1 Introduction
The Ax–Schanuel theorem [Reference AxAx71] is a result regarding the transcendence degrees of fields generated over the complex numbers by power series and their exponentials. Formulated geometrically for the uniformization maps of algebraic tori, it has inspired analogous statements for the uniformization maps of abelian varieties and Shimura varieties. The former, following from another theorem of Ax [Reference AxAx72], has recently been used by Habegger and Pila in their proof of the Zilber–Pink conjecture for curves in abelian varieties [Reference Habegger and PilaHP16].
Habegger and Pila also extended the Pila–Zannier strategy to the Zilber–Pink conjecture for products of modular curves. Their method relies on an Ax–Schanuel conjecture for the
$j$
-function and is conditional on their so-called large Galois orbits conjecture. The purpose of this paper is to show that the Pila–Zannier strategy can be extended to the Zilber–Pink conjecture for general Shimura varieties.
This conjecture can just as easily be stated in the generality of mixed Shimura varieties but, in this article, we will restrict our attention to pure Shimura varieties, though we have no explicit reason to believe that the methods presented here will not extend to the mixed setting. We begin by stating a conjecture of Pink. We note that, throughout this article, unless preceded by the word Shimura, varieties (and, indeed, subvarieties) will be assumed geometrically irreducible.
Conjecture 1.1 (Cf. [Reference PinkPin05b, Conjecture 1.3]).
Let
$\text{Sh}_{K}(G,\mathfrak{X})$
be a Shimura variety and, for any integer
$d$
, let
$\text{Sh}_{K}(G,\mathfrak{X})^{[d]}$
denote the union of the special subvarieties of
$\text{Sh}_{K}(G,\mathfrak{X})$
having codimension at least
$d$
. Let
$V$
be a Hodge generic subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
. Then
$$\begin{eqnarray}\displaystyle V\cap \text{Sh}_{K}(G,\mathfrak{X})^{[1+\dim V]} & & \displaystyle \nonumber\end{eqnarray}$$
is not Zariski dense in
$V$
.
The heuristics of this conjecture are as follows. For two subvarieties
$V$
and
$W$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
, such that the codimension of
$W$
is at least
$1+\dim V$
, we expect
$V\cap W=\emptyset$
. Even if we fix
$V$
and take the union of
$V\cap W$
for countably many
$W$
of codimension at least
$1+\dim V$
, the resulting set should still be rather small in
$V$
unless, of course,
$V$
was not sufficiently generic in
$\text{Sh}_{K}(G,\mathfrak{X})$
. Pink’s conjecture turns this expectation into an explicit statement about the intersection of Hodge generic subvarieties with the special subvarieties of small dimension.
Conjecture 1.1 can also be formulated for algebraic tori, abelian varieties, or even semiabelian varieties, though Conjecture 1.1 for mixed Shimura varieties implies all of these formulations (see [Reference PinkPin05b]). When
$V$
is a curve, defined over a number field, and contained in an algebraic torus, we obtain a theorem of Maurin [Reference MaurinMau08]. We also note that Capuano, Masser, Pila, and Zannier have recently applied the Pila–Zannier method in this setting [Reference Capuano, Masser, Pila and ZannierCMPZ16]. When
$V$
is a curve, defined over a number field, and contained in an abelian variety, we obtain the recent theorem of Habegger and Pila [Reference Habegger and PilaHP16], and it is the ideas presented there that form the basis for this article. Habegger and Pila had given some partial results when
$V$
is a curve, defined over a number field, and contained in the Shimura variety
$\mathbb{C}^{n}$
[Reference Habegger and PilaHab12], and Orr has recently generalized their results to a curve contained in
${\mathcal{A}}_{g}^{2}$
(see [Reference OrrOrr17] for more details).
We should point out that Conjecture 1.1 implies the André–Oort conjecture.
Conjecture 1.2 (André–Oort).
Let
$\text{Sh}_{K}(G,\mathfrak{X})$
be a Shimura variety and let
$V$
be a subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
such that the special points of
$\text{Sh}_{K}(G,\mathfrak{X})$
in
$V$
are Zariski dense in
$V$
. Then
$V$
is a special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
.
To see this, we may assume that
$V$
is Hodge generic in
$\text{Sh}_{K}(G,\mathfrak{X})$
. Then, since special points have codimension
$\dim \text{Sh}_{K}(G,\mathfrak{X})$
, Conjecture 1.1 implies that, either
$\dim V=\dim \text{Sh}_{K}(G,\mathfrak{X})$
, in which case
$V$
is a connected component of
$\text{Sh}_{K}(G,\mathfrak{X})$
and, in particular, a special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
, or the set of special points of
$\text{Sh}_{K}(G,\mathfrak{X})$
in
$V$
are not Zariski dense in
$V$
.
In precisely the same fashion, the Zilber–Pink conjecture for abelian varieties implies the Manin–Mumford conjecture.
The André–Oort conjecture has a rich history of its own. Here, we simply recall that it was recently settled for
${\mathcal{A}}_{g}$
by Pila and Tsimerman [Reference Pila and TsimermanPT14, Reference TsimermanTsi18], thanks to recent progress on the Colmez conjecture due to Andreatta, Goren, Howard and Madapusi Pera [Reference Andreatta, Goren, Howard and Madapusi PeraAGHM18], and Yuan and Zhang [Reference Yuan and ZhangYZ18], and it is known to hold for all Shimura varieties under conjectural lower bounds for Galois orbits of special points due to the work of Orr, Klingler, Ulmo, Yafaev, and the first author [Reference Daw and OrrDO16, Reference Klingler, Ullmo and YafaevKUY16, Reference Ullmo and YafaevUY15]. Furthermore, Gao has generalized these proofs to all mixed Shimura varieties [Reference GaoGao17, Reference GaoGao16].
In his work on Schanuel’s conjecture, Zilber made his own conjecture on unlikely intersections [Reference ZilberZil02], which was closely related to the independent work of Bombieri, Masser and Zannier [Reference Bombieri, Masser and ZannierBMZ07]. To describe Zilber’s formulation, we require the following definition.
Definition 1.3. Let
$\text{Sh}_{K}(G,\mathfrak{X})$
be a Shimura variety and let
$V$
be a subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
. A subvariety
$W$
of
$V$
is called atypical with respect to
$V$
if there is a special subvariety
$T$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
such that
$W$
is an irreducible component of
$V\cap T$
and
$$\begin{eqnarray}\displaystyle \dim W>\dim V+\dim T-\dim \text{Sh}_{K}(G,\mathfrak{X}). & & \displaystyle \nonumber\end{eqnarray}$$
We denote by
$\text{Atyp}(V)$
the union of the subvarieties of
$V$
that are atypical with respect to
$V$
.
Zilber’s conjecture, formulated for Shimura varieties, is then as follows.
Conjecture 1.4 (Cf. [Reference Habegger and PilaHP16, Conjecture 2.2]).
Let
$\text{Sh}_{K}(G,\mathfrak{X})$
be a Shimura variety and let
$V$
be a subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
. Then
$\text{Atyp}(V)$
is equal to a finite union of atypical subvarieties of
$V$
.
Since there are only countably many special subvarieties of
$\text{Sh}_{K}(G,\mathfrak{X})$
, the conjecture is equivalent to the statement that
$V$
contains only finitely many subvarieties that are atypical with respect to
$V$
and maximal with respect to this property.
We will see that Conjecture 1.4 strengthens Conjecture 1.1 and, therefore, it is Conjecture 1.4 that we refer to as the Zilber–Pink conjecture. Habegger and Pila obtained a proof of the Zilber–Pink conjecture for products of modular curves assuming the weak complex Ax conjecture and the large Galois orbits conjecture. Subsequently, Pila and Tsimerman obtained the weak complex Ax conjecture as a corollary to their proof of the Ax–Schanuel conjecture for the
$j$
-function [Reference Pila and TsimermanPT16]. Habegger and Pila had previously verified the large Galois orbits conjecture for so-called asymmetric curves [Reference Habegger and PilaHab12].
This article seeks to generalize the ideas of [Reference Habegger and PilaHP16] to general Shimura varieties. Hence, we will have to make generalizations of the previously mentioned hypotheses. The foremost of which will be the statement from functional transcendence, namely, the hyperbolic Ax–Schanuel conjecture that generalizes the Ax–Schanuel conjecture for the
$j$
-function to general Shimura varieties. Our main result (Theorem 8.3) is that, under the hyperbolic Ax–Schanuel conjecture, the Zilber–Pink conjecture can be reduced to a problem of point counting. However, given that Mok, Pila, and Tsimerman have recently announced a proof of the hyperbolic Ax–Schanuel conjecture [Reference Mok, Pila and TsimermanMPT17], this result is now very likely unconditional. Besides the hyperbolic Ax–Schanuel conjecture, our main input will be the theory of o-minimality and, in particular, the fact that the uniformization map of a Shimura variety is definable in
$\mathbb{R}_{\text{an},\text{exp}}$
when it is restricted to an appropriate fundamental domain.
After establishing the main result, we attempt to tackle the point counting problem using the now famous Pila–Wilkie counting theorem. To do so, we formulate several arithmetic conjectures that are inspired by previous applications of the Pila–Zannier strategy. In this vein, our paper is very much in the spirit of [Reference UllmoUll14], which, at the time, reduced the André–Oort conjecture to a point counting problem and then explained how various conjectural ingredients, namely, the hyperbolic Ax–Lindemann conjecture, lower bounds for Galois orbits of special points, upper bounds for the heights of pre-special points, and the definability of the uniformization map, could be combined to deliver a proof of the André–Oort conjecture.
Our arithmetic hypotheses are: (1) lower bounds for Galois orbits of so-called optimal points (see Definition 3.2), which we also refer to as the large Galois orbits conjecture, and (2) upper bounds for the heights of pre-special subvarieties. Hypothesis (1) generalizes the (in some cases still conjectural) lower bounds for Galois orbits of special points (when such special points are also maximal special subvarieties), and also generalizes the large Galois orbits conjecture of Habegger and Pila. Hypothesis (2) generalizes the upper bounds for heights of pre-special points, which were proved by Orr and the first author [Reference Daw and OrrDO16]. However, we also show that it is possible to replace hypothesis (2) with two other arithmetic hypotheses, namely: (3) upper bounds for the degrees of fields associated with special subvarieties, and (4) upper bounds for the heights of lattice elements. Hypothesis (3) is a replacement for the fact that, for an abelian variety, its abelian subvarieties can be defined over a fixed finite extension of the base field. Hypothesis (4) is an analogue of a known result for abelian varieties. We verify hypotheses (2), (3), and (4) for a product of modular curves.
Conventions.
⋆ Throughout this paper, definable means definable in the o-minimal structure
$\mathbb{R}_{\text{an},\text{exp}}$
.⋆ Unless preceded by the word Shimura, varieties (and, indeed, subvarieties) will be assumed geometrically irreducible.
⋆ By a subvariety, we will always mean a closed subvariety.
Index of notation. We collect here the main symbols appearing in this article.
⋆
$\langle W\rangle$
is the smallest special subvariety containing
$W$
.⋆
$\langle W\rangle _{\text{ws}}$
is the smallest weakly special subvariety containing
$W$
.⋆
$\langle A\rangle _{\text{Zar}}$
is the smallest algebraic subvariety containing
$A$
.⋆
$\langle A\rangle _{\text{geo}}$
is the smallest totally geodesic subvariety containing
$A$
.⋆
$\unicode[STIX]{x1D6FF}(W):=\dim \langle W\rangle -\dim W$
.⋆
$\unicode[STIX]{x1D6FF}_{\text{ws}}(W):=\dim \langle W\rangle _{\text{ws}}-\dim W$
.⋆
$\unicode[STIX]{x1D6FF}_{\text{Zar}}(A):=\dim \langle A\rangle _{\text{Zar}}-\dim A$
.⋆
$\unicode[STIX]{x1D6FF}_{\text{geo}}(A):=\dim \langle A\rangle _{\text{geo}}-\dim A$
.⋆
$\text{Opt}(V)$
is the set of subvarieties of
$V$
that are optimal in
$V$
.⋆
$\text{Opt}_{0}(V)$
is the set of points of
$V$
that are optimal in
$V$
.⋆
$G^{\text{ad}}$
is the adjoint group of
$G$
, i.e., the quotient of
$G$
by its centre.⋆
$G^{\text{der}}$
is derived group of
$G$
.⋆
$G^{\circ }$
is the connected component of
$G$
containing the identity.⋆
$G_{H}:=HZ_{G}(H)^{\circ }$
whenever
$H$
is a subgroup of
$G$
.
2 Special and weakly special subvarieties
Let
$(G,\mathfrak{X})$
be a Shimura datum and let
$K$
be a compact open subgroup of
$G(\mathbb{A}_{f})$
, where
$\mathbb{A}_{f}$
will henceforth denote the finite rational adèles. Let
$\text{Sh}_{K}(G,\mathfrak{X})$
denote the corresponding Shimura variety. By this, we mean the complex quasi-projective algebraic variety such that
$\text{Sh}_{K}(G,\mathfrak{X})(\mathbb{C})$
is equal to the image of
$$\begin{eqnarray}\displaystyle G(\mathbb{Q})\backslash [\mathfrak{X}\times (G(\mathbb{A}_{f})/K)] & & \displaystyle\end{eqnarray}$$
under the canonical embedding into complex projective space given by Baily and Borel [Reference Baily and BorelBB66]. We will identify (2.0.1) with
$\text{Sh}_{K}(G,\mathfrak{X})(\mathbb{C})$
. We recall that, on
$\mathfrak{X}\times (G(\mathbb{A}_{f})/K)$
, the action of
$G(\mathbb{Q})$
is the diagonal one.
Let
$X$
be a connected component of
$\mathfrak{X}$
and let
$G(\mathbb{Q})_{+}$
be the subgroup of
$G(\mathbb{Q})$
acting on it. For any
$g\in G(\mathbb{A}_{f})$
, we obtain a congruence subgroup
$\unicode[STIX]{x1D6E4}_{g}$
of
$G(\mathbb{Q})_{+}$
by intersecting it with
$gKg^{-1}$
. Furthermore, the locally symmetric variety
$\unicode[STIX]{x1D6E4}_{g}\backslash X$
is contained in (2.0.1) via the map that sends the class of
$x$
to the class of
$(x,g)$
. If we take the disjoint union of the
$\unicode[STIX]{x1D6E4}_{g}\backslash X$
over a (finite) set of representatives for
$$\begin{eqnarray}\displaystyle G(\mathbb{Q})_{+}\backslash G(\mathbb{A}_{f})/K, & & \displaystyle \nonumber\end{eqnarray}$$
the corresponding inclusion map is a bijection.
Definition 2.1. For any compact open subgroup
$K^{\prime }$
of
$G(\mathbb{A}_{f})$
contained in
$K$
, we obtain a finite morphism
$$\begin{eqnarray}\displaystyle \text{Sh}_{K^{\prime }}(G,\mathfrak{X})\rightarrow \text{Sh}_{K}(G,\mathfrak{X}), & & \displaystyle \nonumber\end{eqnarray}$$
given by the natural projection. Furthermore, for any
$a\in G(\mathbb{A}_{f})$
, we obtain an isomorphism
$$\begin{eqnarray}\displaystyle \text{Sh}_{K}(G,\mathfrak{X})\rightarrow \text{Sh}_{a^{-1}Ka}(G,\mathfrak{X}) & & \displaystyle \nonumber\end{eqnarray}$$
sending the class of
$(x,g)$
to the class of
$(x,ga)$
. We let
$T_{K,a}$
denote the map on algebraic cycles of
$\text{Sh}_{K}(G,\mathfrak{X})$
given by the algebraic correspondence
$$\begin{eqnarray}\displaystyle \text{Sh}_{K}(G,\mathfrak{X})\leftarrow \text{Sh}_{K\cap aKa^{-1}}(G,\mathfrak{X})\rightarrow \text{Sh}_{a^{-1}Ka\cap K}(G,\mathfrak{X})\rightarrow \text{Sh}_{K}(G,\mathfrak{X}), & & \displaystyle \nonumber\end{eqnarray}$$
where the outer arrows are the natural projections and the middle arrow is the isomorphism given by
$a$
. We refer to a map of this sort as a Hecke correspondence.
Definition 2.2. Let
$(H,\mathfrak{X}_{H})$
be a Shimura subdatum of
$(G,\mathfrak{X})$
and let
$K_{H}$
denote a compact open subgroup of
$H(\mathbb{A}_{f})$
contained in
$K$
. The natural map
$$\begin{eqnarray}\displaystyle H(\mathbb{Q})\backslash [\mathfrak{X}_{H}\times (H(\mathbb{A}_{f})/K_{H})]\rightarrow G(\mathbb{Q})\backslash [\mathfrak{X}\times (G(\mathbb{A}_{f})/K)] & & \displaystyle \nonumber\end{eqnarray}$$
yields a finite morphism of Shimura varieties
$$\begin{eqnarray}\displaystyle \text{Sh}_{K_{H}}(H,\mathfrak{X}_{H})\rightarrow \text{Sh}_{K}(G,\mathfrak{X}) & & \displaystyle \nonumber\end{eqnarray}$$
(see, for example, [Reference PinkPin05a, Facts 2.6]), and we refer to the image of any such morphism as a Shimura subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
.
For any Shimura subvariety
$Z$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
and any
$a\in G(\mathbb{A}_{f})$
, we refer to any irreducible component of
$T_{K,a}(Z)$
as a special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
.
Recall that, by definition,
$\mathfrak{X}$
is a
$G(\mathbb{R})$
conjugacy class of morphisms from
$\mathbb{S}$
to
$G_{\mathbb{R}}$
and the Mumford-Tate group
$\text{MT}(x)$
of
$x\in \mathfrak{X}$
is defined as the smallest
$\mathbb{Q}$
-subgroup
$H$
of
$G$
such that
$x$
factors through
$H_{\mathbb{R}}$
. If we let
$\mathfrak{X}_{M}$
denote the
$M(\mathbb{R})$
conjugacy class of
$x\in X$
, where
$M:=\text{MT}(x)$
, then
$(M,\mathfrak{X}_{M})$
is a Shimura subdatum of
$(G,\mathfrak{X})$
. In particular, if we let
$X_{M}$
denote a connected component of
$\mathfrak{X}_{M}$
contained in
$X$
, then the image of
$X_{M}$
in
$\unicode[STIX]{x1D6E4}_{g}\backslash X$
, for any
$g\in G(\mathbb{A}_{f})$
, is a special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
, and it is easy to see that every special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
arises this way.
Of course, if
$x\in X_{M}$
, then
$X_{M}$
is equal to the
$M(\mathbb{R})^{+}$
conjugacy class of
$x$
. Furthermore, the action of
$M(\mathbb{R})$
on
$\mathfrak{X}_{M}$
factors through
$M^{\text{ad}}(\mathbb{R})$
and the group
$M^{\text{ad}}$
is equal to the direct product of its
$\mathbb{Q}$
-simple factors. Therefore, we can write
$M^{\text{ad}}$
as a product
$$\begin{eqnarray}\displaystyle M^{\text{ad}}=M_{1}\times M_{2} & & \displaystyle \nonumber\end{eqnarray}$$
of two normal
$\mathbb{Q}$
-subgroups, either of which may (by choice or necessity) be trivial, and we thus obtain a corresponding splitting
$$\begin{eqnarray}\displaystyle X_{M}=X_{1}\times X_{2}. & & \displaystyle \nonumber\end{eqnarray}$$
For any such splitting, and any
$x_{1}\in X_{1}$
or
$x_{2}\in X_{2}$
, we refer to the image of
$\{x_{1}\}\,\times \,X_{2}$
or
$X_{1}\,\times \,\{x_{2}\}$
in
$\unicode[STIX]{x1D6E4}_{g}\backslash X$
, for any
$g\in G(\mathbb{A}_{f})$
, as a weakly special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
. In particular, every special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
is a weakly special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
. By [Reference MoonenMoo98, § 4], the weakly special subvarieties of
$\text{Sh}_{K}(G,\mathfrak{X})$
are precisely those subvarieties of
$\text{Sh}_{K}(G,\mathfrak{X})$
that are totally geodesic in
$\text{Sh}_{K}(G,\mathfrak{X})$
. Furthermore, a weakly special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
is a special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
if and only if it contains a special subvariety of dimension zero, henceforth known as a special point.
Remark 2.3. The following observations will facilitate various reductions.
⋆ Let
$K^{\prime }$
be a compact open subgroup of
$G(\mathbb{A}_{f})$
contained in
$K$
. By definition, a subvariety
$Z$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
is a (weakly) special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
if and only if any irreducible component of the inverse image of
$Z$
in
$\text{Sh}_{K^{\prime }}(G,\mathfrak{X})$
is a (weakly) special subvariety of
$\text{Sh}_{K^{\prime }}(G,\mathfrak{X})$
.⋆ For any
$a\in G(\mathbb{A}_{f})$
, a subvariety
$Z$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
is a (weakly) special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
if and only if any irreducible component of
$T_{K,a}(Z)$
is a (weakly) special subvariety of
$\text{Sh}_{K^{\prime }}(G,\mathfrak{X})$
.⋆ If we let
$G^{\text{ad}}$
denote the adjoint group of
$G$
, i.e., the quotient of
$G$
by its centre, we obtain another Shimura datum
$(G^{\text{ad}},\mathfrak{X}^{\text{ad}})$
, known as the adjoint Shimura datum associated with
$(G,\mathfrak{X})$
. For any compact open subgroup
$K^{\text{ad}}$
of
$G^{\text{ad}}(\mathbb{A}_{f})$
containing the image of
$K$
, we obtain a finite morphism As in [Reference Edixhoven and YafaevEY03, Proposition 2.2], a subvariety$$\begin{eqnarray}\displaystyle \text{Sh}_{K}(G,\mathfrak{X})\rightarrow \text{Sh}_{K^{\text{ad}}}(G^{\text{ad}},\mathfrak{X}^{\text{ad}}). & & \displaystyle \nonumber\end{eqnarray}$$
$Z$
of
$\text{Sh}_{K^{\text{ad}}}(G^{\text{ad}},\mathfrak{X}^{\text{ad}})$
is a special subvariety of
$\text{Sh}_{K^{\text{ad}}}(G^{\text{ad}},\mathfrak{X}^{\text{ad}})$
if and only if any irreducible component of its inverse image in
$\text{Sh}_{K}(G,\mathfrak{X})$
is a special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
.
By [Reference PinkPin05a, Remark 4.9], for any subvariety
$W$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
, there exists a smallest weakly special subvariety
$\langle W\rangle _{\text{ws}}$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
containing
$W$
and a smallest special subvariety
$\langle W\rangle$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
containing
$W$
. We note that here, and throughout, our notations and terminology regarding subvarieties often differ from those found in [Reference Habegger and PilaHP16].
3 The Zilber–Pink conjecture
For the remainder of this article, we fix a Shimura datum
$(G,\mathfrak{X})$
and we let
$X$
be a connected component of
$\mathfrak{X}$
. We fix a compact open subgroup
$K$
of
$G(\mathbb{A}_{f})$
and we let
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}:=G(\mathbb{Q})_{+}\cap K, & & \displaystyle \nonumber\end{eqnarray}$$
where
$G(\mathbb{Q})_{+}$
is the subgroup of
$G(\mathbb{Q})$
acting on
$X$
. We denote by
$S$
the connected component
$\unicode[STIX]{x1D6E4}\backslash X$
of
$\text{Sh}_{K}(G,\mathfrak{X})$
.
As in [Reference Habegger and PilaHP16], we will consider an equivalent formulation of Conjecture 1.4 using the language of optimal subvarieties.
Definition 3.1. Let
$W$
be a subvariety of
$S$
. We define the defect of
$W$
to be
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}(W):=\dim \langle W\rangle -\dim W. & & \displaystyle \nonumber\end{eqnarray}$$
Definition 3.2. Let
$V$
be a subvariety of
$S$
and let
$W$
be a subvariety of
$V$
. Then
$W$
is called optimal in
$V$
if, for any subvariety
$Y$
of
$S$
,
$$\begin{eqnarray}\displaystyle W\subsetneq Y\subseteq V\;\Longrightarrow \;\unicode[STIX]{x1D6FF}(Y)>\unicode[STIX]{x1D6FF}(W). & & \displaystyle \nonumber\end{eqnarray}$$
We denote by
$\text{Opt}(V)$
the set of all subvarieties of
$V$
that are optimal in
$V$
.
Remark 3.3. Let
$V$
be a subvariety of
$S$
. First note that
$V\in \text{Opt}(V)$
. Secondly, if
$W\in \text{Opt}(V)$
, then
$W$
is an irreducible component of
$$\begin{eqnarray}\displaystyle \langle W\rangle \cap V. & & \displaystyle \nonumber\end{eqnarray}$$
Conjecture 3.4 (Cf. [Reference Habegger and PilaHP16, Conjecture 2.6]).
Let
$V$
be a subvariety of
$S$
. Then
$\text{Opt}(V)$
is finite.
Observe that a maximal special subvariety of
$V$
is an optimal subvariety of
$V$
. Therefore, Conjecture 3.4 immediately implies that
$V$
contains only finitely many maximal special subvarieties, which is another formulation of the André–Oort conjecture for
$V$
.
Proof. Consider the situation described in the statement of Conjecture 1.4. By Remark 2.3, we suffer no loss in generality if we assume that
$V$
is contained in
$S$
. Then the result follows from [Reference Habegger and PilaHP16, Lemma 2.7].◻
Lemma 3.6. The Zilber–Pink conjecture implies Conjecture 1.1.
Proof. By Lemma 3.5, it suffices to show that Conjecture 3.4 implies Conjecture 1.1.
Consider the situation described in Conjecture 1.1. By Remark 2.3, we suffer no loss in generality if we assume that
$V$
is contained in
$S$
. Let
$P$
be a point belonging to
$$\begin{eqnarray}\displaystyle V\cap \text{Sh}_{K}(G,\mathfrak{X})^{[1+\dim V]}. & & \displaystyle \nonumber\end{eqnarray}$$
Let
$W$
be a subvariety of
$V$
that is optimal in
$V$
and contains
$P$
such that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}(W)\leqslant \unicode[STIX]{x1D6FF}(P)=\dim \langle P\rangle . & & \displaystyle \nonumber\end{eqnarray}$$
Since
$P$
belongs to a special subvariety of codimension at least
$\dim V+1$
and
$V$
is Hodge generic in
$\text{Sh}_{K}(G,\mathfrak{X})$
, we have
$$\begin{eqnarray}\displaystyle \dim \langle P\rangle \leqslant \dim S-\dim V-1=\dim \langle V\rangle -\dim V-1<\unicode[STIX]{x1D6FF}(V). & & \displaystyle \nonumber\end{eqnarray}$$
Therefore,
$\unicode[STIX]{x1D6FF}(W)<\unicode[STIX]{x1D6FF}(V)$
and we conclude that
$W$
is not
$V$
. According to Conjecture 3.4, the union of the subvarieties belonging to
$\text{Opt}(V)\setminus V$
is not Zariski dense in
$V$
.◻
4 The defect condition
In this section, we prove Habegger and Pila’s defect condition for Shimura varieties and thus show that a subvariety that is optimal is weakly optimal.
Definition 4.1. Let
$W$
be a subvariety of
$S$
. We define the weakly special defect of
$W$
to be
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{ws}}(W):=\dim \langle W\rangle _{\text{ws}}-\dim W. & & \displaystyle \nonumber\end{eqnarray}$$
We note that, in [Reference Habegger and PilaHP16], this notion was referred to as geodesic defect.
Definition 4.2. If
$V$
is a subvariety of
$S$
and
$W$
a subvariety of
$V$
, then
$W$
is called weakly optimal in
$V$
if, for any subvariety
$Y$
of
$S$
,
$$\begin{eqnarray}\displaystyle W\subsetneq Y\subseteq V\;\Longrightarrow \;\unicode[STIX]{x1D6FF}_{\text{ws}}(Y)>\unicode[STIX]{x1D6FF}_{\text{ws}}(W). & & \displaystyle \nonumber\end{eqnarray}$$
Remark 4.3. Let
$V$
be a subvariety of
$S$
and
$W$
a subvariety of
$V$
. If
$W$
is weakly optimal in
$V$
, then
$W$
is an irreducible component of
$$\begin{eqnarray}\displaystyle \langle W\rangle _{\text{ws}}\cap V. & & \displaystyle \nonumber\end{eqnarray}$$
Proposition 4.4 (Cf. [Reference Habegger and PilaHP16, Proposition 4.3]).
The following defect condition holds.
Let
$W\subseteq Y$
be two subvarieties of
$S$
. Then
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}(Y)-\unicode[STIX]{x1D6FF}_{\text{ws}}(Y)\leqslant \unicode[STIX]{x1D6FF}(W)-\unicode[STIX]{x1D6FF}_{\text{ws}}(W). & & \displaystyle \nonumber\end{eqnarray}$$
Proof. We need to show that
$$\begin{eqnarray}\displaystyle \dim \langle Y\rangle -\dim \langle Y\rangle _{\text{ws}}\leqslant \dim \langle W\rangle -\dim \langle W\rangle _{\text{ws}}. & & \displaystyle \nonumber\end{eqnarray}$$
By Remark 2.3, we can and do assume that
$G$
is the generic Mumford–Tate group on
$X$
, that it is equal to
$G^{\text{ad}}$
, and that
$Y$
is Hodge generic in
$S$
. By definition, there exists a decomposition
$$\begin{eqnarray}\displaystyle G=G_{1}\times G_{2}, & & \displaystyle \nonumber\end{eqnarray}$$
which induces a splitting
$$\begin{eqnarray}\displaystyle X=X_{1}\times X_{2}, & & \displaystyle \nonumber\end{eqnarray}$$
such that
$\langle Y\rangle _{\text{ws}}$
is equal to the image of
$X_{1}\times \{x_{2}\}$
in
$S$
, for some
$x_{2}\in X_{2}$
.
Let
$\unicode[STIX]{x1D6E4}_{1}:=\text{p}_{1}(\unicode[STIX]{x1D6E4})$
and
$\unicode[STIX]{x1D6E4}_{2}:=\text{p}_{2}(\unicode[STIX]{x1D6E4})$
, where
$\text{p}_{1}$
and
$\text{p}_{2}$
are the projections from
$G$
to
$G_{1}$
and
$G_{2}$
, respectively. Then
$\unicode[STIX]{x1D6E4}^{\prime }:=\unicode[STIX]{x1D6E4}_{1}\,\times \,\unicode[STIX]{x1D6E4}_{2}$
is a congruence subgroup of
$G(\mathbb{Q})_{+}$
containing
$\unicode[STIX]{x1D6E4}$
as a finite index subgroup. Let
$\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6E4}\backslash X\rightarrow \unicode[STIX]{x1D6E4}^{\prime }\backslash X$
denote the natural (finite) morphism. Then
$\unicode[STIX]{x1D719}(W)\subseteq \unicode[STIX]{x1D719}(Y)\subseteq S^{\prime }:=\unicode[STIX]{x1D6E4}^{\prime }\backslash X$
, and we have
$$\begin{eqnarray}\displaystyle & \displaystyle \dim \langle Y\rangle =\dim \langle \unicode[STIX]{x1D719}(Y)\rangle , & \displaystyle \nonumber\\ \displaystyle & \displaystyle \dim \langle W\rangle =\dim \langle \unicode[STIX]{x1D719}(W)\rangle , & \displaystyle \nonumber\\ \displaystyle & \displaystyle \dim \langle Y\rangle _{\text{ws}}=\dim \langle \unicode[STIX]{x1D719}(Y)\rangle _{\text{ws}}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \dim \langle W\rangle _{\text{ws}}=\dim \langle \unicode[STIX]{x1D719}(W)\rangle _{\text{ws}}. & \displaystyle \nonumber\end{eqnarray}$$
Therefore, after replacing
$Y$
,
$W$
, and
$S$
by
$\unicode[STIX]{x1D719}(Y)$
,
$\unicode[STIX]{x1D719}(W)$
, and
$S^{\prime }$
, respectively, we may assume that
$\unicode[STIX]{x1D6E4}$
is of the form
$\unicode[STIX]{x1D6E4}_{1}\times \unicode[STIX]{x1D6E4}_{2}$
, and
$S=\unicode[STIX]{x1D6E4}_{1}\backslash X_{1}\times \unicode[STIX]{x1D6E4}_{2}\backslash X_{2}=S_{1}\times S_{2}$
.
Thus,
$\langle Y\rangle _{\text{ws}}=S_{1}\times \{s_{2}\}$
, where
$s_{2}$
is the image of
$x_{2}$
in
$S_{2}$
,
$Y=Y_{1}\times \{s_{2}\}$
, where
$Y_{1}$
is the projection of
$Y$
to
$S_{1}$
, and
$W=W_{1}\times \{s_{2}\}$
, where
$W_{1}$
is the projection of
$W$
to
$S_{1}$
. In particular, we can take
$$\begin{eqnarray}\displaystyle x:=(x_{1},x_{2})\in X_{1}\times X_{2} & & \displaystyle \nonumber\end{eqnarray}$$
such that
$\langle W\rangle$
is equal to the image in
$S$
of the
$M(\mathbb{R})^{+}$
conjugacy class
$X_{M}$
of
$x$
, where
$M:=\text{MT}(x)$
.
Again, there exists a decomposition
$$\begin{eqnarray}\displaystyle M^{\text{ad}}=M_{1}\times M_{2}, & & \displaystyle \nonumber\end{eqnarray}$$
which induces a splitting
$$\begin{eqnarray}\displaystyle X_{M}=X_{M_{1}}\times X_{M_{2}}\subseteq X=X_{1}\times X_{2} & & \displaystyle \nonumber\end{eqnarray}$$
such that
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of
$X_{M_{1}}\times \{y_{2}\}$
, for some
$y_{2}\in X_{M_{2}}$
.
Since
$\text{MT}(x_{2})$
is equal to
$G_{2}$
, it follows that
$M$
is a subgroup of
$G_{1}\times G_{2}$
that surjects on to the second factor. In particular,
$$\begin{eqnarray}\displaystyle X_{M}=M^{\text{der}}(\mathbb{R})^{+}x & & \displaystyle \nonumber\end{eqnarray}$$
surjects on to
$X_{2}$
. Therefore, let
$M_{1}^{\prime }$
and
$M_{2}^{\prime }$
be two normal semisimple subgroups of
$M^{\text{der}}$
corresponding to
$M_{1}$
and
$M_{2}$
, respectively, so that
$$\begin{eqnarray}\displaystyle M^{\text{der}}(\mathbb{R})^{+}x=M_{1}^{\prime }(\mathbb{R})^{+}M_{2}^{\prime }(\mathbb{R})^{+}x. & & \displaystyle \nonumber\end{eqnarray}$$
Since
$W$
is contained in
$S_{1}\times \{s_{2}\}$
, the projection of
$M_{1}^{\prime }$
to
$G_{2}$
must be trivial. Hence,
$M_{1}^{\prime }(\mathbb{R})^{+}x$
is contained in
$X_{1}\times \{x_{2}\}$
and we conclude that
$M_{2}^{\prime }(\mathbb{R})^{+}x$
surjects on to
$X_{2}$
. Since
$$\begin{eqnarray}\displaystyle M_{2}^{\prime }(\mathbb{R})^{+}x=\{y_{1}\}\times X_{M_{2}}, & & \displaystyle \nonumber\end{eqnarray}$$
for some
$y_{1}\in X_{M_{1}}$
, we have
$$\begin{eqnarray}\displaystyle \dim \langle W\rangle -\dim \langle W\rangle _{\text{ws}}=\dim X_{M_{2}}\geqslant \dim X_{2}=\dim \langle Y\rangle -\dim \langle Y\rangle _{\text{ws}}, & & \displaystyle \nonumber\end{eqnarray}$$
as required. ◻
Corollary 4.5 (Cf. [Reference Habegger and PilaHP16, Proposition 4.5]).
Let
$V$
be a subvariety of
$S$
. A subvariety of
$V$
that is optimal in
$V$
is weakly optimal in
$V$
.
5 The hyperbolic Ax–Schanuel conjecture
In this section, we formulate various conjectures about Shimura varieties that are analogous to the original Ax–Schanuel theorem from functional transcendence theory.
Theorem 5.1 (Cf. [Reference AxAx71, Theorem 1]).
Let
$f_{1},\ldots ,f_{n}\in \mathbb{C}[[t_{1},\ldots ,t_{m}]]$
be power series that are
$\mathbb{Q}$
-linearly independent modulo
$\mathbb{C}$
. Then we have the following inequality
$$\begin{eqnarray}\text{tr}.\text{deg}_{\mathbb{C}}\mathbb{C}(f_{1},\ldots ,f_{n},e(f_{1}),\ldots ,e(f_{n}))\geqslant n+\text{rank}\biggl(\frac{\unicode[STIX]{x2202}f_{i}}{\unicode[STIX]{x2202}t_{j}}\biggr)_{\substack{ i=1,\ldots ,n \\ j=1,\ldots ,m}}\end{eqnarray}$$
where
$e(f)=e^{2\unicode[STIX]{x1D70B}if}\in \mathbb{C}[[t_{1},\ldots ,t_{m}]]$
.
The following theorem is then an immediate corollary.
Theorem 5.2. Let
$f_{1},\ldots ,f_{n}\in \mathbb{C}[[t_{1},\ldots ,t_{m}]]$
as above. Then
$$\begin{eqnarray}\displaystyle \text{tr}.\text{deg}_{\mathbb{C}}\mathbb{C}(f_{1},\ldots ,f_{n})+\text{tr}.\text{deg}_{\mathbb{C}}\mathbb{C}(e(f_{1}),\ldots ,e(f_{n}))\geqslant n+\text{rank}\biggl(\frac{\unicode[STIX]{x2202}f_{i}}{\unicode[STIX]{x2202}t_{j}}\biggr)_{\substack{ i=1,\ldots ,n \\ j=1,\ldots ,m}}. & & \displaystyle \nonumber\end{eqnarray}$$
Let
$\unicode[STIX]{x1D70B}$
denote the uniformization map
$$\begin{eqnarray}\displaystyle \mathbb{C}^{n}\rightarrow (\mathbb{C}^{\times })^{n}:(x_{1},\ldots ,x_{n})\mapsto (e(x_{1}),\ldots ,e(x_{n})) & & \displaystyle \nonumber\end{eqnarray}$$
and let
$D_{n}$
denote its graph in
$\mathbb{C}^{n}\times (\mathbb{C}^{\times })^{n}$
. We can rephrase Theorem 5.1 as follows.
Theorem 5.3 (Cf. [Reference Tsimerman, Jones and WilkieTsi15, Theorem 1.2]).
Let
$V$
be a subvariety of
$\mathbb{C}^{n}\times (\mathbb{C}^{\times })^{n}$
and let
$U$
be an irreducible analytic component of
$V\cap D_{n}$
. Assume that the projection of
$U$
to
$(\mathbb{C}^{\times })^{n}$
is not contained in a coset of a proper subtorus of
$(\mathbb{C}^{\times })^{n}$
. Then
$$\begin{eqnarray}\displaystyle \dim V\geqslant \dim U+n. & & \displaystyle \nonumber\end{eqnarray}$$
Similarly, we can rephrase Theorem 5.2 as follows.
Theorem 5.4. Let
$W$
be a subvariety of
$\mathbb{C}^{n}$
and
$V$
a subvariety of
$(\mathbb{C}^{\times })^{n}$
. Let
$A$
be an irreducible analytic component of
$W\,\cap \,\unicode[STIX]{x1D70B}^{-1}(V)$
. If
$A$
is not contained in
$b+L$
, for any proper
$\mathbb{Q}$
-linear subspace
$L$
of
$\mathbb{C}^{n}$
and any
$b\in \mathbb{C}^{n}$
, then
$$\begin{eqnarray}\displaystyle \dim V+\dim W\geqslant \dim A+n. & & \displaystyle \nonumber\end{eqnarray}$$
Recall that
$X$
is naturally endowed with the structure of a hermitian symmetric domain. In particular, it is a complex manifold. We define an (irreducible algebraic) subvariety of
$X$
as in Appendix B of [Reference Klingler, Ullmo and YafaevKUY16]. In particular, we consider the Harish-Chandra realization of
$X$
, which is a bounded domain in
$\mathbb{C}^{N}$
, for some
$N\in \mathbb{N}$
, and we define an (irreducible algebraic) subvariety of
$X$
to be an irreducible analytic component of the intersection of
$X$
with an algebraic subvariety of
$\mathbb{C}^{N}$
. We define an (irreducible algebraic) subvariety of
$X\times S$
to be an irreducible analytic component of the intersection of
$X\times S$
with an algebraic subvariety of
$\mathbb{C}^{N}\,\times \,S$
. We note, however, that, by [Reference Klingler, Ullmo and YafaevKUY16, Corollary B.2], the algebraic structure that we are putting on
$X$
and
$X\times S$
does not depend on our particular choice of the Harish-Chandra realization of
$X$
; any realization of
$X$
would yield the same algebraic structures.
We are, therefore, able to formulate conjectures for Shimura varieties that are analogous to those above. Let
$\unicode[STIX]{x1D70B}$
henceforth denote the uniformization map
$$\begin{eqnarray}\displaystyle X\rightarrow S & & \displaystyle \nonumber\end{eqnarray}$$
and let
$D_{S}$
denote the graph of
$\unicode[STIX]{x1D70B}$
in
$X\times S$
. The following conjecture generalizes [Reference Pila and TsimermanPT16, Conjecture 1.1].
Conjecture 5.5 (Hyperbolic Ax–Schanuel).
Let
$V$
be a subvariety of
$X\times S$
and let
$U$
be an irreducible analytic component of
$V\,\cap \,D_{S}$
. Assume that the projection of
$U$
to
$S$
is not contained in a weakly special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
strictly contained in
$S$
. Then
$$\begin{eqnarray}\displaystyle \dim V\geqslant \dim U+\dim S. & & \displaystyle \nonumber\end{eqnarray}$$
For
$S=\mathbb{C}^{n}$
, Conjecture 5.5 and its generalization involving derivatives were obtained in [Reference Pila and TsimermanPT16]. Mok, Pila, and Tsimerman have very recently announced a proof of Conjecture 5.5 in full [Reference Mok, Pila and TsimermanMPT17].
For applications to the Zilber–Pink conjecture, only the following weaker version will be needed.
Conjecture 5.6 (Cf. [Reference Habegger and PilaHP16, Conjecture 5.10]).
Let
$W$
be a subvariety of
$X$
and let
$V$
be a subvariety of
$S$
. Let
$A$
be an irreducible analytic component of
$W\cap \unicode[STIX]{x1D70B}^{-1}(V)$
and assume that
$\unicode[STIX]{x1D70B}(A)$
is not contained in a weakly special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
strictly contained in
$S$
. Then
$$\begin{eqnarray}\displaystyle \dim V+\dim W\geqslant \dim A+\dim S. & & \displaystyle \nonumber\end{eqnarray}$$
Proof that Conjecture 5.5 implies Conjecture 5.6.
Consider the situation described in the statement of Conjecture 5.6. Then
$Y:=W\times V$
is an algebraic subvariety of
$X\times S$
and
$$\begin{eqnarray}\displaystyle U:=\{(a,\unicode[STIX]{x1D70B}(a));a\in A\} & & \displaystyle \nonumber\end{eqnarray}$$
is an irreducible analytic component of
$Y\,\cap \,D_{S}$
. Clearly, the projection of
$U$
to
$S$
is not contained in a weakly special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
strictly contained in
$S$
. Therefore, by Conjecture 5.5,
$$\begin{eqnarray}\displaystyle \dim Y\geqslant \dim U+\dim S & & \displaystyle \nonumber\end{eqnarray}$$
and the result follows since
$\dim U=\dim A$
and
$\dim Y=\dim W+\dim V$
.◻
In our applications, we will use a reformulation of Conjecture 5.6. For this reformulation, we will need the following definitions.
Fix a subvariety
$V$
of
$S$
.
Definition 5.7. An intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
is an irreducible analytic component of the intersection of
$\unicode[STIX]{x1D70B}^{-1}(V)$
with a subvariety of
$X$
.
Clearly, for any intersection component
$A$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
, there exists a smallest subvariety
$\langle A\rangle _{\text{Zar}}$
of
$X$
containing
$A$
.
Definition 5.8. Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. We define the Zariski defect of
$A$
to be
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{Zar}}(A):=\dim \langle A\rangle _{\text{Zar}}-\dim A. & & \displaystyle \nonumber\end{eqnarray}$$
Definition 5.9. We say that an intersection component
$A$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
is Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
if, for any intersection component
$B$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
,
$$\begin{eqnarray}\displaystyle A\subsetneq B\subseteq \unicode[STIX]{x1D70B}^{-1}(V)\;\Longrightarrow \;\unicode[STIX]{x1D6FF}_{\text{Zar}}(B)>\unicode[STIX]{x1D6FF}_{\text{Zar}}(A). & & \displaystyle \nonumber\end{eqnarray}$$
Remark 5.10. Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. If A is Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
, then
$A$
is an irreducible analytic component of
$$\begin{eqnarray}\displaystyle \langle A\rangle _{\text{Zar}}\cap \unicode[STIX]{x1D70B}^{-1}(V). & & \displaystyle \nonumber\end{eqnarray}$$
Definition 5.11. Let
$x\in X$
and let
$X_{M}$
denote the
$M(\mathbb{R})^{+}$
conjugacy class of
$x$
in
$X$
, where
$M:=\text{MT}(x)$
. Write
$M^{\text{ad}}$
as a product
$$\begin{eqnarray}\displaystyle M^{\text{ad}}=M_{1}\times M_{2} & & \displaystyle \nonumber\end{eqnarray}$$
of two normal
$\mathbb{Q}$
-subgroups, either of which may be trivial, thus inducing a splitting
$$\begin{eqnarray}\displaystyle X_{M}=X_{1}\times X_{2}. & & \displaystyle \nonumber\end{eqnarray}$$
For any
$x_{1}\in X_{1}$
or
$x_{2}\in X_{2}$
, we obtain a subvariety
$\{x_{1}\}\times X_{2}$
or
$X_{1}\times \{x_{2}\}$
of
$X$
. We refer to any subvariety of
$X$
taking this form as a pre-weakly special subvariety of
$X$
. That is, a weakly special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
contained in
$S$
is, by definition, the image in
$S$
of a pre-weakly special subvariety of
$X$
.
Remark 5.12. Note that pre-weakly special subvarieties of
$X$
are indeed subvarieties of
$X$
(see [Reference GaoGao17, Lemma 6.2], for example). In particular, they are irreducible analytic subsets of
$X$
. As explained in [Reference MoonenMoo98], pre-weakly special subvarieties of
$X$
are totally geodesic subvarieties of
$X$
.
Definition 5.13. An intersection component
$A$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
is called pre-weakly special if it is an irreducible analytic component of
$$\begin{eqnarray}\displaystyle \langle A\rangle _{\text{Zar}}\cap \unicode[STIX]{x1D70B}^{-1}(V) & & \displaystyle \nonumber\end{eqnarray}$$
and
$\langle A\rangle _{\text{Zar}}$
is a pre-weakly special subvariety of
$X$
.
Conjecture 5.14 (Weak hyperbolic Ax–Schanuel).
Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
that is Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
. Then
$A$
is pre-weakly special.
Note that, since
$A$
is assumed to be Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
, it is automatically an irreducible analytic component of
$$\begin{eqnarray}\displaystyle \langle A\rangle _{\text{Zar}}\cap \unicode[STIX]{x1D70B}^{-1}(V). & & \displaystyle \nonumber\end{eqnarray}$$
The content of Conjecture 5.14, therefore, is the claim that
$\langle A\rangle _{\text{Zar}}$
is a pre-weakly special subvariety of
$X$
.
Note that Conjecture 5.14 is a direct generalization of the hyperbolic Ax–Lindemann theorem.
Theorem 5.15 (Hyperbolic Ax–Lindemann).
The maximal subvarieties contained in
$\unicode[STIX]{x1D70B}^{-1}(V)$
are pre-weakly special.
Proof that Conjecture 5.14 implies Theorem 5.15.
The maximal subvarieties contained in
$\unicode[STIX]{x1D70B}^{-1}(V)$
are precisely the intersection components of
$\unicode[STIX]{x1D70B}^{-1}(V)$
that are Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
and whose Zariski defect is zero.◻
Although [Reference Habegger and PilaHP16, § 5.2] is dedicated to products of modular curves, the proof that Formulations A and B of Weak Complex Ax are equivalent is completely general and, when translated into our terminology, yields the following.
We conclude this section with the following consequence of the weak hyperbolic Ax–Schanuel conjecture. Here and elsewhere, we will tacitly make use of the following remark.
Remark 5.17. Let
$W$
be a subvariety of
$S$
and let
$A$
denote an irreducible analytic component of
$\unicode[STIX]{x1D70B}^{-1}(W)$
in
$X$
. Then, since
$W$
is analytically irreducible, every irreducible analytic component of
$\unicode[STIX]{x1D70B}^{-1}(W)$
is equal to a
$\unicode[STIX]{x1D6E4}$
-translate of
$A$
(as mentioned in [Reference Ullmo and YafaevUY11, § 4], for example). In particular,
$\unicode[STIX]{x1D70B}(A)$
is equal to
$W$
.
Lemma 5.18. Assume that the weak hyperbolic Ax–Schanuel conjecture holds for
$V$
and let
$A$
be a Zariski optimal intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. Then
$\unicode[STIX]{x1D70B}(A)$
is a closed irreducible subvariety of
$V$
and, as such, is weakly optimal in
$V$
.
Proof. Clearly, the Zariski closure
$\overline{\unicode[STIX]{x1D70B}(A)}$
of
$\unicode[STIX]{x1D70B}(A)$
is irreducible. Therefore, let
$W$
be a subvariety of
$V$
containing
$\overline{\unicode[STIX]{x1D70B}(A)}$
such that
$\unicode[STIX]{x1D6FF}_{\text{ws}}(W)\leqslant \unicode[STIX]{x1D6FF}_{\text{ws}}(\overline{\unicode[STIX]{x1D70B}(A)})$
. We can and do assume that
$W$
is weakly optimal in
$V$
. Let
$B$
be an irreducible analytic component of
$\unicode[STIX]{x1D70B}^{-1}(W)$
containing
$A$
. We have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{Zar}}(B) & = & \displaystyle \dim \langle B\rangle _{\text{Zar}}-\dim B=\dim \langle B\rangle _{\text{Zar}}-\dim W\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \dim \langle W\rangle _{\text{ws}}-\dim W=\unicode[STIX]{x1D6FF}_{\text{ws}}(W)\leqslant \unicode[STIX]{x1D6FF}_{\text{ws}}(\overline{\unicode[STIX]{x1D70B}(A)})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \dim \langle A\rangle _{\text{Zar}}-\dim A=\unicode[STIX]{x1D6FF}_{\text{Zar}}(A),\nonumber\end{eqnarray}$$
where we use the fact that, by the weak hyperbolic Ax–Schanuel conjecture,
$\langle A\rangle _{\text{Zar}}$
is pre-weakly special. Therefore, we conclude that
$B=A$
. Hence,
$\unicode[STIX]{x1D70B}(A)=\unicode[STIX]{x1D70B}(B)=W$
.◻
6 A finiteness result for weakly optimal subvarieties
In this section, we deduce from the weak hyperbolic Ax–Schanuel conjecture a finiteness statement for the weakly optimal subvarieties of a given subvariety
$V$
.
Definition 6.1. Let
$x\in X$
and let
$X_{M}$
denote the
$M(\mathbb{R})^{+}$
conjugacy class of
$x$
in
$X$
, where
$M:=\text{MT}(x)$
. Then
$X_{M}$
is a subvariety of
$X$
and we refer to any subvariety of
$X$
taking this form as a pre-special subvariety of
$X$
. In particular, a pre-special subvariety of
$X$
is a pre-weakly special subvariety of
$X$
. If
$X_{M}$
is a point, that is, if
$M$
is a torus, we refer to
$X_{M}$
as a pre-special point of
$X$
. A special subvariety of
$\text{Sh}_{K}(G,\mathfrak{X})$
contained in
$S$
is, by definition, the image in
$S$
of a pre-special subvariety of
$X$
.
Definition 6.2. Let
$x\in X$
and let
$X_{M}$
denote the
$M(\mathbb{R})^{+}$
conjugacy class of
$x$
in
$X$
, where
$M:=\text{MT}(x)$
. Decomposing
$M^{\text{ad}}$
as a product
$$\begin{eqnarray}\displaystyle M^{\text{ad}}=M_{1}\times M_{2} & & \displaystyle \nonumber\end{eqnarray}$$
of two normal
$\mathbb{Q}$
-subgroups, either of which may be trivial, induces a splitting
$$\begin{eqnarray}\displaystyle X_{M}=X_{1}\times X_{2}. & & \displaystyle \nonumber\end{eqnarray}$$
For any such splitting, and any
$x_{1}\in X_{1}$
or
$x_{2}\in X_{2}$
, we refer to the pre-weakly special subvariety
$\{x_{1}\}\times X_{2}$
or
$X_{1}\times \{x_{2}\}$
as a fibre of (the pre-special subvariety)
$X_{M}$
. In particular, the points of
$X_{M}$
are all fibres of
$X_{M}$
, and so too is
$X_{M}$
itself.
The main result of this section is the following.
Proposition 6.3 (Cf. [Reference Habegger and PilaHP16, Proposition 6.6]).
Let
$V$
be a subvariety of
$S$
and assume that the weak hyperbolic Ax–Schanuel conjecture is true for
$V$
. Then there exists a finite set
$\unicode[STIX]{x1D6F4}$
of pre-special subvarieties of
$X$
such that the following holds.
Let
$W$
be a subvariety of
$V$
that is weakly optimal in
$V$
. Then there exists
$Y\in \unicode[STIX]{x1D6F4}$
such that
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of a fibre of
$Y$
.
Note that similar theorems also hold for abelian varieties (see [Reference Habegger and PilaHP16, Proposition 6.1] and [Reference RémondRém09, Proposition 3.2]).
Now fix a subvariety
$V$
of
$S$
. Given an intersection component
$A$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
, there is a smallest totally geodesic subvariety
$\langle A\rangle _{\text{geo}}$
of
$X$
that contains
$A$
. In particular, we may make the following definition.
Definition 6.4. Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. We define the geodesic defect of
$A$
to be
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{geo}}(A):=\dim \langle A\rangle _{\text{geo}}-\dim A. & & \displaystyle \nonumber\end{eqnarray}$$
We note that, in [Reference Habegger and PilaHP16], this notion was referred to as the Möbius defect of
$A$
.
Definition 6.5. We say that an intersection component
$A$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
is geodesically optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
if, for any intersection component
$B$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
,
$$\begin{eqnarray}\displaystyle A\subsetneq B\subseteq \unicode[STIX]{x1D70B}^{-1}(V)\;\Longrightarrow \;\unicode[STIX]{x1D6FF}_{\text{geo}}(B)>\unicode[STIX]{x1D6FF}_{\text{geo}}(A). & & \displaystyle \nonumber\end{eqnarray}$$
We note that the terminology geodesically optimal has a different meaning in [Reference Habegger and PilaHP16].
Remark 6.6. Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. If
$A$
is geodesically optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
, then
$A$
is an irreducible analytic component of
$$\begin{eqnarray}\displaystyle \langle A\rangle _{\text{geo}}\cap \unicode[STIX]{x1D70B}^{-1}(V). & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 6.7. Assume that the weak hyperbolic Ax–Schanuel conjecture is true for
$V$
and let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. If
$A$
is geodesically optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
, then
$A$
is Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
.
Proof. Suppose that
$B$
is an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
containing
$A$
such that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{Zar}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{Zar}}(A). & & \displaystyle \nonumber\end{eqnarray}$$
We can and do assume that
$B$
is Zariski optimal and so, by the weak hyperbolic Ax–Schanuel conjecture, it is pre-weakly special. In particular,
$\langle B\rangle _{\text{Zar}}$
is a pre-weakly special subvariety of
$X$
and, therefore, equal to
$\langle B\rangle _{\text{geo}}$
. Then
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{geo}}(B)=\unicode[STIX]{x1D6FF}_{\text{Zar}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{Zar}}(A)\leqslant \unicode[STIX]{x1D6FF}_{\text{geo}}(A), & & \displaystyle \nonumber\end{eqnarray}$$
and, since
$A$
is geodesically optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
, we conclude that
$B=A$
.◻
Lemma 6.8. Assume that the weak hyperbolic Ax–Schanuel conjecture is true for
$V$
and let
$W$
be a subvariety of
$V$
that is weakly optimal in
$V$
. Let
$A$
be an irreducible analytic component of
$\unicode[STIX]{x1D70B}^{-1}(W)$
. Then
$A$
is an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
and is geodesically optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
.
Proof. Clearly,
$A$
is an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
since
$W$
is an irreducible component of
$\langle W\rangle _{\text{ws}}\cap V$
and
$\unicode[STIX]{x1D70B}^{-1}\langle W\rangle _{\text{ws}}$
is equal to the
$\unicode[STIX]{x1D6E4}$
-orbit of a pre-weakly special subvariety of
$X$
.
Therefore, let
$B$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
containing
$A$
such that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{geo}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{geo}}(A). & & \displaystyle \nonumber\end{eqnarray}$$
We can and do assume that
$B$
is geodesically optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
and so, by Lemma 6.7,
$B$
is Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
. Therefore, by the weak hyperbolic Ax–Schanuel conjecture,
$B$
is pre-weakly special, i.e.,
$\langle B\rangle _{\text{Zar}}$
is a pre-weakly special subvariety of
$X$
.
Let
$Z:=\unicode[STIX]{x1D70B}(B)$
(which is a closed irreducible subvariety of
$V$
by Lemma 5.18). We claim that
$\langle Z\rangle _{\text{ws}}=\unicode[STIX]{x1D70B}(\langle B\rangle _{\text{Zar}})$
. To see this, note that
$Z$
is contained in
$\unicode[STIX]{x1D70B}(\langle B\rangle _{\text{Zar}})$
and so
$\langle Z\rangle _{\text{ws}}$
is contained in
$\unicode[STIX]{x1D70B}(\langle B\rangle _{\text{Zar}})$
. On the other hand,
$\langle B\rangle _{\text{Zar}}$
is contained in
$\unicode[STIX]{x1D70B}^{-1}(\langle Z\rangle _{\text{ws}})$
and so
$\unicode[STIX]{x1D70B}(\langle B\rangle _{\text{Zar}})$
is contained in
$\langle Z\rangle _{\text{ws}}$
, which proves the claim. Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{ws}}(Z) & = & \displaystyle \dim \unicode[STIX]{x1D70B}(\langle B\rangle _{\text{Zar}})-\dim Z\nonumber\\ \displaystyle & = & \displaystyle \dim \langle B\rangle _{\text{Zar}}-\dim Z\nonumber\\ \displaystyle & = & \displaystyle \dim \langle B\rangle _{\text{Zar}}-\dim B=\unicode[STIX]{x1D6FF}_{\text{geo}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{geo}}(A)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \dim \langle W\rangle _{\text{ws}}-\dim W=\unicode[STIX]{x1D6FF}_{\text{ws}}(W).\nonumber\end{eqnarray}$$
Since
$W$
is weakly optimal in
$V$
and contained in
$Z$
, we conclude that
$Z=W$
. In particular,
$B$
is contained in
$\unicode[STIX]{x1D70B}^{-1}(W)$
and, therefore,
$B=A$
.◻
Let us briefly summarize the relationship between Zariski optimal and weakly optimal.
Proposition 6.9. Assume that the weak hyperbolic Ax–Schanuel conjecture is true for
$V$
.
If
$A$
is a Zariski optimal intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
, then
$\unicode[STIX]{x1D70B}(A)$
is a closed irreducible subvariety of
$V$
that is weakly optimal in
$V$
.
On the other hand, if
$W$
is a subvariety of
$V$
that is weakly optimal in
$V$
, and
$A$
is an irreducible analytic component of
$\unicode[STIX]{x1D70B}^{-1}(W)$
, then
$A$
is a Zariski optimal intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
.
As explained in [Reference Klingler, Ullmo and YafaevKUY16], we can and do fix, once and for all, an open, semialgebraic fundamental set
${\mathcal{F}}$
in
$X$
for the action of
$\unicode[STIX]{x1D6E4}$
such that the set
${\mathcal{V}}:=\unicode[STIX]{x1D70B}^{-1}(V)\cap {\mathcal{F}}$
is definable. The key step in the proof of Proposition 6.3 is the following.
Proposition 6.10. Assume that the weak hyperbolic Ax–Schanuel theorem is true for
$V$
. There exists a finite set
$\unicode[STIX]{x1D6F4}$
of pre-special subvarieties of
$X$
such that the following holds.
Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
that is pre-weakly special such that, for some
$x\in \langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}$
,
$$\begin{eqnarray}\displaystyle \dim A=\dim _{x}(\langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}). & & \displaystyle \nonumber\end{eqnarray}$$
Then there exists
$Y\in \unicode[STIX]{x1D6F4}$
such that
$\langle A\rangle _{\text{Zar}}$
is equal to a fibre of
$Y$
.
In order to prove Proposition 6.10, we require some further preparations.
Definition 6.11. We say that a real semisimple algebraic group
$F$
is without compact factors if it is equal to an almost direct product of almost simple subgroups whose underlying real Lie groups are not compact. We allow the product to be trivial, i.e., we consider the trivial group as a real semisimple algebraic group without compact factors.
Lemma 6.12. A subvariety of
$X$
that is totally geodesic in
$X$
is of the form
$$\begin{eqnarray}\displaystyle F(\mathbb{R})^{+}x, & & \displaystyle \nonumber\end{eqnarray}$$
where
$F$
is a semisimple algebraic subgroup of
$G_{\mathbb{R}}$
without compact factors and
$x\in X$
factors through
$$\begin{eqnarray}\displaystyle G_{F}:=FZ_{G_{\mathbb{R}}}(F)^{\circ }. & & \displaystyle \nonumber\end{eqnarray}$$
Conversely, if
$F$
is a semisimple algebraic subgroup of
$G_{\mathbb{R}}$
without compact factors and
$x\in X$
factors through
$G_{F}$
, then
$F(\mathbb{R})^{+}x$
is a subvariety of
$X$
that is totally geodesic in
$X$
.
Proof. See [Reference Ullmo and YafaevUY18, Proposition 2.3]. ◻
We let
$\unicode[STIX]{x1D6FA}$
denote a (finite) set of representatives for the
$G(\mathbb{R})$
-conjugacy classes of semisimple algebraic subgroups of
$G_{\mathbb{R}}$
that are without compact factors. It is clear that the set
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{0}:=\{(x,g,F)\in {\mathcal{V}}\times G(\mathbb{R})\times \unicode[STIX]{x1D6FA}:x(\mathbb{S})\subseteq gG_{F}g^{-1}\}, & & \displaystyle \nonumber\end{eqnarray}$$
parametrizing (albeit in a many-to-one fashion) the totally geodesic subvarieties of
$X$
passing through
${\mathcal{V}}$
, is definable.
Recall from [Reference van den DriesVdD98], 1.17 that the local dimension
$\dim _{x}A$
of a definable set
$A$
at a point
$x\in A$
is definable. By [Reference Habegger and PilaHP16, Lemma 6.2], if
$A$
is also a (complex) analytic set, then this dimension is exactly twice the local analytic dimension at
$x$
. Furthermore, if
$A$
is analytically irreducible, then its local dimension at the points of
$A$
is constant. For the remainder of this section, dimensions will be taken in the sense of definable sets.
Consider the two functions
$$\begin{eqnarray}\displaystyle & \displaystyle d(x,g,F):=\dim _{x}(gF(\mathbb{R})^{+}g^{-1}x)=\dim (gF(\mathbb{R})^{+}g^{-1}x), & \displaystyle \nonumber\\ \displaystyle & \displaystyle d_{{\mathcal{V}}}(x,g,F):=\dim _{x}({\mathcal{V}}\cap gF(\mathbb{R})^{+}g^{-1}x), & \displaystyle \nonumber\end{eqnarray}$$
and let
$\unicode[STIX]{x1D6F1}_{1}$
denote the definable set
$$\begin{eqnarray}\displaystyle & & \displaystyle \{(x,g,F)\in \unicode[STIX]{x1D6F1}_{0}:(x,g_{1},F_{1})\in \unicode[STIX]{x1D6F1}_{0},gF(\mathbb{R})^{+}g^{-1}x\subsetneq g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x\nonumber\\ \displaystyle & & \displaystyle \quad \;\Longrightarrow \;d(x,g,F)-d_{{\mathcal{V}}}(x,g,F)<d(x,g_{1},F_{1})-d_{{\mathcal{V}}}(x,g_{1},F_{1})\!\}.\nonumber\end{eqnarray}$$
Finally, let
$\unicode[STIX]{x1D6F1}_{2}$
denote the definable set
$$\begin{eqnarray}\displaystyle & & \displaystyle \{(x,g,F)\in \unicode[STIX]{x1D6F1}_{1}:(x,g_{1},F_{1})\in \unicode[STIX]{x1D6F1}_{0},g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x\subsetneq gF(\mathbb{R})^{+}g^{-1}x\nonumber\\ \displaystyle & & \displaystyle \quad \;\Longrightarrow \;d_{{\mathcal{V}}}(x,g_{1},F_{1})<d_{{\mathcal{V}}}(x,g,F)\!\}.\nonumber\end{eqnarray}$$
The proof of Proposition 6.10 will require the following three lemmas.
Lemma 6.13. Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
that is pre-weakly special such that, for some
$x\in \langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}$
,
$$\begin{eqnarray}\displaystyle \dim A=\dim _{x}(\langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}). & & \displaystyle \nonumber\end{eqnarray}$$
Then we can write
$$\begin{eqnarray}\displaystyle \langle A\rangle _{\text{Zar}}=gF(\mathbb{R})^{+}g^{-1}x, & & \displaystyle \nonumber\end{eqnarray}$$
where
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{2}$
.
Proof. By Lemma 6.12, we can write
$$\begin{eqnarray}\displaystyle \langle A\rangle _{\text{Zar}}=gF(\mathbb{R})^{+}g^{-1}x & & \displaystyle \nonumber\end{eqnarray}$$
for some
$F\in \unicode[STIX]{x1D6FA}$
and some
$x\in {\mathcal{V}}$
that factors through
$gG_{F}g^{-1}$
. In particular,
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{0}$
. By assumption, we can and do choose
$x\in \langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}$
such that
$$\begin{eqnarray}\displaystyle \dim A=\dim _{x}(\langle A\rangle _{\text{Zar}}\cap {\mathcal{V}})=d_{{\mathcal{V}}}(x,g,F). & & \displaystyle \nonumber\end{eqnarray}$$
Suppose that
$(x,g,F)$
does not belong to
$\unicode[STIX]{x1D6F1}_{1}$
, i.e., that there exists
$(x,g_{1},F_{1})\in \unicode[STIX]{x1D6F1}_{0}$
such that
$$\begin{eqnarray}\displaystyle gF(\mathbb{R})^{+}g^{-1}x\subsetneq g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x, & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle d(x,g,F)-d_{{\mathcal{V}}}(x,g,F)\geqslant d(x,g_{1},F_{1})-d_{{\mathcal{V}}}(x,g_{1},F_{1}). & & \displaystyle\end{eqnarray}$$
Let
$B$
be an irreducible analytic component of
$$\begin{eqnarray}\displaystyle g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x\cap \unicode[STIX]{x1D70B}^{-1}(V) & & \displaystyle \nonumber\end{eqnarray}$$
passing through
$x$
such that
$$\begin{eqnarray}\displaystyle \dim B=d_{{\mathcal{V}}}(x,g_{1},F_{1}). & & \displaystyle \nonumber\end{eqnarray}$$
From (6.13.1), we obtain
$\unicode[STIX]{x1D6FF}_{\text{Zar}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{Zar}}(A)$
.
On the other hand, the intersection inequality (see [Reference Grauert and RemmertGR84, ch. 5, § 3]) yields
$$\begin{eqnarray}\displaystyle \dim _{x}(B\cap \langle A\rangle _{\text{Zar}})\geqslant \dim B+\dim \langle A\rangle _{\text{Zar}}-d(x,g_{1},F_{1}) & & \displaystyle \nonumber\end{eqnarray}$$
and, from (6.13.1), we obtain
$$\begin{eqnarray}\displaystyle \dim _{x}(B\cap \langle A\rangle _{\text{Zar}})\geqslant \dim A. & & \displaystyle \nonumber\end{eqnarray}$$
It follows that
$B\,\cap \,\langle A\rangle _{\text{Zar}}$
, and hence
$B$
itself, contains a complex neighbourhood of
$x$
in
$A$
, which implies that
$A$
is contained in
$B$
.
Therefore, since
$A$
is Zariski optimal, we conclude that
$A=B$
. However, this implies that
$$\begin{eqnarray}\displaystyle d(x,g_{1},F_{1})-d_{{\mathcal{V}}}(x,g_{1},F_{1})>2\unicode[STIX]{x1D6FF}_{\text{Zar}}(B)=2\unicode[STIX]{x1D6FF}_{\text{Zar}}(A)=d(x,g,F)-d_{{\mathcal{V}}}(x,g,F), & & \displaystyle \nonumber\end{eqnarray}$$
which contradicts (6.13.1).
Therefore, suppose that
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{1}$
does not belong to
$\unicode[STIX]{x1D6F1}_{2}$
, i.e., that there exists
$(x,g_{1},F_{1})\in \unicode[STIX]{x1D6F1}_{0}$
such that
$$\begin{eqnarray}\displaystyle g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x\subsetneq gF(\mathbb{R})^{+}g^{-1}x, & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle d_{{\mathcal{V}}}(x,g_{1},F_{1})=d_{{\mathcal{V}}}(x,g,F)=\dim A. & & \displaystyle \nonumber\end{eqnarray}$$
But then
$A$
is contained in
$$\begin{eqnarray}\displaystyle g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x\subsetneq \langle A\rangle _{\text{Zar}}, & & \displaystyle \nonumber\end{eqnarray}$$
which is a contradiction. ◻
Lemma 6.14. Assume that the weak hyperbolic Ax–Schanuel conjecture is true for
$V$
. Then, if
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{2}$
, there exists a semisimple subgroup
$F^{\prime }$
of
$G$
defined over
$\mathbb{Q}$
such that
$gFg^{-1}$
is equal to the almost direct product of the almost simple factors of
$F_{\mathbb{R}}^{\prime }$
whose underlying real Lie groups are non-compact.
Proof. By [Reference UllmoUll14, Proposition 3.1], it suffices to show that
$gF(\mathbb{R})^{+}g^{-1}x$
is a pre-weakly special subvariety of
$X$
. Therefore, let
$A$
be an irreducible analytic component of
$$\begin{eqnarray}\displaystyle gF(\mathbb{R})^{+}g^{-1}x\cap \unicode[STIX]{x1D70B}^{-1}(V) & & \displaystyle \nonumber\end{eqnarray}$$
passing through
$x$
such that
$$\begin{eqnarray}\displaystyle \dim A=d_{{\mathcal{V}}}(x,g,F). & & \displaystyle \nonumber\end{eqnarray}$$
Let
$B$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
containing
$A$
such that
$\unicode[STIX]{x1D6FF}_{\text{Zar}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{Zar}}(A)$
. We can and do assume that
$B$
is Zariski optimal and, therefore, by the weak hyperbolic Ax–Schanuel conjecture, pre-weakly special, i.e.,
$B$
is an irreducible component of
$$\begin{eqnarray}\displaystyle \langle B\rangle _{\text{Zar}}\cap \unicode[STIX]{x1D70B}^{-1}(V) & & \displaystyle \nonumber\end{eqnarray}$$
and
$\langle B\rangle _{\text{Zar}}$
is a pre-weakly special subvariety of
$X$
.
Therefore,
$A$
is contained in
$$\begin{eqnarray}\displaystyle gF(\mathbb{R})^{+}g^{-1}x\cap \langle B\rangle _{\text{Zar}} & & \displaystyle \nonumber\end{eqnarray}$$
and we let
$Y$
be an irreducible analytic component of this intersection containing
$A$
. Then
$Y$
is a subvariety of
$X$
that is totally geodesic is
$X$
and, hence, equal to
$g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x$
for some
$(x,g_{1},F_{1})\in \unicode[STIX]{x1D6F1}_{0}$
. Furthermore,
$$\begin{eqnarray}\displaystyle d_{{\mathcal{V}}}(x,g_{1},F_{1})=\dim A=d_{{\mathcal{V}}}(x,g,F) & & \displaystyle \nonumber\end{eqnarray}$$
and, since
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{2}$
, we conclude that
$$\begin{eqnarray}\displaystyle g_{1}F_{1}(\mathbb{R})^{+}g_{1}^{-1}x=gF(\mathbb{R})^{+}g^{-1}x\subseteq \langle B\rangle _{\text{Zar}}. & & \displaystyle \nonumber\end{eqnarray}$$
We also have
$$\begin{eqnarray}\displaystyle \dim \langle B\rangle _{\text{Zar}}-\dim _{x}(\langle B\rangle _{\text{Zar}}\cap {\mathcal{V}}) & {\leqslant} & \displaystyle \unicode[STIX]{x1D6FF}_{\text{Zar}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{Zar}}(A)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle d(x,g,F)-\dim A\nonumber\\ \displaystyle & = & \displaystyle d(x,g,F)-d_{{\mathcal{V}}}(x,g,F),\nonumber\end{eqnarray}$$
and so, since
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{1}$
, we conclude that
$$\begin{eqnarray}\displaystyle gF(\mathbb{R})^{+}g^{-1}x=\langle B\rangle _{\text{Zar}}.\Box & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 6.15. Assume that the weak hyperbolic Ax–Schanuel conjecture is true for
$V$
. Then, the set
$$\begin{eqnarray}\displaystyle \{gFg^{-1}:(x,g,F)\in \unicode[STIX]{x1D6F1}_{2}\} & & \displaystyle \nonumber\end{eqnarray}$$
is finite.
Proof. Decompose
$\unicode[STIX]{x1D6F1}_{2}$
as the finite union of the
$\unicode[STIX]{x1D6F1}_{F}$
, varying over the members
$F$
of
$\unicode[STIX]{x1D6FA}$
, where
$\unicode[STIX]{x1D6F1}_{F}$
denotes the set of tuples
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{2}$
. For each
$F\in \unicode[STIX]{x1D6FA}$
, consider the map
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{F}\rightarrow G(\mathbb{R})/N_{G(\mathbb{R})}(F), & & \displaystyle \nonumber\end{eqnarray}$$
defined by
$$\begin{eqnarray}\displaystyle (x,g,F)\mapsto gN_{G(\mathbb{R})}(F), & & \displaystyle \nonumber\end{eqnarray}$$
whose image, therefore, is in bijection with
$\{gFg^{-1}:(x,g,F)\in \unicode[STIX]{x1D6F1}_{2}\}$
. It is also definable and, by Lemma 6.14, it is countable. Hence, it is finite.◻
Proof of Proposition 6.10.
Let
$A$
be an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
that is pre-weakly special such that, for some
$x\in \langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}$
,
$$\begin{eqnarray}\displaystyle \dim A=\dim _{x}(\langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}). & & \displaystyle \nonumber\end{eqnarray}$$
Then, by Lemma 6.13, we can write
$$\begin{eqnarray}\displaystyle \langle A\rangle _{\text{Zar}}=gF(\mathbb{R})^{+}g^{-1}x, & & \displaystyle \nonumber\end{eqnarray}$$
where
$(x,g,F)\in \unicode[STIX]{x1D6F1}_{2}$
. By Lemma 6.14, there exists a semisimple subgroup
$F^{\prime }$
of
$G$
defined over
$\mathbb{Q}$
such that
$gFg^{-1}$
is equal to the almost direct product of the almost simple factors of
$F_{\mathbb{R}}^{\prime }$
whose underlying real Lie groups are non-compact. In fact, by [Reference UllmoUll14, Proposition 3.1],
$F^{\prime }$
is the smallest subgroup of
$G$
defined over
$\mathbb{Q}$
containing
$gFg^{-1}$
. Since, by Lemma 6.15,
$gFg^{-1}$
comes from a finite set, so too does
$F^{\prime }$
. Therefore, the reductive algebraic group
$$\begin{eqnarray}\displaystyle M:=F^{\prime }Z_{G}(F^{\prime })^{\circ } & & \displaystyle \nonumber\end{eqnarray}$$
is defined over
$\mathbb{Q}$
and belongs to a finite set.
If we write
$M^{\text{nc}}$
for the almost direct product of the almost
$\mathbb{Q}$
-simple factors of
$M$
whose underlying real Lie groups are not compact, then
$x$
factors through
$M_{\mathbb{R}}^{\prime }:=Z(M)_{\mathbb{R}}^{\circ }M_{\mathbb{R}}^{\text{nc}}$
and, if we write
$\mathfrak{X}_{M}$
for the
$M^{\prime }(\mathbb{R})$
conjugacy class of
$x$
in
$\mathfrak{X}$
, then, by [Reference UllmoUll07, Lemme 3.3],
$(M^{\prime },\mathfrak{X}_{M})$
is a Shimura subdatum of
$(G,\mathfrak{X})$
. Furthermore, by [Reference Ullmo and YafaevUY14, Lemma 3.7], the number of Shimura subdatum
$(M^{\prime },\mathfrak{Y})$
is finite. Therefore, since the
$M^{\prime }(\mathbb{R})^{+}$
conjugacy class
$X_{M}$
of
$x$
in
$X$
is a pre-special subvariety of
$X$
and
$\langle A\rangle _{\text{Zar}}$
is a fibre of
$X_{M}$
, the proof is complete.◻
Proof of Proposition 6.3.
Let
$A$
be an irreducible analytic component of
$\unicode[STIX]{x1D70B}^{-1}(W)$
. By Proposition 6.9,
$A$
is an intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
and is Zariski optimal in
$\unicode[STIX]{x1D70B}^{-1}(V)$
. Therefore, by the weak hyperbolic Ax–Schanuel conjecture,
$A$
is pre-weakly special. It follows that the image of
$\langle A\rangle _{\text{Zar}}$
in
$S$
is equal to
$\langle W\rangle _{\text{ws}}$
.
After possibly replacing
$A$
by a
$\unicode[STIX]{x1D6FE}A$
, for some
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$
, we can and do assume that there exists
$x\in \langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}$
such that
$$\begin{eqnarray}\displaystyle \dim (A)=\dim _{x}(\langle A\rangle _{\text{Zar}}\cap {\mathcal{V}}). & & \displaystyle \nonumber\end{eqnarray}$$
By Proposition 6.10,
$\langle A\rangle _{\text{Zar}}$
is a fibre of
$Y\in \unicode[STIX]{x1D6F4}$
, where
$\unicode[STIX]{x1D6F4}$
is a finite set of pre-special subvarieties of
$X$
depending only on
$V$
. ◻
7 Anomalous subvarieties
In this section, we recall the notion of an anomalous subvariety, which is defined by Bombieri, Masser and Zannier [Reference Bombieri, Masser and ZannierBMZ07] for subvarieties of algebraic tori. In fact, we give the more general notion of an
$r$
-anomalous subvariety, as introduced by Rémond [Reference RémondRém09].
Let
$V$
be a subvariety of
$S$
. We will use Proposition 6.3 to show that, under the weak hyperbolic Ax–Schanuel conjecture, the union of the subvarieties of
$V$
that are
$r$
-anomalous in
$V$
constitutes a Zariski closed subset of
$V$
. We will then give a criterion for when it is a proper subset.
Definition 7.1. Let
$r\in \mathbb{Z}$
. A subvariety
$W$
of
$V$
is called
$r$
-anomalous in
$V$
if
$$\begin{eqnarray}\displaystyle \dim W\geqslant \max \{1,r+\dim \langle W\rangle _{\text{ws}}-\dim S\}. & & \displaystyle \nonumber\end{eqnarray}$$
A subvariety of
$V$
is maximal
$r$
-anomalous in
$V$
if it is
$r$
-anomalous in
$V$
and not strictly contained in another subvariety of
$V$
that is also
$r$
-anomalous in
$V$
.
We denote by
$\text{an}(V,r)$
the set of subvarieties of
$V$
that are maximal
$r$
-anomalous in
$V$
and by
$V^{\text{an},r}$
the union of the elements of
$\text{an}(V,r)$
, which is then the union of all the subvarieties of
$V$
that are
$r$
-anomalous in
$V$
.
We say that a subvariety of
$V$
is anomalous if it is
$(1+\dim V)$
-anomalous. We write
$\text{an}(V)$
for
$\text{an}(V,1+\dim V)$
and
$V^{\text{an}}$
for
$V^{\text{an},1+\dim V}$
.
Theorem 7.2. Assume that the weak hyperbolic Ax–Schanuel conjecture is true for
$V$
and let
$r\in \mathbb{Z}$
. Then
$V^{\text{an},r}$
is a Zariski closed subset of
$V$
.
We refer the reader to [Reference Bombieri, Masser and ZannierBMZ07, Reference RémondRém09, Reference Habegger and PilaHP16] for similar results on algebraic tori and abelian varieties. We will require the following facts.
Proposition 7.3 (Cf. [Reference HartshorneHar77, ch. 2, Exercise 3.22(d)]).
Let
$f:W\rightarrow Y$
be a dominant morphism between two integral schemes of finite type over a field and let
$$\begin{eqnarray}\displaystyle e:=\dim W-\dim Y & & \displaystyle \nonumber\end{eqnarray}$$
denote the relative dimension. For
$h\in \mathbb{Z}$
, let
$E_{h}$
denote the set of points
$x\in W$
such that the fibre
$f^{-1}(f(x))$
possesses an irreducible component of dimension at least
$h$
that contains
$x$
. Then we have the following.
(1)
$E_{h}$
is a Zariski closed subset of
$W$
.(2)
$E_{e}=W$
.(3) If
$h>e$
,
$E_{h}$
is not Zariski dense in
$W$
.
Lemma 7.4. Let
$W\in \text{an}(V,r)$
. Then
$W$
is weakly optimal in
$V$
.
Proof. Let
$Y$
be a subvariety of
$V$
containing
$W$
such that
$\unicode[STIX]{x1D6FF}_{\text{ws}}(Y)\leqslant \unicode[STIX]{x1D6FF}_{\text{ws}}(W)$
. We can and do assume that
$Y$
is weakly optimal. Then
$$\begin{eqnarray}\displaystyle \dim Y & = & \displaystyle \dim \langle Y\rangle _{\text{ws}}-\unicode[STIX]{x1D6FF}_{\text{ws}}(Y)\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \dim \langle Y\rangle _{\text{ws}}-\unicode[STIX]{x1D6FF}_{\text{ws}}(W)\nonumber\\ \displaystyle & = & \displaystyle \dim \langle Y\rangle _{\text{ws}}-(\dim \langle W\rangle _{\text{ws}}-\dim W)\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \dim \langle Y\rangle _{\text{ws}}+r-\dim S.\nonumber\end{eqnarray}$$
Since
$Y$
contains
$W$
, we know that
$\dim Y\geqslant 1$
, and so
$Y$
is
$r$
-anomalous in
$V$
. Since
$W$
is maximal
$r$
-anomalous in
$V$
, we conclude that
$Y$
must be equal to
$W$
. Therefore,
$W$
is weakly optimal.◻
Proof of Theorem 7.2.
Let
$\unicode[STIX]{x1D6F4}$
be a finite set of pre-special subvarieties of
$X$
(whose existence is ensured by Proposition 6.3) such that, if
$W$
is a subvariety of
$V$
that is weakly optimal in
$V$
, then there exists
$x\in X$
such that, if
$M:=\text{MT}(x)$
, the
$M(\mathbb{R})^{+}$
conjugacy class
$X_{M}$
of
$x$
in
$X$
belongs to
$\unicode[STIX]{x1D6F4}$
and
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of a fibre of
$X_{M}$
. That is, we may write
$M^{\text{ad}}$
as a product
$M_{1}\times M_{2}$
of two normal
$\mathbb{Q}$
-subgroups, which induces a splitting
$X=X_{1}\times X_{2}$
, such that
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of
$\{x_{1}\}\times X_{2}$
, for some
$x_{1}\in X_{1}$
.
Let
$W\in \text{an}(V,r)$
. By Lemma 7.4, there exists
$X_{M}\in \unicode[STIX]{x1D6F4}$
such that
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of
$\{x_{1}\}\times X_{2}$
, for some
$x_{1}\in X_{1}$
, where
$X_{M}=X_{1}\times X_{2}$
, as above.
Let
$\unicode[STIX]{x1D6E4}_{M}$
be a congruence subgroup of
$M(\mathbb{Q})_{+}$
contained in
$\unicode[STIX]{x1D6E4}$
, where
$M(\mathbb{Q})_{+}$
denotes the subgroup of
$M(\mathbb{Q})$
acting on
$X_{M}$
, and let
$\unicode[STIX]{x1D6E4}_{1}$
denote the image of
$\unicode[STIX]{x1D6E4}$
under the natural maps
$$\begin{eqnarray}\displaystyle M(\mathbb{Q})\rightarrow M^{\text{ad}}(\mathbb{Q})\rightarrow M_{1}(\mathbb{Q}). & & \displaystyle \nonumber\end{eqnarray}$$
We denote by
$f$
the restriction of
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{M}\backslash X_{M}\rightarrow \unicode[STIX]{x1D6E4}_{1}\backslash X_{1} & & \displaystyle \nonumber\end{eqnarray}$$
to an irreducible component
$\widetilde{V}$
of
$\unicode[STIX]{x1D719}^{-1}(V)$
, such that
$\dim \widetilde{V}=\dim V$
, where
$\unicode[STIX]{x1D719}$
denotes the natural map
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{M}\backslash X_{M}\rightarrow \unicode[STIX]{x1D6E4}\backslash X=S. & & \displaystyle \nonumber\end{eqnarray}$$
In particular,
$\unicode[STIX]{x1D719}(\widetilde{V})=V$
. Therefore, by Proposition 7.3(1), the set
$E_{h}$
of points
$z$
in
$\widetilde{V}$
such that the fibre
$f^{-1}(f(z))$
possesses an irreducible component of dimension at least
$h\in \mathbb{Z}$
that contains
$z$
is a Zariski closed subset of
$\widetilde{V}$
. Since
$\unicode[STIX]{x1D719}$
is a closed morphism,
$\unicode[STIX]{x1D719}(E_{h})$
is Zariski closed in
$V$
.
We claim that
$W$
is contained in
$\unicode[STIX]{x1D719}(E_{h})$
, where
$$\begin{eqnarray}\displaystyle h:=\max \{1,r+\dim X_{2}-\dim S\}. & & \displaystyle \nonumber\end{eqnarray}$$
To see this, fix an irreducible component
$\widetilde{W}$
of
$\unicode[STIX]{x1D719}^{-1}(W)$
contained in
$\widetilde{V}$
such that
$\dim \widetilde{W}=\dim W$
. Then
$\langle \widetilde{W}\rangle _{\text{ws}}$
is equal to the image of
$\{x_{1}\}\times X_{2}$
in
$\unicode[STIX]{x1D6E4}_{M}\backslash X_{M}$
and so
$\widetilde{W}$
lies in a fibre of
$f$
. Since
$$\begin{eqnarray}\displaystyle \dim \widetilde{W}=\dim W\geqslant \max \{1,r+\dim \langle W\rangle _{\text{ws}}-\dim S\}=\max \{1,r+\dim X_{2}-\dim S\}, & & \displaystyle \nonumber\end{eqnarray}$$
$\widetilde{W}$
is contained in
$E_{h}$
, which implies that
$W$
is contained in
$\unicode[STIX]{x1D719}(E_{h})$
.
On the other hand, we claim that
$\unicode[STIX]{x1D719}(E_{h})$
is contained in
$V^{\text{an},r}$
. To see this, let
$z\in E_{h}$
and let
$Y$
be an irreducible component of the fibre
$f^{-1}(f(z))$
of dimension at least
$h$
containing
$z$
. Then
$Y$
is contained in the image of
$\{x_{1}\}\times X_{2}$
in
$\unicode[STIX]{x1D6E4}_{M}\backslash X_{M}$
, where
$x_{1}\in X_{1}$
lies above
$f(z)\in \unicode[STIX]{x1D6E4}_{1}\backslash X_{1}$
, and so
$$\begin{eqnarray}\displaystyle \dim \langle Y\rangle _{\text{ws}}\leqslant \dim X_{2}. & & \displaystyle \nonumber\end{eqnarray}$$
Therefore,
$$\begin{eqnarray}\displaystyle \dim \unicode[STIX]{x1D719}(Y)=\dim Y\geqslant h=\max \{1,r+\dim X_{2}-\dim S\}\geqslant \max \{1,r+\dim \langle \unicode[STIX]{x1D719}(Y)\rangle _{\text{ws}}-\dim S\} & & \displaystyle \nonumber\end{eqnarray}$$
and so
$\unicode[STIX]{x1D719}(Y)$
is
$r$
-anomalous in
$V$
.
Hence, if we let
$E$
denote the union of the
$\unicode[STIX]{x1D719}(E_{h})$
as we vary over the finitely many maps
$f$
obtained from the
$X_{M}\in \unicode[STIX]{x1D6F4}$
and their possible splittings, we conclude that
$E=V^{\text{an},r}$
, which finishes the proof.◻
We denote by
$V^{\text{oa}}$
the complement in
$V$
of
$V^{\text{an}}$
. By Theorem 7.2, this is a (possibly empty) open subset of
$V$
. In the literature, it is sometimes referred to as the open-anomalous locus, hence the subscript. We conclude this section by showing that, when
$V$
is sufficiently generic,
$V^{\text{oa}}$
is not empty.
Proposition 7.5. Suppose that
$V$
is Hodge generic in
$S$
. Then
$V^{\text{an}}=V$
if and only if we can write
$G^{\text{ad}}=G_{1}\times G_{2}$
, and thus
$X=X_{1}\times X_{2}$
, such that
$$\begin{eqnarray}\displaystyle \dim f(V)<\text{min}\{\dim V,\dim X_{1}\}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$f$
denotes the projection map
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}\backslash X\rightarrow \unicode[STIX]{x1D6E4}_{1}\backslash X_{1}, & & \displaystyle \nonumber\end{eqnarray}$$
and
$\unicode[STIX]{x1D6E4}_{1}$
denotes the image of
$\unicode[STIX]{x1D6E4}$
under the natural maps
$$\begin{eqnarray}\displaystyle G(\mathbb{Q})\rightarrow G^{\text{ad}}(\mathbb{Q})\rightarrow G_{1}(\mathbb{Q}). & & \displaystyle \nonumber\end{eqnarray}$$
Proof. First suppose that
$V^{\text{an}}=V$
. Then, for any set
$\unicode[STIX]{x1D6F4}$
as in the proof of Theorem 7.2,
$V$
is contained in the (finite) union of the images in
$S$
of the
$X_{M}\in \unicode[STIX]{x1D6F4}$
. Therefore, since
$V$
is assumed to be Hodge generic in
$S$
, it must be that
$X\in \unicode[STIX]{x1D6F4}$
and, furthermore, that there exists
$W\in \text{an}(V)$
such that
$G^{\text{ad}}=G_{1}\times G_{2}$
, and thus
$X=X_{1}\times X_{2}$
, such that
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of
$\{x_{1}\}\times X_{2}$
, for some
$x_{1}\in X_{1}$
.
Let
$f$
denote the projection map
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}\backslash X\rightarrow \unicode[STIX]{x1D6E4}_{1}\backslash X_{1} & & \displaystyle \nonumber\end{eqnarray}$$
and consider its restriction
$$\begin{eqnarray}\displaystyle V\rightarrow \overline{f(V)}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$\overline{f(V)}$
denotes the Zariski closure of
$f(V)$
in
$\unicode[STIX]{x1D6E4}_{1}\backslash X_{1}$
. Since
$V^{\text{an}}=V$
, it follows from Proposition 7.3(3), that
$$\begin{eqnarray}\displaystyle h:=\max \{1,1+\dim V+\dim X_{2}-\dim X\}\leqslant \dim V-\dim f(V). & & \displaystyle \nonumber\end{eqnarray}$$
Hence,
$$\begin{eqnarray}\displaystyle \dim f(V)<\dim X-\dim X_{2}=\dim X_{1} & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \dim f(V)\leqslant \dim V-h\leqslant \dim V-1<\dim V. & & \displaystyle \nonumber\end{eqnarray}$$
Conversely, suppose that
$G^{\text{ad}}=G_{1}\times G_{2}$
, and thus
$X=X_{1}\times X_{2}$
, such that
$$\begin{eqnarray}\displaystyle \dim f(V)<\text{min}\{\dim V,\dim X_{1}\}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$f$
again denotes the projection map
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}\backslash X\rightarrow \unicode[STIX]{x1D6E4}_{1}\backslash X_{1}. & & \displaystyle \nonumber\end{eqnarray}$$
Restricting
$f$
to
$$\begin{eqnarray}\displaystyle V\rightarrow \overline{f(V)}, & & \displaystyle \nonumber\end{eqnarray}$$
as before, we see from Proposition 7.3(2) that the set
$E_{h}$
of points
$z$
in
$V$
such that the fibre
$f^{-1}(f(z))$
possesses an irreducible component of dimension at least
$$\begin{eqnarray}\displaystyle h:=\max \{1,1+\dim V-\dim X_{1}\}\leqslant \dim V-\dim f(V)=\dim V-\dim \overline{f(V)} & & \displaystyle \nonumber\end{eqnarray}$$
that contains
$z$
is equal to
$V$
. However, from the proof of Theorem 7.2, we have seen that
$E_{h}$
is contained in
$V^{\text{an}}$
, so the claim follows.◻
Corollary 7.6. If
$G^{\text{ad}}$
is
$\mathbb{Q}$
-simple and
$V$
is a Hodge generic subvariety in
$S$
, then
$V^{\text{an}}$
is strictly contained in
$V$
. In particular,
$V^{\text{an}}$
is strictly contained in
$V$
whenever
$V$
is a Hodge generic subvariety of
${\mathcal{A}}_{g}$
.
8 Main results (part 1): Reductions to point counting
In this section, we prove our main theorem: under the weak hyperbolic Ax–Schanuel conjecture, the Zilber–Pink conjecture can be reduced to a problem of point counting. We also give a reduction of Pink’s conjecture in the case when the open-anomalous locus is non-empty.
Definition 8.1. Let
$V$
be a subvariety of
$S$
. We denote by
$\text{Opt}_{0}(V)$
the set of all points in
$V$
that are optimal in
$V$
.
Consider the following corollary of the Zilber–Pink conjecture.
Conjecture 8.2. Let
$V$
be a subvariety of
$S$
. Then
$\text{Opt}_{0}(V)$
is finite.
We will later show that, under certain arithmetic hypotheses, one can prove Conjecture 8.2 when
$V$
is a curve. Our main result in this section is that (under the weak hyperbolic Ax–Schanuel conjecture), Conjecture 8.2 implies the Zilber–Pink conjecture.
Theorem 8.3. Assume that the weak hyperbolic Ax–Schanuel conjecture is true and assume that Conjecture 8.2 holds.
Let
$V$
be a subvariety of
$S$
. Then
$\text{Opt}(V)$
is finite.
Proof. We prove Theorem 8.3 by induction on
$\dim V$
. Of course, Theorem 8.3 is trivial when
$\dim V=0$
or
$\dim V=1$
. Therefore, we assume that
$\dim V\geqslant 2$
and that Theorem 8.3 holds whenever the subvariety in question is of lower dimension.
We need to show that the induction hypothesis implies that there are only finitely many subvarieties of positive dimension belonging to
$\text{Opt}(V)$
.
Let
$\unicode[STIX]{x1D6F4}$
be a finite set of pre-special subvarieties of
$X$
, as in the proof of Theorem 7.2, and let
$W\in \text{Opt}(V)$
be of positive dimension.
By Corollary 4.5,
$W$
is weakly optimal and, therefore, there exists
$x\in X$
such that, if
$M:=\text{MT}(x)$
, the
$M(\mathbb{R})^{+}$
conjugacy class
$X_{M}$
of
$x$
in
$X$
belongs to
$\unicode[STIX]{x1D6F4}$
and
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of a fibre of
$X_{M}$
. That is, we may write
$M^{\text{ad}}$
as a product
$$\begin{eqnarray}\displaystyle M^{\text{ad}}=M_{1}\times M_{2} & & \displaystyle \nonumber\end{eqnarray}$$
of two normal
$\mathbb{Q}$
-subgroups, thus inducing a splitting
$$\begin{eqnarray}\displaystyle X_{M}=X_{1}\times X_{2}, & & \displaystyle \nonumber\end{eqnarray}$$
such that
$\langle W\rangle _{\text{ws}}$
is equal to the image in
$S$
of
$\{x_{1}\}\times X_{2}$
, for some
$x_{1}\in X_{1}$
.
Let
$\unicode[STIX]{x1D6E4}_{M}$
be a congruence subgroup of
$M(\mathbb{Q})_{+}$
contained in
$\unicode[STIX]{x1D6E4}$
, where
$M(\mathbb{Q})_{+}$
denotes the subgroup of
$M(\mathbb{Q})$
acting on
$X_{M}$
, such that the image of
$\unicode[STIX]{x1D6E4}_{M}$
under the natural map
$$\begin{eqnarray}\displaystyle M(\mathbb{Q})\rightarrow M^{\text{ad}}(\mathbb{Q})=M_{1}(\mathbb{Q})\times M_{2}(\mathbb{Q}) & & \displaystyle \nonumber\end{eqnarray}$$
is equal to a product
$\unicode[STIX]{x1D6E4}_{1}\times \unicode[STIX]{x1D6E4}_{2}$
. We denote by
$f$
the natural morphism
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{M}\backslash X_{M}\rightarrow \unicode[STIX]{x1D6E4}_{1}\backslash X_{1}, & & \displaystyle \nonumber\end{eqnarray}$$
and by
$\unicode[STIX]{x1D719}$
the finite morphism
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{M}\backslash X_{M}\rightarrow \unicode[STIX]{x1D6E4}\backslash X=S. & & \displaystyle \nonumber\end{eqnarray}$$
Let
$\widetilde{V}$
be an irreducible component of
$\unicode[STIX]{x1D719}^{-1}(V)$
such that
$\dim \widetilde{V}=\dim V$
, and let
$\widetilde{W}$
denote an irreducible component of
$\unicode[STIX]{x1D719}^{-1}(W)$
contained in
$\widetilde{V}$
such that
$\dim \widetilde{W}=\dim W$
. Then
$\widetilde{W}$
is optimal in
$\widetilde{V}$
. On the other hand, by the generic smoothness property, there exists a dense open subset
$V_{0}$
of
$\widetilde{V}$
such that the restriction
$f_{0}$
of
$f$
to
$V_{0}$
is a smooth morphism of relative dimension
$\unicode[STIX]{x1D708}$
. We denote by
$V_{1}$
the Zariski closure of
$f(V_{0})$
in
$\unicode[STIX]{x1D6E4}_{1}\backslash X_{1}$
.
Now suppose that
$$\begin{eqnarray}\displaystyle \widetilde{W}\cap V_{0}=\emptyset . & & \displaystyle\end{eqnarray}$$
Then
$\widetilde{W}$
is a subvariety of some irreducible component
$V^{0}$
of
$\widetilde{V}\setminus V_{0}$
. Furthermore,
$\widetilde{W}$
is optimal in
$V^{0}$
. However, since
$\dim V^{0}$
is strictly less than
$\dim V$
, our induction hypothesis implies that
$\text{Opt}(V^{0})$
is finite.
Therefore, we assume that (8.3.1) does not hold. As an irreducible component of the fibre
$f_{0}^{-1}(z)$
, where
$z$
denotes the image of
$x_{1}$
in
$V_{1}$
, its dimension is equal to
$\unicode[STIX]{x1D708}$
. In particular,
$$\begin{eqnarray}\displaystyle \dim \widetilde{W}=\unicode[STIX]{x1D708}. & & \displaystyle \nonumber\end{eqnarray}$$
We claim that
$z$
is optimal in
$V_{1}$
. To see this, note that
$f(\langle \widetilde{W}\rangle )$
contains
$z$
and is a special subvariety of dimension
$$\begin{eqnarray}\displaystyle \dim \langle \widetilde{W}\rangle -\dim X_{2}=\dim \widetilde{W}+\unicode[STIX]{x1D6FF}(\widetilde{W})-\dim X_{2}=\unicode[STIX]{x1D708}+\unicode[STIX]{x1D6FF}(\widetilde{W})-\dim X_{2}. & & \displaystyle \nonumber\end{eqnarray}$$
Therefore, let
$A$
be a subvariety of
$V_{1}$
containing
$z$
such that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}(A)\leqslant \unicode[STIX]{x1D6FF}(z)\leqslant \unicode[STIX]{x1D708}+\unicode[STIX]{x1D6FF}(\widetilde{W})-\dim X_{2}, & & \displaystyle \nonumber\end{eqnarray}$$
and let
$B$
be an irreducible component of
$f^{-1}(A)$
containing
$\widetilde{W}$
. Since
$V_{0}$
is open in
$\widetilde{V}$
and
$A$
is contained in
$V_{1}$
,
$$\begin{eqnarray}\displaystyle \dim B=\dim A+\unicode[STIX]{x1D708}. & & \displaystyle \nonumber\end{eqnarray}$$
Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}(B)\leqslant \dim \langle A\rangle +\dim X_{2}-\dim B=\unicode[STIX]{x1D6FF}(A)+\dim A+\dim X_{2}-\dim B\leqslant \unicode[STIX]{x1D6FF}(\widetilde{W}) & & \displaystyle \nonumber\end{eqnarray}$$
and, since
$\widetilde{W}$
is optimal in
$\widetilde{V}$
, we conclude that
$B$
is equal to
$\widetilde{W}$
. In particular,
$\widetilde{W}$
is an irreducible component of
$f^{-1}(A)$
but, since it is also contained in
$f^{-1}(z)$
, it must be that
$A$
is equal to
$z$
, proving the claim.
Since
$W$
was assumed to be of positive dimension, so too must be
$X_{2}$
. It follows that
$\dim V_{1}$
is strictly less than
$\dim V$
and so, by the induction hypothesis,
$\text{Opt}(V_{1})$
is finite. Since
$z\in \text{Opt}(V_{1})$
and since
$\unicode[STIX]{x1D6F4}$
and the number of splittings are finite, we are done.◻
We will later prove that the following conjecture is a consequence of the weak hyperbolic Ax–Schanuel conjecture and our arithmetic conjectures. It is inspired by the cited theorem of Habegger and Pila.
Conjecture 8.4 (Cf. [Reference Habegger and PilaHP16, Theorem 9.15(iii)]).
Let
$V$
be a subvariety of
$S$
. Then the set
$V^{\text{oa}}\cap S^{[1+\dim V]}$
is finite.
The importance of Conjecture 8.4 for us is that, when
$V$
is suitably generic, Conjecture 8.4 implies Pink’s conjecture (assuming the weak hyperbolic Ax–Schanuel conjecture).
Theorem 8.5. Assume that the weak hyperbolic Ax–Schanuel conjecture is true and that Conjecture 8.4 holds.
Let
$V$
be a Hodge generic subvariety of
$S$
such that (even after replacing
$\unicode[STIX]{x1D6E4}$
)
$S$
cannot be decomposed as a product
$S_{1}\times S_{2}$
such that
$V$
is contained in
$V^{\prime }\times S_{2}$
, where
$V^{\prime }$
is a proper subvariety of
$S_{1}$
of dimension strictly less than the dimension of
$V$
. Then
$$\begin{eqnarray}\displaystyle V\cap S^{[1+\dim V]} & & \displaystyle \nonumber\end{eqnarray}$$
is not Zariski dense in
$V$
.
Proof. We claim that the assumptions guarantee that
$V^{\text{an}}$
is strictly contained in
$V$
. Otherwise, by Proposition 7.5, we can write
$G^{\text{ad}}=G_{1}\times G_{2}$
, and thus
$X=X_{1}\times X_{2}$
, such that
$$\begin{eqnarray}\displaystyle \dim f(V)<\text{min}\{\dim V,\dim X_{1}\}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$f$
denotes the projection map
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}\backslash X\rightarrow \unicode[STIX]{x1D6E4}_{1}\backslash X_{1}, & & \displaystyle \nonumber\end{eqnarray}$$
and
$\unicode[STIX]{x1D6E4}_{1}$
denotes the image of
$\unicode[STIX]{x1D6E4}$
under the natural maps
$$\begin{eqnarray}\displaystyle G(\mathbb{Q})\rightarrow G^{\text{ad}}(\mathbb{Q})\rightarrow G_{1}(\mathbb{Q}). & & \displaystyle \nonumber\end{eqnarray}$$
Therefore, after replacing
$\unicode[STIX]{x1D6E4}$
, we can write
$S$
as a product
$S_{1}\,\times \,S_{2}$
so that
$f$
is simply the projection on to the first factor and
$V$
is contained in
$V^{\prime }\times S_{2}$
, where
$V^{\prime }$
is Zariski closure of
$f(V)$
in
$S_{1}$
. However, since
$$\begin{eqnarray}\displaystyle \dim V^{\prime }=\dim f(V), & & \displaystyle \nonumber\end{eqnarray}$$
this is a contradiction.
Therefore, by Theorem 7.2,
$V^{\text{an}}$
is a proper Zariski closed subset of
$V$
. On the other hand,
$V\cap S^{[1+\dim V]}$
is contained in
$$\begin{eqnarray}\displaystyle V^{\text{an}}\cup [V^{\text{oa}}\cap S^{[1+\dim V]}] & & \displaystyle \nonumber\end{eqnarray}$$
and so the theorem follows from Conjecture 8.4. ◻
9 The counting theorem
Henceforth, we turn our attention to the counting problems themselves. We will approach these problems using a theorem of Pila and Wilkie concerned with counting points in definable sets. We first recall the notations.
Let
$k\geqslant 1$
be an integer. For any real number
$y$
, we define its
$k$
-height as
$$\begin{eqnarray}\displaystyle \text{H}_{k}(y):=\min \{\max \{|a_{0}|,\ldots ,|a_{k}|\}:a_{i}\in \mathbb{Z},\text{gcd}\{a_{0},\ldots ,a_{k}\}=1,a_{0}y^{k}+\cdots +a_{k}=0\}, & & \displaystyle \nonumber\end{eqnarray}$$
where we use the convention that, if the set is empty, i.e.,
$y$
is not algebraic of degree at most
$k$
, then
$\text{H}_{k}(y)$
is
$+\infty$
. For
$y=(y_{1},\ldots ,y_{m})\in \mathbb{R}^{m}$
, we set
$$\begin{eqnarray}\displaystyle \text{H}_{k}(y):=\max \{\text{H}_{k}(y_{1}),\ldots ,\text{H}_{k}(y_{m})\}. & & \displaystyle \nonumber\end{eqnarray}$$
For any set
$A\subseteq \mathbb{R}^{m}\times \mathbb{R}^{n}$
, and for any real number
$T\geqslant 1$
, we define
$$\begin{eqnarray}\displaystyle A(k,T):=\{(y,z)\in Z:\text{H}_{k}(y)\leqslant T\}. & & \displaystyle \nonumber\end{eqnarray}$$
The counting theorem of Pila and Wilkie is stated as follows.
Theorem 9.1 (Cf. the proof of [Reference Habegger and PilaHP16, Corollary 7.2]).
Let
$D\subseteq \mathbb{R}^{l}\times \mathbb{R}^{m}\times \mathbb{R}^{n}$
be a definable family parametrized by
$\mathbb{R}^{l}$
, let
$k$
be a positive integer, and let
$\unicode[STIX]{x1D716}>0$
. There exists a constant
$c:=c(D,k,\unicode[STIX]{x1D716})>0$
with the following properties.
Let
$x\in \mathbb{R}^{l}$
and let
$$\begin{eqnarray}\displaystyle D_{x}:=\{(y,z)\in \mathbb{R}^{m}\times \mathbb{R}^{n}:(x,y,z)\in D\}. & & \displaystyle \nonumber\end{eqnarray}$$
Let
$\unicode[STIX]{x1D70B}_{1}$
and
$\unicode[STIX]{x1D70B}_{2}$
denote the projections
$\mathbb{R}^{m}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$
and
$\mathbb{R}^{m}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$
, respectively. If
$T\geqslant 1$
and
$\unicode[STIX]{x1D6F4}\subseteq D_{x}(k,T)$
satisfies
$$\begin{eqnarray}\displaystyle \#\unicode[STIX]{x1D70B}_{2}(\unicode[STIX]{x1D6F4})>cT^{\unicode[STIX]{x1D716}}, & & \displaystyle \nonumber\end{eqnarray}$$
there exists a continuous and definable function
$\unicode[STIX]{x1D6FD}:[0,1]\rightarrow D_{x}$
such that the following properties hold.
(1) The composition
$\unicode[STIX]{x1D70B}_{1}\circ \unicode[STIX]{x1D6FD}:[0,1]\rightarrow \mathbb{R}^{m}$
is semialgebraic.(2) The composition
$\unicode[STIX]{x1D70B}_{2}\circ \unicode[STIX]{x1D6FD}:[0,1]\rightarrow \mathbb{R}^{n}$
is non-constant.(3) We have
$\unicode[STIX]{x1D6FD}(0)\in \unicode[STIX]{x1D6F4}$
.(4) The restriction
$\unicode[STIX]{x1D6FD}_{|(0,1)}$
is real analytic.
Note that, although the conclusion
$\unicode[STIX]{x1D6FD}(0)\in \unicode[STIX]{x1D6F4}$
does not appear in the statement of [Reference Habegger and PilaHP16, Corollary 7.2], it is, indeed, established in its proof. The final property holds because
$\mathbb{R}_{\text{an},\text{exp}}$
admits analytic cell decomposition (see [Reference van den Dries and MillerVdDM94]).
10 Complexity
In order to apply the counting theorem, we will need a way of counting special points and, more generally, special subvarieties. Recall that
$S$
is a connected component of the Shimura variety
$\text{Sh}_{K}(G,\mathfrak{X})$
defined by the Shimura datum
$(G,X)$
and the compact open subgroup
$K$
of
$G(\mathbb{A}_{f})$
.
Let
$P$
be a special point in
$S$
and let
$x\in X$
be a pre-special point lying above
$P$
. In particular,
$T:=\text{MT}(x)$
is a torus and we denote by
$D_{T}$
the absolute value of the discriminant of its splitting field. We let
$K_{T}^{m}$
denote the maximal compact open subgroup of
$T(\mathbb{A}_{f})$
and we let
$K_{T}\subseteq K_{T}^{m}$
denote
$K\cap T(\mathbb{A}_{f})$
.
Definition 10.1. The complexity of
$P$
is the natural number
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}(P):=\max \{D_{T},[K_{T}^{m}:K_{T}]\}. & & \displaystyle \nonumber\end{eqnarray}$$
Note that this does not depend on the choice of
$x$
.
Now let
$Z$
be a special subvariety of
$S$
. There exists a Shimura subdatum
$(H,\mathfrak{X}_{H})$
of
$(G,\mathfrak{X})$
, such that
$H$
is the generic Mumford–Tate group on
$\mathfrak{X}_{H}$
, and a connected component
$X_{H}$
of
$\mathfrak{X}_{H}$
contained in
$X$
such that
$Z$
is the image of
$X_{H}$
in
$\unicode[STIX]{x1D6E4}\backslash X$
. In fact, these choices are well defined up to conjugation by
$\unicode[STIX]{x1D6E4}$
.
By the degree
$\deg (Z)$
of
$Z$
, we refer to the degree of the Zariski closure of
$Z$
in the Baily–Borel compactification of
$S$
, defined in [Reference Baily and BorelBB66], which is naturally a projective variety.
Definition 10.2. The complexity of
$Z$
is the natural number
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}(Z):=\max \{\deg (Z),\text{min}\{\unicode[STIX]{x1D6E5}(P):P\in Z\text{ is a special point}\}\hspace{-1.5pt}\}. & & \displaystyle \nonumber\end{eqnarray}$$
Note that when
$Z$
is a special point, this complexity coincides with the former.
This is a natural generalization of the complexities given in [Reference Habegger and PilaHP16, Definitions 3.4 and 3.8]. In order to count special subvarieties, however, it is crucial that the complexity of
$Z$
satisfies the following property.
Conjecture 10.3. For any
$b\geqslant 1$
, we have
$$\begin{eqnarray}\displaystyle \#\{Z\subseteq S:Z\text{ is special and }\unicode[STIX]{x1D6E5}(Z)\leqslant b\}<\infty . & & \displaystyle \nonumber\end{eqnarray}$$
The obstruction to proving that this property holds for a general Shimura variety can be expressed as follows.
Conjecture 10.4. For any
$b\geqslant 1$
, there exists a finite set
$\unicode[STIX]{x1D6FA}$
of semisimple subgroups of
$G$
defined over
$\mathbb{Q}$
such that, if
$Z$
is a special subvariety of
$S$
, and
$\deg (Z)\leqslant b$
, then
$$\begin{eqnarray}\displaystyle H^{\text{der}}=\unicode[STIX]{x1D6FE}F\unicode[STIX]{x1D6FE}^{-1}, & & \displaystyle \nonumber\end{eqnarray}$$
for some
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$
and some
$F\in \unicode[STIX]{x1D6FA}$
.
We will later verify Conjecture 10.4 for a product of modular curves.
Proof that Conjecture 10.4 implies Conjecture 10.3.
By Conjecture 10.4, there exists a finite set
$\unicode[STIX]{x1D6FA}$
of semisimple subgroups of
$G$
defined over
$\mathbb{Q}$
, independent of
$Z$
, such that
$$\begin{eqnarray}\displaystyle H^{\text{der}}=\unicode[STIX]{x1D6FE}F\unicode[STIX]{x1D6FE}^{-1}, & & \displaystyle \nonumber\end{eqnarray}$$
for some
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$
and some
$F\in \unicode[STIX]{x1D6FA}$
.
Let
$P\in Z$
be a special point such that
$\unicode[STIX]{x1D6E5}(P)$
is minimal among all special points in
$Z$
and let
$x\in X$
be a point lying above
$P$
such that
$\text{MT}(x)$
is contained in
$H$
. Therefore,
$Z$
is equal to the image of
$F(\mathbb{R})^{+}\unicode[STIX]{x1D6FE}^{-1}x$
in
$\unicode[STIX]{x1D6E4}\backslash X$
. Furthermore,
$\text{MT}(\unicode[STIX]{x1D6FE}^{-1}x)$
is contained in
$$\begin{eqnarray}\displaystyle G_{F}:=FZ_{G}(F)^{\circ } & & \displaystyle \nonumber\end{eqnarray}$$
and, by [Reference UllmoUll07, Lemme 3.3], if we denote by
$\mathfrak{X}^{\prime }$
the
$G_{F}(\mathbb{R})$
conjugacy class of
$\unicode[STIX]{x1D6FE}^{-1}x$
, we obtain a Shimura subdatum
$(G_{F},\mathfrak{X}^{\prime })$
of
$(G,\mathfrak{X})$
.
Therefore, let
$X^{\prime }$
denote the connected component
$G_{F}(\mathbb{R})^{+}\unicode[STIX]{x1D6FE}^{-1}x$
of
$\mathfrak{X}^{\prime }$
and let
$\unicode[STIX]{x1D6E4}^{\prime }$
denote
$\unicode[STIX]{x1D6E4}\,\cap \,G_{F}(\mathbb{Q})_{+}$
, where
$G_{F}(\mathbb{Q})_{+}$
denotes the subgroup of
$G_{F}(\mathbb{Q})$
acting on
$X^{\prime }$
. By [Reference Ullmo and YafaevUY14, Proposition 3.21] and its proof, there exist only finitely many
$\unicode[STIX]{x1D6E4}^{\prime }$
orbits of pre-special points in
$X^{\prime }$
whose image in
$\unicode[STIX]{x1D6E4}^{\prime }\backslash X^{\prime }$
has complexity at most
$b$
. Therefore, there exists
$\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6E4}^{\prime }$
such that
$\unicode[STIX]{x1D6FE}^{-1}x=\unicode[STIX]{x1D706}y$
, where
$y\in X^{\prime }$
belongs to a finite set. We conclude that
$Z$
is equal to the image of
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}F(\mathbb{R})^{+}\unicode[STIX]{x1D6FE}^{-1}x=\unicode[STIX]{x1D6E4}F(\mathbb{R})^{+}\unicode[STIX]{x1D706}y=\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D706}F(\mathbb{R})^{+}y=\unicode[STIX]{x1D6E4}F(\mathbb{R})^{+}y & & \displaystyle \nonumber\end{eqnarray}$$
in
$\unicode[STIX]{x1D6E4}\backslash X$
, which concludes the proof.◻
11 Galois orbits
In [Reference Habegger and PilaHP16], Habegger and Pila formulated a conjecture about Galois orbits of optimal points in
$\mathbb{C}^{n}$
that in [Reference Habegger and PilaHab12] they had been able to prove for so-called asymmetric curves. In [Reference OrrOrr17], Orr generalized the result to asymmetric curves in
${\mathcal{A}}_{g}^{2}$
.
Recall that
$\text{Sh}_{K}(G,\mathfrak{X})$
possesses a canonical model, defined over a number field
$E$
, which depends only on
$(G,\mathfrak{X})$
. Furthermore,
$S$
is defined over a finite abelian extension
$F$
of
$E$
. In particular, for any extension
$L$
of
$F$
contained in
$\mathbb{C}$
, it makes sense to say that a subvariety
$V$
of
$S$
is defined over
$L$
. Moreover, if
$V$
is such a subvariety, then
$\text{Aut}(\mathbb{C}/L)$
acts on the points of
$V$
.
If
$Z$
is a special subvariety of
$S$
and
$\unicode[STIX]{x1D70E}\in \text{Aut}(\mathbb{C}/F)$
, then
$\unicode[STIX]{x1D70E}(Z)$
is also a special subvariety of
$S$
and its complexity is also
$\unicode[STIX]{x1D6E5}(Z)$
. In particular, if
$V$
is a subvariety of
$S$
, as above, then
$\text{Aut}(\mathbb{C}/L)$
acts on
$\text{Opt}(V)$
and its orbits are finite.
Conjecture 11.1 (Large Galois orbits).
Let
$V$
be a subvariety of
$S$
, defined over a finitely generated extension
$L$
of
$F$
contained in
$\mathbb{C}$
. There exist positive constants
$c_{G}$
and
$\unicode[STIX]{x1D6FF}_{G}$
such that the following holds.
If
$P\in \text{Opt}_{0}(V)$
, then
$$\begin{eqnarray}\displaystyle \#\text{Aut}(\mathbb{C}/L)\cdot P\geqslant c_{G}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D6FF}_{G}}. & & \displaystyle \nonumber\end{eqnarray}$$
Remark 11.2. In the context of the André–Oort conjecture, there is the pioneering hypothesis that Galois orbits of special points should be large. See [Reference Edixhoven, Moonen and OortEMO01, Problem 14] for the formulation for special points in
${\mathcal{A}}_{g}$
and see [Reference YafaevYaf06, Theorem 2.1] for special points in a general Shimura variety. This hypothesis, which was verified by Tsimerman for special points of
${\mathcal{A}}_{g}$
[Reference TsimermanTsi18] via progress on the Colmez conjecture due to Andreatta, Goren, Howard and Madapusi Pera [Reference Andreatta, Goren, Howard and Madapusi PeraAGHM18], and Yuan and Zhang [Reference Yuan and ZhangYZ18], is now the only obstacle in an otherwise unconditional proof of the André–Oort conjecture. The conjecture is that there exist positive constants
$c$
and
$\unicode[STIX]{x1D6FF}$
such that, for any special point
$P\in S$
,
$$\begin{eqnarray}\displaystyle \#\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\cdot P\geqslant c\unicode[STIX]{x1D6E5}(P)^{\unicode[STIX]{x1D6FF}}. & & \displaystyle \nonumber\end{eqnarray}$$
Of course, this conjecture does not follow from Conjecture 11.1 because special points lying in
$V$
need not be optimal in
$V$
. However, the proof of the André–Oort conjecture only requires the bound for special points that are not contained in the positive dimensional special subvarieties contained in
$V$
, i.e., special points contained in
$\text{Opt}_{0}(V)$
(see [Reference DawDaw15] for more details). Furthermore, since special points are defined over number fields, we may also assume in that case that
$V$
is defined over a finite extension of
$F$
. It follows that Conjecture 11.1 is sufficient to prove the André–Oort conjecture.
To prove Conjecture 8.4, however, one only requires the following hypothesis.
Conjecture 11.3. Let
$V$
be a subvariety of
$S$
, defined over a finitely generated extension
$L$
of
$F$
contained in
$\mathbb{C}$
. There exist positive constants
$c_{G}$
and
$\unicode[STIX]{x1D6FF}_{G}$
such that the following holds.
If
$P\in V^{\text{oa}}\cap S^{[1+\dim V]}$
, then
$$\begin{eqnarray}\displaystyle \#\text{Aut}(\mathbb{C}/L)\cdot P\geqslant c_{G}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D6FF}_{G}}. & & \displaystyle \nonumber\end{eqnarray}$$
Remark 11.4. Note that, if
$P\in V^{\text{oa}}\cap S^{[1+\dim V]}$
, then
$P\in \text{Opt}_{0}(V)$
. To see this, let
$W$
be a subvariety of
$V$
containing
$P$
such that
$\unicode[STIX]{x1D6FF}(W)\leqslant \unicode[STIX]{x1D6FF}(P)$
, i.e.,
$$\begin{eqnarray}\displaystyle \dim \langle W\rangle -\dim W\leqslant \dim \langle P\rangle \leqslant \dim S-1-\dim V. & & \displaystyle \nonumber\end{eqnarray}$$
In fact, we can and do assume that
$W$
is optimal. We have
$$\begin{eqnarray}\displaystyle \dim W\geqslant 1+\dim V+\dim \langle W\rangle -\dim S\geqslant 1+\dim V+\dim \langle W\rangle _{\text{ws}}-\dim S, & & \displaystyle \nonumber\end{eqnarray}$$
and so
$\dim W=0$
, as
$P\notin V^{\text{an}}$
, which implies that
$W=P$
, proving the claim. Therefore, Conjecture 11.3 follows from Conjecture 11.1, but the former may turn out to be more tractable. It is worth recalling that, when
$S$
is an abelian variety and
$V$
is a subvariety defined over
$\bar{\mathbb{Q}}$
, Habegger [Reference HabeggerHP09] famously showed that the Néron–Tate height is bounded on
$\bar{\mathbb{Q}}$
-points of
$V^{\text{oa}}\cap S^{[\dim V]}$
.
12 Further arithmetic hypotheses
The principal obstruction to applying the Pila–Wilkie counting theorems to our point counting problems (except for the availability of lower bounds for Galois orbits) is the ability to parametrize pre-special subvarieties of
$S$
using points of bounded height in a definable set.
Definition 12.1. We say that a semisimple algebraic group defined over
$\mathbb{Q}$
is of non-compact type if its almost-simple factors all have the property that their underlying real Lie group is not compact.
Let
$\unicode[STIX]{x1D6FA}$
be a (finite) set of representatives for the semisimple subgroups of
$G$
defined over
$\mathbb{Q}$
of non-compact type modulo the equivalence relation
$$\begin{eqnarray}\displaystyle H_{2}\sim H_{1}\;\Longleftrightarrow \;H_{2,\mathbb{R}}=gH_{1,\mathbb{R}}g^{-1}\quad \text{for some }g\in G(\mathbb{R}). & & \displaystyle \nonumber\end{eqnarray}$$
Add the trivial group to
$\unicode[STIX]{x1D6FA}$
. Our needs can be encapsulated in the following conjecture. Recall that
$X$
is realized as a bounded symmetric domain in
$\mathbb{C}^{N}$
for some
$N\in \mathbb{N}$
, which we identify with
$\mathbb{R}^{2N}$
. Henceforth, we fix an embedding of
$G$
into
$\text{GL}_{n}$
such that
$\unicode[STIX]{x1D6E4}$
is contained in
$\text{GL}_{n}(\mathbb{Z})$
. We consider
$\text{GL}_{n}(\mathbb{R})$
as a subset of
$\mathbb{R}^{n^{2}}$
in the natural way.
Conjecture 12.2 (Cf. [Reference Habegger and PilaHP16, Proposition 6.7]).
There exist positive constants
$d$
,
$c_{{\mathcal{F}}}$
, and
$\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}$
such that, if
$z\in {\mathcal{F}}$
, then the smallest pre-special subvariety of
$X$
containing
$z$
can be written
$gF(\mathbb{R})^{+}g^{-1}x$
, where
$F\in \unicode[STIX]{x1D6FA}$
, and
$g\in G(\mathbb{R})$
and
$x\in X$
satisfy
$$\begin{eqnarray}\displaystyle \text{H}_{d}(g,x)\leqslant c_{{\mathcal{F}}}\unicode[STIX]{x1D6E5}(\langle \unicode[STIX]{x1D70B}(z)\rangle )^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}. & & \displaystyle \nonumber\end{eqnarray}$$
This is seemingly a natural generalization of the following theorem due to Orr and the first author on the heights of pre-special points, which plays a crucial role in the proof of the André–Oort conjecture.
Theorem 12.3 (Cf. [Reference Daw and OrrDO16, Theorem 1.4]).
There exist positive constants
$d$
,
$c_{{\mathcal{F}}}$
and
$\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}$
such that, if
$z\in {\mathcal{F}}$
is a pre-special, then
$$\begin{eqnarray}\displaystyle \text{H}_{d}(z)\leqslant c_{{\mathcal{F}}}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70B}(z))^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}. & & \displaystyle \nonumber\end{eqnarray}$$
We remark that the problem of finding
$d$
as in Conjecture 12.2 poses no obstacle in itself. Indeed a proof of the following theorem will appear in a forthcoming article of Borovoi and the authors.
Theorem 12.4 (Cf. [Reference Borovoi, Daw and RenBDR18, Corollary 0.7]).
There exists a positive constant
$d$
such that, for any two semisimple subgroups
$H_{1}$
and
$H_{2}$
of
$G$
defined over
$\mathbb{Q}$
that are conjugate by an element of
$G(\mathbb{R})$
, there exists a number field
$K$
contained in
$\mathbb{R}$
of degree at most
$d$
, and an element
$g\in G(K)$
, such that
$$\begin{eqnarray}\displaystyle H_{2,K}=gH_{1,K}g^{-1}. & & \displaystyle \nonumber\end{eqnarray}$$
A nice feature of Conjecture 12.2 is that it implies Conjecture 10.3 that there are only finitely many special subvarieties of bounded complexity.
Proof. Let
$Z$
be a special subvariety of
$S$
such that
$\unicode[STIX]{x1D6E5}(Z)\leqslant b$
and let
$P\in Z$
be such that
$\langle P\rangle =Z$
. Let
$z\in {\mathcal{F}}$
be such that
$\unicode[STIX]{x1D70B}(z)=P$
and let
$X_{H}$
be the smallest pre-special subvariety of
$X$
containing
$z$
. Then
$\unicode[STIX]{x1D70B}(X_{H})=Z$
and, by Conjecture 12.2,
$X_{H}=gFg^{-1}x$
, where
$F\in \unicode[STIX]{x1D6FA}$
, and
$g\in G(\mathbb{R})$
and
$x\in X$
satisfy
$$\begin{eqnarray}\displaystyle \text{H}_{d}(g,x)\leqslant c_{{\mathcal{F}}}\unicode[STIX]{x1D6E5}(Z)^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}\leqslant c_{{\mathcal{F}}}b^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}. & & \displaystyle \nonumber\end{eqnarray}$$
The claim follows, therefore, from the fact that there are only finitely many algebraic numbers of bounded degree and height, i.e., Northcott’s property. ◻
Another, albeit longer, approach to our point counting problems can be given by replacing Conjecture 12.2 with two related conjectures, although we will have to additionally assume Conjecture 10.3 in this case. We will also rely on the fact that Theorem 9.1 is uniform in families. The advantage is that the following two conjectures are seemingly more accessible.
Conjecture 12.6. For any
$\unicode[STIX]{x1D705}>0$
, there exists a positive constant
$c_{\unicode[STIX]{x1D705}}$
such that, if
$Z$
is a special subvariety of
$S$
, then there exists a semisimple subgroup
$H$
of
$G$
defined over
$\mathbb{Q}$
of non-compact type, and an extension
$L$
of
$F$
satisfying
$$\begin{eqnarray}\displaystyle [L:F]\leqslant c_{\unicode[STIX]{x1D705}}\unicode[STIX]{x1D6E5}(Z)^{\unicode[STIX]{x1D705}}, & & \displaystyle \nonumber\end{eqnarray}$$
such that, for any
$\unicode[STIX]{x1D70E}\in \text{Aut}(\mathbb{C}/L)$
,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}(Z)=\unicode[STIX]{x1D70B}(H(\mathbb{R})^{+}x_{\unicode[STIX]{x1D70E}}), & & \displaystyle \nonumber\end{eqnarray}$$
where
$H(\mathbb{R})^{+}x_{\unicode[STIX]{x1D70E}}$
is a pre-special subvariety of
$X$
intersecting
${\mathcal{F}}$
.
Recall that, for an abelian variety
$A$
, defined over a field
$K$
, every abelian subvariety of
$A$
can be defined over a fixed, finite extension of
$K$
. The analogue of Conjecture 12.6 is, therefore, trivial. In a Shimura variety, one hopes that the degrees of fields of definition of strongly special subvarieties grow as in Conjecture 12.6. If this were true, Conjecture 12.6 for strongly special subvarieties would follow easily.
Our final conjecture is also inspired by the abelian setting.
Conjecture 12.7 (Cf. [Reference Habegger and PilaHP16, Lemma 3.2]).
There exist positive constants
$c_{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
such that, if
$X_{H}$
is a pre-special subvariety of
$X$
intersecting
${\mathcal{F}}$
and
$z\in {\mathcal{F}}$
belongs to
$\unicode[STIX]{x1D6E4}X_{H}$
, then
$z\in \unicode[STIX]{x1D6FE}X_{H}$
, where
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$
satisfies
$$\begin{eqnarray}\displaystyle \text{H}_{1}(\unicode[STIX]{x1D6FE})\leqslant c_{\unicode[STIX]{x1D6E4}}\deg (\unicode[STIX]{x1D70B}(X_{H}))^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}}. & & \displaystyle \nonumber\end{eqnarray}$$
Conjecture 12.7 has the following useful consequence.
Lemma 12.8. Assume that Conjecture 12.7 holds.
There exist positive constants
$d$
,
$c_{\text{H}}$
, and
$\unicode[STIX]{x1D6FF}_{\text{H}}$
such that, if
$H(\mathbb{R})^{+}x$
is a pre-special subvariety of
$X$
intersecting
${\mathcal{F}}$
, then
$$\begin{eqnarray}\displaystyle H(\mathbb{R})^{+}x=H(\mathbb{R})^{+}y & & \displaystyle \nonumber\end{eqnarray}$$
where
$\text{H}_{d}(y)\leqslant c_{\text{H}}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70B}(H(\mathbb{R})^{+}x))^{\unicode[STIX]{x1D6FF}_{\text{H}}}$
.
Proof. Let
$d$
,
$c_{{\mathcal{F}}}$
, and
$\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}$
be the positive constants afforded to us by Theorem 12.3, and let
$c_{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
be the positive constants afforded to us by Conjecture12.7.
Let
$x^{\prime }\in \unicode[STIX]{x1D6E4}H(\mathbb{R})^{+}x\cap {\mathcal{F}}$
denote a pre-special point such that
$\unicode[STIX]{x1D70B}(x^{\prime })$
is of minimal complexity among the special points of
$\unicode[STIX]{x1D70B}(H(\mathbb{R})^{+}x)$
. By Theorem 12.3, we have
$$\begin{eqnarray}\displaystyle \text{H}_{d}(x^{\prime })\leqslant c_{{\mathcal{F}}}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70B}(H(\mathbb{R})^{+}x))^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}. & & \displaystyle \nonumber\end{eqnarray}$$
On the other hand, by Conjecture 12.7,
$x^{\prime }\in \unicode[STIX]{x1D6FE}H(\mathbb{R})^{+}x$
, where
$$\begin{eqnarray}\displaystyle \text{H}_{1}(\unicode[STIX]{x1D6FE})\leqslant c_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70B}(H(\mathbb{R})^{+}x))^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}}. & & \displaystyle \nonumber\end{eqnarray}$$
It follows easily from the properties of heights that there exist positive constants
$c$
and
$\unicode[STIX]{x1D6FF}$
depending only on the fixed data such that
$$\begin{eqnarray}\displaystyle \text{H}_{d}(\unicode[STIX]{x1D6FE}^{-1}x^{\prime })\leqslant c\text{H}_{1}(\unicode[STIX]{x1D6FE})^{\unicode[STIX]{x1D6FF}}\text{H}_{d}(x^{\prime })^{\unicode[STIX]{x1D6FF}}. & & \displaystyle \nonumber\end{eqnarray}$$
Therefore, the previous remarks show that
$$\begin{eqnarray}\displaystyle y:=\unicode[STIX]{x1D6FE}^{-1}x^{\prime }\in H(\mathbb{R})^{+}x & & \displaystyle \nonumber\end{eqnarray}$$
satisfies the requirements of the lemma. ◻
We will now verify the arithmetic conjectures stated above in an arbitrary product of modular curves.
13 Products of modular curves
Our definition of a Shimura variety allows for the possibility that
$S$
might be a product of modular curves. In that case
$G=\text{GL}_{2}^{n}$
, where
$n$
is the number of modular curves, and
$\mathfrak{X}$
is the
$G(\mathbb{R})$
conjugacy class of the morphism
$\mathbb{S}\rightarrow G_{\mathbb{R}}$
given by
$$\begin{eqnarray}\displaystyle a+ib\mapsto \left(\left(\begin{array}{@{}cc@{}}a & b\\ -b & a\end{array}\right),\ldots ,\left(\begin{array}{@{}cc@{}}a & b\\ -b & a\end{array}\right)\right). & & \displaystyle \nonumber\end{eqnarray}$$
We let
$X$
denote the
$G(\mathbb{R})^{+}$
conjugacy class of this morphism, which one identifies with the
$n$
th cartesian power
$\mathbb{H}^{n}$
of the upper half-plane
$\mathbb{H}$
.
For our purposes, we can and do suppose that
$\unicode[STIX]{x1D6E4}$
is equal to
$\text{SL}_{2}(\mathbb{Z})^{n}$
and we let
${\mathcal{F}}$
denote a fundamental set in
$X$
for the action of
$\unicode[STIX]{x1D6E4}$
, equal to the
$n$
th cartesian power of a fundamental set
${\mathcal{F}}_{\mathbb{H}}$
in
$\mathbb{H}$
for the action of
$\text{SL}_{2}(\mathbb{Z})$
. Note that, as explained in [Reference OrrOrr18, § 1.3], we can and do choose
${\mathcal{F}}_{\mathbb{H}}$
in the image of a Siegel set. Via the
$j$
-function applied to each factor of
$\mathbb{H}^{n}$
, the quotient
$\unicode[STIX]{x1D6E4}\backslash X$
is isomorphic to the algebraic variety
$\mathbb{C}^{n}$
. Special subvarieties have the following well-documented description.
Proposition 13.1 (Cf. [Reference EdixhovenEdi05, Proposition 2.1]).
Let
$I=\{1,\ldots ,n\}$
. A subvariety
$Z$
of
$\mathbb{C}^{n}$
is a special subvariety if and only if there exists a partition
$\unicode[STIX]{x1D6FA}=(I_{1},\ldots ,I_{t})$
of
$I$
, with
$|I_{i}|=n_{i}$
, such that
$Z$
is equal to the product of subvarieties
$Z_{i}$
of
$\mathbb{C}^{n_{i}}$
, where either
⋆
$I_{i}$
is a one element set and
$Z_{i}$
is a special point, or⋆
$Z_{i}$
is the image of
$\mathbb{H}$
in
$\mathbb{C}^{n_{i}}$
under the map sending
$\unicode[STIX]{x1D70F}\in \mathbb{H}$
to the image of
$(g_{j}\unicode[STIX]{x1D70F})_{j\in I_{i}}$
in
$\mathbb{C}^{n_{i}}$
for elements
$g_{j}\in \text{GL}_{2}(\mathbb{Q})^{+}$
.
First note that Conjecture 12.2 for
$\mathbb{C}^{n}$
follows from [Reference Habegger and PilaHP16, Proposition 6.7]. Hence, we will now verify Conjectures 12.6 and 12.7 in that setting.
Proof of Conjecture 12.6 for
$\mathbb{C}^{n}$
.
Let
$Z$
be a special subvariety of
$\mathbb{C}^{n}$
, equal to a product of special subvarieties
$Z_{i}$
of
$\mathbb{C}^{n_{i}}$
, as above. Without loss of generality, we may assume that the product contains only one factor and, by Theorem 12.3, we may assume that it is not a special point. Therefore,
$Z$
is equal to the image of
$\mathbb{H}$
in
$\mathbb{C}^{n}$
under the map sending
$\unicode[STIX]{x1D70F}\in \mathbb{H}$
to the image of
$(g_{j}\unicode[STIX]{x1D70F})_{j=1}^{n}$
in
$\mathbb{C}^{n}$
for elements
$g_{j}\in \text{GL}_{2}(\mathbb{Q})^{+}$
.
In other words, we have a morphism of Shimura data from
$(\text{GL}_{2},\mathbb{H}^{\pm })$
to
$(G,\mathfrak{X})$
, where
$\mathbb{H}^{\pm }$
is the union of the upper and lower half-planes (or, rather, the conjugacy class we associate with it, as above), induced by the morphism
$$\begin{eqnarray}\displaystyle \text{GL}_{2}\rightarrow \text{GL}_{2}^{n}:g\mapsto (g_{j}gg_{j}^{-1})_{j=1}^{n}, & & \displaystyle \nonumber\end{eqnarray}$$
such that
$Z$
is equal to the image of
$\mathbb{H}\times \{1\}$
under the corresponding morphism
$$\begin{eqnarray}\displaystyle \text{Sh}_{K}(\text{GL}_{2},\mathbb{H}^{\pm })\rightarrow \text{Sh}_{\text{GL}_{2}(\hat{\mathbb{Z}})^{n}}(G,\mathfrak{X}), & & \displaystyle\end{eqnarray}$$
where
$K$
is the product of the groups
$$\begin{eqnarray}\displaystyle K_{p}:=g_{1}\text{GL}_{2}(\mathbb{Z}_{p})g_{1}^{-1}\cap \cdots \cap g_{n}\text{GL}_{2}(\mathbb{Z}_{p})g_{n}^{-1} & & \displaystyle \nonumber\end{eqnarray}$$
over all primes
$p$
.
Since (13.1.1) is defined over
$E(\text{GL}_{2},\mathbb{H}^{\pm })=\mathbb{Q}$
, it suffices to bound the size of
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}_{0}(\text{Sh}_{K}(\text{GL}_{2},\mathbb{H}^{\pm })), & & \displaystyle \nonumber\end{eqnarray}$$
which, by [Reference MilneMil05, Theorem 5.17], is in bijection with
$$\begin{eqnarray}\displaystyle \mathbb{Q}_{{>}0}\backslash \mathbb{A}_{f}^{\times }/\unicode[STIX]{x1D708}(K), & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D708}$
is the determinant map on
$\text{GL}_{2}$
. However, since
$\mathbb{A}_{f}^{\times }$
is equal to the direct product
$\mathbb{Q}_{{>}0}\hat{\mathbb{Z}}^{\times }$
, it suffices to bound the size of
$\hat{\mathbb{Z}}^{\times }/\unicode[STIX]{x1D708}(K)$
.
To that end, let
$\unicode[STIX]{x1D6F4}$
denote the (finite) set of primes
$p$
such that
$g_{j}\notin \text{GL}_{2}(\mathbb{Z}_{p})$
, for some
$j\in \{1,\ldots ,n\}$
. In particular,
$$\begin{eqnarray}\displaystyle [\hat{\mathbb{Z}}^{\times }:\unicode[STIX]{x1D708}(K)]=\mathop{\prod }_{p\in \unicode[STIX]{x1D6F4}}[\mathbb{Z}_{p}^{\times }:\unicode[STIX]{x1D708}(K_{p})] & & \displaystyle \nonumber\end{eqnarray}$$
and, since
$K_{p}$
contains the elements
$\text{diag}(a,a)$
, where
$a\in \mathbb{Z}_{p}^{\times }$
,
$$\begin{eqnarray}\displaystyle [\mathbb{Z}_{p}^{\times }:\unicode[STIX]{x1D708}(K_{p})]\leqslant [\mathbb{Z}_{p}^{\times }:\mathbb{Z}_{p}^{\times 2}]\leqslant 4, & & \displaystyle \nonumber\end{eqnarray}$$
where
$\mathbb{Z}_{p}^{\times 2}$
denotes the squares in
$\mathbb{Z}_{p}^{\times }$
.
On the other hand, by [Reference DawDaw12],
$$\begin{eqnarray}\displaystyle \deg (Z)\geqslant \mathop{\prod }_{p\in \unicode[STIX]{x1D6F4}}p, & & \displaystyle \nonumber\end{eqnarray}$$
and the conjecture follows easily from the following classical fact regarding primorials.
Lemma 13.2. Let
$n\in \mathbb{N}$
. The product of the first
$n$
prime numbers is equal to
$$\begin{eqnarray}\displaystyle e^{(1+o(1))n\log n}.\Box & & \displaystyle \nonumber\end{eqnarray}$$
Proof of Conjecture 12.7 for
$\mathbb{C}^{n}$
.
Let
$X_{H}$
be a product of spaces
$X_{i}\subseteq \mathbb{H}^{n_{i}}$
each equal to either a pre-special point or to the image of
$\mathbb{H}$
given by the map sending
$\unicode[STIX]{x1D70F}$
to
$(g_{j}\unicode[STIX]{x1D70F})_{j=1}^{n_{i}}$
for elements
$g_{j}\in \text{GL}_{2}(\mathbb{Q})^{+}$
. Without loss of generality, we may assume that the product contains only one factor. If
$X_{H}$
is a pre-special point contained in
${\mathcal{F}}$
, then the claim follows from the fact that
$$\begin{eqnarray}\displaystyle \{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6FE}{\mathcal{F}}\cap {\mathcal{F}}\neq \emptyset \} & & \displaystyle \nonumber\end{eqnarray}$$
is finite. Therefore, assume that
$X$
is equal to the image of
$\mathbb{H}$
in
$\mathbb{H}^{n}$
given by the map sending
$\unicode[STIX]{x1D70F}$
to
$(g_{j}\unicode[STIX]{x1D70F})_{j=1}^{n}$
for elements
$g_{j}\in \text{GL}_{2}(\mathbb{Q})^{+}$
. We can and do assume that
$g_{1}$
is equal to the identity element and that all of the
$g_{j}$
have coprime integer entries.
As in the statement of Conjecture 12.7, we assume that
$X$
intersects
${\mathcal{F}}$
, and we let
$x\in {\mathcal{F}}\,\cap \,X$
. Therefore,
$$\begin{eqnarray}\displaystyle x=(g_{j}\unicode[STIX]{x1D70F}_{x})_{j=1}^{n}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D70F}_{x}\in {\mathcal{F}}_{\mathbb{H}}$
. By [Reference OrrOrr18, Theorem 1.2] (cf. [Reference Habegger and PilaHab12, Lemma 5.2]),
$\text{H}_{1}(g_{j})\leqslant c_{1}\det (g_{j})^{2}$
, for all
$j\in \{1,\ldots ,n\}$
, where
$c_{1}$
is a positive constant not depending on
$Z$
. In particular,
$$\begin{eqnarray}\displaystyle \text{H}_{1}(g_{j}^{-1})\leqslant \det (g_{j})\text{H}_{1}(g_{j})\leqslant c_{1}\det (g_{j})^{3}, & & \displaystyle \nonumber\end{eqnarray}$$
for each
$j\in \{1,\ldots ,n\}$
.
Now let
$z:=(z_{j})_{j=1}^{n}\in {\mathcal{F}}$
be a point belonging to
$\unicode[STIX]{x1D6E4}X$
. For each
$j\in \{1,\ldots ,n\}$
,
$$\begin{eqnarray}\displaystyle z_{j}=\unicode[STIX]{x1D6FE}_{j}g_{j}g\unicode[STIX]{x1D70F}_{x}, & & \displaystyle \nonumber\end{eqnarray}$$
for some
$g\in \text{GL}_{2}(\mathbb{R})^{+}$
and some
$\unicode[STIX]{x1D6FE}_{j}\in \text{SL}_{2}(\mathbb{Z})$
. Therefore, let
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}:=\mathop{\bigcap }_{j=1}^{n}g_{j}^{-1}\text{SL}_{2}(\mathbb{Z})g_{j} & & \displaystyle \nonumber\end{eqnarray}$$
and let
${\mathcal{C}}$
denote a set of representatives in
$\text{SL}_{2}(\mathbb{Z})$
for
$\unicode[STIX]{x1D6EC}\backslash \text{SL}_{2}(\mathbb{Z})$
. Note that, if we define
$$\begin{eqnarray}\displaystyle m_{j}:=\det (g_{j}), & & \displaystyle \nonumber\end{eqnarray}$$
then, for any multiple
$N$
of
$m_{j}$
, the principal congruence subgroup
$\unicode[STIX]{x1D6E4}(N)$
is contained in
$g_{j}^{-1}\text{SL}_{2}(\mathbb{Z})g_{j}$
. In particular, if we define
$N$
to be the lowest common multiple of the
$m_{j}$
, then
$\unicode[STIX]{x1D6E4}(N)$
is contained in
$\unicode[STIX]{x1D6EC}$
. It follows that any subset of
$\text{SL}_{2}(\mathbb{Z})$
mapping bijectively to
$\text{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$
contains a set
${\mathcal{C}}$
, as above. Via the procedure outlined in [Reference Diamond and ShurmanDS05, Exercise 1.2.2], it is straightforward to verify that we can (and do) choose
${\mathcal{C}}$
such that, for any
$c\in {\mathcal{C}}$
,
$$\begin{eqnarray}\displaystyle H(c)\leqslant 7N^{5} & & \displaystyle \nonumber\end{eqnarray}$$
(though we certainly do not claim that this is the best possible bound; any polynomial bound would suffice for our purposes).
The union
$$\begin{eqnarray}\displaystyle \mathop{\bigcup }_{c\in {\mathcal{C}}}c{\mathcal{F}}_{\mathbb{H}} & & \displaystyle \nonumber\end{eqnarray}$$
constitutes a fundamental set in
$\mathbb{H}$
for the action of
$\unicode[STIX]{x1D6EC}$
. Hence, there exists
$c\in {\mathcal{C}}$
and
$\unicode[STIX]{x1D706}_{g}\in \unicode[STIX]{x1D6EC}$
such that
$$\begin{eqnarray}\displaystyle c^{-1}\unicode[STIX]{x1D706}_{g}g\unicode[STIX]{x1D70F}_{x}\in {\mathcal{F}}_{\mathbb{H}}. & & \displaystyle \nonumber\end{eqnarray}$$
Furthermore, for each
$j\in \{1,\ldots ,n\}$
, we can write
$\unicode[STIX]{x1D706}_{g}=g_{j}^{-1}\unicode[STIX]{x1D706}_{j}g_{j}$
, for some
$\unicode[STIX]{x1D706}_{j}\in \text{SL}_{2}(\mathbb{Z})$
and, hence,
$$\begin{eqnarray}\displaystyle z_{j}=\unicode[STIX]{x1D6FE}_{j}g_{j}g\unicode[STIX]{x1D70F}_{x}=\unicode[STIX]{x1D6FE}_{j}g_{j}\unicode[STIX]{x1D706}_{g}^{-1}c\cdot c^{-1}\unicode[STIX]{x1D706}_{g}g\unicode[STIX]{x1D70F}_{x}=\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1}g_{j}c\cdot c^{-1}\unicode[STIX]{x1D706}_{g}g\unicode[STIX]{x1D70F}_{x}. & & \displaystyle\end{eqnarray}$$
Therefore, by [Reference OrrOrr18, Theorem 1.2], we have
$\text{H}_{1}(\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1}g_{j}c)\leqslant c_{1}m_{j}^{2}$
, for all
$j\in \{1,\ldots ,n\}$
.
We write
$$\begin{eqnarray}\displaystyle \text{H}_{1}(\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1})=\text{H}_{1}(\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1}g_{j}c\cdot c^{-1}g_{j}^{-1})\leqslant c_{2}\text{H}_{1}(\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1}g_{j}c)^{\unicode[STIX]{x1D6FF}}\text{H}_{1}(c^{-1})^{\unicode[STIX]{x1D6FF}}\text{H}_{1}(g_{j}^{-1})^{\unicode[STIX]{x1D6FF}}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$c_{2}$
and
$\unicode[STIX]{x1D6FF}$
are positive constants not depending on
$Z$
, and we obtain
$$\begin{eqnarray}\displaystyle \text{H}_{1}(\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1})\leqslant c_{3}N^{10\unicode[STIX]{x1D6FF}}, & & \displaystyle \nonumber\end{eqnarray}$$
for each
$j\in \{1,\ldots ,n\}$
, where
$c_{3}$
is a positive constant not depending on
$Z$
.
Conversely, by [Reference DawDaw12, § 2],
$$\begin{eqnarray}\displaystyle \deg (Z)\geqslant [\text{SL}_{2}(\mathbb{Z}):\unicode[STIX]{x1D6EC}] & & \displaystyle \nonumber\end{eqnarray}$$
and, by writing the
$g_{j}$
in Smith normal form, i.e.,
$$\begin{eqnarray}\displaystyle g_{j}=\unicode[STIX]{x1D6FE}_{j,1}\left(\begin{array}{@{}cc@{}}1 & 0\\ 0 & m_{j}\end{array}\right)\unicode[STIX]{x1D6FE}_{j,2}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FE}_{j,1},\unicode[STIX]{x1D6FE}_{j,2}\in \text{SL}_{2}(\mathbb{Z})$
, we conclude that
$$\begin{eqnarray}\displaystyle \text{SL}_{2}(\mathbb{Z})\cap g_{j}^{-1}\text{SL}_{2}(\mathbb{Z})g_{j}=\unicode[STIX]{x1D6FE}_{j,2}^{-1}\unicode[STIX]{x1D6E4}_{0}(m_{j})\unicode[STIX]{x1D6FE}_{j,2}, & & \displaystyle \nonumber\end{eqnarray}$$
whose index in
$\text{SL}_{2}(\mathbb{Z})$
is the same as
$\unicode[STIX]{x1D6E4}_{0}(m_{j})$
, which is
$m_{j}\unicode[STIX]{x1D711}(m_{j})$
. It follows that
$$\begin{eqnarray}\displaystyle [\text{SL}_{2}(\mathbb{Z}):\unicode[STIX]{x1D6EC}]\geqslant N, & & \displaystyle \nonumber\end{eqnarray}$$
and so
$$\begin{eqnarray}\displaystyle \text{H}_{1}(\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1})\leqslant c_{3}\deg (Z)^{10\unicode[STIX]{x1D6FF}}. & & \displaystyle \nonumber\end{eqnarray}$$
Therefore, we let
$\unicode[STIX]{x1D6FE}:=(\unicode[STIX]{x1D6FE}_{j}\unicode[STIX]{x1D706}_{j}^{-1})_{j=1}^{n}\in \unicode[STIX]{x1D6E4}$
. By (13.2.1),
$z\in \unicode[STIX]{x1D6FE}X$
, and the result follows.◻
Finally, we will verify Conjecture 10.4 in this case.
Proof of Conjecture 10.4 for
$\mathbb{C}^{n}$
.
Let
$Z$
be a special subvariety of
$\mathbb{C}^{n}$
. Then
$Z$
is a product of special subvarieties. Since there are only finitely many partitions of
$\{1,\ldots ,n\}$
, we may assume that the product contains only one factor. If
$Z$
is a special point,
$H^{\text{der}}$
is trivial. Therefore, we assume that
$Z$
is equal to the image of
$\mathbb{H}$
in
$\mathbb{C}^{n}$
under the map sending
$\unicode[STIX]{x1D70F}\in \mathbb{H}$
to the image of
$(g_{j}\cdot \unicode[STIX]{x1D70F})_{j=1}^{n}$
in
$\mathbb{C}^{n}$
for elements
$g_{j}\in \text{GL}_{2}(\mathbb{Q})^{+}$
. Then
$H^{\text{der}}$
is the image of
$\text{SL}_{2}$
under the morphism
$$\begin{eqnarray}\displaystyle \text{SL}_{2}\rightarrow \text{GL}_{2}^{n}:g\mapsto (g_{j}gg_{j}^{-1})_{j=1}^{n}. & & \displaystyle \nonumber\end{eqnarray}$$
We see from the calculations in the previous proof that the bound
$\deg (Z)\leqslant b$
implies that the
$g_{j}$
come from the union of finitely many double cosets
$\unicode[STIX]{x1D6E4}g\unicode[STIX]{x1D6E4}$
for
$g\in \text{GL}_{2}(\mathbb{Q})^{+}$
. Since each such double coset is equal to a finite union of single cosets
$\unicode[STIX]{x1D6E4}h$
for
$h\in \text{GL}_{2}(\mathbb{Q})^{+}$
, the result follows. ◻
14 Main results (part 2): Conditional solutions to the counting problems
We conclude by demonstrating how our arithmetic conjectures might be used to resolve the counting problems stated in § 8. In our applications of the counting theorem, we will need the following.
Lemma 14.1. Let
$\unicode[STIX]{x1D6FD}:[0,1]\rightarrow G(\mathbb{R})\times X$
be semialgebraic. Then
$\text{Im}(\unicode[STIX]{x1D6FD})$
is contained in a complex algebraic subset
$B$
of
$G(\mathbb{C})\times \mathbb{C}^{N}$
of dimension at most
$1$
.
Proof. Let
$Y$
denote the real Zariski closure of
$\text{Im}(\unicode[STIX]{x1D6FD})$
in
$G(\mathbb{R})\times \mathbb{R}^{2N}$
. In particular,
$\dim Y\leqslant 1$
. Without loss of generality, we can and do assume that
$Y$
is irreducible. If
$Y$
is a point then there is nothing to prove. Therefore, we can and do assume that
$Y$
is an irreducible real algebraic curve. In particular, the complexification
$Y_{\mathbb{C}}$
of
$Y$
in
$G(\mathbb{C})\times \mathbb{C}^{2N}$
is an irreducible complex algebraic curve.
Let
$g_{1},\ldots ,g_{n^{2}},x_{1},y_{1},\ldots ,x_{N},y_{N}$
denote the real coordinate functions on
$G(\mathbb{R})\times X$
and let
$z_{j}=x_{j}+iy_{j}$
denote the coordinate functions on
$\mathbb{C}^{N}=\mathbb{R}^{2N}$
. If all of the coordinates functions on
$\mathbb{R}^{2N}$
are constant on
$Y$
, the result is obvious. Therefore, without loss of generality, we can and do assume that
$x_{1}$
is not constant on
$Y$
.
We claim that each of the coordinate functions
$x_{2},y_{2},\ldots ,x_{N},y_{N}$
on
$\mathbb{C}^{2N}$
is algebraic over the field
$\mathbb{C}(z_{1})$
, considered as a field of functions on
$Y_{\mathbb{C}}$
. To see this, note that
$z_{1}$
is non-constant on
$Y_{\mathbb{C}}$
, and so
$\mathbb{C}(z_{1})$
has transcendence degree at least
$1$
. On the other hand,
$\mathbb{C}(z_{1})$
is contained in
$\mathbb{C}(x_{1},y_{1})$
, which is algebraic over
$\mathbb{C}(x_{1})$
.
In particular, each of the functions
$x_{2}+iy_{2},\ldots ,x_{N}+iy_{N}$
is algebraic over the field
$\mathbb{C}(z_{1})$
. It follows that, for each
$j\geqslant 2$
, there exists a polynomial
$f_{j}(z_{1},z_{j})\in \mathbb{C}[z_{1},z_{j}]$
, non-trivial in
$z_{j}$
, such that
$f_{j}(z_{1},z_{j})=0$
on
$Y$
. Similarly, for each
$k=1,\ldots ,n^{2}$
, there exists a polynomial
$f_{j}(z_{1},g_{k})\in \mathbb{C}[z_{1},g_{k}]$
, non-trivial in
$g_{k}$
, such that
$f_{k}(z_{1},g_{k})=0$
on
$Y$
. In particular,
$Y$
is contained in the vanishing locus of the
$f_{j}$
and the
$f_{k}$
, which define a complex algebraic curve in
$G(\mathbb{C})\times \mathbb{C}^{N}$
.◻
We denote by
$X^{\vee }$
the compact dual of
$X$
, which is a complex algebraic variety on which
$G(\mathbb{C})$
acts via an algebraic morphism
$$\begin{eqnarray}\displaystyle G(\mathbb{C})\times X^{\vee }\rightarrow X^{\vee }. & & \displaystyle \nonumber\end{eqnarray}$$
Furthermore,
$X$
naturally embeds into
$X^{\vee }$
and the embedding factors through an embedding of
$\mathbb{C}^{N}$
, i.e., the Harish-Chandra realization, into
$X^{\vee }$
. We could have defined subvarieties of
$X$
using
$X^{\vee }$
in the place of
$\mathbb{C}^{N}$
but, as mentioned previously, the two notions coincide. If we have a decomposition
$G^{\text{ad}}=G_{1}\times G_{2}$
, and thus
$X=X_{1}\times X_{2}$
, we have a natural decomposition
$$\begin{eqnarray}\displaystyle X^{\vee }=X_{1}^{\vee }\times X_{2}^{\vee }. & & \displaystyle \nonumber\end{eqnarray}$$
Furthermore, if
$(H,\mathfrak{X}_{H})$
denotes a Shimura subdatum of
$(G,\mathfrak{X})$
and
$X_{H}$
is a connected component of
$\mathfrak{X}_{H}$
contained in
$X$
, then
$X_{H}^{\vee }$
is naturally contained in
$X^{\vee }$
. We refer the reader to [Reference Ullmo and YafaevUY11, § 3] for more details.
Theorem 14.2. Assume that Conjecture 11.1 holds and assume that either
Then Conjecture 8.2 is true for curves, i.e., if
$V$
is a curve contained in
$S$
, then the set
$\text{Opt}_{0}(V)$
is finite.
Proof. We will assume that Conjecture 12.2 holds. The proof in the case that Conjectures 10.3, 12.6, and 12.7 hold is very similar, hence we omit it. To elucidate the use of Conjectures 10.3, 12.6, and 12.7 we will use them in the proof of Theorem 14.3, at the expense of making the proof longer. We suffer no loss of generality if we assume, as we will, that
$V$
is Hodge generic.
Let
$\unicode[STIX]{x1D6FA}$
denote a finite set of semisimple subgroups of
$G$
defined over
$\mathbb{Q}$
as in § 12 and let
$d$
,
$c_{{\mathcal{F}}}$
, and
$\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}$
be the constants afforded to us by Conjecture 12.2. Let
$L$
be a finitely generated extension of
$F$
contained in
$\mathbb{C}$
over which
$V$
is defined and let
$c_{G}$
and
$\unicode[STIX]{x1D6FF}_{G}$
be the constants afforded to us by Conjecture 11.1. Let
$\unicode[STIX]{x1D705}:=2\unicode[STIX]{x1D6FF}_{G}/3\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}$
.
We claim that there exists a positive constant
$c$
such that, for any
$P\in \text{Opt}_{0}(V)$
, we have
$$\begin{eqnarray}\displaystyle \#\text{Aut}(\mathbb{C}/L)\cdot P\leqslant cc_{{\mathcal{F}}}^{\unicode[STIX]{x1D705}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}. & & \displaystyle \nonumber\end{eqnarray}$$
This would be sufficient to prove Theorem 14.2 since then, by Conjecture 11.1, we obtain
$$\begin{eqnarray}\displaystyle c_{G}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D6FF}_{G}}\leqslant \#\text{Aut}(\mathbb{C}/L)\cdot P\leqslant cc_{{\mathcal{F}}}^{\unicode[STIX]{x1D705}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}} & & \displaystyle \nonumber\end{eqnarray}$$
and, rearranging this expression, we obtain
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}(\langle P\rangle )\leqslant (c_{3}c_{{\mathcal{F}}}^{\unicode[STIX]{x1D705}}c_{G}^{-1})^{3}, & & \displaystyle \nonumber\end{eqnarray}$$
which is a bound independent of
$P$
. We remind the reader that
$P$
is one of only finitely many irreducible components of
$\langle P\rangle \cap V$
. Hence, Theorem 14.2 would follow from Lemma 12.5 and, therefore, it remains only to prove the claim.
To that end, for each
$\unicode[STIX]{x1D70E}\in \text{Gal}(\mathbb{C}/L)$
, let
$z_{\unicode[STIX]{x1D70E}}\in {\mathcal{V}}$
be a point in
$\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D70E}(P))$
. Therefore, by Conjecture 12.2, the smallest pre-special subvariety of
$X$
containing
$z_{\unicode[STIX]{x1D70E}}$
can be written
$g_{\unicode[STIX]{x1D70E}}F_{\unicode[STIX]{x1D70E}}(\mathbb{R})^{+}g_{\unicode[STIX]{x1D70E}}^{-1}x_{\unicode[STIX]{x1D70E}}$
, where
$F_{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D6FA}$
, and
$g_{\unicode[STIX]{x1D70E}}\in G(\mathbb{R})$
and
$x_{\unicode[STIX]{x1D70E}}\in X$
satisfy
$$\begin{eqnarray}\displaystyle \text{H}_{d}(g_{\unicode[STIX]{x1D70E}},x_{\unicode[STIX]{x1D70E}})\leqslant c_{{\mathcal{F}}}\unicode[STIX]{x1D6E5}(\langle \unicode[STIX]{x1D70E}(P)\rangle )^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}=c_{{\mathcal{F}}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}. & & \displaystyle \nonumber\end{eqnarray}$$
Without loss of generality, we can and do assume that
$F:=F_{\unicode[STIX]{x1D70E}}$
is fixed. Therefore, for each
$\unicode[STIX]{x1D70E}\in \text{Gal}(\mathbb{C}/L)$
, the tuple
$(g_{\unicode[STIX]{x1D70E}},x_{\unicode[STIX]{x1D70E}},z_{\unicode[STIX]{x1D70E}})$
belongs to the definable set
$D$
of tuples
$$\begin{eqnarray}\displaystyle (g,x,z)\in G(\mathbb{R})\times X\times X\subseteq \mathbb{R}^{n^{2}+2N}\times \mathbb{R}^{2N}, & & \displaystyle \nonumber\end{eqnarray}$$
such that
$z\in {\mathcal{V}}\cap gF(\mathbb{R})^{+}g^{-1}x$
and
$x(\mathbb{S})\subseteq gG_{F}g^{-1}$
. We consider
$D$
as a family over a point in an omitted parameter space and choose for
$c$
the constant
$c(D,d,\unicode[STIX]{x1D705})$
afforded to us by Theorem 9.1 applied to
$D$
. Since
$\unicode[STIX]{x1D6FA}$
is finite, we can and do assume that
$c$
does not depend on
$F$
. We let
$\unicode[STIX]{x1D6F4}$
denote the union over
$\text{Aut}(\mathbb{C}/L)$
of the tuples
$(g_{\unicode[STIX]{x1D70E}},x_{\unicode[STIX]{x1D70E}},z_{\unicode[STIX]{x1D70E}})\in D$
. In particular,
$\unicode[STIX]{x1D6F4}$
is contained in the subset
$$\begin{eqnarray}\displaystyle D(d,c_{{\mathcal{F}}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}). & & \displaystyle \nonumber\end{eqnarray}$$
Let
$\unicode[STIX]{x1D70B}_{1}$
and
$\unicode[STIX]{x1D70B}_{2}$
be the projection maps from
$\mathbb{R}^{n^{2}+2N}\times \mathbb{R}^{2N}$
to
$\mathbb{R}^{n^{2}+2N}$
and
$\mathbb{R}^{2N}$
, respectively, and suppose, for the sake of obtaining a contradiction, that
$$\begin{eqnarray}\displaystyle \#\text{Aut}(\mathbb{C}/L)\cdot P=\#\unicode[STIX]{x1D70B}_{2}(\unicode[STIX]{x1D6F4})>cc_{{\mathcal{F}}}^{\unicode[STIX]{x1D705}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}_{{\mathcal{F}}}}. & & \displaystyle \nonumber\end{eqnarray}$$
Then, by Theorem 9.1, there exists a continuous definable function
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}:[0,1]\rightarrow D, & & \displaystyle \nonumber\end{eqnarray}$$
such that
$\unicode[STIX]{x1D6FD}_{1}:=\unicode[STIX]{x1D70B}_{1}\circ \unicode[STIX]{x1D6FD}$
is semialgebraic,
$\unicode[STIX]{x1D6FD}_{2}:=\unicode[STIX]{x1D70B}_{2}\circ \unicode[STIX]{x1D6FD}$
is non-constant,
$\unicode[STIX]{x1D6FD}(0)\in \unicode[STIX]{x1D6F4}$
, and
$\unicode[STIX]{x1D6FD}_{|(0,1)}$
is real analytic. Let
$z_{0}:=\unicode[STIX]{x1D6FD}_{2}(0)$
and let
$P_{0}:=\unicode[STIX]{x1D70B}(z_{0})$
. To obtain a contradiction, we will closely imitate arguments found in [Reference OrrOrr17].
It follows from the Global Decomposition Theorem (see [Reference Grauert and RemmertGR84, p. 172]) that there exists
$0<t\leqslant 1$
such that
$\unicode[STIX]{x1D6FD}_{2}([0,t))$
intersects only finitely many of the irreducible analytic components of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. In fact, since
$\unicode[STIX]{x1D6FD}_{2|(0,t)}$
is real analytic,
$\unicode[STIX]{x1D6FD}_{2}((0,t))$
must be wholly contained in one such component
$V_{1}$
. Since
$V_{1}$
is closed, we conclude from the fact that
$\unicode[STIX]{x1D6FD}$
is continuous that
$V_{1}$
contains
$\unicode[STIX]{x1D6FD}_{2}([0,t])$
.
By [Reference Ullmo and YafaevUY18, Theorem 1.3] (the inverse Ax–Lindemann conjecture),
$\langle V_{1}\rangle _{\text{Zar}}$
is pre-weakly special and so, since
$V$
is Hodge generic in
$S$
, we can decompose
$G^{\text{ad}}=G_{1}\times G_{2}$
, and thus
$X=X_{1}\times X_{2}$
, so that
$$\begin{eqnarray}\displaystyle \langle V_{1}\rangle _{\text{Zar}}=X_{1}\times \{x_{2}\}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$x_{2}\in X_{2}$
is Hodge generic. By abuse of notation, we denote by
$\unicode[STIX]{x1D70B}_{2}$
both the projection from
$G$
to
$G_{2}$
and from
$X^{\vee }$
to
$X_{2}^{\vee }$
.
Note that, for any
$(g,x)\in \text{Im}(\unicode[STIX]{x1D6FD}_{1})$
, we have
$(g^{-1}x)(\mathbb{S})\subseteq G_{F,\mathbb{R}}$
. If we write
$G_{F}^{\prime }$
for the largest normal subgroup of
$G_{F}$
of non-compact type, then the properties of Shimura data imply that
$g^{-1}x$
factors through
$G_{F,\mathbb{R}}^{\prime }$
and, if we write
$\mathfrak{X}^{\prime }$
for the
$G_{F}^{\prime }(\mathbb{R})$
conjugacy class of
$g^{-1}x$
in
$\mathfrak{X}$
, then, by [Reference UllmoUll07, Lemme 3.3],
$(G_{F}^{\prime },\mathfrak{X}^{\prime })$
is a Shimura subdatum of
$(G,\mathfrak{X})$
. Furthermore, by [Reference Ullmo and YafaevUY14, Lemma 3.7], the number of Shimura subdata
$(G_{F}^{\prime },\mathfrak{Y})$
of
$(G,\mathfrak{X})$
is finite and, by [Reference MilneMil05, Corollary 5.3], the number of connected components
$Y$
of
$\mathfrak{Y}$
is also finite. It follows that, after possibly replacing
$t$
, we can and do assume that
$g^{-1}x$
belongs to one such component
$Y$
, which we write as
$Y_{1}\times Y_{2}$
, such that
$F(\mathbb{R})^{+}$
acts transitively on
$Y_{1}$
. In particular,
$$\begin{eqnarray}\displaystyle \dim Y_{1}=\dim \langle P_{0}\rangle . & & \displaystyle \nonumber\end{eqnarray}$$
We let
$p_{2}$
denote the projection from
$Y^{\vee }$
to
$Y_{2}^{\vee }$
.
Let
$B$
denote the complex algebraic subset of
$G(\mathbb{C})\times X^{\vee }$
of dimension at most
$1$
containing
$\text{Im}(\unicode[STIX]{x1D6FD}_{1})$
afforded to us by Lemma 14.1. For any
$(g,x)\in B$
, we have
$g^{-1}x\in Y^{\vee }$
.
Let
$\overline{V}_{1}$
denote the Zariski closure of
$V_{1}$
in
$X^{\vee }$
and consider the complex algebraic set
$$\begin{eqnarray}\displaystyle W_{B}:=\{(g,x,y)\in B\times Y^{\vee }:p_{2}(y)=p_{2}(g^{-1}x),gy\in \overline{V}_{1}\}. & & \displaystyle \nonumber\end{eqnarray}$$
Let
$V_{B}$
denote the Zariski closure in
$X^{\vee }$
of the set
$$\begin{eqnarray}\displaystyle \{gy:(g,x,y)\in W_{B}\}. & & \displaystyle \nonumber\end{eqnarray}$$
Since the latter is the image of
$W_{B}$
under an algebraic morphism, we have
$\dim V_{B}\leqslant \dim W_{B}$
.
Since
$V_{1}$
is an irreducible complex analytic curve having uncountable intersection with
$V_{B}$
, it follows that
$V_{1}$
is contained in
$V_{B}$
. Therefore,
$\langle V_{1}\rangle _{\text{Zar}}$
is contained in
$V_{B}$
also, and so
$$\begin{eqnarray}\displaystyle \dim X_{1}=\dim \langle V_{1}\rangle _{\text{Zar}}\leqslant \dim V_{B}\leqslant \dim W_{B}. & & \displaystyle\end{eqnarray}$$
Now, for each
$(g,x)\in B$
, consider the fibre
$W_{(g,x)}$
of
$W_{B}$
over
$(g,x)$
, i.e., the set
$$\begin{eqnarray}\displaystyle \{y\in Y^{\vee }:p_{2}(y)=p_{2}(g^{-1}x),\unicode[STIX]{x1D70B}_{2}(y)=\unicode[STIX]{x1D70B}_{2}(g)^{-1}x_{2}\}. & & \displaystyle \nonumber\end{eqnarray}$$
Since
$P_{0}\in V$
, it follows that
$\unicode[STIX]{x1D70B}_{2}(F)=G_{2}$
and so, for any
$y\in Y_{2}^{\vee }$
, the natural projection
$$\begin{eqnarray}\displaystyle Y_{1}^{\vee }\times \{y\}\rightarrow X_{2}^{\vee } & & \displaystyle \nonumber\end{eqnarray}$$
is an equivariant morphism of
$F(\mathbb{C})$
-homogeneous spaces. In particular, its fibres are equidimensional of dimension
$$\begin{eqnarray}\displaystyle \dim Y_{1}^{\vee }-\dim X_{2}^{\vee }=\dim Y_{1}-\dim X_{2}. & & \displaystyle \nonumber\end{eqnarray}$$
Since
$W_{(g,x)}$
is contained in such a fibre, we have
$$\begin{eqnarray}\displaystyle \dim W_{(g,x)}\leqslant \dim Y_{1}-\dim X_{2}\leqslant \dim X-2-\dim X_{2}=\dim X_{1}-2, & & \displaystyle \nonumber\end{eqnarray}$$
where we use the fact that
$P_{0}\in \text{Opt}_{0}(V)$
, hence,
$$\begin{eqnarray}\displaystyle \dim Y_{1}=\unicode[STIX]{x1D6FF}(P_{0})\leqslant \unicode[STIX]{x1D6FF}(V)-1=\dim X-2. & & \displaystyle \nonumber\end{eqnarray}$$
Since this holds for all
$(g,x)\in B$
and
$\dim B\leqslant 1$
, we conclude that
$$\begin{eqnarray}\displaystyle \dim W_{B}\leqslant \dim X_{1}-1, & & \displaystyle \nonumber\end{eqnarray}$$
which contradicts (14.2.1). ◻
Of course, Theorem 14.2 is not really satisfactory in the sense that it only deals with curves. One would hope that, for
$V$
of arbitrary dimension, a path such as
$\unicode[STIX]{x1D6FD}$
would yield, via the weak hyperbolic Ax–Schanuel conjecture, a positive dimensional subvariety of
$V$
, containing a conjugate of
$P$
, having defect at most
$\unicode[STIX]{x1D6FF}(P)$
, thus contradicting the optimality of
$P$
. However, the authors have not been able to carry out this procedure. Instead, the very same idea appears to work when one attempts to contradict the membership of a point in the open-anomalous locus. The difference is that we are only required to bound the weakly special defect, as opposed to the defect itself.
Theorem 14.3. Assume that Conjecture 11.3 holds and assume that the weak hyperbolic Ax–Schanuel conjecture is true. Assume also that either
Then, Conjecture 8.4 is true, i.e., if
$V$
is a subvariety of
$S$
, then the set
$$\begin{eqnarray}\displaystyle V^{\text{oa}}\cap S^{[1+\dim V]} & & \displaystyle \nonumber\end{eqnarray}$$
is finite.
Proof. We will assume that Conjectures 10.3, 12.6, and 12.7 hold. The proof in the case that Conjecture 12.2 holds is very similar, hence we omit it. We used Conjecture 12.2 in the proof of Theorem 14.2.
Let
$\unicode[STIX]{x1D6FA}$
denote a finite set of semisimple subgroups of
$G$
defined over
$\mathbb{Q}$
as in § 12. Let
$c_{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
be the constants afforded to us by Conjecture 12.7, let
$d$
,
$c_{\text{H}}$
, and
$\unicode[STIX]{x1D6FF}_{\text{H}}$
be the constants afforded to us by Lemma 12.8, and let
$$\begin{eqnarray}\displaystyle c:=\max \{c_{\text{H}},c_{\unicode[STIX]{x1D6E4}}\}\quad \text{and}\quad \unicode[STIX]{x1D6FF}:=\max \{\unicode[STIX]{x1D6FF}_{\text{H}},\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}\}. & & \displaystyle \nonumber\end{eqnarray}$$
Let
$L^{\prime }$
be a finitely generated extension of
$F$
contained in
$\mathbb{C}$
over which
$V$
is defined and let
$c_{G}$
and
$\unicode[STIX]{x1D6FF}_{G}$
be the constants afforded to us by Conjecture 11.1. Let
$\unicode[STIX]{x1D705}:=\unicode[STIX]{x1D6FF}_{G}/3$
, and let
$c_{\unicode[STIX]{x1D705}}$
be the constant afforded to us by Conjecture 12.6. Let
$$\begin{eqnarray}\displaystyle P\in V^{\text{oa}}\cap S^{[1+\dim V]} & & \displaystyle \nonumber\end{eqnarray}$$
and let
$L$
and
$H$
be, respectively, the finite field extension of
$F$
and the semisimple subgroup of
$G$
defined over
$\mathbb{Q}$
of non-compact type afforded to us by Conjecture 12.6 applied to
$\langle P\rangle$
. Replacing
$L$
by its compositum with
$L^{\prime }$
, we have
$$\begin{eqnarray}\displaystyle [L:L^{\prime }]\leqslant c_{\unicode[STIX]{x1D705}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D705}}. & & \displaystyle \nonumber\end{eqnarray}$$
We claim that there exists a positive constant
$c_{3}$
, independent of
$P$
, such that
$$\begin{eqnarray}\displaystyle \#\text{Aut}(\mathbb{C}/L)\cdot P\leqslant c_{3}c^{\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D705}}. & & \displaystyle \nonumber\end{eqnarray}$$
This would be sufficient to prove Theorem 14.3 since then, by Conjecture 11.3, we obtain
$$\begin{eqnarray}\displaystyle \frac{c_{G}}{c_{\unicode[STIX]{x1D705}}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{2\unicode[STIX]{x1D705}}\leqslant \frac{1}{[L:L^{\prime }]}\#\text{Aut}(\mathbb{C}/L^{\prime })\cdot P=\#\text{Aut}(\mathbb{C}/L)\cdot P\leqslant c_{3}c^{\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D705}} & & \displaystyle \nonumber\end{eqnarray}$$
and, rearranging this expression, we obtain
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}(\langle P\rangle )\leqslant (c_{3}c^{\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FF}}c_{\unicode[STIX]{x1D705}}c_{G}^{-1})^{1/\unicode[STIX]{x1D705}}, & & \displaystyle \nonumber\end{eqnarray}$$
which is a bound independent of
$P$
. We remind the reader that, as explained in Remark 11.4,
$P\in \text{Opt}_{0}(V)$
and, therefore,
$P$
is one of only finitely many irreducible components of
$\langle P\rangle \cap V$
. Hence, Theorem 14.3 would follow from Conjecture 10.3 and it remains only, therefore, to prove the claim.
By Conjecture 12.6, for each
$\unicode[STIX]{x1D70E}\in \text{Aut}(\mathbb{C}/L)$
,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}(\langle P\rangle )=\unicode[STIX]{x1D70B}(H(\mathbb{R})^{+}x_{\unicode[STIX]{x1D70E}}), & & \displaystyle \nonumber\end{eqnarray}$$
where
$H(\mathbb{R})^{+}x_{\unicode[STIX]{x1D70E}}$
is a pre-special subvariety of
$X$
intersecting
${\mathcal{F}}$
. By Lemma 12.8, we can and do assume that
$$\begin{eqnarray}\displaystyle \text{H}_{d}(x_{\unicode[STIX]{x1D70E}})\leqslant c_{\text{H}}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70E}(\langle P\rangle ))^{\unicode[STIX]{x1D6FF}_{\text{H}}}=c_{\text{H}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D6FF}_{\text{H}}}. & & \displaystyle \nonumber\end{eqnarray}$$
We let
$z_{\unicode[STIX]{x1D70E}}\in {\mathcal{V}}$
be a point in
$\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D70E}(P))$
, so that
$$\begin{eqnarray}\displaystyle z_{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D6E4}H(\mathbb{R})^{+}x_{\unicode[STIX]{x1D70E}} & & \displaystyle \nonumber\end{eqnarray}$$
and so, by Conjecture 12.7, there exists
$\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D6E4}$
satisfying
$$\begin{eqnarray}\displaystyle \text{H}_{1}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70E}})\leqslant c_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6E5}(\langle P\rangle )^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}} & & \displaystyle \nonumber\end{eqnarray}$$
such that
$z_{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70E}}H(\mathbb{R})^{+}x_{\unicode[STIX]{x1D70E}}$
.
By definition, there exists
$F\in \unicode[STIX]{x1D6FA}$
and
$g\in G(\mathbb{R})$
such that
$H_{\mathbb{R}}$
is equal to
$gF_{\mathbb{R}}g^{-1}$
. In particular, for each
$\unicode[STIX]{x1D70E}\in \text{Aut}(\mathbb{C}/L)$
, the tuple
$(g,(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70E}},x_{\unicode[STIX]{x1D70E}}),z_{\unicode[STIX]{x1D70E}})$
belongs to the definable family
$D$
of tuples
$$\begin{eqnarray}\displaystyle (g,(\unicode[STIX]{x1D6FE},x),z)\in G(\mathbb{R})\times [G(\mathbb{R})\times X]\times X\subseteq \mathbb{R}^{n^{2}}\times \mathbb{R}^{n^{2}+2N}\times \mathbb{R}^{2N}, & & \displaystyle \nonumber\end{eqnarray}$$
parametrized by
$G(\mathbb{R})$
, such that
$$\begin{eqnarray}\displaystyle z\in {\mathcal{V}}\cap \unicode[STIX]{x1D6FE}gF(\mathbb{R})^{+}g^{-1}x\quad \text{and}\quad x(\mathbb{S})\subseteq gG_{F}g^{-1}. & & \displaystyle \nonumber\end{eqnarray}$$
We choose, then, for
$c_{3}$
the constant
$c(D,d,\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FF})$
afforded to us by Theorem 9.1 applied to
$D$
. Since,
$\unicode[STIX]{x1D6FA}$
is finite, we can and do assume that
$c_{3}$
does not depend on
$F$
. We let
$\unicode[STIX]{x1D6F4}$
denote the union over
$\text{Aut}(\mathbb{C}/L)$
of the tuples
$((\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70E}},x_{\unicode[STIX]{x1D70E}}),z_{\unicode[STIX]{x1D70E}})\in D_{g}$
(to use the notation of § 9). In particular,
$\unicode[STIX]{x1D6F4}$
is contained in the subset
$$\begin{eqnarray}\displaystyle D_{g}(d,c\unicode[STIX]{x1D6E5}(Z)^{\unicode[STIX]{x1D6FF}}). & & \displaystyle \nonumber\end{eqnarray}$$
Let
$\unicode[STIX]{x1D70B}_{1}$
and
$\unicode[STIX]{x1D70B}_{2}$
be the projection maps from
$\mathbb{R}^{n^{2}+2N}\times \mathbb{R}^{2N}$
to
$\mathbb{R}^{n^{2}+2N}$
and
$\mathbb{R}^{2N}$
, respectively, and suppose, for the sake of obtaining a contradiction, that
$$\begin{eqnarray}\displaystyle \#\text{Aut}(\mathbb{C}/L)\cdot P=\#\unicode[STIX]{x1D70B}_{2}(\unicode[STIX]{x1D6F4})>c_{3}c^{\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6E5}(Z)^{\unicode[STIX]{x1D705}}. & & \displaystyle \nonumber\end{eqnarray}$$
Then, by Theorem 9.1, there exists a continuous definable function
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}:[0,1]\rightarrow D_{g}, & & \displaystyle \nonumber\end{eqnarray}$$
such that
$\unicode[STIX]{x1D6FD}_{1}:=\unicode[STIX]{x1D70B}_{1}\circ \unicode[STIX]{x1D6FD}$
is semialgebraic,
$\unicode[STIX]{x1D6FD}_{2}:=\unicode[STIX]{x1D70B}_{2}\circ \unicode[STIX]{x1D6FD}$
is non-constant,
$\unicode[STIX]{x1D6FD}(0)\in \unicode[STIX]{x1D6F4}$
, and
$\unicode[STIX]{x1D6FD}_{|(0,1)}$
is real analytic. Let
$z_{0}:=\unicode[STIX]{x1D6FD}_{2}(0)$
and
$(\unicode[STIX]{x1D6FE}_{0},x_{0}):=\unicode[STIX]{x1D6FD}_{1}(0)$
. Denote by
$P_{0}$
the point
$\unicode[STIX]{x1D70B}(z_{0})$
and denote by
$X_{0}$
the pre-special subvariety
$\unicode[STIX]{x1D6FE}_{0}H(\mathbb{R})^{+}x_{0}$
.
We claim that there exists a positive dimensional intersection component
$A$
of
$\unicode[STIX]{x1D70B}^{-1}(V)$
containing
$z_{0}$
. To see this, let
$W$
denote the union of the totally geodesic subvarieties
$\unicode[STIX]{x1D6FE}H(\mathbb{R})^{+}x$
of
$X$
, where
$(\unicode[STIX]{x1D6FE},x)$
varies over
$\text{Im}(\unicode[STIX]{x1D6FD}_{1})$
, and let
$\overline{W}$
denote the Zariski closure of
$W$
in
$X^{\vee }$
. The irreducible analytic components of
$\overline{W}\cap \unicode[STIX]{x1D70B}^{-1}(V)$
are, by definition, intersection components of
$\unicode[STIX]{x1D70B}^{-1}(V)$
. It follows from the global decomposition theorem (see [Reference Grauert and RemmertGR84, p. 172]) that there exists
$0<t\leqslant 1$
such that
$\unicode[STIX]{x1D6FD}_{2}([0,t))$
intersects only finitely many of said components. In fact, since
$\unicode[STIX]{x1D6FD}_{2|(0,t)}$
is real analytic,
$\unicode[STIX]{x1D6FD}_{2}((0,t))$
must be wholly contained in one such component
$A$
. Since
$A$
is closed, we conclude from the fact that
$\unicode[STIX]{x1D6FD}$
is continuous that
$A$
contains
$\unicode[STIX]{x1D6FD}_{2}([0,t])$
, which proves the claim.
Let
$B$
denote a Zariski optimal intersection component of
$\unicode[STIX]{x1D70B}^{-1}(V)$
containing
$A$
such that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\text{Zar}}(B)\leqslant \unicode[STIX]{x1D6FF}_{\text{Zar}}(A), & & \displaystyle \nonumber\end{eqnarray}$$
and let
$Z$
denote the Zariski closure of
$\unicode[STIX]{x1D70B}(B)$
in
$S$
. By the weak hyperbolic Ax–Schanuel conjecture,
$\langle B\rangle _{\text{Zar}}$
is pre-weakly special and, as in the proof of Lemma 6.8,
$$\begin{eqnarray}\displaystyle \langle Z\rangle _{\text{ws}}=\unicode[STIX]{x1D70B}(\langle B\rangle _{\text{Zar}}). & & \displaystyle \nonumber\end{eqnarray}$$
Therefore, we have
$\dim Z\geqslant 1$
and, also,
$$\begin{eqnarray}\displaystyle \dim Z\geqslant \dim B\geqslant \dim \langle B\rangle _{\text{Zar}}-\unicode[STIX]{x1D6FF}_{\text{Zar}}(A)\geqslant \dim \langle Z\rangle _{\text{ws}}-(\dim \overline{W}-1), & & \displaystyle \nonumber\end{eqnarray}$$
where we use the fact that
$\unicode[STIX]{x1D6FF}_{\text{Zar}}(A)$
is at most
$\dim \overline{W}-1$
. We claim that
$\dim \overline{W}-1\leqslant \dim X_{0}$
, which would conclude the proof as
$$\begin{eqnarray}\displaystyle \dim X_{0}\leqslant \dim S-\dim V-1 & & \displaystyle \nonumber\end{eqnarray}$$
and this would imply that
$Z\in \text{an}(V)$
, which is not allowed as
$P_{0}\in Z$
.
Therefore, it remains to prove the claim. However, this is easy to prove working with complex duals and using the methods explained in the proof of Theorem 14.2. ◻
15 A brief note on special anomalous subvarieties
In their paper [Reference Bombieri, Masser and ZannierBMZ07], Bombieri et al. also defined what they referred to as a torsion anomalous subvariety. We will make the analogous definition in the context of Shimura varieties. Let
$V$
be a subvariety of
$S$
.
Definition 15.1. A subvariety
$W$
of
$V$
is called special anomalous in
$V$
if
$$\begin{eqnarray}\displaystyle \dim W\geqslant \max \{1,1+\dim V+\dim \langle W\rangle -\dim S\}. & & \displaystyle \nonumber\end{eqnarray}$$
A subvariety of
$V$
is maximal special anomalous in
$V$
if it is special anomalous in
$V$
and not strictly contained in another subvariety of
$V$
that is also special anomalous in
$V$
.
The similarity with the definition of an atypical subvariety is clear. Indeed, it is immediate that a positive dimensional subvariety of
$V$
that is atypical with respect to
$V$
is special anomalous in
$V$
. However, since a subvariety
$W$
of
$V$
that is special anomalous in
$V$
is not necessarily an irreducible component of
$V\,\cap \,\langle W\rangle$
, it is not necessarily the case that
$W$
is atypical with respect to
$V$
. Nonetheless, it follows that the properties of being maximal special anomalous and atypical are equivalent for positive dimensional subvarieties of
$V$
. In particular, Conjecture 1.4 implies the following analogue of the torsion openness conjecture of Bombieri, Masser, and Zannier.
Conjecture 15.2. There are only finitely many subvarieties of
$V$
that are maximal special anomalous in
$V$
.
Of course, one would more naturally translate the torsion openness conjecture as follows.
Conjecture 15.3. The complement
$V^{\text{sa}}$
in
$V$
of the subvarieties of
$V$
that are special anomalous in
$V$
is open in
$V$
.
However, these two formulations are equivalent. Indeed, the statement that Conjecture 15.2 implies Conjecture 15.3 is obvious. On the other hand, suppose that Conjecture 15.3 were true. Then the union of all subvarieties of
$V$
that are positive dimensional and atypical with respect to
$V$
would be closed in
$V$
. In particular, we could write it as a finite union of subvarieties of
$V$
. However, since there are only countably many subvarieties of
$V$
that are atypical with respect to
$V$
, it follows that each member of the aforementioned union would be atypical with respect to
$V$
.
Of course, if one could prove Conjecture 15.2, one would reduce the Zilber–Pink conjecture to the following analogue of the torsion finiteness conjecture of Bombieri, Masser, and Zannier.
Conjecture 15.4. There are only finitely many points in
$V$
that are maximal atypical with respect to
$V$
.
In this article, we have concerned ourselves with optimal subvarieties. Now, it is straightforward to verify that a subvariety
$W$
of
$V$
that is optimal in
$V$
is atypical with respect to
$V$
. However, it is not necessarily the case that
$W$
is maximal atypical. On the other hand, a subvariety of
$V$
that is maximal atypical with respect to
$V$
is optimal. In particular, the points in
$V$
that are maximal atypical with respect to
$V$
constitute a (possibly proper) subset of
$\text{Opt}_{0}(V)$
.
Again, it would be more natural to translate the torsion finiteness conjecture as follows.
Conjecture 15.5. For any integer
$d$
, let
$S^{[d]}$
denote the union of the special subvarieties contained in
$S$
having codimension at least
$d$
. Then
$$\begin{eqnarray}\displaystyle V^{\text{sa}}\cap S^{[1+\dim V]} & & \displaystyle \nonumber\end{eqnarray}$$
is finite. Equivalently, there are only finitely many points
$P\in V^{\text{sa}}$
such that
$$\begin{eqnarray}\displaystyle \dim \langle P\rangle \leqslant \dim S-\dim V-1. & & \displaystyle \nonumber\end{eqnarray}$$
However, the two formulations are also equivalent. Indeed, Conjecture 15.5 implies Conjecture 15.4 because a point
$P\in V$
that is maximal atypical with respect to
$V$
is contained in
$V^{\text{sa}}$
and
$$\begin{eqnarray}\displaystyle \dim \langle P\rangle \leqslant \dim S-\dim V-1. & & \displaystyle \nonumber\end{eqnarray}$$
On the other hand, suppose that Conjecture 15.4 were true and consider a point
$P\in V^{\text{sa}}$
such that
$$\begin{eqnarray}\displaystyle \dim \langle P\rangle \leqslant \dim S-\dim V-1. & & \displaystyle \nonumber\end{eqnarray}$$
Then
$P$
is a component of
$\langle P\rangle \cap V$
. Otherwise, such a component
$W$
containing
$P$
would be special anomalous in
$V$
, which would contradict the fact that
$P\in V^{\text{sa}}$
. Therefore,
$P$
is atypical with respect to
$V$
and, in fact, maximal atypical with respect to
$V$
.
In their article [Reference Bombieri, Masser and ZannierBMZ07], Bombieri et al. showed that, in fact, the torsion openness conjecture implies the torsion finiteness conjecture. We imitate their argument to show the following.
Proposition 15.6. Let
$Y(1)$
denote the modular curve associated with
$\text{SL}_{2}(\mathbb{Z})$
. If Conjecture 15.3 is true for
$S\times Y(1)$
, then Conjecture 15.5 is true for
$S$
.
Proof. We denote by
$U$
the complement in
$V^{\text{sa}}$
of
$V^{\text{sa}}\cap S^{[\dim V+1]}$
. Regarding
$V\times Y(1)$
as a subvariety of the Shimura variety
$S\times Y(1)$
, we claim that
$$\begin{eqnarray}\displaystyle (V\times Y(1))^{\text{sa}}=U\times Y(1). & & \displaystyle\end{eqnarray}$$
To see this, first let
${\hat{Y}}$
be a special anomalous subvariety of
$V\times Y(1)$
. Then
$$\begin{eqnarray}\displaystyle \dim {\hat{Y}} & {\geqslant} & \displaystyle 1+\dim \langle {\hat{Y}}\rangle +\dim (V\times Y(1))-\dim (S\times Y(1))\nonumber\\ \displaystyle & = & \displaystyle 1+\dim \langle {\hat{Y}}\rangle +(1+\dim V)-(1+\dim S)\nonumber\\ \displaystyle & = & \displaystyle 1+\dim \langle {\hat{Y}}\rangle +\dim V-\dim S.\nonumber\end{eqnarray}$$
We denote by
$\unicode[STIX]{x1D70B}_{S}$
the projection
$\unicode[STIX]{x1D70B}_{S}:S\times Y(1)\rightarrow S$
. Then the Zariski closure
$Y$
of
$\unicode[STIX]{x1D70B}_{S}({\hat{Y}})$
lies in the special subvariety
$\unicode[STIX]{x1D70B}_{S}(\langle {\hat{Y}}\rangle )$
of
$S$
. If
$\dim Y=\dim {\hat{Y}}$
, then
$$\begin{eqnarray}\displaystyle \dim Y & = & \displaystyle \dim {\hat{Y}}\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 1+\dim \langle {\hat{Y}}\rangle +\dim V-\dim S\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 1+\dim \unicode[STIX]{x1D70B}_{S}(\langle {\hat{Y}}\rangle )+\dim V-\dim S\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 1+\dim \langle Y\rangle +\dim V-\dim S\nonumber\end{eqnarray}$$
and
$Y$
is special anomalous in
$V$
. If
$1\leqslant \dim Y<\dim {\hat{Y}}$
, then
$\dim Y=\dim {\hat{Y}}\,-\,1$
and
${\hat{Y}}=Y\,\times \,Y(1)$
, which implies
$\langle {\hat{Y}}\rangle =\langle Y\rangle \times Y(1)$
. Therefore,
$$\begin{eqnarray}\displaystyle \dim Y & = & \displaystyle \dim {\hat{Y}}-1\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 1+\dim \langle {\hat{Y}}\rangle +\dim V-\dim S-1\nonumber\\ \displaystyle & = & \displaystyle 1+\dim \langle Y\rangle +1+\dim V-\dim S-1\nonumber\\ \displaystyle & = & \displaystyle 1+\dim \langle Y\rangle +\dim V-\dim S\nonumber\end{eqnarray}$$
and
$Y$
is special anomalous in
$V$
. In both cases,
${\hat{Y}}$
is contained in
$$\begin{eqnarray}\displaystyle Y\times Y(1)\subseteq (V\backslash V^{\text{sa}})\times Y(1)\subseteq (V\backslash U)\times Y(1)=(V\times Y(1))\backslash (U\times Y(1)). & & \displaystyle \nonumber\end{eqnarray}$$
Finally, if
$\dim Y=0$
, then
${\hat{Y}}=\{P\}\times Y(1)$
for some
$P\in S$
. Since
${\hat{Y}}$
is special anomalous, we have
$$\begin{eqnarray}\displaystyle 1 & = & \displaystyle \dim {\hat{Y}}\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 1+\dim \langle {\hat{Y}}\rangle +(1+\dim V)-(1+\dim S)\nonumber\\ \displaystyle & = & \displaystyle 1+\dim \langle P\rangle +1+\dim V-\dim S\nonumber\end{eqnarray}$$
and
$P\in S^{[1+\dim V]}$
. Therefore,
$$\begin{eqnarray}\displaystyle {\hat{Y}} & \subseteq & \displaystyle V\backslash (V-S^{[1+\dim V]})\times Y(1)\nonumber\\ \displaystyle & \subseteq & \displaystyle V\backslash (V^{\text{sa}}-S^{[1+\dim V]})\times Y(1)\nonumber\\ \displaystyle & = & \displaystyle (V\backslash U)\times Y(1)\nonumber\\ \displaystyle & = & \displaystyle (V\times Y(1))\backslash (U\times Y(1)).\nonumber\end{eqnarray}$$
We conclude that
$(V\times Y(1))^{\text{sa}}\subseteq (U\times Y(1))$
.
On the other hand, for any
$P\in V\backslash U=V\backslash (V^{\text{sa}}-S^{[1+\dim V]})$
, we have either
$P\notin V^{\text{sa}}$
or
$P\in V^{\text{sa}}\cap S^{[1+\dim V]}$
.
If
$P\notin V^{\text{sa}}$
, then
$P$
is contained in a special anomalous
$Y$
of
$V$
. Therefore,
$$\begin{eqnarray}\displaystyle \dim (Y\times Y(1)) & = & \displaystyle 1+\dim Y\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 1+1+\dim \langle Y\rangle +\dim V-\dim S\nonumber\\ \displaystyle & = & \displaystyle 1+\dim \langle Y\times Y(1)\rangle +\dim (V\times Y(1))-\dim (S\times Y(1))\nonumber\end{eqnarray}$$
and
$Y\times Y(1)\subset (V\times Y(1))\backslash (U\times Y(1))$
is special anomalous.
If
$P\in V^{\text{sa}}\cap S^{[1+\dim V]}$
, we let
${\hat{Y}}=\{P\}\times Y(1)$
. Then
$$\begin{eqnarray}\displaystyle \dim {\hat{Y}} & = & \displaystyle 1\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 1+1+\dim \langle P\rangle +\dim V-\dim S\nonumber\\ \displaystyle & = & \displaystyle 1+\dim \langle {\hat{Y}}\rangle +\dim (V\times Y(1))-\dim (S\times Y(1))\nonumber\end{eqnarray}$$
and
${\hat{Y}}\subseteq (V\,\times \,Y(1))\backslash (U\,\times \,Y(1))$
is special anomalous. We conclude that
$(U\,\times \,Y(1))\subseteq (V\,\times \,Y(1))^{\text{sa}}$
, which proves (15.6.1).
Therefore, if Conjecture 15.3 is true for
$S\times Y(1)$
, we conclude that
$U\times Y(1)$
is open in
$V\times Y(1)$
. Therefore,
$U$
is open in
$V$
. On the other hand,
$V^{\text{sa}}$
is also open in
$V$
, whereas
$V^{\text{sa}}\cap S^{[1+\dim V]}$
is at most countable since there are only countably many special subvarieties and each point in
$V^{\text{sa}}\,\cap \,S^{[1+\dim V]}$
is an irreducible component of
$V\,\cap \,Z$
for some special subvariety
$Z$
of codimension at most
$1+\dim V$
. It follows that
$V^{\text{sa}}\cap S^{[1+\dim V]}$
is finite.◻
Acknowledgements
The first author would like to thank the EPSRC, as well as Jonathan Pila, for the opportunity to be part of the project Model Theory, Functional Transcendence, and Diophantine Geometry as a postdoctoral research assistant. He would like to thank Linacre College, Oxford, the Mathematical Institute at the University of Oxford, and the Department of Mathematics and Statistics at the University of Reading, all for providing excellent working conditions. Finally, he would like to thank Martin Orr, Jonathan Pila, Harry Schmidt, Emmanuel Ullmo, and Andrei Yafaev for several valuable discussions. The second author is grateful to the Institut des Hautes Études Scientifiques and the Université Paris Saclay for providing great environments in which to work. He would like to thank his supervisor Emmanuel Ullmo for regular discussions and constant support during the preparation of this article and he would like to thank Mikhail Borovoi, Philipp Habegger, Ziyang Gao, Martin Orr, and Jonathan Pila for several useful discussions. His work was supported by grants from Région l’Île de France. Both authors would like to thank Martin Orr, for sharing drafts of his preprint [Reference OrrOrr17], as well as the anonymous referee, for a long list of invaluable comments.
