APPENDIX

CHRISTOPHE CORNUT

1. Introduction

Let M = Sh_{K}(G, X) be a Shimura variety. An analytic subvariety Y of X is special if it is a connected component of Y for some sub Shimura datum (H, Y) of (G, X). An algebraic subvariety Z of M is special if there exists an element g ∈ G(A_{f}) and a special subvariety Y of X such that Z is the image of gK × Y in

M(C) = G(Q)\ (G(A_{f})/K × X) .

The special subvarieties of dimension zero are the special points, which correspond to sub Shimura datum (H, Y) for which H is a Q-subtorus of G. The following proposition is well known, see for instance [8, Chapter 13].

Proposition 1.1. Any special subvariety contains a dense set of special points.

Conjecture 1.2. (André-Oort) Any irreducible subvariety Z of M which contains a Zariski dense set of special points is special.

This conjecture has been proven under GRH [7], as well as unconditionally in many special cases [1, 5, 6, 7]. For various applications [2, 3], one has to list, or at least parametrize in a more explicit way the set of all special subvarieties of a given Shimura variety. We give such a parametrization for M a product of Shimura curves, following Bas Edixhoven’s treatment of products of modular curves [5].

2. Statement

Let F be totally real number field, B a quaternion algebra over F and G = Res_{F/Q}GL_{1}(B). Thus G is a reductive group over Q with center Z = Res_{F/Q}G_{m,F}. Let d = [F : Q] and denote by {τ_{1}, ⋯, τ_{d}} the embeddings F ↪ R. We assume that B_{j} = B ⊗_{F,τ_{j}} R splits only for j = 1. Put S = Res_{C/R}G_{m,C} and let X be the G(R)-conjugacy class of morphisms S → G_{R} ≃ ∏_{j=1}^{d} GL_{1}(B_{j}) whose projection to GL_{1}(B_{j}) is trivial for j ≠ 1, and whose projection S → GL_{1}(B_{1}) is induced by an embedding of R-algebras C ↪ B_{1} ≃ M_{2}(R). Then (G, X) is a Shimura datum with reflex field τ_{1}(F) ⊂ C and its associated Shimura varieties have dimension one: these are the Shimura curves. The classification of all special subvarieties of a given product of Shimura curves, as sketched without proof in [4], is easily derived from the following proposition.

Proposition 2.1. Let I ≠ ∅ be a finite set. For any special subvariety Y of X^{I}, there exists

(1) a partition I = J′ ∐ J, J = ∐_{α∈A} J(α),

(2) for each j ∈ J′, a special point y_{j} ∈ X,

(3) for each j ∈ J, an element σ_{j} ∈ G(Q),

(4) for each α ∈ A, a connected component X_{α} of X such that

Y = {(y_{j})_{j∈J′}} ×_{∏α∈A}Δ_{α}(X_{α}) in X^{I} = X^{J′} × ∏_{α∈A} X^{J(α)},

where Δ_{α} : X → X^{J(α)} maps x to (σ_{j}x)_{j∈J(α)}.

3. Preliminaries

In the sequel, we view F as a subfield of R ⊂ C through τ_{1} : F ↪ R. For a subfield L of the algebraic closure \overline{Q} of Q in C, we put Γ_{L} = Gal(\overline{Q}/L). For a torus T over a subfield of C, we denote by X_{⋆}(T) (resp. X^{⋆}(T)) the group of cocharacters G_{m,C} → T_{C} (resp. characters T_{C} → G_{m,C}). If T is defined over L, Γ_{L} acts on these groups and

X_{⋆}(Res_{L/Q}T) = Ind_{Γ_{L}}^{Γ_{Q}}X_{⋆}(T) X^{⋆}(Res_{L/Q}T) = Ind_{Γ_{L}}^{Γ_{Q}}X^{⋆}(T).

Lemma 3.1. The maximal Q-subtori of G are in bijection with the quadratic (maximal) F -algebras K inside B via K ↦ T_{K} = Res_{F/Q}GL_{1}(K).

Proof. Let T be a maximal Q-subtorus of G. Then T is the centralizer in G of some regular element γ ∈ T (Q) ⊂ G(Q) = B^{×}. Let K be the commutant of γ in B. Then K is a maximal (quadratic) commutative F -subalgebra of B and T = T_{K}. □

For any algebraic Q-subgroup H of G, we denote by X (H) the set of all x ∈ X whose corresponding morphism h_{x} : S → G_{R} factors through H_{R}.

Proposition 3.2. For a connected Q-subgroup H of G, X (H) contains a special point if and only if H = G or H = T_{K} for some totally imaginary quadratic extension K of F inside B.

Proof. Let x ∈ X (H) be a special point, T a maximal Q-subtorus of G such that h_{x} : S → G_{R} factors through T_{R} and S the connected component of T ∩ H. Thus S is a maximal Q-subtorus of H and h_{x} : S → H_{R} factors through S_{R}.

Write T = T_{K} for some quadratic F -subalgebra K of B. Put K_{j} = K ⊗_{F,τ_{j}} R, so that T_{R} ≃ ∏_{j=1}^{d} GL_{1}(K_{j}/R). For j ≠ 1, the projection of h_{x} : S → T_{R} to GL_{1}(K_{j}/R) is trivial. For j = 1, it is induced by an R-algebra isomorphism K_{1} ≃ C (corresponding to an extension of τ_{1} to K ↪ C). Since also K_{j} ≃ C for j ≠ 1 (because B_{j} does not split), K is a totally imaginary extension of F.

Let η : G_{m,R} → G_{R} be the restriction of h_{x} : S → G_{R} to G_{m,R} ⊂ S. Then η factors through Z_{R}, and its Galois orbit Γ_{Q} · η spans X_{⋆}(Z) ≃ Ind_{Γ_{F}}^{Γ_{Q}}Z. Thus any Q-subtorus of G through which η factors, such as S, contains Z. Put \overline{S} = S/Z, \overline{T} = T/Z and \overline{S} = \overline{S}/G_{m,R}. Then h_{x} : S → T_{R} induces a morphism \overline{h}_{x} : \overline{S} → \overline{T}_{R}. Since \overline{S}_{C} ≃ G_{m,C}, we may view \overline{h}_{x} as a cocharacter of \overline{T}_{R}. One checks again that its Galois orbit Γ_{Q} · \overline{h}_{x} spans X_{⋆}(\overline{T}) ≃ Ind_{Γ_{K}}^{Γ_{Q}} Z/Ind_{Γ_{F}}^{Γ_{Q}}Z, so that any Q-subtorus of \overline{T} through which \overline{h}_{x} factors, such as \overline{S} = S/Z, actually equals \overline{T}. Thus S = T.

Now h_{x}(i) ∈ H(R) yields a Cartan involution Adh_{x}(i) of G_{R}, and thus also one of H_{R}. Therefore H is a reductive group. As such, it is now entirely characterized by the set of roots of T in LieH, which has to be a Galois stable subset of the roots of T in LieG. Since Γ_{Q} acts transitively on the latter, H = T or H = G. This finishes the proof of the non-obvious part of the proposition. □

Proposition 3.3. Let \overline{G} = G/Z be the adjoint group of G. Then (1) \overline{G} is Q-simple: any normal Q-subgroup of \overline{G} equals 1 or \overline{G}.

(2) Any Q-automorphism of \overline{G} is inner.

(3) The projection G(Q) → \overline{G}(Q) is surjective.

Proof. (1) The Dynkin diagram D of \overline{G} has no edges, and its Γ_{Q}-set of vertices is isomorphic to Γ_{Q}/Γ_{F}. Since Γ_{Q} acts transitively on D, \overline{G} is almost Q-simple: its connected normal Q-subgroup are 1 and \overline{G}. Since \overline{G} is also adjoint, it is Qsimple. (2) Any Q-automorphism σ of \overline{G} induces an automorphism of D which (a) commutes with the transitive action of Γ_{Q} and (b) fixes the node of D which corresponds to the unique non-compact simple component of \overline{G}_{R}. Thus σ acts trivially on D, and therefore is inner. (3) There is an exact sequence

1 → Z(Q) → G(Q) → \overline{G}(Q) → H^{1}(Q, Z).

By Shapiro’s lemma and Hilbert’s theorem 90, H^{1}(Q, Z) = H^{1}(F, G_{m}) = 0. Thus G(Q) → \overline{G}(Q) is indeed surjective. □

Proposition 3.4. Let \overline{G} be a Q-simple group. Let I be a finite set and let H be an algebraic Q-subgroup of \overline{G}^{I} all of whose projections to \overline{G} are surjective. Then there exists a partition I = ∐_{α∈A} I(α) and elements σ_{i} ∈ Aut_{Q}\overline{G} (for all i ∈ I) such that

H = ∏_{α∈A} Δ_{α}(\overline{G}) in \overline{G}^{I} = ∏_{α∈A} \overline{G}^{I(α)},

where Δ_{α} : \overline{G} → \overline{G}^{I(α)} maps g to (σ_{i}(g))_{i∈I(α)}.

Proof. Call special the subgroups which are of this form. We argue by induction on |I|. If |I| = 1, there is nothing to prove. If |I| = 2, let p_{1/2} : \overline{G}^{2} → \overline{G} be the projections and let H_{2/1} be the kernel of the restriction of p_{1/2} to H, so that H_{1} × H_{2} is a normal subgroup of H and

p_{2}H/p_{2}H_{2} ≃ H/H_{1} × H_{2} ≃ p_{1}H/p_{1}H_{1}.

Since p_{1}H = p_{2}H = \overline{G}, either p_{1}H_{1} = p_{2}H_{2} = \overline{G} (in which case H = \overline{G}^{2}) or p_{1}H_{1} = p_{2}H_{2} = 1, in which case H = Δ(\overline{G}) with Δ(g) = (g, θ(g)) for θ = (p_{2}|_{H}) ∘ (p_{1}|_{H})^{−1} ∈ Aut_{Q}(\overline{G}). In both cases, H is indeed special.

Suppose now that |I| ≥ 3. For any subset J ≠ ∅ of I, let p_{J} : \overline{G}^{I} → \overline{G}^{J} be the projection. By induction, we know already that p_{J}H is special for every proper subset J ≠ ∅ of I. Suppose that there is such a J for which p_{J}H ≠ \overline{G}^{J}. This means that there exists i_{1} ≠ i_{2} ∈ I and σ_{1}, σ_{2} ∈ Aut_{Q}\overline{G} such that H is contained in the image of the morphism

\overline{G}^{I−\{i_{1},i_{2}\}} ×\overline{G} → \overline{G}^{I} = \overline{G}^{I−\{i_{1},i_{2}\}} × \overline{G}^{\{i_{1},i_{2}\}}
((g_{i}) , g) ↦ ((g_{i}) , (σ_{1}(g), σ_{2}(g)))

By induction, the preimage of H under this morphism is special, and so is H.

Finally, suppose that for any proper subset J ≠ ∅ of I, p_{J}H = \overline{G}^{J}. Then H = \overline{G}^{I} (which is special). Indeed, it is sufficient to show that H(\overline{Q}) contains every factor \overline{G}(\overline{Q}) of \overline{G}^{I}(\overline{Q}). Thus, fix j ∈ I, g ∈ \overline{G}(\overline{Q}), and also pick two other elements j_{±} ∈ I (recall that |I| ≥ 3). Since \overline{G} is semi-simple, there exists g_{±} ∈ \overline{G}(\overline{Q}) such that g = [g_{+}, g_{−}]. For ϵ ∈ \{±1\}, we know that H(\overline{Q}) contains an element h_{ϵ} with p_{j}(h_{ϵ}) = g_{ϵ} and p_{i}(h_{ϵ}) = 1 for all i ∉ {j, j_{ϵ}}. Then p_{i}([h_{+}, h_{−}]) = g if i = j and 1 otherwise. This proves our claim. □

4. Proof of Proposition 2.1

Let (H, Y) be a sub Shimura datum of (G^{I}, X^{I}) such that Y is a connected component of Y. For any subset J ≠ ∅ of I, let p_{J} : G^{I} → G^{J} and q_{J} : X^{I} → X^{J} be the projections. Define

J = {i ∈ I | p_{i}H = G} and J′ = {i ∈ I | p_{i}H ≠ G}.

By Propositions 3.4 and 3.3, there exist a partition J = ∐_{α∈A} J(α) and elements σ_{j} ∈ G(Q) (for all j ∈ J) such that the two subgroups p_{J}H and H_{J} = ∏_{α∈A} Δ_{α}(G) of G^{J} = ∏_{α∈A} G^{J(α)} have the same projection to (G/Z)^{J}, where

Δ_{α} : G → G^{J(α)} maps g to (σ_{j}gσ_{j}^{−1})_{j∈J(α)}.

According to Proposition 1.1, there exists at least one special point y ∈ Y. For all i ∈ I, put y_{i} = q_{i}(y) and for each α ∈ A, denote by X_{α} the connected component of X which contains x_{α} = σ_{j(α)}^{−1} · y_{j(α)} for some arbitrary (but fixed) choice of an element j(α) in J(α) (we will see that x_{α} does not depend upon that choice).

For any i ∈ I, y_{i} = q_{i}(y) is a special point of q_{i}(Y) ⊂ X (p_{i}H). Thus by Proposition 3.2, for any j ∈ J′, p_{j}H is a (CM type) maximal subtori of G. Then X (p_{j}H) is a finite (two points) set and its connected subset q_{j}(Y) reduces to {y_{j}}. This already shows that

Y = {(y_{j})_{j∈J′}} × q_{J}(Y) in X^{I} = X^{J′} × X^{J}.

Put y_{J} = q_{J}(y) and y_{α} = q_{J(α)}(y). Since Y = H(R)^{0} · y,

q_{J}(Y) = p_{J}(H(R)^{0}) · y_{J} = (p_{J}H)(R)^{0} · y_{J} = H_{J}(R)^{0} · y_{J}

because p_{J}(H(R)^{0}) = (p_{J}H)(R)^{0} and Z^{J} · p_{J}H = Z^{J} · H_{J}. Therefore

q_{J}(Y) = _{∏α∈A}q_{J(α)}(Y) with q_{J(α)}(Y) = Δ_{α}(G(R)^{0}) · y_{α}.

Now y_{α} = q_{J(α)}(y) ∈ X^{J(α)} corresponds to a morphim h_{α} : S → G_{R}^{J(α)} which factors through (p_{J(α)}H)_{R} and therefore also through

(Z^{J(α)} · Δ_{α}(G))_{R} = (Z^{J(α)} · p_{J(α)}H)_{R}.

Put Z_{α} = {1} × Z^{J(α)−{j(α)}} ⊂ Z^{J(α)}, so that Z^{J(α)} · Δ_{α}(G) = Z_{α} × Δ_{α}(G) and h_{α} = h_{α}^{Z} · h_{α}^{Δ} with h_{α}^{Z} : S → (Z_{α})_{R} and h_{α}^{Δ} : S → Δ_{α}(G)_{R}. Write h_{α}^{Δ} = Δ_{α} ∘ h for some h : S → G_{R}. Then

Ad(σ_{j(α)}) ∘ h = p_{j(α)} ∘ h_{α}^{Δ} = p_{j(α)} ∘ h_{α} = h_{y_{j(α)}} = Ad(σ_{j(α)}) ∘ h_{x_{α}}

so that h = h_{x_{α}}. For each j ∈ J(α), choose g_{j} ∈ G(R) such that g_{j} · y_{j} = σ_{j} · x_{α} in X. Then for all s ∈ S,

h_{α}^{Z}(s) = h_{α}(s)h_{α}^{Δ}(s)^{−1} = (h_{y_{j}}(s)·h_{σ_{j}x_{α}}(s)^{−1})_{j∈J(α)} = (h_{y_{j}}(s)g_{j}h_{y_{j}}(s)^{−1}g_{j}^{−1})_{j∈J(α)}

Thus h_{α}^{Z} : S → (Z_{α})_{R} ↪ Z_{R}^{J(α)} ↪ G_{R}^{J(α)} factors through the derived group of G_{R}^{J(α)}. Since the latter has finite intersection with Z_{R}^{J(α)} while S is connected, h_{α}^{Z} = 1 and h_{α} = h_{α}^{Δ} = Δ_{α} ∘ h_{x_{α}}. This simply means that y_{α} = Δ_{α}(x_{α}), so that

q_{J(α)}(Y) = Δ_{α}(G(R)^{0}) · y_{α} = Δ_{α}(G(R)^{0} · x_{α}) = Δ_{α}(X_{α}).

Thus Y = {(y_{j})_{j∈J′}} × ∏_{α∈A} Δ_{α}(X_{α}), which proves the Proposition.

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