2.1 Definitions

Let $(G,X)$ be a Shimura datum. We always assume, as in [Reference Ullmo and YafaevUY14], that our Shimura datum is normalised so that $G$ is the generic Mumford–Tate group of $X$.

Let $x_0$ be a point of $X$ and let $M \leq G$ be the Mumford–Tate group of $x_0$. Recall that $x_0$ is a morphism ${\mathbb {S}} :=\operatorname {Res}_{{\mathbb {C}}/{\mathbb {R}}}(GL(1))\longrightarrow G_{{\mathbb {R}}}$ and that $M = x_0({\mathbb {S}})^{Zar, {\mathbb {Q}}}$ is the smallest ${\mathbb {Q}}$-algebraic subgroup of $G$ containing $x_0({\mathbb {S}})$. In the rest of the paper we denote the identity monomorphism $M \hookrightarrow G$ by $\phi _0$.

In the following definition $\mbox {Hom}(M,G)$ denotes the set of algebraic group morphisms defined over ${\mathbb {Q}}$.

Definition 2.1 (Generalised Hecke orbit)

We define the Generalised Hecke orbit $\mathcal {H}(x_0)$ of $x_0$ in $X$ as

\begin{align*} \mathcal{H}(x_0) := X\cap \{ \phi \circ x_0 : \phi \in \mbox{Hom}(M,G)\}. \end{align*}

Let $X_M=M({\mathbb {R}})\cdot x_0\subset X$. Then $(M,X_M)$ is a Shimura datum, and $\phi \in \mbox {Hom}(M,G)$ such that $\phi \circ x_0\in X$ are precisely those giving rise to a morphism of Shimura data $(M,X_M)\to (G,X)$. In particular, $\phi (X_M)\subseteq X$.

Let $K$ be a compact open subgroup of $G({\mathbb {A}}_f)$ and ${\rm Sh}_K(G,X)$ be the Shimura variety associated to these data. There is a natural map

\[ X \times G({\mathbb{A}}_f) \longrightarrow {\rm Sh}_K(G,X) \]

and we denote the image of a point $(x,g)$ by $[x,g]$.

Definition 2.2 We define the generalised Hecke orbit $\mathcal {H}([x_0, g_0])$ of $[x_0, g_0]$ in ${\rm Sh}_K(G,X)$ by

\[ \mathcal{H}([x_0,g_0]) := \{ [x,g] : x\in \mathcal{H}(x_0), g\in G({\mathbb{A}}_f) \}. \]

Let $W=G\cdot \phi _0$ be the conjugacy class of $\phi _0$ which we view as an algebraic variety defined over ${\mathbb {Q}}$. Denoting by $Z_G(M)$ the centraliser of $M$ in $G$, we will identify $G/Z_G(M)\simeq W$. The set $W(\overline {{\mathbb {Q}}})$ is the $G(\overline {{\mathbb {Q}}})$-conjugacy class of $\phi _0$ in $\mbox {Hom}(M_{\overline {{\mathbb {Q}}}},G_{\overline {{\mathbb {Q}}}})$, and the points in $W({\mathbb {Q}})$ are the ${\mathbb {Q}}$-defined homomorphisms $\phi \in \mbox {Hom}(M,G)$ which are conjugated to $\phi _0$ by elements of $G(\overline {{\mathbb {Q}}})$.

In Definition 2.1, if we replace $\mbox {Hom}(M,G)$ by its subset $W({\mathbb {Q}})$, we obtain a more restrictive definition: that of a geometric Hecke orbit.

Definition 2.3 We define the geometric Hecke orbit $\mathcal {H}^g(x_0)$ of $x_0$ by

\begin{align*} \mathcal{H}^g(x_0) =X\cap \{ \phi \circ x_0 : \phi \in W({\mathbb{Q}})\} \subset \mathcal{H}(x_0) \end{align*}

and the geometric Hecke orbit of $[x_0,g_0]$ by

\[ \mathcal{H}^g([x_0,g_0]) = \{ [x,g] : x\in \mathcal{H}^g(x_0), g\in G({\mathbb{A}}_f) \}. \]

The main result of this section is the following.

Proof. One can check directly that

\[ H:=\{m\in M({\mathbb{C}}) : \phi(m)=\phi'(m)\} \]

is a subgroup of $M({\mathbb {C}})$ (it is the ‘equaliser’ of $\phi$ and $\phi '$). It is algebraic and defined over ${\mathbb {Q}}$ because $\phi$ and $\phi '$ are. It contains the image $x_0({\mathbb {C}})$ by hypothesis. But $M$ is the Mumford–Tate group of $x_0$: there is no proper ${\mathbb {Q}}$-algebraic subgroup of $M$ containing $x_0({\mathbb {C}})$. Therefore, $H=M$. Thus, $\phi =\phi '$.

The algebraic variety $W$ is our central object in this article. We will use the notation

(2a)\begin{align} W({\mathbb{R}})^+&=G({\mathbb{R}})/Z_G(M)({\mathbb{R}})\nonumber\\ &=\{\phi\in W({\mathbb{R}}):\phi\circ x_0\in X\} \end{align}

and

(2b)\begin{align} W({\mathbb{Q}})^+&=W({\mathbb{R}})^+\cap W({\mathbb{Q}})\nonumber\\ &=\{\phi\in W({\mathbb{Q}}):\phi\circ x_0\in \mathcal{H}^g(x_0)\}. \end{align}

The subset $W({\mathbb {R}})^+\subset W({\mathbb {R}})$ is a union of some connected components of $W({\mathbb {R}})$. With this notation, Lemma 2.5 implies that we have a bijection

(3)\begin{equation} \begin{aligned} W({\mathbb{Q}})^+ & \stackrel{\sim}{\to}\mathcal{H}^g(x_0)\\ \phi & \mapsto \phi\circ x_0. \end{aligned} \end{equation}

2.2 Functoriality of generalised and geometric Hecke orbits

2.2.1 Restriction to special subvarieties

The following is a set-theoretic tautology.

Proposition 2.6 Let $(G',X')$ be a Shimura datum with $M\leq G'\leq G$ and $X_M\subset X'\subset X$, and define $K'=G'({\mathbb {A}}_f)\cap K$.

1. (i) Let $\mathcal {H}'(x_0)$ be the generalised Hecke orbit of $x_0$ viewed as a point of $X'$.

   Then

   \[ \mathcal{H}'(x_0)= \mathcal{H}(x_0)\cap X'. \]

2. (ii) Let $\mathcal {H}'([x_0,1])$ be the generalised Hecke orbit of $[x_0,1]$ viewed as a point of $Sh_{K'}(G',X')$, and $S'$ the image of

   \[ f:=Sh(\iota):Sh_{K'}(G',X')\to Sh_{K}(G,X) \]
   where $\iota :G'\to G$ is the inclusion. Then
   \[ \mathcal{H}([x_0,1])\cap S' = f(\mathcal{H}'([x_0,1]))\text{ and }\mathcal{H}'([x_0,1]) =\stackrel{-1}{f}(\mathcal{H}([x_0,1])). \]

The following corollary can be deduced by combining Lemma 2.5 with Theorem 2.4 (it can also be deduced from [Reference RichardsonRic67]).

Corollary 2.7 We keep previous notation. Then

\[ \mathcal{H}^g(x_0)\cap X' \]

is a finite union of geometric Hecke orbits in $X'$.

Accordingly, $\stackrel {-1}{f}(\mathcal {H}^g([x_0,1]))$ is the image of finitely many geometric Hecke orbits in $Sh_{K'}(G',X')$.

2.2.2 Compatibility to products

A useful property of geometric Hecke orbits is the compatibility with respect to products of Shimura data.

Lemma 2.8 Let $(G,X)$ be an adjoint Shimura datum, and factor $G=G_1\times \cdots \times G_f$ as a product of its ${\mathbb {Q}}$-defined simple normal subgroups, and assume $K=K_1\times \cdots \times K_f$ for compact open subgroups $K_i\leq G_i({\mathbb {A}}_f)$. We use $X=X_1\times \cdots \times X_f$ to denote the corresponding factorisation, and choose $x_0= (x_1,\ldots,x_f)\in X_1\times \cdots \times X_f$. We use $\mathcal {H}^g(x_i)$ to denote the geometric Hecke orbit of $x_i$ with respect to the Shimura datum $(G_i,X_i)$.

With respect to the corresponding factorisation of Shimura varieties

\[ Sh_K(G,X)=Sh_{K_1}(G_1,X_1)\times \cdots \times Sh_{K_f}(G_f,X_f), \]

we have

\[ \mathcal{H}^g(x_0)=\mathcal{H}^g(x_1)\times \cdots \times \mathcal{H}^g(x_f). \]

It follows from Lemma 2.8 that, at the level of Shimura varieties,

\[ \mathcal{H}^g([x_0,1])=\mathcal{H}^g([x_1,1])\times \cdots \times \mathcal{H}^g([x_f,1] ). \]

Proof. Since $G$ is adjoint, we have a factorisation

\[ X=X_1\times \cdots \times X_f. \]

Let $M$ be the Mumford–Tate group of $x_0$ and let $\phi _0=(\phi _1,\ldots,\phi _f):M\to G=G_1\times \cdots \times G_f$ be the inclusion. As the conjugacy class in a product is the product of conjugacy classes, we have

\[ G\cdot\phi_0=G_1\cdot \phi_1\times \cdots \times G_f\cdot \phi_f. \]

The Mumford–Tate group of $x_i$ is $M_i:=\phi _i(M)$. Because $x_0({\mathbb {S}})$ is Zariski dense over ${\mathbb {Q}}$ in $M$ so is $x_i({\mathbb {S}})$ in $M_i$. Let $\phi _i':M_i\to G_i$ be the identity map. We can identify $G_i\cdot \phi _i\simeq G_i\cdot \phi '_i$, and have

\[ \mathcal{H}^g(x_i)=\{g\cdot \phi'_i\circ x_i : g\in G_i\}\cap X_i=\{g\cdot \phi_i\circ x_i : g\in G_i\}\cap X_i. \]

The rest follows from the definition of geometric Hecke orbits.

2.2.3 Passing to the adjoint Shimura datum

The following property is used to reduce the proof of Conjecture 1.1 and Theorem 1.2 to the case where $G$ is adjoint.

Lemma 2.9 Let $ad:(G,X)\to (G^{\rm ad},X^{\rm ad})$ be the map of Shimura dataFootnote ⁷ induced by the natural morphism $ad:G\to G^{\rm ad}$ and choose a compact open subgroup $K^{\rm ad}\leq G^{\rm ad}({\mathbb {A}}_f)$ containing $ad(K)$. Let $ad:x\mapsto x^{\rm ad}:=ad\circ x$ be the map $X\to X^{\rm ad}$ and

\[ Sh(ad): Sh_K(G,X)\to Sh_{K^{\rm ad}}(G^{\rm ad},X^{\rm ad}) \]

the corresponding morphism of Shimura varieties.

Let $x_0\in X$. Recall that $\mathcal {H}^g(x_0)$ and $\mathcal {H}^g(x_0^{\rm ad})$ denote the geometric Hecke orbit of $x_0$ and $x_0^{\rm ad}$ with respect to $G$ and $G^{\rm ad}$.

We have

(4)\begin{equation} ad(\mathcal{H}^g(x_0)) \subseteq ad(X)\cap \mathcal{H}^g(x_0^{\rm ad}). \end{equation}

Lemma 2.9 implies the inclusion

\[ ad(\mathcal{H}^g(x_0)) \times G({\mathbb{A}}_f) \subseteq \mathcal{H}^g(x_0^{\rm ad}) \times G^{\rm ad}({\mathbb{A}}_f). \]

Passing to the quotient, we obtain the following.

Corollary 2.10 We have $Sh(ad)(\mathcal {H}^g([x_0,1]))\subseteq \mathcal {H}^g([x_0^{\rm ad},1]).$

We now prove Lemma 2.9.

Proof. Choose $x\in \mathcal {H}^g(x_0)$. Clearly $x':=ad(x) \in ad(X) \subset X^{\rm ad}$.

The Mumford–Tate group of $x'_0:=ad(x_0)$ is $M':=ad(M)$. We denote by $\phi _0' \colon M' \to G^{\rm ad}$ the natural injection. We can write $x=\phi \circ x_0$ with $\phi = g \phi _0 g^{-1}$ and $g \in G(\overline {{\mathbb {Q}}})$. Then $\phi ' := ad(g) \phi '_0 ad(g)^{-1}$ is defined over ${\mathbb {Q}}$ because the map $G \cdot \phi _0\to G^{\rm ad} \cdot \phi '_0$ between conjugacy classes is a morphism of varieties defined over ${\mathbb {Q}}$. One computes $x'=ad(gx_0g^{-1})=ad(g)ad(x_0)ad(g)^{-1}=\phi '\circ x_0'$, where $x_0'\in X^{\rm ad}$, and $\phi$ is defined over ${\mathbb {Q}}$ and conjugated to $\phi _0'$ over $\overline {{\mathbb {Q}}}$; that is, $x'\in \mathcal {H}^g(x_0')$.

2.3 Rational conjugacy of linear representations

The following notable fact will be used at several places in this article. We believe this property is also of independent interest.

It follows from the theory of linear representations for which references are for example [Reference HumphreysHum75, Chapter XI] for $\overline {{\mathbb {Q}}}$ and [Reference Borel and TitsBT65, § 12] over ${\mathbb {Q}}$. We will only need the case where $M$ is connected and reductive, and this case can be found, for instance, in [Reference Borel and TitsBT65, § 12.3, third paragraph]. They give a Galois cohomology argument, and the same Galois cohomology argument works in general with a reference to [Reference KneserKne69, 1.7 Example 1, p. 16] instead. For reductive groups, it is also possible to reduce the result to Skolem–Noether theorem. For tori, it can be reduced to the fact that any matrix is rationally conjugated to its canonical companion form.

2.4 Proof of the finiteness Theorem 2.4

The strategy will combine an argument for semisimple groups and another for algebraic tori.

For simplicity, we will only give an argument which assumes $M$ is Zariski connected, which is the case considered in the proof of Theorem 2.4.

We prove Theorem 2.4 combining [Reference Ullmo and YafaevUY14, Lemma 2.6] with Proposition 2.13.

Proof. Because the set of characters is invariant under conjugation, the condition is necessary. We prove that this condition is also sufficient.

We know that two representations of a torus $T$ are conjugated if and only if they have the same weights, with same multiplicities. As the weights belongs to $F$, and the dimension $d$ is fixed, there are only finitely many possibilities for these weights and multiplicities. Hence, $\{\rho \restriction _T:\rho \in \Sigma \}$ is contained in at most finitely many conjugacy classes $GL(d)\cdot \rho _1\restriction _T,\ldots,GL(d)\cdot \rho _c\restriction _T$. Without loss of generality we may assume that there is only one conjugacy class, say $GL(d)\cdot \rho _{1}\restriction _{T}$.

We want to prove that

(5)\begin{equation} \text{there are finitely many $\rho\in\Sigma$, up to $GL(d)$-conjugation.} \end{equation}

Possibly after conjugating, we may assume $\rho \restriction _T=\rho _1\restriction _T$. Because $M$ is connected, one has $M=M^{\rm der}\cdot T$. Thus,

(6)\begin{equation} \text{$\rho$ is determined by $\rho\restriction_{M^{\rm der}}$ and $\rho\restriction_{T}$.} \end{equation}

As $M^{\rm der}$ and $T$ commute with each other, $\rho \restriction _{M^{\rm der}}:M^{\rm der}\to GL(d)$ factors through $G':=Z_{GL(d)}(\rho _1(T))$. As $T$ is reductive, so is $G'$.

By Proposition 2.12, these $\rho \restriction _{M^{\rm der}}$ belong to finitely many conjugacy classes $G'\cdot \rho _{1,1}\restriction _{M^{\rm der}} ,\ldots,G'\cdot \rho _{1,e}\restriction _{M^{\rm der}}$. Possibly after conjugating $\rho$ by some $g\in G'$, which does not change $\rho \restriction _T$, we have

\[ \rho\restriction_T=\rho_1\restriction_T\text{ and }\rho\restriction_{M^{\rm der}} \in \{\rho_{1,1}\restriction_{M^{\rm der}} ;\ldots;\rho_{1,e}\restriction_{M^{\rm der}}\}. \]

In light of (6), this proves (5) and the conclusion follows.

2.5 Relation to other notions of Hecke orbits

The following is not used in the rest of this article, however it clarifies the relation between different notions of Hecke orbits and we believe it to be of independent interest. We compare our generalised and geometric Hecke orbits to the classical Hecke orbits and another notion of ‘generalised Hecke’ orbit found in the literature.

2.5.1 Relation to the classical definition of Hecke orbit

Let us recall the notion of the classical Hecke orbit.

Definition 2.14 (classical Hecke orbit)

Define the classical Hecke orbit of $x_0$ as follows:

\[ \mathcal{H}^c(x_0) = \{ \phi \circ x_0 \in X : \phi \in G({\mathbb{Q}})/Z_G(M)({\mathbb{Q}}) \} \subset \mathcal{H}(x_0) \]

and the classical Hecke orbit of $[x_0,1]$ as

\[ \mathcal{H}^c(x_0) = \{ [x,g] : x\in\mathcal{H}^c(x_0), g\in G({\mathbb{A}}_f)\}. \]

We have a chain of inclusions:

(7)$$\begin{gather} \mathcal{H}^c(x_0) \subset \mathcal{H}^g(x_0) \subset \mathcal{H}(x_0) \end{gather}$$

(8)$$\begin{gather}\mathcal{H}^c(s_0) \subset \mathcal{H}^g(s_0) \subset \mathcal{H}(s_0). \end{gather}$$

In general, $\mathcal {H}^g(x_0)$ is not a finite union of classical Hecke orbits, even when $G$ is of adjoint type.

Hecke correspondences

Recall that the classical Hecke orbit can be described using Hecke correspondences. For $g\in G({\mathbb {Q}})$, the points $s_0=[x_0,1]$ and $s_g=[g\cdot x_0,1]$ have a common inverse image by the left, respectively, right, finite map in

\[ Sh_K(G,X) \xleftarrow{Sh(Ad_1)} Sh_{K\cap gKg^{-1}}(G,X) \xrightarrow{Sh(Ad_g)} Sh_K(G,X), \]

where $Sh(Ad_g)$ the right map is the Shimura morphism associated to the map of Shimura data $AD_g:(G,X)\to (G,X)$ induced by the conjugation $AD_g:G\to G$ and $Sh(Ad_1)$ is induced by the identity map $AD_1:G\to G$.

Likewise generalised Hecke orbits can be interpreted using finite correspondences between Shimura varieties. For a point $\phi \circ x_0\in \mathcal {H}(x_0)$, the point $s_0$ and $s_\phi =[\phi \circ x_0,1]$ have a common inverse image in

\[ Sh_K(G,X) \xleftarrow{Sh(\phi_0)} Sh_{K\cap \stackrel{{\scriptscriptstyle{-1}}}{\phi} (K)}(M,X_M)\xrightarrow{Sh(\phi)} Sh_K(G,X). \]

This time the correspondence is induced by a correspondence from the image of $Sh(\phi _0)$ to that of $Sh(\phi )$. These are also the smallest special subvarieties containing $s_0$, respectively, $s_\phi$.

2.5.2 Relation to the usual definition of the generalised Hecke orbit

We compare our notion of generalised Hecke to the ‘generalised Hecke orbits’ used in [Reference Klingler and YafaevKY14] and [Reference Edixhoven and YafaevEY03, Reference PinkPin05, Reference OrrOrr15, Reference Ullmo and YafaevUY13]. The latter is defined in terms of linear representations.

For any faithful representation $\rho :G\to GL(N)$ over ${\mathbb {Q}}$, let the ‘$\rho$-Hecke orbit’ be

\[ \mathcal{H}^\rho(x_0) := \{ \phi \circ x_0 \in X : \phi \in \mbox{Hom}(M,G)({\mathbb{Q}}), \rho\circ \phi \in GL(N,{\mathbb{Q}})\cdot \rho\circ \phi_0\}. \]

By Theorem 2.11, we also have

\[ \mathcal{H}^\rho(x_0) = \{ \phi \circ x_0 \in X : \phi \in \mbox{Hom}(M,G)({\mathbb{Q}}), \rho\circ \phi \in GL(N,\overline{{\mathbb{Q}}})\cdot \rho\circ \phi_0\}. \]

Proposition 2.15 The $\rho$-Hecke orbit $\mathcal {H}^\rho (x_0)$ is contained in the generalised Hecke orbit $\mathcal {H}(x_0)$.

The $\rho$-Hecke orbit $\mathcal {H}^\rho (x_0)$ is a finite union of geometric Hecke orbits $\mathcal {H}^g(x_0)\cup \cdots \cup \mathcal {H}^\rho (x_k)$.

The first statement is clear from the definition of $\mathcal {H}^\rho (x_0)$. The second statement follows from the second definition of $\mathcal {H}^\rho (x_0)$ and [Reference RichardsonRic67].

The number of geometric Hecke orbits is bounded independently from $\rho$ thanks to Theorem 2.4. It is unclear whether we can achieve $\mathcal {H}^\rho (x_0)=\mathcal {H}(x_0)$ for a sufficiently general representation $\rho$.

3.1 Galois representations

Our statements will use the following terminology.

Definition 3.1 (Galois representations)

Let $(M,X_M)$ be a Shimura datum, let $x_0$ be a point in $X_M$, and let $E\leq {\mathbb {C}}$ be a subfield containing the reflex field $E(M,X_M)$.

We say that a continuous homomorphism

(9)\begin{equation} \rho=\rho_{x_0}:{\rm Gal}(\overline{E}/E)\to M({\mathbb{A}}_f) \end{equation}

is a Galois representation (defined over $E$) for $x_0$ (in $X_M$) if: for any compact open subgroup $K'\leq M({\mathbb {A}}_f)$, denoting $[x_0,1]'$ the image of $(x_0,1)$ in $Sh_{K'}(M,X_M)$, we have $[x_0,1]'\in Sh_{K'}(M,X_M)(\overline {E})$ and

(10)\begin{equation} \forall\,\sigma\in {\rm Gal}(\overline{E}/E), \sigma([x_0,1]')=[x_0,\rho_{x_0}(\sigma)]'. \end{equation}

In the important case of moduli spaces of abelian varieties, a representation $\rho _{x_0}$ can be directly constructed from the linear Galois action on the Tate module (see [Reference Ullmo and YafaevUY13, Reference Cadoret and MoonenCM20]).

Here we only need the existence of a $\rho _{x_0}$.

The main ingredient in this proposition is the following, which is part of the definition of canonical models: for any $[x_0,m_0]$, any $m\in M({\mathbb {A}}_f)$ and $\sigma \in Aut({\mathbb {C}}/E(M,X_M))$,

(11)\begin{equation} \text{if }\sigma([x_0,m_0])=[x',m']\text{ then }\sigma([x_0,m_0\cdot m])=[x',m'\cdot m]. \end{equation}

The continuity of $\rho _{x_0}$ is used in the following lemma.

Lemma 3.3 Let $K$ be an open subgroup of $M({\mathbb {A}}_f)$. Then, after possibly replacing $E$ by a finite extension, we have

(12)\begin{equation} \rho_{x_0}({\rm Gal}(\overline{E}/E))\leq K. \end{equation}

3.2 Functoriality of the Galois representation

In the next statement we denote by $E(G,X)$ the reflex field of a Shimura datum $(G,X)$. It is a number field over which $Sh(G,X)$ (and, hence, all the $Sh_K(G,X)$) admits a canonical model.

Proposition 3.4 (Functoriality)

Let $\phi :(M,X_M)\to (G,X)$ be a morphism of Shimura data, and $x_0$ a point in $X_M$.

If $\rho _{x_0}$ is a Galois representation defined over a field $E$ for $x_0$, then

\[ \phi\circ \rho_{x_0}|_{Gal(\overline{E}/E\cdot E(G,X))} \]

is a Galois representation defined over $E\cdot E(G,X)$ for $\phi (x_0)$ in $X$.

This follows from the definition and the identity

\[ \sigma([\phi\circ x,\phi(g)])=[\phi\circ x',\phi(g')]\quad \text{for }[x',g']=\sigma([x,g]), \]

which holds when $\sigma \in \operatorname {Aut}({\mathbb {C}}/E(M,X_M)E(G,X))$. Equivalently, the Shimura morphisms induced by $\phi$ are defined over $E(M,X_M)E(G,X)$. (See [Reference DeligneDel71, 1.14, 5.1].)

The compositum field $E\cdot E(G,X)\leq {\mathbb {C}}$ is a finite extension of $E$ which does not depend on the morphism $\phi$. With our definition, it also does not depend on the compact open subgroups. As a consequence, Galois representations for points in the same generalised Hecke orbit can be deduced from each other, after passing to the same finite extension $E\cdot E(G,X)/E$.

For future reference we summarise the above statements as follows.

Proposition 3.5 We keep the same notation. For any $\sigma \in Gal(\overline {E}/E\cdot E(G,X))$, any $g\in G({\mathbb {A}}_f)$, and any $\gamma \in G({\mathbb {Q}})$, we have

\[ \sigma([\gamma\cdot \phi(x_0),g])= [\gamma\cdot \phi(x_0), \rho'(\sigma)\cdot g], \]

where

\[ \rho':=Ad_{\gamma}\circ \phi\circ \rho_{x_0}:\sigma\mapsto \gamma\cdot \phi\circ \rho_{x_0}(\sigma)\cdot \gamma^{-1} \]

is a Galois representation defined over $E\cdot E(G,X)$ for $\gamma \cdot \phi (x_0)$ in $X$.

3.3 Galois orbits versus Adelic orbits

Let $U=\rho _{x_0}(Gal(\overline {E}/E))$. By definition, we have

\begin{align*} Gal(\overline{E}/E)\cdot [\phi\circ x_0,g]&= [\phi\circ x_0,\phi\circ\rho_{x_0}(Gal(\overline{E}/E))\cdot g]\\ &= G({\mathbb{Q}})\backslash G({\mathbb{Q}})\cdot \big(\{\phi\circ x_0\}\times \phi(U)\cdot g\big)\cdot K\cdot /K. \end{align*}

The next proposition reduces the estimation of the size of the Galois orbit to that of the $\phi (U)$-orbit $\phi (U)\cdot g\cdot K\cdot /K$.

Proposition 3.6 There is a real number $C\in {\mathbb {R}}_{>0}$ such that

\[ \forall \, (\phi\circ x_0,g)\in \mathcal{H}(x_0)\times G({\mathbb{A}}_f),\quad \frac{1}{C}\leq \frac{\lvert Gal(\overline{E}/E)\cdot [x_0,g] \rvert}{[\phi(U):\phi(U)\cap K]}\leq 1. \]

After possibly passing to a finite extension of $E$, we may choose $C=1$.

Proof. We want to bound the cardinality of the fibres of the map

(13)\begin{equation} \phi(U)\cdot g\cdot K/K\to G({\mathbb{Q}})\backslash G({\mathbb{Q}})\cdot (\{\phi\circ x_0\}\times \phi(U)\cdot g)\cdot K\cdot /K. \end{equation}

We first describe the fibres. Let $Z_\phi :=Z_G(\phi (M))$. The classical description of Hecke orbits gives an identity

\begin{align*} G({\mathbb{Q}})\backslash G({\mathbb{Q}})\cdot \{\phi\circ x_0\}\times G({\mathbb{A}}_f) &\simeq Z_{\phi}({\mathbb{Q}})\backslash \{\phi\circ x_0\}\times G({\mathbb{A}}_f)\\ &\simeq \{\phi\circ x_0\}\times Z_{\phi}({\mathbb{Q}})\backslash G({\mathbb{A}}_f). \end{align*}

(This follows from $G({\mathbb {Q}})\cap Stab_{G({\mathbb {R}})}(\phi \circ x_0)=Z_\phi ({\mathbb {Q}})$ in $G({\mathbb {R}})$. We have embedded $Z_{\phi }({\mathbb {Q}})$ in $G({\mathbb {A}})$ in the first line, and in $G({\mathbb {A}}_f)$ in the second line.)

Define

\[ \Gamma= Z_\phi({\mathbb{Q}})\cap \phi(U). \]

The map (13) can be written as a quotient map

\[ \phi(U)\cdot g\cdot K/K\to \Gamma\backslash(\phi(U)\cdot g\cdot K/K). \]

It will suffice to bound the order $\lvert \Gamma \rvert$.

The group $Z_\phi ({\mathbb {Q}})$ is discrete in $G({\mathbb {A}}_f)$ because $Z_\phi ({\mathbb {R}})$ is compact modulo $Z(G)({\mathbb {R}})$ and $Z(G)({\mathbb {Q}})$ is discrete in $G({\mathbb {A}}_f)$ (see [Reference Ullmo and YafaevUY13, Appendix Lemma 5.13]), where $Z(G)$ is the centre of $G$. As usual, we assume that $G$ is the generic Mumford–Tate group on $X$. Therefore, $\Gamma$ is compact and discrete, and thus is finite.

We will realise $\Gamma$ as a finite arithmetic group. We choose a faithful representation $G\to GL(N)$ defined over ${\mathbb {Q}}$, and identify $M$ and $G$ with their images in $GL(N)$.

We let $K[m]=\ker (GL(N,\widehat {{\mathbb {Z}}})\to GL(N,{{\mathbb {Z}}}/(m))$ for $m\in {\mathbb {Z}}$.

There is a maximal compact subgroup $K'$ of $GL(N,{\mathbb {A}}_f)$ which contains $K$. In $GL(N,{\mathbb {A}}_f)$ all maximal compact subgroups are conjugated: $K'$ is of the form $h\cdot GL(N,\widehat {{\mathbb {Z}}})\cdot h^{-1}$ with $h\in GL(N,{\mathbb {A}}_f)$. We may even choose $h\in GL(N,{\mathbb {Q}})$ (this is a consequence of the fact that the class number of $GL(N)/{\mathbb {Q}}$ is one).

Conjugating the representation by $h^{-1}$ we may assume $h=1$: we have

\[ U\leq K\leq GL(N,\widehat{{\mathbb{Z}}}). \]

If $m=3$ we pass to the finite extension of $E$ corresponding to the subgroup $\stackrel {-1}{\rho _{x_0}}(U\cap K[m])$ of ${\rm Gal}(\overline {E}/E)$. In any case we may assume

\[ U\leq K\cap K[m]\leq K[m]. \]

From Proposition 3.5, we know that $\phi =\gamma \phi _0 \gamma ^{-1}$ for some $\gamma \in GL(N,{\mathbb {Q}})$. It follows that

\[ \phi(U)\leq \gamma K[m] \gamma^{-1}, \]

and, thus,

\[ \Gamma = Z_\phi({\mathbb{Q}})\cap \phi(U)\leq GL(N,{\mathbb{Q}})\cap \gamma K[m]\gamma^{-1}. \]

Conjugating by $\gamma ^{-1}$ yields

\begin{align*} \gamma^{-1}\cdot \Gamma\cdot \gamma &\leq \gamma^{-1}GL(N,{\mathbb{Q}})\gamma\cap K[m]\\ &=GL(N,{\mathbb{Q}})\cap K[m]\\ &= \begin{cases} GL(N,{\mathbb{Z}}) & \text{if $m=1$,}\\ \ker(GL(N,{\mathbb{Z}})\to GL(N,{\mathbb{Z}}/(3))) & \text{if $m=3$.} \end{cases} \end{align*}

Recall that $\lvert \Gamma \rvert =\lvert \gamma ^{-1}\Gamma \gamma \rvert$. We may thus conclude by applying the following lemma to $\gamma ^{-1}\cdot \Gamma \cdot \gamma$. It follows that for $m=1$, $|\Gamma |$ is bounded independently of $\phi$ and for $m=3$, $|\Gamma |=1$.

Lemma 3.7 For every $N$, there is a real number $C(N)$ such that, for every finite subgroup $\Gamma \leq GL(N,{\mathbb {Z}})$ we have

\[ \lvert \Gamma \rvert\leq C(N), \]

and if $\Gamma \leq \ker (GL(N,{\mathbb {Z}})\to GL(N,{\mathbb {Z}}/(3)))$, then $\Gamma =1$.

Proof. From [Reference Platonov and RapinchukPR94, Lemma 4.19.(Minkowski), p. 232] the kernel has no nontrivial torsion. This implies the second assertion.

This also implies that the reduction map $GL(N,{\mathbb {Z}})\to GL(N,{\mathbb {Z}}/(3))$ is injective on $\Gamma$, thus inducing an embedding of $\Gamma$ in $GL(N,{\mathbb {Z}}/(3))$. The first conclusion follows with

\[ C(N)=\lvert GL(N,{\mathbb{Z}}/(3)) \rvert=\prod_{i=0}^{N-1}(3^N-3^i). \]

4.1 Height functions

4.1.1 Local affine height functions over ${\mathbb {R}}$ or ${\mathbb {Q}}_p$

Let $W$ be an affine variety over $K={\mathbb {R}}$ or $K={\mathbb {Q}}_p$. For every affine embedding defined over $K$

\[ \iota_K:W\to {\mathbb{A}}^N_K \]

there is an associated affine local Weil height function $H_{\iota _K}:W(K)\to {\mathbb {R}}_{\geq 0}$ given by

(14)\begin{equation} H_{\iota_K}(w)=\max\{1;\lvert w_1 \rvert_K;\ldots;\lvert w_N \rvert_K\}, \end{equation}

where $\lvert - \rvert _K$ is the standard absolute value on $K$.

4.1.2 Affine height functions over ${\mathbb {Q}}$

When $W$ and $\iota :=\iota _K$ are defined over ${\mathbb {Q}}$, we can define, for $w\in W({\mathbb {Q}})$,

(15)\begin{align} H_\iota(w)&=H_{\iota\otimes{\mathbb{R}}}(w)\cdot H_{\iota,f}(w), \end{align}

(16)\begin{align} \text{with }H_{\iota,f}(w)&=\prod_{p} H_{\iota\otimes{\mathbb{Q}}_p}(w). \end{align}

We define more generally, for $w=(w_p)_p\in W({\mathbb {A}}_f)$,

(17)\begin{equation} H_{\iota,f}(w)=\prod_p H_{\iota\otimes {\mathbb{Q}}_p}(w_p). \end{equation}

When $W$ and the embedding $\iota _{{\mathbb {R}}}$, respectively, $\iota _{{\mathbb {Q}}_p}$, respectively, $\iota$ are clear from the context, we will simply write

(18)\begin{equation} H_{{\mathbb{R}}}=H_{\iota_{\mathbb{R}}},\quad H_p=H_{\iota_{{\mathbb{Q}}_p}},\quad H_W=H_\iota\text{ and }H_f=H_{\iota,f}. \end{equation}

Then (15) becomes

(19)\begin{equation} H_W=H_{\mathbb{R}}\cdot H_f. \end{equation}

4.2 Polynomial equivalence and functoriality of heights

We recall the functoriality properties of heights. See [Reference SerreSer97] or [Reference Bombieri and GublerBG06] for corresponding statements about projective Weil heights. See Definition 1.7 for the symbols $\preccurlyeq$ and $\approx$.

Theorem 4.1 (Functoriality of heights)

Let $\phi :V\to V'$ be a morphism of affine varieties over ${\mathbb {R}}$, respectively, ${\mathbb {Q}}_p$, respectively, ${\mathbb {Q}}$, and let

\[ \text{$\iota_{{\mathbb{R}}}:V\to {\mathbb{A}}^N_{\mathbb{R}}$, respectively, $\iota_{{\mathbb{Q}}_p}:V\to {\mathbb{A}}^N_{{\mathbb{Q}}_p}$, respectively, $\iota:V\to {\mathbb{A}}^N_{\mathbb{Q}}$} \]

be an affine embedding of $V$, and let $\iota '_{{\mathbb {R}}}:V'\to {\mathbb {A}}^{N'}_{\mathbb {R}}$, respectively, $\iota '_{{\mathbb {Q}}_p}:V'\to {\mathbb {A}}^{N'}_{{\mathbb {Q}}_p}$, respectively, $\iota ':V'\to {\mathbb {A}}^{N'}_{\mathbb {Q}}$ be an affine embedding of $V'$.

Then, as functions on $V({\mathbb {R}})$, respectively, $V({\mathbb {Q}}_p)$, respectively, $V({\mathbb {Q}})$ and $V({\mathbb {A}}_f)$,

\begin{gather*} H_{\iota'_{\mathbb{R}}}\circ \phi \preccurlyeq H_{\iota_{\mathbb{R}}} , \text{respectively, }H_{\iota'_{{\mathbb{Q}}_p}}\circ \phi \preccurlyeq H_{\iota_{{\mathbb{Q}}_p}},\\ \text{respectively, }H_{\iota'}\circ \phi \preccurlyeq H_{\iota}\text{ and }H_{\iota',f}\circ \phi \preccurlyeq H_{\iota,f}. \end{gather*}

Corollary 4.2 Let $V$ be an affine algebraic variety over ${\mathbb {R}}$, respectively, ${\mathbb {Q}}_p$, respectively, ${\mathbb {Q}}$. Let

\begin{gather*} \text{$\iota_{{\mathbb{R}}}:V\to {\mathbb{A}}^{N}_{\mathbb{R}}$ and $\iota'_{{\mathbb{R}}}:V\to {\mathbb{A}}^{N'}_{\mathbb{R}}$,}\\ \text{respectively, $\iota_{{\mathbb{Q}}_p}:V\to {\mathbb{A}}^N_{{\mathbb{Q}}_p}$ and $\iota'_{{\mathbb{Q}}_p}:V\to {\mathbb{A}}^{N'}_{{\mathbb{Q}}_p}$,}\\ \text{respectively, $\iota:V\to {\mathbb{A}}^N_{\mathbb{Q}}$ and $\iota:V'\to {\mathbb{A}}^{N'}_{\mathbb{Q}}$} \end{gather*}

be affine embeddings of $V'$.

Then, as functions on $V({\mathbb {R}})$, respectively, $V({\mathbb {Q}}_p)$, respectively, $V({\mathbb {Q}})$ and $V({\mathbb {A}}_f)$,

\[ H_{\iota'_{\mathbb{R}}}\approx H_{\iota_{\mathbb{R}}} , \text{respectively, } H_{\iota'_{{\mathbb{Q}}_p}}\approx H_{\iota_{{\mathbb{Q}}_p}} , \text{respectively, } H_{\iota'}\approx H_{\iota} \text{ and } H_{\iota',f}\approx H_{\iota,f}. \]

4.3 Galois invariant height on the Hecke orbit

Let $S=Sh_K(G,X)$ and $x_0$ be as in § 1.1 and $\rho _{x_0}:{\rm Gal}(\overline {E}/E)\to M({\mathbb {A}}_f)$ be as in (9). Let $W=G\cdot \phi _0\subseteq \mbox {Hom}(M,G)$ be the algebraic variety defined in § 2.1. We have $W\simeq G/Z_G(M)$. Let $\mbox {Hom}(\mathfrak {m},\mathfrak {g})$ be affine algebraic variety of linear maps $\mathfrak {m}\to \mathfrak {g}$. As $M$ is connected, we have an embedding

\[ \phi\mapsto d\phi:W\hookrightarrow \mbox{Hom}(\mathfrak{m},\mathfrak{g}). \]

As $M$ is reductive, the image is closed, by [Reference RichardsonRic67].

We choose a lattice $\mathfrak {g}_{\mathbb {Z}}\leq \mathfrak {g}$ such that

\[ \mathfrak{g}_{\mathbb{Z}}\otimes\widehat{{\mathbb{Z}}}\leq \mathfrak{g}\otimes{\mathbb{A}}_f \]

is stable under the action of $K\leq G({\mathbb {A}}_f)$. We define $\mathfrak {m}_{\mathbb {Z}}=\mathfrak {g}_{\mathbb {Z}}\cap \mathfrak {m}$. We choose a basis of $\mathfrak {g}$ which generates $\mathfrak {g}_{\mathbb {Z}}$ and a basis of $\mathfrak {m}$ which generates $\mathfrak {m}_{\mathbb {Z}}$. This choice induces an isomorphism

\[ j:\mbox{Hom}(\mathfrak{m},\mathfrak{g})\xrightarrow{\sim} {\mathbb{Q}}^{\dim(M)\cdot \dim(G)}. \]

This induces an affine embedding

\[ \iota:=j\circ d: W=G\cdot \phi_0\hookrightarrow \mbox{Hom}(\mathfrak{m},\mathfrak{g})\to {\mathbb{A}}^{\dim(M)\cdot \dim(G)}_{\mathbb{Q}} \]

by first mapping $\phi$ to $d\phi$ and then to its matrix with respect to the bases we have chosen.

We denote by $H_f:W({\mathbb {A}}_f)\to {\mathbb {Z}}_{\geq 1}$ and $H'_f:\mbox {Hom}(\mathfrak {m}_{{\mathbb {A}}_f},\mathfrak {g}_{{\mathbb {A}}_f})\to {\mathbb {Z}}_{\geq 1}$ the functions given by (17) with respect to the embeddings $\iota$ and $j$.

Proposition 4.3 (Galois invariance)

Let $\phi _1,\phi _2\in W({\mathbb {Q}})$ be such that $s_1=[\phi _1\circ x_0,g_1]$ and $s_2=[\phi _2\circ x_0,g_2]$ define points in $\mathcal {H}^g(s_0)$, where $g_1,g_2\in G({\mathbb {A}}_f)$, and assume that there exists a $\sigma \in {\rm Gal}(\overline {E}/E)$ such that

(20)\begin{align} \sigma(s_1)=s_2. \end{align}

Then

\[ H_{f}({g_1}^{-1}\phi_1 g_1)=H_{f}({g_2}^{-1}\phi_2 g_2). \]

We first remark, from the formula

(21)\begin{equation} \forall \, \psi\in \mbox{Hom}(\mathfrak{m}_{{\mathbb{A}}_f},\mathfrak{g}_{{\mathbb{A}}_f}),\ H'_{f}(\psi)=\min\{n\in{\mathbb{Z}}_{\geq1} : n\cdot \psi(\mathfrak{m}_{\widehat{{\mathbb{Z}}}})\subset \mathfrak{g}_{\widehat{{\mathbb{Z}}}}\}, \end{equation}

that for every $\widehat {{\mathbb {Z}}}$-module automorphism $u:\mathfrak {m}_{\widehat {{\mathbb {Z}}}}\to \mathfrak {m}_{\widehat {{\mathbb {Z}}}}$ and $k:\mathfrak {g}_{\widehat {{\mathbb {Z}}}}\to \mathfrak {g}_{\widehat {{\mathbb {Z}}}}$, we have

(22)\begin{equation} H'_{f}(k\circ \psi\circ u)=H'_{f}(\psi). \end{equation}

When $\psi =d\phi$, and $k=ad_{k'}:\mathfrak {g}\to \mathfrak {g}$ with $k'\in K$ and $u=ad_{u'}:\mathfrak {m}\to \mathfrak {m}$ with $u'\in K\cap M({\mathbb {A}}_f)$, this gives, with $AD_{k'}\circ \phi \circ AD_{u'}:m\mapsto k'\phi (u'mu'^{-1})k'^{-1}$,

(23)\begin{equation} H_{f}(AD_{k'}\circ \phi\circ AD_{u'})=H_{f}(\psi). \end{equation}

Proof of Proposition 4.3 We define $u=\rho _{x_0}(\sigma )\in M({\mathbb {A}}_f)\cap K$. From (20) and the functoriality of Galois action Propositions 3.4 and 3.5, we have

\[ [\phi_1\circ x_0,\phi_1(u)\cdot g_1]=[\phi_2\circ x_0,g_2]. \]

Hence, there exists $\gamma \in G({\mathbb {Q}})$ and $k\in K$ such that

\[ (\gamma\cdot \phi_1\circ x_0,\gamma\cdot \phi_1(u)g_1 k)=(\phi_2\circ x_0,g_2). \]

By Lemma 2.5, we also have $\gamma \cdot \phi _1\cdot \gamma ^{-1}=\phi _2$. Thus,

\begin{align*} {g_2}^{-1}\phi_2 g_2&= (k^{-1}{g_1}^{-1}\phi_1(u)^{-1}\gamma^{-1}) \cdot (\gamma\cdot \phi_1\cdot \gamma^{-1}) \cdot (\gamma\cdot \phi_1(u)g_1 k)\\ &= k^{-1}{g_1}^{-1}\phi_1(u^{-1})\cdot \phi_1 \cdot \phi_1(u)g_1 k. \end{align*}

We have

\[ \forall \, m\in M({\mathbb{A}}_f), \phi_1(u)^{-1}\cdot \phi_1(m) \cdot \phi_1(u)= \phi_1(u^{-1}mu)=\phi_1\circ AD_{u^{-1}}(m) \]

and, hence,

\[ k^{-1}{g_1}^{-1}\phi_1(u)^{-1}\cdot \phi_1 \cdot \phi_1(u)g_1 k=AD_{k^{-1}}\circ({g_1}^{-1}\cdot \phi_1\cdot {g_1})\circ AD_{u^{-1}}. \]

We finally have, using (23),

\[ H_{f}({g_2}^{-1}\phi_2 g_2)=H_{f}(AD_{k^{-1}}\circ({g_1}^{-1}\cdot \phi_1\cdot {g_1})\circ AD_{u^{-1}}) =H_{f}({g_1}^{-1}\phi_1 g_1). \]

4.3.1 Height function on the generalised Hecke orbit $\mathcal {H}([x_0,1])$

The function $H'_f$ on $\mbox {Hom}(\mathfrak {m}_{{\mathbb {A}}_f},\mathfrak {g}_{{\mathbb {A}}_f})$ induces, at the level of the $Sh_K(G,X)$, a function $H_{s_0}$ on the generalised Hecke orbit $\mathcal {H}(s_0)$ of $s_0:=[x_0,1]$, given as follows. For $\phi \in \mbox {Hom}(\mathfrak {m}_{\mathbb {Q}},\mathfrak {g}_{\mathbb {Q}})$ such that $\phi \circ x_0\in X$ and $g\in G({\mathbb {A}}_f)$, we define

\[ H_{s_0}([\phi\circ x_0, g])=H'_{f}(d(g^{-1}\phi g)). \]

The function $H_{s_0}$ depends on the choices we have made, but different choices will produce the same function, up to a bounded factor.

The case $\sigma =1$ of Proposition 4.3 implies that $H_{s_0}$ is well defined. Proposition 4.3 can then be rephrased as follows.

Lemma 4.4 For every $\sigma \in {\rm Gal}(\overline {E}/E)$ and $s\in \mathcal {H}(s_0)$ we have

(24)\begin{equation} H_{s_0}(\sigma(s))=H_{s_0}(s). \end{equation}

5.1 Polynomial equivalence and archimedean height

We use Definition 1.7.

Lemma 5.1 Let $A\subset {\mathbb {R}}^n$ be a semialgebraic subset, and let $f,g \colon A \longrightarrow {\mathbb {R}}_{\geq 0}$ be semialgebraic and continuous functions. Assume that $f$ is a proper map.

Then

\[ g \preccurlyeq f. \]

Proof. We claim that the following function

\begin{align*} h \colon ]\inf_A(f), \infty[ &\to {\mathbb{R}}_{\geq 0}\\ t &\mapsto \sup \{g(a): a\in A, f(a) \leq t\} \end{align*}

is well defined. Fix an arbitrary $t$ be in $]\inf _A(f), \infty [$. The set $\{ a\in A: f(a) \leq t\}$ is compact since $f$ is proper. It is nonempty since $t> \inf _A(f)$. As $g$ is continuous, $\{g(a): a\in A, f(a) \leq t\}$ is compact and nonempty, and its maximum belongs to ${\mathbb {R}}_{\geq 0}$, which proves the claim.

The function $h$ is also semialgebraic (see [Reference Bochnak, Coste and RoyBCR98, Proposition 2.2.4.]). By [Reference van den DriesvdDri98, § 4.1 ‘Notes and comments’ and references therein], $h$ is polynomially bounded. The conclusion follows.

The following uses Lemma 5.1 for $f$ and $g$, and again after swapping $f$ and $g$.

We will also encounter the following situation.

Lemma 5.4 Let $V$ be an affine algebraic variety over ${\mathbb {R}}$. Let $\phi :V\to {\mathbb {A}}^N$, and $\phi ':V\to {\mathbb {A}}^M$ be two closed embeddings, and let $H_\phi$ and $H_{\phi '}$ be defined as in (14).

Then $H_\phi$ and $H_{\phi '}$ are semialgebraic continuous proper functions, and

\[ H_{\phi} \approx H_{\phi'}. \]

Lemma 5.5 Let $p:U\to V$ be a morphism of affine algebraic varieties over ${\mathbb {R}}$, and $\phi _U:U\to {\mathbb {A}}_{\mathbb {R}}^N$ and $\phi _V:V\to {\mathbb {A}}^M_{\mathbb {R}}$ be closed embeddings. Let $H_{\phi _U}$ and $H_{\phi _V}$ be defined as in (14).

Let $A\subset U({\mathbb {R}})$ be a semialgebraic subset such that $p|_A:A\to V({\mathbb {R}})$ is proper. Then, as functions $A\to {\mathbb {R}}_{\geq 0}$,

\[ H_{\phi_U}|_A\approx H_{\phi_V}\circ p|_A. \]

5.2 Comparison of archimedean and finite height

Proof. Thanks to Corollary 4.2, we may substitute $\iota$ with the closed embedding

(25)\begin{equation} \iota_N: {{\mathbb{G}}_m}^N\xrightarrow{(t_1,\ldots,t_N)\mapsto (t_1,{t_1}^{-1},\ldots,t_N,{t_N}^{-1})} {\mathbb{A}}^{2N}. \end{equation}

We start with the case $N=1$. We write an element $t\in {\mathbb {Q}}^\times$ as a reduced fraction $n/m$. We can compute

\[ H_{\iota_1\otimes{\mathbb{R}}}(t)=\max\{|t|;|1/t|\}\text{ and }H_{\iota_1,f}(t)=|n\cdot m|. \]

It follows $H_{\iota _1\otimes {\mathbb {R}}}(t)\leq H_{\iota _1,f}(t)$.

For general $\vec {t}=(t_1,\ldots,t_N)\in {{\mathbb {Q}}^\times }^N$ there is some $1\leq i\leq N$ such that

\[ H_{\iota_N\otimes{\mathbb{R}}}(\vec{t}\,)=\max\{|t_1|;|1/t_1|;\ldots; |t_N|;|1/t_N|\}=\max\{|t_i|;|1/t_i|\}. \]

By the previous computation we have

\[ H_{\iota_N\otimes{\mathbb{R}}}({\vec{t}\,})=H_{\iota_1\otimes{\mathbb{R}}}(t_i)\leq H_{\iota_1,f}(t_i). \]

We conclude by observing that

\[ H_{\iota_1,f}(t_i)\leq H_{\iota_N,f}(\vec{t}), \]

as can be seen prime by prime.

Proof. By Corollary 4.2, we may assume that $\iota _V$ is $\iota _N$ of (25), and that $\iota _W$ is the identity map. We can again reduce the problem to the case $N=1$. We write $t_i=n/m$ as an irreducible fraction and then we have

\[ H_{\iota_V,f}(n/m)=\lvert n\cdot m \rvert\leq \max\{\lvert n \rvert,\lvert m \rvert\}^2=H_{\iota_W}(n/m)^2. \]

Corollary 5.8 Let $C\in {\mathbb {R}}_{>0}$. We have

\[ H_{\iota_V,f} \preccurlyeq H_{\iota_W,f}\text{ on }({\mathbb{Q}}^\times\cap [-C;C])^N. \]

We establish the following.

Proposition 5.9 Let $W$ be an affine variety over ${\mathbb {Q}}$ and let $p\colon W \longrightarrow {\mathbb {A}}^r$ be an algebraic map and $\mathfrak {S} \subset W({\mathbb {R}})$ be a semialgebraic closed subset such that:

1. (i) we have $p(\mathfrak {S}) \subseteq ({\mathbb {R}}^\times )^r$;

2. (ii) the restriction $p |_{\mathfrak {S}} \colon \mathfrak {S} \longrightarrow {{\mathbb {R}}^\times }^r$ is a proper map;

3. (iii) the image $p(\mathfrak {S})$ is bounded in ${\mathbb {R}}^r$.

We fix an affine embedding $\iota$ of $W$ and use notation (18). Then

\[ H_{{\mathbb{R}}}|_{\mathfrak{S} \cap W({\mathbb{Q}})} \preccurlyeq H_{f} |_{\mathfrak{S} \cap W({\mathbb{Q}})}. \]

In particular,

\[ H_{W} |_{\mathfrak{S} \cap W({\mathbb{Q}})} \approx H_{f} |_{\mathfrak{S} \cap W({\mathbb{Q}})}. \]

Proof. We denote by ${{\mathbb {G}}_m}^r\subset {\mathbb {A}}^r$ the affine open subset on which every coordinate is invertible.

We fix affine embeddings $\iota _W$ of $W$, and $\iota _{{{\mathbb {G}}_m}^r}$ of ${{\mathbb {G}}_m}^r$ and $\iota _{{\mathbb {A}}^r}$ of ${\mathbb {A}}^r$.

Because $p|_{\mathfrak {S}}$ is continuous real algebraic, and (as a function to ${\mathbb {R}}^r$) is proper, by Lemma 5.5 we have

(26)\begin{equation} H_{\iota_W\otimes{\mathbb{R}}}|_{\mathfrak{S}}\approx H_{\iota_{{{\mathbb{G}}_m}^r}\otimes{\mathbb{R}}}\circ p|_\mathfrak{S}. \end{equation}

By functoriality of heights, Theorem 4.1, we have, on $W({\mathbb {Q}})$,

(27)\begin{equation} H_{\iota_{{{\mathbb{A}}}^r},f}\circ p \preccurlyeq H_{\iota_W,f}. \end{equation}

As $p(\mathfrak {S})$ is bounded in ${\mathbb {R}}^r$, we have, by Lemma 5.6,

(28)\begin{equation} H_{\iota_{{{\mathbb{G}}_m}^r}\otimes{\mathbb{R}}} \preccurlyeq H_{\iota_{{{\mathbb{G}}_m}^r},f}. \end{equation}

By hypothesis (iii) we can use Corollary 5.8 and get

(29)\begin{equation} H_{\iota_{{{\mathbb{G}}_m}^r},f}|_{p(\mathfrak{S})\cap {{\mathbb{Q}}^\times}^r} \preccurlyeq H_{\iota_{{{\mathbb{A}}}^r},f}|_{p(\mathfrak{S})\cap {{\mathbb{Q}}^\times}^r}. \end{equation}

Combining these we get, using (26), (28), (29), and then (27),

\begin{align*} H_{\iota_W\otimes{\mathbb{R}}}|_{\mathfrak{S}\cap W({\mathbb{Q}})} &\approx H_{\iota_{{{\mathbb{G}}_M}^r}\otimes{\mathbb{R}}}\circ p|_{\mathfrak{S}\cap W({\mathbb{Q}})}\\ & \preccurlyeq H_{\iota_{{{\mathbb{G}}_M}^r},f}\circ p|_{\mathfrak{S}\cap W({\mathbb{Q}})} \preccurlyeq H_{\iota_{{\mathbb{A}}^r},f}\circ p|_{\mathfrak{S}\cap W({\mathbb{Q}})} \preccurlyeq H_{\iota_W,f}|_{\mathfrak{S}\cap W({\mathbb{Q}})}. \end{align*}

5.3 Construction of Siegel sets

We start by recalling some facts about parabolic subgroups and Siegel sets. A general reference is [Reference Borel and JiBJ06].

Let $G_{\mathbb {Q}}$ be a semisimple ${\mathbb {Q}}$-algebraic group of adjoint type. We fix a minimal ${\mathbb {Q}}$-defined parabolicFootnote ¹¹ subgroup $P_{\mathbb {Q}}$. Let $G_{\mathbb {R}}$ and $P_{\mathbb {R}}$ be the corresponding ${\mathbb {R}}$-algebraic groups.

Let $X$ be the associated symmetric space,Footnote ¹² and choose $x \in X$ and let $\Theta :G_{{\mathbb {R}}}\to G_{{\mathbb {R}}}$ be the Cartan involution associated with $x$. The orbit map $g\mapsto g\cdot x$ induces the identification $G({\mathbb {R}})/K\simeq X$ where $K$ is the maximal compact subgroup $\{g\in G({\mathbb {R}}): g=\Theta (g)\}$. We denote by $K_\infty =K^+$ the neutral component.

We let $N_{\mathbb {Q}}$ be the unipotent radical of $P_{\mathbb {Q}}$: thus $P_{\mathbb {Q}}/N_{\mathbb {Q}}$ is the maximal reductive quotient of $P_{\mathbb {Q}}$. The ${\mathbb {R}}$-algebraic group

\[ L:= P_{{\mathbb{R}}} \cap \Theta(P_{{\mathbb{R}}}) \]

is a maximal ${\mathbb {R}}$-algebraic reductive subgroup of $P_{{\mathbb {R}}}$ (cf. [Reference Borel and JiBJ06, § III.1.9]), not necessarily defined over ${\mathbb {Q}}$, and the map $L\to P_{\mathbb {R}}\to (P_{\mathbb {Q}}/N_{\mathbb {Q}})_{\mathbb {R}}$ is an isomorphism. We denote by $A'_{\mathbb {Q}}$ the maximal central ${\mathbb {Q}}$-split torus of $P_{\mathbb {Q}}/N_{\mathbb {Q}}$, and define $A\leq L$ as the inverse image of $A_{\mathbb {R}}$ in $L$. We denote by $A^+=A({\mathbb {R}})^+$ the neutral component as a real Lie group.

We denote by $\Phi$ the set of non-zero weights of the adjoint action of $A$ on $\mathfrak {g}\otimes {\mathbb {R}}$ (the ‘(relative) roots’), and $\Phi ^+$ the subset of weights of the action on $\mathfrak {n}\otimes {\mathbb {R}}$ (the ‘positive’ ones). The eigenspaces are not necessarily defined over ${\mathbb {Q}}$. There exists a unique subset $\Delta =\{\alpha _1;\ldots ;\alpha _r\}\subset \Phi ^+$ such that $\alpha _1,\ldots,\alpha _r$ is a basis of $X(A)=\mbox {Hom}(A,{{\mathbb {G}}_m}_{\mathbb {R}})$ and $\Phi ^+\subset \alpha _1\cdot {\mathbb {Z}}_{\geq 0} + \cdots +\alpha _r\cdot {\mathbb {Z}}_{\geq 0}$. The $\alpha _i$ are known as the (relative) simple roots, and $r$ is equal to the ${\mathbb {Q}}$-rank of $G_{\mathbb {Q}}$.

The positive Weyl chamber in $A^+$ is

(30)\begin{equation} A^+_{\geq 0}=\{a\in A^+ : \forall \, 1\leq i\leq r,\ \alpha_i(a)\geq 1\}. \end{equation}

We define

(31)\begin{equation} H_P:= \bigcap_{\chi \in X(P_{\mathbb{Q}})} \ker(\chi^2). \end{equation}

We note that, for every one-dimensional representation ${\mathbb {Q}}\cdot \eta$ of $H_P$, we have

(32)\begin{equation} \forall \, h\in H_P({\mathbb{R}}),\ h\cdot \eta\in \{+\eta;-\eta\}. \end{equation}

We first define Siegel sets in $G_{\mathbb {Q}}({\mathbb {R}})$.

Definition 5.10 (Siegel set)

A ${\mathbb {Q}}$-Siegel set $\mathfrak {S}$ in $G_{\mathbb {Q}}({\mathbb {R}})$ with respect to $P_{\mathbb {Q}}$ and $x$ is a subset $\mathfrak {S}\subseteq G({\mathbb {R}})$ of the following form.

There is a nonempty open and relatively compact subset $\Omega \subseteq P_{\mathbb {Q}}({\mathbb {R}})$ and an element $a\in A^+$ such that

\[ \mathfrak{S} = \Omega \cdot A^+_{\geq 0} \cdot a \cdot K_{\infty}. \]

We will always assume that $\Omega \subseteq H_{\mathbb {Q}}({\mathbb {R}})$ and that $\Omega$ is semialgebraic.

Usually Siegel sets are defined in $G_{\mathbb {Q}}({\mathbb {R}})$ or in $X=G_{\mathbb {Q}}({\mathbb {R}})/K$. Working with geometric Hecke orbits as defined in Definition 2.3, we use the variety $W({\mathbb {R}})^+=G({\mathbb {R}})/Z_G(M)({\mathbb {R}})$. We can view $W({\mathbb {R}})^+$ as an intermediary space in the sequence of maps $G({\mathbb {R}})\to W=G({\mathbb {R}})/Z_G(M)({\mathbb {R}}) \to X$. The following definition allows us to work with Siegel sets in a greater generality.

We note that if $Z$ is defined over ${\mathbb {Q}}$ then so is $W$ and we can consider the subset $W^+({\mathbb {Q}})\cap \mathfrak {S}_W$.

5.4 Divergence in Siegel sets

In the rest of this section we use the notation $G=G_{\mathbb {Q}}$.

5.4.2

We will use the closure of a Siegel set.

Proposition 5.12 Consider $\mathfrak {S}=\Omega \cdot A^+_{\geq 0}\cdot a\cdot K_{\infty }$ as in Definition 5.10.

Then its closure in $G({\mathbb {R}})$ is given by

(33)\begin{equation} \overline{\mathfrak{S}}=\overline{\Omega}\cdot A^+_{\geq 0}\cdot a\cdot K_{\infty}. \end{equation}

and $\overline {\mathfrak {S}}$ is contained in a ${\mathbb {Q}}$-Siegel set $\mathfrak {S}'$ in $G({\mathbb {R}})$ with respect to $P$ and $x$.

Proof. The set $\mathfrak {S}$ is obviously dense in the right-hand side of (33). It is the image of the proper map in Lemma 5.13, and thus it is a closed subset in $G({\mathbb {R}})$. This proves the first assertion. Let $U$ be a nonempty relatively compact semialgebraic open neighbourhood of $1$ in $H({\mathbb {R}})$, for instance a small euclidean open ball in a faithful representation $H\to GL(N)$. Then $\Omega '=U\cdot \overline {\Omega }$ is an open relatively compact semialgebraic open neighbourhood of $\overline {\Omega }$ in $H({\mathbb {R}})$, and the Siegel set $\Omega '\cdot A^+_{\geq 0}\cdot a\cdot K$ contains $\overline {\mathfrak {S}}$.

We used the following.

Lemma 5.13 The map

\begin{align*} (\omega, a, k)\mapsto \omega\cdot a\cdot k: \overline{\Omega}\times A^+_{\geq 0}\cdot a\times K\to G({\mathbb{R}}) \end{align*}

is proper.

Proof. It suffices to prove that the image of every divergent sequence in the left-hand side is not a convergent sequence in the right-hand side. We prove the contrapositive.

Let $(\omega _n, a_n, k_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ be a sequence in the left-hand side such that $(\omega _n\cdot a_n\cdot k_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ is a convergent sequence in $G({\mathbb {R}})$. Because $\overline {\Omega }$ and $K$ are compact, after possibly extracting a subsequence we may assume that $(\omega _n)_{n\in {\mathbb {Z}}_{\geq 1}}$ and $(k_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ are convergent sequences. It follows that $(a_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ is a convergent subsequence. We recall that $(\alpha _1,\ldots,\alpha _r):A^+\to {{\mathbb {R}}_{>0}}^r$ is an isomorphism. It follows that $A^+_{\geq 0}$ is closed in $A^+$. Because $A({\mathbb {R}})$ is a closed subgroup of $G({\mathbb {R}})$, that $A^+$ is a closed subgroup of $A({\mathbb {R}})$, this $A_{\geq 0}^+$ is closed in $G({\mathbb {R}})$. One deduces that $A_{\geq 0}^+\cdot a$ is closed in $G({\mathbb {R}})$ and that the limit of $(a_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ in $G({\mathbb {R}})$ belongs to $A_{\geq 0}^+\cdot a$.

We proved that the original sequence $(\omega _n, a_n, k_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ contains a convergent infinite subsequence. Thus, it is not a divergent sequence.

These results have the following consequence.

Corollary 5.14 A sequence $(\omega _n\cdot a_n\cdot k_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ is divergent in $\overline {\mathfrak {S}}$ if and only if $a_n$ is divergent in $A^+_{\geq 0}$.

Proof. Because $\overline {\mathfrak {S}}$ is closed, $(\omega _n\cdot a_n\cdot k_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ is also divergent in $G({\mathbb {R}})$. It follows that the sequence $(\omega _n, a_n, k_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ contains no convergent subsequence. Because $\overline {\Omega }$ and $K$ are compact, the projection map

\begin{align*} \overline{\Omega}\times A^+_{\geq 0}\cdot a\times K\to A^+_{\geq 0}\cdot a \end{align*}

is proper. It follows that the image sequence $(a_n)_{n\in {\mathbb {Z}}_{\geq 1}}$ is divergent in $A^+_{\geq 0}\cdot a$.

5.4.3

Let $P_{1},\ldots,P_{r}$ be the maximal ${\mathbb {Q}}$-defined properFootnote ¹³ parabolic subgroups of $G$ containing $P$. We denote by $N_{i}$ their unipotent radicals, and $\mathfrak {n}_i$ the (${\mathbb {Q}}$-linear) Lie algebra of $N_{i}$. The adjoint representation of $G$ induces an action of $G$ on the ${\mathbb {Q}}$-vector space $V_i=\bigwedge ^{\dim (N_i)}\mathfrak {g}$. Then the ${\mathbb {Q}}$-vector subspace $\bigwedge ^{\dim (N_i)}\mathfrak {n}_i\leq V_i$ is of dimension $1$, and we choose a generator $\eta _i$ of this ${\mathbb {Q}}$-line.

Then the ${\mathbb {R}}$-line ${\mathbb {R}} \cdot \eta _i$ is an eigenspace of $A$ acting on $V_i\otimes {\mathbb {R}}$, and this eigenspace is defined over ${\mathbb {Q}}$. Let $\chi _i$ be the corresponding eigencharacters of $A$: we have

(34)\begin{equation} \forall \, a\in A({\mathbb{R}}),\ \forall \, 1\leq i\leq r,\ \forall a\cdot \eta_i =\chi_i(a)\cdot \eta_i. \end{equation}

For $1\leq i\leq r$ the $\chi _i$ are positive multiples $k_1\cdot \omega _1,\ldots,k_r\cdot \omega _r$ of the (relative) fundamental weightsFootnote ¹⁴ $\omega _1,\ldots,\omega _r\in X(A)\otimes {\mathbb {Q}}$. In particular,

(35)\begin{equation} \forall \, \alpha\in A^+_{\geq 0}, \forall \, 1\leq i\leq r, \quad \chi_i(\alpha)^{-1}\leq 1. \end{equation}

One knows that the fundamental weights are positive ${\mathbb {Q}}$-linear combinations of $\alpha _i$ and that they form a basis of $X(A)\otimes {\mathbb {Q}}$. The same holds for $\chi _i$. We deduce the following.

Lemma 5.15 Let $(a_n)_{n\in {\mathbb {Z}}_{\geq 0}}$ be a sequence in $A^+_{\geq 0}\cdot a$. Then the sequence is divergent (no infinite subsequence is convergent) if and only if

(36)\begin{equation} \lim_{n\to\infty} \min_{1\leq i\leq r} \chi_i(a_n)^{-1}=0. \end{equation}

Proof. If $(a_n)_{n\in {\mathbb {Z}}_{\geq 0}}$ contains a convergent infinite subsequence, then the sequence $(\min _{1\leq i\leq r} \chi _i(a_n)^{-1})_{n\in {\mathbb {Z}}_{\geq 0}}$ contains a convergent infinite subsequence in ${\mathbb {R}}_{>0}$ and we cannot have (36).

This proves one implication and we now prove the other.

Assume that (36) fails. Equivalently,

\[ L:=\limsup_{n\to\infty} \min_{1\leq i\leq r} \chi_i(a_n)^{-1}>0. \]

After possibly extracting a subsequence, we have

(37)\begin{equation} \lim_{n\to\infty} \min_{1\leq i\leq r} \chi_i(a_n)^{-1}=L\neq 0. \end{equation}

Because the $a_n$ belong to $A^+_{\geq 0}\cdot a$ we have $\sup _{n\in {\mathbb {Z}}_{\geq 0}}\alpha _i(a_n)^{-1}\leq \alpha _i(a)^{-1}$ for every $1\leq i\leq r$. Because the $\chi _i$ are positive linear combination of the $\alpha _i$, the $\chi _i(a_n)^{-1}$ are bounded above. According to (37) they are bounded below in ${\mathbb {R}}_{>0}$. Because the $\chi _i$ form a basis of $X(A)\otimes {\mathbb {Q}}$, the $\alpha _i$ are linear combination of the $\chi _i$. Hence, the $\alpha _i(a_n)$ are bounded above and below in ${\mathbb {R}}_{>0}$. Equivalently $(a_n)_{n\in {\mathbb {Z}}_{\geq 0}}$ is bounded in $A^+$. Hence $(a_n)_{n\in {\mathbb {Z}}_{\geq 0}}$ is not divergent.

This proves the other implication.

5.5 Height on Siegel sets

The following statement is the main objective of § 5.

Theorem 5.16 Let $\mathfrak {S}_W$ as in Definition 5.11 with $Z$ defined over ${\mathbb {Q}}$, let $\iota :G/Z\to {\mathbb {A}}_{\mathbb {Q}}^N$ be an affine embedding and let $H_W=H_{{\mathbb {R}}}\cdot H_{f}$ be as in (19). Then, as functions $\overline {\mathfrak {S}_W}\cap W({\mathbb {Q}})\to {\mathbb {R}}_{\geq 0}$ we have

\[ H_W \approx H_{f}. \]

This will be deduced from Proposition 5.9. We first construct the map $p$ to which we apply the proposition, and then verify the assumptions of the proposition.

5.5.1 Construction of the morphism $p$

Let $\eta _i\in V_i=\bigwedge ^{\dim {\mathfrak {n}_i}} \mathfrak {g}$ and $\chi _i$ be as in § 5.4.3.

For each $1\leq i\leq r$, we choose a positive-definite quadratic form

\[ Q'_i \colon V_i \longrightarrow {\mathbb{Q}}. \]

We denote by $dz$ the Haar probability measure on $Z({\mathbb {R}})$ we define the real quadratic form

(38)\begin{equation} Q_i(v) = \int_{Z({\mathbb{R}})} Q'_i(z\cdot v)\, dz:V_i\otimes{\mathbb{R}}\to {\mathbb{R}}. \end{equation}

The following is central in our argument.

Lemma 5.17 The quadratic form $Q_i$ is invariant under $Z({\mathbb {R}})$, is positive definite, and is defined over ${\mathbb {Q}}$.

Proof. The two first properties are immediate from (38). We prove that $Q_i$ is defined over ${\mathbb {Q}}$. Let $V$ be the ${\mathbb {Q}}$-vector space of quadratic forms, as a representation of $Z$, and $V^Z$ be the subspace of elements fixed by $Z$. As $Z({\mathbb {R}})$ is compact, the ${\mathbb {Q}}$-group $Z$ is reductive, and there is a $Z$-stable ${\mathbb {Q}}$-subspace $W$ such that we can decompose

\begin{align*} V=V^Z\oplus W. \end{align*}

Let us write correspondingly

\begin{align*} Q'_i=Q'_Z+Q'_W \end{align*}

with $Q'_Z$ in $V^Z$ and $Q'_W$ in $W$.

Because $Q_Z$ is invariant under $Z({\mathbb {R}})$ and $W\otimes {\mathbb {R}}$ is stable under $Z({\mathbb {R}})$,

\[ \int_{Z({\mathbb{R}})} z \cdot Q_Z\,d z=Q_Z\quad\text{and}\quad\int_{Z({\mathbb{R}})} z \cdot Q_W\,d z\in W\otimes{\mathbb{R}}. \]

By construction, $\int _{Z({\mathbb {R}})} z \cdot Q_W\,d z$ is fixed by $Z({\mathbb {R}})$ and, thus, it belongs to $W\otimes {\mathbb {R}}\cap V^Z\otimes {\mathbb {R}}=\{0\}$. We compute

\[ Q_i =\int_{Z({\mathbb{R}})} z \cdot Q'_i\,d z =\int_{Z({\mathbb{R}})} z \cdot Q_Z\,d z + \int_{Z({\mathbb{R}})} z \cdot Q_W\,d z =Q_Z+0. \]

Because $Q_Z$ is defined over ${\mathbb {Q}}$, so is $Q_i$.

We can now define $p:W\to {\mathbb {A}}_{\mathbb {Q}}^r$ by

(39)\begin{equation} p(gZ)= (Q_1(g^{-1}\cdot \eta_1) , \ldots , Q_r(g^{-1}\cdot \eta_r)). \end{equation}

As the $Q_i$ are defined over ${\mathbb {Q}}$ so is $p$, and as the $Q_i$ are $Z$-invariant, $p$ is well defined.

5.5.2 Properties of the morphism $p$

Our next task is to verify the assumptions of Proposition 5.9.

1. (i) We have $p(\overline {\mathfrak {S}_W}) \subset ({\mathbb {R}}^\times )^r$.

2. (ii) As a map $\overline {\mathfrak {S}_W}\to ({\mathbb {R}}^\times )^r$, $p|_{\overline {\mathfrak {S}_W}}$ is proper with respect to the real topologies.

3. (iii) The image $p(\overline {\mathfrak {S}})$ is bounded in ${\mathbb {R}}^r$.

Proof of assumption (i) Every point of $\overline {\mathfrak {S}_W}$ is of the form $gZ$ with $g\in G({\mathbb {R}})$. The vector $g\cdot \eta _i$ is thus in $V_i\otimes {\mathbb {R}}$. As $\eta _i\neq 0$ and $g$ is invertible, $g^{-1}\cdot \eta _i\neq 0$. As $Q_i$ is positive definite by Lemma 5.17, we have $p_i(g):=Q_i(g^{-1}\eta _i)\in {\mathbb {R}}_{>0}$.

We will use that there exists $C\in {\mathbb {R}}_{>0}$ such that for every $1\leq i\leq r$,

(40)\begin{equation} \forall \, (h,\alpha,k)\in H_P\times A^+_{\geq 0}\times K_\infty,\quad 0\leq p_i(h\cdot a\cdot \alpha\cdot k)\leq C\cdot \chi_i(\alpha)^{-2}. \end{equation}

Proof of (40) We write $\sigma =h\cdot a\cdot \alpha \cdot k$. By (32), we have $h^{-1}\cdot \eta _i=\pm \eta _i$. Thus,

\[ \sigma^{-1}\cdot \eta_i=\pm k^{-1}\cdot \alpha^{-1}\cdot a^{-1}\cdot \eta_i=\pm k^{-1} \cdot \chi_i(\alpha)^{-1}\cdot \eta_i. \]

Because $K$ is compact there exists a $K$-invariant euclidean norm $\lVert - \rVert$ on $V_i\otimes {\mathbb {R}}$. The two norms $\sqrt {Q_i}$ and $\lVert - \rVert$ on $V_i\otimes {\mathbb {R}}$ are comparable: there is $C_i\in {\mathbb {R}}_{>0}$ such that for any $v\in V_i\otimes {\mathbb {R}}$,

\[ Q_i(v)\leq C_i\cdot \lVert v \rVert^2. \]

We deduce (40) with $C=\max _{i\in \{1;\ldots ;r\}} C_i\cdot \lVert a^{-1}\cdot \eta _i \rVert ^2$ from

\begin{align*} p_i(\sigma)&=Q_i(\sigma^{-1}\cdot \eta_i)\\ &\leq C_i\cdot \lVert \pm k^{-1} \cdot \chi_i(\alpha)^{-1}\cdot a^{-1}\cdot \eta_i \rVert^2\\ &= C_i\cdot \lVert \pm\chi_i(\alpha)^{-1}\cdot a^{-1}\cdot \eta_i \rVert^2\\ &=C_i\cdot \chi_i(\alpha)^{-2} \cdot \lVert a^{-1}\cdot \eta_i \rVert^2\geq 0. \end{align*}

Proof of assumption (ii) For a divergent sequence $\sigma _n=\omega _n a\cdot \alpha _n k_n$ in $\overline {\mathfrak {S}_W}$, Corollary 5.14 and Lemma 5.15 imply that $\min _{1\leq i \leq r} \chi _i(\alpha _n) \to 0$. Using (40), we deduce that

(41)\begin{equation} \min_{1\leq i \leq r} p_i(\omega_n a_n a k_n) \to 0 \end{equation}

and, thus, $p(\omega _n a_n k_n)$ is divergent in ${{\mathbb {R}}^\times }^r$. This proves the properness.

Proof of assumption (iii) For $\sigma =h\cdot \alpha \cdot a\cdot k\in \overline {\mathfrak {S}}$, we have $0\leq \chi _1(\alpha ),\ldots,\chi _r(\alpha )\leq 1$ by (35), and deduce from (40) that

\[ \max_{1\leq i \leq r} \lvert p_i(\omega_n a_n a k_n) \rvert\leq C. \]

We now use Proposition 5.9 with $\mathfrak {S}=\overline {\mathfrak {S}_W}$. This concludes the proof of Theorem 5.16.

6.1 The Mumford–Tate hypothesis

We start with a property applicable in more general situations.

Definition 6.1 Let $M$ be a linear algebraic group over ${\mathbb {Q}}$, let $K_M\leq M({\mathbb {A}}_f)$ be a compact open subgroup, and let $U\leq M({\mathbb {A}}_f)$ be a compact subgroup.

We say that $U$ is MT in $M$ if the indices

(42)\begin{equation} [K_M \cap M({\mathbb{Q}}_p):U\cap K_M\cap M({\mathbb{Q}}_p)] \end{equation}

are finite and uniformly bounded as $p$ ranges through primes, where $M({\mathbb {Q}}_p)\leq M({\mathbb {A}}_f)$ is understood as a factor subgroup of $M({\mathbb {A}}_f)$.

Note that the definition does not depend on the choice of $K_M$, as any two compact open subgroups are commensurable. We may always enlarge $K_M$ so that it takes the product form $K_M=\prod _p K_p$ in which case the indices become

\[ [K_p:U\cap K_p]. \]

Likewise if $U'\leq M({\mathbb {A}}_f)$ is a compact subgroup commensurable to $U$, then $U$ is MT in $M$ if and only if $U'$ is MT in $M$. Note (and this is very important) that the condition that $U$ is MT in $M$ does not imply that $U$ is open in $M({\mathbb {A}}_f)$.

The following observation is an immediate consequence of the definition.

Lemma 6.2 In the Definition 6.1, let

(43)\begin{equation} \text{$U_p:=U\cap K_M\cap M({\mathbb{Q}}_p)$ and $U'=\prod_p U_p\leq \prod_p M({\mathbb{Q}}_p)$.} \end{equation}

Then $U$ is MT in $M$ if and only if $U'$ is MT in $M$.

We specialise the above definition to the context of images of Galois representations.

6.2 Lower bounds for Galois orbits in terms of finite heights under the MT hypothesis

The following statement is an essential ingredient in the proof of the main theorem (see § 7). We also believe it to be of independent interest.

First, let us reduce the second assertion to the first.

Proof. We identify $G$ with its image by a faithful representation $G\to GL(N)$. We may replace $K$ by a commensurable group, and assume $K$ is a maximal compact subgroup of $G({\mathbb {A}}_f)$. For any maximal compact subgroup $K'$ of $GL(N,{\mathbb {A}}_f)$ such that $K\leq K'\leq GL(N,{\mathbb {A}}_f)$, we then have

(46)\begin{equation} K=K'\cap G({\mathbb{A}}_f). \end{equation}

We choose such a $K'$, and, possibly conjugating by an element of $GL(N,{\mathbb {Q}})$, we may assume $K'=GL(N,\widehat {{\mathbb {Z}}})$.

Consider $\phi :M\to G$ in (45). From $\phi (U)\leq G({\mathbb {A}}_f)$ and (46), we deduce

(47)\begin{equation} [\phi(U):\phi(U) \cap K]=[\phi(U):\phi(U) \cap K']=[\phi(U):\phi(U) \cap GL(N,\widehat{{\mathbb{Z}}})]. \end{equation}

We have identified the left-hand side of (47) with the left-hand side of (45).

It will be enough to identify the right-hand sides. We will show that a height function $H_f$ on the $GL(N)$-conjugacy class of $\phi$, when restricted to the $G$-conjugacy class, is a height function on this $G$-conjugacy class.

If $H_f:GL(N,{\mathbb {A}}_f)\cdot \phi \to {\mathbb {Z}}_{\geq 1}$ is associated to $\iota :GL(N)\cdot \phi \to {\mathbb {A}}^N_{\mathbb {Q}}$, then its restriction to $GL(N,{\mathbb {A}}_f)\cdot \phi$ is associated to $\iota ':G\cdot \phi \to GL(N)\cdot \phi \to {\mathbb {A}}^N_{\mathbb {Q}}$, provided $\iota '$ is a closed embedding.

It is equivalent to proving that $G\cdot \phi \subseteq GL(N,{\mathbb {A}}_f)\cdot \phi$ is Zariski closed.

To do this, we choose the map

\[ \iota:GL(N)\cdot \phi\xrightarrow{\phi'\mapsto d\phi'} \mbox{Hom}(\mathfrak{m},\mathfrak{gl}(N)). \]

By assumption, $M$ is Zariski connected. This map is thus injective. As $M$ is reductive, according to [Reference RichardsonRic88, Theorem 3.6], the image of $G\cdot \phi$ is closed in $\mbox {Hom}(\mathfrak {m},\mathfrak {gl}(N))$, and thus $G\cdot \phi \subseteq GL(N,{\mathbb {A}}_f)\cdot \phi$ is Zariski closed.

We now reduce the first assertion to Corollary B.2, Theorems B.1 and B.4.

Proof. Writing $K=GL(N,\widehat {{\mathbb {Z}}})$ for short, we may rewrite the left-hand side of (44) as

\[ [\phi(U):\phi(U) \cap K]=\lvert \phi(U)\cdot K/K \rvert=\lvert U/\stackrel{-1}{\phi}(K) \rvert= [U:\stackrel{-1}{\phi}(K)]. \]

Theorem C.1 implies $[\phi (U):\phi (U) \cap K] \preccurlyeq H_f(\phi )$. We now prove $H_f(\phi ) \preccurlyeq [\phi (U):\phi (U) \cap K]$.

It is enough to obtain a lower bound after replacing $U$ by the smaller group $U'\leq U$ as in Lemma 6.2: without loss of generality we may assume $U=U'$. We thus assume that $U=\prod _p U_p$ as in (43).

The left-hand side is the product of $K_p=GL(N,{\mathbb {Z}}_p)$, hence we have

\[ [\phi(U):\phi(U) \cap K] =\prod_p [\phi(U_p):\phi(U_p) \cap K_p]. \]

We apply Definition 6.1 for $K_M=M(\widehat {{\mathbb {Z}}})=M({\mathbb {A}}_f)\cap GL(d,\widehat {{\mathbb {Z}}})$: the upper bound $C=\sup _p[M({\mathbb {Z}}_p):U_p]$ is finite. Using (B.6) we have

(48)\begin{equation} [\phi(M(U_p)):\phi(M(U_p))\cap GL(N,{{\mathbb{Z}}}_p)]\geq\frac{H_p(d\phi)}{c\cdot C}. \end{equation}

As in the proof of (B.1) of Theorem B.1, we can deduce

(49)\begin{equation} [\phi(U):\phi(U)\cap GL(N,\widehat{{\mathbb{Z}}})] \geq \frac{1}{(c\cdot C)^{\omega(H_f(d\phi))}}\cdot H_f(d\phi). \end{equation}

Arguing as in the proof of (B.2) and (B.3) of Corollary B.2, we obtain

(50)\begin{equation} H_{W,f}(\phi)\approx H_f(d\phi) \preccurlyeq [\phi(U):\phi(U)\cap GL(N,\widehat{{\mathbb{Z}}})]. \end{equation}

6.3 Functoriality properties of the MT hypothesis

The following uses general properties of adelic topologies on algebraic groups. A good reference is [Reference Platonov and RapinchukPR94].