CM points on products of Drinfeld modular curves

Florian Breuer

April 26, 2004

Abstract

We prove an analogue of the André-Oort conjecture for subvarieties of products of Drinfeld modular curves over general base rings A of odd characteristic. More precisely, we classify those subvarieties containing a Zariski-dense set of CM points.

1 Introduction

1.1 Basic notations

The following notations will be used throughout this paper. Let 𝔽_{q} denote the finite field with q elements, where q is a power of the odd prime p. Let k be a global function field with field of constants 𝔽_{q}. Fix a place ∞ of k, and denote by k_{∞} the completion of k at ∞, and by ℂ_{∞} = ˆ k̄_{∞} the completion of an algebraic closure of k. Let A = {x ∈ k | x is regular outside ∞}, it is a Dedekind domain of finite class number. We denote by | · | the absolute value corresponding to ∞, and note that for a ∈ A, we have |a| = q^{deg(a)} = |A/⟨a⟩|.

Unless stated otherwise, a Drinfeld module always means a rank 2 Drinfeld A-module defined over ℂ_{∞} of generic characteristic. See [8] and [10] for an overview of Drinfeld modules.

We denote by Â = \underleftarrow{lim} A/𝔫 the profinite completion of A, and by 𝔸_{f} = Â ⊗_{A} k the ring of finite adèles of k. Let K be a subgroup of finite index (i.e. an open subgroup) of GL_{2}(Â), then we denote by M_{A}^{2}(K) the coarse moduli scheme parametrizing rank 2 Drinfeld A-modules with level K-structure, it is an affine curve over Spec(A), which is not in general irreducible. A Drinfeld modular curve over ℂ_{∞} is an irreducible component of some M_{A}^{2}(K)_{ℂ_{∞}} = M_{A}^{2}(K)×_{A} ℂ_{∞}. If K = GL_{2}(Â) we will just write M_{A}^{2} = M_{A}^{2}(GL_{2}(Â)), which is the coarse moduli space of Drinfeld modules without level structure.

Let Ω := ℙ^{1}(ℂ_{∞}) − ℙ^{1}(k_{∞}) denote the Drinfeld upper half-plane, which has a rigid analytic structure. The group GL_{2}(k_{∞}) acts on Ω via fractional linear transformations. Let X be a Drinfeld modular curve, then it is known that, as rigid analytic varieties, we have

X(ℂ_{∞})^{an} ≅ Γ\Ω  (1.1)

for some arithmetic subgroup Γ ⊂ GL_{2}(k). We denote by [ω] ∈ X(ℂ_{∞}) the point corresponding to ω ∈ Ω.

1.2 Main results

Let X = X_{1} × ⋯ × X_{n} be a product of Drinfeld modular curves. A point x ∈ X(ℂ_{∞}) is called a CM point if it corresponds to an n-tuple of Drinfeld modules with complex multiplication. These in turn correspond to points ω ∈ Ω with [k(ω) : k] = 2 via (1.1).

A subcurve Y ⊂ X is called a special subcurve if there exists a partition {1, …, n} = I_{0} ∐ I_{1}, I_{1} ≠ ∅, and elements g_{i} ∈ GL_{2}(k), for i ∈ I_{1}, such that Y = {x} × Y′, where x ∈ ∏_{i∈I_{0}} X_{i}(ℂ_{∞}) is a CM point and Y′ ⊂ ∏_{i∈I_{1}} X_{i} is a curve such that Y′(ℂ_{∞}) is the image of the map

Ω ⟶ ∏_{i∈I_{1}} X_{i}(ℂ_{∞}); ω ⟼ ([g_{i}(ω)])_{i∈I_{1}}.

The curve Y is called a pure special subcurve if I_{1} = ∅, i.e. if the projections p_{i} : Y → X_{i} are surjective for all i = 1, …, n.

A subvariety Y ⊂ X is called a special subvariety if there exists a partition {1, …, n} = ∐_{j=0}^{g} I_{j} such that Y = {x} × ∏_{j=1}^{g} Y_{j}, where x ∈ ∏_{i∈I_{0}} X_{i}(ℂ_{∞}) is a CM point, and each Y_{j} ⊂ ∏_{i∈I_{j}} X_{i} is a pure special subcurve, for j = 1, …, g. We see that CM points are just special subvarieties of dimension zero.

The aim of this paper is to prove the following result.

Theorem 1.2 Let X = X_{1} × ⋯ × X_{n} be a product of Drinfeld modular curves. Then an irreducible algebraic subvariety Y ⊂ X contains a Zariski-dense subset of CM points if and only if Y is a special subvariety.

It is easy to see that a special subvariety contains a Zariski-dense set of CM points, the hard part is to prove the converse. In the special case where Y is a curve, we actually prove an effective result, namely that Y is special if and only if Y contains a CM point of sufficiently large CM height, see Theorem 3.21.

Theorem 1.2 is an analogue of the André-Oort Conjecture for products of classical modular curves, see [1] and [4]. In fact, our proof is closely modelled on Edixhoven’s approach [4].

Theorem 1.2 was proved in [3] for the special case A = 𝔽_{q}[T]. In this paper we show how to adapt the arguments of [3] to the case of general A (but still of odd characteristic). This result, which was stated as Conjecture 7.1 in [2], is also the main step towards extending the non-triviality result for higher Heegner points on elliptic curves over rational function fields [2, Thm. 1] to arbitrary global fields, at least of odd characteristic, see [2, Thm. 2]. The remaining step, concerning the non-triviality of certain linear combinations of automorphic forms, will be treated in a separate paper.

2 Some preliminaries

We begin by gathering some basic results which will be needed in the proof of Theorem 1.2.

2.1 Complex multiplication and CM heights

Let ϕ be a Drinfeld module with complex multiplication by the ring R, and let K be the quotient field of R. Then K/k is a quadratic imaginary extension, which means that only one prime of K lies above ∞, which we again call ∞. Denote by O_{K} the ring of integers of K, i.e. the integral closure of A in K. Then A ⊂ R ⊂ O_{K}, and R is a projective A-module of rank 2, hence by the invariant factor theorem, R = A + 𝔣O_{K}, for some ideal 𝔣 ⊂ A, which we will call the conductor of R. Note that 𝔣O_{K} is the largest O_{K}-ideal which is also an R-ideal, which is the definition of conductor usually found in the literature.

The ring R is an order in K, and is not in general integrally closed. However, one may still view ϕ as a Drinfeld R-module of rank 1, after Hayes [9], and in fact we have M_{R}^{1} = Spec(O_{H_{R}}), where O_{H_{R}} is the ring of integers of the class field H_{R} corresponding to the class group K^{×}\𝔸_{f,K}^{×}/R̂^{×} ≅ Pic(R). The field H_{R} is also known as the ring class field of R, and H_{R}/K is unramified outside 𝔣O_{K}. Note that we only deal with the finite adèles 𝔸_{f,K}, so that H_{R}/K splits completely at ∞. The action of 𝔸_{f,K}^{×} on Spec(O_{H_{R}}) coincides with the action of Gal(H_{R}/K) on M_{R}^{1} given by class field theory. Hence ϕ is defined over H_{R} and isogenies act like Galois. In particular, we have

Proposition 2.1 Let ϕ be a Drinfeld module with complex multiplication by R. Let 𝔫 ⊂ A be a non-zero ideal such that every prime factor of 𝔫 splits in R, and let σ_{𝔫} = (𝔫, H_{R}/k) ∈ Gal(H_{R}/k) be the corresponding Frobenius element. Then ϕ and ϕ^{σ_{𝔫}} are linked by a cyclic isogeny of degree 𝔫.

Proof. As every prime factor of 𝔫 splits in R, we may choose an ideal 𝔑 of R such that R/𝔑 ≅ A/𝔫. Now an isogeny of ϕ as a rank 2 Drinfeld A-module with kernel A/𝔫 is also an isogeny of ϕ as a rank 1 Drinfeld R-module with kernel R/𝔑. The result now follows from the above discussion. Note that σ_{𝔫} = (𝔑, H_{R}/K), which is well-defined. □

Denote by |𝔣| = |A/𝔣|, then we define the CM height of ϕ by

H_{CM}(ϕ) := q^{g}|𝔣|,  (2.2)

where g is the genus of K. Note that this definition differs from [3, Def. 3.7] by a power of 1/2. If x = (x_{1}, …, x_{n}) ∈ M_{A}^{2}(K_{1})(ℂ_{∞}) × ⋯ × M_{A}^{2}(K_{n})(ℂ_{∞}), then we define

H_{CM}(x) := max (H_{CM}(x_{1}), …, H_{CM}(x_{n})).

The following result shows that H_{CM} is a counting function on the set of CM points in M_{A}^{2}(ℂ_{∞}), which justifies the terminology.

Proposition 2.3 For every B > 0, the set

{ϕ ∈ M_{A}^{2}(ℂ_{∞}) | ϕ is CM and H_{CM}(ϕ) < B}

is finite.

Proof. For a given g ≥ 0 there are only finitely many global function fields K with genus g and field of constants contained in 𝔽_{q^{2}}. For each such field K, there are only finitely many orders of the form R = A + 𝔣O_{K} with bounded 𝔣. And for each such R, there are only |Pic(R)| Drinfeld modules ϕ with End(ϕ) ≅ R. □

We will need the following class-number estimate

Proposition 2.4 Let ϕ be a Drinfeld module with complex multiplication by R. Then for every ε > 0 there exists a computable constant C_{ε} > 0 such that

|Pic(R)| > C_{ε}H_{CM}(ϕ)^{1−ε}.

Proof. Let K be the quotient field of R, and g its genus. Firstly, we have [3, Prop. 3.1]

|Pic(O_{K})| = h(K) ≥ \frac{(q − 1)(q^{2g} − 2gq^{g} + 1)}{2g(q^{g+1} − 1)}.

Secondly, the exact sequence [13, §I.12]

1 → O_{K}^{×}/R^{×} ⟶ (O_{K}/𝔣O_{K})^{×}/(R/𝔣R)^{×} ⟶ Pic(R) ⟶ Pic(O_{K}) → 1

leads, as in the classical case, to the pleasing expression

|Pic(R)| = \frac{|Pic(O_{K})|}{[O_{K}^{×} : R^{×}]} · |𝔣| ∏_{𝔭|𝔣}(1 − χ(𝔭)|𝔭|^{−1}),  (2.5)

where

χ(𝔭) = \{ \begin{matrix}1 & if 𝔭 splits in K/k \\ −1 & if 𝔭 is inert in K/k \\ 0 & if 𝔭 is ramified in K/k. \end{matrix}

Here one uses the fact that R/𝔣R ≅ A/𝔣. The estimate now follows easily. □

2.2 Drinfeld modular curves

Let K ⊂ GL_{2}(Â) be a subgroup of finite index, and recall that M_{A}^{2}(K) denotes the coarse moduli scheme parametrizing Drinfeld modules with level-K structure. Identifying Drinfeld modules over ℂ_{∞} with their associated rank 2 lattices in ℂ_{∞}, and parametrizing the space of such lattices adèlically, one arrives at the following analytic parametrization (see e.g. [7]).

M_{A}^{2}(K)(ℂ_{∞})^{an} ≅ GL_{2}(k)\Ω × GL_{2}(𝔸_{f})/K
≅ ∐_{s∈S} Γ_{s}\Ω,  (2.6)

where S ⊂ GL_{2}(𝔸_{f}) denotes a (finite) set of representatives for GL_{2}(k)\GL_{2}(𝔸_{f})/K, and Γ_{s} = sKs^{−1} ∩ GL_{2}(k).

The determinant map induces an isomorphism

det : GL_{2}(k)\GL_{2}(𝔸_{f})/K ⟶^{∼} k^{×}\𝔸_{f}^{×}/ det(K) ( ≅ Pic(A) if det(K) = Â^{×}).

We next describe a scheme-theoretic version of the determinant map. Let f ∈ A, and denote by K(f) ⊂ GL_{2}(Â) the kernel of reduction mod f. The Weil pairing for Drinfeld modules [12, Chap. 5] gives us a map of A[1/f]-schemes

M_{A}^{2}(K(f)) ⟶^{w_{f}} M_{A}^{1}(det(K(f))) = Spec(O_{H_{f}}),  (2.7)

where O_{H_{f}} denotes the ring of integers of the class field H_{f} of k corresponding to the class group k^{×}\𝔸_{f}^{×}/ det(K(f)) = k^{×}\𝔸_{f}^{×}/{x ∈ Â^{×} | x ≡ 1 mod f}. Under this map, the 𝔸_{f}^{×}GL_{2}(Â)-action on M_{A}^{2}(K(f)) corresponds to its determinant 𝔸_{f}^{×}-action on Spec(O_{H_{f}}), which in turn corresponds to the Gal(k^{ab}/k)-action on both sides given by the reciprocity map. It follows that M_{A}^{2}(K(f)) × k is defined over k, and all its k̄-irreducible components are defined over H_{f}.

Suppose now that K contains K(f) for some f ∈ A, so K is a congruence subgroup. Then we may divide out by the action of K and det(K) in the left and right hand sides of (2.7), respectively. As M_{A}^{2}(K) = M_{A}^{2}(K(f))/K, we thus get a map of A[1/f]-schemes

M_{A}^{2}(K) ⟶^{w_{K}} M_{A}^{1}(det(K)) = Spec(O_{H_{K}}),  (2.8)

where this time H_{K} is the class field corresponding to k^{×}\𝔸_{f}^{×}/ det(K). As before, it follows that M_{A}^{2}(K) × k is defined over k, and its k̄-irreducible components are defined over H_{K}.

We point out that, analytically, the maps w_{f} and w_{K} above send the rank 2 lattice L to the rank 1 lattice ∧^{2}L.

In the case K = GL_{2}(Â), recall that we use the notation M_{A}^{2} := M_{A}^{2}(GL_{2}(Â)). Let H denote the Hilbert class field of k, corresponding to the class group k^{×}\𝔸_{f}^{×}/Â^{×} ≅ Pic(A). It is the maximal unramified abelian extension of k in which ∞ splits completely. Then we see that the irreducible components of M_{A,ℂ_{∞}}^{2} are defined over H. We also see that its set of irreducible components corresponds to Pic(A) in such a way that the component corresponding to [𝔞] ∈ Pic(A) parametrizes lattices isomorphic to A ⊕ 𝔞 as projective A-modules.

We may choose our representatives S such that 1 ∈ S, in which case Γ_{1} = GL_{2}(A), and one of the irreducible components of M_{A,ℂ_{∞}}^{2} thus corresponds to GL_{2}(A)\Ω via (2.6). We call it the identity component, and denote it by 𝕄. It is the simplest modular curve. If A = 𝔽_{q}[T] then in fact 𝕄 ≅ 𝔸^{1} is the affine line. In general, we have seen that 𝕄 is a smooth irreducible affine curve defined over H.

2.3 The curves Y (𝔫), Y_{0}(𝔫) and Y_{2}(𝔫)

We denote by Z the center of the algebraic group GL_{2}. Let 𝔫 ⊂ A be a non-zero ideal, and consider the following open subgroups of GL_{2}(Â).

K(𝔫) = ker (GL_{2}(Â) → GL_{2}(Â/𝔫)),
K_{0}(𝔫) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Â) | c ∈ 𝔫\},
K_{2}(𝔫) = {γ ∈ GL_{2}(Â) | (γ mod 𝔫) ∈ Z(Â/𝔫)}.

These lead to coarse moduli schemes M_{A}^{2}(K(𝔫)) (also denoted M_{A}^{r}(𝔫) in the literature), M_{A}^{2}(K_{0}(𝔫)) and M_{A}^{2}(K_{2}(𝔫)). Similarly to the previous section, we define the Drinfeld modular curves Y (𝔫), Y_{0}(𝔫) and Y_{2}(𝔫) as the identity components of the respective moduli schemes tensored with ℂ_{∞}.

We have isomorphisms of rigid analytic varieties

Y (𝔫)(ℂ_{∞})^{an} ≅ Γ(𝔫)\Ω, Y_{0}(𝔫)(ℂ_{∞})^{an} ≅ Γ_{0}(𝔫)\Ω, Y_{2}(𝔫)(ℂ_{∞})^{an} ≅ Γ_{2}(𝔫)\Ω,

for the arithmetic groups

Γ(𝔫) = {γ ∈ GL_{2}(A) | γ ≡ 1 mod 𝔫},
Γ_{0}(𝔫) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(A) | c ∈ 𝔫\},
Γ_{2}(𝔫) = {γ ∈ GL_{2}(A) | (γ mod 𝔫) ∈ Z(A/𝔫)}.

As in the classical case, Y_{0}(𝔫) parametrizes pairs of Drinfeld modules linked by cyclic 𝔫-isogenies.

As K(𝔫), K_{0}(𝔫) and K_{2}(𝔫) are all congruence subgroups of GL_{2}(Â) (pick any 0 ≠ f ∈ 𝔫), it follows from the Weil-pairing (2.8) that Y_{0}(𝔫) is defined over H, whereas the curves Y (𝔫) and Y_{2}(𝔫) are defined over the class fields corresponding to

k^{×}\𝔸_{f}^{×}/{x ∈ Â^{×} | x ≡ 1 mod 𝔫}, and
k^{×}\𝔸_{f}^{×}/{x ∈ Â^{×} | x is a square mod 𝔫},

respectively.

For a ring R ⊃ 𝔽_{q}, and algebraic group G, we define

G^{1}(R) := {g ∈ G(R) | det(g) ∈ 𝔽_{q}^{×}}.

The degree of an ideal 𝔫 ⊂ A is deg(𝔫) := log_{q} |𝔫| = log_{q} |A/𝔫|.

Proposition 2.9 We have covers Y (𝔫) → Y_{2}(𝔫) → Y_{0}(𝔫) → 𝕄. Moreover, Y (𝔫) and Y_{2}(𝔫) are Galois covers of 𝕄 with Galois groups

Gal(Y (𝔫)/𝕄) ≅ GL_{2}^{1}(A/𝔫)/Z(𝔽_{q})
Gal(Y_{2}(𝔫)/𝕄) ≅ GL_{2}^{1}(A/𝔫)/Z^{1}(A/𝔫) ≅ PGL_{2}^{1}(A/𝔫).

Note that PGL_{2}^{1}(A/𝔫) ≅ PSL_{2}(A/𝔫) if every prime factor of 𝔫 has even degree.

Proof. Clearly, Γ(𝔫) ⊂ Γ_{2}(𝔫) ⊂ Γ_{0}(𝔫), whence the coverings. Furthermore, Γ(𝔫) and Γ_{2}(𝔫) are normal subgroups of GL_{2}(A), so the respective coverings are Galois with Galois groups

Gal(Y (𝔫)/𝕄) ≅ GL_{2}(A)/Γ(𝔫) ≅ GL_{2}^{1}(A/𝔫)/Z(𝔽_{q}), and
Gal(Y_{2}(𝔫)/𝕄) ≅ GL_{2}(A)/Γ_{2}(𝔫) ≅ GL_{2}^{1}(A/𝔫)/Z^{1}(A/𝔫) ≅ PGL_{2}^{1}(A/𝔫).

Lastly, if every prime factor of 𝔫 has even degree, then every element of 𝔽_{q} is a square in A/𝔫 and PGL_{2}^{1}(A/𝔫) = PSL_{2}(A/𝔫). □

2.4 Special subcurves of (M_{A,ℂ_{∞}}^{2})^{n}

In the Introduction we defined the notion of (pure) special subcurves of a product of n Drinfeld modular curves, in particular in 𝕄^{n}.

For a subset I ⊂ {1, …, m} we denote by p_{I} : 𝕄^{n} → 𝕄^{I} the projection onto the coordinates listed in I. We will also write p_{{i}} = p_{i} and p_{{i,j}} = p_{i,j}.

The curve Y_{0}(𝔫) parametrizes Drinfeld modules linked by a cyclic isogeny of degree 𝔫. If (ϕ, ϕ′) is such a pair, with ϕ ∈ 𝕄(ℂ_{∞}), then ϕ′ ∈ 𝕄(ℂ_{∞}) if and only if 𝔫 is a principal ideal. Thus, for 𝔫 = ⟨N⟩ principal, we obtain a map Y_{0}(𝔫) → 𝕄^{2} whose image is denoted by Y′_{0}(N). Analytically, this map is given by

Γ_{0}(𝔫)\Ω ⟶ (GL_{2}(A)\Ω) × (GL_{2}(A)\Ω)
[ω] ⟼ ([ω], [Nω]).  (2.10)

In particular, we see that Y′_{0}(N) is an irreducible pure special subcurve of 𝕄^{2}. Moreover, every irreducible pure special subcurve of 𝕄^{2} is of the form Y′_{0}(N) for some N ∈ A. Indeed, for any g ∈ GL_{2}(k), we denote by N = deg(g) the determinant of a·g, with a ∈ A chosen in such a way that the entries of a · g are in A and have no common factor. This N is defined up to the square of a unit, and is called the degree of g. Now, let g_{1}, g_{2} ∈ GL_{2}(k) and N := deg(g_{2}g_{1}^{−1}). Then the special curve corresponding to the map ω ↦ ([g_{1}(ω)], [g_{2}(ω)]) is just Y′_{0}(N) ⊂ 𝕄^{2}.

Notice that an irreducible curve Y ⊂ 𝕄^{n} is a pure special subcurve if and only if p_{i,j}(Y) = Y′_{0}(N_{ij}) for some N_{ij} ∈ A for all i ≠ j.

Next, it will be convenient to study the space SC_{n} of all special subcurves in (M_{A,ℂ_{∞}}^{2})^{n}. Analytically, this will turn out to be the following double coset space

SC_{n}(ℂ_{∞})^{an} := GL_{2}(k)\GL_{2}(𝔸_{f})^{n} × Ω/(Z(𝔸_{f})GL_{2}(Â))^{n},  (2.11)

where (Z(𝔸_{f})GL_{2}(Â))^{n} acts from the right on GL_{2}(𝔸_{f})^{n} in the usual way, and trivially on Ω, while GL_{2}(k) acts from the left on Ω in the usual way, and diagonally on GL_{2}(𝔸_{f})^{n}.

Choose a set of representatives T ⊂ GL_{2}(𝔸_{f})^{n} for the double quotient

GL_{2}(k)\GL_{2}(𝔸_{f})^{n}/(Z(𝔸_{f})GL_{2}(Â))^{n}.  (2.12)

Note that T is infinite, and we again choose 1 ∈ T. For each t = (t_{1}, …, t_{n}) ∈ T let Γ_{t} = ∩_{i=1}^{n}t_{i}GL_{2}(Â)t_{i}^{−1} ∩ GL_{2}(k). Then we have a canonical bijection

GL_{2}(k)\GL_{2}(𝔸_{f})^{n} × Ω/(Z(𝔸_{f})GL_{2}(Â))^{n} ⟶^{∼} ∐_{t∈T} Γ_{t}\Ω
[h, ω]  ⟼ [g^{−1}(ω)]_{t},

where t ∈ T is such that [t] = [h] in (2.12), and g ∈ GL_{2}(k) is such that h = gta, for some a ∈ (Z(𝔸_{f})GL_{2}(Â))^{n}. We see that SC_{n} is the disjoint union of an infinite family of Drinfeld modular curves, parametrized by T.

Next, we describe a map from SC_{n}(ℂ_{∞}) into (M_{A}^{2}(ℂ_{∞}))^{n}.

SC_{n}(ℂ_{∞}) ⟶^{θ} (M_{A}^{2}(ℂ_{∞}))^{n}
GL_{2}(k)\GL_{2}(𝔸_{f})^{n} × Ω/(Z(𝔸_{f})GL_{2}(Â))^{n} ⟶ GL_{2}(k)^{n}\GL_{2}(𝔸_{f})^{n} × Ω^{n}/(Z(𝔸_{f})GL_{2}(Â))^{n}
= (GL_{2}(k)\GL_{2}(𝔸_{f}) × Ω/GL_{2}(Â))^{n}
[(h_{1}, …, h_{n}), ω] ⟼ ([h_{1}, ω], …, [h_{n}, ω]);
∐_{t∈T} Γ_{t}\Ω ⟶ ∐_{s∈S^{n}} (Γ_{s_{i}}\Ω)_{i=1}^{n}
[ω]_{t} ⟼ ([g_{i}^{−1}(ω)]_{s_{i}})_{i=1}^{n},  (2.13)

where s_{i} ∈ S such that [s_{i}] = [t_{i}] in GL_{2}(k)\GL_{2}(𝔸_{f})/GL_{2}(Â), and g_{i} ∈ GL_{2}(k) such that g_{i}s_{i}a_{i} = t_{i} for some a_{i} ∈ GL_{2}(Â), for i = 1, …, n.

We note that GL_{2}(k)\GL_{2}(𝔸_{f})/(Z(𝔸_{f})GL_{2}(Â)) = GL_{2}(k)\GL_{2}(𝔸_{f})/GL_{2}(Â), as for any x ∈ 𝔸_{f} we have ( \begin{matrix}x & 0 \\ 0 & x \end{matrix} ) ≡ ∏_{𝔭} ( \begin{matrix}π_{𝔭}^{n𝔭} & 0 \\ 0 & π_{𝔭}^{n𝔭} \end{matrix} ) mod GL_{2}(Â), where π_{𝔭} ∈ k denotes a chosen uniformizer at 𝔭, and n_{𝔭} = min(0, v_{𝔭}(x)) is zero for almost all 𝔭. This latter scalar is in GL_{2}(k), and, as scalars commute in GL_{2}(𝔸_{f}), is killed by the left GL_{2}(k) action. The same principle does not hold when we have n > 1 copies of GL_{2}(𝔸_{f}) and GL_{2}(Â) but only one copy of GL_{2}(k), which is why the Z(𝔸_{f}) appears explicitly in the definition of SC_{n}(ℂ_{∞})^{an}.

Let

T^{0} := ker (T → GL_{2}(k)^{n}\GL_{2}(𝔸_{f})^{n}/GL_{2}(Â)^{n}).

Then for every t ∈ T^{0}, the Drinfeld modular curve Γ_{t}\Ω is mapped into 𝕄^{n}(ℂ_{∞}) by θ, where its image is a pure special curve, as defined in the Introduction.

Conversely, all pure special curves in 𝕄^{n} arise in this way:

Proposition 2.14 We have bijections

T^{0} ⟷ (Z(k)GL_{2}(A))^{n}\GL_{2}(k)^{n}/GL_{2}(k) ⟷ {Isomorphism classes of pure

special curves in 𝕄^{n}}

t ⟼ [g] = [(g_{1}, …, g_{n})] ⟼ Pure special curve defined by ω ↦ [(g_{1}(ω), …, g_{n}(ω))],

with g ∈ GL_{2}(k)^{n} such that g = at^{−1} for some a ∈ (Z(𝔸_{f})GL_{2}(Â))^{n}.

Proof. In the first bijection, let t ∈ T^{0}, then by definition there exist g ∈ GL_{2}(k)^{n} and a^{−1} ∈ (Z(𝔸_{f})GL_{2}(Â))^{n} such that gta^{−1} = 1. Then [g] = [at^{−1}] is well-defined, since if also g′ = a′t^{−1}, then g′ = a′a^{−1}g with a′a^{−1} ∈ (Z(𝔸_{f})GL_{2}(Â))^{n} ∩ GL_{2}(k)^{n} = (Z(k)GL_{2}(A))^{n}. Moreover, given g ∈ GL_{2}(k)^{n}, there exist h ∈ GL_{2}(k), t ∈ T^{0} and a ∈ (Z(𝔸_{f})GL_{2}(Â))^{n} such that g^{−1} = hta, hence the map is surjective. If also g^{−1} = h′t′a′, then it is clear that t = t′, by definition of T.

The second bijection follows as (g_{1}, …, g_{n}) and (g′_{1}, …, g′_{n}) in GL_{2}(k)^{n} define the same pure special curve in 𝕄^{n} if and only if g′_{i} = γ_{i}g_{i}σ for all i = 1, …, n, for some (γ_{1}, …, γ_{n}) ∈ (Z(k)GL_{2}(A))^{n} and σ ∈ GL_{2}(k). □

This proposition justifies our definition of SC_{n} as the space of all pure special subcurves of (M_{A,ℂ_{∞}}^{2})^{n}. We also denote by SC_{n}^{0} ⊂ SC_{n} the subfamily of those Drinfeld modular curves corresponding to T^{0}, i.e. SC_{n}^{0}(ℂ_{∞})^{an} ≅ ∐_{t∈T^{0}} Γ_{t}\Ω, which is the space of all pure special subcurves of 𝕄^{n}.

Lastly, we point out that special subvarieties are simply products of CM points and pure special subcurves of subproducts.

2.5 Pure special curves and trees

Let 𝔭 ⊂ A be a prime. Recall (e.g. [14]) that the Brûhat-Tits tree T_{𝔭} of GL_{2}(k_{𝔭}) is the (|𝔭| + 1)-regular tree whose vertices represent homothety classes of A_{𝔭}-lattices in k_{𝔭}^{2}, and two vertices representing classes Λ and Λ′ are joined by an (unoriented) edge if there exist representative lattices L ∈ Λ and L′ ∈ Λ′ such that L′ ⊂ L and L/L′ ≅ A/𝔭. The group GL_{2}(k_{𝔭}) acts transitively on T_{𝔭} and the stabilizer of the “origin” v_{o,𝔭}, which is the vertex corresponding to the lattice A_{𝔭}^{2}, is Z(k_{𝔭})GL_{2}(A_{𝔭}). Thus we get a bijection T_{𝔭} ↔ GL_{2}(k_{𝔭})/Z(k_{𝔭})GL_{2}(A_{𝔭}).

Hence we have bijections

T ⟷ GL_{2}(k)\GL_{2}(𝔸_{f})^{n}/(Z(𝔸_{f})GL_{2}(Â))^{n} ⟷ GL_{2}(k)\∏_{𝔭}′T_{𝔭}^{n}.  (2.15)

Here the restricted product ∏′_{𝔭} denotes the subset of those families (v_{1,𝔭}, …, v_{n,𝔭})_{𝔭} of ntuples of vertices v_{i,𝔭} ∈ T_{𝔭} such that v_{1,𝔭} = ⋯ = v_{n,𝔭} = v_{o,𝔭} for almost all primes 𝔭.

We also have the following non-canonical bijection

GL_{2}(k)\∏_{𝔭}′T_{𝔭}^{n} ⟷ S × ∏_{𝔭}′GL_{2}(k)\T_{𝔭}^{n} ( ↔ Pic(A) × ∏_{𝔭}′GL_{2}(k_{𝔭})\T_{𝔭}^{n} )  (2.16)

which arises from

GL_{2}(k)\GL_{2}(𝔸_{f})^{n}/(Z(𝔸_{f})GL_{2}(Â))^{n} ⟶^{∼}

GL_{2}(k)\GL_{2}(𝔸_{f})/Z(𝔸_{f})GL_{2}(Â) × GL_{2}(𝔸_{f})\GL_{2}(𝔸_{f})^{n}/(Z(𝔸_{f})GL_{2}(Â))^{n};

[g_{1}, …, g_{n}] ⟼ ([g_{1}], [1, g_{1}^{−1}g_{2}, …, g_{1}^{−1}g_{n}]),

which is readily verified. This bijection may also be of help to the reader puzzling over [3, §1.3].

The moduli interpretation of (2.15) and (2.16) is the following.

We may view T_{𝔭} as the tree of 𝔭-isogenies of Drinfeld modules as in the elliptic curve case, but with the added subtlety that different vertices will map to different irreducible components of (M_{A,ℂ_{∞}}^{2})^{n} when 𝔭 ⊂ A is not principal. Now let t ∈ T correspond to the

pure special subcurve C_{t} ↪^{θ} (M_{A,ℂ_{∞}}^{2})^{n}. Let (s, \underline{v}) ∈ S × _{∏𝔭}′GL_{2}(k)\T_{𝔭}^{n} correspond to t via (2.16). We may choose a representative family (v_{1,𝔭}, …, v_{n,𝔭})_{𝔭} of the class \underline{v} such that v_{1,𝔭} = v_{o,𝔭} for all 𝔭. Then a typical point x = (x_{1}, …, x_{n}) ∈ C_{t}(ℂ_{∞}) satisfies: x_{1} lies on the s-component of M_{A,ℂ_{∞}}^{2}, and the isogenies x_{1} → x_{2} → ⋯ → x_{n} correspond to (v_{2,𝔭}, …, v_{n,𝔭})_{𝔭}, respectively.

In the case n = 3 we have a particularly pleasing combinatorial description of GL_{2}(k)\T_{𝔭}^{3}. Any triple of vertices (v_{1}, v_{2}, v_{3}) ∈ T_{𝔭}^{3} has a well-defined center v_{c}, defined by the property that the three paths (possibly of length zero) from v_{c} to the v_{i} are disjoint. Denote by (n_{1}, n_{2}, n_{3}) ∈ ℕ_{0}^{3} the lengths of these paths. Furthermore, GL_{2}(k_{𝔭}) acts 3-transitively on the set of ends of T_{𝔭}, and it follows that a triple of vertices (v_{1}, v_{2}, v_{3}) with corresponding center v_{c} and triple of path-lengths (n_{1}, n_{2}, n_{3}) is mapped by GL_{2}(k) to another triple (v′_{1}, v′_{2}, v′_{3}) with center v′_{c} and path-lengths (n′_{1}, n′_{2}, n′_{3}) if and only if (n_{1}, n_{2}, n_{3}) = (n′_{1}, n′_{2}, n′_{3}). Thus we have bijections

GL_{2}(k)\T_{𝔭}^{3} ⟷ ℕ_{0}^{3}; ∏_{𝔭}′GL_{2}(k)\T_{𝔭}^{3} ⟷ I_{A}^{3}; T ⟷ Pic(A) × I_{A}^{3},  (2.17)

where I_{A} denotes the semigroup of non-zero A-ideals. The second bijection is given by

(v_{1,𝔭}, v_{2,𝔭}, v_{3,𝔭})_{𝔭} ↦ (n_{1,𝔭}, n_{2,𝔭}, n_{3,𝔭})_{𝔭} ↦ (𝔫_{1}, 𝔫_{2}, 𝔫_{3}) = ∏_{𝔭}(𝔭^{n_{1,𝔭}}, 𝔭^{n_{2,𝔭}}, 𝔭^{n_{3,𝔭}}).

Let Y be the special curve corresponding to the data ([𝔞], (𝔫_{1}, 𝔫_{2}, 𝔫_{3})) ∈ Pic(A) × I_{A}^{3}. Then Y maps into 𝕄^{3} if and only if [𝔞] = 1 and 𝔫_{i} = ⟨N_{i}⟩ is principal for i = 1, 2, 3. In this case we have p_{i,j}(Y) = Y′_{0}(N_{i}N_{j}) ⊂ 𝕄^{2}. In general, if a pure modular curve Y ⊂ (M_{A,ℂ_{∞}}^{2})^{3} corresponds to a triple (𝔫_{1}, 𝔫_{2}, 𝔫_{3}) ∈ I_{A}^{3}, then p_{i,j}(Y) is isomorphic to a suitable irreducible component of M_{A}^{2}(K_{0}(𝔫_{i}𝔫_{j}))_{ℂ_{∞}}.

The following result will be useful later on.

Proposition 2.18 Let Y ⊂ 𝕄^{3} be a pure special curve corresponding to a triple (𝔫_{1}, 𝔫_{2}, 𝔫_{3}), and fix (x_{1}, x_{2}) ∈ p_{1,2}(Y (ℂ_{∞})). Then

|p_{1,2}^{−1}(x_{1}, x_{2}) ∩ Y (ℂ_{∞})| ≥ ∏_{𝔭|𝔫_{3}} \frac{(|𝔭| − 1)(|𝔭| + 1)^{n_{3,𝔭}−1}}{2n_{3,𝔭} + 1} ,

where 𝔫_{i} = ∏_{𝔭} 𝔭^{n_{i,𝔭}}.

Proof. Let (v_{1,𝔭}, v_{2,𝔭}, v_{3,𝔭})_{𝔭} ∈ ∏′_{𝔭} T_{𝔭}^{3} be a representative family of tuples corresponding to the special curve Y, and denote by (v_{c,𝔭})_{𝔭} its family of centers. Fixing x_{1} and x_{2} amounts to fixing v_{1,𝔭}, v_{2,𝔭} for all 𝔭. We now count the possible vertex families (v_{3,𝔭})_{𝔭} for which (v_{1,𝔭}, v_{2,𝔭}, v_{3,𝔭})_{𝔭} represents Y. As (𝔫_{1}, 𝔫_{2}, 𝔫_{3}) is prescribed, so is v_{c,𝔭} for every 𝔭, and from (2.17) follows that we must count the paths of length n_{3,𝔭} from v_{c,𝔭} which are disjoint from the two paths leading to v_{1,𝔭} and v_{2,𝔭}, which gives at least (|𝔭| − 1)(|𝔭| + 1)^{n_{3,𝔭}−1} for each 𝔭. Thus the number of valid cyclic 𝔫_{3}-isogenies from x_{c} (corresponding to (v_{c,𝔭})_{𝔭}) to x_{3} is given by ∏_{𝔭|𝔫_{3}}(|𝔭| − 1)(|𝔭| + 1)^{n_{3,𝔭}−1}.

But if x_{c} has complex multiplication by an order in K, then distinct isogenies may well lead to the same x_{3}, they correspond to non-trivial endomorphisms f ∈ End(x_{c}) of norm N_{K/k}(f) = 𝔫_{3}^{2}. There are at most ∏_{𝔭}(2n_{3,𝔭} + 1) such endormorphisms, which completes the proof. □

2.6 Main results and reduction to 𝕄^{n}

Let K_{i} ⊂ GL_{2}(Â) be subgroups of finite index, for i = 1, …, n. With a bit more effort, we could give a general treatment of pure special subcurves of ∏_{i=1}^{n} M_{A}^{2}(K_{i})_{ℂ_{∞}} as in the previous two sections. But the following definition is much easier:

Consider the morphism

π : ∏_{i=1}^{n} M_{A}^{2}(K_{i})_{ℂ_{∞}} ⟶ (M_{A,ℂ_{∞}}^{2})^{n}

induced by the inclusions K_{i} ↪ GL_{2}(Â). Then an irreducible subvariety Y ⊂ ∏_{i=1}^{n} M_{A}^{2}(K_{i})_{ℂ_{∞}} is a special subvariety if and only if the image π(Y) is a special subvariety of (M_{A,ℂ_{∞}}^{2})^{n}. Similarly for (pure) special subcurves.

We restate our main result as follows

Theorem 1.2′ Let K_{i} ⊂ GL_{2}(Â) be subgroups of finite index for i = 1, …, n. Let Y ⊂ ∏_{i=1}^{n} M_{A}^{2}(K_{i})_{ℂ_{∞}} be an irreducible subvariety. Then Y (ℂ_{∞}) contains a Zariski-dense subset of CM points if and only if Y is a special subvariety.

As every irreducible component of M_{A}^{2}(K_{i})_{ℂ_{∞}} is a Drinfeld modular curve corresponding to Γ_{s}\Ω, for the arithmetic subgroup Γ_{s} = sK_{i}s^{−1} ∩ GL_{2}(k) with some s ∈ GL_{2}(𝔸_{f}), we see that the above Theorem is equivalent to Theorem 1.2.

Next we want to show that it suffices to prove Theorem 1.2 for X = 𝕄^{n}.

Proposition 2.19 Let Γ′_{i} ⊂ Γ_{i} ⊂ GL_{2}(k) be arithmetic subgroups corresponding to Drinfeld modular curves X′_{i} and X_{i}, respectively, for i = 1, …, n. Let X = X_{1} × ⋯ × X_{n} and X′ = X′_{1} × ⋯ × X′_{n}. Then the canonical map f : X′ → X preserves special subvarieties. In other words, a subvariety Y ⊂ X′ is special if and only if f(Y) is special.

Proof. This is immediate, as special subvarieties are defined purely in terms of isogeny relations between coordinates, and the property that any constant projections Y → X′_{i} have CM points as images. These properties are independent of any level structures. □

Now let Γ_{i} ⊂ GL_{2}(k) be an arithmetic subgroup such that X_{i}(ℂ_{∞})^{an} ≅ Γ_{i}\Ω, for each i = 1, …, n. Then each Γ′_{i} = GL_{2}(A) ∩ Γ_{i} is again an arithmetic subgroup, and we have maps

Γ′_{i}\Ω ⟶ Γ_{i}\Ω, Γ′_{i}\Ω ⟶ GL_{2}(A)\Ω

corresponding to morphisms of Drinfeld modular curves

f_{i} : X′_{i} ⟶ X_{i}, g_{i} : X′_{i} ⟶ 𝕄,

for each i = 1, …, n.

Applying Proposition 2.19 to the maps ∏_{i} f_{i} and ∏_{i} g_{i} shows that Theorem 1.2 holds for X_{1} × ⋯ × X_{n} if and only if it holds for 𝕄^{n}.

With this reduction step out of the way, the rest of our proof of Theorem 1.2 will follow [3] very closely, with 𝕄^{n} playing the role of 𝔸^{n}. The main geometric aspect of 𝔸^{n} used in [3] is its structure as the product of affine lines, and it turns out that the product structure of 𝕄^{n} will suffice, with some caveats. Firstly, 𝕄 is not defined over k but over H, the Hilbert class field of k. (Classically, the elliptic modular curve Y (1) is defined over the Hilbert class field of ℚ, which is just ℚ itself. In our case, k ≠ H in general, which illustrates the “metamathematical” phenomenon of splitting of the two different roles of ℚ into two different fields (k and H) in characteristic p. See [8, Preface] for a discussion).

Secondly, in general the isomorphism

GL_{2}(A)\Ω ⟶^{∼} 𝕄(ℂ_{∞})^{an}

is no longer induced by a well-behaved “j-invariant” j : Ω → ℂ_{∞}, as is the case for A = 𝔽_{q}[T]. Consequently, we cannot take advantage of the analytic properties of the j-function, and will say nothing about the Weil heights of CM points.

3 Hecke correspondences and CM points

3.1 The general formalism

We briefly describe Hecke correspondences on M_{A}^{2}(K) in general. Let g ∈ GL_{2}(𝔸_{f}), and set K_{g} = K ∩ g^{−1}Kg. Then g acts from the left on M_{A}^{2}(K_{g}) (by letting g^{−1} act from the right on Drinfeld modules with full level structures, see [7, II.3]). Combined with the standard projection π : M_{A}^{2}(K_{g}) → M_{A}^{2}(K) this gives rise to the Hecke correspondence on M_{A}^{2}(K):

T_{g} : M_{A}^{2}(K) ⟵^{π} M_{A}^{2}(K_{g}) ⟶^{π∘g} M_{A}^{2}(K).  (3.1)

This is an algebraic correspondence of finite degree [K : K_{g}].

Analytically, this correspondence is described as follows. The group GL_{2}(𝔸_{f}) acts from the left on GL_{2}(𝔸_{f}) × Ω by g · (h, ω) = (hg^{−1}, ω). This gives rise to the correspondence

\xymatrix{GL_{2}(𝔸_{f}) × Ω \ar[r]^{g} \ar[d] & GL_{2}(𝔸_{f}) × Ω \ar[d] \\ GL_{2}(k)\GL_{2}(𝔸_{f}) × Ω/K & GL_{2}(k)\GL_{2}(𝔸_{f}) × Ω/K,}

which is easily seen to factor through GL_{2}(k)×K_{g}, thus describing the finite correspondence

\xymatrix{ & GL_{2}(k)\GL_{2}(𝔸_{f}) × Ω/K_{g} \ar@{-->}[dl]_{π} \ar@{-->}[dr]^{π∘g} \\ GL_{2}(k)\GL_{2}(𝔸_{f}) × Ω/K & T_{g} \ar@{.>}[l] \ar@{.>}[r] & GL_{2}(k)\GL_{2}(𝔸_{f}) × Ω/K.}

(3.2)

Explicitly, this correspondence acts as follows. Choose a finite set {g_{i} ∈ GL_{2}(𝔸_{f}) | i ∈ I_{g}} of representatives for the cosets K\KgK. Let [h, ω] ∈ GL_{2}(k)\GL_{2}(𝔸_{f}) × Ω/K ≅ M_{A}^{2}(K)(ℂ_{∞}). Then T_{g} maps the point [h, ω] to the set {[hg_{i}^{−1}, ω] | i ∈ I_{g}}.

At first sight, the g-action does not seem to effect the Ω part of the above spaces, but we will show next how T_{g} acts on M_{A}^{2}(K)(ℂ_{∞})^{an} ≅ ∐_{s∈S} Γ_{s}\Ω. For simplicity, we assume that det(K_{g}) = det(K). That way, M_{A}^{2}(K_{g})_{ℂ_{∞}} and M_{A}^{2}(K)_{ℂ_{∞}} have the same number of irreducible components, and we may choose the same set 1 ∈ S ⊂ GL_{2}(𝔸_{f}) of representatives for GL_{2}(k)\GL_{2}(𝔸_{f})/K and for GL_{2}(k)\GL_{2}(𝔸_{f})/K_{g}. We may also assume that h ∈ S.

For each i ∈ I_{g}, there exist f_{h,g,i} ∈ GL_{2}(k) and k_{h,g,i} ∈ K such that f_{h,g,i}hg_{i}^{−1}k_{h,g,i} = s_{h,g} ∈ S. Now [hg_{i}^{−1}, ω] = [s_{h,g}, f_{h,g,i}(ω)], and this element maps to [f_{h,g,i}(ω)] ∈ Γ_{s_{h,g}}\Ω. Moreover, we can check that s_{h,g} here only depends on h and on g, not on i, so the correspondence T_{g} maps the h-component Γ_{h}\Ω to the s_{h,g}-component Γ_{s_{h,g}}\Ω. Within each component, T_{g} maps the point [ω] to the set {[f_{h,g,i}(ω)] | i ∈ I_{g}}, and in fact the set {f_{h,g,i} | i ∈ I_{g}} is a set of representatives for Γ_{s_{h,g}}\Γ_{s_{h,g}}f_{h,g}Γ_{h}, where f_{h,g} ∈ GL_{2}(k) is such that f_{h,g}hg^{−1}k_{h,g} = s_{h,g} ∈ S for some k_{h,g} ∈ K. Hence, for each h ∈ S, the correspondence T_{g} on M_{A}^{2}(K) induces the following correspondence from the h-component to the s_{h,g}-component of M_{A}^{2}(K)(ℂ)^{an}:s

\xymatrix{Ω \ar[d] \ar[r]^{f_{h,g}} & Ω \ar[d] \\ Γ_{h}\Ω & Γ_{s_{h,g}}\Ω}

which factors through

Γ_{h} ∩ f_{h,g}^{−1}Γ_{s_{h,g}}f_{h,g} = hK_{g}h^{−1} ∩ GL_{2}(k) = Γ_{h} ∩ Γ_{hg},

thus giving the correspondence

\xymatrix{ & (Γ_{h} ∩ Γ_{hg})\Ω \ar[dl]_{π} \ar[dr]^{π∘f_{h,g}} \\ Γ_{h}\Ω &  & Γ_{s_{h,g}}\Ω}  (3.3)

which has finite degree [Γ_{h} : Γ_{h} ∩ Γ_{hg}] = [K : K_{g}]. Thus the correspondence (3.3) gives a component by component description of the correspondence T_{g} on M_{A}^{2}(K)_{ℂ_{∞}} of (3.1).

Remark 3.4 Let s_{1}, s_{2} ∈ S, and set g = s_{2}^{−1}s_{1}. Then f_{s_{1},g} = k_{s_{1},g} = 1, and so T_{g} maps the s_{1}-component of M_{A}^{2}(K)_{ℂ_{∞}} to the s_{2}-component, while leaving the ω unchanged in that component.

We also notice that s_{h,g} = h if and only if g ∈ Kh^{−1}GL_{2}(k)h.

3.2 Hecke correspondences on 𝕄^{n}

In the remainder of this section, we closely follow [3, §2]. From now on, we will assume that K = GL_{2}(Â) and g = diag(N, 1, …, 1) for some N ∈ A with |N| > 1. In this case, we see that K_{g} = K_{0}(⟨N⟩) =: K_{0}(N), Γ_{1} = GL_{2}(A), Γ_{1} ∩ Γ_{g} = Γ_{0}(N), and furthermore, T_{N} := T_{g} acts trivially on the set of irreducible components on M_{A,ℂ_{∞}}^{2}, in particular, it induces a correspondence, again denoted T_{N}, on 𝕄. The correspondence is also described by the inclusion Y′_{0}(N) ⊂ 𝕄^{2} from (2.10), and it maps the point x ∈ 𝕄(ℂ_{∞}) to the set of those points y ∈ 𝕄(ℂ_{∞}) corresponding to Drinfeld modules linked to x via cyclic ⟨N⟩-isogenies.

We define T_{𝕄^{n},N} to be the correspondence on 𝕄^{n} which is the product of the correspondences T_{N} on each factor 𝕄. When there is no risk of confusion, we also denote it by T_{N}. We say T_{𝕄^{n},N} stabilizes an algebraic subvariety Y ⊂ 𝕄^{n} if Y ⊂ T_{𝕄^{n},N}(Y). In this case, we define the restriction of T_{N} to Y, denoted T_{Y,N}, to be the union of the irreducible components of T_{𝕄^{n},N} ∩ (Y × Y) of maximal dimension. It is a correspondence on Y which is still surjective, in the sense that the two projections T_{Y,N} → Y are surjective. Whenever we mention T_{Y,N}, it is implicit that T_{N} stabilizes Y.

Let x ∈ 𝕄(ℂ_{∞}), and suppose that x ∈ T_{N}(x). Then x admits a cyclic endomorphism, hence is a CM point. For a given N ∈ A, |N| > 1, there are only finitely many points stabilized by T_{N}, which correspond to the points of the diagonal in 𝕄^{2} which intersect Y′_{0}(N). On the other hand, given a CM point x ∈ 𝕄(ℂ_{∞}) with R = End(x), there are infinitely many N ∈ A such that x ∈ T_{N}(x), namely all those N composed of primes 𝔭 ⊂ A for which 𝔭R is a product of two distinct principal prime ideals of R. Equivalently, these primes split completely in the class field corresponding to Pic(R), hence the set of such primes has density 1/2|Pic(R)|, by Čebotarev. Notice also that such primes 𝔭 are necessarily principal in A.

3.3 Some intersection theory

𝕄 is affine, so we may fix an embedding 𝕄 ⊂ 𝔸_{ℂ_{∞}}^{m}. Then we obtain embeddings 𝕄^{n} ⊂ 𝔸_{ℂ_{∞}}^{mn} and \overline{𝕄^{n}} ⊂ ℙ_{ℂ_{∞}}^{mn}. (Here ¯· denotes the Zariski-closure).

For any irreducible Y ⊂ 𝕄^{n}, we denote by deg(Y) the degree of \overline{Y} ⊂ ℙ^{mn} in the usual sense. If Y = ∪_{i}Y_{i} is a union of irreducible components, then deg(Y) := ∑_{i} deg(Y_{i}), (regardless of their dimensions).

We denote by ψ(N) = [GL_{2}(A) : Γ_{0}(N)] = ∏_{𝔭|N}(1 + |𝔭|^{−1}) the degree of the correspondence T_{N} on 𝕄. In particular, ψ(N) ≤ deg(Y′_{0}(N)) ≤ 2ψ(N).

We collect the following facts:

Proposition 3.5 Let Y ⊂ 𝕄^{n} be an algebraic subvariety.

1. Y has at most deg(Y) irreducible components.

2. If Y′ ⊂ 𝕄^{n} is another subvariety, then deg(Y ∩ Y′) ≤ deg(Y) deg(Y′).

3. deg(T_{𝕄^{n},N}(Y)) ≤ 2^{n}ψ(N)^{n} deg(Y).

4. There are only finitely many pure special subcurves of 𝕄^{n} of degree less than a given bound.

Proof. (1) is trivial, (2) and (3) follow from [6, 8.4.6], and (4) follows from the fact that a curve Y ⊂ 𝕄^{n} is pure special if and only if p_{i,j}(Y) = Y′_{0}(N_{ij}) for some N_{ij} ∈ A for all i ≠ j, and deg(Y′_{0}(N_{ij})) ≥ ψ(N_{ij}). □

3.4 Preimages in Ω^{n}

Denote by π the rigid analytic map π : Ω^{n} → 𝕄^{n}(ℂ_{∞})^{an}, which is the quotient for the GL_{2}(A)^{n}-action. For each irreducible component Y_{i} of Y, we choose an irreducible rigid analytic variety Z_{i} ⊂ Ω^{n} with π(Z_{i}) = Y_{i}(ℂ_{∞})^{an}. The group GL_{2}(k_{∞})^{n} acts on Ω^{n}, and the GL_{2}(A)^{n}-orbit of Z_{i} is π^{−1}(Y_{i}). We next describe the action of T_{N} in Ω^{n}.

We may choose a set T ⊂ GL_{2}(k) of representatives for GL_{2}(A)\GL_{2}(A)( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} )GL_{2}(A) such that the following holds: For every pair of principal ideals 𝔞, 𝔡 ⊂ A with 𝔞𝔡 = ⟨N⟩, and set of representatives {b_{1}, …, b_{r}} ⊂ A of A/𝔡, there exist a, d ∈ A with 𝔞 = ⟨a⟩ and 𝔡 = ⟨d⟩ such that

(\begin{matrix}a & b_{j} \\ 0 & d \end{matrix}) ∈ T for j = 1, …, r.  (3.6)

One easily verifies that these elements indeed represent distinct coset classes. We write T^{n} = {t_{j} ∈ GL_{2}(k)^{n} | j ∈ I^{n}}, where I = {1, …, ψ(N)}.

For each Z_{i} we define J_{Z_{i}} ⊂ I^{n} as the set of those indices j for which t_{j}(Z_{i}) ⊂ π^{−1}(Y_{i}). Now let y ∈ Y_{i}(ℂ_{∞}) and choose some z ∈ Z_{i} with π(z) = y. Then

T_{𝕄^{n},N}(y) = {π(t_{j}(z)) | j ∈ I^{n}},  (3.7)
T_{Y,N}(y) = {π(t_{j}(z)) | j ∈ J_{Z_{i}}}.  (3.8)

In particular, we see that each J_{Z_{i}} is non-empty, as T_{Y,N} is a surjective correspondence. In fact, under suitable conditions, the sets J_{Z_{i}} are fairly large:

Proposition 3.9 Let Y ⊂ 𝕄^{n} be a subvariety all of whose irreducible components have the same dimension, and suppose that Y ⊂ T_{𝕄^{n},N}(Y) for some N ∈ A such that ⟨N⟩ is a product of distinct primes 𝔭 ⊂ A of even degree satisfying |𝔭| ≥ max(13, deg(Y)). Let Y_{i} be an irreducible component of Y for which the projection p_{I} : Y_{i} → 𝕄^{I} is dominant, for some I ⊂ {1, …, n}, and choose a preimage Z_{i} ⊂ Ω^{n} of Y_{i}(ℂ_{∞})^{an}. Then the projection

p_{I} : J_{Z_{i}} ⟶ I^{I}

is surjective.

Proof. This is [3, Thm. 4], the proof is exactly the same. We briefly explain the hypotheses on N: we use the fact that 𝕄 has a PSL_{2}(A/NA) ≅ ∏_{𝔭} PSL_{2}(A/𝔭)-covering Y_{2}(N), of which Y_{0}(N) is a subcover (Proposition 2.9), and that this group has no proper subgroups of index deg(Y) or less when |𝔭| ≥ max(13, deg(Y)). □

3.5 An interlude in group theory

We remark that the GL_{2}(k_{∞}) action on Ω induces a PGL_{2}(k_{∞}) action, as the centre acts trivially. Until now we have found it more convenient to work with GL_{2}, but in order to continue we will need a number of group-theoretic results, and here it will be simpler to work with PGL_{2}, as had been done throughout [3].

For the convenience of the reader, we recall here some basic properties of the groups PGL_{2}(k_{∞}), which we will need. These results were used implicitly in [3], and thus may also help the reader with that paper.

Lemma 3.10 k_{∞}^{×}/k_{∞}^{×2} ≅ (ℤ/2ℤ)^{2}.

Proof. This is trivial, as k_{∞} ≅ 𝔽_{q}((ϖ)) for a uniformizer ϖ. □

Proposition 3.11 Every non-trivial normal subgroup of PGL_{2}(k_{∞}) contains PSL_{2}(k_{∞}). In particular, PGL_{2}(k_{∞}) has no non-trivial discrete normal subgroups.

Proof. Let H ◁ PGL_{2}(k_{∞}) be a normal subgroup. Then, as PSL_{2}(k_{∞}) is simple, either PSL_{2}(k_{∞}) ⊂ H, or H ∩ PGL_{2}(k_{∞}) = {1}. In the latter case we get an embedding H ↪ PGL_{2}(k_{∞})/PSL_{2}(k_{∞}) ≅ k_{∞}^{×}/k_{∞}^{×2} ≅ (ℤ/2ℤ)^{2}. It remains to show that PGL_{2}(k_{∞}) has no normal subgroups isomorphic to ℤ/2ℤ or (ℤ/2ℤ)^{2}. This may be verified with explicit calculations, by conjugating an element of order 2 with ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). □

Proposition 3.12 Let H be a subgroup of finite index in G = PGL_{2}(k_{∞}). Then H is normal and contains PSL_{2}(k_{∞}). In particular, if H is simple then H = PSL_{2}(k_{∞}).

Proof. G acts on the cosets G/H, giving a representation ρ : G → Aut(G/H) ≅ S_{n} where n = [G : H]. Then K = ker(ρ) is a normal subgroup of G contained in H. As K is infinite, K must intersect non-trivially with PSL_{2}(k_{∞}), hence contains PSL_{2}(k_{∞}), as the latter is simple. We will not need the fact that H is normal, but it is known that all the subgroups between PSL_{2}(k_{∞}) and PGL_{2}(k_{∞}) are normal in G. □

Corollary 3.13 Let PSL_{2}(k_{∞}) ⊂ H ⊂ PGL_{2}(k_{∞}) and suppose that f : H ↪ PGL_{2}(k_{∞}) is a monomorphism with image of finite index. Then f|_{PSL_{2}(k_{∞})} in an automorphism of PSL_{2}(k_{∞}).

Proof. The image f(PSL_{2}(k_{∞})) has finite index in PGL_{2}(k_{∞}) and is simple, hence f(PSL_{2}(k_{∞})) = PSL_{2}(k_{∞}). □

The above results may easily be generalized to PGL_{2} over other (infinite) fields. Mostly we have just used the fact that k_{∞}^{×}/k_{∞}^{×2} is finite.

Proposition 3.14 (Goursat’s Lemma) Let G_{1} and G_{2} be groups, and H ⊂ G_{1} × G_{2} a subgroup such that the two projections pr_{i} : H → G_{i} are surjective. Then K_{i} = ker(pr_{i}) can be considered a normal subgroup of G_{j}, for i ≠ j. Then H is the inverse image of the graph of an isomorphism ρ : G_{1}/K_{2} →^{∼} G_{2}/K_{1}.

Proof. This is straight forward. The map

ρ : G_{1}/K_{2} ⟶ G_{2}/K_{1}
g_{1} ⟼ g_{2} with (g_{1}, g_{2}) ∈ H

is easily checked to be a well-defined isomorphism. Now H is the inverse image of the graph of ρ. □

3.6 Curves stabilized by Hecke correspondences

Our next goal is to characterize special subvarieties of 𝕄^{n} by their property of being stabilized by suitable Hecke correspondences. The hard part is to prove this for curves.

Theorem 3.15 Let Y ⊂ 𝕄^{2} be an irreducible algebraic curve such that both projections Y → 𝕄 are dominant, and suppose Y ⊂ T_{𝕄^{2},N}(Y) for some N ∈ A such that ⟨N⟩ is a product of distinct primes 𝔭 ⊂ A of even degree satisfying |𝔭| ≥ max(13, deg(Y)). Then Y = Y′_{0}(N′) for some N′ ∈ A.

Note that we cannot deduce N′ from N. Indeed Y′_{0}(N′) is stabilized by T_{𝕄^{2},N} for all N ∈ A coprime to N′.

Proof. Suppose that the hypotheses of Theorem 3.15 are satisfied. The group G := PGL_{2}(k_{∞})^{2} acts on Ω^{2}, and we define the following subgroups: S := PSL_{2}(k_{∞})^{2}, Γ := PGL_{2}(A)^{2} and Σ := PSL_{2}(A)^{2}. We fix an irreducible rigid analytic curve Z ⊂ Ω^{2} with π(Z) = Y (ℂ_{∞})^{an}. Let G_{Z} ⊂ G be the stabilizer of Z, which is a closed analytic subgroup of G. We also define the subgroups S_{Z} = G_{Z} ∩ S, Γ_{Z} = G_{Z} ∩ Γ and Σ_{Z} = G_{Z} ∩ Σ. We intend to prove Theorem 3.15 by investigating the structure of G_{Z} and S_{Z}.

Denote by pr_{i} : G_{Z} → PGL_{2}(k_{∞}) the two projections for i = 1, 2.

Lemma 3.16 The projection pr_{i} : G_{Z} → PGL_{2}(k_{∞}) is injective, and pr_{i}(Γ_{Z}) has finite index in PGL_{2}(A), for i = 1, 2.

Proof. Exactly the same as [3, Lemma 2.11]. Here one uses Proposition 3.11. □

Let i ∈ J_{Z}, then t_{i}(Z) ⊂ π^{−1}(Y), so there is some γ_{i} ∈ Γ such that γ_{i}t_{i} ∈ G_{Z}. As the projections pr_{1}, pr_{2} : J_{Z} → I are surjective (Proposition 3.9), we get many non-trivial elements of G_{Z} this way.

Let H_{1} = pr_{1}(G_{Z}). Our first goal is to show that PSL_{2}(k_{∞}) ⊂ H_{1}. By a slight abuse of notation, we also denote the elements of pr_{1}(T) ⊂ GL_{2}(k) by t_{i}, for i ∈ I. We have seen that, for every i ∈ pr_{1}(J_{Z}) = I, there exists some γ_{i} ∈ PGL_{2}(A) such that g_{i} := γ_{i}t_{i} ∈ H_{1}. Lemma 3.16 says that H_{1} ∩ PGL_{2}(A) has finite index in PGL_{2}(A), and we choose a finite set R ⊂ PGL_{2}(A) of representatives for PGL_{2}(A)/(PGL_{2}(A) ∩ H_{1}).

We claim that given any string i_{1}, …, i_{n} of elements of I, and any a ∈ GL_{2}(A), there exists γ ∈ R such that

γt_{i_{n}}t_{i_{n−1}} ⋯ t_{i_{1}}a ∈ H_{1}.

Indeed, by induction it suffices to prove the claim for n = 1. Let i_{1} ∈ I and a ∈ GL_{2}(A) be given. GL_{2}(A) acts from the right on the set of left cosets GL_{2}(A)\GL_{2}(A)( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} )GL_{2}(A), so we let a act from the right on GL_{2}(A) · t_{i_{1}}, obtaining GL_{2}(A) · t_{i_{1}}a = GL_{2}(A) · t_{j} for some j ∈ I. Thus t_{i_{1}}a = γ′_{j}t_{j}, and for suitable γ ∈ R and γ′ ∈ H_{1} ∩ PSL_{2}(A) we have

γt_{i_{1}}a = γγ′_{j}t_{j} = γ(γ′_{j}γ_{j}^{−1})γ_{j}t_{j} = γ′γ_{j}t_{j} = γ′g_{j} ∈ H_{1}.

This proves the claim.

Now, multiplying by a suitable power of ( \begin{matrix}N & 0 \\ 0 & N \end{matrix} ), we see from (3.6) that for any x ∈ A[1/N] and a ∈ GL_{2}(A), there exists some γ_{x,a} ∈ R such that γ_{x,a}( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )a ∈ H_{1}. Denote by E ⊂ PSL_{2}(A[1/N]) the subgroup generated by elementary matrices. We have shown that H_{1} ∩ E has finite index in E. As A[1/N] is dense in k_{∞}, and PSL_{2}(k_{∞}) is generated by elementary matrices, it follows that E is dense in PSL_{2}(k_{∞}).

Next, we see that H_{1} is a closed subgroup of PGL_{2}(k_{∞}), exactly the same way as [3, Lemma 2.12 and the following paragraph], where we need only replace PSL_{2}(A[1/𝔪]) by E. It follows that H_{1} ∩ PSL_{2}(k_{∞}) has finite index in PSL_{2}(k_{∞}), so by Proposition 3.12

PSL_{2}(k_{∞}) ⊂ H_{1}, and similarly, PSL_{2}(k_{∞}) ⊂ H_{2} = pr_{2}(G_{Z}).  (3.17)

Now, since the projections pr_{1}, pr_{2} : G_{Z} → PGL_{2}(k_{∞}) are injective, Proposition 3.14 implies that

G_{Z} = {(g, ρ(g)) | g ∈ H_{1}},

where ρ : H_{1} →^{∼} H_{2} is an isomorphism. From (3.17) and Corollary 3.13 follows that

S_{Z} = G_{Z} ∩ PSL_{2}(k_{∞})^{2} = {(g, ρ(g)) | g ∈ PSL_{2}(k_{∞})},  (3.18)

where ρ|_{PSL_{2}(k_{∞})} is an automorphism of PSL_{2}(k_{∞}).

It is known that the automorphisms of PSL_{2}(k_{∞}) are all of the form g ↦ hg^{σ}h^{−1} for some h ∈ PGL_{2}(k_{∞}) and σ ∈ Aut(k_{∞}), see [11]. By the definition of Σ_{Z} and (3.18), we see that h · pr_{1}(Σ_{Z})^{σ} · h^{−1} ⊂ PSL_{2}(A). On the other hand, Lemma 3.16 tells us that pr_{1}(Σ_{Z}) has finite index in PSL_{2}(A). This in turn tells us a lot about h and σ. Fix some ϖ ∈ A so that ϖ^{−1} is a uniformizer for k_{∞}, i.e. k_{∞} = 𝔽_{q}((ϖ^{−1})).

Proposition 3.19 Let G be a subgroup of finite index in PGL_{2}(A), and suppose that hG^{σ}h^{−1} ⊂ PGL_{2}(k), for some h ∈ PGL_{2}(k_{∞}) and σ ∈ Aut(k_{∞}). Then h ∈ PGL_{2}(k), σ|_{𝔽_{q}} ∈ Gal(𝔽_{q}/𝔽_{p}), and σ(ϖ^{−1}) = uϖ^{−1} + v for some u ∈ 𝔽_{q}^{∗} and v ∈ 𝔽_{q}.

Proof. Let R = 𝔽_{q}[[ϖ^{−1}]] be the unique valuation ring in k_{∞}. We show that h ∈ PGL_{2}(k), σ(k) ⊂ k and σ(R) ⊂ R exactly as in [3, Prop. 2.13].

It follows that σ(A) ⊂ A and σ(A^{×}) ⊂ A^{×}, hence σ(𝔽_{q}) ⊂ 𝔽_{q}. Furthermore, σ(ϖ^{−1}) ∈ A^{−1} must again be a uniformizer for k_{∞}. But all other uniformizers of k_{∞} that lie in A^{−1} are of the form uϖ^{−1} + v for some u ∈ 𝔽_{q}^{∗} and v ∈ 𝔽_{q}. This completes the proof. □

Now that we have assembled enough ingredients, the proof of Theorem 3.15 follows exactly as in [3, §2.7]. We provide a sketch. We have seen that

S_{Z} = {(g, hg^{σ}h^{−1}) | g ∈ PSL_{2}(k_{∞})},

where h ∈ PGL_{2}(k) and σ ∈ Aut(k_{∞}) is as in Proposition 3.19. In particular, σ(α) = α^{q^{t}}, for all α ∈ 𝔽_{q} and some t ≥ 0. Let f = (ϖ^{−q} − ϖ^{−1})^{q−1}, and F := 𝔽_{p}((f)), which is a complete subfield of k_{∞} on which σ acts trivially. Fix some non-square α ∈ 𝔽_{q}, and set

P = {z ∈ Ω | z^{2} = αe, e ∈ F } ⊂ Ω.

Pick any z_{1} ∈ P. Then S_{1} = Stab_{PSL_{2}(F)}(z_{1}) is a one-dimensional Lie group (so far we have made essential use of the assumption that p be odd).

Now let z_{2} ∈ Ω such that (z_{1}, z_{2}) ∈ Z. Then the “S_{1}-orbit”

{(g(z_{1}), hg^{σ}h^{−1}(z_{2})) | g ∈ S_{1}} ⊂ Z ∩ ({z_{1}} × Ω)

is discrete, but S_{1} is not, so there exists 1 ≠ g ∈ S_{1} such that hg^{σ}h^{−1} fixes both z_{2} and h′(z_{1}), where h′ = h ∘ ( \begin{matrix}α^{(p^{t}−1)/2} & 0 \\ 0 & 1 \end{matrix} ) ∈ PGL_{2}(k). This means that z_{2} and h′(z_{1}) are conjugate over k_{∞}, i.e. z_{2} = ±h′(z_{1}). It follows that either π(z_{1}, h′(z_{1})) ∈ Y (ℂ_{∞}) or π(−z_{1}, h′(−z_{1})) ∈ Y (ℂ_{∞}). Both of these points also lie on Y′_{0}(N′), where N′ = deg(h′) is independent of (z_{1}, z_{2}). As P is uncountable, whereas the fibres of π are countable, it follows that Y = Y′_{0}(N′). □

To extend Theorem 3.15 to subvarieties Y ⊂ 𝕄^{n} of higher dimension we may follow [3, §2.8 and Corollaries 2.8 and 2.9] almost verbatim, no new ingredients are required. We must just keep in mind that 𝕄 is defined over H, the Hilbert class field of k. We also use Galois action on CM points as in the following section.

Theorem 3.20 Let F be a field lying between H and ℂ_{∞}. Let Y ⊂ 𝕄^{n} be an F -irreducible algebraic subvariety, containing a CM point x ∈ Y (ℂ_{∞}). Suppose that Y ⊂ T_{𝕄^{n},N}(Y) for some N ∈ A such that ⟨N⟩ is a product of distinct primes 𝔭 ⊂ A of even degree satisfying |𝔭| ≥ max(13, deg(Y)). Then Y ⊂ 𝕄^{n} is a special subvariety. □

3.7 CM points on curves

We may now start proving our main results, by exploiting the behavior of CM points under Galois action (Proposition 2.1) in conjunction with Theorems 3.15 and 3.20. We first treat the case of curves, where our results are effective.

Theorem 3.21 Let X = X_{1} × ⋯ × X_{n} be a product of Drinfeld modular curves. Let F/H be a finite extension, and d ∈ ℕ. Then there exists an absolutely computable constant B = B(X, F, d) > 0 such that the following holds. Let Y ⊂ X be an irreducible algebraic subcurve of degree d and defined over F. Then Y is a special subcurve if and only if Y (ℂ_{∞}) contains a CM point x satisfying H_{CM}(x) > B.

This indeed implies Theorem 1.2 for curves, since if Y ⊂ X contains a Zariski-dense set of CM points, then it contains CM points of arbitrary CM height and Y is defined over some finite extension F/k, as the CM points are all defined over k̄.

Proof. We again follow [3, §3.4] very closely, but will provide full details here for the benefit of the reader.

It follows from Proposition 2.19 that we may assume that X = 𝕄^{n}. Furthermore, we may assume that none of the projections p_{i} : Y → 𝕄 are constant. Using the fact that Y ⊂ 𝕄^{n} is a pure special subcurve if and only if p_{i,j}(Y) ⊂ 𝕄^{2} is special for all i < j, we have reduced the problem to the case n = 2.

Let x = (x_{1}, x_{2}) ∈ Y (ℂ_{∞}) be a CM point. For i = 1, 2 we write R_{i} = End(x_{i}) = A + 𝔣_{i}O_{K_{i}}, an order of conductor 𝔣_{i} ⊂ A in the CM field K_{i}, which has genus g_{i}. Furthermore, we denote by K_{i}(x_{i}) := H_{R_{i}} the ring class field of R_{i}, which is a field of definition for x_{i}, and we have Gal(K_{i}(x_{i})/K_{i}) ≅ Pic(R_{i}), and K_{i}(x_{i})/K_{i} is unramified outside 𝔣_{i}. We denote by K = K_{1}K_{2} and K(x_{1}, x_{2}) = K_{1}(x_{1})K_{2}(x_{2}) the composite fields. Denote by F_{s} the separable closure of k in F (which contains H), and by L the Galois closure of F_{s}K(x_{1}, x_{2}) over k.

Let 𝔭 be a prime of k of even degree which splits completely in F_{s}K (in particular, 𝔭 is principal, as it splits in H), and suppose 𝔭 ∤ 𝔣_{1}𝔣_{2}. Let 𝔓 be a prime of L lying above 𝔭, and denote by 𝔓_{i} its restriction to K_{i}(x_{i}). Denote by σ ∈ Aut(F L/F K) an extension of the Frobenius element (𝔓, L/k), and let σ_{i} = σ|_{K_{i}(x_{i})} = (𝔓_{i}, K_{i}(x_{i})/K_{i}) (remember that 𝔭 splits in K and is unramified in L).

It follows from Proposition 2.1 that x_{i} and x_{i}^{σ} correspond to Drinfeld modules linked by cyclic 𝔭-isogenies, and hence

(x_{1}, x_{2}) ∈ Y ∩ T_{𝕄^{2},𝔭}(Y^{σ}) = Y ∩ T_{𝕄^{2},𝔭}(Y).  (3.22)

Moreover, the whole Gal(F K(x_{1}, x_{2})/F)-orbit of (x_{1}, x_{2}) lies in this intersection, which thus contains at least max (|Pic(R_{1})|, |Pic(R_{2})|)/[F : k] points. On the other hand, from Proposition 3.5, deg(Y ∩ T_{𝕄^{2},𝔭}(Y)) ≤ 4(|𝔭| + 1)^{2} deg(Y)^{2}. Therefor, if

|Pic(R_{i})|/[F : k] > 4(|𝔭| + 1)^{2} deg(Y)^{2}, i = 1 or i = 2,  (3.23)

then the intersection is improper, Y ⊂ T_{𝕄^{2},𝔭}(Y) and hence Y is special (Theorem 3.15), provided also |𝔭| ≥ max(13, deg(Y)).

It remains to show that such a suitable prime 𝔭 indeed exists, if H_{CM}(x) is sufficiently large. Let M be the Galois closure of F_{s}K over k, and set

π_{M}(t) = #{𝔭 ⊂ A | prime, split in M and |𝔭| = q^{t}}.

Let T ∈ k be a transcendental element such that k is a finite separable geometric extension of 𝔽_{q}(T), and let e = [k : 𝔽_{q}(T)]. Let 𝔽 be the algebraic closure of 𝔽_{q} in M, let n_{c} = [𝔽 : 𝔽_{q}] be the constant extension degree and n_{g} = [M : 𝔽k] the geometric extension degree of M/k. The Čebotarev Theorem for function fields [5, Prop. 5.16] says

if n_{c}|t, then |π_{M}(t) − \frac{1}{n_{g}} q^{t}/t| < 4(e^{2} + g_{M}(e + 1)/2 + g_{k} + 1)q^{t/2},  (3.24)

where g_{M} and g_{k} are the genus of M and k, respectively. We may bound g_{M} in terms of g_{1}, g_{2} and g_{k} using the Castelnuovo inequality [15, III.10.3], and eventually obtain

π_{M}(t) > C_{1}q^{t}/t − (C_{2}(g_{1} + g_{2}) + C_{3})q^{t/2},  (3.25)

where C_{1}, C_{2} and C_{3} are absolutely computable positive constants, depending on k and F.

We want π_{M}(t) > log_{q} |𝔣_{1}𝔣_{2}|, q^{t} ≥ max(13, deg(Y)) and 2n_{c}|t, so that there exists a prime 𝔭 which splits in M (and thus in F_{s}K), does not divide 𝔣_{1}𝔣_{2}, and satisfies the hypotheses of Theorem 3.15. We also want (3.23) to hold, for which we employ Proposition 2.4:

|Pic(R_{i})| > C_{ε}H_{CM}(x_{i})^{1−ε} = C_{ε}(q^{g_{i}}|𝔣_{i}|)^{1−ε}, for any ε > 0.

In summary, we need a simultaneous solution t ∈ 2n_{c}ℕ to the three inequalities

q^{t} ≥ max(13, deg(Y)),
C_{1}q^{t}/t − (C_{2}(g_{1} + g_{2}) + C_{3})q^{t/2} > log_{q} |𝔣_{1}𝔣_{2}|,

and

C_{ε}(q^{g_{i}}|𝔣_{i}|)^{1−ε} > 4[F : k](q^{t} + 1)^{2} deg(Y)^{2} for some ε > 0, i = 1 or i = 2.

Such a solution will always exist if H_{CM}(x) = max(H_{CM}(x_{1}), H_{CM}(x_{2})) is sufficiently large. (Intuitively, q^{t} must be large compared to log_{q} |𝔣_{i}| and g_{i}, and small compared to |𝔣_{i}| and q^{g_{i}}). □

3.8 Completing the proof of Theorem 1.2

The proof of Theorem 1.2 now follows exactly as in [3, §3.5]. We sketch the proof here for completeness.

Firstly, as Y (ℂ_{∞}) contains a Zariski-dense subset S of CM points, which are defined over k^{sep}, there exists a finite Galois extension F/k over which Y is defined. We will use induction on d = dim(Y), the case d = 1 following from Theorem 3.21, so we assume that d ≥ 2 and n ≥ 3. We may furthermore assume that Y ⊂ 𝕄^{n} is a hypersurface, as it is an irreducible component of

⋂_{\begin{matrix}I⊂\{1,...,n\} \\ |I|=d+1 \end{matrix}} p_{I}^{−1}p_{I}(Y).

Lastly, we may assume that all the projections p_{i} : Y → 𝕄 are dominant.

Step 1. For a given constant B > 0, we may assume that every x = (x_{1}, …, x_{n}) ∈ S satisfies H_{CM}(x_{i}) > B, ∀i = 1, …, n.

Pick one such x ∈ S. Suppose that there exist primes 𝔭_{1}, …, 𝔭_{d−1} ⊂ A of even degree satisfying the following conditions:

1. Each 𝔭_{j} splits in F and in End(x_{i}), for all i = 1, …, n.

2. |𝔭_{1}| ≥ max(13, deg(Y)).

3. |𝔭_{j+1}| ≥ (deg(Y))^{2^{j}} ∏_{m=1}^{j}(2|𝔭_{m}| + 2)^{n2^{j−m}}, for j = 1, …, d − 2.

4. |Pic(End(x_{i}))| > [F : k]|𝔭_{d−1}|^{2}(2|𝔭_{d−1}| + 2)^{n}, for all i = 1, …, n.

Then one shows, again using Galois action on x together with Theorem 3.20, that there exists a pure special subvariety Y_{x} ⊂ Y.

As the points in S are Zariski-dense, it follows that there exists a Zariski-dense family C of pure special subcurves C ⊂ Y, C ∈ C.

Step 2. We now show that Y is special. Choose a CM point x_{1} ∈ 𝕄(ℂ_{∞}), and consider the slice

Y_{1} = Y ∩ ({x_{1}} × 𝕄^{n−1}).

Each pure special curve C ∈ C intersects Y_{1} in at least one CM point, and we denote by Y′ the Zariski-closure of these intersection points:

Y′ = \overline{∪_{C∈C}(C ∩ Y_{1})}

We must have dim(Y′) < dim(Y), so by the induction hypothesis, Y′ is special. We write Y′ = Y′_{1} ∪ ⋯ ∪ Y′_{r} as a union of irreducible components. Replacing C by a Zariski-dense subfamily and renumbering if necessary, we may assume that Y′_{1} contains at least 1/r of the points of C ∩ Y_{1} for all C ∈ C.

Now, either Y′_{1} ≅ {y} × 𝕄^{m} for some CM point y ∈ 𝕄^{n−m}(ℂ_{∞}), in which case Y = Y′_{0}(N′) × 𝕄^{n−2} is special, or else at least one pure special curve appears as a factor of Y′_{1}.

It follows that there exist indices 1 < i < j such that p_{i,j}(Y′_{1}) = Y′_{0}(M) for some fixed M ∈ A. Fix C ∈ C and let the pure special curve p_{{1,i,j}}(C) ⊂ 𝕄^{3} correspond to the triple (N_{C,1}, N_{C,i}, N_{C,j}) ∈ A^{3} via (2.17). Again restricting C if necessary, we may assume that |N_{c,i}| ≤ |N_{C,j}| for all C ∈ C.

Now we fix C ∈ C and x_{1}, x_{i} ∈ 𝕄(ℂ_{∞}). Then the number of distinct x_{j} ∈ 𝕄(ℂ_{∞}) such that (x_{1}, x_{i}, x_{j}) ∈ p_{{1,i,j}}(C) is bounded from below by an increasing function in |N_{C,j}| (Proposition 2.18). But at least 1/r of these points x_{j} must also satisfy (x_{i}, x_{j}) ∈ Y′_{0}(M), of which there can be at most ψ(M). It follows that |N_{C,i}| and |N_{C,j}| are bounded independently of C ∈ C. Restricting C once again, we may assume that p_{i,j}(C) = Y′_{0}(N_{0}) for all C ∈ C.

Now one can show that Y ≅ Y′_{0}(N′) × 𝕄^{n−2}, which is special.

Step 3. It remains to show that the primes 𝔭_{1}, …, 𝔭_{d−1} ⊂ A satisfying (1)-(4) above actually exist, if the constant B > 0 is chosen sufficiently large. Let x = (x_{1}, …, x_{n}) ∈ S such that H_{CM}(x_{i}) = q^{g_{i}}|𝔣_{i}| > B for all i = 1, …, n.

As before, the problem boils down to finding simultaneous solutions t_{1}, …, t_{d−1} ∈ 2n_{c}ℕ to the following four inequalities:

q^{t_{1}} ≥ max(13, deg(Y)),
C_{ε}(q^{g_{i}}|𝔣_{i}|)^{1−ε} > [F : k]q^{2t_{d−1}}(2q^{t_{d−1}} + 2)^{n}, for some ε > 0 and some 1 ≤ i ≤ n,

q^{t_{j+1}} ≥ (deg(Y))^{2^{j}} ∏_{m=1}^{j}(2q^{t_{m}} + 2)^{n2^{j−m}}, for all 1 ≤ j ≤ d − 1,

C_{1}q^{t_{j}}/t_{j} − (C_{2}(g_{1} + ⋯ + g_{n}) + C_{3})q^{t_{j}/2} > log_{q} |𝔣_{1} ⋯ 𝔣_{n}|, for all 1 ≤ j ≤ d − 1.

If B > 0 is sufficiently large, then such solutions exist. This completes the proof of Theorem 1.2. □

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany. and Department of Mathematics, University of Stellenbosch, Stellenbosch 7600, South Africa. flo@math.jussieu.fr