2.1 The results of Gabber and Nori

In [Reference Katz16, Ch. 12], Katz presents the proof of the following result of Gabber.

Unfortunately, the implicit constant depends on $p$ in an ineffective way. A similar result is shown by Nori [Reference Nori33], with the same limitations.

For the applications in analytic number theory that we would like to consider, however, it would be necessary that (4) holds for all $\ell$ large enough, independently from  $p$ .

2.2 Consequences of the works of Larsen–Pink

By results of Larsen and Pink (see [Reference Larsen28, Theorem 3.17] and the applications in [Reference Katz, Goldfeld, Jorgenson, Jones, Ramakrishnan, Ribet and Tate18, §7], [Reference Kowalski25, pp. 155–156], [Reference Kowalski23, p. 29] and [Reference Kowalski24, p. 7]), the monodromy result of Katz over $\mathbb{C}$ (Theorem 1.3) implies that for every  $p$ , there exists a set $\unicode[STIX]{x1D6EC}(n,p)$ of primes of natural density 1 such that for all $\ell \in \unicode[STIX]{x1D6EC}(n,p)$ , the result (5) holds. Indeed, for fixed $n,q$ and varying $\unicode[STIX]{x1D706}$ , Kloosterman sheaves form a compatible system (see [Reference Katz16, 8.9]).

However, as for the results of Gabber and Nori, the way $\unicode[STIX]{x1D6EC}(n,p)$ is constructed is highly dependent on  $p$ . This is not a problem for the applications of Kowalski mentioned above, but issues arise if we need to take $\ell ,p\rightarrow \infty$ with some restrictions on the range as in [Reference Kowalski22].

2.3 (Invariant) generation of $\operatorname{SL}_{n}(\mathbb{F}_{\ell })$

In §4, we will see that $G_{\text{geom}}({\mathcal{K}}l_{n})$ contains elements conjugate to

(6) $$\begin{eqnarray}m=\left(\begin{array}{@{}ccccc@{}}0 & 0 & \cdots \, & 0 & (-1)^{n+1}\\ 1 & 0 & \cdots \, & 0 & 0\\ 0 & 1 & \cdots \, & 0 & 0\\ \vdots & & \ddots & \vdots & \vdots \\ 0 & & & 1 & 0\end{array}\right)\quad \text{and}\quad u=\left(\begin{array}{@{}ccccc@{}}1 & 1 & & & \\ & 1 & 1 & & \\ & & 1 & \ddots & \\ & & & \ddots & 1\\ & & & & 1\end{array}\right).\end{eqnarray}$$

As an indication that the monodromy group for $n$ odd should be $\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ , we have the following proposition.

However, we do not know whether these elements are invariant generators, namely whether any two conjugates are still generators. Without that, we may not conclude anything concerning Theorem 1.6.

2.4 The case $n=2$ and $\mathbb{F}_{\unicode[STIX]{x1D706}}=\mathbb{F}_{\ell }$

Hall [Reference Hall12] proved a classification theorem that generalizes a theorem of Yu on the $\mathbb{F}_{\ell }$ -monodromy of hyperelliptic curves, and also applies to show big monodromy results for families of twists of elliptic curves, as needed in [Reference Kowalski22].

A particular case is the following theorem, deduced from the classification of linear groups generated by transvections by Zalesski and Serezkin (a well-known result of Dickson when $n=2$ ).

This can be applied to the sheaf ${\mathcal{K}}l_{2}$ as follows.

However, this argument does not generalize to $n\geqslant 3$ , since the image of the inertia at $0$ in ${\mathcal{K}}l_{n}$ contains a transvection only when $n=2$ (see Proposition 4.1). Moreover, we cannot handle the case $\mathbb{F}_{\unicode[STIX]{x1D706}}\neq \mathbb{F}_{\ell }$ , since Hall considers only reduction of sheaves of $\mathbb{Z}_{\ell }$ -modules (and not $\mathbb{Z}[\unicode[STIX]{x1D701}_{4p}]_{\unicode[STIX]{x1D706}}$ -modules as for Kloosterman sheaves).

3.1 Equivalence of large ${\mathcal{O}}_{\unicode[STIX]{x1D706}}$ and $\mathbb{F}_{\unicode[STIX]{x1D706}}$ -monodromy

Understandably, we prefer to work with finite groups of Lie type in (5), rather than with groups over complete rings in (4).

3.2 Classification theorem over $\mathbb{F}_{\unicode[STIX]{x1D706}}$

The strategy of the proof of Theorem 1.6 is then as follows: if the conclusion does not hold, there exists a maximal (proper) subgroup

$$\begin{eqnarray}G_{\text{geom}}(\widehat{{\mathcal{K}}l}_{n})\leqslant H\lneq \left\{\begin{array}{@{}l@{}}\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}}):n~\text{odd},\quad \\ \operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}}):n~\text{even}.\quad \end{array}\right.\end{eqnarray}$$

By using the classification of maximal subgroups of the classical groups of Aschbacher and Kleidman–Liebeck, we will show the following general classification theorem.

Theorem 3.2. Let $n\geqslant 2$ , let $\mathbb{F}_{\unicode[STIX]{x1D706}}$ be a field of characteristic $\ell$ and let

$$\begin{eqnarray}H\leqslant \left\{\begin{array}{@{}l@{}}\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}}):n~\text{odd},\quad \\ \operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}}):n~\text{even},\quad \end{array}\right.\end{eqnarray}$$

be a maximal (proper) subgroup such that:

1. (1) the action of $H$ on $\mathbb{F}_{\unicode[STIX]{x1D706}}^{n}$ is irreducible;

2. (2) $H$ contains a unipotent element with a single Jordan block.

Then, for $\ell \gg _{n}1$ , we have either:

1. (1) $H=N_{\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})}(\operatorname{SO}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}}))$ for $n\geqslant 3$ oddFootnote ⁴ ;

2. (2) $H=N_{\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})}(\operatorname{SL}_{n}(\mathbb{F}^{\prime }))$ for $n\geqslant 3$ odd or $H=N_{\operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})}(\operatorname{Sp}_{n}(\mathbb{F}^{\prime }))$ for $n$ even, if $\mathbb{F}^{\prime }\leqslant \mathbb{F}_{\unicode[STIX]{x1D706}}$ is a subfield of prime index;

3. (3) $H=N_{\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})}(\operatorname{SU}_{n}(\mathbb{F}^{\prime }))$ for $n\geqslant 3$ odd, if $\mathbb{F}^{\prime }\leqslant \mathbb{F}_{\unicode[STIX]{x1D706}}$ is a subfield of index  $2$ .

More precisely, a theorem of Larsen–Pink on finite subgroups of algebraic groups and results on representations of finite groups of Lie type in various characteristics allows descent of a classification theorem of Suprunenko [Reference Suprunenko39, Theorem (1.9)] about subgroups of classical groups over algebraically closed fields containing regular unipotent elements (i.e. a unipotent element with a single Jordan block in the case of  $\operatorname{SL}_{n}$ ).

By the properties of Kloosterman sheaves, $H$ verifies the hypotheses of Theorem 3.2, while at the same time conclusions (1)–(3) can be excluded. This would give Theorem 1.6.

3.3 Comparison with Katz’s method over $\mathbb{C}$

The above strategy is parallel to the approach Katz uses to prove his Theorem 1.3.

Indeed, by using the semisimplicity of $G_{\text{geom}}^{0}({\mathcal{K}}l_{n})\leqslant \operatorname{SL}_{n}(\mathbb{C})$ and the presence of a unipotent element with a single Jordan block, Katz reduces to the following classification theorem.

Theorem 3.3 [Reference Katz16, Classification Theorem 11.6].

Let $n\geqslant 2$ be an integer and ${\mathcal{G}}$ be a simple Lie algebra over $\mathbb{C}$ given with a faithful irreducible representation

$$\begin{eqnarray}\unicode[STIX]{x1D70C}:{\mathcal{G}}{\hookrightarrow}\mathfrak{sl}_{n}(\mathbb{C}).\end{eqnarray}$$

Suppose that there exists a nilpotent element $N\in {\mathcal{G}}$ such that $\unicode[STIX]{x1D70C}(N)$ has a single Jordan block. Then the pair $({\mathcal{G}},\unicode[STIX]{x1D70C})$ is isomorphic to one of the following:

1. (1) ${\mathcal{G}}=\mathfrak{sl}_{n}(\mathbb{C})~(n\geqslant 2)$ , $\mathfrak{sp}_{n}(\mathbb{C})~(n\geqslant 4~\text{even})$ or $\mathfrak{so}_{n}(\mathbb{C})~(n\geqslant 5~\text{odd})$ , with the standard $n$ -dimensional representation;

2. (2) ${\mathcal{G}}=\mathfrak{g}_{2}(\mathbb{C})$ with its unique seven-dimensional irreducible representation.

The extraneous cases are then excluded by using properties of Kloosterman sheaves.

Comparing the classification Theorems 3.2 and 3.3, it is interesting to note how the maximality assumption replaces the simplicity (it cannot be assumed a priori that $G_{\text{geom}}(\widehat{{\mathcal{K}}l}_{n})\leqslant \operatorname{GL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ is quasisimple), and how additional conclusions arise in Theorem 3.2, calling for new arithmetic inputs to exclude them.

4.1 Results from [Reference Katz16]

By Proposition 4.1(1), note that the geometric monodromy group contains an element conjugate to the Jordan block $u$ from equation (6).

4.2 Explicit local monodromy at $\infty$

The local monodromy at $\infty$ of ${\mathcal{K}}l_{n}$ is determined explicitly in [Reference Katz16] (as $P_{\infty }$ -representation) and [Reference Kowalski, Michel and Sawin26] (more precisely as $I_{\infty }$ -representation), and more generally for hypergeometric sheaves in [Reference Fu7, Proposition 0.7]. We make this even more concrete by finding a matrix form of the representation.

Proposition 4.2. Under the same notations as in Proposition 4.1, we assume that $k=\mathbb{F}_{q}$ contains a primitive $2n$ th root of unity $\unicode[STIX]{x1D701}_{2n}$ . Let $Z\in \overline{k(T)}$ be a solution to $Z^{2n}=T$ and $W\in \overline{k(Z)}=\overline{k(T)}$ be a solution to

$$\begin{eqnarray}W^{|k|}-W=-Z^{2}.\end{eqnarray}$$

Then the restriction $\unicode[STIX]{x1D70C}|_{I_{\infty }}:I_{\infty }\rightarrow \operatorname{GL}_{n}(A)$ is isomorphic (over $\mathbb{F}_{\unicode[STIX]{x1D706}}$ ) to the representation

$$\begin{eqnarray}\unicode[STIX]{x1D70E}\mapsto \biggl((-1)^{(n+1)((j-i+i_{0})/n)}e\biggl(\frac{n\operatorname{tr}_{k/\mathbb{F}_{p}}(a_{0}\unicode[STIX]{x1D701}_{n}^{i})}{p}\biggr)\unicode[STIX]{x1D6FF}_{i-j\equiv i_{0}~(\text{mod}~n)}\biggr)_{1\leqslant i,j\leqslant n,}\end{eqnarray}$$

where $i_{0}\in \mathbb{Z}/2n$ and $a_{0}\in k$ are such that $\unicode[STIX]{x1D70E}(Z)=\unicode[STIX]{x1D701}_{2n}^{i_{0}}Z$ and $\unicode[STIX]{x1D70E}(W)=\unicode[STIX]{x1D701}_{2n}^{2i_{0}}$ $(W+a_{0})$ .

Remark 4.3. Assuming that $k$ contains an $n$ th root of unity is not a constraint for our purpose. Indeed, if $L/k$ is a finite extension, we have a commutative diagram

Proof. By [Reference Katz16, 10.4.5] and [Reference Kowalski, Michel and Sawin26, Lemma 4.9], the representation of $I_{\infty }$ corresponding to ${\mathcal{K}}l_{n}$ is isomorphic to

$$\begin{eqnarray}[x\mapsto x^{n}]_{\ast }({\mathcal{L}}_{\unicode[STIX]{x1D712}_{2}}^{n+1}\otimes {\mathcal{L}}_{\unicode[STIX]{x1D713}(xn)}),\end{eqnarray}$$

where $\unicode[STIX]{x1D712}_{2}$ is the character of order 2 of $\overline{\mathbb{F}}_{p}^{\times }$ and $\unicode[STIX]{x1D713}(x)=e(\operatorname{tr}(x)/p)$ . In other words, it is isomorphic to

$$\begin{eqnarray}\operatorname{Ind}_{I_{\infty ,n}}^{I_{\infty }}({\mathcal{L}}_{\unicode[STIX]{x1D712}_{2}}^{n+1}\otimes {\mathcal{L}}_{\unicode[STIX]{x1D713}(xn)}),\end{eqnarray}$$

where $I_{\infty ,n}$ is the unique subgroup of index $n$ in $I_{\infty }$ (see [Reference Katz16, 1.13]).

The extension $k(Z,W)/k(T)$ is Galois, and we have a split exact sequence

so an isomorphism

$$\begin{eqnarray}\displaystyle k\rtimes \mathbb{Z}/2n & \rightarrow & \displaystyle G=\operatorname{Gal}(k(Z,W)/k(T))\nonumber\\ \displaystyle (a,i) & \mapsto & \displaystyle \unicode[STIX]{x1D70E}(a,i),\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}(a,i)$ is such that $W\mapsto \unicode[STIX]{x1D701}_{2n}^{2i}(W+a)$ and $Z\mapsto \unicode[STIX]{x1D701}_{2n}^{i}Z$ . For every $(a,i)\in G$ , there exists an element of $I_{\infty ,n}$ extending $\unicode[STIX]{x1D70E}(a,i)$ , that we will again denote by $\unicode[STIX]{x1D70E}(a,i)$ .

We have $I_{\infty ,n}=I_{\infty }\cap \unicode[STIX]{x1D70B}_{1,q}^{(n)}$ , where $\unicode[STIX]{x1D70B}_{1,q}^{(n)}=\operatorname{Gal}(\overline{K(T)}/K(Y))$ is the subgroup of index $n$ in $\unicode[STIX]{x1D70B}_{1,q}$ . Indeed,

$$\begin{eqnarray}\displaystyle I_{\infty }/(I_{\infty }\cap \unicode[STIX]{x1D70B}_{1,q}^{(n)}) & = & \displaystyle I_{\infty }\unicode[STIX]{x1D70B}_{1,q}^{(n)}/\unicode[STIX]{x1D70B}_{1,q}^{(n)}=\unicode[STIX]{x1D70B}_{1,q}/\unicode[STIX]{x1D70B}_{1,q}^{(n)}\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Gal}(k(Y)/k(T))\cong \unicode[STIX]{x1D707}_{n}(k)\cong \mathbb{Z}/n.\nonumber\end{eqnarray}$$

Note that $(\unicode[STIX]{x1D70E}_{i})_{1\leqslant i\leqslant n}$ is a complete reduced system of representatives of $I_{\infty }/I_{\infty ,n}$ , where we abbreviate $\unicode[STIX]{x1D70E}_{i}=\unicode[STIX]{x1D70E}(0,i)$ .

By definition (or properties) of induced representations, a matrix form of the representation $I_{\infty }\rightarrow \operatorname{GL}_{n}(A)$ evaluated at $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}(a_{0},i_{0})\in I_{\infty }$ is then

$$\begin{eqnarray}(({\mathcal{L}}_{\unicode[STIX]{x1D712}_{2}}^{n+1}\otimes {\mathcal{L}}_{\unicode[STIX]{x1D713}(xn)})(\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E}))\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E})\in I_{\infty ,n}})_{1\leqslant i,j\leqslant n},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D70E}_{i}^{-1}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70E}_{j}$ . It remains to note that $\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E})\in I_{\infty ,n}$ if and only if $2(i-j)\equiv 2i_{0}~(\text{mod}~2n)$ , in which case:

1. – by definition of the Artin–Schreier representation,

   $$\begin{eqnarray}{\mathcal{L}}_{\unicode[STIX]{x1D713}(xn)}(\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E}))=\unicode[STIX]{x1D713}((\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E})(W)-W)n),\end{eqnarray}$$
   and $\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E})(W)-W=\unicode[STIX]{x1D701}_{2n}^{2(j+i_{0})}a_{0}$ ;

2. – by definition of the Kummer representation, if $n$ is even,

   $$\begin{eqnarray}{\mathcal{L}}_{\unicode[STIX]{x1D712}_{2}}^{n+1}(\unicode[STIX]{x1D70E}_{i,j}(\unicode[STIX]{x1D70E}))=\unicode[STIX]{x1D712}_{2}(\unicode[STIX]{x1D701}_{n}^{(j-i+i_{0})/2})=(-1)^{(j-i+i_{0})/n}.\end{eqnarray}$$

Since the two representations are defined over $\mathbb{F}_{\unicode[STIX]{x1D706}}$ , the same holds for the isomorphism by the rational canonical form.◻

4.2.1 Fields generated by traces

The following will be useful to deal with subfield subgroups; it shows that we still recover conjugacy-invariant arithmetic information (the subfield generated by the traces of the Frobenius) in the geometric monodromy group.

Proposition 4.5. Let $n\geqslant 2$ , let $\unicode[STIX]{x1D706}$ be an $\ell$ -adic valuation on

$$\begin{eqnarray}{\mathcal{O}}=\left\{\begin{array}{@{}ll@{}}\mathbb{Z}[\unicode[STIX]{x1D701}_{p}]\quad & n~\text{odd or}~p\equiv 1~(\text{mod}~4),\\ \mathbb{Z}[\unicode[STIX]{x1D701}_{4p}]\quad & \text{otherwise, with}~\ell \equiv 1~(\text{mod}~4)\end{array}\right.\end{eqnarray}$$

(see (3)), and let $\unicode[STIX]{x1D70C}_{n}:\unicode[STIX]{x1D70B}_{1,q}\rightarrow \operatorname{GL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ be the representation corresponding to the Kloosterman sheaf ${\mathcal{K}}l_{n}$ of $\mathbb{F}_{\unicode[STIX]{x1D706}}$ -modules over $\mathbb{P}^{1}/\mathbb{F}_{q}$ . Then

$$\begin{eqnarray}\mathbb{F}_{\ell }(\operatorname{Kl}_{n,q}(a):a\in \mathbb{F}_{q}^{\times })=\mathbb{F}_{\ell }(\operatorname{tr}\unicode[STIX]{x1D70C}_{n}(I_{\infty })),\end{eqnarray}$$

with index $(f,n)$ in $\mathbb{F}_{\unicode[STIX]{x1D706}}$ .

Proof. Under the hypotheses, $\mathbb{F}_{\unicode[STIX]{x1D706}}=\mathbb{F}_{\ell }(\unicode[STIX]{x1D701}_{p})$ (since $\unicode[STIX]{x1D701}_{4}\in \mathbb{F}_{\ell }$ if $\ell \equiv 1~(\text{mod}~4)$ ) and $f:=[\mathbb{F}_{\unicode[STIX]{x1D706}}:\mathbb{F}_{\ell }]=\operatorname{ord}(\ell \in \mathbb{F}_{p}^{\times })$ . Fisher [Reference Fisher6, Proposition 2.8] showed that for $\mathbb{Q}(\unicode[STIX]{x1D701}_{p})$ -valued Kloosterman sums,

$$\begin{eqnarray}\mathbb{Q}(q^{(n-1)/2}\operatorname{Kl}_{n,q}(a):a\in \mathbb{F}_{q}^{\times })=\mathbb{Q}(\unicode[STIX]{x1D701}_{p})^{\operatorname{Gal}(\mathbb{Q}(\unicode[STIX]{x1D701}_{p})/\mathbb{Q})[n]}.\end{eqnarray}$$

We proceed similarly to show that for $G=\operatorname{Gal}(\mathbb{F}_{\ell }(\unicode[STIX]{x1D701}_{p})/\mathbb{F}_{\ell })$ ,

$$\begin{eqnarray}\left.\begin{array}{@{}l@{}}L_{1}:=\mathbb{F}_{\ell }(\operatorname{Kl}_{n,q}(a):a\in \mathbb{F}_{q}^{\times })\\ L_{2}:=\mathbb{F}_{\ell }(\operatorname{tr}\unicode[STIX]{x1D70C}_{n}(I_{\infty }))\end{array}\right\}=\mathbb{F}_{\ell }(\unicode[STIX]{x1D701}_{p})^{G[n]}.\end{eqnarray}$$

For $\unicode[STIX]{x1D70E}\in G$ , let $u_{\unicode[STIX]{x1D70E}}\in \mathbb{F}_{p}^{\times }$ be such that $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D701}_{p})=\unicode[STIX]{x1D701}_{p}^{u_{\unicode[STIX]{x1D70E}}}$ , and note that for $a\in \mathbb{F}_{q}^{\times }$ ,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}(\operatorname{Kl}_{n,q}(a)) & = & \displaystyle \operatorname{Kl}_{n,q}(au_{\unicode[STIX]{x1D70E}}^{n})\nonumber\\ \displaystyle \unicode[STIX]{x1D70E}(\operatorname{tr}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70E}(a,0))) & = & \displaystyle \operatorname{tr}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70E}(au_{\unicode[STIX]{x1D70E}},0))=\mathop{\sum }_{i=1}^{n}e\biggl(\frac{n\operatorname{tr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}(au_{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D701}_{n}^{i})}{p}\biggr),\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}(a,0)$ is as defined in the proof of Proposition 4.2. Hence, $G[n]\leqslant \operatorname{Gal}(\mathbb{F}_{\ell }(\unicode[STIX]{x1D701}_{p})/L_{i})$ for $i=1,2$ .

On the other hand, let us assume that $\unicode[STIX]{x1D70E}\in \operatorname{Gal}(\mathbb{F}_{\ell }(\unicode[STIX]{x1D701}_{p})/L_{2})$ . For every character $\unicode[STIX]{x1D6EC}:\mathbb{F}_{q}^{\times }\rightarrow \overline{\mathbb{F}}_{\ell }^{\times }$ , we define

$$\begin{eqnarray}\displaystyle S_{2}(\unicode[STIX]{x1D6EC}) & = & \displaystyle \mathop{\sum }_{a\in \mathbb{F}_{q}^{\times }}\operatorname{tr}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70E}(a,0))\unicode[STIX]{x1D6EC}(a)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=1}^{n}\overline{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D701}_{n}^{i})G_{n}(\unicode[STIX]{x1D6EC})=n\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}\mid _{\unicode[STIX]{x1D707}_{n}}=1}G_{n}(\unicode[STIX]{x1D6EC}),\nonumber\end{eqnarray}$$

where $G_{n}(\unicode[STIX]{x1D6EC}):=\sum _{a\in \mathbb{F}_{q}^{\times }}e((n\operatorname{tr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}(a))/p)\unicode[STIX]{x1D6EC}(a)\neq 0$ since $G_{n}(\unicode[STIX]{x1D6EC})G_{-n}(\overline{\unicode[STIX]{x1D6EC}})=q\in \mathbb{F}_{\ell }^{\times }$ . Then, since $\unicode[STIX]{x1D70E}|_{L_{2}}=\operatorname{id}$ , we have $S_{2}(\unicode[STIX]{x1D6EC})=\overline{\unicode[STIX]{x1D6EC}}(u_{\unicode[STIX]{x1D70E}})S_{2}(\unicode[STIX]{x1D6EC})$ , which yields that $\unicode[STIX]{x1D6EC}(u_{\unicode[STIX]{x1D70E}})=1$ whenever $\unicode[STIX]{x1D6EC}|_{\unicode[STIX]{x1D707}_{n}}=1$ . Thus, $\sum _{\unicode[STIX]{x1D6EC}\in \widehat{\mathbb{F}_{q}^{\times }/\unicode[STIX]{x1D707}_{n}}}\unicode[STIX]{x1D6EC}(u_{\unicode[STIX]{x1D70E}})=(q-1)/n$ , so that $u_{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D707}_{n}$ , i.e. $\unicode[STIX]{x1D70E}\in G[n]$ .

Finally, $\operatorname{Gal}(\mathbb{F}_{\ell }(\unicode[STIX]{x1D701}_{p})/L_{1})\leqslant \operatorname{Gal}(\mathbb{F}_{\ell }(\unicode[STIX]{x1D701}_{p})/L_{2})$ by Chebotarev’s density theorem (see [Reference Serre38, I.2.2, Corollary 2a]).

The claim on the index follows from $|G[n]|=|(\mathbb{Z}/f)[n]|=(n,f)$ . ◻

6.1 Reminder on the classification of maximal subgroups of classical groups

As already mentioned, an essential input to prove Theorem 3.2 is the classification of maximal subgroups of classical groups, which originated from the work of Aschbacher [Reference Aschbacher1], and was then expanded by Kleidman and Liebeck [Reference Kleidman and Liebeck20]. Another proof was given by Liebeck and Seitz [Reference Liebeck and Seitz31] through an analogous result over an algebraically closed field and using descent (as in the treatment of finite groups of Lie type with Steinberg isomorphisms). A good exposition of these results can be found in [Reference Malle and Testerman32, II.18.1, III.27–28].

In the remainder of this section, we survey this classification in a way adapted to our needs.

We now recall the definition of the classes ${\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{7}$ , along with some convenient properties we will use.

Let $V=\mathbb{F}_{\unicode[STIX]{x1D706}}^{n}$ , $\boldsymbol{V}=\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}}^{n}$ , and $\operatorname{Frob}\in \operatorname{Gal}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}}/\mathbb{F}_{\unicode[STIX]{x1D706}})$ be the arithmetic Frobenius $x\mapsto x^{|\mathbb{F}_{\unicode[STIX]{x1D706}}|}$ . We will write $\boldsymbol{G}=\operatorname{SL}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})$ (respectively $\operatorname{Sp}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})$ ). Let $\unicode[STIX]{x1D6FD}$ be the zero bilinear form on $V$ if $G=\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ or the symplectic form associated to $G$ if $G=\operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ . The classes appearing in (1) of Theorem 6.1 are the following.

1. – Class ${\mathcal{C}}_{1}$ (subspace stabilizers):

   $$\begin{eqnarray}H=\operatorname{Stab}_{G}(W)\end{eqnarray}$$
   with $0\neq W\lneq V$ totally singular or non-degenerate with respect to $\unicode[STIX]{x1D6FD}$ . Note that $W\leqslant V$ is an $H$ -submodule, so this case does not arise if $H$ acts on $V$ irreducibly.

2. – Class ${\mathcal{C}}_{2}$ (stabilizers of orthogonal decompositions):

   $$\begin{eqnarray}\displaystyle \boldsymbol{V} & = & \displaystyle \boldsymbol{V}_{1}\bot \cdots \bot \boldsymbol{V}_{t}\nonumber\\ \displaystyle & & \displaystyle (t\geqslant 2,\text{all the}~\boldsymbol{V}_{i}~\text{isometric},n=\dim (\boldsymbol{V}_{1})t),\nonumber\\ \displaystyle \boldsymbol{M} & = & \displaystyle \operatorname{Stab}_{\boldsymbol{G}}(\boldsymbol{V}_{1}\bot \cdots \bot \boldsymbol{V}_{t}),\nonumber\\ \displaystyle H & {\leqslant} & \displaystyle \boldsymbol{M}^{\operatorname{Frob}}.\nonumber\end{eqnarray}$$
   In other words, the elements of $\boldsymbol{M}$ are the $g\in \boldsymbol{G}$ such that there exists a permutation $\unicode[STIX]{x1D70E}\in \mathfrak{S}_{t}$ with $g\boldsymbol{V}_{i}=\boldsymbol{V}_{\unicode[STIX]{x1D70E}(i)}$ for all $1\leqslant i\leqslant t$ .

3. – Class ${\mathcal{C}}_{3}$ (stabilizers of totally singular decompositions): if $G=\operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ ,

   $$\begin{eqnarray}\displaystyle \boldsymbol{V} & = & \displaystyle \boldsymbol{V}_{1}\oplus \boldsymbol{V}_{2}\quad (\boldsymbol{V}_{i}~\text{maximal totally isotropic:}~\unicode[STIX]{x1D6FD}|_{\boldsymbol{V}_{i}}=0),\nonumber\\ \displaystyle \boldsymbol{M} & = & \displaystyle \operatorname{Stab}_{\boldsymbol{G}}(\boldsymbol{V}_{1}\oplus \boldsymbol{V}_{2}),\nonumber\\ \displaystyle H & {\leqslant} & \displaystyle \boldsymbol{M}^{\operatorname{Frob}}.\nonumber\end{eqnarray}$$
   In other words, the elements of $M$ are the $g\in \boldsymbol{G}$ such that there exists a permutation $\unicode[STIX]{x1D70E}\in \mathfrak{S}_{2}$ with $g\boldsymbol{V}_{i}=\boldsymbol{V}_{\unicode[STIX]{x1D70E}(i)}$ for $i=1,2$ . In particular, $\dim (\boldsymbol{V}_{1})=\dim (\boldsymbol{V}_{2})=n/2$ .

4. – Class ${\mathcal{C}}_{4}$ (stabilizers of tensor product decompositions):

   $$\begin{eqnarray}\displaystyle \boldsymbol{V} & = & \displaystyle \boldsymbol{V}_{1}\otimes \cdots \otimes \boldsymbol{V}_{t}\quad (\dim \boldsymbol{V}_{i}\geqslant 2,t\geqslant 2),\nonumber\\ \displaystyle \boldsymbol{L} & = & \displaystyle \boldsymbol{G}(\boldsymbol{V}_{1})\times \cdots \times \boldsymbol{G}(\boldsymbol{V}_{t}),\text{acting on}~\boldsymbol{V}~\text{by tensor product},\nonumber\\ \displaystyle \boldsymbol{M} & = & \displaystyle N_{\operatorname{GL}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})}(\boldsymbol{L})\cap \boldsymbol{G},\nonumber\\ \displaystyle H & {\leqslant} & \displaystyle \boldsymbol{M}^{\operatorname{Frob}},\nonumber\end{eqnarray}$$
   with $t=2$ if the $\boldsymbol{V}_{i}$ are not mutually isomorphic, where we write $\boldsymbol{G}(\boldsymbol{V}_{i})$ for the classical group of type $\boldsymbol{G}$ on the vector space $\boldsymbol{V}_{i}$ . Note that $n=\dim (\boldsymbol{V}_{1})\cdots \dim (\boldsymbol{V}_{t})$ , and $n=\dim (\boldsymbol{V}_{1})^{t}$ if the $\boldsymbol{V}_{i}$ are mutually isomorphic. We have
   $$\begin{eqnarray}\unicode[STIX]{x1D70B}(N_{\operatorname{GL}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})}(\boldsymbol{L})\cap \boldsymbol{G})\leqslant N_{\operatorname{PGL}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})}(\unicode[STIX]{x1D70B}(\boldsymbol{L}))\cap \unicode[STIX]{x1D70B}(\boldsymbol{G})=\boldsymbol{N}.\end{eqnarray}$$
   Since $\unicode[STIX]{x1D70B}(\boldsymbol{G})$ has trivial center, there is a morphism from $\boldsymbol{N}$ to
   $$\begin{eqnarray}\displaystyle \operatorname{Aut}(\unicode[STIX]{x1D70B}(\boldsymbol{L})) & = & \displaystyle \operatorname{Aut}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{1}))\times \cdots \times \unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{t})))\nonumber\\ \displaystyle & \cong & \displaystyle \left\{\begin{array}{@{}l@{}}\operatorname{Aut}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{1})))\times \operatorname{Aut}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{2}))):t=2\quad \\ \operatorname{Aut}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{1})))\wr \mathfrak{S}_{t}:t>2\quad \end{array}\right.\nonumber\\ \displaystyle & \rightarrow & \displaystyle \left\{\begin{array}{@{}l@{}}\operatorname{Out}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{1})))\times \operatorname{Out}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{2}))):t=2\quad \\ \operatorname{Out}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{1})))\wr \mathfrak{S}_{t}:t>2,\quad \end{array}\right.\nonumber\end{eqnarray}$$
   with kernel isomorphic to $\unicode[STIX]{x1D70B}(\boldsymbol{L})\cap \unicode[STIX]{x1D70B}(\boldsymbol{G})$ . The isomorphism on the second line follows from the lemma below.

Lemma 6.2. Let $G_{1},\ldots ,G_{t}$ be non-abelian simple groups. Then $\operatorname{Aut}(G_{1}\times \cdots \times G_{t})$ is isomorphic to

$$\begin{eqnarray}\operatorname{Aut}(G_{1})\times \cdots \times \operatorname{Aut}(G_{t})\end{eqnarray}$$

if there is no isomorphism among the $G_{i}$ , respectively $\operatorname{Aut}(G_{1})\wr \mathfrak{S}_{t}$ if the $G_{i}$ are mutually isomorphic.

1. – Class ${\mathcal{C}}_{5}$ (symplectic-type $r$ -subgroups):

   $$\begin{eqnarray}\unicode[STIX]{x1D70B}(H)\cong \left\{\begin{array}{@{}l@{}}\mathbb{Z}/r^{2m}.\operatorname{Sp}_{2m}(\mathbb{F}_{r}):G=\operatorname{SL}_{n}(\mathbb{F}_{r^{m}}),\quad \\ \mathbb{Z}/2^{2m}.\operatorname{GO}_{2m}^{-}(\mathbb{F}_{2}):G=\operatorname{Sp}_{n}(\mathbb{F}_{2^{m}}),\quad \end{array}\right.\end{eqnarray}$$
   with $n=r^{m}$ , $r\neq \ell$ prime, $\ell \equiv 1~(\text{mod}~r(2,r))$ . Here, we have only given the classification of the subgroups that arise in the class; for more details about the latter, see [Reference Kleidman and Liebeck20, §4.6].

2. – Class ${\mathcal{C}}_{6}$ (normalizers of classical groups): for $G=\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ with $n$ odd (since $\ell \neq 2$ , this class does not arise in the symplectic case) and $\mathbb{F}^{\prime }\leqslant \mathbb{F}_{\unicode[STIX]{x1D706}}$ a subfield such that $|\mathbb{F}^{\prime }|=|\mathbb{F}_{\unicode[STIX]{x1D706}}|^{1/2}$ ,

   $$\begin{eqnarray}H=N_{G}(\operatorname{SO}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}}))\quad \text{or}\quad N_{G}(\operatorname{SU}_{n}(\mathbb{F}^{\prime })).\end{eqnarray}$$

3. – Class ${\mathcal{C}}_{7}$ (subfield subgroups): for $\mathbb{F}^{\prime }\leqslant \mathbb{F}_{\unicode[STIX]{x1D706}}$ of prime index,

   $$\begin{eqnarray}H=N_{G}(G(\mathbb{F}^{\prime })).\end{eqnarray}$$

Note that the unitary cases of classes ${\mathcal{C}}_{6}$ and ${\mathcal{C}}_{7}$ do not arise if $\mathbb{F}_{\unicode[STIX]{x1D706}}=\mathbb{F}_{\ell }$ .

6.2 Review on automorphism groups

We briefly recall the following results about automorphisms of Lie algebras/Lie groups/finite groups of Lie type that will be useful several times, in particular to handle class ${\mathcal{C}}_{4}$ and case (2) of Theorem 6.1.

In the finite case, this becomes the following.

Proposition 6.5. If $G$ is a finite simple group of Lie type defined over a finite field $k$ , every automorphism can be written as the product of an inner, graph, diagonal, and field automorphism. More precisely,

$$\begin{eqnarray}\operatorname{Out}(G)\cong (\text{Diag}(G)\operatorname{Aut}(k)).\text{Graph}(G),\end{eqnarray}$$

where $\text{Diag}(G)$ is the group of diagonal automorphisms.

6.3 Strategy to prove Theorem 3.2

As already mentioned, we apply Theorem 6.1 to the situation of Theorem 3.2. Let us first sketch the general approach.

6.3.2 Almost-simple subgroups

On the other hand, if $H$ satisfies (2) of Theorem 6.1, there exists a non-abelian simple group $S$ such that

$$\begin{eqnarray}S=\unicode[STIX]{x1D70B}(T)\unlhd \tilde{H}=\unicode[STIX]{x1D70B}(H)\leqslant \operatorname{Aut}(S)\end{eqnarray}$$

with $T\unlhd H\leqslant \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ and $T$ acting irreducibly on $\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}}^{n}$ , for $\unicode[STIX]{x1D70B}:\operatorname{SL}_{n}\rightarrow \operatorname{PSL}_{n}$ the projection.

The first step is to reduce to the case where $S$ is a group of Lie type in characteristic $\ell$ , applying the following with the regular unipotent element contained in  $H$ .

Proposition 6.8. In the above setting, if $H$ contains a non-scalar element of order ${\geqslant}\ell$ , then either $\ell \ll _{n}1$ or $S$ is a group of Lie type in characteristic  $\ell$ .

Our first approach relied on the classification of finite simple groups, but this is also a particular case of a recent powerful theorem of Larsen and Pink, independent from the classification. We will give both arguments.

For groups of Lie type in characteristic $\ell$ , we have the following.

Proposition 6.9. In the above setting, if $S=\unicode[STIX]{x1D70B}(T)$ is a simple group of Lie type in characteristic $\ell \gg _{n}1$ and $H$ contains a regular unipotent element, then either:

1. (1) $T=\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ or $T=\operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ ; or

2. (2) the inclusion $T\rightarrow \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ fixes a non-degenerate bilinear or unitary form on $\mathbb{F}_{\unicode[STIX]{x1D706}}^{n}$ , and if $n\geqslant 4$ , the image of $T$ is not contained in $\operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ .

To prove this, we first show that the degree of the field of definition must be small, according to results of Liebeck on the minimal dimension of faithful irreducible modular representations of simple groups of Lie type. This implies that the group of outer automorphisms is small, and then that $S$ must contain the regular unipotent element of $\tilde{H}$ for $\ell \gg _{n}1$ . Over an algebraically closed field, the irreducible representations of a semisimple algebraic group with central kernel whose images contain an element with a single Jordan block are classified by a result of Suprunenko. The absolute irreducibility of the action of $T$ allows the descent to finite groups of Lie type by a result of Seitz–Testerman.

Remark 6.10. The strategy to exclude alternating groups and groups of Lie type in cross-characteristic in the almost simple case of the characterization is quite standard (see e.g. [Reference Malle and Testerman32, Ch. 28]). The mentioned results of Liebeck are notably used in [Reference Kantor and Lubotzky15] to determine the probability that two random elements of $\operatorname{PSL}_{n}(\mathbb{F}_{\ell })$ are generators, in the case $\ell \leqslant 9$ .

6.4 Excluding members of classes ${\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{5}$ (Proposition 6.7)

We prove Proposition 6.7, assuming throughout that $\ell \gg _{n}1$ to apply Lemma 6.12.

6.4.2 Class ${\mathcal{C}}_{2}$

By assumption, there exists a permutation $\unicode[STIX]{x1D70E}\in \mathfrak{S}_{t}$ such that $u\boldsymbol{V}_{i}=\boldsymbol{V}_{\unicode[STIX]{x1D70E}(i)}$ for all $i$ . Note that there is at most one $\boldsymbol{V}_{i}$ which is $u$ -stable, since the $\boldsymbol{V}_{i}$ are disjoint with equal dimension, and $u$ has exactly only invariant subspace of each dimension $0\leqslant d\leqslant n$ . In particular, $\unicode[STIX]{x1D70E}$ has at most one fixed point. Write $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{1}\ldots \unicode[STIX]{x1D70E}_{k}$ where $\unicode[STIX]{x1D70E}_{1},\ldots ,\unicode[STIX]{x1D70E}_{k}$ are disjoint cycles, with $\unicode[STIX]{x1D70E}_{j}$ of length $2\leqslant s_{j}\leqslant n$ . Since $u$ has order $\ell$ , we have

$$\begin{eqnarray}\boldsymbol{V}_{i}=u^{\ell }\boldsymbol{V}_{i}=\boldsymbol{V}_{\unicode[STIX]{x1D70E}^{\ell }(i)},\end{eqnarray}$$

so $\unicode[STIX]{x1D70E}^{\ell }=\operatorname{id}$ and either:

1. – $\unicode[STIX]{x1D70E}=\operatorname{id}$ , which implies that all $\boldsymbol{V}_{i}$ are $u$ -stable, a contradiction;

2. – $\operatorname{ord}(\unicode[STIX]{x1D70E})=\operatorname{lcm}(s_{1},\ldots ,s_{k})=\ell$ . Hence $s_{j}=\ell$ for all $j$ , thus $k\ell =$ $n-|\operatorname{fix}(\unicode[STIX]{x1D70E})|$ , i.e. $\ell \mid n$ or $\ell \mid n-1$ . This can be excluded if $\ell >n$ .

6.4.4 Class ${\mathcal{C}}_{4}$

Consider the morphism

$$\begin{eqnarray}N_{\operatorname{PGL}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})}(\unicode[STIX]{x1D70B}(\boldsymbol{L}))\cap \unicode[STIX]{x1D70B}(\boldsymbol{G})\rightarrow \left\{\begin{array}{@{}l@{}}\operatorname{Out}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{1})))\times \operatorname{Out}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{2}))):t=2,\quad \\ \operatorname{Out}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{1})))\wr \mathfrak{S}_{t}:t>2,\quad \end{array}\right.\end{eqnarray}$$

with kernel $\unicode[STIX]{x1D70B}(\boldsymbol{L})\cap \unicode[STIX]{x1D70B}(\boldsymbol{G})$ . If $\unicode[STIX]{x1D70B}(u)\not \in \unicode[STIX]{x1D70B}(\boldsymbol{L})$ , then the order of the image of $\unicode[STIX]{x1D70B}(u)$ is

$$\begin{eqnarray}\ell \mid |\operatorname{Out}(\unicode[STIX]{x1D70B}(\boldsymbol{G}(\boldsymbol{V}_{i})))|^{t}t!=2^{t}t!\end{eqnarray}$$

for some $i\in \{1,2\}$ , by Propositions 6.4 and 6.6. Thus $\ell \ll _{n}1$ since $t\leqslant n$ . On the other hand, elements in $\unicode[STIX]{x1D70B}(\boldsymbol{L})$ have at least two Jordan blocksFootnote ⁵ , which also rules out the case $\unicode[STIX]{x1D70B}(u)\in \unicode[STIX]{x1D70B}(\boldsymbol{L})$ . Indeed, if $g_{1}\in \boldsymbol{G}(\boldsymbol{V}_{1})$ and $g_{2}\in \boldsymbol{G}(\boldsymbol{V}_{2})$ are two Jordan blocks, then $g_{1}\otimes g_{2}$ fixes the linearly independent vectors $v_{1,1}\otimes v_{2,1}$ and $v_{1,2}\otimes v_{2,1}-v_{1,1}\otimes v_{2,2}$ , where $v_{i,1},v_{i,2}$ are the first two elements of the standard basis of $\boldsymbol{V}_{i}$ ( $i=1,2$ ).

6.4.5 Class ${\mathcal{C}}_{5}$

Consider the morphism

$$\begin{eqnarray}H\rightarrow \unicode[STIX]{x1D70B}(H)\rightarrow \unicode[STIX]{x1D70B}(H)/(\mathbb{Z}/r^{m})\cong \operatorname{Sp}_{2m}(\mathbb{F}_{r}).\end{eqnarray}$$

Since $u$ has order $\ell \neq r$ , the image of $u$ in $\operatorname{Sp}_{2m}(\mathbb{F}_{r})$ still has order $\ell$ , hence

$$\begin{eqnarray}\ell \mid |\operatorname{Sp}_{2m}(\mathbb{F}_{r})|=r^{m^{2}}(r^{2}-1)(r^{4}-1)\cdots (r^{2m}-1)\leqslant r^{m(2m+1)},\end{eqnarray}$$

which implies that $\ell \ll _{n}1$ because $n=r^{m}$ .◻

6.5 Excluding almost simple groups (Propositions 6.8 and 6.9)

Let us now assume there exists a non-abelian simple group $S$ such that

$$\begin{eqnarray}S=\unicode[STIX]{x1D70B}(T)\unlhd \unicode[STIX]{x1D70B}(H)=\tilde{H}\leqslant \operatorname{Aut}(S),\end{eqnarray}$$

with $T\unlhd H\leqslant \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ and $T$ acting irreducibly on $\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}}^{n}$ .

6.5.2 Groups of Lie type in characteristic $\ell$

We finally prove Proposition 6.9. Let us assume that $S$ is a group of Lie type of rank $l$ over $\mathbb{F}_{r}$ , with $r=\ell ^{a}$ . We continue to assume that $\ell \gg _{n}1$ so that the regular unipotent element $u$ has order  $\ell$ .

The first difficulty to overcome is that we do not know a priori whether $S$ itself contains a regular unipotent element. However, we can show the following.

Proposition 6.14. If $\ell \gg _{n}1$ , then $\unicode[STIX]{x1D70B}(u)\in S$ and $T$ also contains an element with a single Jordan block.

We start by proving this in the following paragraphs. Recall that we have an exact sequence

$$\begin{eqnarray}1\rightarrow S\cong \operatorname{Inn}(S)\rightarrow \operatorname{Aut}(S)\rightarrow \operatorname{Out}(S)\rightarrow 1,\end{eqnarray}$$

an inclusion $S\leqslant \tilde{H}\leqslant \operatorname{Aut}(S)$ , and $\unicode[STIX]{x1D70B}(u)\in \tilde{H}$ of order  $\ell$ . If $\unicode[STIX]{x1D70B}(u)\not \in S$ , then its image in $\operatorname{Out}(S)$ has order $\ell$ and so $\ell$ divides $|\operatorname{Out}(S)|$ . Thus, it suffices to show that $|\operatorname{Out}(S)|\ll _{n}1$ to rule out this possibility.

Lemma 6.15. We have

$$\begin{eqnarray}|\operatorname{Out}(S)|=Na\end{eqnarray}$$

with $N\in \{1,2,6,8,12\}$ , unless:

1. – $S=A_{l}(r)$ with $l\geqslant 3$ odd, where we have $|\operatorname{Out}(S)|=2a(l+1,r-1)$ ; or

2. – $S=\text{}^{2}A_{l}(r)$ with $l\geqslant 3$ odd, where we have $|\operatorname{Out}(S)|=2a(l+1,r+1)$ .

Proof. See Propositions 6.5, 6.6 and Table 2: there are $a$ field automorphisms, one, two or three graph automorphisms, and less than four diagonal automorphisms, except for $A_{l}$ and $\text{}^{2}A_{l}$ which have respectively $(l+1,r-1)$ and $(l+1,r+1)$ diagonal automorphisms.◻

Letting $m(S)$ be the minimal dimension of a faithful irreducible projective representation of $S$ over an algebraically closed field of characteristic  $\ell$ , the following result lets us bound the rank of $S$ and the degree of its defining field.

Lemma 6.16. We have

$$\begin{eqnarray}l\leqslant m(S)\leqslant n^{([\mathbb{F}_{\unicode[STIX]{x1D706}}:\mathbb{F}_{\ell }],a)/a},\end{eqnarray}$$

whence $l,a\ll _{n}1$ . In particular, for $n$ fixed, there is only a finite number of possibilities for  $S$ .

Proof. The bounds follow from [Reference Liebeck30, (2.1)–(2.2)] and the fact that $S\leqslant \tilde{H}\leqslant \operatorname{PSL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ . Since $m(S)\geqslant 2$ (see Table 1), we have $a\leqslant \log n/([\mathbb{F}_{\unicode[STIX]{x1D706}}:\mathbb{F}_{\ell }]\log 2)\ll _{n}1$ , so that $l,a\ll _{n}1$ .◻

Table 1

Minimal dimension of a faithful irreducible projective representation of a simple group of Lie type over an algebraically closed field in the same characteristic $p>3$ , according to [Reference Liebeck30, Table 2] or [Reference Kantor and Lubotzky15, Table 1].

By Lemmas 6.15 and 6.16,

$$\begin{eqnarray}|\operatorname{Out}(S)|\leqslant 12a(l+1)\ll _{n}1,\end{eqnarray}$$

which concludes the proof of Proposition 6.14. ◻

Back to Proposition 6.9, the next difficulty is that we do not know whether the action of $T$ on $\mathbb{F}_{\unicode[STIX]{x1D706}}^{n}$ as a group of Lie type is the action induced by the inclusion $T\leqslant H\leqslant \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ . However, thanks to Proposition 6.14 and the irreducibility of the action of $T$ on $\boldsymbol{V}$ , the representation can be determined by the following.

Theorem 6.17. Let $T$ be a finite group of Lie type in characteristic $\ell \geqslant 5$ , of simply connected type, with a faithful absolutely irreducible representation $\unicode[STIX]{x1D711}:T\rightarrow \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ . Assume that $\unicode[STIX]{x1D711}(T)$ contains an element with a single Jordan block. Then either (up to conjugacy):

1. (1) $T=\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ , $\operatorname{Spin}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ for $n$ odd or $\operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ for $n$ even, with the standard embedding in $\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ ;

2. (2) $T=\operatorname{SU}_{n}(\mathbb{F}^{\prime })$ with the standard embedding in $\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ , for $\mathbb{F}^{\prime }\leqslant \mathbb{F}_{\unicode[STIX]{x1D706}}$ a subfield such that $|\mathbb{F}^{\prime }|=|\mathbb{F}_{\unicode[STIX]{x1D706}}|^{1/2}$ ;

3. (3) $T=G_{2}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ and $n=7$ , with $\unicode[STIX]{x1D711}$ the unique seven-dimensional irreducible representation.

Remark 6.18. This is to be compared with the fact that the only non-trivial irreducible complex representations of $\operatorname{SL}_{n}(\mathbb{C})$ of dimension ${\leqslant}n$ are the standard representation and its dual.

Theorem 6.17 is a version of [Reference Suprunenko39, Theorem (1.9)] for finite groups of Lie typeFootnote ⁶ . To prove this variant, we use the lifting theorem of Seitz and Testerman.

Theorem 6.19 [Reference Seitz and Testerman37, Theorem 1, case $G=\operatorname{SL}_{n}$ ]Footnote ⁷ .

Let $\boldsymbol{H}$ be a simple algebraic group over $\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}}$ , with a Steinberg endomorphism $F:\boldsymbol{H}\rightarrow \boldsymbol{H}$ , and $X=[\boldsymbol{H}^{F},\boldsymbol{H}^{F}]$ perfect. If $\unicode[STIX]{x1D711}:X\rightarrow \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ is a morphism such that $\unicode[STIX]{x1D711}(X)$ lies in no proper $F$ -stable parabolic subgroup of $\operatorname{SL}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})$ , then $\unicode[STIX]{x1D711}$ can be extended to a morphism of algebraic groups $\unicode[STIX]{x1D753}:\boldsymbol{H}\rightarrow \operatorname{SL}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})$ with $\unicode[STIX]{x1D753}\mid _{X}=\unicode[STIX]{x1D711}$ .

Proof of Theorem 6.17.

By hypothesis, $T=\boldsymbol{T}^{F}$ for $\boldsymbol{T}$ a simple algebraic group over $\overline{\mathbb{F}}_{\ell }$ and $F:\boldsymbol{T}\rightarrow \boldsymbol{T}$ a Steinberg endomorphism. We consider the absolutely irreducible representation $\unicode[STIX]{x1D711}:T\rightarrow \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ . By irreducibility, the image of $T$ is not contained in a proper parabolic subgroup of $\operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ . Theorem 6.19 thus shows the existence of a morphism $\unicode[STIX]{x1D753}:\boldsymbol{T}\rightarrow \operatorname{SL}_{n}(\overline{\mathbb{F}}_{\ell })$ extending $\unicode[STIX]{x1D711}$ and which is still an irreducible representation. We can then apply [Reference Suprunenko39, Theorem (1.9)], and the classification of Steinberg endomorphisms [Reference Malle and Testerman32, 22.1–22.2] gives the result. ◻

It remains to recall that:

1. – $\operatorname{Spin}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ , $G_{2}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ and $\operatorname{SU}_{n}(\mathbb{F}^{\prime })$ fix a non-degenerate bilinear or unitary form on $V$ (see the proof of Theorem 1.6 p. 13; note that here we use that $T\rightarrow \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ is the natural representation);

2. – $\operatorname{SU}_{n}(\mathbb{F}^{\prime })$ is not contained in $\operatorname{Sp}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ if $n\geqslant 4$ (with respect to the standard inclusion $\operatorname{SU}_{n}(\mathbb{F}^{\prime })\leqslant \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ ): take an orthogonal matrix corresponding to an even permutation exchanging the indices of two distinct entries of the matrix of the symplectic form.

This concludes the proof of Proposition 6.9. ◻

Table 2

Outer automorphism groups of finite simple groups of Lie type over $\mathbb{F}_{r}$ , with $r=q^{a}$ odd, $q>3$ .

6.6 Further classification theorems

Let $K$ be an algebraically closed field of characteristic $\ell \geqslant 0$ and let $G$ be a classical group over $K$ (e.g. $G=\operatorname{SL}_{n}(K)$ or $G=\operatorname{Sp}_{n}(K)$ ) with associated vector space  $V$ .

Saxl and Seitz [Reference Saxl and Seitz36] classified maximal closed subgroups $H\leqslant G$ of positive dimension acting irreducibly on $V$ and containing a regular unipotent element of  $G$ , using the generalization of Aschbacher’s result to algebraically closed fields by Liebeck and Seitz [Reference Liebeck and Seitz31]. In particular, this generalizes [Reference Suprunenko39, Theorem (1.9)]. Note that Proposition 6.7 is analogous to [Reference Saxl and Seitz36, Proposition 2.1].

We remark that our classification Theorem 3.2 over $\mathbb{F}_{\unicode[STIX]{x1D706}}$ cannot be simply deduced from [Reference Saxl and Seitz36, Theorem B] by descent. Indeed, taking a maximal (proper) subgroup $H\leqslant \operatorname{SL}_{n}(\mathbb{F}_{\unicode[STIX]{x1D706}})$ containing a regular unipotent element and acting irreducibly on $\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}}^{n}$ , we do not know whether there exists a positive-dimensional closed subgroup $H^{\prime }\leqslant \operatorname{SL}_{n}(\overline{\mathbb{F}}_{\unicode[STIX]{x1D706}})$ containing  $H$ . Showing this would actually be more or less equivalent to the proof of Theorem 3.2: if $H$ is allowed to be 0-dimensional in [Reference Saxl and Seitz36, Theorem B], one has to consider almost simple subgroups and not only simple ones, which is the additional difficulty we need to deal with in the proof of Theorem 3.2.

More generally, Testerman and Zalesski [Reference Testerman and Zalesski40, Theorem 1.2] show that connected reductive linear algebraic groups containing a unipotent element with a single Jordan block are irreducible. Combined with [Reference Saxl and Seitz36], this gives a classification of semisimple subgroups $H$ of simple algebraic groups $G$ containing a regular unipotent element of $G$ [Reference Testerman and Zalesski40, Theorem 1.4].