Subvarieties of Shimura varieties.^{∗}

Bas Edixhoven  Andrei Yafaev

May 29, 2001

Contents

1 Introduction. 2

2 The strategy. 4

3 Some preliminaries. 6 3.1 Mumford-Tate groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Variations of ℤ-Hodge structure on Shimura varieties. . . . . . . . . . . . . . . . 7 3.3 Representations of tori. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Lower bounds for Galois orbits. 9 4.2 Galois orbits and Mumford-Tate groups. . . . . . . . . . . . . . . . . . . . . . . 10 4.3 Getting rid of G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.4 Proof of Proposition 4.3.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Images under Hecke correspondences. 14

6 Density of Hecke orbits. 16

7 Proof of the main result. 19 7.3 The case where i is bounded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7.4 The case where i is not bounded. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

∗Both authors were partially supported by the European Research Training Network Contract HPRN-CT-2000-00120 “arithmetic algebraic geometry”. Adresses: IRMAR, Campus de Beaulieu, 35042 Rennes cedex, France; edix@maths.univ-rennes1.fr, ayafaev@maths.univ-rennes1.fr

1 Introduction.

The aim of this article is to prove a special case of the following conjecture of André and Oort on subvarieties of Shimura varieties. For the terminology, notation, history and results obtained so far we refer to the introduction of [8], and the references therein.

Conjecture 1.1 (André-Oort) Let (G, X) be a Shimura datum. Let K be a compact open subgroup of G(𝔸_{f}) and let S be a set of special points in Sh_{K}(G, X)(ℂ). Then every irreducible component of the Zariski closure of S in Sh_{K}(G, X)_{ℂ} is a subvariety of Hodge type.

The choice of the special case that we will prove is motivated by work of Wolfart [20], (see also Cohen and Wüstholz [4]) on algebraicity of values of hypergeometric functions at algebraic numbers. The hypergeometric functions considered in [20] are the multi-valued holomorphic F (a, b, c) on ℙ^{1}(ℂ) − {0, 1, ∞} defined by:

F (a, b, c)(z) = 1 + \frac{ab}{c}z + \frac{a(a + 1)b(b + 1)}{c(c + 1)} \frac{z^{2}}{2!} + ⋯, z ∈ ℂ, |z| < 1,

with a, b and c rational numbers, −c not in ℕ. For the following properties of the F (a, b, c) the reader is referred to [20]. The functions F (a, b, c) satisfy the differential equations:

z(z − 1)F (a, b, c)′′ + ((a + b + 1)z − c)F (a, b, c)′ + abF (a, b, c) = 0.

Suppose from now on that a, b, c, a − c and b − c are all not integers. Then, up to a factor in \overline{ℚ}^{∗}, F (a, b, c)(z) is a quotient of a certain period P (a, b, c)(z) of a certain sub abelian variety A(a, b, c, z) of the jacobian of the smooth projective model of the curve given by the equation y^{N} = x^{A}(1 − x)^{B}(1 − zx)^{C}, for suitable N, A, B, and C depending on (a, b, c), by a period P_{0}(a, b, c) depending only on (a, b, c). More precisely, P (a, b, c)(z) is the integral of the differential form y^{−1}dx (all of whose residues are zero) over a suitable cycle. The fixed period P_{0}(a, b, c) is a period of an abelian variety of CM type. Using a theorem of Wüstholz [21, Thm. 5] (on Grothendieck’s conjecture on the relation between relations between periods and correspondences (see [1]) in the case of abelian varieties) it follows that if z and one value of F (a, b, c) at z are algebraic, then A(a, b, c, z) is also of CM type, with the same type as P_{0}(a, b, c), and hence all values of F (a, b, c) at z are algebraic. Because of this, it makes sense to ask the following question:

under what conditions on (a, b, c) is the set E(a, b, c) of z in ℙ^{1}(\overline{ℚ}) − \{0, 1, ∞\} such
that F (a, b, c)(z) is in \overline{ℚ} finite?

Wolfart proposes in [20, Theorem] that the answer should depend only on the monodromy group Δ(a, b, c) (with its two-dimensional representation) of the differential equation. If this monodromy group is finite, then F (a, b, c) is algebraic over \overline{ℚ}(z), and hence E(a, b, c) equals ℙ^{1}(\overline{ℚ}) − \{0, 1, ∞\}. Suppose now that the monodromy group is infinite. Then one distinguishes two cases: Δ(a, b, c) is arithmetic or not. In terms of Shimura varieties, these two cases correspond to the image of ℙ^{1}(ℂ)−{0, 1, ∞} in a suitable moduli space of polarized abelian varieties under the map that sends z to the isomorphism class of A(a, b, c, z) being of Hodge type or not. If this image is of Hodge type, then the set of z in \overline{ℚ} such that A(a, b, c, z) is of CM type with type that of P_{0}(a, b, c) is dense in ℙ^{1}(ℂ) (even for the Archimedean topology), and Wolfart shows that under certain additional conditions on (a, b, c) the set E(a, b, c) is infinite. Suppose now that the image of ℙ^{1}(ℂ) − {0, 1, ∞} is not of Hodge type. Then Wolfart’s Theorem claims that E(a, b, c) is finite. But Walter Gubler has pointed out an error in Wolfart’s proof: in [20, §9], there is no reason that the group \overline{Δ} fixes the image of ℍ in the product of copies of the unit disk under the product of the maps D_{ω}, and therefore the identity δ^{σ}(f(w)) = f(δ(w)) for δ in \overline{Δ} is not proved.

Since by Wüstholz’s theorem all A(a, b, c, z) with z in E(a, b, c) are of a fixed CM type, hence isogeneous and hence contained in one Hecke orbit, the following theorem completes Wolfart’s program.

Theorem 1.2 Let (G, X) be a Shimura datum and let K be a compact open subgroup of G(𝔸_{f}). Let V be a finite dimensional faithful representation of G, and for h in X let V_{h} be the corresponding ℚ-Hodge structure. For x = \overline{(h, g)} in Sh_{K}(G, X)(ℂ), let [V_{x}] denote the isomorphism class of V_{h}. Let Z be an irreducible closed algebraic curve contained in Sh_{K}(G, X)_{ℂ} such that Z(ℂ) contains an infinite set of special points x such that all [V_{x}] are equal. Then Z is of Hodge type. In particular, if Z is an irreducible closed algebraic curve contained in Sh_{K}(G, X)_{ℂ} such that Z(ℂ) contains an infinite set of special points that lie in one Hecke orbit, then Z is of Hodge type.

This theorem, in the case where the special points in question lie in one Hecke orbit, was first proved in the second author’s thesis [22], in which one chapter (providing a lower bound for Galois orbits) was written by the first author. The main difference between this article and the thesis is that now we consider isomorphism classes of ℚ-Hodge structures in stead of Hecke orbits. This makes it possible to reduce the proof of the theorem to the case where Z is Hodge generic and G of adjoint type (the proof in the thesis could not achieve this and was therefore more difficult to follow).

The proof given in this article is nice because it is entirely in “(G, X)-language”; the main tools are algebraic groups and their groups of adelic points. But it is not completely satisfactory in the sense that it should be possible to proceed as in [8], i.e., without distinguishing the two cases as we do in Section 7. On the other hand, the proof in the first of these two cases can lead to a generalization to arbitrary Shimura varieties of Moonen’s result in [14, §5] (Conjecture 1.1 for moduli spaces of abelian varieties, and sets of special points for which there is a prime at which they are all “canonical”). Finally, it would be nice to replace the condition that all [V_{x}] are the same in the theorem above by the condition that all associated Mumford-Tate groups are isomorphic (this would give a statement that does not depend on the choice of a representation).

2 The strategy.

The aim of this section is to explain the strategy of the proof of Theorem 1.2. In Sections 4–6 we will prove the necessary ingredients, that will then be put together in Section 7.

We observe that the compact open subgroup K in Conjecture 1.1 is irrelevant: for (G, X) a Shimura datum, K and K′ open compact in G(𝔸_{f}) with K ⊂ K′, an irreducible subvariety Z of Sh_{K}(G, X)_{ℂ} is of Hodge type if and only if its image in Sh_{K′}(G, X)_{ℂ} is. A bit more general: if moreover g is in G(𝔸_{f}) and T_{g} is the correspondence from Sh_{K}(G, X)_{ℂ} to Sh_{K′}(G, X)_{ℂ} induced by g, then Z is of Hodge type if and only if one (or equivalently, all) of the irreducible components of T_{g}Z is (are) of Hodge type. Another observation is that the irreducible components of intersections of subvarieties of Hodge type are again of Hodge type (this is clear from the interpretation of subvarieties of Hodge type as loci where certain classes are Hodge classes). Hence there does exist a smallest subvariety of Hodge type of Sh_{K}(G, X)_{ℂ} that contains Z; our first concern is now to describe that subvariety.

Proposition 2.1 Let (G, X) be a Shimura datum, K a compact open subgroup of G(𝔸_{f}) and let Z be a closed irreducible subvariety of Sh_{K}(G, X)_{ℂ}. Let s in Z be Hodge generic. Let (x, g) in X × G(𝔸_{f}) lie over s, and let G′ be the Mumford-Tate group of x. Then we have a morphism of Shimura data from (G′, X′) to (G, X) with X′ the G′(ℝ)-conjugacy class of x. Let K′ be the intersection of G′(𝔸_{f}) and gKg^{−1}. Then the inclusion of G′ in G, followed by right multiplication by g induces a morphism f: Sh_{K′}(G′, X′)_{ℂ} → Sh_{K}(G, X)_{ℂ}. This morphism is finite and its image contains Z. Let Z′ be an irreducible component of f^{−1}Z. Then Z is of Hodge type if and only if Z′ is.

Proof. This follows from Proposition 2.8 and Section 2.9 of [13]. □

Proposition 2.1 shows that Conjecture 1.1 is true if and only if it is true for S whose Zariski closure is irreducible and Hodge generic. Similarly, Proposition 2.1 reduces the proof of Theorem 1.2 to the case where Z is Hodge generic. We note that even if Z(ℂ) has an infinite intersection with the Hecke orbit of a special point, this is not necessarily so for Z′, because the inverse image in Sh_{K′}(G′, X′)_{ℂ} of a Hecke orbit in Sh_{K}(G, X)_{ℂ} is a disjoint union of a possibly infinite number of Hecke orbits. This explains why we work with equivalence classes of ℚ-Hodge structures.

Proposition 2.2 Let (G, X) be a Shimura datum, let G^{ad} be the quotient of G by its center, and let X^{ad} be the G^{ad}(ℝ)-conjugacy class of morphisms from 𝕊 to G_{ℝ}^{ad} that contains the image of X. Let K^{ad} be a compact open subgroup of G^{ad}(𝔸_{f}), and let K be a compact open subgroup of G(𝔸_{f}) whose image in G^{ad}(𝔸_{f}) is contained in K^{ad}. Then the induced morphism from Sh_{K}(G, X)_{ℂ} to Sh_{K^{ad}}(G^{ad}, X^{ad})_{ℂ} is finite. Let Z be a closed irreducible subvariety of Sh_{K}(G, X)_{ℂ}, and let Z^{ad} be its image in Sh_{K^{ad}}(G^{ad}, X^{ad})_{ℂ}. Then Z is of Hodge type if and only if Z^{ad} is.

Proof. By [13, §2.1], X is just a union of connected components of X^{ad}. Let S and S^{ad} be the connected components of Sh_{K}(G, X)_{ℂ} and of Sh_{K^{ad}}(G^{ad}, X^{ad})_{ℂ} that contain Z and Z^{ad}, respectively. Let X^{+} be a connected component of X and let g in G(𝔸_{f}) be such that S is the image in Sh_{K}(G, X)_{ℂ} of X^{+} × {g}. Then the inverse images of Z and Z^{ad} in X^{+} × {g} are equal, hence the property of being of Hodge type for them is equivalent. □

We want to use Proposition 2.2 to reduce the proof of Theorem 1.2 to the case where G is semi-simple of adjoint type. In order to do that, all we need to do is to construct a faithful representation W of G^{ad} such that Z^{ad}(ℂ) contains a Zariski dense set of special points x with all [W_{x}] equal.

Construction 2.3 Let G be a reductive algebraic group over ℚ and let V be a faithful finite dimensional representation. Let E be a finite extension of ℚ such that the representation V_{E} of G_{E} is a direct sum V_{1} ⊕ ⋯ ⊕ V_{r} with each V_{i} absolutely irreducible. Let C be the center of G; then C_{E} acts via a character χ_{i} on V_{i}. For each i, we let d_{i} = dim_{E}(V_{i}), and we define W_{i} := V_{i}^{⊗_{E}d_{i}} ⊗_{E} det_{E}(V_{i})^{∗}. Then W := W_{1} ⊕ ⋯ ⊕ W_{r} is a faithful representation of G_{E}^{ad}. We get a faithful representation of G^{ad} on W as ℚ-vector space via the sequence of injective morphisms of algebraic groups:

G^{ad} → Res_{E/ℚ}G_{E}^{ad} → Res_{E/ℚ}GL_{E}(W) → GL_{ℚ}(W).

Suppose now that Σ is a Zariski dense subset of X such that the ℚ-Hodge structures V_{x} with x in Σ are all isomorphic to a fixed ℚ-Hodge structure H. Then the ℚ-Hodge structures with E-coefficients E ⊗ V_{x} are all isomorphic to E ⊗ H. Now E ⊗ H is a direct sum of finitely many simple ℚ-Hodge structures with E-coefficients. Hence there are, up to isomorphism, only finitely many ways to decompose E ⊗ H into a direct sum of r terms. Hence Σ is a finite disjoint union of subsets Σ_{i} such that for each i the E ⊗ V_{x} = V_{1,x} ⊕ ⋯ ⊕ V_{r,x} with x in Σ_{i} are all isomorphic term by term. It follows that the W_{x} with x in Σ_{i} are all isomorphic. So it remains to prove Theorem 1.2 in the case where G is semi-simple of adjoint type and Z Hodge generic.

At this point we can describe the strategy of the proof of the main result. So let the notation now be as in Theorem 1.2, and suppose that G is semi-simple of adjoint type, that Z is Hodge generic, and that K is neat. Let Σ be an infinite subset of special points of Z such that all [V_{x}] with x in Σ are equal. Let S be the irreducible component of Sh_{K}(G, X) that contains Z. We will show that there exist g in G(𝔸_{f}) such that an irreducible component T_{g}^{0} of the Hecke correspondence on S induced by g has the property that T_{g}^{0}Z = Z = T_{g^{−1}}^{0}Z (with T_{g^{−1}}^{0} the transpose of T_{g}^{0}) and is such that all the T_{g}^{0} + T_{g^{−1}}^{0}-orbits in S are dense (for the Archimedean topology). This clearly implies that Z = S, so that Z is of Hodge type.

To find such a g, we proceed as follows. We will take g in G(ℚ_{p}), for some prime number p. For all but finitely many p, the image of Z under each irreducible component of any T_{g} with g in G(ℚ_{p}) is either empty or irreducible. The proof of this will be given in Section 5, its main ingredient is Theorem 5.2 by Nori. The density of all T_{g}^{0}+T_{g^{−1}}^{0}-orbits will be proved in Section 6, under the assumption that no image of g under projection to a simple factor H of G is contained in a compact subgroup of H(ℚ_{p}). To get the equalities T_{g}^{0}Z = Z and T_{g^{−1}}^{0}Z = Z we try to find g such that T_{g}Z ∩ Z contains a large number of the given special points, compared to the degree of the correspondence T_{g}. In doing this, we distinguish two cases. In one case, the intersection will contain at least one big Galois orbit. In the other case, it contains infinitely many of the given special points. The main ingredient here is the description of the Galois action on special points, plus a lower bound on the number of points in the Galois orbits of our given special points that will be established in Section 4.

3 Some preliminaries.

3.1 Mumford-Tate groups.

For V a free ℤ-module of finite rank, we define GL(V) to be the group scheme given by GL(V)(A) = GL_{A}(V_{A}) for all rings A. For h a ℤ-Hodge structure, i.e., a free ℤ-module of finite rank, together with a morphism h: 𝕊 → GL(V)_{ℝ}, we let MT(V, h) be the Zariski closure in GL(V) of the usual Mumford-Tate group MT(V_{ℚ}, h) in GL(V)_{ℚ}.

3.2 Variations of ℤ-Hodge structure on Shimura varieties.

Let (G, X) be a Shimura datum, K a neat compact open subgroup of G(𝔸_{f}) and ρ: G → GL_{n} a representation that factors through G → G^{ad}, such that ρ(K) is in GL_{n}(ℤ̂). Then there is a variation of ℤ-Hodge structures V on Sh_{K}(G, X) constructed as follows. On X × G(𝔸_{f})/K, we consider the variation of ℤ-Hodge structure V_{1} whose restriction to X ×\{\overline{g}\} is ℚ^{n}∩ρ(g)(ℤ̂^{n})×X (with the ℚ-Hodge structure on ℚ^{n} × {x} is given by the morphism ρ_{ℝ} ∘ x from 𝕊 to GL_{n,ℝ}). Then G(ℚ) acts on V_{1}, and the quotient is the V that we want (for each (x, \overline{g}) in X × G(𝔸_{f})/K and q in G(ℚ) that stabilizes (x, \overline{g}), the image of q in G^{ad}(ℚ) is trivial, hence ρ(q) is the identity).

A more conceptual way to describe V is as follows. We consider two actions of G(ℚ)×G(𝔸_{f}) on 𝔸_{f}^{n} × X × G(𝔸_{f}) given by:

(q, k) ∗_{1} (v, x, g) = (qv, qx, qgk),
(q, k) ∗_{2} (v, x, g) = (k^{−1}v, qx, qgk).

The first action stabilizes ℚ^{n}×X×G(𝔸_{f}), and the second stabilizes ℤ̂^{n}×X×G(𝔸_{f}). The quotient by the first action gives a locally constant sheaf V_{ℚ} of ℚ-vector spaces on Sh_{K}(G, X)(ℂ), and the second one a locally constant sheaf Vℤ̂ of ℤ̂-modules. The automorphism:

𝔸_{f}^{n} × X × G(𝔸_{f}) ⟶ 𝔸_{f}^{n} × X × G(𝔸_{f}), (v, x, g) ↦ (g^{−1}v, x, g)

transforms the first action into the second, hence gives an isomorphism between the two locally constant sheaves of 𝔸_{f}-modules on Sh_{K}(G, X)(ℂ). Then V is the “intersection” of V_{ℚ} and Vℤ̂ in V_{𝔸_{f}}.

3.3 Representations of tori.

A torus over a scheme S is an S-group scheme T that is of the form 𝔾_{mS}^{r}, locally for the fpqc topology on S ([7, Exp. IX, Déf. 1.3]). If S is normal and noetherian, then a torus T/S is split over a suitable surjective finite etale cover of S′ → S, i.e., T_{S′} is isomorphic to some 𝔾_{mS′}^{r} ([7, Exp. IX, Thm. 5.16]); one may take S′ → S Galois and S′ connected. If S is integral normal and noetherian, with generic point η, then any isomorphism f: T_{1,η} → T_{2,η} with T_{1} and T_{2} tori over S extends uniquely to an isomorphism over S (use [7, Exp. X, Cor. 1.2]).

For S a connected scheme, T ≅ 𝔾_{mS}^{r} a split torus and V an O_{S}-module, it is equivalent to give a T -action on V or an X-grading on V, with X = Hom(T, 𝔾_{mS}) the character group of T ([7, Exp. I, §4.7]).

Suppose now that S is an integral normal noetherian scheme, that T is a torus over S and that π: S′ → S is a connected finite etale Galois cover with group Γ over which T is split. Let X be the character group of T_{S′}; then X is a free ℤ-module of finite rank with a Γ-action. Then, to give an action of T on a quasi-coherent O_{S}-module V is equivalent to giving an X-grading V_{S′} = π^{∗}V = ⊕_{x}V_{S′,x} such that for all γ in Γ and all x in X one has γV_{S′,x} = V_{S′,γx} (to see this, use finite etale descent of quasi-coherent modules as in [3, §6.2]). If V and W are two representations of T on locally free O_{S}-modules of finite rank, then V and W are isomorphic, locally for the Zariski topology on S, if and only if for all x in X the ranks of V_{S′,x} and W_{S′,x} are equal (use that Hom_{O_{S}}(V, W)^{T} is a direct summand of Hom_{O_{S}}(V, W), whose formation commutes with base change).

Lemma 3.3.1 Let p be a prime number, let T be a torus over ℤ_{p}, let V be a free ℤ_{p}-module of finite rank equipped with a faithful action of T_{ℚ_{p}} on V_{ℚ_{p}}. Let T′ be the scheme theoretic closure of T_{ℚ_{p}} in GL(V). Then the following conditions are equivalent:

1. T′_{𝔽_{p}} is a torus;

2. T′ is a torus;

3. the action of T_{ℚ_{p}} on V_{ℚ_{p}} extends to an action of T on V;

4. T stabilizes the lattice V in the sense that for all finite extensions K of ℚ_{p} the lattice V_{O_{K}} is fixed by all elements of T (O_{K}).

The set of ℤ_{p}-lattices in V_{ℚ_{p}} that are fixed by T form exactly one orbit under C(ℚ_{p}), where C denotes the centralizer of T in GL(V).

Proof. Suppose first that T′_{𝔽_{p}} is a torus. Then T′, being a flat group scheme affine and of finite type over ℤ_{p} whose fibers over 𝔽_{p} and ℚ_{p} are tori, is a torus by [7, Exp. X, Cor. 4.9].

Suppose that T′ is a torus. Then T′ = T by [7, Exp. X, Cor. 1.2]). Hence the action of T_{ℚ_{p}} on V_{ℚ_{p}} extends to an action of T on V.

Now suppose that the action of T_{ℚ_{p}} on V_{ℚ_{p}} extends to an action of T on V. Then T stabilizes V, by definition. Also, the description above of representations of tori shows that T acts faithfully on V, so that T is a closed subscheme of GL(V), flat over ℤ_{p}, and hence equal to the scheme theoretic closure of its generic fiber. So T′ = T and T′_{𝔽_{p}} is a torus.

Finally, suppose that T stabilizes V. Let K be the splitting field of T_{ℚ_{p}}. Then T_{O_{K}} is a split torus, and the action of T_{ℚ_{p}} on V_{ℚ_{p}} is given by an X-grading of V_{K}, where X is the character group of T_{K}. Let m be an integer that is prime to p such that the characters x in X with V_{K,x} ≠ 0 have distinct images in X/mX. Since T stabilizes V, the m-torsion subgroup scheme T [m] of T acts on V. This action corresponds to an X/mX-grading on V_{O_{K}} that is compatible with the X-grading on V_{K}. Hence the X-grading on V_{K} extends to an X-grading on V_{O_{K}}, which shows that the action of T_{ℚ_{p}} on V_{ℚ_{p}} extends to an action of T on V.

Finally, let S be the set of ℤ_{p}-lattices in V_{ℚ_{p}} that are fixed by T. Let W be any ℤ_{p}-lattice in V_{ℚ_{p}}. The T_{ℚ_{p}}-action on W_{ℚ_{p}} = V_{ℚ_{p}} corresponds to the X-grading on V_{K}. By finite etale descent, the O_{K}-submodule ⊕_{x}(W_{O_{K}} ∩V_{K,x}) of W_{O_{K}} is of the form W′_{O_{K}} for a unique ℤ_{p}-lattice W′ contained in W. Then W′_{O_{K}} is the direct sum of the W′_{O_{K}} ∩ V_{K,x}, hence is a representation of T. Hence W′ is in S; in fact, W′ is the largest sublattice of W that is fixed by T. In particular, S is not empty. Let now V_{1} and V_{2} be two elements of S. Then both are representations of T. Since for each x the V_{i,O_{K},x} are of equal rank, V_{1} and V_{2} are isomorphic as representations of T. Let g: V_{1} → V_{2} be an isomorphism. Then g is an element of C(ℚ_{p}) that sends V_{1} to V_{2}. □

4 Lower bounds for Galois orbits.

The aim of this section is to give certain lower bounds for the sizes of Galois orbits of special points on a Shimura variety. To be precise, we will prove the following theorem.

Theorem 4.1 Let (G, X) be a Shimura datum, with G semi-simple of adjoint type, and let K be a neat compact open subgroup of G(𝔸_{f}). Via a suitable faithful representation we view G as a closed algebraic subgroup of GL_{n,ℚ}, such that K is in GL_{n}(ℤ̂). Let V be the induced variation of ℤ-Hodge structures on Sh_{K}(G, X)_{ℂ}. Let s_{0} be a special point of Sh_{K}(G, X)_{ℂ}. Let F ⊂ ℂ be a number field over which the Shimura variety Sh_{K}(G, X)_{ℂ} has a canonical model Sh_{K}(G, X)_{F}. Then there exist real numbers c_{1} > 0 and c_{2} > 0 such that for all s in Sh_{K}(G, X)_{F} (\overline{ℚ}) such that the ℚ-Hodge structure V_{s,ℚ} is isomorphic to V_{s_{0},ℚ}, we have:

|Gal(\overline{ℚ}/F)·s| > c_{1} ∏_{\{p prime | MT(V_{s})_{𝔽_{p}} is not a torus\}} c_{2}p.

Let us note that varying F and K does not affect the statement of the theorem: if F′ and K′ satisfy the same hypotheses as F and K, then the sizes of the Galois orbits differ by a bounded factor, and K′ ∩ K has finite index in both K and K′. In the course of the proof of Theorem 4.1 we will assume that F is the splitting field of M_{ℚ}, with M the Mumford-Tate group of s_{0}.

We note that M(ℝ) is compact: the kernel of the action of G(ℝ) on X consists precisely of the product of the compact factors, i.e., if G_{ℝ} is the product of simple G_{i}’s, then the kernel is the product of the G_{i}(ℝ) that are compact, and M(ℝ) stabilizes a point in the hermitian manifold X. It follows that M(ℚ) is discrete in M(𝔸_{f}).

4.2 Galois orbits and Mumford-Tate groups.

Let the notation be as in Theorem 4.1. We choose a set of representatives R in G(𝔸_{f}) for the quotient G(ℚ)\G(𝔸_{f})/K; note that R is finite. Then for s in Sh_{K}(G, X)_{F} (ℂ) there exists a unique g_{s} in R and an element s̃ in X unique up to Γ_{s} := G(ℚ) ∩ g_{s}Kg_{s}^{−1}, such that s = \overline{(s̃, g_{s})}. We fix a choice for s̃_{0}.

Let s = \overline{(s̃, g_{s})} be in Sh_{K}(G, X)_{F} (\overline{ℚ}) such that the ℚ-Hodge structure V_{s,ℚ} is isomorphic to V_{s_{0},ℚ}. Then s̃ gives an embedding of the Mumford-Tate group MT(s)_{ℚ} in G, and an inclusion of Shimura data from (MT(s)_{ℚ}, {s̃}) in (G, X). Note that MT(s)_{ℚ} is isomorphic to M_{ℚ}, hence has splitting field F. This gives morphisms of Shimura varieties over F:

(4.2.1)  Sh_{K_{s}∩MT(s)(𝔸_{f})}(MT(s)_{ℚ})_{F} ⟶ Sh_{K_{s}}(G, X)_{F} ⟶^{·g_{s}} Sh_{K}(G, X)_{F} ,

with the first one given by the inclusion, the second one by right multiplication by g_{s}, and with K_{s} = K ∩ g_{s}Kg_{s}^{−1}. By construction, V_{s,ℚ} and V_{s_{0},ℚ} are both ℚ^{n} with Hodge structures s̃ and s̃_{0} : 𝕊 → GL_{n,ℝ}, respectively. Let h in GL_{n}(ℚ) be an automorphism of ℚ^{n} that is an isomorphism from V_{s_{0},ℚ} to V_{s,ℚ}. Then we have s̃ = inn_{h} ∘ s̃_{0}. It follows that the reciprocity morphisms rs̃ and rs̃_{0}, viewed as morphisms of tori over ℚ from Res_{F/ℚ}𝔾_{mF} to G with images MT(s)_{ℚ} and M_{ℚ} are related by: rs̃ = inn_{h} ∘ rs̃_{0}. In particular, the isomorphism from M_{ℚ} to MT(s)_{ℚ} induced by inn_{h} gives an isomorphism of Shimura varieties over F:

(4.2.2)  Sh(M_{ℚ})_{F} ⟶ Sh(MT(s)_{ℚ})_{F}

The Galois group Gal(\overline{ℚ}/F) acts on Sh(M_{ℚ}) = M(ℚ)\M(𝔸_{f}) via its maximal abelian quotient, which we view, via class field theory, as a quotient of (𝔸 ⊗ F)^{∗}/(ℝ ⊗ F)^{∗,+}. The action of Gal(\overline{ℚ}/F) is then given by rs̃_{0} : Res_{F/ℚ}𝔾_{mF} → M_{ℚ}. We note that (𝔸 ⊗ F)^{∗}/(ℝ ⊗ F)^{∗,+} is the product of (𝔸_{f} ⊗ F)^{∗} and the finite group of connected components of (ℝ ⊗ F)^{∗}. Hence the size of the Gal(\overline{ℚ}/F)-orbit of s in the Sh_{K}(G, X)_{F} (\overline{ℚ}) is, up to a bounded factor which is independent of s, the size of the (𝔸_{f} ⊗ F)^{∗}-orbit. Since F^{∗,+} in (𝔸_{f} ⊗ F)^{∗} acts trivially, and since the class group F^{∗}\(𝔸_{f} ⊗ F)^{∗}/(ℤ̂ ⊗ O_{F})^{∗} is finite, proving the lower bound we want for the (𝔸_{f} ⊗ F)^{∗}-orbits is equivalent to proving it for the (ℤ̂ ⊗ O_{F})^{∗}-orbits. Moreover, since the set R is finite, and since each ·g_{s}: Sh_{K_{s}}(G, X)_{F} → Sh_{K}(G, X)_{F} is finite, it is enough to prove the lower bound for the (ℤ̂ ⊗ O_{F})^{∗}-orbit in Sh_{K_{s}}(G, X)_{F} (\overline{ℚ}).

4.3 Getting rid of G.

In order to simplify our task (i.e., proving Theorem 4.1) we introduce the pair (GL_{n,ℚ}, Y), with Y the GL_{n}(ℝ) conjugacy class in Hom_{ℝ}(𝕊, GL_{n,ℝ}) that contains the image of X under ρ. Of course, this pair is not a Shimura datum (if n > 2).

For s = \overline{(s̃, g_{s})} in Sh_{K}(G, X)_{F} (\overline{ℚ}) and h in GL_{n}(ℚ) as in the previous section, we consider the following commutative diagram (of sets):

Sh_{K_{s}∩MT(s)(𝔸_{f})}(MT(s)_{ℚ})(ℂ) ⟶  Sh_{K_{s}}(G, X)(ℂ)⟶^{·g_{s}} Sh_{K}(G, X)(ℂ)
i_{s}↑  ↓
Sh_{K′_{s}}(M_{ℚ})(ℂ)  ⟶^{f_{s}} GL_{n}(ℚ)\ (Y × GL_{n}(𝔸_{f})) /GL_{n}(ℤ̂),

where K′_{s} is the subgroup of M(𝔸_{f}) that corresponds to K_{s} ∩ MT(s)(𝔸_{f}) via inn_{h}, so that i_{s} is bijective, and where f_{s} is induced by the morphism (M_{ℚ}, {s_{0}}) → (GL_{n,ℚ}, Y) given by the inclusion of M_{ℚ} in GL_{n,ℚ} followed by inn_{h}, and s_{0} ↦ inn_{h} ∘ ρ ∘ s_{0}.

The set GL_{n}(ℚ)\(Y × GL_{n}(𝔸_{f}))/GL_{n}(ℤ̂) is the set of isomorphism classes of ℤ-Hodge structures W such that W_{ℝ} is isomorphic to V_{s_{0},ℝ} (to (y, g) in Y × GL_{n}(𝔸_{f}) one associates the Hodge structure y: 𝕊 → GL_{n,ℝ} on the lattice ℚ^{n} ∩ gℤ̂^{n}). Its subset of isomorphism classes of W such that W_{ℚ} is isomorphic to V_{s_{0},ℚ} is in bijection with S = C(ℚ)\GL_{n}(𝔸_{f})/GL_{n}(ℤ̂), where C_{ℚ} is the centralizer in GL_{n,ℚ} of M_{ℚ}. Being the centralizer of a torus, C_{ℚ} is connected and reductive (actually, C\overline{ℚ} is isomorphic to a product of GL_{d_{i},\overline{ℚ}}’s).

Note that M(𝔸_{f}) acts on S by left-multiplication. The image in S of (ℤ̂ ⊗ O_{F})^{∗}·s is now simply the (ℤ̂⊗O_{F})^{∗}-orbit of the class of h, where (ℤ̂⊗O_{F})^{∗} acts via rs̃_{0} : (ℤ̂⊗O_{F})^{∗} → M(𝔸_{f}) and left multiplication by M(𝔸_{f}). Let L = GL_{n}(𝔸_{f})/GL_{n}(ℤ̂) be the set of ℤ-lattices in ℚ^{n} (or, equivalently, ℤ̂-lattices in 𝔸_{f}^{n}). The following lemma can be seen as a comparison between the sizes of the M(ℤ̂)-orbits M(ℤ̂)\overline{h}^{S} and M(ℤ̂)\overline{h}^{L} of h in L and in S.

Lemma 4.3.1 There exists an integer m ≥ 1 such that for all h in GL_{n}(𝔸_{f}) we have:

|M(ℤ̂) \overline{h}^{S}| ≥ |M(ℤ̂) \overline{h}^{L}|, where M(ℤ̂) acts on L via mth powers.

Proof. The fact that C(ℝ) is not necessarily compact gives us some trouble, and so we decompose C_{ℚ} up to isogeny. There exists a connected reductive subgroup M′_{ℚ} of C_{ℚ} such that we have a short exact sequence:

(4.3.2)  1 ⟶ F_{ℚ} ⟶ M_{ℚ} × M′_{ℚ} ⟶ C_{ℚ} ⟶ 1,

with F_{ℚ} = M_{ℚ} ∩ M′_{ℚ} a finite group scheme. To get such an M′_{ℚ}, consider the decomposition up to isogeny of C_{ℚ} into its center (that contains M_{ℚ}) and its semi-simple part, and use that the center decomposes up to isogeny into M_{ℚ} and another factor. All groups in (4.3.2) are closed subgroup schemes of GL_{n,ℚ}, which gives each of them a ℤ-structure. There is a non-empty open part Spec(ℤ[1/N]) of Spec(ℤ) over which F is finite etale, M and M′ tori, C reductive and the sequence exact (for the etale topology). Let e ≥ 1 be an integer that annihilates F_{ℚ} and M(ℚ) ∩ U, where U is the maximal compact (open) subgroup of M(𝔸_{f}) (see [17, Prop. 3.16]).

Let h be in GL_{n}(𝔸_{f}). Let x be in M(ℤ̂) and suppose that x stabilizes \overline{h} in S. Then there exist q in C(ℚ) and k in GL_{n}(ℤ̂) such that in GL_{n}(𝔸_{f}) we have:

(4.3.3)  xh = qhk, i.e., x = q·hkh^{−1}.

Since x and q commute, we have:

(4.3.4)  x^{e} = q^{e}·hk^{e}h^{−1}.

As H^{1}(Gal(\overline{ℚ}/ℚ), F (\overline{ℚ})) is annihilated by e, there exist q_{1} in M(ℚ) and q_{2} in M′(ℚ) such that q^{e} = q_{1}q_{2} in C(ℚ). By (4.3.3), hkh^{−1} is in C(𝔸_{f}). Since the sequence (4.3.2) extends as indicated over ℤ[1/N], there are k_{1} and k_{2} in GL_{n}(𝔸_{f}) such that:

(4.3.5)  hk^{e}h^{−1} = hk_{1}h^{−1}·hk_{2}h^{−1}, with hk_{1}h^{−1} in M(𝔸_{f}) and hk_{2}h^{−1} in M′(𝔸_{f}).

Rewriting (4.3.4) gives:

(4.3.6)  x^{e} = q_{1}q_{2}·hk_{1}h^{−1}·hk_{2}h^{−1} = q_{1}·hk_{1}h^{−1}·q_{2}hk_{2}h^{−1}.

It follows that q_{2}hk_{2}h^{−1} is in F (𝔸_{f}), hence that:

(4.3.7)  x^{e^{2}} = q_{1}^{e}·hk_{1}^{e}h^{−1}, in M(𝔸_{f}).

This identity shows that q_{1}^{e} is in U ∩ M(ℚ). Hence q_{1} is in U ∩ M(ℚ), and since e annihilates U ∩ M(ℚ), we have:

(4.3.8)  x^{e^{2}} = hk_{1}^{e}h^{−1}, in M(𝔸_{f}).

We conclude that for m = e^{2}, we have, for x in M(ℤ̂) stabilizing \overline{h}^{S}, that x^{m} stabilizes \overline{h}^{L} (for the usual action of M(𝔸_{f}) on L). □

Lemma 4.3.1 implies that in order to prove Theorem 4.1 we may as well prove the lower bound in question for the action of M(ℤ̂) on the set L, if we replace the reciprocity morphism rs̃_{0} by mrs̃_{0}, for a suitable m ≥ 1. Note that mrs̃_{0} is still surjective, as morphism of tori over ℚ.

For h in GL_{n}(𝔸_{f}), its class in L is a ℤ-lattice V_{h} in ℚ^{n}, and ℚ^{n} is equipped with an action of M_{ℚ}. Let MT(V_{h}) be the scheme theoretic closure of M_{ℚ} in GL(V_{h}). In this situation, Lemma 3.3.1 says that MT(V_{h})_{𝔽_{p}} is a torus if and only if M fixes V_{h}.

We have reduced the proof of Theorem 4.1 to the following statement, that will be proved in the next section.

Proposition 4.3.9 Let n be a positive integer. Let T be a torus over ℤ[1/n], acting on a free ℤ[1/n]-module V of finite rank. Let S = GL(V_{𝔸_{f}})/GL(V_{ℤ̂[1/n]}) be the set of ℤ[1/n]-lattices in V_{ℚ}. Let T (ℤ̂[1/n]) act on S via left multiplications. For W in S, let P_{W} be the set of primes p that do not divide n and such that T_{ℤ_{p}} does not fix W_{ℤ_{p}}. Then there exists a positive real number c such that for each W in S we have:

|T (ℤ̂[1/n])·W | ≥ ∏_{p∈P_{W}} cp.

Equivalently, as S is a restricted product over the primes not dividing n, there is a positive real number c such that for all p not dividing n, and all ℤ_{p}-lattices W in V_{ℚ_{p}} that are not fixed by T_{ℤ_{p}}, we have:

|T (ℤ_{p})·W | ≥ cp.

4.4 Proof of Proposition 4.3.9.

We will now prove Proposition 4.3.9. Let K be a splitting field of T_{ℚ}, and let X be the group of characters of T_{K}. Let p be a prime number that does not divide n, and let W be a ℤ_{p}-lattice in V_{ℚ_{p}} that is not fixed by T_{ℤ_{p}}. Let W′ be the largest sublattice of W that is fixed by T_{ℤ_{p}}, as constructed in the last part of the proof of Lemma 3.3.1. Let L be the kernel of multiplication by p on W/W′. We view L as a sub 𝔽_{p}-vector space of W′_{𝔽_{p}} via L = (W ∩ p^{−1}W′)/W′ = (pW ∩ W′)/pW′, where the second equality comes from multiplication by p.

An element of T (ℤ_{p}) that stabilizes W, also stabilizes L. Hence T (ℤ_{p})·W has at least as many elements as T (𝔽_{p})·L, where we let T (𝔽_{p}) act on the set of subspaces of W′_{𝔽_{p}}. By construction, L is non-zero, and has intersection zero with W′_{\overline{𝔽}_{p},x} for each x in X, where W′_{\overline{𝔽}_{p}} = ⊕_{x}W′_{\overline{𝔽}_{p},x} is the X-grading corresponding to the T_{𝔽_{p}}-action on W′_{𝔽_{p}}. This implies that L, viewed as an 𝔽_{p}-valued point in some Grassmannian, is not fixed by the action of T_{𝔽_{p}}.

Let T′_{𝔽_{p}} be the stabilizer of L in T_{𝔽_{p}}. Lemma 4.4.1 below says that the order of the group of connected components of T′_{\overline{𝔽}_{p}} is bounded independently of p and W. Put T′′_{𝔽_{p}} := T_{𝔽_{p}}/T′_{𝔽_{p}}. Then T′′_{𝔽_{p}} is a non-trivial torus over 𝔽_{p}, and by Lemma 4.4.2, the morphism T (𝔽_{p}) → T′′_{𝔽_{p}}(𝔽_{p}) has its cokernel of order bounded independently of p and W. The proof of Proposition 4.3.9 is now finished by noting that T′′_{𝔽_{p}}(𝔽_{p}) has at least p−1 elements (see [15, Lemma 3.5]).

Lemma 4.4.1 Let k be an algebraically closed field, and T a k-torus. Let V be a finite dimensional k-vector space with an action by T. Then the set of stabilizers T_{W}, for W running through the set of subspaces of V, is finite. The set of groups of connected components of these stabilizers, up to isomorphism, is finite, and bounded in terms of the dimension of V and the set of characters of T that do occur in V.

Proof. Let us consider the set S of subspaces W of a fixed dimension, call it d. Then we have a natural injection from S into ℙ(Λ^{d}(V)), compatible with the T (k)-action. The image of a W under this map is the line generated by w_{1} ∧⋯∧w_{d}, where w is any k-basis of W. Hence the set of stabilizers of the elements of S is contained in the set of stabilizers of elements of ℙ(Λ^{d}(V)). This reduces the proof of the lemma to the case of one-dimensional subspaces (we replace V by Λ^{d}(V)).

Let X be the character group of T, and let V = ⊕_{x}V_{x} be the X-grading of V given by the T -action. Of course, almost all V_{x} are zero. For v in V, we have v = ∑_{x} v_{x}, and we let Supp(v) be the set of x with v_{x} ≠ 0. For v non-zero, the stabilizer in T of the element kv in ℙ(V) is the intersection of the kernels of the x − x′ with x and x′ in Supp(v). Since the set of such differences is finite, the claim follows. □

Lemma 4.4.2 Let T′ be the kernel of a surjective morphism T → T′′ of tori over k, with k a finite field. Let Φ be the group of connected components of T ′ ; \overline{k}. Then we have the exact sequence:

0 ⟶ T′(k) ⟶ T (k) ⟶ T′′(k) ⟶ H^{1}(Gal(\overline{k}/k), Φ),

and the upper bound:

|H^{1}(Gal(\overline{k}/k), Φ)| ≤ |Φ|.

Proof. One has, of course, the long exact sequence coming from Galois cohomology:

0 ⟶ T′(k) ⟶ T (k) ⟶ T′′(k) ⟶ H^{1}(Gal(\overline{k}/k), T′(\overline{k})).

We combine this with the sequence:

H^{1}(Gal(\overline{k}/k), (T′)^{0}(\overline{k})) ⟶ H^{1}(Gal(\overline{k}/k), T′(\overline{k})) ⟶ H^{1}(Gal(\overline{k}/k), Φ)

coming from the short exact sequence:

0 ⟶ (T′)^{0} ⟶ T′ ⟶ T′/(T′)^{0} ⟶ 0.

Lang’s Theorem ([17, Thm. 6.1]) implies that H^{1}(Gal(\overline{k}/k), (T′)^{0}(\overline{k})) = 0. The upper bound for |H^{1}(Gal(\overline{k}/k), Φ)| follows from the fact that this cohomology group is just the group of coinvariants for the action of Gal(\overline{k}/k) on Φ. □

5 Images under Hecke correspondences.

In this section we prove that the images under irreducible components of certain Hecke correspondences of an irreducible Hodge generic subvariety of a Shimura variety defined by a semisimple algebraic group of adjoint type are irreducible.

Theorem 5.1 Consider a Shimura variety defined by a Shimura datum (G, X) where G is a semi-simple algebraic group of adjoint type. Let K be a neat compact open subgroup of G(𝔸_{f}) that is the product of compact open subgroups K_{p} of G(ℚ_{p}). Let X^{+} be a connected component of X and let S be the image of X^{+} ×{1} in Sh_{K}(G, X)_{ℂ}. Let Z be an irreducible Hodge generic subvariety of S containing a non-singular special point. Then there exists a nonzero integer n such that if q is an element of G(ℚ)^{+} := G(ℚ)∩G(ℝ)^{+} whose image in G(ℚ_{l}) is in K_{l} for every l dividing n, then T_{q}(Z) is irreducible, with T_{q} the correspondence on S given by the action of q on X^{+}.

Proof. By [6, 2.1.2] we have S = Γ\X^{+} with Γ = G(ℚ)^{+} ∩ K. For q in G(ℚ)^{+} the correspondence T_{q} is defined as follows. Consider the diagram:

S ⟵^{π} X^{+} ⟶^{q·} X^{+} ⟶^{π} S,

with π the quotient map for the action of Γ. The morphism π ∘ q· is the quotient for the action of q^{−1}Γq, hence π and π ∘ q· both factor through the quotient S_{q} = Γ_{q}\X^{+} with Γ_{q} = Γ ∩ q^{−1}Γq. With these notations, T_{q} is the correspondence:

S ⟵^{π_{q,1}} S_{q} ⟶^{π_{q,2}} S.

In particular, T_{q}Z = π_{q,2}(π_{q,1}^{−1}Z). In order to show that T_{q}Z is irreducible, it suffices to show that Z_{q} := π_{q,1}^{−1}Z is. Since π_{q,1} is a covering, restriction to the smooth loci in Z and Z_{q} gives a covering π_{q,1} : Z_{q}^{sm} → Z^{sm}. Since Z_{q}^{sm} is Zariski-dense in Z_{q}, it is enough to prove that Z_{q}^{sm} is irreducible.

In order to find an integer n as in the Theorem, we choose a faithful representation ξ of G on a ℚ-vector space V. Then ξ gives a polarizable variation of Hodge structures on the constant sheaf V_{X} ([12, 1.8]). Since K is neat, Γ acts freely on X^{+}, hence V_{X^{+}} descends to S. Let Vℤ̂ be a K-invariant ℤ̂-lattice in V_{𝔸_{f}}, and let V_{ℤ} = Vℤ̂ ∩ V. Then V_{ℤ} is a Γ-invariant lattice in V. This gives us a polarizable variation V_{ℤ}(ξ) of ℤ-Hodge structure on S, hence on Z^{sm}. Let s be a Hodge generic point in Z^{sm} and x a point of X^{+} lying above it. This gives an isomorphism between the fiber V_{ℤ}(ξ)_{s} of V_{ℤ}(ξ) at s and V_{ℤ}, such that the Mumford-Tate group of V_{ℤ}(ξ)_{s} corresponds to ξ(G).

On the other hand, the fundamental group π_{1}(Z^{sm}, s) acts on V_{ℤ}(ξ)_{s} and hence on V_{ℤ}; let Π be its image in GL(V_{ℤ}). Then Π is a finitely generated subgroup of GL(V_{ℤ}). By a theorem of André (see [13, Theorem 1.4]), the connected component of the Zariski closure of Π in the algebraic group GL(V) is ξ(G). Since Γ is an arithmetic subgroup of G(ℚ), it is Zariski dense in G, and the Zariski closure of ξ(Γ) is ξ(G).

Since V_{ℤ}(ξ) exists over S, the image Π of π_{1}(Z^{sm}, s) is contained in that of π_{1}(S, s), i.e., in ξ(Γ). The inclusion Π ⊂ ξ(Γ) →^{ξ^{−1}} Γ makes Γ into a π_{1}(Z^{sm}, s)-set. For q in G(ℚ)^{+}, the π_{1}(Z^{sm}, s)-set that corresponds to the covering π_{q,1} : Z_{q}^{sm} → Z^{sm} is isomorphic to Γ/q^{−1}Γq ∩ Γ, hence Z^{sm} is connected (and hence irreducible) if and only if Γ/q^{−1}Γq ∩ Γ is transitive. Now since π_{1}(Z^{sm}, s) and Γ have the same Zariski closure in the ℤ-group scheme GL(V_{ℤ}), Nori’s Theorem 5.2 below gives a non-zero integer n, such that for all non-zero integers m prime to n, π_{1}(Z^{sm}, s) and Γ have the same image in GL(V_{ℤ/mℤ}). (Indeed, G(ℂ) is a semi-simple complex Lie group, hence π_{1}(G(ℂ) is finite by [18, thm. 2(c)].) Let now q be in G(ℚ)^{+}, such that q_{l} is in K_{l} for all l dividing n. Then all we have to show is that there is a non-zero integer m = ∏_{l} l^{m_{l}}, prime to n, such that Γ ∩ q^{−1}Γq contains the kernel of the natural map Γ → GL(V_{ℤ/mℤ}). We have:

q^{−1}Γq ∩ Γ = q^{−1}Kq ∩ K ∩ G(ℚ)^{+} = (∏_{l} q^{−1}K_{l}q ∩ K_{l}) ∩ G(ℚ)^{+}.

For l with q_{l} ∈ K_{l} (in particular, for l dividing n), we have q^{−1}K_{l}q ∩ K_{l} = K_{l}, and we put m_{l} = 0. For the remaining finitely many l, we take m_{l} sufficiently large such that:

ker (K_{l} ⟶ GL(V_{ℤ/l^{m_{l}}ℤ})) ⊂ q^{−1}K_{l}q.

Then we have the last inclusion for all l, so by taking the product over all l we have:

ker (K ⟶ GL(V_{ℤ/mℤ})) ⊂ q^{−1}Kq, hence ker (Γ ⟶ GL(V_{ℤ/mℤ})) ⊂ q^{−1}Γq.
□

Theorem 5.2 (Nori (Thm. 5.3 of [15])) Let H be a finitely generated subgroup of GL_{n}(ℤ). Let \overline{H} be its Zariski closure in GL_{n,ℤ}. Suppose that \overline{H}(ℂ) has a finite fundamental group, then the closure of H in GL_{n}(ℤ̂) is open in the closure of \overline{H}(ℤ) in GL_{n}(ℤ̂).

Remark 5.3 A careful reader might be worried by our use of Theorem 5.3 of Nori’s article [15], because Theorem 5.2 of the same article is clearly wrong. Probably, the problem with Theorem 5.2 is only typographical: should [A, A](R̂) be replaced by [A, A](R)^{∧}, the closure of [A, A](R) in GL_{n}(R̂)? Or by [A(R), A(R)]^{∧}? Anyway, Theorem 5.3 can be deduced from Theorem 5.4, which in turn can be found in other references ([17, Thm. 7.14], [16]).

6 Density of Hecke orbits.

We will now prove the following result about the density of Hecke orbits in Shimura varieties.

Theorem 6.1 Let (G, X) be a Shimura datum with G = G_{1} × ⋯ × G_{r} a semi-simple algebraic group of adjoint type with simple factors G_{1}, …, G_{r}. Let K be a compact open subgroup of G(𝔸_{f}) that is the product of compact open subgroups K_{p} of G(ℚ_{p}). Let X^{+} be a connected component of X, let S be the image of X^{+} × {1} in Sh_{K}(G, X)_{ℂ}, and put Γ = G(ℚ)^{+} ∩ K. Then there exists an integer n such that for all prime numbers p ≥ n the following holds.

Let q be an element of G(ℚ_{p}) such that for all j the projection of q in G_{j}(ℚ_{p}) is not contained in a compact subgroup. The connected components of the correspondence that T_{q} induces on S are the T_{q_{i}}, induced by q_{i} in G(ℚ)^{+} acting on X^{+}, such that:

G(ℚ)^{+} ∩ KqK = ∐_{i} Γq_{i}^{−1}Γ.

Then, for all i and for all s in S, the T_{q_{i}} + T_{q_{i}^{−1}}-orbit ∪_{n≥0}(T_{q_{i}} + T_{q_{i}^{−1}})^{n}s is dense in S for the Archimedean topology.

Proof. Let p_{j}: G → G_{j} denote the jth projection. For each j, let n_{j} be as in Proposition 6.2 applied to G_{j} and p_{j}(K), and let n be the maximum of the n_{j}. Let p ≥ n be prime, and let q in G(ℚ_{p}) and the q_{i} in G(ℚ) be as in the Theorem. For each i, let Γ_{i} denote the subgroup of G(ℚ)^{+} generated by Γ and q_{i}. Then we have S = Γ\X^{+}, and for s in S, the T_{q_{i}} + T_{q_{i}^{−1}}-orbit of s is the image in S of a Γ_{i}-orbit in X^{+}. Hence it suffices to prove that each Γ_{i} is dense in G(ℝ)^{+}.

Let H be the closure in G(ℝ)^{+} of say Γ_{1}. Then H is a Lie subgroup of G(ℝ). We let H^{+} be the connected component of the identity. Now Γ normalizes H^{+}, hence its Lie algebra. Since Γ is Zariski dense in G_{ℝ} ([17, Thm. 4.10], note that no G_{i}(ℝ) is compact), this implies that H^{+} is a product of simple factors of G(ℝ)^{+}. We claim that if H^{+} contains a simple factor of G_{i}(ℝ)^{+}, then it contains all of G_{i}(ℝ)^{+}. This follows from two facts. One: H^{+} ∩ G(ℚ) is dense in H^{+}. Two: for every i, and for every factor G_{i,j} of G_{i,\overline{ℚ}} the map G_{i}(ℚ) → G_{i,j}(\overline{ℚ}) is injective. The second fact is a direct consequence of the statement ([2, 6.21(ii)]) that G_{i} = Res_{F/ℚ}G′_{i} for some finite extension F of ℚ and some absolutely simple group G′_{i} over F.

Let now I be the subset of {1, …, r} of the i such that H^{+} does not contain G_{i}(ℝ)^{+}, let G_{I} be the product of the G_{i} with i in I, and let f: G(ℝ)^{+} → G_{I}(ℝ)^{+} be the projection. Then H^{+} is the kernel of f, hence f(H) is discrete in G_{I}(ℝ). But then f(Γ_{1}), being a subgroup of f(H), is discrete. Now suppose that I is not empty. By Proposition 6.2, f(Γ) has infinite index in f(Γ_{1}), which contradicts the fact that f(Γ)\G_{I}(ℝ)^{+} has finite volume (see [17, Thm. 4.13]). Hence I is empty, and H = G(ℝ)^{+}. □

Proposition 6.2 Let G be a simple (hence adjoint) algebraic group over ℚ with G(ℝ) not compact. Let K be a compact open subgroup of G(𝔸_{f}) and put Γ = G(ℚ)^{+} ∩ K. Then there is an integer n such that for all prime numbers p ≥ n the following holds.

Let g be an element of G(ℚ_{p}) that is not contained in a compact subgroup. Let q be an element of G(ℚ) ∩ KgK. Then ΓqΓ contains an element q′ such that Γ has infinite index in the group generated by Γ and q′.

Proof. Let V be a finite dimensional absolutely irreducible faithful representation of G_{E}, where E is a suitable finite extension of ℚ. As Γ is Zariski dense in G\overline{ℚ} ([17, Thm. 4.10]), V is an absolutely irreducible representation of Γ. By the double centralizer theorem ([10, XVII, Cor. 3.5]), End_{E}(V) is spanned (over E) by the images of the γ in Γ. Let O_{E} be the ring of integers of E, and let V_{O_{E}} be a Γ-invariant O_{E}-lattice in V. Let γ_{1}, …, γ_{m} be in Γ whose images \overline{γ_{1}}, …, \overline{γ_{m}} span End_{E}(V). Let n be the index of O_{E}\overline{γ_{1}} + ⋯ + O_{E}\overline{γ_{m}} in End_{O_{E}}(V_{O_{E}}). For p prime put O_{E,p} = ℤ_{p}⊗O_{E}, and E_{p} = ℚ_{p}⊗E. For p > n we have O_{E,p}\overline{γ_{1}}+⋯+O_{E,p}\overline{γ_{m}} = End_{O_{E,p}}(V_{O_{E,p}}).

Let now p > n be prime, and let g be an element of G(ℚ_{p}) that is not contained in a compact group. Then not all eigenvalues of g acting on \overline{ℚ}_{p} ⊗_{ℚ} V (as \overline{ℚ}_{p}-vector space) have absolute value 1. Replacing g, q and q′ by their inverses if necessary, we may suppose that at least one of the eigenvalues of g has absolute value > 1. Then the matrix of g with respect to an O_{E,p}-basis of V_{O_{E,p}} has at least one coefficient that is not in O_{E,p}.

Let q be an element of KgK. Then we have, in G(ℚ_{p}), q = k_{1}gk_{2}, with k_{1} and k_{2} in the image K_{p} of K. So in GL_{E_{p}}(V_{E_{p}}) we have \overline{q} = \overline{k_{1}}\overline{g}\overline{k_{2}} with \overline{k_{1}} and \overline{k_{2}} in GL_{O_{E,p}}(V_{O_{E,p}}). But then we have \overline{g} = \overline{k_{1}}^{−1}\overline{q}\overline{k_{2}}^{−1}. It follows that all u\overline{g}v with u and v in End_{O_{E,p}}(V_{O_{E,p}}) are O_{E_{p}}-linear combinations of elements of the form \overline{γ}\overline{q}\overline{γ′} with γ and γ′ in Γ. Hence there are such γ and γ′ such that the trace of q′ = γqγ′ is not in O_{E,p}. But then q′ has an eigenvalue (in \overline{ℚ}_{p}) of absolute value > 1, hence q′ is not contained in a compact subgroup of G(ℚ_{p}). If q is in G(ℚ) ∩ KgK, then we get a q′ in ΓqΓ of which no non-trivial power lies in Γ. □

One would like to have a generalization of Proposition 6.2 that requires only a weaker hypothesis on q and that gives a better understanding of the primes p that are to be excluded. The following Proposition, that will not be used in the rest of this article, gives such a result in the simply connected case. It would be useful to have a version of it in the adjoint case (maybe the results in [11] can be used here).

Proposition 6.3 Let G be a semi-simple algebraic group over ℚ whose adjoint is simple. Suppose that G(ℝ) is not compact. Let K be a compact open subgroup of G(𝔸_{f}). For p prime let K_{p} be the image of K_{p} under projection to G(ℚ_{p}). Let p be a prime for which K_{p} is a maximal compact subgroup of G(ℚ_{p}) and let q be an element of G(ℚ) such that the image of q in G(ℚ_{p}) is not in K_{p}. Then Γ = G(ℚ) ∩ K is of infinite index in the group Γ_{q} generated by Γ and q.

Proof. By strong approximation ([17, Thm. 7.12]), G(ℚ) is dense in G(𝔸_{f}). Hence Γ is dense in K. Suppose that Γ_{q}/Γ is finite. Let \overline{Γ} and \overline{Γ_{q}} be the closures of Γ and Γ_{q} in G(ℚ_{p}). We have \overline{Γ} = K_{p}. Since Γ_{q} is a finite union of Γ-cosets, \overline{Γ_{q}} is a finite union of \overline{Γ}-cosets. Hence \overline{Γ_{q}} is compact. As q is not in K_{p}, and K_{p} is a maximal compact open subgroup of G(ℚ_{p}), we have a contradiction. □

7 Proof of the main result.

The aim of this section is to prove Theorem 1.2. So let (G, X) be a Shimura datum, let K be a compact open subgroup of G(𝔸_{f}), let V be a faithful finite dimensional representation of G and let Z be a closed irreducible curve in Sh_{K}(G, X)_{ℂ} that contains an infinite set Σ of special points x such that all [V_{x}] with x in Σ are equal.

As we have seen in Section 2, it suffices to prove the result in the case where G is semi-simple and adjoint, K is neat and Z Hodge generic. So we suppose that. We write G = G_{1} × ⋯ × G_{r} with the G_{i} simple. We will also suppose that K is the product of compact open subgroups K_{p} of the G(ℚ_{p}). We let S be the connected component of Sh_{K}(G, X) that contains Z. We assume that we have a ℤ-structure on V given by the choice of a K-invariant ℤ̂-lattice in V_{𝔸_{f}}. Then V induces a variation of ℤ-Hodge structure on S. For each s in S we have its Mumford-Tate group MT(V_{s}) which is a closed sub group scheme of the ℤ-group scheme GL(V_{s}).

Let X^{+} be a connected component of X. After replacing Z by an irreducible component of its image under a suitable Hecke correspondence, we may suppose that S is the image of X^{+} × {1} in Sh_{K}(G, X).

Theorem 7.1 Assume that p is a prime and m an element of G(ℚ_{p}), such that:

1. p does not divide the integer n of Theorem 5.1, applied to the subvariety Z of Sh_{K}(G, X);

2. p ≥ n with n as in Theorem 6.1 applied to (G, X), K and X^{+}, and for every ℚ-simple factor G_{i} of G, the image of m in G_{i}(ℚ_{p}) is not contained in a compact subgroup;

3. Z ⊂ T_{m}Z, with T_{m} the correspondence induced by m on Sh_{K}(G, X).

Then Z = S, hence Z is of Hodge type.

Proof. Assume that p and m satisfy the three conditions. We have S = Γ\X^{+}, with Γ the intersection of K and G(ℚ)^{+}. The connected components of the correspondence that T_{m} induces on S are the T_{m_{i}}, induced by m_{i} in G(ℚ)^{+} acting on X^{+}, such that:

G(ℚ)^{+} ∩ KmK = ∐_{i} Γm_{i}^{−1}Γ.

Let q be one of the m_{i} such that Z ⊂ T_{q}Z. Since q^{−1} is in KmK, and m in G(ℚ_{p}), the image of q in G(ℚ_{l}) is in K_{l} for all l ≠ p, in particular, for all l not dividing n. By Theorem 5.1, T_{q}Z and T_{q^{−1}}Z are irreducible, and hence equal to Z. By Theorem 6.1, all T_{q} + T_{q^{−1}}-orbits in S are dense, hence Z = S. □

For each s in Σ we choose an element s̃ of X^{+} such that s = \overline{(s̃, 1)}. Then we have for each s in Σ the Mumford-Tate group MT(V_{s}) = MT(s̃) as a closed sub group scheme of GL(V). We give G a ℤ-structure G_{ℤ} by taking its Zariski closure in GL(V).

Since all V_{s,ℚ} for s in Σ are isomorphic, all MT(s̃)_{ℚ} are isomorphic. Let F ⊂ ℂ be a finite extension of ℚ that splits all these tori. Then F contains the reflex field of (G, X). By taking F large enough, we may and do assume that Z is defined over F (as an absolutely irreducible closed F -subscheme Z_{F} of Sh_{K}(G, X)_{F}).

In the rest of the proof of Theorem 1.2 we will distinguish two cases, depending on the behavior of the MT(V_{s}) for s in Σ.

Definition 7.2 For s in Σ, let i(s) be the number of prime numbers p such that MT(V_{s})_{𝔽_{p}} is not a torus.

This function i: Σ → ℤ depends on the choice of the ℤ-structure on V, but an other choice would give a function that differs from i by a bounded function. Also, replacing Z by an irreducible component of the image of Z under a Hecke correspondence only changes the function i by a bounded function. The two cases that we will distinguish depend on whether or not i is bounded.

7.3 The case where i is bounded.

In this section we assume that the function i : Σ → ℤ defined above is bounded. It follows that for all but finitely many prime numbers p, the subset Σ_{p} of s in Σ with MT(V_{s})_{𝔽_{p}} a torus is Zariski dense in Z.

Proposition 7.3.1 Let p be a prime such that G_{𝔽_{p}} is smooth over 𝔽_{p}. Then the set of subtori MT(s̃)_{ℤ_{p}} of G_{ℤ_{p}} for s in Σ_{p} meets only a finite number of G_{ℤ}(ℤ_{p})-conjugacy classes of subtori of G_{ℤ_{p}}.

Proof. Let p be prime. Let s_{1} and s_{2} be in Σ_{p}. Let h in GL(V_{ℚ}) be an isomorphism from (V_{ℚ}, s̃_{1}) to (V_{ℚ}, s̃_{2}). Then h induces an isomorphism of ℤ-Hodge structures from (h^{−1}V, s̃_{1}) to (V, s̃_{2}). Hence h^{−1}V_{ℤ_{p}} is a ℤ_{p}-lattice in V_{ℚ_{p}} such that the Zariski closure of MT(s̃_{1})_{ℚ_{p}} in GL(h^{−1}V_{ℤ_{p}}) is a torus. Lemma 3.3.1 (applied to T = MT(V, s̃_{1})_{ℤ_{p}} and the free ℤ_{p}-module h^{−1}V_{ℤ_{p}}) gives an element c in GL(V_{ℚ_{p}}) centralizing MT(V, s̃_{1})_{ℚ_{p}} such that h^{−1}V_{ℤ_{p}} = cV_{ℤ_{p}}. Hence there exists k in GL(V_{ℤ_{p}}) such that h^{−1} = ck. We have MT(s̃_{2})_{ℤ_{p}} = kMT(s̃_{1})_{ℤ_{p}}k^{−1}. So we have that all MT(s̃)_{ℤ_{p}} for s in Σ_{p} lie in one GL(V_{ℤ_{p}})-orbit.

The set of MT(s̃)_{𝔽_{p}} for s in Σ_{p} is contained in one GL(V_{𝔽_{p}})-orbit, and hence is a finite set. If s_{1} and s_{2} in Σ_{p} are such that MT(s̃_{1})_{𝔽_{p}} = MT(s̃_{2})_{𝔽_{p}}, then MT(s̃_{1})_{ℤ_{p}} and MT(s̃_{2})_{ℤ_{p}} are conjugated by an element of G(ℤ_{p}) by [7, Exp. XI, Cor. 5.2], which says that the “transporteur” in G(ℤ_{p}) between MT(s̃_{1})_{ℤ_{p}} and MT(s̃_{2})_{ℤ_{p}} is smooth over ℤ_{p}. □

Let p be a prime with the following properties:

1. p satisfies the conditions of Theorem 7.1;

2. Σ_{p} is Zariski dense in Z;

3. G_{𝔽_{p}} is smooth over 𝔽_{p};

4. K_{p} = G_{ℤ}(ℤ_{p});

5. all the MT(s̃)_{ℚ_{p}} for s in Σ are split.

Indeed, the first four conditions only exclude finitely many primes, and the last condition is verified by a set of primes of positive density (Chebotarev). We replace Σ by a suitable Zariski dense subset of Σ_{p} such that the MT(s̃)_{ℤ_{p}} lie in one G_{ℤ}(ℤ_{p})-conjugacy class.

Let s_{0} be an element of Σ, and let M = MT(s̃_{0}). The reciprocity morphism r from Res_{F/ℚ}𝔾_{mF} to M_{ℚ} is surjective, hence r((ℚ_{p} ⊗ F)^{∗}) is of finite index, say e, in M(ℚ_{p}). For each i, the image M_{i} of M in G_{i} is non-trivial by the axioms for what constitutes a Shimura datum (condition (2.1.1.3) in [6, 2.1]). By hypothesis, M_{ℚ_{p}} and hence the M_{i,ℚ_{p}} are split tori. For a split torus T over ℚ_{p}, T (ℚ_{p}) modulo its maximal compact subgroup is a free ℤ-module of rank the dimension of T. Since M(ℚ_{p}) modulo its maximal compact subgroup is not equal to a finite union of proper sub ℤ-modules, we can take an element m of M(ℚ_{p}) that satisfies the following conditions:

1. m is in the image of multiplication by e on M(ℚ_{p});

2. for every simple factor G_{i} of G, the image of m in G_{i}(ℚ_{p}) is not contained in a compact subgroup.

We will show that Z ⊂ T_{m}Z, so that Z is of Hodge type by Theorem 7.1. Let s be in Σ. We have:

T_{m}s = \{\overline{(s̃, k_{1}mk_{2})} | k_{1} ∈ K_{p}, k_{2} ∈ K_{p}\}.

By hypothesis, there exists an h in G_{ℤ}(ℤ_{p}) such that MT(s̃)_{ℤ_{p}} = hM_{ℤ_{p}}h^{−1}. It follows that hmh^{−1} is in MT(s̃)(ℚ_{p}), and even in the image of the reciprocity morphism for s̃, so that:

\overline{(s̃, hmh^{−1})} is in Gal(\overline{ℚ}/F)·s.

We conclude that \overline{(s̃, hmh^{−1})} is in the intersection of T_{m}s and Gal(\overline{ℚ}/F)·s. As T_{m}s is contained in T_{m}Z, and Gal(\overline{ℚ}/F)·s is contained in Z, \overline{(s̃, hmh^{−1})} is in Z ∩ T_{m}Z. As both Z and T_{m}Z are defined over F, we have:

Gal(\overline{ℚ}/F)·s ⊂ Z ∩ T_{m}Z.

It follows that Z is contained in T_{m}Z. This ends the proof of Theorem 1.2 in the case where i is bounded.

Remark 7.3.2 The proof of Theorem 1.2 that we have just given, in the case where the function i is bounded, does not use that Z is a curve. Hence it proves Theorem 1.2 without the condition that Z is a curve, but with the extra condition that i is bounded.

This kind of result can be useful. For example Vatsal and Cornut (see [19] and [5]) consider sets of Heegner points on products of modular curves where the discriminants of the endomorphism rings of the elliptic curves in question are products of a fixed finite set of prime numbers (of course, the André-Oort conjecture in this case was already proved by Moonen in [14, §5], since it concerns a Shimura datum related to moduli of abelian varieties).

7.4 The case where i is not bounded.

In this section we assume that i(s) is not bounded when s ranges through the points of Σ. The strategy for proving Theorem 1.2 in this case is as follows. For s in Σ with i(s) big enough, we show that there exists a prime number p and an element m in G(ℚ_{p}) that satisfy the first two conditions in Theorem 7.1, such that Z ∩T_{m}Z contains Gal(\overline{ℚ}/F)·s and such that |Gal(\overline{ℚ}/F)·s| exceeds the “intersection number” of Z and T_{m}Z. Then it follows that Z and T_{m}Z do not intersect properly, and hence that Z is contained in T_{m}Z (it is here that we use that Z is a curve). Theorem 7.1 then says that Z is of Hodge type.

We cite the following result (see [8, Thm. 7.2]) that bounds the intersection of Z and its images under Hecke correspondences, if finite.

Theorem 7.4.1 Let (G, X) be a Shimura datum, let K_{1} and K_{2} be compact open subgroups of G(𝔸_{f}), and let Z_{1} and Z_{2} be closed subvarieties of the Shimura varieties S_{1} := Sh_{K_{1}}(G, X)_{ℂ} and S_{2} := Sh_{K_{2}}(G, X)_{ℂ}, respectively. Suppose that Z_{1} or Z_{2} is of dimension at most one. Then there exists an integer c such that for all g in G(𝔸_{f}) for which T_{g}Z_{1} ∩ Z_{2} is finite, one has:

|T_{g}Z_{1} ∩ Z_{2}| ≤ c deg(π_{1} : S_{g} → S_{1}),

where S_{g} = Sh_{K_{g}}(G, X)_{ℂ} with K_{g} = K_{1} ∩ gK_{2}g^{−1}, and with T_{g} and π_{1} the morphism that is induced by the inclusion of K_{g} in K_{1}.

Applied in our situation, this theorem gives the following result.

Corollary 7.4.2 There exists an integer c such that for all m in G(𝔸_{f}) with Z ∩ T_{m}Z finite one has:

|Z ∩ T_{m}Z| < c |K/K ∩ mKm^{−1}|.

We note that if p is a prime such that K_{p} = G(ℤ_{p}), and if m is in G(ℚ_{p}), then K/K ∩ mKm^{−1} is the K_{p}-orbit of the lattice mV_{ℤ_{p}} in the set of ℤ_{p}-lattices in V_{ℚ_{p}}, and hence |K/K ∩ mKm^{−1}| is at most |GL(V_{ℤ_{p}})·mV_{ℤ_{p}}|. In order to get prime numbers p and elements m in G(ℚ_{p}) that verify the first two conditions in Theorem 7.1 and are such that |K/K ∩ mKm^{−1}| is not too big, we prove the following.

Proposition 7.4.3 Let n, r, e and B be nonnegative integers. There exists an integer k with the following property. Let p be a prime number and let M be a split subtorus of GL_{n,ℚ_{p}}, such that M_{ℤ_{p}} (obtained from Zariski closure in GL_{n,ℤ_{p}}) is a torus. Suppose that with respect to a suitable ℤ-basis of the character group X^{∗}(M_{ℚ_{p}}) all coordinates of the differences of the characters that intervene in ℚ_{p}^{n} have absolute value at most B. Let q_{i}: M → M_{i} be r quotients of M, with each M_{i} non-trivial. Then there exists an element m in M(ℚ_{p}) such that no q_{i}(m) lies in a compact subgroup of M_{i}(ℚ_{p}), and such that |GL(V_{ℤ_{p}})·m^{e}V_{ℤ_{p}}| < p^{k}.

Proof. Let n, r, e and B be given. Let p be a prime number, let M be a split subtorus of GL_{n,ℚ_{p}}, such that M_{ℤ_{p}} a torus. Then we have a direct sum decomposition into character spaces:

ℤ_{p}^{n} = ⊕_{χ∈X^{∗}(M)}L_{χ}.

Let d be the dimension of M, and let f: ℤ^{d} → X^{∗}(M) be an isomorphism such that for each pair (χ_{1}, χ_{2}) with L_{χ_{1}} ≠ 0 ≠ L_{χ_{2}} one has ∥f^{−1}(χ_{1} − χ_{2})∥ ≤ B, where ∥·∥ denotes the maximum norm on ℤ^{d}. We identify X^{∗}(M) and its dual X_{∗}(M) with ℤ^{d} via f and its dual. Then the kernels of the X_{∗}(q_{i}) give us r subgroups S_{i} of ℤ^{d} of rank less than d. For T a split torus over ℚ_{p} one has T (ℚ_{p}) = X_{∗}(T) ⊗ ℚ_{p}^{∗}, and hence the valuation map v_{p}: ℚ_{p}^{∗} → ℤ induces an isomorphism from T (ℚ_{p}) modulo its maximal compact subgroup to X_{∗}(T). It follows that for an element m of M(ℚ_{p}) no q_{i}(m) is in a compact subgroup if and only if the image of m in ℤ^{d} avoids all S_{i}.

The quotient ℤ^{d}/(r + 2)ℤ^{d} has (r + 2)^{d} elements, whereas the union of the images of the S_{i} has at most (r + 2)^{d−1}r elements, which is less than (r + 2)^{d}. Let x be an element of ℤ^{d} with ∥x∥ < r + 2 whose image in ℤ^{d}/(r + 2)ℤ^{d} is not in the union of the images of the S_{i}. Let m be an element in M(ℚ_{p}) with image x in ℤ^{d}. Let a and b be the smallest and largest integers such that p^{a}ℤ_{p}^{n} ⊂ m^{e}ℤ_{p}^{n} ⊂ p^{b}ℤ_{p}^{n}. Then the GL_{n}(ℤ_{p})-orbit of m^{e}ℤ_{p}^{n} is contained in the set of ℤ_{p}-lattices between p^{a}ℤ_{p}^{n} and p^{b}ℤ_{p}^{n}. The number of such lattices is the number of subgroups of (ℤ/p^{a−b}ℤ)^{d}, hence is at most p^{(a−b)d^{2}} (use that every subgroup is generated by d elements).

It remains to bound a − b. On L_{χ}, m^{e} acts, up to an element of ℤ_{p}^{∗}, by multiplication by p^{⟨f^{−1}χ,ex⟩}, where ⟨·, ·⟩ denotes the standard pairing. Hence a and b are the maximum and minimum of the ⟨f^{−1}χ, ex⟩, with L_{χ} ≠ 0. It follows that a − b ≤ edB(r + 1). As d ≤ n, we can take k = en^{3}B(r + 1). □

Applied to our situation, we get the following consequence (note that the MT(s̃)_{ℚ} with s in Σ lie in one GL(V_{ℚ})-conjugacy class).

Corollary 7.4.4 There exists an integer k with the following property. Let s be in Σ, and let p be a prime such that MT(s̃)_{ℚ_{p}} is split, and MT(s̃)_{𝔽_{p}} is a torus. Then there exists an element m in MT(s̃)(ℚ_{p}) such that:

1. m is in the image of the reciprocity morphism rs̃: (ℚ_{p} ⊗ F)^{∗} → MT(s̃)(ℚ_{p});

2. for all i, the image of m in G_{i}(ℚ_{p}) is not in a compact subgroup;

3. if Z ∩ T_{m}Z is finite, then |Z ∩ T_{m}Z| ≤ p^{k}.

Theorem 4.1 provides us with positive reals numbers c_{1} and c_{2} such that for all s in Σ we have:

|Gal(\overline{ℚ}/F)·s| > c_{1}c_{2}^{i(s)}i(s)!.

For x in ℝ, let π_{F,1}(x) be the number of prime numbers p ≤ x such that F is split over ℚ_{p}. Chebotarev’s density theorem (see [9, Ch. VIII, §4]) says that for x in ℝ large enough, one has:

π_{F,1}(x) ≥ \frac{1}{2[F : ℚ]}·\frac{x}{log(x)}.

Elementary calculus shows that for all y in ℤ large enough, there exists x in ℝ such that:

\frac{1}{2[F : ℚ]}·\frac{x}{log(x)} > y and x^{k} < c_{1}c_{2}^{y}y!.

As the function i: Σ → ℤ is not bounded, we conclude that there exists an s in Σ, a prime number p and an element m in MT(s̃)(ℚ_{p}) such that m and p satisfy the first two conditions of Theorem 7.1, and even the last one because Z ∩ T_{m}Z contains Gal(\overline{ℚ}/F)·s.

Acknowledgements

We would like to thank Paula Cohen for pointing out to us that it would be interesting to prove Conjecture 1.1 in the case of a set of special points that is contained in one Hecke orbit. We thank Paula Cohen and Gisbert Wüstholz for making preliminary versions of their preprint [4] available to us, and for useful conversations and correspondence. We thank Jürgen Wolfart for explaining to us the problem in his article [20]. We thank Yves André and Richard Pink for the work they have done as referees of the thesis of the second author. We thank Rutger Noot for useful discussions on Shimura varieties.

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