Transport of Zariski density in compatible collections of $G$-representations

In this item, let us assume all the standard conjectures on pure motives. Let $X$ be a connected normal scheme of finite type over $\mathbf{Z}$. Consider a collection $\{\rho_{\ell}\colon\pi_{1}(X_{\mathbf{Z}[1/\ell]})\to\operatorname{GL}_{n}(\mathbf{Q}_{\ell})\}_{\ell}$ of (continuous) representations of the étale fundamental group of $X$. Suppose $\{\rho_{\ell}\}_{\ell}$ is of geometric origin, e.g. $\rho_{\ell}$ arises as the monodromy representation of the $\ell$-adic local system $R^{i}f_{*}\mathbf{Q}_{\ell}$ for a fixed smooth proper family $f$ over $X$. Tate’s conjecture then has the following consequence: the algebraic monodromy groups $M_{\ell}\coloneqq\overline{\operatorname{Img}(\rho_{\ell})}{}^{\mathrm{Zar}}$ are independent of $\ell$ in the sense that $M_{\ell}\cong G_{\mathbf{Q}_{\ell}}$ for some algebraic group $G$ over $\mathbf{Q}$ [And04, Proposition 7.3.2.1]. This $G$ is the “motivic Galois group” of the family $f$,¹¹1 More precisely, of the pure numerical motive over the function field of $X$ defined by $f$. and a theorem of Jannsen [Jan92] predicts that $G$ is a (possibly non-connected) reductive group.

On the other hand, by the Riemann hypothesis for varieties over finite fields, $\{\rho_{\ell}\}_{\ell}$ is compatible in the sense that for any closed point $s$ of $X$, the characteristic polynomial of $\rho_{\ell}(\operatorname{Frob}_{s})$, an element of $\mathbf{Q}_{\ell}[t]$, lives in $\mathbf{Q}[t]$ and is independent of $\ell$ when $\ell\neq\operatorname{char}(\kappa_{s})$.

Assuming Tate’s conjecture, the Tannakian formalism allows one to upgrade $\{\rho_{\ell}\}_{\ell}$ to a collection $\{\widetilde{\rho}_{\ell}\colon\pi_{1}(X_{\mathbf{Z}[1/\ell]})\to G(\mathbf{Q}_{\ell})\}_{\ell}$ of $G$-representations all having Zariski-dense image which is compatible in the sense that $\{\xi\circ\widetilde{\rho}_{\ell}\}_{\ell}$ is compatible for any $\mathbf{Q}$-representation $\xi\colon G\to\operatorname{GL}_{r}$. If $G$ is connected, this is equivalent to the following apparently stronger form of compatibility: For each closed point $s$ of $X$, there exists an element $g\in G(\overline{\mathbf{Q}})$ whose conjugacy class is defined over $\mathbf{Q}$ (i.e. $\operatorname{Gal}_{\mathbf{Q}}$-stable) such that the semisimple part of $\rho_{\ell}(\operatorname{Frob}_{s})$ is conjugate in $G(\overline{\mathbf{Q}_{\ell}})$ to $g$.²²2 We have to fix some embeddings $\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}_{\ell}}$, but nothing will depend on them. Note that, in general, a conjugacy class defined over $\mathbf{Q}$ will not contain an element defined over $\mathbf{Q}$, i.e. we cannot take $g\in G(\mathbf{Q})$. The equivalence stated here is an immediate consequence of the facts that the ring of class functions on a connected reductive group is generated by the characters of irreducible representations, separates semisimple conjugacy classes, and detects the field of definition of a semisimple conjugacy class. An alternative statement of compatibility uses the variety $G/\!\!/G$ of semisimple conjugacy classes, defined as the GIT quotient $\operatorname{Spec}(\mathcal{O}(G)^{G})$ with respect to the conjugation action. For an algebraically closed field $F$, the map $[-]\colon G\to G/\!\!/G$ induces a bijection between $(G/\!\!/G)(F)$ and the set of semisimple conjugacy classes in $G(F)$. The compatibility of $\{\widetilde{\rho}_{\ell}\}_{\ell}$ means that the elements $[\widetilde{\rho}_{\ell}(\operatorname{Frob}_{s})]\in(G/\!\!/G)(\mathbf{Q}_{\ell})$, for $\ell\neq\operatorname{char}(\kappa_{s})$, arise from a common element of $(G/\!\!/G)(\mathbf{Q})$.

Due to versions of Tate’s conjecture for Abelian varieties proven by Faltings, much stronger results are known for $(G,X)$ of Abelian type; see e.g. the work of Cadoret–Kret [CK16, Theorem A]. Thus Corollary 1.3.2 is novel only when $(G,X)$ is not of Abelian type, in which case the $\rho_{\ell}$ are not known, but conjectured, to be of geometric origin. As a concrete example, we obtain from these “non-Abelian Shimura varieties” compatible collections of representations $\{\operatorname{Gal}_{F}\to G(\mathbf{Q}_{\ell})\}$, where $F$ is a number field and $G$ is an adjoint group of type $\mathrm{E}_{6(-14)}$ or $\mathrm{E}_{7(-25)}$, with Zariski-dense image along a set of primes of Dirichlet density $1$.

Fix a connected reductive³³3In fact, with the exception of Lemma 2.3.5, the final arguments will reduce immediately to the case when $G$ is semisimple, so the reader loses little in assuming this to be so for all of §2 and replacing all root data with root systems. group $G$ over $K$ and a maximal torus $T_{0}$ of $G_{{K^{\mathrm{s}}}}$ (which need not be defined over $K$, though we will later assume it to be). Let $\Psi_{0}$ be the root datum attached to $T_{0}$. Denote by $\operatorname{W}(\Psi_{0})$ and $\operatorname{Aut}(\Psi_{0})$, respectively, the Weyl and automorphism groups of $\Psi_{0}$ in the sense of root data, and by $\operatorname{Out}(\Psi_{0})$ the quotient $\operatorname{Aut}(\Psi_{0})/{\operatorname{W}(\Psi_{0})}$. Finally, for any subset $\Omega\subseteq\operatorname{Aut}(\Psi_{0})$ stable under conjugation by $\operatorname{W}(\Psi_{0})$, we will denote by $[\Omega]$ the set of orbits of the conjugaction action of $\operatorname{W}(\Psi_{0})$ on $\Omega$.

Let $T$ be any maximal torus of $G$. The Galois action on $T_{K^{\mathrm{s}}}$ induces a homomorphism $\varphi_{T}\colon{\operatorname{Gal}_{K}}\to\operatorname{GL}(\mathrm{X}^{*}(T_{K^{\mathrm{s}}}))$. If we fix an element $a\in G({K^{\mathrm{s}}})$ satisfying $aT_{K^{\mathrm{s}}}a^{-1}=T_{0}$, it induces an isomorphism $\theta_{a}\colon{\operatorname{GL}(\mathrm{X}^{*}(T_{K^{\mathrm{s}}}))}\overset{\sim}{\to}\operatorname{GL}(\mathrm{X}^{*}(T_{0}))$, and the image of $\theta_{a}\circ\varphi_{T}$ lands in $\operatorname{Aut}(\Psi_{0})$. The isomorphism $\theta_{a}$ depends on $a$ only up to $\operatorname{W}(\Psi_{0})$-conjugation, so the $\operatorname{W}(\Psi_{0})$-conjugacy class $[\varphi_{T}]$ of $\theta_{a}\circ\varphi_{T}$ depends only on $T$ (and the fixed objects $G$ and $T_{0}$). Below, we will conflate $[\varphi_{T}]$ with the function $\operatorname{Gal}_{K}\to[\operatorname{Aut}(\Psi_{0})]$ valued in the set of $\operatorname{W}(\Psi_{0})$-conjugacy classes of $\operatorname{Aut}(\Psi_{0})$.

The composition

$$\operatorname{Gal}_{K}\xrightarrow{\theta_{a}\circ\varphi_{T}}\operatorname{Aut}(\Psi_{0})\mathrel{\mathrlap{\rightarrow}\mkern-2.25mu\rightarrow}\operatorname{Out}(\Psi_{0})$$

is therefore also well defined, and one checks that it is independent of $T$. Denote it by $\varphi_{G}$. (This keeps track of action on $G^{\mathrm{Ab}}$ and the “$*$-action” on the Dynkin diagram induced by $\operatorname{Gal}_{K}$.) Thus each maximal torus of $G$ corresponds to a $\operatorname{W}(\Psi_{0})$-conjugacy class of lifts of $\varphi_{G}$.

If $T$ and $T^{\prime}$ are maximal tori of $G$ which are conjugate by an element of $G(K)$, then $[\varphi_{T}]=[\varphi_{T^{\prime}}]$. Indeed, we have $\theta_{b}\circ\varphi_{T^{\prime}}=\varphi_{T}$ for any $b\in G(K)$ satisfying $bT^{\prime}b^{-1}=T$.

Given an abstract root datum $\Psi$ and a homomorphism $\operatorname{Gal}_{K}\to\operatorname{Out}(\Psi)$, there exists a quasisplit connected reductive group over $K$, unique up to isomorphism, realizing this data in the sense of (2.1.1–2.1.2).

Let $g\in G({K^{\mathrm{s}}})$ be an element whose $G({K^{\mathrm{s}}})$-conjugacy class $[g]$ is defined over $K$ (i.e. stable under the $\operatorname{Gal}_{K}$-action on $G({K^{\mathrm{s}}})$). For any $K$-representation $\xi\colon G\to\operatorname{GL}_{d}$, the $\operatorname{GL}_{d}({K^{\mathrm{s}}})$-conjugacy class of $\xi(g)$ is defined over $K$, so the characteristic polynomial $P$ of $\xi(g)$ has coefficients in $K$. If $\xi$ is faithful, we will call the splitting field $F$ of $P$ over $K$ the splitting field of $[g]$ over $K$. It is well defined because if $T$ is a maximal torus of $G_{K^{\mathrm{s}}}$ containing the semisimple part $g_{\mathrm{ss}}$ of $g$, then $F$ is the subfield of ${K^{\mathrm{s}}}$ generated by the values $\chi(g_{\mathrm{ss}})$ where $\chi\in\mathrm{X}^{*}(T)$.

We now describe the action of $\operatorname{Gal}_{K}$ on the roots of $P$ in a way intrinsic to $G$. To do so, we may assume (after conjugating $g$ and replacing it by $g_{\mathrm{ss}}$) that it is contained in a maximal torus $T$ defined over $K$. For simplicity, we will consider the case when $G$ is an adjoint group (i.e. has trivial center), $\xi$ is its adjoint representation, and the values $\chi_{1}(g),\dots,\chi_{m}(g)$ are distinct, where $\chi_{1},\dots,\chi_{m}$ are the nontrivial characters of $T_{K^{\mathrm{s}}}$ making up $\xi|_{T}$. (Such $g$ are called strongly regular.)

In this case, the values $\chi_{1}(g),\dots,\chi_{m}(g)$ are the roots of $P$ different from $1$, so the Galois action on the roots of $P$ permutes the set of characters $\{\chi_{1},\dots,\chi_{m}\}$.

Two easy observations about the $\varphi_{T,g}$ of Lemma 2.2.3:

1. (a)

   Since $\varphi_{T,g}(\sigma)$ differs from $\varphi_{T}(\sigma)$ by $\operatorname{inn}(a_{\sigma})$, the projection of $\theta_{a}\circ\varphi_{T,g}$ to $\operatorname{Out}(\Psi_{0})$ is again equal to $\varphi_{G}$ for any $a\in G({K^{\mathrm{s}}})$ satisfying $aT_{K^{\mathrm{s}}}a^{-1}=T_{0}$.

2. (b)

   If $g\in G(K)$, then $\varphi_{T,g}=\varphi_{T}$.

Following the notation of (2.1.2), we write $[\varphi_{T,g}]$ for the $\operatorname{W}(\Psi_{0})$-conjugacy class of $\theta_{a}\circ\varphi_{T,g}$.

Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$. As above we have a connected reductive group $G$, and from now on we (for convenience) assume $T_{0}$ is a maximal torus of $G$ rather than $G_{\overline{\mathbf{Q}}}$ (so $\Psi_{0}$ is now the root datum attached to $T_{0,\overline{\mathbf{Q}}}$, etc.). Let $E|\mathbf{Q}$ be the splitting extension of $\varphi_{G}$, i.e. the extension for which $\operatorname{Ker}(\varphi_{G})=\operatorname{Gal}_{E}$.

For each prime $\ell$, fix an algebraic closure $\overline{\mathbf{Q}_{\ell}}$ of $\mathbf{Q}_{\ell}$. Let $\iota_{\ell}$ be a field embedding $\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}_{\ell}}$. It induces an embedding $\operatorname{Gal}_{\mathbf{Q}_{\ell}}\hookrightarrow\operatorname{Gal}_{\mathbf{Q}}$, and given a torus $T$ over $\mathbf{Q}$, maps $\mathrm{X}^{*}(T_{\overline{\mathbf{Q}}})\to\mathrm{X}^{*}(T_{\overline{\mathbf{Q}_{\ell}}})$ and ${\operatorname{GL}(\mathrm{X}^{*}(T_{\overline{\mathbf{Q}}}))\to\operatorname{GL}(\mathrm{X}^{*}(T_{\overline{\mathbf{Q}_{\ell}}}))}$ via base change. By choosing an isomorphism $T_{\overline{\mathbf{Q}}}\cong\mathbf{G}_{\mathrm{m}}^{r}$, one sees that the latter are isomorphisms (but depend on $\iota_{\ell}$). If $T$ is a maximal torus of $G$, then $\iota_{\ell}$ induces isomorphisms

$$\operatorname{Aut}(\Psi)\overset{\sim}{\to}\operatorname{Aut}(\Psi_{\ell})\quad\text{and}\quad\operatorname{W}(\Psi)\overset{\sim}{\to}\operatorname{W}(\Psi_{\ell}),$$

(2.3.2.1)

where $\Psi$ and $\Psi_{\ell}$ are, respectively, the root data attached to $T_{\overline{\mathbf{Q}}}$ and $T_{\overline{\mathbf{Q}_{\ell}}}$.

Having picked a embedding $\iota_{\ell}$, we are safe to forget it in the notation and identify all the $\mathbf{Q}$- and $\mathbf{Q}_{\ell}$-versions of the above objects, because of (2.3.2.1) and because the following diagrams commute:

where $T$ is any torus over $\mathbf{Q}$,

(2.3.2.2)

where $T$ is any maximal torus of $G$ and $h\in T(\overline{\mathbf{Q}})$ any regular semisimple element, and

(2.3.2.3)

where $T,T^{\prime}$ are any maximal tori of $G$ and $a\in G(\overline{\mathbf{Q}})$ is any element satisfying $aT^{\prime}_{\overline{\mathbf{Q}}}a^{-1}=T_{\overline{\mathbf{Q}}}$. Thus, for example, if $T_{\ell}$ is a maximal torus of $G_{\mathbf{Q}_{\ell}}$, then (via $\iota_{\ell}$) we view $[\varphi_{T_{\ell}}]$ as a function $\operatorname{Gal}_{\mathbf{Q}_{\ell}}\to[\operatorname{Aut}(\Psi_{0})]$ and its domain as a subgroup of $\operatorname{Gal}_{\mathbf{Q}}$.

Given a prime $\ell$, we will let $\operatorname{Frob}_{\ell}\subseteq\operatorname{Gal}_{\mathbf{Q}_{\ell}}$ denote the Frobenius coset of $\mathbf{Q}_{\ell}$. In accordance with (2.3.2), if $\iota_{\ell}$ has been chosen, we view $\operatorname{Frob}_{\ell}$ as a subset of $\operatorname{Gal}_{\mathbf{Q}}$. Then for a finite Galois extension $F|\mathbf{Q}$ unramified at $\ell$, there is a Frobenius element in $\operatorname{Gal}(F|\mathbf{Q})$ defined by $\operatorname{Frob}_{\ell}$. Moreover if $\sigma$ is any element of the Frobenius conjugacy class of $\ell$ in $\operatorname{Gal}(F|\mathbf{Q})$, one can pick $\iota_{\ell}$ so that the image of $\operatorname{Frob}_{\ell}$ in $\operatorname{Gal}(F|\mathbf{Q})$ is $\{\sigma\}$.

Fix a prime $\ell$ unramified in $E|\mathbf{Q}$ and an embedding $\iota_{\ell}$. Let $\Omega$ be the preimage of $\varphi_{G}(\operatorname{Frob}_{\ell})$ in $\operatorname{Aut}(\Psi_{0})$; it is a coset of $\operatorname{W}(\Psi_{0})$. If $T$ is an unramified maximal torus of $G_{\mathbf{Q}_{\ell}}$, i.e. $\varphi_{T}$ factors through the Galois group of the maximal unramified extension of $\mathbf{Q}_{\ell}$, then $[\varphi_{T}](\operatorname{Frob}_{\ell})$ is a well defined element $\omega\in[\Omega]$.⁴⁴4 Recall that $[\Omega]$ is the set of orbits under the action of $\operatorname{W}(\Psi_{0})$ on $\Omega$ by conjugation. This makes sense because $\operatorname{W}(\Psi_{0})$ is normal in $\operatorname{Aut}(\Psi_{0})$. Say, in this case, that the torus $T$ corresponds to $\omega$.