Algebraic flows on Shimura varieties

In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images of totally geodesic subvarieties of symmetric spaces uniformising Shimura varieties. This is a special case of our conjectures.

In the paper [15] we formulated certain conjectures about algebraic flows on abelian varieties and proved certain cases of these conjectures. The purpose of this paper is two-fold. We first prove the 'logarithmic Ax-Lindemann theorem' (see details below). We then prove a result analogous to one of the main results of [15] in the hyperbolic (Shimura) case about the topological closure of the images of totally geodesic subvarieties of the symmetric spaces uniformising Shimura varieties.
Let (G, X) be a Shimura datum and X + be a connected component of X. Recall from [16], section 2.1 that a realisation X of X + is a complex quasi-projective variety X with a transitive holomorphic action of G(R) + such that for any x 0 ∈ X , the orbit map ψ x 0 : G(R) + −→ X mapping g to gx 0 is semi-algebraic. There is a natural notion of a morphism of realisations. By [16], lemma 2.1, any realisation of X + has a canonical semi-algebraic structure and any morphism of realisations is semi-algebraic.
In what follows we fix a realisation X of X + and by a slight abuse of language still call this realisation X + . It is an immediate consequence of Lemma 2.1 of [16] that all the conjectures and statements that follow are independent of the chosen realisation.
In view of the lemma B1 of [7], we may define an algebraic subset Y of X + to be a closed analytic, semi-algebraic subset of X + . Given an irreducible analytic subset Θ ⊂ X + , we define the Zariski closure of Θ to be the analytic component containing Θ of the smallest algebraic subset of X + containing Θ.
We can now state some results and conjectures. The classical formulation of the hyperbolic Ax-Lindemann theorem is as follows: Theorem 1.1 (Hyperbolic Ax-Lindemann theorem). Let S be a Shimura variety and π : X + −→ S be the uniformisation map. Let Z be an algebraic subvariety of S and Y a maximal algebraic subvariety of π −1 (Z). Then π(Y ) is a weakly special subvariety of S.
We will see (see proposition 5.1) that this is equivalent to: Theorem 1.2 (Hyperbolic Ax-Lindemann theorem, version 2.). Let Z be any irreducible algebraic subvariety of X + then the Zariski closure of π(Z) is weakly special.
The hyperbolic Ax-Lindemann conjecture has been proven in full generality in [7].
In the second section we define a notion of a weakly special subvariety of X + . This is a complex analytic subset Θ of X + such that there exists a semi-simple algebraic subgroup F of G(R) + and a point x ∈ X + satisfying certain conditions such that Θ = F · x.
In Section 3 of this paper we prove a 'logarithmic' Ax-Lindemann theorem (a question asked by D. Bertrand). Theorem 1.3 (Logarithmic Ax-Lindemann). Let π : X + −→ S be the uniformisation map. Let Y be an algebraic subvariety of S and let Y ′ be an analytic component of π −1 (Y ). The Zariski closure of Y ′ is a weakly special subvariety.
In [15], we formulated two conjectures on algebraic flows on abelian varieties and proved partial results towards these conjectures. An attempt to formulate conjectures of this type in the context of Shimura varieties displays new phenomena that we intend to investigate in the future. We however prove a result which may be seen as a generalisation in the context of Shimura varieties of one of the main results of [15]. To state our result we need to introduce a few notations.
Consider an algebraic subset Θ of X + . In general, instead of (as in the hyperbolic Ax-Lindemann case) being interested in the Zariski closure of π(Θ), we look at the usual topological closure π(Θ). We define a notion of real weakly special subvariety roughly as the image of H(R)·x where H is a semisimple subgroup of G satisfying certain conditions and x is a point of X + . Let K x be the stabiliser of x in G(R) + . In the case where H(R) + ∩ K x is a maximal compact subgroup, a real weakly special subvariety of S is a real totally geodesic subvariety of S. Notice that in this case the homogeneous space H(R) + /H(R) + ∩ K x is a real symmetric space. In the case where x viewed as a morphism from S to G R factors through H R , the corresponding real weakly special subvariety has Hermitian structure and in fact is a weakly special subvariety in the usual sense. We also note that given a real weakly special subvariety Z of S, there is a canonical probability measure µ Z attached to Z which is the pushforward of the Haar measure on H(R) + , suitably normalised to make it a probability measure.
In this paper we prove the following theorem.
Theorem 1.4. Let Θ be a complex totally geodesic subvariety of X.
Then the components of the topological closure π(Θ) are real weakly special subvarieties.
Recall that a complex totally geodesic subvariety of X + is of the form F · x where F is a semisimple real Lie group subject to certain conditions and x is a point of X such that F ∩K x is a maximal compact subgroup of F .
In certain cases, for example when the centraliser of F in G(R) is trivial, we are able to show that π(Θ) is actually a (complex) weakly special subvariety. This condition is satisfied in many cases. For example in the case of SL 2 (R) diagonally embedded into a product of copies of SL 2 (R). In particular this answers in the affirmative the question of Jonathan Pila which was the following. Consider the subset Z of The proof of Theorem 1.4 relies on the results of Ratner (see [13]) on closure of unioptent one parameter subgroups in homogeneous spaces.

Acknowledgements.
We thank Jonathan Pila for discussions around the topic of the second part of this paper. We also thank Daniel Bertrand who raised the question of Logarithmic Ax-Lindemann theorem. We are very grateful to Ngaiming Mok for many stimulations discussions.
2.1. Monodromy. Let (G, X) be a Shimura datum. Recall that G is a reductive group over Q such that G ad has no Q-simple factor whose real points are compact and X is a G(R)-conjugacy class of a morphism x : S −→ G R where S = Res C/R G m,C . The morphism x is required to satify Deligne's conditions which imply that components of X are Hermitian symmetric domains. There is a natural notion of morphisms of Shimura data. We fix a connected component X + of X and we let Γ = G(Q) + ∩ K where G(Q) + is the stabiliser of X + in G(Q). Let S be Γ\X + and π : X + −→ S be the natural morphism.
To (G, X), one associates the adjoint Shimura datum (G ad , X ad ) with a natural morphism (G, X) −→ (G ad , X ad ) induced by the natural map G −→ G ad . Notice that the this map identifies X + with a connected component of X ad . We have the following description of weakly special (or totally geodesic) subvarieties (see Moonen [10]): Note that X ad,+ M = X + 1 × X + 2 (with a suitable choice of connected components) is a subspace of X + .
We can without loss of any generality assume the group Γ to be neat, i.e. the stabiliser of each point of X + in Γ to be trivial (replacing Γ by a subgroup of finite index changes nothing to the property of a subvareity to be weakly special). Fix a point x of the smooth locus Z sm and x ∈ π −1 (x)∩Z sm . This gives rise to the monodromy representation ρ m : π 1 (Z sm , x) −→ Γ whose image we denote by Γ m . By Theorem 1.4 (due to Deligne and André) of [10] This can all be summarised in the following theorem. Let Γ m ⊂ M der (Q) ∩ Γ be the monodromy group attached to Z as described above. Let where Z 1 is a subvariety of S 1 whose monodromy is Zariski dense in M 1 and z is a point of S 2 .

2.2.
Weakly special subvarieties of X + . In this section we give a precise description of totally geodesic (weakly special) subvarieties of X + .
Let (G, X) be a Shimura datum and X + a connected component of X. For the purposes of this section, we can without loss of generality assume that G is a semi-simple group of adjoint type. This is because there is a natural identification between connected components of X + and a connected component of X ad . We will now describe totally geodesic subvarieties of X + (that we will naturally call weakly special).
The group G has no Q-simple factors whose real points are compact and there is a morphism x 0 : S −→ G R satisfying the following Deligne's conditions such that We have the following: Proposition 2.3. Let Z be a totally geodesic complex subvariety of X + . There exists a semi-simple real algebraic subgroup F of G R without compact factors and some x ∈ X such that x factors through F Z G (F ) 0 such that Z = F (R) + .x. Conversely, let F be a semi-simple real algebraic subgroup of G R without compact factors and let x ∈ X such that x factors through F Z G (F ) 0 . Then F (R) + .x is a totally geodesic subvariety of X + .
Proof. let F be a semi-simple real algebraic subgroup of G R without compact factors and let x ∈ X such that x factors through H : (H(R)). By using [17] lemma 3.13 we see that H is reductive.
Then the proof of [17] lemma 3.3 shows that X H := H(R) + .x is an hermitian symmetric subspace of X + . We give the arguments to be as self contained as possible.
As Then H R is the almost direct product H R ≃ F F nc 1 F c 1 where F 1 is either trivial or semi-simple without compact factors and F c 1 is reductive is a product of hermitian subspaces and we have the natural identification of X + F with X + F × {x 1 } where x 1 is the projection of x on X + F nc 1 . In any case X + F is hermitian symmetric and totally geodesic in X + . Conversely a totally geodesic subvariety of X + is of the form X + F = F (R) + .x for a semi-simple subgroup F R of G R without compact factors. Let T x (X + F ) ⊂ T x (X + ) be the tangent space of X + F at x. Let U 1 ⊂ S be the unit circle. The complex structure on T x (X + ) is given by the adjoint action of x(U 1 ). If X F is a complex subvariety, then T x (X + F ) is stable by x(U 1 ). Using Cartan decomposition we see that x(U 1 ) = x(S) normalizes F . Let F 1 = x(S)F , then F 1 is reductive and is contained in F Z G (F ) 0 . It follows that x factors through F Z G (F ) 0 . Definition 2.4. An algebraic group H over Q is said to be of type H if its radical is unipotent and if H/R u (H) is an almost direct product of Q simple factors H i with H i (R) non-compact. Furthermore we assume that at least one of those factors not to be trivial.
Let H ⊂ G be a subgroup of type H and let us assume that G is of adjoint type. We will now explain how to attach a hermitian symmetric space X H to a group of type H and explain that X H is independent of the choice of a Levi subgroup in H.
The domain X + is the set of maximal compact subgroups of G(R) + . Let x ∈ X + , we denote by K x the associated maximal compact subgroup of G(R) + . Let H be a subgroup of type H and let L be a Levi subgroup of H. We have a Levi decomposition H = R u (H).L. Assume that K x ∩ L(R) + is a maximal compact subgroup of L(R) + . Then X + L = L(R) + .x ⊂ X + is the symmetric space associated to L and is independent of the choice of x ∈ X + such that K x ∩L(R) + is a maximal compact subgroup of L(R) + . Let X + H := R u (H)X L (R) + , then X + H is independent of the chosen Levi decomposition of H. This can be seen as follows. The Levi subgroups of H are conjugate by an element of R u (H). Let L ′ be a Levi of H and w ∈ R u (H) such that L ′ = wLw −1 .
This shows that the space X + H is independent of the choice of the Levi. Definition 2.5. A real weakly special subvariety of S is a real analytic subset of S of the form where H is an algebraic subgroup of G of type H and x ∈ X + .
In the case where K x ∩ L(R) + is a maximal compact subgroup of L(R) + for some Levi subgroup of H, H(R) + /K x ∩ H(R) + is a real symmetric space.
We have the following proposition. Proposition 2.6. Let Z be a real weakly special subvariety of S. Then the Zariski closure Z Zar of Z is weakly special.
Proof. By definition, Z is of the form Z = H(R) + · x where H is a group of type H. Let S M be as in Theorem 2.2 the smallest special subvariety containing Z Zar .
Let S 1 × S 2 be the product of Shimura varieties as in Theorem 2.2 such that the image of Z Zar in S 1 × S 2 is of the form Z 1 × {z} where Z 1 is a subvariety of S 1 whose monodromy Γ m 1 is Zariski dense in M 1 and z is a Hodge generic point of S 2 .
To prove that Z Zar is weakly special, it is enough to show that Z 1 = S 1 . In what follows, we replace S by S 1 and Z by Z 1 .
For any q ∈ H(Q) + , we have that Z ⊂ T q Z, therefore and therefore, for each q ∈ H(Q) we have Let T be a non-trivial subtorus of H. We define the Nori constant C(Z Zar ) of Z Zar as in [19], section 4. Let p > C(Z Zar ) and q ∈ T (Q) given by Lemma 6.1 of [19]. Then T q (Z Zar ) is irreducible and the orbits of T q +T q −1 are dense in S. This implies that Z Zar = S as required.

Logarithmic Ax-Lindemann.
Let S = Γ\X + as before and consider a realisation X + ⊂ C n (in the sense of [17]). In particular X + is a semi-algebraic set and the action of G(R) + is semi-algebraic.
Let Y be a complex analytic subset of X + . Then the Zariski closure Y Zar in C n is an algebraic subset of C n and Y Zar ∩ X + has finitely many analytic components. By slight abuse of notation, we refer to Y Zar ∩ X + as Zariski closure of Y . These components are algebraic in the sense of the definition given in the Appendix B of [7]. Theorem 3.1 (Logarithmic Ax-Lindemann). Let π : X + −→ S be the uniformisation map. Let Y be an algebraic subvariety of S and let Y ′ be an analytic component of π −1 (Y ). The Zariski closure of Y ′ is a weakly special subvariety.
Proof. Let Y be an analytic component of Y ′ . As in the previous section we can replace S by S 1 and Y by Y 1 given by the Proposition 2.2. In doing this we attach the monodromy to a point y ∈ Y sm and y ∈ Y ′ . Let Γ Y be the monodromy group attached to Y . Notice that Γ Y is the stabiliser of Y ′ in Γ. Then, with our assumptions, Γ Y is Zariski dense in G.
Let α ∈ Γ Y . We have We also have αY ′ Zar ⊃ αY ′ and since αY ′ Zar is algebraic, we have The same argument with α −1 instead of α shows that the reverse inclusion holds and therefore Since Y ′ Zar is semialgebraic and the action of G(R) + on X + is semi-algebraic, G Y is semialgebraic. Furthermore, G Y is analytically closed and hence is a real algebraic group. Since G Y contains Γ Y which is Zariski dense in G R , we see that G Y = G(R) + . It follows that Y ′ Zar = S as required.

Facts from ergodic theory: Ratner's theory.
In this section we recall some results from ergodic theory of homogeneous varieties to be used in the next section. The contents of this section are mainly taken from Section 3 of [3]. We present results in the way they are presented in [16].
Let G be a semi-simple algebraic group over Q. We assume that G has no Q-simple simple factors that are anisotropic over R. This condition is satisfied by all groups defining Shimura data.
Let Γ be an arithmetic lattice in G(R) + and let Ω = Γ\G(R) + . We have already defined a subgroup H ⊂ G of type H, we now define a group of type K. Definition 4.1. Let F ⊂ G(R) be a closed connected Lie subgroup. We say that F is of type K if (1) F ∩ Γ is a lattice in F . In particular F ∩ Γ\F is closed in Γ\G(R) + . We denote by µ F the F -invariant normalised measure on Γ\G(R) + . (2) The subgroup L(F ) generated by one-parameter unipotent subgroups of F acts ergodically on F ∩ Γ\F with respect to µ F . For the purposes of this section, we in addition assume F to be semisimple.
The relation between types K and H is as follows (see [2], lemme 3.1 and 3.2): (1) If H is of type H, then H(R) + is of type K. (2) It F is a closed Lie subgroup of G(R) + of type K, then there exists a Q subgroup F Q of G of type H such that F = F (R) + .
For a subset E of G(R), we define the Mumford-Tate group MT (E) of E as the smallest Q-subgroup of G whose R-points contain E. If F is a Lie subgroup of G(R) + of type K , then by (2) of the above lemma, MT (F ) = F Q and it is of type H.
We will make use of the following lemma, which is Lemma 2.4 of [16]. Let Ω = Γ\G(R) + . Note that Ω carries a natural probability measure, the pushforward of the Haar measure on G(R) + , normalised to be a probability measure (the volume of Ω is finite). For each F of type K, there is a natural probability measure µ F attached to F .
The following theorem is a consequence of results of Ratner.  We first give a reformulation of the hyperbolic Ax-Lindemann conjecture in terms of algebraic flows.
Proposition 5.1. The hyperbolic Ax-Lindemann conjecture is equivalent to the following statement. Let Z be any irreducible algebraic subvariety of X + then the Zariski closure of π(Z) is weakly special.
Proof. Let us assume that the hyperbolic Ax-Lindemann conjecture holds true. Let A be an irreducible algebraic subvariety of X + and V be the Zariski closure of π(A). Let A ′ be a maximal irreducible algebraic subvariety of π −1 (V ) containing A. By the hyperbolic Ax-Lindemann conjecture π(A ′ ) is a weakly special subvariety of V . As A ⊂ π(A ′ ) ⊂ V and as π(A ′ ) is irreducible algebraic we have V = π(A ′ ). Therefore V is weakly special.
Let us assume that the statement of the proposition holds true. Let V be an irreducible algebraic subvariety of S. Let Y be a maximal irreducible algebraic subvariety of π −1 (V ). Then the Zariski closure V ′ of π(Y ) is weakly special. Moreover V ′ ⊂ V . Let W be an analytic component of π −1 (V ′ ) containing Y . As V ′ is weakly special, W is irreducible algebraic. By maximality of Y we have Y = W . Therefore π(Y ) = V ′ is weakly special.

5.2.
Application of Ratner's theory. Let (G, X) be a Shimura datum and X + a connected component of X. We assume that G is semi-simple of adjoint type, which we do.
We now consider a totally geodesic (weakly special) subvariety Z of X + . Recall that there exists a semi-simple subgroup F (R) + of G without almost simple compact factors and a point x such that x factors through F Z G (F ) 0 .
Let α be the natural map G(R) + −→ Γ\G(R) + and π x be the map Γ\G(R) + −→ Γ\X + sending x to gx. Recall that π : X + −→ Γ\X + is the uniformisation map. We have We let H be the Mumford-Tate group of F (R) + . Recall that it is defined to be the smallest connected subgroup of G (hence defined over Q) whose extension to R contains F (R) + .
By [14], Prop 7.6, the group F (R) + is generated by its one-parameter unipotent subgroups.

Closure in S.
From the fact that the map π x is proper and Proposition 5.2, we immediately deduce the following Theorem 5.3. The closure of π(Z) in S is V , the image of H(R) + · x i.e. it is a real weakly special subvariety.
In this section we examine cases where we can actually make a stronger conclusion, namely: (1) The variety V from Theorem 5.3 is locally symmetric and hence real totally geodesic. (2) It has a Hermitian structure i.e. is a weakly special subvariety.
Proof. It is enough to show that H(R) + ∩ K x is a maximal compact subgroup of H(R) + .
Notice that since Z G (F ) fixes x, we have We follow Section 3.2 of [16].
Since K x is a maximal compact subgroup of G(R) + such that F (R) + ∩ K x is a maximal compact subgroup of F (R) + , we have two Cartan decompositions: G(R) + = P x K x and F = (P x ∩ F ) · (K x ∩ F ) for a suitable parabolic subgroup P x of G(R) + . We now apply Proposition 3.10 of [16] in out situation. We have a connected semi-simple group H such that F ⊂ H R . According to Proposition 3.10 of [16], there exists a Cartan decomposition H(R) = (P x ∩ H(R)) · (K x ∩ H(R)) This, in particular implies that K x ∩ H(R) is a maximal compact subgroup of H(R) + as required.
Theorem 5.5. Assume that Z G (F ) is trivial. Then V is a weakly special subvariety.
Proof. In this case, x factors through F and therefore through H R . Let X H be the H(R)-orbit of x. By lemma 3.3 of [16], (H, X H ) is a Shimura subdatum of (G, X) and therefore V is a weakly special subvariety.
Example 5.6. We give examples where Z G (F ) is neither trivial nor compact, but the closure of π(Z) is nevertheless hermitian.
Let G be an almost simple group over Q. A typical example is G = Res K/Q SL 2,K where K is a totally real field of degree n. Let F be an Rsimple factor of G R . In the above case F could be for example SL 2 (R) embedded as SL 2 (R) × {1} × · · ·× {1}. Then the centraliser of F is not compact. However, by Lemma 2.4 of [16], the Mumford-Tate group of F is G and for any point x of X + , the image of F · x in S is G. Let Γ = SL 2 (Z) × SL 2 (Z) and π : H × H −→ Γ\X + .
Then, if g ∈ G(Q), then the closure of π(Z) is a special subvariety. It is the modular curve Y 0 (n) for some n.
Example 5.8 (Rank one groups). Here is another quite general example where Z G (F ) is trivial and hence the closure of the image of F (R) + · x is a weakly special subvariety.
Suppose that the groups G is U(n, 1). In this case X + is an open ball in C n . The real rank of G is one. Let F be the subgroup U(m, 1) of U(n, 1) (with m ≤ n). Then the centraliser Z G (F ) is trivial. Indeed, as the split torus is already contained in F , the centraliser must be compact. Bibliography.