Algebraic flows on Shimura varieties

In this paper we prove two results on algebraic flows on Shimura varieties. One is the so-called ‘logarithmic’ Ax-Lindemann theorem. The other concerns the closure of the image of a totally geodesic sub-variety of a symmetric space by the uniformisation map.


Introduction
In the paper [11] we formulated some conjectures about algebraic flows on abelian varieties and proved certain cases of these conjectures. The purpose of this paper is twofold. We first prove the 'inverse Ax-Lindemann theorem' (see details below). We then prove a result analogous to one of the main results of [11] 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 . This X + is a Hermitian symmetric domain. Recall from [12], Sect. 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 [12], 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 [12] that all the conjectures and statements that follow are independent of the chosen realisation.
In view of the lemma B1 of [4], we define an algebraic subset Y of X + to be a closed analytic, semi-algebraic subset of X + . Given an irreducible analytic subset ⊂ X + , we define the Zariski closure of to be the analytic component containing of the smallest algebraic subset of X + containing . We denote this closure by Zar( ). Furthermore, for a subset of a complex algebraic variety, we denote by Zar( ) an irreducible component of the Zariski closure of . For a subset of C n , we denote by the closure of for the Archimedean topology.
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.
For compact Shimura varieties this theorem was proved by the present authors (see [10]). Pila and Tsimerman (see [6]) proved this theorem for A g . It was finally proved in full generality by Klingler and the present authors (see [14]). We will see (see Proposition 5.1) that Theorem 1.1 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.
In the second section we define a notion of a weakly special subvariety of X + . This is a complex analytic subset 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 = F · x.
In Sect. 3 of this paper we prove an 'inverse' Ax-Lindemann theorem (a question asked by D. Bertrand). Theorem 1.3. (The inverse 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 Z ar (Y ) of Y is a weakly special subvariety.
The idea of the proof is as follows. We first reduce ourselves to the case where Y is not contained in a proper weakly special subvariety of X + . We write S = \X + where is an arithmetic subgroup of G(Q) acting 'algebraically' on X + . Using a classical theorem about monodromy, we show that Zar(Y ) is stable by a Zariskidense subgroup of G which implies that Zar(Y ) = X + and thus concludes the proof.
In [11], 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 [11]. To state our result we need to introduce a few notations.
Consider an algebraic subset of X + . In general, instead of (as in the hyperbolic Ax-Lindemann case) being interested in the Zariski closure of π( ), we look at the usual topological closure π( ). 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. 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.
The proof of Theorem 1.4 is a more-or-less direct consequence of Ratner's theorem adapted to Shimura varieties by Clozel and Ullmo. This theory is explained in detail in Sect. 4. In certain cases, for example when the centraliser of F in G(R) is trivial, we are able to show that π( ) 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 H × H which is where g ∈ SL 2 (R)\SL 2 (Q). Is the image of Z dense in C × C?
The proof of Theorem 1.4 relies on the results of Ratner (see [7]) on closure of unioptent one parameter subgroups in homogeneous spaces.

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 satisfy Deligne's conditions (see 2.1.1 of [3]) which imply that connected 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 = G(Q) + ∩ K where G(Q) + is the stabiliser of X + in G(Q). Let S be \X + and π : X + −→ S be the natural morphism.
To (G, X ), one associates the adjoint Shimura datum (G ad , Notice that 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 [5], Theorem 4.3):

Theorem 2.1. A subvariety Z of S is totally geodesic if and only if there exists a sub-datum (M, X M ) of (G, X ) and a product decomposition
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 to be neat, i.e. the stabiliser of each point of X + in to be trivial (replacing by a subgroup of finite index changes nothing to the property of a subvariety to be weakly special). We also assume the group K to be a product of compact open subgroups K p of G(Q p ). This also causes no loss of generality. Fix a point x of the smooth locus Z sm and x ∈ π −1 (x)∩ Z sm . We let M be the Mumford-Tate group of x and call it the generic Mumford-Tate group on Z . This gives rise to the monodromy representation ρ m : π 1 (Z sm , x) −→ whose image we denote by m . As a consequence of Theorem 1.4 of [5], we have an inclusion m ⊂ M der (Q)∩ . Furthermore the Zariski closure of m is a normal subgroup of M der . We call m the monodromy group attached to Z .
We summarise the situation in the following theorem. Let m ⊂ M der (Q) ∩ be the monodromy group attached to Z as described above.
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 .

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 .
The group G has no Q-simple factors whose real points are compact and there is a morphism x 0 : S −→ G R such that X + = G(R) + · x 0 . Furthermore x 0 satisfies the following conditions such that We will now describe totally geodesic subvarieties of X + (that we will naturally call weakly special). 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 := F Z G (F) 0 . Then (H (R)). By using [13] lemma 3.13 we see that H is reductive.
Then the proof of [13] 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.
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 is a product of Hermitian subspaces and we have the natural identification of In any case X + F is Hermitian symmetric and totally geodesic in X + . Conversely a totally geodesic subvariety of X + is of the form

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 these 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 . Let x = w·x. Then K x is a maximal compact subgroup of G(R) + such that K x ∩ L (R) + is a maximal compact subgroup of L (R) + and 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. Let S 1 × S 2 be the product of Shimura varieties as in Theorem 2.2 such that the image of Zar(Z ) in S 1 × S 2 is of the form Z 1 × {z} where Z 1 is a subvariety of S 1 whose monodromy m 1 is Zariski dense in M 1 and z is a Hodge generic point of S 2 .
To prove that Zar(Z ) 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 . The monodromy of Zar(Z ) is hence now Zariski dense in G.
For any q ∈ H (Q) + , we have that Z ⊂ T q Z , therefore Z ⊂ Zar(Z ) ∩ T q (Zar(Z )).
Since Zar(Z ) ∩ T q (Zar(Z )) is algebraic, we have and therefore, for each q ∈ H (Q) we have

Zar(Z ) ⊂ T q (Zar(Z )).
Let T be a non-trivial subtorus of H . Let us recall the notion of the Nori constant C(V ) of a Hodge generic subvariety V of a Shimura variety S as in Sect. 2.1 such that the monodromy of V is Zariski dense in G. We refer to [15], Sect. 4 for details.
There exists an integer C(V ) > 0 such that the following holds. Let g ∈ G(Q) + and p > C(V ). Assume that for all l = p, g l is in K l . Then T g (V ) is irreducible.
We apply this for V = Zar(Z ). Let p > C(Zar(Z )) and q ∈ T (Q) given by Lemma 6.1 of [15]. Then T q (Zar(Z )) is irreducible and the orbits of T q + T q −1 are dense in S. Therefore Zar(Z ) = T q (Zar(Z )) and Zar(Z ) contains a dense subset of S. This implies that Zar(Z ) = S as required.

The inverse Ax-Lindemann
Let S = \X + as before and consider a realisation X + ⊂ C n (in the sense of [13]). 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 Zar( Y ) of Y in C n is an algebraic subset of C n and Zar( Y ) ∩ X + has finitely many analytic components. By slight abuse of notation, we refer to Zar( Y ) ∩ X + as the Zariski closure of Y and still denote it by Zar( Y ).
The components of Zar( Y ) are algebraic in the sense that they are analytic and semi-algebraic subsets of X + . We refer to the Appendix B of [4] for more on these notions. 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 Y be the monodromy group attached to Y . Notice that Y is the stabiliser of Y in . Then, with our assumptions, Therefore, We also have α Zar(Y ) ⊃ αY .
The same argument with α −1 instead of α shows that the reverse inclusion holds and therefore It follows that Zar(Y ) is stabilised by Y . Consider the stabiliser G Y of Zar(Y ) in G(R). Since Zar(Y ) is semi-algebraic and the action of G(R) + on X + is semi-algebraic, G Y is semi-algebraic. Furthermore, G Y is analytically closed and hence is a real algebraic group. Since G Y contains Y which is Zariski dense in G R , we see that G Y = G(R) + . It follows that Zar(Y ) = X + 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. This is known as Ratner's theory. The orginal paper by Ratner is [9]. Ratner's theory has been first applied to Shimura varieties by Clozel and Ullmo, see [2]. The contents of this section are mainly based on Section 3 of [2,12].
Let G be a semi-simple algebraic group over Q. We can 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 be an arithmetic lattice in G(R) + and let = \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 ∩ is a lattice in F. In particular F ∩ \F is closed in \G(R) + . We denote by μ F the F-invariant normalised measure on \G(R) + . 2. The subgroup L(F) generated by one-parameter unipotent subgroups of F acts ergodically on F ∩ \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 [1], lemme 3.1 and 3. 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 [12]. = \G(R) + . Note that carries a natural probability measure, the pushforward of the Haar measure on G(R) + , normalised to be a probability measure (the volume of 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.

Reformulation of the hyperbolic Ax-Lindemann theorem
Let (G, X ) be a Shimura datum. Let K be a compact open subgroup of G(A f ), = G(Q) + ∩ G(A f ) and S = \X + . Let π : X + → S be the uniformizing map. Without loss of any generality, in this section we assume that the group G is semisimple of adjoint type.
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.

An 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) + −→ \G(R) + and π x be the map \G(R) + −→ \X + sending x to gx. Recall that π : X + −→ \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 [8], Prop 7.6, the group F(R) + is generated by its one-parameter unipotent subgroups.
By Theorem 4.4, we conclude the following:

The 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 derive a stronger conclusion, namely: for a suitable parabolic subgroup P x of G(R) + .
We now apply Proposition 3.10 of [12] in our situation. We have a connected semi-simple group H such that F ⊂ H R . According to Proposition 3.10 of [12], there exists a Cartan decomposition H (R) = (P x ∩ H (R)) · (K x ∩ H (R)).
In particular K x ∩ H (R) is a maximal compact subgroup of H (R) + . This finishes the proof.
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 [12], (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 R-simple 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 [12], the Mumford-Tate group of F is G and for any point x of X + , the image of F · x in S is G. 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.
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