Contents

This paper provides a proof of a technical result (Corol- lary 2.10 of Theorem 2.9) which is an essential ingredient in our proof of Mazur's conjecture over totally real number fields (3).


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 ǫ = ±1, while section 3.3.2provides 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 ̟ 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.

Quaternion algebras.
Let S be a finite set of finite places of F such that S1 ∀v ∈ S, B is unramified at v.
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 ).

Algebraic groups
We put 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 × .

Adelic groups
Let A f denote the finite adeles of Q.We shall consider the following locally compact, totally discontinuous groups: 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: 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.
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 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) ) are the maximal order of K v .For such v's we may choose the isomorphism For the remaining v's not in S, we only require the second condition: ∈S satisfies (1) and (2).

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 where 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 are respectively induced by nr : 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.

Galois actions
The profinite commutative group 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 : K , we obtain a continuous, G(A f )-equivariant and faithful action of Gal ab K on CM.1 Similarly, Artin's reciprocity map rec F : F allows one to view Z as a principal homogeneous Gal ab F -space.From this point of view, c : CM → Z is a Gal ab K -equivariant map in the sense that for x ∈ CM and

Further objects
For technical purposes, we will also need to consider the following objects: The composite map c S • q S : X S → Z is induced by nr 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).

Measures
The group G 1 (S) = ker(nr ′ S ) (resp.G 1 S (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.Proposition 2.5 The above actions are transitive and for each z ∈ Z, (1) there exists a unique G 1 S (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 −1 S (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 −1 S (z) maps one to the other: this is why we use the same notation µ z for both measures.Similarly, for any

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: Note that X H (S) and Z H are finite spaces, being discrete and compact.We have ∈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) = 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 −1 S (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 ab K and consider the sequence where • 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 The Galois group Gal ab K acts diagonally on Z(S, R) = S,σ Z (through its quotient Gal ab F ) and the composite map C • Red : CM → Z(S, R) is Gal ab Kequivariant.For x ∈ CM, we shall frequently write x = C • Red(x).Explicitly:

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 ǫ > 0, there exists a compact subset C(ǫ) of CM such that |f (x) − a| ≤ ǫ for all x ∈ H \ C(ǫ).
Remark 2.7 This definition can be somewhat clarified if we introduce the Alexandroff "one point" compactification CM = CM ∪ {∞} of the locally compact space CM.It is easy to see that the point ∞ ∈ CM 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 ab K is P -rational if σ = rec K (λ) for some λ ∈ K × whose P -component λ P belongs to the subgroup K × • F × P of K × P .We denote by Gal P −rat K ⊂ Gal ab K the subgroup of all P -rational elements.
In the above definition, rec K : K × ։ Gal ab K 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 ab K consists of elements which are pairwise distinct modulo Gal P −rat K .Let H ⊂ CM be a P -isogeny class and let G be a compact open subgroup of Gal ab K with Haar measure dg.Then for every continuous function f : f dµ g•x goes to 0 as x goes to infinity in H.

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 where 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 H be the image of H in CM H .

Equidistribution
When H = R × for some Eichler order R in B, we can furthermore specify the asymptotic behavior (as where s is a fixed point in X H (S, R).To state our result, we first need to define a few constants.
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(B S ) = Q∈Ram f (BS ) ( Q − 1), • Ω(N S ) = N S • Q|NS ( 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 ǫ > 0, there exists a finite set C(ǫ) ⊂ H such that for all s ∈ X H (S, R) and x ∈ H \ C(ǫ), 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.

Notations
For a continuous function f : X (S, R) → C and x ∈ CM, we put Then the theorem says that for all ǫ > 0, there exists a compact subset C(ǫ) ⊂ CM such that, We claim that the functions x → A(f, x) and x → B(f, x) are well-defined.This is clear for ) 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 If the theorem were known for f ′ , we could find a compact subset For a nonzero nilpotent element N ∈ B P , the formula u(t) = 1 + tN defines a group isomorphism u : 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 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 ab K , 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 κn , we obtain: 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( Fix ǫ > 0 and choose N ≥ 0 such that Reduction of Corollaries 2.10 and 2.11 For x ∈ CM, we easily obtain: The main theorem asserts that for all ǫ > 0, there exists a compact subset C(ǫ) of CM such that for all x ∈ H \ C(ǫ) (where H and C(ǫ) are the images of H and C(ǫ) in CM H ), 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.
In particular, I(s) > 0 and if H = R × with R as above, Thus we obtain Corollaries 2.10 and 2.11.

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.

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.
The fiber of c S • q S above z is the image of and the stabilizer of We break this up into a series of steps.
Step 1: . This is easy.See for instance the proof of Corollary 3.10.

A computation.
Any Haar measure z 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 It follows that these measures assign the same volume λ to each fiber of c S • q S , and µ 1 z = 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 where The x i,j 's then form a set of representatives for X S /H S and since (x i,j H S ) ni j=1 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 = G S (Q)g and put k=1 be a set of representatives for the left hand side of (2), with 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 where k(g, H S ) and q(g, H S ) are respectively the kernel and cokernel of When H S = R × S for some Eichler order R S in B S , the following simplifications occur: • The following commutative diagram with exact rows .
Combining this, (1), ( 3) and [20, Corollaire 2.3 p. 142], we obtain: 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: ) and q(g, H S ) are as above, with Ω(F ), Ω(B S ) and Ω(N S ) as in section 2.2.4.

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 This action is transitive and the stabilizer of Proof.
S,P acts transitively on (c S •q S ) −1 (z)/H.An easy computation shows that Γ S (x) = g −1 P Γ S g P with Γ S as above.
(2) surjective by the strong approximation theorem.In particular, U ∩G 1 S (Q) is a discrete and cocompact subgroup of U .Since U = gHg −1 × B 1 S,P (with gHg −1 compact), the projection Γ S of U ∩ G 1 S (Q) to B 1 S,P is indeed discrete and cocompact in B 1 S,P .Finally, since the compact open subgroups of G 1 S (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 Lemma 2.20 This action is transitive and the stabilizer of x ∈ c −1 S (z)/H is a discrete and cocompact subgroup Γ S (x) of B 1 S,P .For x = G(S, Q)gH with g in G(S), Γ S (x) = g −1 P Γ S g P where Γ S = Γ S (gHg −1 ) is the projection to B 1 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 In particular, the map b → x • b induces a B 1 S,P -equivariant homeomorphism S,P ≃ SL 2 (F P ), there exists a unique B 1 S,P -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 −1 S (z) → c −1 S (z)/H: the latter is indeed yet another B 1 S,P -invariant Borel probability measure.
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 (x).Then for any continuous function f : Proposition 2.22 Under the assumptions of Theorem 2.9, Red(γ for almost all γ ∈ Gal ab K .

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 (x)/H(S, R).For our purposes, it will be sufficient to assume that f is right 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 (x) induces a right action of S,σ B 1 S,P on C −1 (x)/H(S, R) which together with the isomorphism S,σ φ S,P : ( yields a right action of ( where Γ(x, H) is the stabilizer of Red(x) • H(S, R) in (B 1 P ) S×R .Note that Γ(x, H) equals S,σ Γ S,σ (x, H) where for each S ∈ S and σ ∈ R, is a discrete and cocompact subgroup of B 1 P ≃ SL 2 (F P ).Under this equivariant homeomorphism, , where ∆ : B 1 P → (B 1 P ) S×R is the diagonal map; • the image of µ x on C −1 (x)/H(S, R) corresponds to the (unique) (B 1 P ) S×R -invariant Borel probability measure on Γ(x, H)\(B 1 P ) 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.

Reduction of Proposition 2.22
We keep the above notations and choose: For S ∈ S and σ ∈ R, we thus obtain (using Lemma 2.20): where λ σ,P and g P are the P -components of λ σ and g while Γ 0 S,σ (x, H) is the inverse image (through φ S,P : For a subgroup Γ of B • λ P λ σ,P g P where λ P is the P -component of λ.On the other hand, the stabilizer of 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 is the disjoint union of countably many cosets of Gal P −rat K in Gal ab K .
Proof.Fix (S, σ) = (S ′ , σ ′ ) in S × R. We have to show that (under the assumptions of Theorem 2.9) the image of 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 To see this, suppose that [Γ 0 , 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 −1 S,P (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 = 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 for w(t) = λ σ,P φ S (g P u(t)g −1 P )λ −1 σ,P .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) But now looking at the form of b(t) shows that w(t) = 1 and (λ σ,P λ −1 σ ′ ,P ) ∈ F × P K × , which contradicts the fact that σ ≡ σ ′ mod Gal P −rat K .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 Proof.Left to the reader.

K
and a compact open subgroup H ⊂ G(A f ).We choose an element x 0 ∈ H such that x 0 = T (Q) • g 0 for some g 0 ∈ G(A f ) whose P -component equals 1.Let K G P be the kernel of This is an open subgroup of finite index in K × P .Let N be a positive integer such that We denote by ) and put I = I 1 × {0, 1} × I 2 .For i = (λ, ǫ, r) ∈ I, we put We finally put κ = 1 + P N +1 O FP , a compact open subgroup of F × P .The following result gives the proof of Proposition 2.12. 1.For every x ∈ X, there exists • an L-invariant Borel probability measure µ on X supported on 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 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 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 , where ∆ : 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 c∆(G)c −1 ⊂ Σ.
Proof.This is Corollary 4 of Theorem 6 in [15] when F P = Q p (note that the centralizer of ∆(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: where Proof.This is a slight generalization of Proposition 3.10 of [2].According to the latter, there exists a partition Taking the derived group on both sides gives α ∆ Iα (G) = [Ω : Ω] ⊂ Ω.

Documenta Mathematica 10 (2005) 263-309
The equivalence relation ∼ on {1, • • • , r} which is defined by the above partition can easily be retrieved from x • ∆(U ) = x • Σ by the following rule: for 1 ≤ i, j ≤ r, i ∼ j if and only if the projection 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: Proof.The map ω → y • ω induces a continuous bijection θ : To prove the converse, it is sufficient to show that θ is an homeomorphism when i Γ i g i , and Γ α \G is compact if and only if g −1 i Γ i g i is commensurable with g −1 j 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.
Proposition 2.35 The following conditions are equivalent: 2. For all 1 ≤ i = j ≤ r, g −1 i Γ i g i and g −1 j 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 3 The case of Shimura curves 3.1 Shimura Curves

Definitions
We start by defining the Shimura curves.Let ) 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 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 τi ⊗ R A) × .Fix ǫ = ±1 and let X be the G(R)conjugacy class of the morphism from We have used an isomorphism of R-algebras B τ1 ≃ 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 ǫ, cf.section 3.

below). For every compact open subgroup
It is a smooth curve over F (the reflex field) whose underlying Riemann surface M H (C) equals M an H .

CM points
Among the models of M an H , 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 an H 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 an H 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 an H = 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 3.2.4below).

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 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 = 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 = lim ← − M H .

Reduction maps
Let us choose a place v of K ab above v, with ring of integers O(v) ⊂ K ab and residue field F(v), an algebraic closure of the residue field F(v) of v. Consider the specialization maps: 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 Let M ss H (v) be the set of supersingular points in M H (F(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 The connected part of the special fiber E 0 | RED v (x) therefore inherits a K v -module structure.Since End Fv (Σ 0 1 ) ≃ F v , this connected part can not be isomorphic to Σ 0 1 unless v splits in K.

Connected components
We now want to define yet another type of "reduction map".Recall from Shimura's theory that the natural map from M an H to its set of connected components π 0 (M an H ) 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 Documenta Mathematica 10 (2005) 263-309

CM points
From [4, Proposition 2.1.10]or [14,Theorem 5.27], 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 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 Moreover, τ 1 : K ֒→ C induces an isomorphism between K τ1 and C which allows us to identify T 1 and S.There are exactly two morphisms s and s : S → T R whose composite with ι R : T R ֒→ G R belongs to X.They are characterized by Finally, there exists an element b this holds more generally for any number field).Therefore, ) is a locally closed, hence closed subgroup of T (A f ).Our claim easily follows.

Connected components
Let G(R) + and Z(R) + be the identity components of G(R) and Z(R) and put It follows that Z H = π 0 (M an H ) ≃ G(Q) + \G(A f )/H.On the other hand, the reduced norm nr : B → F induces a surjective morphism nr : [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 H , we thus obtain: ) factors through the reduced norm and yields a bijection where Z(Q) + is the closure of Z(Q) + in Z(A f ).

Supersingular points
Proposition 3.9 (1) The right action of G(A f ) on X (v) def = lim ← − X H (v) is transitive and factors through the surjective group homomorphism 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 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.
Corollary 3.10 The map g → x • g factors through φ x (v) and induces a bijection where Proof.We have to show that [20, p. 139]).The map nr v : v is open and surjective with a compact kernel: it is therefore a closed map, and so is (1, nr and the other inclusion is trivial.

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 is induced by z ⊗ R a → (za, za) for z ∈ C and a in some C-algebra A).The isomorphism 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 = ǫ 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 sends z to z ǫ .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 ) = K × , the action of σ on CM ≃ T (Q)\G(A f ) is given by multiplication on the left by λ ǫ .

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 ) = F × , the action of σ on Z ≃ Z(Q) + \Z(A f ) is given by multiplication by λ ǫ .
In particular, the pro-étale F -scheme M def = 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) ǫ ), while M def

= 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 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) Remark 3. 14 The assumption nr(H v ) = O × v holds true when H = R × for some Eichler order R ⊂ B.

Conclusion
Putting lemmas 3.7, 3.8 and Corollary 3.10 together, we obtain a commutative diagram where ( 1) is induced by in which the middle vertical arrow is surjective (and a bijection when H = R × 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).

On the parameter ǫ = ±1
Let us fix an isomorphism of R-algebras between B τ1 and M 2 (R), thus obtaining an isomorphism of group schemes over R between G 1 and GL 2 (R).Let X ǫ be the G(R)-conjugacy class of the morphism h and let {M H (ǫ)} be the corresponding collection of Shimura curves.We thus have a compatible system of isomorphisms ψ H (ǫ) : The topology, the differentiable structure and the real analytic structure of X ǫ are induced from those of G(R) through the map g → gh ǫ g −1 .For h ∈ X ǫ 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 ǫ of X ǫ at h.The almost complex structure on X ǫ is characterized by the fact that T h (adh(z)) acts by multiplication by z/z on T h X ǫ for all h ∈ X ǫ and z ∈ C × .This almost complex structure is known to be integrable.
This corresponds to ǫ = 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/z.On the other hand, our main reference [1] on Shimura curves very explicitly uses ǫ = −1.While it seems clear that Carayol's constructions could easily be transfered to the ǫ = 1 case, we will show below that the choice of ǫ is, in fact, irrelevant.
From the above discussion, we know that the G(R)-equivariant map Φ : X ǫ → X −ǫ 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 an H (ǫ) and M an H (−ǫ) and an antilinear isomorphism between M alg H (ǫ) and M alg H (−ǫ), namely an isomorphism of schemes Φ : is commutative (τ =complex conjugation).
For any scheme X over Spec(C), we denote by τ X → 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 alg H (ǫ) and τ M alg H (−ǫ) which together with ψ H (ǫ) and ψ H (−ǫ) induces an isomorphism 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 Then µ 0 is defined over F and is the identity map.Since µ h • µh = µ ǫ 0 , µh is also defined over E h and Our claim now easily follows from the uniqueness of canonical models.
As a scheme over F , the twist M H (ǫ) ′ of M H (ǫ) by ρ −ǫ may be constructed as the quotient of M H (ǫ) × Spec(F ) Spec(F ′ H ) by the (right) action of Gal(F ′ H /F ) which maps σ to the F -automorphism α Documenta Mathematica 10 (2005) 263-309 Lemma 3.17 Suppose that H = H where g → ḡ is the anticommutative involution of G(A f ) which is induced by the canonical involution of B. Then M H (ǫ) ′ is isomorphic to M H (ǫ). In particular, M H (ǫ) ≃ M H (−ǫ).
Let A be an F -algebra and let z = (x, y) be an A-valued point of the F -scheme M H (ǫ) × 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 −ǫ ) • y.This defines an F -morphism z → γ(z) from M H (ǫ) × 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 = H, we thus obtain an F -isomorphism between M H (ǫ) and M H (−ǫ). On the level of complex points, such an isomorphism is given by Note that the condition H = H defines a cofinal subset of the set of all compact open subgroups H of G(A f ).Also, H = H when H = R × 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 .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 P −rat K 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 P −rat K is a dense but negligible (i.e.measurable with trivial measure) subgroup of Gal ab K .The map σ = rec K (λ) → λ P yields a bijection between Gal ab K /Gal P −rat K 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 ab K 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 = R × for some maximal order R ⊂ B. For any CM point • 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: 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 .
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: by the norm theorem [20, Théorème 4.1 p. 80]).
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 1 S,P acts on c −1 S (z)/H and we have the following lemma.

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 = ̟ −n P κ so that λ(κ n ) → ∞ as n → ∞.For γ ∈ Gal ab K and t ∈ F P ,

Lemma 2 .
be the set of all O FP -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 FP -order in K P and therefore equals O n = O FP + P n O KP 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 ̟ 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.

Remark 3 .
15 Most authors replace X ǫ by C−R with G(R) acting through the projection on the first component G 1

3. 3 . 2 P
-rational elements of Gal ab K .It may seem rather surprising that the bizarre subgroup Gal P −rat K of P -rational elements in Gal ab K should play any role in the theory of CM points.For instance, Documenta Mathematica 10 (2005) 263-309Gal P −rat K is not a closed subgroup of Gal ab K , although it contains the closed subgroup Gal(K ab /K[P ∞ ]) = rec K λ ∈ O × K , λ P ∈ O × FP .The Galois group Gal(K[P ∞ ]/K) is topologically isomorphic to G 0 × Z [FP :Qp] p

x
= [g] ∈ CM H = T (Q)\G(A f )/H, Documenta Mathematica 10 (2005) 263-309 the stabilizer of x in Gal ab K 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 ℓP (x) for some integral ideal C 0 (x) which is relatively prime to P .By construction, x → C(x) is constant on Gal ab K -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 treeT = F × P \B × P /R × P ≃ F × P \GL 2 (F P )/GL 2 (O FP ).Indeed, the group K × P acts on the left on T by isometries, and forv = [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 FP -order in K P .Just as above, there exists a unique integern def = n(v) ∈ N such that O(v) = O n with O n def = O FP + P n O KP (O n is the completion of O K,C0P 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.
starting with any isomorphism φ ?v : R v → R S,v , we obtain two optimal embeddings φ ?v • ι and ι S of O Kv 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 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 H S where H(S) = φ S (H) ⊂ G(S) and H S = π −1S (H(S)) ⊂ G S .The Galois group Gal ab K still acts continuously on the (now discrete) spaces CM H and Z H , and c is a Gal ab K -equivariant map.2.2 Main theorems: the statements2.2.1 Simultaneous reduction mapsLet 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 1 P , we denote by [Γ] the commensurability class of Γ in B 1 P , namely the set of all subgroups of B 1 P 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 0 S ] of Γ 0 S,σ (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