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On the André–Oort conjecture.

Conjecture. 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.

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Shimura varieties.

𝕊 := Res_{ℂ/ℝ}𝔾_{m,ℂ}, algebraic group over ℝ. Equivalence: representations of 𝕊 and ℝ-Hodge structures.

G := a reductive affine algebraic group over ℚ.

X := a G(ℝ)-conjugacy class in Hom(𝕊, G_{ℝ}), satisfying Deligne’s three conditions.

Sh_{K}(G, X)(ℂ) := G(ℚ)\(X × G(𝔸_{f})/K), for K ⊂ G(𝔸_{f}) compact open subgroup.

Sh_{K}(G, X)_{ℂ} := the corresponding complex algebraic variety.

Sh(G, X)_{ℂ} := lim_{K} Sh_{K}(G, X)_{ℂ}, has G(𝔸_{f})-action.

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Subvarieties of Hodge type.

A closed subvariety Z ⊂ Sh_{K}(G, X)_{ℂ} is of Hodge type if it is an irreducible component of the image of some:

Sh(G′, X′)_{ℂ} ⟶^{f} Sh(G, X)_{ℂ} ⟶^{g} Sh(G, X)_{ℂ}
↓
Sh_{K}(G, X)_{ℂ}

with f: G′ → G and g in G(𝔸_{f}).

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Special points.

For h in X, MT(h) := the smallest H ⊂ G such that h(𝕊) ⊂ H_{ℝ}. h in X is special iff MT(h) is commutative. x in Sh_{K}(G, X)_{ℂ} is special iff x is of Hodge type.

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Conjecture. 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.

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Results.

Moonen: moduli space of abelian varieties; true for S for which there exists a prime number p at which all s in S have an ordinary reduction of which they are the canonical lift.

Edixhoven: moduli space of pairs of elliptic curves, i.e., products of two modular curves, assuming GRH.

André: products of two modular curves.

Yafaev: products of two Shimura curves associated to quaternion algebras over ℚ (GRH).

Belhaj Dahman: work in progress on jacobians of y^{n} = x(x − 1)(x − λ).

Edixhoven: Hilbert modular surfaces, products of modular curves (GRH).

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Two principles.

1. Level structures don’t matter: for K ⊂ K′, Z ⊂ Sh_{K}(G, X)_{ℂ} is of Hodge type if and only if its image in Sh_{K′}(G, X)_{ℂ} is.

2. Irreducible components of intersections of subvarieties of Hodge type are again of Hodge type.

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The case of (ℙ^{1})^{n}.

The subvarieties of Hodge type of (ℙ^{1})^{n} are products of:

1. a CM-point in ℙ^{1};

2. closure of the image of ℍ^{±} → (ℙ^{1})^{m} under τ ↦ (j(g_{1}τ), …, j(g_{m}τ)), with the g_{i} in GL_{2}(ℚ).

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Theorem. Assume GRH. Let n ≥ 0, let d ≥ 0 and X ⊂ (ℙ_{ℂ}^{1})^{n} a subvariety of dimension d, containing a dense subset of CM points. Then X is of Hodge type.

The proof is by induction on n and d.

X is an irreducible component of: ⋂_{|I|=d+1^{I⊂{1,... ,n}}} p_{I}^{−1}p_{I}X

This reduces to the case: d = n − 1 and all p_{I} with |I| = d are surjective.

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The case n = 2.

X_{ℚ} := the image of X in (ℙ_{ℚ}^{1})^{2}. d_{1}, d_{2} := the degrees of the projections X_{ℚ} → ℙ_{ℚ}^{1}. Take x = (x_{1}, x_{2}) in X CM with max(|discr(End(x_{i}))|) big with respect to d_{1} and d_{2}. Take (GRH) a prime number p such that:

1. p ≥ 5, p ≥ d_{i};

2. p split in End(x_{i});

3. p small w.r.t. max(|discr(End(x_{i}))|). Then Gal(\overline{ℚ}/ℚ) · x ⊂ X_{ℚ} ∩ T_{p}X_{ℚ}, and |Gal(\overline{ℚ}/ℚ) · x| > X_{ℚ} · T_{p}X_{ℚ}, hence X_{ℚ} ⊂ T_{p}X_{ℚ}. Then one shows that X is of Hodge type.

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The case n = 3.

We have d = 2, all p_{I} with |I| = 2 surjective. We should get a contradiction.

As before: X_{ℚ}, d_{1}, d_{2}, d_{3}.

Take x in X CM with Gal(\overline{ℚ}/ℚ) · x big w.r.t. the d_{i}.

Take (GRH) p_{1} prime such that:

1. p_{1} ≥ 5, p_{1} ≥ d_{i};

2. Gal(\overline{ℚ}/ℚ) · x ⊂ X_{ℚ} ∩ T_{p_{1}}X_{ℚ};

3. p_{1} small w.r.t. |Gal(\overline{ℚ}/ℚ) · x|.

If X_{ℚ} ⊂ T_{p_{1}}X_{ℚ}, get a contradiction (use T_{p_{1}}X_{ℚ} irreducible, and density of Hecke orbits). Hence suppose that X_{ℚ} ⊄ T_{p_{1}}X_{ℚ}.

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The case n = 3, continued.

Take X_{1} an irreducible component of X_{ℚ} ∩ T_{p_{1}}X_{ℚ} containing x (X_{1} is a curve).

For a suitable p_{2} (GRH) one has X_{1} ⊂ T_{p_{2}}X_{1}, hence X_{1} is of Hodge type.

This gives a Zariski dense set of curves of Hodge type in X.

One finishes by classifying such curves. Those which project surjectively under all p_{i} are of the form:

C_{(n_{1},n_{2},n_{3})} := {(E_{1}, E_{2}, E_{3}) | ∃ E, f_{i}: E → E_{i}}

with ker(f_{i}) ≅ ℤ/n_{i}ℤ for a fixed triplet (n_{1}, n_{2}, n_{3}).

deg(p_{i}: C_{(n_{1},n_{2},n_{3})} → p_{i}C_{(n_{1},n_{2},n_{3})}) ≥ ϕ(n_{i}) gives the desired contradiction.

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The case n > 3.

More successive intersections, the end is more difficult.

As for n = 3, get a dense set of curves of Hodge type. Then consider X ∩ {x} × (ℙ^{1})^{n−1} with x CM.

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The case of Hilbert modular surfaces.

K := a real quadratic extension of ℚ. G := Res_{O_{K}/ℤ}(GL_{2}(O_{K})). G(ℝ) = GL_{2}(ℝ ⊗ O_{K}) = GL_{2}(ℝ)^{2}. S_{ℂ} := G(ℚ)\((ℍ^{±})^{2} × G(𝔸_{f})/G(ℤ̂)) S_{ℚ} := the canonical model over ℚ; it is the moduli space for abelian surfaces with O_{K}-action. Theorem. Assume GRH. Let C ⊂ S_{ℂ} be an irreducible curve containing infinitely many CM points. Then C is of Hodge type.

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Generic Mumford-Tate groups.

The generic Mumford-Tate group of S_{ℂ} is the group G′ given by:

G′ ↪  G_{ℚ}
↓ □  ↓det
𝔾_{m}_{ℚ} ↪ Res_{K/ℚ}𝔾_{m}_{K}

M := the generic Mumford-Tate group of C.

If M ≠ G′, then C is of Hodge type, because of dimensions.

Hence suppose that M = G′. Then C is not of Hodge type, and we have to get a contradiction.

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Algebraic monodromy.

H := the connected component of the Zariski closure of the image of π_{1}(C′) (C′ a suitable finite cover of C), under the monodromy map.

Theorem (André). H = M^{der} (= G^{der} = Res_{K/ℚ}SL_{2,K}).

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Finite monodromy.

Theorem (Nori). The closure of the image of π_{1}(C′) in H(ℤ̂) is open.

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Irreducibility.

For p prime, let T_{p} be the Hecke correspondence on S_{ℚ}, induced by ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) in G(ℚ_{p}). Corollary. For p big enough, T_{p}C is irreducible.

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Galois action.

Let A/ℂ be an abelian surface with O_{K}-action, and with CM.

Then R := End_{O_{K}} (A) is an order in a CM field L of degree 2 over K.

The set of A′/ℂ with End_{O_{K}} (A′) = R and with the same CM type as A is a Pic(R)-torsor.

E := the reflex field of A with its O_{K} action.

Gal(\overline{ℚ}/E) acts via the reflex type norm: E^{∗}\𝔸_{f,E}^{∗} ⟶ L^{∗}\𝔸_{f,L}^{∗}

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Analytic number theory.

Theorem (Siegel). |Pic(R)| = |discr(R)|^{1/2+o(1)} as |discr(R)| → ∞.

Corollary. There exists ε > 0 and c > 0 such that for x on S a CM point we have

|Gal(\overline{ℚ}/ℚ) · x| ≥ c|discr(R)|^{ε}.

Theorem (L.O.M.). Assume GRH. For x with |discr(R)| ≫ 0 there are many p split in R with p at most polynomial in log |discr(R)|.

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End of the proof.

For x with |discr(R)| big enough, there is a p such that:

T_{p}x ∋ σx, T_{p}C is irreducible

for some σ in Gal(\overline{ℚ}/ℚ).

Hence T_{p}C_{ℚ} = C_{ℚ}, which contradicts the density of the T_{p}-orbit of x in its connected component of S(ℂ).