A Characterisation of Special Subvarieties

A CHARACTERISATION OF SPECIAL SUBVARIETIES.

EMMANUEL ULLMO AND ANDREI YAFAEV

1. Introduction. The aim of this note is to obtain a new characterisation of special subvarieties of Shimura varieties. For generalities on Shimura varieties we refer to [3], [4] or [12]. Let (G, X) be a Shimura datum and X+ a connected component of X. We let K be a compact open subgroup of G(Af ) and Γ := + G(Q)+ ∩ K where G(Q)+ denotes the stabiliser in G(Q) of X . We let + S := Γ\X , a connected component of ShK (G, X). A special subvariety of S is a subvariety of Hodge type in the sense of [15]. In section 2 we give a description of slightly more general notion of weakly special subvarieties in terms of sub-Shimura data of (G, X). In [15] Moonen proves that a subvariety of S is weakly special if and only if it is a totally geodesic submanifold of S. A special point is a special subvariety of dimension zero and a weakly special subvariety containing a special point is special. Special subvarieties are interesting for many reasons, one of which is the following conjecture Conjecture 1.1 (André-Oort). Let Z be an irreducible subvariety of ShK (G, X) containing a Zariski-dense set of special points. Then Z is special. This conjecture has recently been proved under the assumption of the Generalised Riemann Hypothsis for CM fields (see [22] and [11]). Part of the strategy consisted in establishing a geometric characterisation of special subvarieties of Shimura varieties. This criterion says roughly that subvarieties contained in their image by certain Hecke correspondences are special. Very recently Pila came up with a new and very promsing strategy for attacking the André-Oortconjecture unconditionally (see [18] and [19]). A step in this strategy consists in establishing a criterion for a

Date: July 6, 2010. 1 2 EMMANUEL ULLMO AND ANDREI YAFAEV subvariety of S to contain special subvarieties involving certain algebraicity properties of their preimages in X+. Let us explain this in more detail. The Borel embedding of X+ into its compact dual X+∨ (see section 3) gives meaning to the notion of algebraicity of subsets of X+. Namely, a subset Y of X+ is algebraic if there exists an algebraic subset Z of X∨ such that Y = Z ∩ X. Let π : X+ −→ S be the natural projection. Let Z be an irreducible subvariety of S and let Ze := π−1(Z). Let Zealg be the union of maximal algebraic subsets (in the sense explained above) of Ze. Pila conjectures that Zealg consists exactly of preimages of weakly special subvarieties of Z. He establishes this conjecture in the case where the Shimura variety is a product of modular curves. The general case, however seems to be very hard. In this paper we establish the following criterion for a subvariety to be special which implies Pila’s conjecture in the case where Z is a curve. Theorem 1.2. An irreducible subvariety Z of S is weakly special if and only if some (equivalently any) analytic component of π−1Z is algebraic in the sense explained above. Suppose that Z is a curve. Let Y be an analytic component of Ze := π−1(Z). Suppose that Y is algebraic, hence Y is a component of Zalg. As dim(Y ) = 1, we have π(Y ) = Z and our theorem shows that Z is special, as predicted by Pila’s conjecture. The main ingredient of the proof is a well-known statement about the image of the monodromy representation associated to the smooth locus of Z due to Deligne [5] and André[1]. The compact dual X+∨ of X+ has a natural model over Q (see section 3.3). We say that a point of X+ is algebraic if its image by the Borel embedding is in X+∨(Q). With this definition we check that CM points of X+ are algebraic (proposition 3.7). Let Y be an irreducible algebraic subvariety of X+. So Y = X+ ∩ Z for an irreducible algebraic subvariety of X+∨. We say that Y is defined over Q if Z is defined over Q. We will check that if Z is a special subvariety of S then any analytic component of π−1Z is defined over Q. Note also that S has a canonical model over Q and that special subvarieties of S are defined over Q. We recall the following result due to Cohen [2] and Shiga-Wolfart [21]. Theorem 1.3. Let π : X+ −→ S := Γ\X+ be the map uniformising the Shimura variety S. Assume that S is a Shimura variety of abelian type. Let x be a point of X+(Q) such that π(x) ∈ S(Q). Then x is a CM point. A CHARACTERISATION OF SPECIAL SUBVARIETIES. 3

In [2] the theorem is only given for Shimura varieties of Hodge type (special subvarieties of some moduli space of polarised abelian varieties with some level structure) but the extension to Shimura varieties of abelian type is straightforward. See [16] 2.10 for the definition and the group theoretic description of Shimura varieties of abelian type. A corollary of our main result is the following generalisation. Theorem 1.4. Let π : X+ −→ S := Γ\X+ be the map uniformising the Shimura variety S. Assume that S is a Shimura variety of abelian type. Let Y˜ be an algebraic subvariety of X+ defined over Q such that Y := π(Y˜ ) is an algebraic subvariety of S defined over Q. Then Y is a special subvariety of S. In the last section we indicate the easy analog of the main result in the context of abelian varieties. The principle of the proof is the same as in the Shimura case. The monodromy theorem of Deligne and Andréin this case is replaced by an elementary property of the Albanese variety. We would like to thank Jonathan Pila for interesting discussions about his method that motivated us to write this article.

2. Weakly special subvarieties. Definition 2.1. Let (G, X) be a Shimura datum and let K be a compact open subgroup of G(Af ). An algebraic subvariety Z of ShK (G, X) is weakly special if there exists a sub-Shimura datum (H,XH ) of (G, X) ad ad and a decomposition (H ,XH ) = (H1,X1) × (H2,X2) and a point + y2 ∈ X2 such that Z is the image of X1 × {y2} in ShK (G, X). In this situation, Z is special (or Hodge type) if and only if y2 is a special point of X2. In particular Z is special if and only if it contains a special point. + The image of X1 × {y2} in ShK (G, X) here is defined as follows. We ad have an inclusion XH ⊂ XH = X1 ×X2 which induces an identification + + + + XH = X1 × X2 . Via this identification we consider X1 × {y2} as a subset of XH and take its image in ShK (G, X). In [15], Moonen proves that an irreducible subvariety Z of ShK (G, X) is weakly special if and only if it is totally geodesic. The following properties follow immediately from the definition. Proposition 2.2. (1) Let K0 ⊂ K be a compact open subgroup and let f : ShK0 (G, X) −→ ShK (G, X) be the map induced by the inclusion. A subvariety Z of ShK (G, X) is weakly special if one (equivalently any) component of its preimage in ShK0 (G, X) is weakly special. 4 EMMANUEL ULLMO AND ANDREI YAFAEV

(2) Let g ∈ G(Af ) and Tg be the corresponding Hecke correspon- dence. A subvariety Z of ShK (G, X) is weakly special if and only if components of its image by Tg are weakly special. (3) Let (Gad,Xad) be the adjoint Shimura datum associated to (G, X). Choose a compact open subgroup Kad containing the image of K ad ad and let α: ShK (G, X) −→ ShKad (G ,X ) be the correspond- ad ad ing morphism. A subvariety Z of ShKad (G ,X ) is weakly special if one (equivalently any) component of its preimage in ShK (G, X) is weakly special. For the third property, it suﬃces to notice that the components of X and Xad are the same.

3. Algebraic structure of hermitian symmetric domains. 3.1. Borel and Harish-Chandra’s embeddings. Let (G, X) be a Shimura datum and let (Gad,Xad) be the associated adjoint Shimura datum. As connected components of X and Xad are the same, we can and do assume that G = Gad. This does not cause any loss of generality. For simplicity of notations, we still denote by X a connected component of X. Choose a point x ∈ X and let Px be the stabiliser of hx in G(C). The group Px is a parabolic subgroup of GC. Let ∨ X = G(C)/Px(C) This is the compact dual of X and it has a natural structure of projec- tive complex algebraic variety. Let

K∞ = G(R) ∩ Px(C). This is a maximal compact subgroup of G(R) and

X = G(R)/K∞. The Borel embedding is given by the natural inclusion ∨ (1) X = G(R)/K∞ ,→ X = G(C)/Px(C) which is G(R) equivariant. Definition 3.1. View X as a subset of X∨. A subset Y of X is called irreducible algebraic if there exists an irreducible closed algebraic subset Z of X∨ of dimension at least one such that Y = Z ∩ X. An algebraic subset of X is a finite union of irreducible algebraic subsets. To clarify the picture it is useful to recall briefly the link between the Harish-Chandra and the Borel embeddings. We refer to [9] VIII-7 + and [14] V for a detailed discussion. Fix a point x0 of X. Let p be A CHARACTERISATION OF SPECIAL SUBVARIETIES. 5 the holomorphic tangent bundle at x0 in X. Then by results of Harish- Chandra X can be canonically realised as a bounded symmetric domain in p+ ' CN . Moreover there exists a map (2) η : p+ −→ X∨ which is an embedding onto a dense open subset of X∨. The restriction of η to X = G(R)/K∞ is the Borel embedding (1) or (5). Remark 3.2. We could define algebraic subsets of positive dimension of X as intersections of closed algebraic subvarieties of p+ of positive dimension with X. Using the previous discussion, we see that the two definitions coincide. Moreover the notion of algebraic subvarieties of X is independent of the choice of x0. The next lemma shows that algebraic varieties of hermitain symmetric domains have a good behaviour when we pass to a hermitian symmetric subdomain. This will be used in the proof of the theorem 3.4. Lemma 3.3. Assume that X is realised as a bounded symmetric do- 0 main in the holomorphic tangent bundle p+ at some x0 ∈ X. Let X 0+ be a bounded symmetric subdomain of X containing x0. Let p be the 0 holomorphic tangent bundle at x0 ∈ X . Then an algebraic subvariety of X0 in p0+ is the intersection of an algebraic subvariety of X in p+ with X0. Proof. Let Y ⊂ X0 be an algebraic subvariety. Then Y = X0 ∩Z for an algebraic subvariety of p0+. Using [20] ch.II prop 8.1, we see that the inclusion map X0 ⊂ X is just the restriction of the inclusion map p0+ ⊂ p+. Therefore Z is an algebraic subvariety of p+ and Y = X0 ∩ (X ∩ Z) + 0 is the intersection of an algebraic subvariety of X in p with X . Explicit realisations of the 4 classical families of irreducible bounded symmetric domains have been given by E.´ Cartan. The reader can consult ([9] ch. X ex D.1) for a summary and ([14] ch. IV) for proofs. Let us mention the two following examples. Let p and q be 2 positive integers. Then I t pq (3) Dp,q := {Z ∈ Mp,q(C),Z Z − Iq < 0} ⊂ Mp,q(C) ' C is a bounded symmetric realisation of SU(p, q)/SU(p + q) and the in- I pq clusion Dp,q ⊂ C is the Harish-Chandra embedding. t Let n be a positive integer, Sn(C) := {Z ∈ Mn,n(C),Z = Z} be the set of symmetric complex matrices in Mn,n(C). Then n(n+1) III t 2 (4) Dn := {Z ∈ Sn(C),Z Z − In < 0} ⊂ Sn(C) ' C 6 EMMANUEL ULLMO AND ANDREI YAFAEV is a bounded symmetric realisation of Sp(n, R)/U(n) and the inclusion III n(n+1) Dn ⊂ C 2 is the Harish-Chandra embedding. In addition to the 4 classical families of irreducible bounded symmetric domains there are 2 irreducible bounded symmetric domains of exceptional type and any bounded symmetric domain is a product of irreducible ones.

3.2. Unbounded realisations. It is sometimes useful in the theory of hermitian symmetric domains to use unbounded realisations. If X is a hermitian symmetric domain and X is realised as an open subset of some CN . We may deﬁne a notion of algebraic subvarieties of X by taking an intersection of an algebraic subvariety of CN as in the Harish- Chandra realisation of X. We would like to know that the algebraic subvarieties of X are in fact independent of the realisation (bounded or unbounded). It may be possible to study this question using the results of Piatetskii-Shapiro [17] and Kor´anyi-Wolf [10] (see also [20] ch. III). We will check this in the case of the generalised Siegel upper half plane which is the most relevant in the theory of Shimura varieties. Let X = Sp(n, R)/U(n), then the bounded symmetric realisation is III Dn (4) and X has the unbounded realisation

t n(n+1) Hn := {Z ∈ Mn,n(C),Z = Z, =(Z) > 0} ⊂ Sn(C) ' C 2 as a Siegel space. n(n+1) n(n+1) The rational map Φ : C 2 → C 2 √ √ −1 Z 7→ (In + −1Z)(In − −1Z)

III induces a biholomorphic transformation from Hn to Dn . If V is an n(n+1) algebraic subvariety of C 2 then III Φ(V ∩ Hn) = Φ(V ) ∩ Dn . The map Φ establishes a bijection between the algebraic subvarieties III of Hn and of Dn . A consequence of the main theorem 3.4 is the following:

Theorem 3.4. Let Ag be the moduli space of principally polarised abelian varieties of dimension g and let π : Hg −→ Ag be the uniformising map. An irreducible subvariety Z of Ag is weakly special if and only if some (equivalently any) analytic component of π−1Z is algebraic subvariety of Hg. A CHARACTERISATION OF SPECIAL SUBVARIETIES. 7

3.3. Hodge theoretic interpretation and rationality. Let x be a point of X. Let µx : Gm,C → GC be the cocharacter associated to x. Let MX be the G(C)-conjugacy class of µx. Let V be a faithful representation of G on a Q-vector space of finite dimension. Then µx defines a filtration ? Fx VC := {· · · ⊃ FpVC ⊃ Fp+1VC ⊃ ... }p∈Z by C-vector subspaces of VC. Fix x ∈ X and let Q be the subgroup of GL(V ) stabilising F ? V . 0 C x0 C ∨ There exists a surjective map MX → X sending µx to the associated ? ∨ filtration Fx VC and we can realise this way X as a subvariety of the flag variety ΘC := GL(V )/Q. The Borel embedding (1) is the map X → X∨ given by ∗ (5) x 7→ Fx VC.

Note that ΘC has a natural model Θ over Q. For any extension L of Q a point z ∈ Θ(L) defines a filtration ? Fz VL := {· · · ⊃ FpVL ⊃ Fp+1VL ⊃ ... }p∈Z ? by L-vector subspaces of VL. In this situation the stabiliser of Fz VL is a parabolic subgroup of GLVL conjugate in GL(VC) to Q. ∨ Then X , as a subvariety of ΘC is defined over the reflex field E(G, X) of (G, X). This is a direct consequence of the definition of the reflex field. See [13] III-1 for details about these constructions. ∨ Definition 3.5. View X as a subset of X ⊂ ΘC. A closed point P of X is said to be algebraic if P ∈ X∨(Q). 3.4. Complex multiplication and rationality. Definition 3.6. Let x ∈ X, the Mumford-Tate group MT (x) of x is the smallest Q subgroup of G such that there is a factorisation of x

x : S → MT (x)R ,→ GR. A point x ∈ X is said to be special or CM if MT (x) is a torus. ∨ Proposition 3.7. View X as a subset of X ⊂ ΘC. Then a CM point x ∈ X is an algebraic point. Let x be a CM point of X. Then we have a factorization of the associated cocharacter µ : → MT (x) ,→ G ,→ GL(V ). x GmC C C C

We can choose a Q- torus Tx maximal in GL(VQ) such that we have a factorization µ : → T ,→ GL(V ). x GmC x,C C 8 EMMANUEL ULLMO AND ANDREI YAFAEV

The cocharacter µx is deﬁned over a number ﬁeld therefore we have a factorization

µx : m → T ,→ GL(V ). G Q x,Q Q ? Therefore the associated filtration Fx VC is fixed by Tx(Q). We just need to prove Lemma 3.8. Let y ∈ Θ( ) and T be a maximal -torus in GL(V ) C y Q Q ? such that the filtration Fy VC is fixed by Ty(Q). Then y ∈ Θ(Q). ? Let Qy be the stabiliser of Fy VC in GL(VC). Then Qy is a parabolic subgroup of GL(VC) containing Ty,C. We have a decomposition Qy = RM as an almost direct product with R the unipotent radical of Qy and

M a Levi subgroup of Qy. Then Ty,C is contained in some conjugate of M and replacing M by this conjugate we may assume that Ty,C ⊂ M. Let Z be the connected centre of M then as Z commutes with Ty,C and as Ty,C is maximal in GL(VC) we see that Z ⊂ Ty,C. The subtori of ∗ TC correspond to sub-Z-modules of the group of character X (Ty,C) = X∗(T ) and are therefore defined over . Therefore Z is defined over y,Q Q Q. As M is the centraliser in GL(VC) of Z, M is defined over Q. As Qy is the normaliser of M, Qy is defined over Q. The associated point of the flag variety is therefore algebraic. Remark 3.9. The proposition 3.7 is well known for the classical irre- I III ducible bounded symmetric domains such as Dp,q or Dn . Note that in these cases the bounded symmetric domains are explicitly realised as sets matrices with complex coefficients and an algebraic point is just given by a matrix with coefficients in Q. It is maybe possible to check this property for the exceptional ones as well. The proof given above is independent of the classification.

4. Proof of the main result. In this section we prove the theorems 3.4 and 1.4. Let (G, X) be a Shimura datum and K a compact open subgroup of + G(Af ). Let X be a connected component of X and let Γ be G(Q)+∩K + + where G(Q)+ is the stabiliser of X in G(Q). We let S be Γ\X . This + is a connected component of ShK (G, X). We also let π : X −→ S be the natural projection. With these notations, our result is the following: Theorem 4.1. Let Y be an algebraic subvariety of S. The variety Y is weakly special if and only if one (equivalently any) component of π−1(Y ) is algebraic. A CHARACTERISATION OF SPECIAL SUBVARIETIES. 9

Remark 4.2. As the assumptions and conclusions of the theorem are invariant under translation by Hecke operators, the result is true for any component of ShK (G, X). Proof. The Borel embedding X → X∨ is G(R)-invariant and the image of an algebraic subvariety of X∨ by an element of G(R) is an algebraic subvariety of X∨. Let Y˜ be an algebraic component of π−1(Y ). Let Z be an algebraic subvariety of X∨ such that Y˜ = X ∩ Z. Let γ ∈ Γ ⊂ G(R) then γY˜ = γ(X ∩ Z) ⊂ X ∩ γ.Z = γγ−1(X ∩ γ.Z) ⊂ γY.˜ Therefore γY˜ = X ∩ γ.Z is algebraic. As the analytic components of π−1(Y ) are permuted transitively by Γ, we see that if one component is algebraic then so is any other. Next, we reduce the situation to the case where Y is Hodge generic in S. Let SY be the smallest special subvariety containing Y . Such a variety SY exists in view of the fact that components of intersections of special subvarieties are special (this is a consequence of the interpretation of special subvarieties as loci of Hodge classes). By [22], Lemma 2.1 and its proof, there exists a sub-Shimura datum (H,XH ) where H is the generic Mumford-Tate group on XH such that SY is the + + image ΓH \XH in S where XH is a connected component of XH and ΓH = Γ ∩ H(Q). By lemma 3.3 a component Ye of the preimage of Y + in XH is still algebraic. We replace (G, X) by (H,XH ) and S by SY and hence assume that Y is Hodge generic in S. Using proposition 2.2-(3) we may and do assume that the group G is semi-simple of adjoint type. Let

(G, X) = (G1,X1) × (G2,X2) be the decomposition of the Shimura datum (G, X) associated to Y as in section 3.6 of [15]. In this decomposition, G1 is the Zariski closure of the algebraic monodromy group ΓY associated to Y . The existence of such a decomposition is a consequence of a theorem of Deligne [5] and Andr´e[1] on the monodromy groups associated to variations of (mixed) Hodge structures (see section 3 of [15] for details and explanations). As the group K can be chosen as small as needed, we assume using proposition 2.2-(1) that Γ := K ∩ G(Af ) is neat and that Γ = Γ1 × Γ2 where Γi are arithmetic subgroups of Gi(Q)+. The subvariety Y is now of the form

(6) Y1 × {y2} + where Y1 is a subvariety of Γ1\X1 and y2 is a Hodge generic point of Γ2\X2 (see [15], prop 3.7). To show that Y is weakly special we 10 EMMANUEL ULLMO AND ANDREI YAFAEV

+ need to show that Y1 = Γ1\X1 . Replacing Y by Y1 and (G, X) by (H1,X1) does not change our assumptions, hence we now assume that the algebraic monodromy group ΓY of Y is Zariski dense in G. We are now in the following situation. Write Ye = Z ∩ X where Z is ∨ an algebraic subvariety of X = G(C)/Px(C). Fix a point y ∈ Ye. As ΓY is Zariski dense in GQ, it is Zariski dense in GC and consequently ∨ the orbit ΓY · y is Zariski dense in X . As ΓY · y ⊂ Y ⊂ Z, it follows that Z is Zariski dense in X∨. As Z is an algebraic subvariety of X∨, ∨ we find that Z = X and therefore Y = Z ∩ X = X. We can now give the proof of theorem 1.4. In the previous situation if we moreover assume that Y and Y˜ are defined over Q, then the point y2 of equation (6) is defined over Q. Moreover we have a decomposition ˜ ˜ 0 ˜ Y = Y1 ×{y˜2} withy ˜2 ∈ Y mapping to y2. As Y is defined over Q then y˜2 is defined over Q. If S is a Shimura variety of abelian type then by the result of Cohen and Shiga-Wolfart (theorem 1.3)y ˜2 is a CM point. The theorem 1.4 is therefore a consequence of theorem 2.1.

5. The case of abelian varieties. In this section we prove the analog of theorem 3.4 for complex abelian varieties. Let A be an abelian variety over C. Then A is of the form A = VA/H1(A, Z) for a C-vector space VA of dimension g = dim(A). A subvariety Y of A is said to be weakly special if Y = P + B for a point P ∈ A and an abelian subvariety B of A. Let π : VA → A be the uniformising map. ˜ Proposition 5.1. Let Y be a subvariety of A. Let Y ⊂ VA be an analytic component of π−1(Y ). Then Y is weakly special if and only if ˜ g Y is an algebraic subvariety of VA ' C . −1 The group H1(A, Z) acts transitively on the components of π (Y ). As this action is algebraic, if some component Y˜ of π−1(Y ) is algebraic, the same property holds for any component of π−1(Y ). Note also that Y is weakly special if and only if for any point Q ∈ A(C), the translate Q + Y of Y is weakly special. The components of π−1(Y ) are algebraic if and only if those of π−1(Q + Y ) are. Hence we may and do assume that the origin O of A is in Y . Weakly special subvarieties of A containing O are just abelian subvarieties of A and are of the form π(V 0) for a C-subvector space V 0 of V . We therefore get the “only if” part of the theorem and we just need to prove the other direction. Let Alb(Y ) be the Albanese variety of Y normalised by the choice of the point O ∈ Y and let a : Y → Alb(Y ) be the associated Albanese A CHARACTERISATION OF SPECIAL SUBVARIETIES. 11 morphism. Let ι : Y → A be the inclusion map. Using the functorial property of the Albanese map, we know that there exists a morphism of abelian variety φ : Alb(Y ) → A such that ι = φa. Let B := φ(Alb(Y )), then B is an abelian subvariety of A. By [23] lemma 12.11, a(Y ) generates Alb(Y ) as a group. Therefore B is in fact the smallest abelian subvariety of A containing Y . Let ΓY ⊂ H1(A, Z) be the image of π1(Y,O) in H1(A, Z). Then ΓY = ι?H1(Y, Z). Moreover

ΓY = (φa)?H1(Y, Z) = φ?H1(Alb(Y ), Z) is of ﬁnite index in H1(B, Z). There exists a C-vector subspace VB of VA such that H1(B, Z) = ˜ VB ∩ H1(A, Z) and B = VB/H1(B, Z). There exists a component Y of −1 −1 ˜ π (Y ) contained in VB = π (B). Lety ˜ be a point of Y mapping to O. ˜ Then as ΓY is of ﬁnite index in H1(B, Z), ΓY .y˜ ⊂ Y is Zariski dense ˜ ˜ in VB. If Y is algebraic then Y = VB and Y = B is weakly special.

Bibliography. [1] Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82 (1992), no. 1, 1–24. [2] P. Cohen, Humbert surfaces and transcendence properties of automorphic functions. Symposium on Diophantine Problems (Boulder, CO, 1994). Rocky Mountain J. Math. 26 (1996), no. 3, 987–1001. [3] P. Deligne Travaux de Shimura, SéminaireBourbaki, Exposé389, Fevrier 1971, Lecture Notes in Maths. 244, Springer-Verlag, Berlin 1971, p. 123-165. [4] P. Deligne Variétésde Shimura: interprétationmodulaire et techniques de con- struction de modèlescanoniques, dans Automorphic Forms, Representations, and L-functions part. 2; Editeurs: A. Borel et W Casselman; Proc. of Symp. in Pure Math. 33, American Mathematical Society, (1979), p. 247–290. [5] P. Deligne, La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206–226. [6] B. Edixhoven Special points on products of two modular curves. Compositio Math 114, 1998, n. 3, 315-328 [7] B.Edixhoven The André-Oortconjecture for Hilbert modular surfaces. In ‘Mod- uli of abelian varieties (Texel Island, 1999). 133-155. Progress In Math. 195, Birkhauser, Basel, 2001 [8] B. Edixhoven, A. Yafaev Subvarieties of Shimura varieties. Annals of Mathe- matics 157, 2003, n. 2, 621-645. [9] S. Helgason Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. xv+628 pp. [10] A. Korányi, J.A. Wolf Generalized Cayley transformation of bounded symmetric domains. Amer. J. Math. 87, (1965), 899–939. [11] B. Klingler, A. Yafaev The André-Oortconjecture. Preprint. 12 EMMANUEL ULLMO AND ANDREI YAFAEV

[12] J. Milne Introduction to Shimura varieties. Lecture notes, available on author’s web-page. [13] J.S. Milne Canonical models of (mixed) Shimura varieties and automorphic vector bundles. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 283–414, Perspect. Math., 10, Academic Press, Boston, MA, 1990. [14] N. Mok Metric rigidity theorems on Hermitian locally symmetric submanifolds. Series in Pure Maths. 6, World Scientific (1989). [15] B. Moonen Linearity properties of Shimura varieties I. Journal of Algebraic Geometry, 7, 1998, n.3, 539-567. [16] B. Moonen Models of Shimura varieties in mixed characteristics in Galois representations in arithmetic algebraic geometry (Durham, 1996), 267–350, Lon- don Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press, Cambridge, 1998. [17] I.I. Piatetskii-Shapiro Geometry of Classical Domains and Theory of Automor- phic functions. (Russian), Fizmatgiz, Moscow 1961; (French transl.) Dunot, Paris, 1966; (English translation) Gordon and Breach, New-York, 1969. [18] J. Pila Rational points of definable sets and results of Andr-Oort-Manin- Mumford type. Int. Math. Res. Not. IMRN 2009, no. 13, 2476–2507 [19] J. Pila O-minimality and the Andre-Oort conjecture for Cn. Preprint. 2010. [20] I. Satake Algebraic structures of bounded symmetric spaces. Iwanami Shoten and Princeton University Press (1980). [21] H. Shiga; J. Wolfart Criteria for complex multiplication and transcendence properties of automorphic functions. J. Reine Angew. Math. 463 (1995), 1–25. [22] E. Ullmo, A. Yafaev Galois orbits and equidistribution: towards the André- Oort conjecture. Preprint. [23] C. Voisin, Théorie de Hodge et géométrie algébrique complexe. Cours Spécialisés, 10. SociétéMathématique de France, Paris, (2002) viii+595 pp.

Emmanuel Ullmo : Universit´ede Paris-Sud; Orsay, Departement de Math´ematique email : ullmo@ math.u-psud.fr Andrei Yafaev : University College London, Department of Mathematics. email : [email protected]