·B_{S,P}^{1} . The continuous map U ∩G_{S}^{1}(Q)\U ↪ G_{S}^{1}(Q)\G_{S}^{1}(A_{f}) is (1) open since U is open in G_{S}^{1}(A_{f}) and (2) surjective by the strong approximation theorem. In particular, U∩G_{S}^{1}(Q) is a discrete and cocompact subgroup of U. Since U = gHg^{−1} × B_{S,P}^{1} (with gHg^{−1} compact), the projection Γ_{S} of U ∩ G_{S}^{1}(Q) to B_{S,P}^{1} is indeed discrete and cocompact in B_{S,P}^{1} . Finally, since the compact open subgroups of G_{S}^{1}(A_{f})^{P} are all commensurable, neither the commensurability class of Γ_{S} nor its commensurator in B_{S,P}^{×} depends upon g or H. When g = 1 and H = R̂^{×}∩G_{S}^{1}(A_{f})^{P} for some Eichler order R ⊂ B_{S}, Γ_{S} is the image in B_{S,P}^{1} of the subgroup {x ∈ R[1/P]^{×} | nr_{S}(x) = 1} of B_{S}^{×}. The commensurator of Γ_{S} in B_{S,P}^{×} then equals F_{P}^{×}B_{S}^{×} by [20, Corollaire 1.5, p. 106].

Similarly, let H be a compact open subgroup of G^{1}(S)^{P} = {x ∈ G^{1}(S) | x_{P} = 1}. Then B_{S,P}^{1} acts on c_{S}^{−1}(z)/H and we have the following lemma.

Lemma 2.20 This action is transitive and the stabilizer of x ∈ c_{S}^{−1}(z)/H is a discrete and cocompact subgroup Γ_{S}(x) of B_{S,P}^{1} . For x = \overline{G(S, Q)}gH with g in G(S), Γ_{S}(x) = g_{P}^{−1}Γ_{S}g_{P} where Γ_{S} = Γ_{S}(gHg^{−1}) is the projection to B_{S,P}^{1} of G^{1}(S, Q) ∩ {gHg^{−1} · B_{S,P}^{1} } ⊂ G^{1}(S). The commensurator of Γ_{S} in B_{S,P}^{×} equals F_{P}^{×}B_{S}^{×}.

Proof. The proof is similar, using Lemma 2.17 instead of 2.16. Alternatively, we may deduce the results for c_{S} from those for c_{S} ∘ q_{S} as follows. Put H′ = π_{S}^{−1}(H). Then H′ is a compact open subgroup of G_{S}^{1}(A_{f}) and q_{S} induces a B_{S,P}^{1} -equivariant homeomorphism between (c_{S} ∘ q_{S})^{−1}(z)/H′ and c_{S}^{−1}(z)/H.

In particular, the map b ↦ x · b induces a B_{S,P}^{1} -equivariant homeomorphism

Γ_{S}(x)\B_{S,P}^{1} ⟶^{≃} c_{S}^{−1}(z)/H.

Since Γ_{S}(x) is discrete and cocompact in B_{S,P}^{1} ≃ SL_{2}(F_{P}), there exists a unique B_{S,P}^{1} -invariant Borel probability measure on the left hand side. It corresponds on the right hand side to the image of the measure μ_{z} through the (proper) map c_{S}^{−1}(z) → c_{S}^{−1}(z)/H: the latter is indeed yet another B_{S,P}^{1} -invariant Borel probability measure.

2.5 Reduction of Proposition 2.13 to Ratner’s theorem

Let us fix a point x ∈ CM, a one parameter unipotent subgroup U = {u(t)} in B_{P}^{×}, a compact open subgroup κ in F_{P}^{×} and a Haar measure λ = dt on F_{P}. For n ≥ 0, we put κ_{n} = ϖ_{P}^{−n}κ so that λ(κ_{n}) → ∞ as n → ∞. For γ ∈ Gal_{K}^{ab} and t ∈ F_{P},

C ∘ Red(γ · x · u(t)) = γ · x̄ with x̄ = C ∘ Red(x) ∈ Z(𝔖, ℜ)

where 𝔖, ℜ, C and Red are as in section 2.2.1. Our aim is to prove the following two propositions, which together obviously imply Proposition 2.13.

Proposition 2.21 Suppose that Red(x · U) is dense in C^{−1}(x̄). Then for any continuous function f : C^{−1}(x̄) → C,

lim_{n→∞} \frac{1}{λ(κ_{n})} ∫_{κ_{n}} f ∘ Red(x · u(t))dt = ∫_{C^{−1}(x̄)} fdμx̄.

Proposition 2.22 Under the assumptions of Theorem 2.9, Red(γ · x · U) is dense in C^{−1}(γ · x̄) for almost all γ ∈ Gal_{K}^{ab}.

2.5.1 Reduction of Proposition 2.21

We may assume that f is locally constant (by the same argument that we already used in section 2.3). In this case, there exists a compact open subgroup H of G^{1}(A_{f}) such that f factors through C^{−1}(x̄)/H(𝔖, ℜ). For our purposes, it will be sufficient to assume that f is right H(𝔖, ℜ)-invariant when H is a compact open subgroup of G^{1}(A_{f})^{P} = {x ∈ G^{1}(A_{f}) | x_{P} = 1}. Here, H(𝔖, ℜ) = ∏_{S,σ} H(S) with H(S) = ϕ_{S}(H) as usual. For such an H, the right action of G^{1}(𝔖, ℜ) on C^{−1}(x̄) induces a right action of ∏_{S,σ} B_{S,P}^{1} on C^{−1}(x̄)/H(𝔖, ℜ) which together with the isomorphism

∏_{S,σ} ϕ_{S,P} : (B_{P}^{1} )^{𝔖×ℜ} ⟶^{≃} _{∏S,σ}B_{S,P}^{1}

yields a right action of (B_{P}^{1} )^{𝔖×ℜ} on C^{−1}(x̄)/H(𝔖, ℜ). By Lemma 2.20, the map (b_{S,σ}) ↦ Red(x) · (ϕ_{S,P} (b_{S,σ})) yields a (B_{P}^{1} )^{𝔖×ℜ}- equivariant homeomorphism

Γ(x, H)\(B_{P}^{1} )^{𝔖×ℜ} ⟶^{≃} C^{−1}(x̄)/H(𝔖, ℜ)  (4)

where Γ(x, H) is the stabilizer of Red(x) · H(𝔖, ℜ) in (B_{P}^{1} )^{𝔖×ℜ}. Note that Γ(x, H) equals ∏_{S,σ} Γ_{S,σ}(x, H) where for each S ∈ 𝔖 and σ ∈ ℜ,

Γ_{S,σ}(x, H) = ϕ_{S,P}^{−1} {Stab_{B_{S,P}^{1}} (RED_{S}(σ · x) · H(S))}

is a discrete and cocompact subgroup of B_{P}^{1} ≃ SL_{2}(F_{P}). Under this equivariant homeomorphism,

• the image of t ↦ Red(x · u(t)) in C^{−1}(x̄)/H(𝔖, ℜ) corresponds to the image of t ↦ Δ ∘ u(t) in Γ(x, H)\(B_{P}^{1} )^{𝔖×ℜ}, where Δ : B_{P}^{1} → (B_{P}^{1} )^{𝔖×ℜ} is the diagonal map;

• the image of μx̄ on C^{−1}(x̄)/H(𝔖, ℜ) corresponds to the (unique) (B_{P}^{1} )^{𝔖×ℜ}-invariant Borel probability measure on Γ(x, H)\(B_{P}^{1} )^{𝔖×ℜ}.

Writing μ_{Γ(x,H)} for the latter measure, the above discussion shows that Proposition 2.21 is a consequence of the following purely P -adic statement, itself a special case of a theorem of Ratner, Margulis, and Tomanov.

Proposition 2.23 Suppose that Γ(x, H) · Δ(U) is dense in (B_{P}^{1} )^{𝔖×ℜ}. Then for any continuous function f : Γ(x, H)\(B_{P}^{1} )^{𝔖×ℜ} → C ,

lim_{n→∞} \frac{1}{λ(κ_{n})} ∫_{κ_{n}} f(Δ ∘ u(t))dt = ∫_{Γ(x,H)\(B_{P}^{1} )^{𝔖×ℜ}} fdμ_{Γ(x,H)}.

Proof. See section 2.7.

2.5.2 Reduction of Proposition 2.22

We keep the above notations and choose:

• an element g ∈ G(A_{f}) such that x = \overline{T (Q)}g in CM;

• for each σ ∈ ℜ, an element λ_{σ} ∈ T (A_{f}) such that σ = rec_{K}(λ_{σ}) ∈ Gal_{K}^{ab}.

For S ∈ 𝔖 and σ ∈ ℜ, we thus obtain (using Lemma 2.20):

• RED_{S}(σ · x) = \overline{G(S, Q)}ϕ_{S}(λ_{σ}g) and

• Γ_{S,σ}(x, H) = g_{P}^{−1}λ_{σ,P}^{−1} Γ_{S,σ}^{0}(x, H)λ_{σ,P} g_{P}

where λ_{σ,P} and g_{P} are the P -components of λ_{σ} and g while Γ_{S,σ}^{0}(x, H) is the inverse image (through ϕ_{S,P} : B_{P}^{1} → B_{S,P}^{1} ) of the projection to B_{S,P}^{1} of

G^{1}(S, Q) ∩ {ϕ_{S} (λ_{σ}gHg^{−1}λ_{σ}^{−1}) · B_{S,P}^{1} } ⊂ G^{1}(S).

For a subgroup Γ of B_{P}^{1} , we denote by [Γ] the commensurability class of Γ in B_{P}^{1} , namely the set of all subgroups of B_{P}^{1} which are commensurable with Γ. The group B_{P}^{×} acts on the right on the set of all commensurability classes (by [Γ] · b = [b^{−1}Γb]) and the stabilizer of [Γ] for this action is nothing but the commensurator of Γ in B_{P}^{×}. Since the compact open subgroups of G^{1}(A_{f})^{P} are all commensurable, the commensurability class [Γ_{S}^{0}] of Γ_{S,σ}^{0}(x, H) does not depend upon H, x or σ (but it does depend on S). Similarly, the commensurability class [Γ_{S,σ}(x)] of Γ_{S,σ}(x, H) does not depend upon H and [Γ_{S,σ}(x)] = [Γ_{S}^{0}] · λ_{σ,P} g_{P}. Changing x to γ · x (γ ∈ Gal_{K}^{ab}) changes g to λg, where λ is an element of T (A_{f}) such that γ = rec_{K}(λ). In particular,

[Γ_{S,σ}(γ · x)] = [Γ_{S}^{0}] · λ_{σ,P} λ_{P} g_{P} = [Γ_{S}^{0}] · λ_{P} λ_{σ,P} g_{P}

where λ_{P} is the P -component of λ. On the other hand, the stabilizer of [Γ_{S}^{0}] in B_{P}^{×} equals F_{P}^{×}ϕ_{S}^{−1}(B_{S}^{×}) by Lemma 2.20. Since K_{P}^{×} ∩ F_{P}^{×}ϕ_{S}^{−1}(B_{S}^{×}) = F_{P}^{×}K^{×},

[Γ_{S,σ}(γ · x)] = [Γ_{S,σ}(γ′ · x)] ⟺ γ ≡ γ′ mod Gal_{K}^{P −rat}.

With these notations, we have

Proposition 2.24 Under the assumptions of Theorem 2.9, for (S, σ) and (S′, σ′) in 𝔖 × ℜ with (S, σ) ≠ (S′, σ′), the set

𝔅((S, σ), (S′, σ′)) = {γ ∈ Gal_{K}^{ab}; [Γ_{S,σ}(γ · x)] · U = [Γ_{S′,σ′}(γ · x)] · U}

is the disjoint union of countably many cosets of Gal_{K}^{P −rat} in Gal_{K}^{ab}.

Proof. Fix (S, σ) ≠ (S′, σ′) in 𝔖 × ℜ. We have to show that (under the assumptions of Theorem 2.9) the image of

𝔅′ = {λ_{P} ∈ K_{P}^{×}; [Γ_{S}^{0}] · λ_{σ,P} λ_{P} g_{P} · U = [Γ_{S′}^{0}] · λ_{σ′,P} λ_{P} g_{P} · U}

in K_{P}^{×}/F_{P}^{×}K^{×} is at most countable. For that purpose we may as well consider the image of 𝔅′ in K_{P}^{×}/F_{P}^{×}.

We first consider the case where S ≠ S′. In this case, we claim that 𝔅′ is empty. In fact: For S ≠ S′, we claim that

[Γ_{S}^{0}] · B_{P}^{×} ≠ [Γ_{S′}^{0}] · B_{P}^{×}.

To see this, suppose that [Γ_{S′}^{0}] = [Γ_{S}^{0}] · b for some b ∈ B_{P}^{×}. Then b^{−1}F_{P}^{×}ϕ_{S,P}^{−1} (B_{S}^{×})b = F_{P}^{×}ϕ_{S′,P}^{−1} (B_{S′}^{×}), so that F_{P}^{×}B_{S′}^{×} = F_{P}^{×}ϕ(B_{S}^{×}) in B_{S′,P}^{×}, where ϕ : B_{S,P} → B_{S′,P} is the isomorphism of F_{P} -algebras which sends α to ϕ_{S′,P} (b^{−1}ϕ_{S,P}^{−1} (α)b). Since F_{P} B_{S} = F_{P}^{×}B_{S}^{×} ∪ {0} and similarly for B_{S′}, F_{P} B_{S′} = F_{P} ϕ(B_{S}). We contend that ϕ maps B_{S} to B_{S′}. Indeed, suppose that α belongs to B_{S} and choose η ∈ F such that Tr(α + η) = Tr(α) + 2η ≠ 0. Since α + η belongs to B_{S}, there exists μ ∈ F_{P} and β ∈ B_{S′} such that ϕ(α + η) = μβ. Taking traces on both sides we obtain μ = \frac{Tr(α+η)}{Tr(β)} ∈ F, so that ϕ(α + η) = ϕ(α) + η belongs to B_{S′}, and so does ϕ(α). By symmetry, ϕ^{−1}(B_{S′}) ⊂ B_{S} and ϕ yields an isomorphism of F -algebras between B_{S} and B_{S′}. This is a contradiction, since B_{S} and B_{S′} are nonisomorphic quaternion algebras over F when S ≠ S′. This proves the proposition when S ≠ S′. Next we consider the case where S = S′ but σ ≠ σ′. In this case, an element λ_{P} in K_{P}^{×} belongs to 𝔅′ if and only if there exists t ∈ F_{P} such that

b(t) = λ_{P} w(t)λ_{P}^{−1}(λ_{σ,P} λ_{σ′,P}^{−1} ) ∈ F_{P}^{×}B_{S}^{×},  (5)

for w(t) = λ_{σ,P} ϕ_{S}(g_{P} u(t)g_{P}^{−1})λ_{σ,P}^{−1} . We contend that this condition can only be satisfied for countably many λ_{P} modulo F_{P}^{×}. Suppose first that K_{P}^{×} normalizes the unipotent subgroup W of elements of the form w(t), for all t ∈ F_{P}. In this situation, K_{P}^{×} is a split torus, and we claim that (5) never holds for any λ_{P} and t. To see this, observe that if k ∈ K_{p} is arbitrary, then, in view of the representation of elements of K_{P}^{×} and W by triangular matrices, the commutator [k, b(t)] is unipotent. (This also follows from standard facts about Borel subgroups.) Since b(t) ∈ F_{P}^{×}B_{S}^{×}, we can apply this to elements of K_{P} ∩ B_{S} = K, to conclude that either B_{S}^{×} contains nontrivial unipotent elements, or that [k, b(t)] is trivial for all k. The former is impossible, since B_{S} is a definite quaternion algebra, so we conclude that b(t) commutes with K^{×} which implies that b(t) ∈ K_{P}. It follows that b(t) ∈ F_{P}^{×}B_{S}^{×} ∩ K_{P} = F_{P}^{×}K^{×}. But now looking at the form of b(t) shows that w(t) = 1 and (λ_{σ,P} λ_{σ′,P}^{−1} ) ∈ F_{P}^{×}K^{×}, which contradicts the fact that σ ≢ σ′ mod Gal_{K}^{P −rat}. It remains to dispose of the situation where K_{P} fails to normalize W. In this case, we may argue as follows. Since w(t) is unipotent, the left-hand-side of (5) has norm independent of λ_{P}. On the other hand, the set F_{P}^{×}B_{S}^{×} contains only countably many elements of given norm. It follows that there are only countably many possibilities for the left-hand-side of (5). Thus consider a given element α in F_{P}^{×}B_{S}^{×}. We want to count the number of cosets λ_{P} F_{P}^{×} ∈ K_{P}^{×}/F_{P}^{×} such that there there exists t ∈ F_{P} with

λ_{P} w(t)λ_{P}^{−1} = α(λ_{σ′,P} λ_{σ,P}^{−1} ).  (6)

Note that since λ_{σ′,P} λ_{σ,P}^{−1} is not an element of Gal_{K}^{P −rat} by assumption, any such t is neccesarily nontrivial. But since the normalizer of W in K_{P}^{×} is precisely F_{P}^{×}, we see that if λ_{P} and λ′_{P} belong to different F_{P}^{×}-cosets, then the conjugates of W by λ_{P} and λ′_{P} have trivial intersection. It follows that for each α, there is at most a unique coset in λ_{P} F_{P}^{×} ∈ K_{P}^{×}/F_{P}^{×} such that (6) holds for some t. Since there are only countably many possibilities for α, our contention follows. □

We may now prove Proposition 2.22. Put

𝔅 = ⋃_{(S,σ)≠(S′,σ′)} 𝔅((S, σ), (S′, σ′))

so that 𝔅 is again the disjoint union of countably many cosets of Gal_{K}^{P −rat} in Gal_{K}^{ab}. Since any such coset is negligible, so is 𝔅. We claim that Red(γ · x · U) is dense in C^{−1}(γ · x̄) if γ belongs to Gal_{K}^{ab} \ 𝔅. In fact:

Lemma 2.25 For γ ∈ Gal_{K}^{ab}, the following conditions are equivalent:

1. Red(γ · x · U) is dense in C^{−1}(γ · x̄).

2. Γ(γ · x, H) · Δ(U) is dense in (B_{P}^{1} )^{𝔖×ℜ} (∀H compact open in G^{1}(A_{f})^{P}).

3. Γ(γ · x, H) · Δ(U) is dense in (B_{P}^{1} )^{𝔖×ℜ} (∃H compact open in G^{1}(A_{f})^{P}).

4. γ does not belong to 𝔅.

Proof. Lemma 2.17 implies that a subset Z of C^{−1}(γ̄·x) is everywhere dense if and only if for every compact open subgroup H of G^{1}(A_{f})^{P}, the image of Z in the quotient C^{−1}(γ̄ · x)/H(𝔖, ℜ) is everywhere dense. Applying this to Z = Red(γ·x·U) yields (1) ⇔ (2). But now (2) implies (3) and (4) is equivalent to (3) for any H by Proposition 2.35 below. □

In summary, the arguments of this section show that Proposition 2.13 follows from Proposition 2.23, together with Proposition 2.35 below.

2.6 Proof of Proposition 2.12.

Let V be a simple left B_{P} -module, so that V ≃ F_{P}^{2} as an F_{P} -module and V ≃ K_{P} as a K_{P} -module. We fix a K_{P} -basis e of V and an O_{F_{P}} -basis (1, ω) of O_{K_{P}}. Then (e, ωe) is an F_{P} -basis of V, which we use to identify B_{P} ≃ End_{F_{P}} (V) with M_{2}(F_{P}). Under this identification, the element x = α + βω of K_{P} corresponds to the matrix ( \begin{matrix}α & −βθ \\ β & α+βτ \end{matrix} ) with θ = nr(ω) and τ = Tr(ω).

Let L be the set of all O_{F_{P}} -lattices in V. To each L in L, we may attach an integer n(L) as follows. The set O(L) = {λ ∈ K_{P}; λL ⊂ L} is an O_{F_{P}} -order in K_{P} and therefore equals O_{n} = O_{F_{P}} + P^{n}O_{K_{P}} for a unique integer n: we take n(L) = n. From a matrix point of view, n(L) is the smallest integer n ≥ 0

such that ϖ_{P}^{n} ( \begin{matrix}0 & −θ \\ 1 & τ \end{matrix} ) L ⊂ L.

Lemma 2.26 The map L ↦ n(L) induces a bijection K_{P}^{×}\L → N.

Proof. For λ ∈ K_{P}^{×} and L ∈ L, O(λL) = O(L), so that n(λ · L) = n(L): our map is well-defined. Conversely, suppose that n(L) = n(L′) = n for L, L′ ∈ L. Since both L and L′ are free, rank one O_{n}-submodules of V = K_{P} · e, there exists λ ∈ K_{P}^{×} such that λ · L = L′: our map is injective. It is also surjective, since n(O_{n} · e) = n for all n ∈ N.

Put L_{0} = O_{0} · e, R = End(L_{0}) = M_{2}(O_{F_{P}}), δ = ( \begin{matrix}ϖ_{P} & 0 \\ 0 & 1 \end{matrix} ) and u(t) = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) for

t ∈ F_{P}. Then:

Lemma 2.27 For n ≥ 0 and t ∈ O_{F_{P}}^{×},

n(u(ϖ_{P}^{−n}t) · L_{0}) = 2n and n(u(ϖ_{P}^{−n}t) · δL_{0}) = 2n + 1.

Proof. Left to the reader.

Let us consider a P -isogeny class H ⊂ CM, a compact open subgroup G ⊂ Gal_{K}^{ab} and a compact open subgroup H ⊂ G(A_{f}). We choose an element x_{0} ∈ H such that x_{0} = \overline{T (Q)} · g_{0} for some g_{0} ∈ G(A_{f}) whose P -component equals 1. Let K_{P}^{G} be the kernel of

K_{P}^{×} ↪ T (A_{f}) ⟶^{rec_{K}} Gal_{K}^{ab} → Gal_{K}^{ab}/G.

This is an open subgroup of finite index in K_{P}^{×}. Let N be a positive integer such that

• 1 + P^{N}O_{K_{P}}^{×} ⊂ K_{P}^{G} and

• H contains the image of R(N)^{×} = 1 + P^{N}R ⊂ B_{P}^{×} in G(A_{f}).

We denote by I_{1} ⊂ K_{P}^{×} (resp. I_{2} ⊂ R^{×}) a chosen set of representatives for K_{P}^{×}/K_{P}^{G} (resp. R^{×}/R(N)^{×}) and put I = I_{1} × {0, 1} × I_{2}. For i = (λ, ϵ, r) ∈ I, we put

x_{i} = x_{0} · λδ^{ϵ}r ∈ H  and  u_{i}(t) = (δ^{ϵ}r)^{−1}u(t)(δ^{ϵ}r) (t ∈ F_{P}).

We finally put κ = 1 + P^{N+1}O_{F_{P}}, a compact open subgroup of F_{P}^{×}. The following result gives the proof of Proposition 2.12.

Proposition 2.28 With notations as above, 1. G · H · H = ∪_{i∈I} ∪_{n≥0} G · x_{i}u_{i}(κ_{n}) · H and

2. ∀i ∈ I and ∀n ≥ 0, G · x_{i}u_{i}(κ_{n}) · H = G · x_{i}u_{i,n} · H

where κ_{n} = ϖ_{P}^{−n}κ and u_{i,n} = u(ϖ_{P}^{−n}) ⊂ u_{i}(κ_{n}).

Proof. Note that x_{i} · u_{i}(t) = x_{0} · λu(t)δ^{ϵ}r.

(1) We have to show that any element x of H belongs to G · x_{i}u_{i}(κ_{n}) · H for some i ∈ I and n ≥ 0. Write x = x_{0} · b with b ∈ B_{P}^{×}. Consider the lattice b · L_{0} ⊂ V and write n(b · L_{0}) = 2n + ϵ with ϵ ∈ {0, 1}. By Lemma 2.26 and 2.27, there exists λ_{0} ∈ K_{P}^{×} such that b·L_{0} = λ_{0}·u(ϖ_{P}^{−n})δ^{ϵ}·L_{0}, hence b = λ_{0} · u(ϖ_{P}^{−n})δ^{ϵ} · r_{0} for some r_{0} ∈ R^{×}. By definition of I_{1} and I_{2}, there exists λ ∈ I_{1}, k ∈ K_{P}^{G}, r ∈ I_{2} and h ∈ R(N)^{×} such that λ_{0} = k · λ and r_{0} = rh. Put i = (λ, ϵ, r) ∈ I, t = ϖ_{P}^{−n} ∈ κ_{n} and σ = rec_{K}(k) ∈ G. Since x_{0} · k = σ · x_{0}, we obtain

x = x_{0} · b = σ · x_{0} · λu(t)δ^{ϵ}rh = σ · (x_{i}u_{i}(t)) · h ∈ G · x_{i} · u_{i}(κ_{n}) · H.

(2) We have to show that for i = (λ, ϵ, r) ∈ I, n ≥ 0 and a ∈ κ,

x_{i} · u_{i}(ϖ_{P}^{−n}a) ∈ G · x_{i}u_{i,n} · H.

Put y_{a} = 1 − a^{−1}, λ_{n,a} = 1 + ϖ_{P}^{n} y_{a}ω ∈ K_{P}^{×} and σ_{n,a} = rec_{K}(λ_{n,a}). Since a belongs to κ = 1 + P^{N+1}O_{F_{P}}^{×}, y_{a} belongs to P^{N+1}, λ_{n,a} belongs to K_{P}^{G} and σ_{n,a} belongs to G. As a matrix,

λ_{n,a} = ( \begin{matrix}1 & −θϖ_{P}^{n} y_{a} \\ ϖ_{P}^{n} y_{a} & 1 + τϖ_{P}^{n} y_{a} \end{matrix} ) ∈ K_{P}^{×} ⊂ GL_{2}(F_{P}).

In particular,

δ^{−ϵ}u(−ϖ_{P}^{−n})·λ_{n,a}·u(ϖ_{P}^{−n}a)δ^{ϵ} = (\begin{matrix}1 − y_{a} & −(θϖ_{P}^{n} + τ)ϖ_{P}^{−ϵ}y_{a} \\ ϖ_{P}^{n+ϵ}y_{a} & 1 + (a + τϖ_{P}^{n})y_{a} \end{matrix}) ≡ 1 mod P^{N}.

In other words: there exists r′ ∈ R(N)^{×} such that λ_{n,a}u(ϖ_{P}^{−n}a)δ^{ϵ} = u(ϖ_{P}^{−n})δ^{ϵ}r′. We thus obtain:

σ_{n,a} · x_{i} · u_{i}(ϖ^{−n}a) = x_{0} · λλ_{n,a}u(ϖ_{P}^{−n}a)δ^{ϵ}r
= x_{0} · λu(ϖ_{P}^{−n})δ^{ϵ}r′r
= x_{i} · u_{i,n} · h

with h = r^{−1}r′r ∈ R(N)^{×} ⊂ H. QED.

2.7 An application of Ratner’s Theorem

In this section, we study the distribution of certain unipotent flows on X = Γ\G^{r}, where G = SL_{2}(F_{P}), r is a positive integer and Γ = Γ_{1} × ⋯ × Γ_{r} is a product of cocompact lattices in G. Our key tool is the following special case of a theorem of Margulis and Tomanov [11, Theorem 11.2] (see also Ratner’s Theorem 3 in [15]). We fix a Haar measure λ on F_{P}.

Theorem 2.29 (Uniform Distribution) Let V = {v(t)} be a oneparameter unipotent subgroup of G^{r}.

1. For every x ∈ X, there exists

• a closed subgroup L ⊃ V of G^{r} such that \overline{x · V} = x · L, and

• an L-invariant Borel probability measure μ on X supported on \overline{x · V}.

2. With x and μ as above, for every continuous function f on X and every compact set κ of F_{P} with positive measure, we have

lim_{|s|→∞} \frac{1}{λ(s · κ)} ∫_{s·κ} f(x · v(t))dλ(t) = ∫_{X} f(y)dμ(y).

Here λ denotes a choice of Haar measure on F_{P}.

The measure μ is uniquely determined by x and V. On the other hand, we may replace the closed subgroup L ⊃ V of G^{r} by Σ = {g ∈ G^{r} | μ is g-invariant}. Indeed, Σ is a closed subgroup of G^{r} which contains L and therefore also V. Since μ is Σ-invariant, so is its support \overline{x · V} = x·L. In particular, \overline{x · V} = x·Σ. Suppose now that V = Δ(U), where Δ : G → G^{r} is the diagonal map and U = {u(t)} is a (non-trivial) one-parameter unipotent subgroup of G. In this case, a result of M. Ratner shows that Σ contains some “twisted” diagonal:

Lemma 2.30 There exists an element c ∈ U^{r} such that cΔ(G)c^{−1} ⊂ Σ.

Proof. This is Corollary 4 of Theorem 6 in [15] when F_{P} = Q_{p} (note that the centralizer of Δ(U) in G^{r} equals {±U}^{r}). The case of general F_{P} seems to be well-known to the experts, see for instance the notes of N. Shah [17].

This leaves only finitely many possible values for Ω = c^{−1}Σc. Indeed:

Lemma 2.31 For any subgroup Ω of G^{r} such that Δ(G) ⊂ Ω, there exists a partition {I_{α}} of {1, ⋯, r} such that

∏_{α} Δ^{I_{α}}(G) ⊂ Ω ⊂ {±1}^{r} · ∏_{α} Δ^{I_{α}}(G)

where Δ^{I_{α}}(G) is the diagonal subgroup of {(g_{i}) ∈ G^{r}; ∀i ∉ I_{α}, g_{i} = 1}.

Proof. This is a slight generalization of Proposition 3.10 of [2]. According to the latter, there exists a partition {I_{α}} of {1, ⋯, r} such that

{±1}^{r} · Ω = {±1}^{r} ·_{∏α}Δ^{I_{α}}(G).

Taking the derived group on both sides gives ∏_{α} Δ^{I_{α}}(G) = [Ω : Ω] ⊂ Ω.

The equivalence relation ∼ on {1, ⋯, r} which is defined by the above partition can easily be retrieved from \overline{x · Δ(U)} = x · Σ by the following rule: for 1 ≤ i, j ≤ r, i ∼ j if and only if the projection

\overline{x · Δ(U)} ⊂ X → Γ_{i}\G × Γ_{j}\G

is not surjective. On the other hand, this equivalence relation can also be used to characterize those Ω-orbits which are closed subsets of X:

Lemma 2.32 For g = (g_{i}) ∈ G^{r}, the Ω-orbit of y = Γ · g is closed in X if and only for all 1 ≤ i, j ≤ r with i ∼ j, g_{i}^{−1}Γ_{i}g_{i} and g_{j}^{−1}Γ_{j}g_{j} are commensurable in G.

Proof. The map ω ↦ y ·ω induces a continuous bijection θ : g^{−1}Γg ∩Ω\Ω → y·Ω. We first claim that y·Ω is closed in X if and only if g^{−1}Γg∩Ω\Ω is compact. The if part is trivial: if g^{−1}Γg ∩ Ω\Ω is compact, so is θ(g^{−1}Γg ∩ Ω\Ω) = y · Ω. To prove the converse, it is sufficient to show that θ is an homeomorphism when y · Ω is closed (hence compact). Now if y · Ω is a closed subset of X, Γ · g · Ω is a closed subset of G^{r} and g^{−1}Γg · Ω is a Baire space. Since Γ is countable (being discrete in a σ-compact space), it follows that Ω is open in g^{−1}Γg · Ω and θ is indeed an homeomorphism. Put Ω′ = ∏_{α} Δ^{I_{α}}(G). Since Ω′ ⊂ Ω ⊂ {±1}^{r}·Ω′, g^{−1}Γg∩Ω is cocompact in Ω if and only if g^{−1}Γg∩Ω′ is cocompact in Ω′. Note that g^{−1}Γg∩Ω′\Ω′ ≃ ∏_{α} Γ_{α}\G where Γ_{α} = ∩_{i∈I_{α}}g_{i}^{−1}Γ_{i}g_{i}, and Γ_{α}\G is compact if and only if g_{i}^{−1}Γ_{i}g_{i} is commensurable with g_{j}^{−1}Γ_{j}g_{j} for all i, j ∈ I_{α}. This finishes the proof of the lemma.

We thus obtain a second characterization of the equivalence relation ∼.

Definition 2.33 We say that two subgroups Γ and Γ′ of G are Ucommensurable if there exists u ∈ U such that Γ and u^{−1}Γu are commensurable.

Corollary 2.34 Write x = Γ · g with g = (g_{i}) ∈ G^{r}. For 1 ≤ i, j ≤ r, i ∼ j if and only if g_{i}^{−1}Γ_{i}g_{i} and g_{j}^{−1}Γ_{j}g_{j} are U-commensurable in G.

Proof. Write c = (c_{i}) ∈ U^{r} and put y = x · c = Γ · gc. Then y · Ω = x · Σ = \overline{x · V} is a closed subset of X. The lemma implies that (g_{i}c_{i})^{−1}Γ_{i}(g_{i}c_{i}) and (g_{j}c_{j})^{−1}Γ_{j}(g_{j}c_{j}) are commensurable in G when i ∼ j. Conversely, suppose that g_{i}^{−1}Γ_{i}g_{i} and α^{−1}g_{j}^{−1}Γ_{j}g_{j}α are commensurable for some α ∈ U. Put Γ′ = Γ_{i} × Γ_{j}, X′ = Γ′\G^{2}, c′ = (1, α) and Δ′(g) = (g, g) for g ∈ G. Let p : X → X′ be the obvious projection. The lemma implies that p(x) · c′Δ′(G)c^{′−1} is closed in X′, so that

p(\overline{x · Δ(U)}) ⊂ \overline{p(x) · Δ′(U)} ⊂ p(x) · c′Δ′(G)c^{′−1}.

In particular, p(\overline{x · Δ(U)}) ≠ X′ and i ∼ j.

Proposition 2.35 The following conditions are equivalent:

1. \overline{x · Δ(U)} = X.

2. For all 1 ≤ i ≠ j ≤ r, g_{i}^{−1}Γ_{i}g_{i} and g_{j}^{−1}Γ_{j}g_{j} are not U-commensurable.

The measure μ of Theorem 2.29 is then the (unique) G^{r}-invariant Borel probability measure on X.

Proof. Both conditions are equivalent to the assertion that the partition {I_{α}} of {1, ⋯, r} is trivial. In that case, Ω = G^{r} = Σ and μ is G^{r}-invariant.

3 The case of Shimura curves

3.1 Shimura Curves

3.1.1 Definitions

We start by defining the Shimura curves. Let {τ_{1}, ⋯, τ_{d}} = Hom_{Q}(F, R) be the set of real embeddings of F. We shall always view F as a subfield of R or C through τ_{1}. Let S be a set of finite primes such that |S|+d is odd, and let B denote the quaternion algebra over F which ramifies precisely at S∪{τ_{2}, ⋯, τ_{d}} (a finite set of even order). Let G be the reductive group over Q whose set of points on a commutative Q-algebra A is given by G(A) = (B ⊗ A)^{×}. In particular, G_{R} ≃ G_{1} × ⋯ × G_{d} where B_{τ_{i}} = B ⊗_{F,τ_{i}} R and G_{i} is the algebraic group over R whose set of points on a commutative R-algebra A is given by G_{i}(A) = (B_{τ_{i}} ⊗_{R} A)^{×}. Fix ϵ = ±1 and let X be the G(R)- conjugacy class of the morphism from 𝕊 =^{def} Res_{C/R}(𝔾_{m,C}) to G_{R} which maps z = x + iy ∈ 𝕊(R) = C^{×} to

[( \begin{matrix}x & y \\ −y & x \end{matrix} )^{ϵ} , 1, ⋯, 1] ∈ G_{1}(R) × ⋯ × G_{d}(R) ≃ G(R).

We have used an isomorphism of R-algebras B_{τ_{1}} ≃ M_{2}(R) to identify G_{1} and GL_{2,R}; the resulting conjugacy class X does not depend upon this choice (but it does depend on ϵ, cf. section 3.3.1 below). For every compact open subgroup H of G(A_{f}), the quotient of G(A_{f})/H × X by the diagonal left action of G(Q) is a Riemann surface

M_{H}^{an} =^{def} G(Q)\ (G(A_{f})/H × X)

which is compact unless d = 1 (F = Q) and S = ∅ (G = GL_{2}). The Shimura curve M_{H} is Shimura’s canonical model for M_{H}^{an}. It is a smooth curve over F (the reflex field) whose underlying Riemann surface M_{H}(C) equals M_{H}^{an}.

3.1.2 CM points

Among the models of M_{H}^{an}, the Shimura curve M_{H} is characterized by specifying the action of Galois (the “reciprocity law”) on certain special points. A morphism h : 𝕊 → G_{R} in X is special if it factors through the real locus of some Q-rational subtorus of G and a point x in M_{H}^{an} is special if x = [g, h] with h special (and g in G(A_{f})). Now let K be an imaginary quadratic extension of F such that there exists some embedding K → B. Put T = Res_{K/Q}(𝔾_{m,K}). Any embedding K ↪ B yields an embedding T ↪ G. In the sequel, we shall fix an embedding of K in B, and study those special points in X or M_{H}^{an} for which h : 𝕊 → G_{R} factors through the morphism T_{R} ↪ G_{R} which is induced by the fixed F -embedding K ↪ B. We shall refer to such points as CM points. We denote by CM_{H} the set of CM points in M_{H}^{an} = M_{H}(C). It is clear that this set is nonempty. Furthermore, Shimura’s theory implies that any CM point is algebraic, defined over the maximal abelian extension K^{ab} of K (see section 3.2.4 below).

3.1.3 Integral models and supersingular points

Let v be a finite place of F where B is split and put S = Spec O(v) where O(v) is the local ring of F at v. We denote by F_{v} and O_{v} the completion of F at v and its ring of integers. For simplicity, we shall only consider level structures H ⊂ G(A_{f}) which decompose as H = H^{v}H_{v} where H^{v} (resp. H_{v}) is a compact open subgroup of

G(A_{f})^{v} = {g ∈ G(A_{f}) | g_{v} = 1} (resp. B_{v}^{×} ⊂ G(A_{f})).

In the non-compact (classical) case where F = Q and G = GL_{2}, it is wellknown that M_{H} is a coarse moduli space which classifies elliptic curves (with level structures) over extensions of Q. Extending the moduli problem to elliptic curves over S-schemes, we obtain a regular model M_{H}/S of M_{H}. A geometric point in the special fiber of M_{H} is supersingular if it corresponds to (the class of) a supersingular elliptic curve. In the general (compact) case, the Shimura curve M_{H} may not be a moduli space. However, provided that H^{v} is sufficiently small (a condition depending upon H_{v}), Carayol describes in [1] a proper and regular model M_{H}/S of M_{H}, which is smooth when H_{v} is a maximal compact open subgroup of B_{v}^{∗}. When H^{v} fails to be sufficiently small in the sense of [1], we let M_{H}/S be the quotient of M_{H′} by the S-linear right action of H/H′, where H′ = H^{′v}H_{v} for a sufficiently small compact, open and normal subgroup H^{′v} of H^{v}. Then M_{H}/S is again a proper and regular model of M_{H} which is smooth when H_{v} is maximal (cf. [9, p. 508]), and it does not depend upon the choice of H^{′v}. These models form a projective system {M_{H}}_{H} of proper S-schemes with finite flat transition maps, whose limit M = \underleftarrow{lim} M_{H} has a right action of G(A_{f}) and carries an O_{v}-divisible module E of height 2 (cf. [1, Appendice] for the definition and basic properties of O_{v}-divisible modules). A geometric point x in the special fiber of M is said to be ordinary if E | x is isomorphic to the product of the O_{v}-divisible constant module F_{v}/O_{v} with Σ_{1}, the unique O_{v}-formal module of height 1. Otherwise, x is supersingular and E | x is isomorphic to Σ_{2}, the unique O_{v}-formal module of height 2. A supersingular point in the special fiber of M_{H} is one which lifts to a supersingular point in M. In the classical case, the supersingular points also have such a description, with E equal to the relevant Barsotti-Tate group in the universal elliptic curve on M = \underleftarrow{lim} M_{H}.

3.1.4 Reduction maps

Let us choose a place v̄ of K^{ab} above v, with ring of integers O(v̄) ⊂ K^{ab} and residue field F(v̄), an algebraic closure of the residue field F(v) of v. Consider the specialization maps:

M_{H}(K^{ab}) = M_{H}(K^{ab}) ← M_{H}(O(v̄)) → M_{H}(F(v̄))

In the compact case, M_{H} is proper over S and the first of these two maps is a bijection by the valuative criterion of properness. In the classical non-compact case, the first map is still injective (M_{H} is separated over S); by [16, Theorem 6], its image contains CM_{H}. In both cases, we obtain a reduction map

RED_{v} : CM_{H} → M_{H}(F(v̄)).

Let M_{H}^{ss}(v) be the set of supersingular points in M_{H}(F(v̄)).

Lemma 3.1 If v does not split in K, RED_{v}(CM_{H}) ⊂ M_{H}^{ss}(v).

Proof. (Sketch) Let E^{0} be the O_{v}-divisible module E “up to isogeny”. There is an F_{v}-linear right action of G(A_{f}) on E^{0} covering the right action of G(A_{f}) on M (see [1, 7.5] for the compact case). For any point x on M, we thus obtain an F_{v}-linear right action of Stab_{G(A_{f})}(x) on E^{0} | x. If x is a CM point, say x = [g, h] for some g ∈ G(A_{f}) and h : 𝕊 → T_{R} ↪ G_{R} in X, g^{−1}T (Q)g ⊂ Stab_{G(A_{f})}(x) and the induced F_{v}-linear action of T (Q) = K^{∗} on E^{0} | x (or its inverse, depending upon ϵ) arises from an F_{v}-linear right K_{v}-module structure on E^{0} | x. The connected part of the special fiber E^{0} | RED_{v}(x) therefore inherits a K_{v}-module structure. Since End_{F_{v}}(Σ_{1}^{0}) ≃ F_{v}, this connected part can not be isomorphic to Σ_{1}^{0} unless v splits in K.

3.1.5 Connected components

We now want to define yet another type of “reduction map”. Recall from Shimura’s theory that the natural map from M_{H}^{an} to its set of connected components π_{0}(M_{H}^{an}) corresponds to an F -morphism c : M_{H} → M_{H} between the Shimura curve M_{H} and a zero-dimensional Shimura variety M_{H} over F whose finitely many points are algebraic over the maximal abelian extension F^{ab} ⊂ K^{ab} of F. Since M_{H} is regular (hence normal), this morphism extends over S to a morphism c : M_{H} → M_{H} between M_{H} and the normalization M_{H} of S in M_{H}, a finite and regular S-scheme. With Z_{H} =^{def} π_{0}(M_{H}^{an}) = M_{H}(K^{ab}) and Z_{H}(v) =^{def} M_{H}(F(\overline{v})), the following diagram is commutative:

CM_{H} ⟶^{RED_{v}} M_{H}^{ss}(v)
c ↓  ↓ c
Z_{H}  ⟶^{RED_{v}} Z_{H}(v)

Put X_{H}(v) = M_{H}^{ss}(v) ×_{Z_{H}(v)} Z_{H}. If v does not split in K, we thus obtain a reduction map and a connected component map

CM_{H} ⟶^{RED_{v}}  X_{H}(v)  ⟶^{c_{v}} Z_{H}
x  ⟼ (RED_{v}(x), c(x)) ⟼ c(x) (7)

The composite map c = c_{v} ∘ RED_{v} : CM_{H} → Z_{H} does not depend upon v and commutes with the action of Gal_{K}^{ab} on both sides.

Remark 3.2 In the compact case, M_{H} ⟶^{c} M_{H} → S is the Stein factorization

M_{H} → Spec(Γ(M_{H}, O_{M_{H}})) → S

of the proper morphism M_{H} → S. Indeed, since M_{H} is affine, c factors through an S-morphism α : Spec(Γ(M_{H}, O_{M_{H}})) → M_{H}. Over the generic point of S, α_{/F} : Spec(Γ(M_{H}, O_{M_{H}})) → M_{H} is a morphism between finite étale F -schemes which induces a bijection on complex points: it is therefore an isomorphism. Since M_{H} is a regular scheme which is proper and flat over S, Spec(Γ(M_{H}, O_{M_{H}})) is a normal scheme which is finite and flat over S. It follows that α is an isomorphism.

3.1.6 Simultaneous reduction maps

Let 𝔖 be a finite set of finite places of F which are non-split in K and away from S: for each v ∈ 𝔖, K_{v} is a field and B_{v} ≃ M_{2}(F_{v}). Let also ℜ be a finite set of Galois elements in Gal_{K}^{ab}. We put

X_{H}(𝔖, ℜ) =^{def} ∏_{v∈𝔖,σ∈ℜ} X_{H}(v) and Z_{H}(𝔖, ℜ) =^{def} ∏_{v∈𝔖,σ∈ℜ} Z_{H}(v)

and define a simultaneous reduction map and a connected component map

Red : CM_{H} → X_{H}(𝔖, ℜ) and C : X_{H}(𝔖, ℜ) → Z_{H}(𝔖, ℜ)

by Red(x) = (RED_{v}(σx))_{v∈𝔖,σ∈ℜ} and C(x_{v,σ}) = (c_{v}(x_{v,σ}))_{v∈𝔖,σ∈ℜ}. For x ∈ CM_{H} and τ ∈ Gal_{K}^{ab}, we have

C ∘ Red(τx) = (τσc(x))_{v∈𝔖,σ∈ℜ}.

3.1.7 Main theorem

Let P be a maximal ideal of O_{F} and suppose that P ∉ S ∪ 𝔖. In particular, B_{P} ≃ M_{2}(F_{P}). We make no assumptions on P relative to K: P may either split, ramify or be inert in K. Then the following definitions reprise the ones already made in the first section of the paper: we have chosen to repeat them here for the convenience of the reader.

Definition 3.3 We say that two points x and y ∈ M_{H}^{an} are P -isogeneous if x = [g, h] and y = [g′, h] for some h ∈ X and g, g′ ∈ G(A_{f}) such that g_{w} = g′_{w} for every finite place w ≠ P of F.

Note that a point which is P -isogeneous to a CM point is again a CM point.

Definition 3.4 An element σ ∈ Gal_{K}^{ab} is P -rational if σ = rec_{K}(λ) for some λ ∈ K̂^{×} whose P -component λ_{P} belongs to the subgroup K^{×} · F_{P}^{×} of K_{P}^{×}. We denote by Gal_{K}^{P −rat} ⊂ Gal_{K}^{ab} the subgroup of all P -rational elements.

In the above definition, rec_{K} : K̂^{×} ↠ Gal_{K}^{ab} is Artin’s reciprocity map. We normalize the latter by specifying that it sends local uniformizers to geometric Frobeniuses.

Theorem 3.5 Suppose that the finite subset ℜ of Gal_{K}^{ab} consists of elements which are pairwise distinct modulo Gal_{K}^{P −rat}. Let H ⊂ CM_{H} be a P -isogeny class of CM points and let G be a compact open subgroup of Gal_{K}^{ab}. Then for all but finitely many points x ∈ H,

Red(G · x) = C^{−1}(C ∘ Red(G · x)).

Remark 3.6 When the level structure H arises from an Eichler order in B, our proof of this surjectivity statement yields a little bit more: for any y ∈ X_{H}(𝔖, ℜ), we can compute the asymptotic behavior of the probability that Red(g·x) = y for some g ∈ G, as x goes to infinity inside H (see Corollary 2.11).

3.2 Uniformization

Write CM, M^{ss}(v), Z, Z(v) and X (v) = M^{ss}(v) ×_{Z(v)} Z for the projective limits of {CM_{H}}, {M_{H}^{ss}(v)}, {Z_{H}}, {Z_{H}(v)} and {X_{H}(v)}. These sets now have a right action of G(A_{f}). For X ∈ {CM, M^{ss}(v), Z, Z(v)}, the natural map X/H → X_{H} is a bijection while X (v)/H → X_{H}(v) is surjective. The projective limit of (7) yields G(A_{f})-equivariant maps

CM ⟶^{RED_{v}} X (v) ⟶^{c_{v}} Z

which we shall now compute.

3.2.1 CM points

From [4, Proposition 2.1.10] or [14, Theorem 5.27],

M^{an} =^{def} \underleftarrow{lim} M_{H}^{an} = G(Q)\ (G(A_{f})/\overline{Z(Q)} × X)

where Z = Res_{F/Q}(𝔾_{m,F}) is the center of G and \overline{Z(Q)} is the closure of Z(Q)

in Z(A_{f}). Inside M^{an}, CM =^{def} \underleftarrow{lim} CM_{H} corresponds to those elements which can be represented by (g, h) with g ∈ G(A_{f}) and h a CM point in X. Let us construct such an h and show that any other CM point belongs to the same G(Q)-orbit. Since K_{v} is a field for all v ∈ S ∪ {τ_{2}, ⋯, τ_{d}} (the set of places of F where B ramifies), there exists an F -embedding ι : K ↪ B. Moreover, any other F -embedding K ↪ B is conjugated to ι by an element of B^{×} = G(Q). We use ι to identify T as a Q-rational subtorus of G and also chose an extension τ_{1} : K ↪ C of our distinguished embedding τ_{1} : F ↪ R. In the sequel, we shall always view K as a subfield of C through τ_{1}. Put T_{i} = Res_{K_{τ_{i}} /R}(𝔾_{m,K_{τ_{i}}}) (with K_{τ_{i}} = K ⊗_{F,τ_{i}} R), so that T_{R} ≃ T_{1} ×⋯×T_{d} and this decomposition is compatible with the decomposition G_{R} ≃ G_{1} × ⋯ × G_{d} of section 3.1.1. Moreover, τ_{1} : K ↪ C induces an isomorphism between K_{τ_{1}} and C which allows us to identify T_{1} and 𝕊. There are exactly two morphisms s and s̄: 𝕊 → T_{R} whose composite with ι_{R} : T_{R} ↪ G_{R} belongs to X. They are characterized by

s(z) = (z^{ϵ}, 1, ⋯, 1) and s̄(z) = (z̄^{ϵ}, 1, ⋯, 1) for z ∈ C^{×} = 𝕊(R).

Finally, there exists an element b ∈ B^{×} = G(Q) such that bι(λ)b^{−1} = ι(λ̄) for all λ ∈ K (where λ ↦ λ̄ is the non-trivial F -automorphism of K). But then b(ι_{R} ∘ s̄)b^{−1} = ι_{R} ∘ s, so that h = ι_{R} ∘ s and h̄ = ι_{R} ∘ s̄ belong to the same G(Q)-orbit in X. Since the centralizer of h in G(Q) equals T (Q), we obtain:

Lemma 3.7 The map g ↦ [1, h] · g = [g, h] induces a bijection

\overline{T (Q)}\G(A_{f}) ⟶^{≃} CM

where \overline{T (Q)} is the closure of T (Q) in T (A_{f}).

Proof. The above discussion gives a bijection T (Q)\G(A_{f})/\overline{Z(Q)} ≃ CM. We claim that T (Q)\overline{Z(Q)} = \overline{T (Q)}. Indeed, \overline{Z(Q)} is the product of Z(Q) = F^{×} with the closure of O_{F}^{×} in Ô_{F}^{×} ⊂ Z(A_{f}) = F̂^{×} (this holds more generally for any number field). Therefore, T (Q)\overline{Z(Q)} = K^{×}\overline{O_{F}^{×}} and

T (Q)\overline{Z(Q)} ∩ Ô_{K}^{×} = O_{K}^{×} \overline{O_{F}^{×}} = ∪_{α∈O_{K}^{×}/O_{F}^{×}}α\overline{O_{F}^{×}}.

Since [O_{K}^{×} : O_{F}^{×}] is finite, T (Q)\overline{Z(Q)} is a locally closed, hence closed subgroup of T (A_{f}). Our claim easily follows.

3.2.2 Connected components

Let G(R)^{+} and Z(R)^{+} be the identity components of G(R) and Z(R) and put G(Q)^{+} = G(Q) ∩ G(R)^{+} and Z(Q)^{+} = Z(Q) ∩ Z(R)^{+}. Thus, G(R)^{+} is the set of elements in G(R) whose projection to G_{1}(R) ≃ GL_{2}(R) has a positive determinant while Z(Q)^{+} is the subgroup of totally positive elements in Z(Q) = F^{×}. Let X^{+} = G(R)^{+} · h be the connected component of h in X. Since G(Q) is dense in G(R), G(Q) · X^{+} = X and

M_{H}^{an} ≃ G(Q)^{+}\ (G(A_{f})/H × X^{+}) .

It follows that Z_{H} = π_{0}(M_{H}^{an}) ≃ G(Q)^{+}\G(A_{f})/H. On the other hand, the reduced norm nr : B → F induces a surjective morphism nr : G → Z whose kernel G^{1} ⊂ G is the derived group of G. The norm theorem (nr(G(Q)^{+}) = Z(Q)^{+}, [20, p. 80]) and the strong approximation theorem (G^{1}(Q) is dense in G^{1}(A_{f}), [20, p. 81]) together imply that the reduced norm induces a bijection between G(Q)^{+}\G(A_{f})/H and Z(Q)^{+}\Z(A_{f})/nr(H). With Z =^{def} \underleftarrow{lim} Z_{H}, we thus obtain:

Lemma 3.8 The map g ↦ c([1, h]) · g = c([g, h]) factors through the reduced norm and yields a bijection

\overline{Z(Q)^{+}}\Z(A_{f}) ⟶^{≃} Z

where \overline{Z(Q)^{+}} is the closure of Z(Q)^{+} in Z(A_{f}).

3.2.3 Supersingular points

Proposition 3.9 (1) The right action of G(A_{f}) on X (v) =^{def} \underleftarrow{lim} X_{H}(v) is transitive and factors through the surjective group homomorphism

(1, nr_{v}) : G(A_{f}) = G(A_{f})^{v} × B_{v}^{×} → G(A_{f})^{v} × F_{v}^{×}

where G(A_{f})^{v} = {g ∈ G(A_{f}); g_{v} = 1}.

(2) For any point x ∈ X (v) (such as x = RED_{v}([1, h]) if v does not split in K), the stabilizer of x in G(A_{f})^{v} × F_{v}^{×} may be computed as follows. Let B′ be the quaternion algebra over F which is obtained from B by changing the invariants at v and τ_{1}: B′ is totally definite and Ram_{f}B′ = Ram_{f}B ∪ {v}. Put G′ = Res_{F/Q}(B^{′×}), a reductive group over Q with center Z. There exists an isomorphism ϕ_{x}^{v} : G(A_{f})^{v} ⟶^{≃} G′(A_{f})^{v} such that (ϕ_{x}^{v}, 1) : G(A_{f})^{v}×F_{v}^{×} ⟶^{≃} G′(A_{f})^{v} × F_{v}^{×} maps Stab(x) to the image of G′(Q)\overline{Z(Q)} ⊂ G′(A_{f}) through the (surjective) map

(1, nr_{v}) : G′(A_{f}) = G′(A_{f})^{v} × B′_{v}^{×} → G′(A_{f})^{v} × F_{v}^{×}.

Proof. In the compact case, this is exactly how Carayol describes the action of G(A_{f}) on a set which he denotes by S, cf. Proposition 11.2 of [1]. The fact that Carayol’s set S equals our X (v) follows from the discussion of [1, Section 10.1]. The non-compact case is similar.

Define

G(v) =^{def} G′(A_{f})^{v} × F_{v}^{×},

ϕ_{x}(v) =^{def} (ϕ_{x}^{v}, nr_{v}) : G(A_{f}) ↠ G(v),

and let G(Q, v) be the image of G′(Q) in G(v):

G(Q, v) = (1, nr_{v})(G′(Q)).

Corollary 3.10 The map g ↦ x · g factors through ϕ_{x}(v) and induces a bijection

\overline{G(Q, v)}\G(v) ⟶^{≃} X (v)

where \overline{G(Q, v)} is the closure of G(Q, v) in G(v).

Proof. We have to show that \overline{G(Q, v)} = (1, nr_{v})(G′(Q)\overline{Z(Q)}). We first claim that G′(Q)\overline{Z(Q)} is locally closed (hence closed) in G′(A_{f}). Indeed, let R be a maximal O_{F} -order in B′. As \overline{Z(Q)} = F^{×}\overline{O_{F}^{×}} ,

(G′(Q)\overline{Z(Q)}) ∩ R̂^{×} = (B^{′×}\overline{O_{F}^{×}}) ∩ R̂^{×} = R^{×}\overline{O_{F}^{×}} = ∪_{α∈R^{×}/O_{F}^{×}}α\overline{O_{F}^{×}}

is closed because [R^{×} : O_{F}^{×}] is finite (use [20, p. 139]). The map nr_{v} : B′_{v}^{×} → F_{v}^{×} is open and surjective with a compact kernel: it is therefore a closed map, and so is (1, nr_{v}) : G′(A_{f}) → G(v). In particular, (1, nr_{v})(G′(Q)\overline{Z(Q)}) is closed in G(v), so that \overline{G(Q, v)} ⊂ (1, nr_{v})(G′(Q)\overline{Z(Q)}) and the other inclusion is trivial.

3.2.4 Reciprocity laws

We now want to describe the reciprocity laws for CM points and connected components, following [14] instead of [4] (see the remark at the end of [14, §12]). In particular: (1) reciprocity maps send uniformizers to geometric Frobenius;

(2) Galois actions on geometric points are left actions. Let μ : 𝔾_{m,C} → T_{C} be the cocharacter which is defined by μ(z) = s ∘ r(z), where r : 𝔾_{m,C} → 𝕊_{C} ≃ 𝔾_{m,C}×𝔾_{m,C} maps z to (z, 1) (and 𝕊_{C} ≃ 𝔾_{m,C}×𝔾_{m,C} is induced by z ⊗_{R} a ↦ (za, z̄a) for z ∈ C and a in some C-algebra A). The isomorphism

\xymatrix{T_{C} \ar[r]^{λ⊗a↦(τ(λ)a)_{τ}} & 𝔾_{m,C}^{Hom(K,C)}}

yields a bijection between the set of cocharacters of T and Z^{Hom(K,C)}, with σ ∈ Aut(C) acting on the latter set by (n_{τ})_{τ} · σ = (n_{στ})_{τ}. The cocharacter μ corresponds to n_{τ} = ϵ if τ = τ_{1} and n_{τ} = 0 otherwise. In particular, the field of definition of μ equals τ_{1}(K) ≃ K and the morphism

\xymatrix{T = Res_{K/Q}(𝔾_{m,K}) \ar[r]^{Res_{K/Q}(μ)} & Res_{K/Q}(T_{K}) \ar[r]^{^{Norm^{K/Q}}} & T}

sends z to z^{ϵ}. We thus obtain:

Lemma 3.11 The CM points are algebraic, defined over the maximal abelian extension K^{ab} of K. For σ = rec_{K}(λ) with λ ∈ T (A_{f}) = K̂^{×}, the action of σ on CM ≃ \overline{T (Q)}\G(A_{f}) is given by multiplication on the left by λ^{ϵ}.

Similarly:

Lemma 3.12 The connected components are defined over the maximal abelian extension F^{ab} of F. For σ = rec_{F} (λ) with λ ∈ Z(A_{f}) = F̂^{×}, the action of σ on Z ≃ \overline{Z(Q)}^{+}\Z(A_{f}) is given by multiplication by λ^{ϵ}.

In particular, the pro-étale F -scheme M =^{def} \underleftarrow{lim} M_{H} together with its right action of G(A_{f}) is (non-canonically) isomorphic to Spec(F^{ab}) on which G(A_{f}) acts through g ↦ Spec(σ) with σ = rec_{F} (nr(g)^{ϵ}), while M =^{def} \underleftarrow{lim} M_{H} is (non-canonically) isomorphic to the spectrum of the ring of v-integers in F^{ab}. It follows that the reduction map Z = M(F^{ab}) → Z(v) = M(F(\overline{v})) identifies Z(v) with Z/O_{v}^{×} (viewing O_{v}^{×} as a subgroup of Z(A_{f}) = Z(A_{f})^{v} × F_{v}^{×}). Since O_{v}^{×} also acts trivially on M^{ss}(v) = \underleftarrow{lim} M_{H}^{ss}(v) (cf. [1, section 11.2]), the projection X (v) → M^{ss}(v) also identifies M^{ss}(v) with X (v)/O_{v}^{×} (viewing now O_{v}^{×} as a subgroup of G(v) = G′(A_{f})^{v} × F_{v}^{×}).

Corollary 3.13 If nr(H_{v}) = O_{v}^{×} (1) M_{H} is a finite étale S-scheme, (2) Z_{H} ≃ Z_{H}(v) and (3) X (v)/H ≃ X_{H}(v) ≃ M_{H}^{ss}(v).

Proof. In general, M_{H} is isomorphic to the spectrum of the ring of v-integers in the abelian extension F_{H} of F which is cut out by rec_{K}(nr(H)). If O_{v}^{×} ⊂ nr(H), F_{H} is unramified at v and M_{H} is therefore a finite étale S-scheme. This proves (1) and (2), and (2) implies that X_{H}(v) = M_{H}^{ss}(v)×_{Z_{H}(v)} Z_{H} ≃ M_{H}^{ss}(v). Finally, since X (v)/O_{v}^{×} ≃ M^{ss}(v), X (v)/H ≃ M^{ss}(v)/H ≃ M_{H}^{ss}(v).

Remark 3.14 The assumption nr(H_{v}) = O_{v}^{×} holds true when H = R̂^{×} for some Eichler order R ⊂ B.

3.2.5 Conclusion

Putting lemmas 3.7, 3.8 and Corollary 3.10 together, we obtain a commutative diagram

\overline{T (Q)}\G(A_{f}) ⟶^{(1)} \overline{G(Q, v)}\G(v) ⟶^{(2)} \overline{Z(Q)^{+}}\Z(A_{f})
≃↓  ≃↓  ≃↓
CM  ⟶^{RED_{v}}  X (v)  ⟶^{c_{v}}  Z

where (1) is induced by ϕ_{x}(v) : G(A_{f}) → G(v) (with x = RED_{v}([1, h])) while (2) is induced by the morphism

G′(A_{f})^{v} × F_{v}^{×} = G(v) ⟶ Z(A_{f}) = Z(A_{f})^{v} × F_{v}^{×}
(g^{v}, λ_{v}) ⟼ (nr(g^{v}), λ_{v})

For a compact open subgroup H of G(A_{f}), put H(v) = ϕ_{x}(v)(H) ⊂ G(v). We thus obtain a diagram

T (Q)\G(A_{f})/H ⟶ G(Q, v)\G(v)/H(v) ⟶ Z(Q)^{+}\Z(A_{f})/nr(H)
≃↓  ↓  ≃↓
CM_{H}  ⟶^{RED_{v}}  X_{H}(v)  ⟶^{c_{v}}  Z_{H}

in which the middle vertical arrow is surjective (and a bijection when H = R̂^{×} for some Eichler order R ⊂ B). Theorem 3.5 is therefore a consequence of a special case (S) of Theorem 2.9, corresponding to the situation where 𝔖 (in the notations of Theorem 2.9) equals {{v}, v ∈ 𝔖} (in the notations of Theorem 3.5).

3.3 Complements

3.3.1 On the parameter ϵ = ±1

Let us fix an isomorphism of R-algebras between B_{τ_{1}} and M_{2}(R), thus obtaining an isomorphism of group schemes over R between G_{1} and GL_{2}(R). Let X_{ϵ} be the G(R)-conjugacy class of the morphism h_{ϵ} : 𝕊 → G_{R} which sends z = x + iy to

h_{ϵ}(z) = [( \begin{matrix}x & y \\ −y & x \end{matrix} )^{ϵ} , 1, ⋯, 1] ∈ G_{1}(R) × ⋯ × G_{d}(R) ≃ G(R)

and let {M_{H}(ϵ)} be the corresponding collection of Shimura curves. We thus have a compatible system of isomorphisms ψ_{H}(ϵ) : M_{H}(ϵ) ×_{F} C → M_{H}^{alg}(ϵ), where M_{H}^{alg}(ϵ) is the algebraic curve over C whose underlying Riemann surface equals

M_{H}^{an}(ϵ) = G(Q)\ (G(A_{f})/H × X_{ϵ}) .

The topology, the differentiable structure and the real analytic structure of X_{ϵ} are induced from those of G(R) through the map g ↦ gh_{ϵ}g^{−1}. For h ∈ X_{ϵ} and z ∈ C^{×} = 𝕊(R), the map x ↦ h(z)xh(z)^{−1} fixes h and therefore induces an R-linear map T_{h}(adh(z)) on the tangent space T_{h}X_{ϵ} of X_{ϵ} at h. The almost complex structure on X_{ϵ} is characterized by the fact that T_{h}(adh(z)) acts by multiplication by z/z̄ on T_{h}X_{ϵ} for all h ∈ X_{ϵ} and z ∈ C^{×}. This almost complex structure is known to be integrable.

Remark 3.15 Most authors replace X_{ϵ} by C−R with G(R) acting through the

projection on the first component G_{1}(R) ≃ GL_{2}(R), by ( \begin{matrix}a & b \\ c & d \end{matrix} ) · λ = \frac{aλ+b}{cλ+d}

(λ ∈ C − R). This corresponds to ϵ = 1. Indeed, the map gh_{1}g^{−1} ↦ g · i yields a diffeomorphism between X_{1} and C − R and for z ∈ C^{×}, the derivative of λ ↦ gh_{1}(z)g^{−1} · λ at λ = g · i equals z/z̄. On the other hand, our main reference [1] on Shimura curves very explicitly uses ϵ = −1. While it seems clear that Carayol’s constructions could easily be transfered to the ϵ = 1 case, we will show below that the choice of ϵ is, in fact, irrelevant.

From the above discussion, we know that the G(R)-equivariant map Φ : X_{ϵ} → X_{−ϵ} which sends h to h^{−1} is an antiholomorphic diffeomorphism. For any compact open subgroup H of G(A_{f}), Φ therefore induces an antiholomorphic diffeomorphism between M_{H}^{an}(ϵ) and M_{H}^{an}(−ϵ) and an antilinear isomorphism between M_{H}^{alg}(ϵ) and M_{H}^{alg}(−ϵ), namely an isomorphism of schemes Φ : M_{H}^{alg}(ϵ) → M_{H}^{alg}(−ϵ) such that the diagram

M_{H}^{alg}(ϵ) ⟶^{Φ} M_{H}^{alg}(−ϵ)
↓  ↓
Spec(C) ⟶^{Spec(τ)} Spec(C)

is commutative (τ=complex conjugation). For any scheme X over Spec(C), we denote by τX → Spec(C) the pull-back of X → Spec(C) through Spec(τ) : Spec(C) → Spec(C). The above diagram thus yields an isomorphism of complex curves between M_{H}^{alg}(ϵ) and τM_{H}^{alg}(−ϵ) which together with ψ_{H}(ϵ) and ψ_{H}(−ϵ) induces an isomorphism

Φ′ : M_{H}(ϵ) ×_{F} C → M_{H}(−ϵ) ×_{F} C ≃ τ(M_{H}(−ϵ) ×_{F} C)

(recall that F is embedded in C through τ_{1} : F ↪ R). In other words, M_{H}(−ϵ) is a twist of M_{H}(ϵ). We shall now determine this twist. For σ ∈ Aut(C/F), let ρ(σ) be the F -automorphism of M_{H}(ϵ) such that ρ(σ) · [g, h] = [gλ, h] for g ∈ G(A_{f}) and h ∈ X_{ϵ}, where λ is any element of Z(A_{f}) (the center of G(A_{f})) such that rec_{F} (λ) = σ in Gal_{F}^{ab}. One easily checks that σ ↦ ρ(σ) is a well-defined group homomorphism ρ : Aut(C/F) → Aut_{F} (M_{H}(ϵ)) which factors through Gal(F′_{H}/F) where F′_{H} is the abelian extension of F corresponding to the subgroup F^{×} · (Z(A_{f}) ∩ H) of Z(A_{f}) = F̂^{×}.

Lemma 3.16 Φ′ realizes M_{H}(−ϵ) as the twist of M_{H}(ϵ) by ρ^{−ϵ}.

Proof. On the level of complex points, Φ′ is the composite of Φ with the action of complex conjugation. The latter is described by a conjecture of Langlands [10], proven in [13]. We obtain: for x = [g, h] ∈ M_{H}^{an}(ϵ) (with g ∈ G(A_{f}) and h ∈ X_{ϵ}), Φ′(x) = [g, h̄^{−1}] ∈ M_{H}^{an}(−ϵ) where h̄ : 𝕊 → G_{R} maps z to h(z̄). Note that h ↦ h̄ is indeed an involution of X_{ϵ} since

h̄_{ϵ}(x + iy) = [( \begin{matrix}x & −y \\ y & x \end{matrix} )^{ϵ} , 1, ⋯, 1] = ωh_{ϵ}(x + iy)ω^{−1}

where ω = [( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , 1, ⋯, 1] ∈ G(R).

If h is a special point of X_{ϵ}, h̄^{−1} is a special point of X_{−ϵ}. More precisely, suppose that h : 𝕊 → G_{R} factors through T′_{R} for some maximal Q-rational subtorus T′ ⊂ G. Let μ_{h} : 𝔾_{m,C} → T′_{C} be the induced cocharacter (μ_{h}(z) = h(z, 1)), let E_{h} ⊂ C be the field of definition of μ_{h} (so that F ⊂ E_{h}) and put rec_{h} = Norm_{E_{h}/Q} ∘ Res_{E_{h}/Q}(μ_{h}) :

rec_{h} : Res_{E_{h}/Q}(𝔾_{m,E_{h}}) → Res_{E_{h}/Q}(T′_{E_{h}}) → T′.

Let also μ_{0} : 𝔾_{m,C} → Z_{C} ⊂ T′_{C} be the cocharacter defined by

μ_{0}(z) = [( \begin{matrix}z \\ & z \end{matrix} ) , 1, ⋯, 1] ∈ Z(R) ⊂ G(R) ≃ G_{1}(R) × ⋯ × G_{d}(R).

Then μ_{0} is defined over F and

Z = Res_{F/Q}(𝔾_{m,F}) ⟶^{Res_{F/Q}(μ_{0})} Res_{F/Q}(Z_{F}) ⟶^{Norm_{F/Q}} Z

is the identity map. Since μ_{h} · μh̄ = μ_{0}^{ϵ}, μh̄ is also defined over E_{h} and

rec_{h} · rech̄ = Norm_{E_{h}/F}^{ϵ} : Res_{E_{h}/Q}(𝔾_{m,E_{h}}) → Res_{F/Q}(𝔾_{m,F}) = Z ⊂ T′.

It follows that (1) for g ∈ G(A_{f}), both x = [g, h] and Φ′(x) = [g, h̄^{−1}] are defined over the maximal abelian extension E_{h}^{ab} ⊂ C of E_{h}; (2) for λ ∈ Ê_{h}^{×} = Res_{E_{h}/Q}(A_{f}) and σ = rec_{E_{h}}(λ) ∈ Gal_{E_{h}}^{ab},

Φ′(ρ(σ)^{−ϵ}(σ · x)) = [rec_{h}(λ)gNorm_{E_{h}/F}^{−ϵ} (λ), h̄^{−1}]
= [rec_{h}(λ)Norm_{E_{h}/F}^{−ϵ} (λ)g, h̄^{−1}]
= [rech̄^{−1}(λ)g, h̄^{−1}]
= σ · Φ′(x).

Our claim now easily follows from the uniqueness of canonical models.

As a scheme over F, the twist M_{H}(ϵ)′ of M_{H}(ϵ) by ρ^{−ϵ} may be constructed as the quotient of M_{H}(ϵ) ×_{Spec(F)} Spec(F′_{H}) by the (right) action of Gal(F′_{H}/F) which maps σ to the F -automorphism α(σ) = (ρ(σ)^{ϵ}, Spec(σ)) of M_{H}(ϵ) ×_{F} Spec(F′_{H}).

Lemma 3.17 Suppose that H = H̄ where g ↦ ḡ is the anticommutative involution of G(A_{f}) which is induced by the canonical involution of B. Then M_{H}(ϵ)′ is isomorphic to M_{H}(ϵ). In particular, M_{H}(ϵ) ≃ M_{H}(−ϵ).

Proof. We shall construct an involution θ of M_{H}(ϵ) with the property that for all σ ∈ Γ_{H} =^{def} Gal(F′_{H}/F),

θ ∘ α(σ) = (1 × Spec(σ)) ∘ θ on M_{H}(ϵ) ×_{F} Spec(F′_{H}).

Such a θ induces an F -isomorphism between M_{H}(ϵ)′ and M_{H}(ϵ). Recall that M_{H}(ϵ) = Spec (Γ(O_{M_{H}(ϵ)}, M_{H}(ϵ))) is non-canonically isomorphic to Spec(F_{H}) where F_{H} is the abelian extension of F cut out by F_{>0}^{×} · nr(H) ⊂ F̂^{×}. Since (F^{×}·H ∩F̂^{×})^{2} ⊂ F_{>0}^{×}·nr(H), there is a well defined group homomorphism κ : Γ_{H} → Gal(F_{H}/F) given by κ(rec_{F} (λ)) = rec_{F} (λ^{2}) for λ ∈ F̂^{×}. It follows from the discussion after Lemma 3.12 that for σ ∈ Γ_{H} and x ∈ M_{H}(ϵ)(C),

c(ρ(σ)(x)) = κ(σ)^{ϵ} · c(x) in M_{H}(ϵ)(C) = M_{H}(ϵ)(F_{H}).

On the other hand, F′_{H} is a subfield of F_{H} whenever F_{>0}^{×} ·nr(H) ⊂ F^{×} ·H ∩F̂^{×}. This is indeed the case when H̄ = H. In particular, we may choose an F - morphism M_{H}(ϵ) → Spec(F′_{H}), thus obtaining an F -morphism c′ : M_{H}(ϵ) → Spec(F′_{H}) such that

∀σ ∈ Γ_{H}, c′ ∘ ρ(σ) = Spec(σ^{2ϵ}) ∘ c′.

Let A be an F -algebra and let z = (x, y) be an A-valued point of the F -scheme M_{H}(ϵ) ×_{F} Spec(F′_{H}). Then c′(x) and y are A-valued points of Spec(F′_{H}). If Spec(A) is connected, there exists a unique element γ =^{def} γ(z) in Γ_{H} such that c′(x) = Spec(γ^{−ϵ}) ∘ y. This defines an F -morphism z ↦ γ(z) from M_{H}(ϵ) ×_{F} Spec(F′_{H}) to the constant F -scheme Γ_{H}. For z = (x, y) as above, we put θ(z) = (ρ(γ(z))(x), y). One easily checks that θ has the required properties.

When H = H̄, we thus obtain an F -isomorphism between M_{H}(ϵ) and M_{H}(−ϵ). On the level of complex points, such an isomorphism is given by

[g, h] ∈ M_{H}^{an}(ϵ) ↦ [ḡ^{−1}, h̄^{−1}] ∈ M_{H}^{an}(−ϵ).

Note that the condition H = H̄ defines a cofinal subset of the set of all compact open subgroups H of G(A_{f}). Also, H = H̄ when H = R̂^{×} for some Eichler order R in B, in which case F_{H} and F′_{H} are respectively the Hilbert class field and the narrow Hilbert class field of F.

3.3.2 P -rational elements of Gal_{K}^{ab}.

It may seem rather surprising that the bizarre subgroup Gal_{K}^{P −rat} of P -rational elements in Gal_{K}^{ab} should play any role in the theory of CM points. For instance, Gal_{K}^{P −rat} is not a closed subgroup of Gal_{K}^{ab}, although it contains the closed subgroup

Gal(K^{ab}/K[P^{∞}]) = rec_{K} {λ ∈ Ô_{K}^{×}, λ_{P} ∈ O_{F_{P}}^{×} } .

The Galois group Gal(K[P^{∞}]/K) is topologically isomorphic to G_{0} × Z_{p}^{[F_{P}:Q_{p}]} where p is the residue characteristic of P and G_{0} is a finite group, the torsion subgroup of Gal(K[P^{∞}]/K). The subfield of K[P^{∞}] which is fixed by G_{0} is the composite of all Z_{p}-extensions of K which are unramified outside P and Galois and dihedral over F. The image of Gal_{K}^{P −rat} in Gal(K[P^{∞}]/K) is a dense but countable subgroup which is generated by the Frobeniuses of those primes of K which are not above P (the intersection of this subgroup with G_{0} plays a key role in [3], where it is denoted by G_{1}). In particular, Gal_{K}^{P −rat} is a dense but negligible (i.e. measurable with trivial measure) subgroup of Gal_{K}^{ab}. The map σ = rec_{K}(λ) ↦ λ_{P} yields a bijection between Gal_{K}^{ab}/Gal_{K}^{P −rat} and K_{P}^{×}/K^{×}F_{P}^{×}. However, the appearance of rational elements is perhaps less surprising when one recalls that the present work originated in the study of elliptic curves over anticyclotomic towers of number fields, since the distinction between suitably defined rational and irrational elements of Galois groups occurs quite frequently in the context of Iwasawa theory. For instance, the celebrated theorems of Ferrero and Washington on the growth of class numbers in Z_{p} extensions of abelian fields rely crucially on the fact that nontrivial roots of unity are irrational. Another example of this occurs in recent work of Hida [7], [8] on anticyclotomic families of Hecke characters, where the key observation is the irrationality of certain Galois actions on Serre-Tate deformation spaces. In fact, the irrationality arguments given by Ferrero and Washington were the original motivation for the introduction in [18] of rational and irrational elements to the study of CM points. In this section, we shall provide some further evidence for the relevance of P -rational elements by relating them to the André-Oort conjecture:

Proposition 3.18 For σ ∈ Gal_{K}^{ab} and x ∈ CM_{H}, put δ(x) = (x, σx) ∈ M_{H}(C)^{2}. The following conditions are equivalent.

1. σ is a P -rational element.

2. For any collection E ⊂ CM_{H} of P -isogeneous CM points, the Zariski closure of δ(E) in (M_{H} ×_{F} C)^{2} has dimension ≤ 1.

3. For some collection E ⊂ CM_{H} of P -isogeneous CM points, the Zariski closure of δ(E) has dimension 1.

For the proof of this proposition, we may and do assume that H = R̂^{×} for some maximal order R ⊂ B. For any CM point

x = [g] ∈ CM_{H} = T (Q)\G(A_{f})/H, the stabilizer of x in Gal_{K}^{ab} then equals rec_{K}(K^{×}O(x)^{×}) where O(x) = K ∩ gHg^{−1} is an O_{F} -order in B. Moreover, there exists a unique integral ideal C ⊂ O_{F} such that O(x) = O_{K,C} =^{def} O_{F} + CO_{K}. We refer to C =^{def} C(x) as the conductor of x and denote by ℓ_{P} (x) ≥ 0 the exponent of P in C(x), so that C(x) = C_{0}(x)P^{ℓ_{P} (x)} for some integral ideal C_{0}(x) which is relatively prime to P. By construction, x ↦ C(x) is constant on Gal_{K}^{ab}-orbits while x ↦ C_{0}(x) is constant on P -isogeny classes. It follows from [20, pp. 42-44] that the fibers of C are finite. In particular:

Lemma 3.19 The function x ↦ ℓ_{P} (x) has finite fibers on any P -isogeny class.

This function is related to the usual distance d on the Bruhat-Tits tree

T = F_{P}^{×}\B_{P}^{×}/R_{P}^{×} ≃ F_{P}^{×}\GL_{2}(F_{P})/GL_{2}(O_{F_{P}}).

Indeed, the group K_{P}^{×} acts on the left on T by isometries, and for v = [b] ∈ T (with b ∈ B_{P}^{×}), the stabilizer of v in K_{P}^{×} equals F_{P}^{×}O(v)^{×}where O(v) = K_{P} ∩ bR_{P} b^{−1} is an O_{F_{P}} -order in K_{P}. Just as above, there exists a unique integer n =^{def} n(v) ∈ N such that O(v) = O_{n} with O_{n} =^{def} O_{F_{P}} + P^{n}O_{K_{P}} (O_{n} is the completion of O_{K,C_{0}P^{n}} at P for any integral ideal C_{0} ⊂ O_{F} which is relatively prime to P). It is clear that for a CM point x = [g] ∈ CM_{H}, ℓ_{P} (x) = n(v) where v = [g_{P}] (g_{P} ∈ B_{P}^{×} is the P -component of g ∈ G(A_{f})). It is well-known that

• The map v ↦ n(v) yields a bijection between K_{P}^{×}\T and N.

• The subset T_{0} = {v ∈ T ; n(v) = 0} of T consists of a vertex, two adjacent vertices or the set of vertices on a line in T , depending upon whether P is inert, ramifies or splits in K.

• For any v ∈ T , n(v) is also the distance between v and T_{0}.

In particular, suppose that (v_{n}, v_{n−1}, ⋯, v_{0}) and (w_{m}, w_{m−1}, ⋯, w_{0}) are geodesics in T from v = v_{n} and w = w_{m} to T_{0}. Then n(v_{i}) = i for 0 ≤ i ≤ n and n(w_{j}) = j for 0 ≤ j ≤ m. The geodesic γ between v and w may then be computed as follows:

• if v_{0} ≠ w_{0}, γ = (v_{n}, v_{n−1}, ⋯, v_{0}, u_{1}, ⋯, u_{r−1}, w_{0}, w_{1}, ⋯, w_{m}) where (v_{0}, u_{1}, ⋯, u_{r−1}, w_{0}) is the geodesic between v_{0} and w_{0} inside the connected subtree T_{0} of T .

• if v_{0} = w_{0}, γ = (v_{n}, v_{n−1}, ⋯, v_{c} = w_{c}, w_{c+1}, ⋯, w_{m}) where c is the largest integer ≤ n, m such that v_{c} = w_{c}.

In the special case where w = λv for some λ ∈ K_{P}^{×}, n = m = n(v) and w_{i} = λv_{i} for 0 ≤ i ≤ n. If moreover d(v, λv) ≤ 2n, it thus must be that v_{0} = w_{0}. With c as above, d(v, w) = 2(n − c) and v_{c} = w_{c} = λv_{c}, so that λ belongs to F_{P}^{×}O_{c}^{×}. We have obtained:

Lemma 3.20 Suppose that d(v, λv) ≤ 2n(v) (with v ∈ T and λ ∈ K_{P}^{×}). Then d(v, λv) = 2k for some k ∈ {0, ⋯, n(v)} and λ belongs to F_{P}^{×}O_{n(v)−k}^{×}.

We may now sketch the proof of Proposition 3.18. Of course, (2) implies (3).

Proof that (1) implies (2)

We have to show that for any P -isogeny class H ⊂ CM_{H}, δ(H) is contained in a one dimensional subscheme of (M_{H} ×_{F} C)^{2} if σ = rec_{K}(λ) for some λ ∈ T (A_{f}) = K̂^{×} whose P -component λ_{P} belongs to K^{×}F_{P}^{×}. Choose g_{0} ∈ G(A_{f}) such that

H = T (Q)\T (Q)B_{P}^{×}g_{0}H/H inside CM_{H} = T (Q)\G(A_{f})/H.

Using Lemma 3.11, we find that

δ(H) = {([bg_{0}], [λ^{ϵ}bg_{0}]) ; b ∈ B_{P}^{×}}
= {([bg_{0}], [bg_{0}γ]) ; b ∈ B_{P}^{×}}

where γ = g_{0}^{−1}λ^{ϵ}g_{0}. Indeed, bg_{0}γ = bλ^{ϵ}g_{0} = λ^{ϵ}bg_{0} for any b ∈ B_{P}^{×} as λ_{P}^{ϵ} belongs to F_{P}^{×}. In particular, δ(H) is contained in the 1-dimensional image of the (algebraic!) morphism M_{H∩γHγ^{−1}} → M_{H}^{2} which sends [g, h] to ([g, h], [gγ, h]) (g ∈ G(A_{f}), h ∈ X).

Proof that (3) implies (1)

Write σ = rec_{K}(λ) with λ ∈ K̂^{×}. Suppose that the Zariski closure of δ(E) in (M_{H} ×_{F} C)^{2} has dimension 1 for some (infinite) collection E of P -isogeneous CM points. We have to show that the P -component λ_{P} ∈ K_{P}^{×} of λ belongs to F_{P}^{×}K^{×}. By a proven case of the André-Oort conjecture [6, Theorem 1.2] there exists an infinite subset E′ ⊂ E and some element γ ∈ G(A_{f}) such that δ(E′) is contained in the image of a morphism M_{H∩γHγ^{−1}} → M_{H}^{2} as above. Fix x = [g_{0}] ∈ E′ and let {g_{1}, ⋯, g_{s}} ⊂ H be a set of representatives for H/H ∩ γHγ^{−1}. For each x′ = [bg_{0}] ∈ E′ (with b ∈ B_{P}^{×}), we know that x = [bg_{0}g_{i}] for any i ∈ {1, ⋯, s} while

σ · x ∈ {[bg_{0}g_{1}γ], ⋯, [bg_{0}g_{s}γ]} .

Replacing g_{0} by g_{0}g_{i} for a suitable 1 ≤ i ≤ s and using lemmas 3.11 and 3.19, we obtain: there exists a sequence b_{n} ∈ B_{P}^{×} such that (a) φ(n) =^{def} ℓ_{P} ([b_{n}g_{0}]) goes to infinity with n and (b) [λ^{ϵ}b_{n}g_{0}] = [b_{n}g_{0}γ] for all n ≥ 0. By (b), there exists λ_{n} ∈ T (Q) = K^{×} and h_{n} ∈ H such that for all n ≥ 0,

λ_{n}λ^{ϵ}b_{n}g_{0} = b_{n}g_{0}γh_{n} in G(A_{f}).

Put v_{n} = [b_{n}g_{0,P}] ∈ T and μ_{n} = λ_{n}λ^{ϵ}. Since μ_{n,P} ·v_{n} = [b_{n}g_{0,P} γ_{P}], d(v_{n}, μ_{n,P} · v_{n}) = d_{0} does not depend upon n. Pick N ≥ 0 such that ∀n ≥ N, d_{0} ≤ 2φ(n). By Lemma 3.20, d_{0} = 2k and

∀n ≥ N : μ_{n,P} ∈ F_{P}^{×}O_{φ(n)−k}^{×}.  (8)

On the other hand, μ_{n}μ_{N}^{−1} = b_{n}g_{0}γh_{n}g_{0}^{−1}b_{n}^{−1}b_{N}g_{0}h_{N}^{−1}γ^{−1}g_{0}^{−1}b_{N}^{−1}. Away from P, this equation simplifies to (μ_{n}μ_{N}^{−1})^{P} = (g_{0}γh_{n}h_{N}^{−1}γ^{−1}g_{0}^{−1})^{P}, so that

(μ_{n}μ_{N}^{−1})_{Q} ∈ K_{Q}^{×} ∩ (g_{0}γ)_{Q}R_{Q}^{×}(g_{0}γ)_{Q}^{−1} ⊂ O_{K_{Q}}^{×}  (9)

for all Q ≠ P. Let U_{F} ⊂ U_{K} be the groups of all elements z ∈ F^{×} (resp. z ∈ K^{×}) which are units away from P. Since K is a totally imaginary quadratic extension of F, rank_{Z}U_{K} = rank_{Z}U_{F} if P does not split in K and rank_{Z}U_{K} = rank_{Z}U_{F} + 1 otherwise. Let U′_{K} be the subgroup of U_{K} defined by U′_{K} = U_{K} ∩ F_{P}^{×}O_{K_{P}}^{×}. Then U_{F} ⊂ U′_{K} ⊂ U_{K}, and [U_{F} : U′_{K}] is finite. Let R ⊂ U′_{K} be a set of representatives for U′_{K}/U_{F}. By (9), μ_{n}μ_{N}^{−1} = λ_{n}λ_{N}^{−1} belongs to U_{K} for all n ≥ 0. Then (8) shows that μ_{n}μ_{N}^{−1} belongs to U′_{K} for all n ≥ N. For such n’s, we may thus write

λ_{n} = λ_{N}r(n)u(n) with r(n) ∈ R and u(n) ∈ U_{F} .

Using (8) again, we find that λ_{N}r(n)λ_{P}^{ϵ} belongs to F_{P}^{×}O_{φ(n)−k}^{×} for all n ≥ N. Choosing a subsequence on which r(n) = r is constant, we finally obtain:

λ_{N}rλ_{P}^{ϵ} ∈ F_{P}^{×} = ∩_{n≥0}F_{P}^{×}O_{n}^{×}.

Since λ_{N}r belongs to K^{×}, λ_{P} indeed belongs to K^{×}F_{P}^{×}.

Remark 3.21 More generally, it may be shown that for any infinite collection E ⊂ CM_{H} of P -isogeneous CM points and any finite subset R = {σ_{1}, ⋯, σ_{r}} of Gal_{K}^{ab}, the Zariski closure of {(σ_{1}x, ⋯, σ_{r}x); x ∈ E} in V = (M_{H} ×_{F} C)^{r} contains a connected component of V if and only if the σ_{i}’s are pairwise distinct modulo Gal_{K}^{P −rat} (Hint: use section 7.3 of [6] and a variant of Proposition 2.1 of [5]).

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