1 Introduction

Let k be a field, which we assume is imperfect of characteristic p. In this case, there exist pseudo-reductive k-groups which are not reductive. Recall that a smooth, connected, affine algebraic k-group G is pseudo-reductive if the largest smooth connected unipotent normal k-subgroup \({\mathscr {R}}_{u,k}(G)\) is trivial.

We are interested in the structure of the geometric unipotent radical \(R:={\mathscr {R}}_u(G_{{\bar{k}}})={\mathscr {R}}_{u,{\bar{k}}}(G_{{\bar{k}}})\) of G. Since R is a p-group, it makes sense to study its exponent: the minimal integer s such that the \(p^s\)-power map on the geometric unipotent radical factors through the trivial group. We denote the exponent of R by e(R).

M. Bate, B. Martin, G. Röhrle, and the third author previously gave some bounds for e(R) in [1]. For example, [1, Lem. 4.1] implies that if \(k'/k\) is a simple purely inseparable field extension with \((k')^{p^e}\subseteq k\), then \(e(R)\le e\).

The monographs [4] and [5] contain a classification of pseudo-reductive groups. They are all related in some way or other to the Weil restrictions of reductive groups, which are themselves pseudo-reductive. We focus on groups which are Weil restrictions, since general pseudo-reductive groups contain central pseudo-reductive subgroups whose classification is thought to be intractable. Thus we assume unless stated otherwise that \(G=\mathrm {R}_{k'/k}(G')\) for some reductive \(k'\)-group \(G'\) where \(k'\) is a finite non-zero reduced k-algebra. Since we are interested in e(R), by the remarks in [1, §4], we may as well consider the case that k is separably closed and \(k'\) is a purely inseparable field extension of k. In this case, \(G'\) is a split reductive group and as such is classified by its root datum.

Recall that the exponent of the extension \(k'/k\) is the smallest integer e such that \((k')^{p^e}\subseteq k\). To describe our results, we need more sensitive data about the structure of \(k'/k\). Since \(k'/k\) is purely inseparable, we can appeal to the results in [10] and [11], which guarantee a normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) of \(k'\) over k with certain properties—see Definition 2.6 below for more details. In particular, these elements come with certain exponents which are invariants of \(k'/k\). These are the integers \(e_1\ge \cdots \ge e_l\) such that \(e_i\) is minimal subject to

$$\begin{aligned} \alpha _i^{p^{e_i}}\in k(\alpha _1^{p^{e_i}},\ldots ,\alpha _{i-1}^{p^{e_i}}). \end{aligned}$$

It is immediate that \(e_1=e\) and it follows from the tower law that \([k':k]=\prod p^{e_i}\). Let \(r\in {\mathbb {N}}\). We define integers

$$\begin{aligned} {{\,\mathrm{m}\,}}(k'/k)&:=\sum _{i=1}^l(p^{e_i}-1)+1=\sum _{i=1}^{l}p^{e_i}-l+1,\\ {{\,\mathrm{em}\,}}(k'/k)&:=\lceil \log _p({{\,\mathrm{m}\,}}(k'/k))\rceil ,\\ {{\,\mathrm{mr}\,}}(k'/k,r)&:=\sum _{i=1}^r p^{e_i}\text {,\quad where we take\,} e_i\,=\,0~\text {for}\, i>l,\\ {{\,\mathrm{emr}\,}}(k'/k,r)&:=\lceil \log _p({{\,\mathrm{mr}\,}}(k'/k,r))\rceil ,\\ \text {and finally } {{\,\mathrm{E}\,}}(k'/k,r)&:=\min \{{{\,\mathrm{em}\,}}(k'/k),{{\,\mathrm{emr}\,}}(k'/k,r)\}. \end{aligned}$$

In other words, \({{\,\mathrm{em}\,}}(k'/k)\) is the minimal s such that \(p^s\ge {{\,\mathrm{m}\,}}(k'/k)\), and similarly for \({{\,\mathrm{emr}\,}}\).

In answering a question of [1] on the exponent of the intersection of R with a pseudo-Borel subgroup, we discovered that these numbers give lower bounds on e(R) and for large enough rank determine the exponent of R exactly. We first state our results in the case that \(G'\) is a general linear group.

Theorem 1.1

Let \(k'/k\) be a finite purely inseparable field extension of a separably closed field k of characteristic p. Suppose \(k'/k\) has exponent e and a normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) with sequence of exponents \(e_1,\ldots ,e_l\). For \(r\in {\mathbb {N}}\), let \(G':={{\,\mathrm{GL}\,}}_r\), \(G:=\mathrm {R}_{k'/k}(G')\), and \(R:={\mathscr {R}}_u(G_{{\bar{k}}})\). Furthermore let \(B'\) denote a Borel subgroup of \(G'\) and \(B:=\mathrm {R}_{k'/k}(B')\subseteq G\) the corresponding pseudo-Borel subgroup of G. The following hold:

  1. (i)

    if s satisfies \(p^s\ge p^e r\), then e(R) is at most s; i.e. \(e(R)\le \lceil e+\log _p(r)\rceil \);

  2. (ii)

    we have \(e(R\cap B_{{\bar{k}}})={{\,\mathrm{E}\,}}(k'/k,r)\);

  3. (iii)

    in particular, \(e(R)\ge {{\,\mathrm{E}\,}}(k'/k,r)\);

  4. (iv)

    if \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\), then \(e(R)={{\,\mathrm{E}\,}}(k'/k,r)={{\,\mathrm{em}\,}}(k'/k)\).

Remark 1.2

  1. (i)

    The bounds in Theorem 1.1 are both arithmetic and Lie-theoretic in character: the integers \({{\,\mathrm{m}\,}}(k'/k)\) and \({{\,\mathrm{em}\,}}(k'/k)\) depend only on the arithmetic structure of \(k'/k\) but \({{\,\mathrm{mr}\,}}(k'/k,r)\), \({{\,\mathrm{emr}\,}}(k'/k,r)\), and \({{\,\mathrm{E}\,}}(k'/k,r)\) depend on both \(k'/k\) and the rank r of \(G'\).

  2. (ii)

    Part (i) of Theorem 1.1 improves [1, Lem. 4.4], which required \(p^s\ge r^2(p^e-1)\). As a corollary of (i) and (iv), if \(G=\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r)\) for \(p\ge r\ge 2\), then the exponent of \({\mathscr {R}}_u(G_{\bar{k}})\) is e if \(k'/k\) is a simple extension and is \(e+1\) otherwise. If \(r=1\), then \(e(R)=e\) as \(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_1)\) is sent into the canonical \({{\,\mathrm{GL}\,}}_1\) k-subgroup by the \(p^e\)-power map, since it also sends \(k'\) into k.

  3. (iii)

    In [1, Rem. 4.5(iv)], it was asked if the exponent e(R) always coincides with the exponent \(e(R\cap B_{{\bar{k}}})\). Clearly Theorem 1.1(ii)-(iv) answers this question in the affirmative when \(G'={{\,\mathrm{GL}\,}}_r\) of large enough rank. We give an example to show that in general the answer is ‘no’ (see Proposition 4.3), coming from \({{\,\mathrm{SL}\,}}_2\) in characteristic 2. In fact, in Section 4, we give a complete description of e(R) when \(G'\) is \({{\,\mathrm{SL}\,}}_2\), \({{\,\mathrm{PGL}\,}}_2\), or \({{\,\mathrm{GL}\,}}_2\).

Together with the rank 1 results in Section 4, the following (proved in Section 5) confirms the large rank behaviour in all simple groups, with mild conditions on the characteristic.

Corollary 1.3

Let \(k'/k\) be a finite, purely inseparable field extension of a separably closed field k of characteristic p. Suppose \(k'/k\) has exponent e and a normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) with sequence of exponents \(e_1,\ldots ,e_l\). Let \(G'\) be a (split) simple algebraic k-group of rank \(r\ge 2\), \(G:=R_{k'/k}(G')\), and let \(R:={\mathscr {R}}_{u}(G_{{\bar{k}}})\). If \(G'\) is type BCDF, then assume \(p\ne 2\) or \(G'\) is \({{\,\mathrm{SO}\,}}_{n}\) or \({{\,\mathrm{Sp}\,}}_{2n}\), and if G is of type \(E_6\), then assume \(p\ne 3\). Then the exponent \(e(R)\ge {{\,\mathrm{E}\,}}(k'/k,r)\); if \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\), then in fact, \(e(R)={{\,\mathrm{E}\,}}(k'/k,r)={{\,\mathrm{em}\,}}(k'/k)\).

2 Preliminaries and notation

Let k be a field of characteristic p and G be a k-group, by which we mean an affine algebraic group scheme of finite type over k. The k-unipotent radical is the maximal smooth connected normal unipotent k-subgroup of G and we denote this by \({\mathscr {R}}_{u,k}(G)\). The geometric unipotent radical is the \(\bar{k}\)-unipotent radical of the base change \(G_{\bar{k}}\) of G to \({\bar{k}}\). This is denoted by \({\mathscr {R}}_{u}(G_{\bar{k}})\). We call a smooth connected k-group G reductive if \({\mathscr {R}}_{u}(G_{\bar{k}})=1\) and pseudo-reductive if \({\mathscr {R}}_{u,k}(G)=1\). From now on, G will always be pseudo-reductive. We recall the definition and key properties of Weil restriction from [4, A.5].

Definition 2.1

Let \(B\rightarrow B'\) be a finite flat map of Noetherian rings, and \(X'\) a quasi-projective \(B'\)-scheme. The Weil restriction is a finite type B-scheme satisfying the universal property

$$\begin{aligned} \mathrm {R}_{B'/B}(X')(A)=X'(B'\otimes _B A) \end{aligned}$$

for all B-algebras A.

The following proposition makes use of the natural map

$$\begin{aligned} q_{G'}:\mathrm {R}_{k'/k}(G')_{k'}\rightarrow G', \end{aligned}$$
(2.2)

which is induced by \(k'\otimes _k A \rightarrow A, a\otimes b \mapsto ab\) for any \(k'\)-algebra A.

Proposition 2.3

([4, Prop. A.5.11, Thm. 1.6.2], [1, §2], [1, Lem. 3.4]). Let \(k'/k\) be a finite and purely inseparable field extension, \(G'\) a non-trivial smooth connected \(k'\)-group, and \(G=\mathrm {R}_{k'/k}(G')\).

  1. (i)

    The kernel of \(q_{G'}\) is a smooth connected unipotent \(k'\)-subgroup and thus is contained in \({\mathscr {R}}_{u,k'}(G_{k'})\).

  2. (ii)

    If \(G'\) is reductive over \(k'\), then the kernel of \(q_{G'}\) has field of definition over k equal to \(k'\subseteq \bar{k}\). Thus the kernel \(\ker (q_{G'})\) coincides with \({\mathscr {R}}_{u,k'}(G_{k'})\), which is a \(k'\)-descent of \({\mathscr {R}}_u(G_{\bar{k}})\) (i.e. \({\mathscr {R}}_{u,k'}(G_{k'})_{{\bar{k}}}\cong {\mathscr {R}}_u(G_{\bar{k}})\)).

  3. (iii)

    If \(H'\) is a reductive subgroup of \(G'\), then the geometric unipotent radical of \(\mathrm {R}_{k'/k}(H')\) is a subgroup of the geometric unipotent radical of \(\mathrm {R}_{k'/k}(G')\).

  4. (iv)

    If \(f:G_1\rightarrow G_2\) is an étale isogeny of reductive \(k'\)-groups, then \(\mathrm {R}_{k'/k}(f)_{{\bar{k}}}\) induces an isomorphism of geometric unipotent radicals of the respective Weil restrictions.

We are interested in the following invariant of a unipotent k-group:

Definition 2.4

Let k be a field of characteristic p and U a unipotent k-group. The exponent \(\exp (U)\) of U is the minimal s such that the \(p^s\)-power map on U factors through the trivial group.

Clearly, this definition is insensitive to base change. As in [1, §4], we may therefore assume k is separably closed for the remainder of the article.

When calculating with matrices, we make use of the fact that if \(U(k)\subset U\) is dense (e.g. if k is perfect [6, Thm. 34.4]), then a map factors through the trivial group if and only if it maps to the identity on U(k). Since \(R={\mathscr {R}}_u(G_{{\bar{k}}})\) descends to \(R':={\mathscr {R}}_{u,k'}(G_{k'})\) whose \(k'\)-points are dense (by the fact that \(k'\) is separably closed), we have that e(R) is the smallest s such that \(p^s\) kills \(R'(k')\).

Following [9], let \({\mathfrak {B}}:=k'\otimes _k k'\). Then \({\mathfrak {B}}\) is a local ring with maximal ideal

$$\begin{aligned} {\mathfrak {m}}=\langle 1\otimes x - x \otimes 1 \mid x\in k' \rangle =\ker (\phi ), \end{aligned}$$

where \(\phi :k'\otimes _k k' \rightarrow k', a\otimes b \mapsto ab\). Moreover, \(\mathrm {R}_{k'/k}(G')_{k'}\) can be identified with \(\mathrm {R}_{{\mathfrak {B}}/k'}(G'_{\mathfrak {B}})\) and the \(k'\)-points of the \(k'\)-unipotent radical of the former is the kernel \(G'({\mathfrak {m}})\) of the map \(G'({\mathfrak {B}})\rightarrow G'(k)\) induced by \(\phi \). Suppose now \(G'={{\,\mathrm{GL}\,}}_r\) and \(G=\mathrm {R}_{k'/k}(G')\). As the entries of any \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\) are nilpotent elements of \({\mathfrak {B}}\), it follows that \(I_r+M\) has invertible determinant, hence is an element of \(G'({\mathfrak {m}})\). Therefore

$$\begin{aligned} {\mathscr {R}}_{u,k}(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r)_{k'})(k')=\{I_r+M\mid M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\}. \end{aligned}$$

As \(I_r\) and M commute, we have \(A^{p^s}=I_r+M^{p^s}=I_r\) is equivalent to \(M^{p^s}=0\). Hence:

Lemma 2.5

The exponent of the geometric unipotent radical of \(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r)\) is the smallest s such that \(M^{p^s}=0\) for all \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\).

It was observed in [1, Lem. 4.1] that as \({\mathfrak {m}}^n=0\) for some \(n\in {\mathbb {N}}\), one knows that s in the lemma above is bounded above by the minimal t such that \(p^t\ge n\): the entries of \(M^{p^t}\) are homogeneous polynomials of degree \(p^t\) in elements of \({\mathfrak {m}}\) and so must vanish.

Definition 2.6

Let \(k'/k''/k\) be a tower of finite purely inseparable field extensions.

  1. (i)

    If \(x\in k'\), then the minimal \(s\in {\mathbb {N}}\) such \(x^{p^s}\in k''\) is called the exponent of x with respect to \(k''\) and denoted by \(e_{k''}(x)\).

  2. (ii)

    We say that \(x\in k'\) is normal in \(k'/k''\) if \(e_{k''}(x)\ge e_{k''}(y)\) for all \(y\in k'\).

  3. (iii)

    A sequence \(\alpha _1,\ldots , \alpha _l\) in \(k'\) is called a normal sequence if for every \(1\le i \le l\) and \(k_i=k[\alpha _1,\ldots , \alpha _i]\), we have that \(\alpha _i\) is normal in \(k'/k_{i-1}\), and \(\alpha _i \notin k_{i-1}\).

  4. (iv)

    A normal sequence \(\alpha _1,\ldots , \alpha _l\) is called a normal generating sequence of \(k'/k\) if \(k[\alpha _1,\ldots , \alpha _l]=k'\).

  5. (v)

    For a normal generating sequence \(\alpha _1,\ldots , \alpha _l\) of \(k'/k\), we set \(e_i=e_{k_{i-1}}(\alpha _i)\). We call \(e_1,\ldots ,e_l\) a sequence of exponents for \(k'/k\)

Note that if \(e_1,\ldots ,e_l\) is a sequence of exponents for \(k'/k\), then we have \(e_1\ge e_2 \ge \cdots \ge e_l\) due to [11, Prop. 5]. Crucially, by loc. cit., we also have

$$\begin{aligned} \alpha _i^{p^{e_i}}\in k(\alpha _{1}^{p^{e_i}},\ldots ,\alpha _{i-1}^{p^{e_i}}) \text { for all }1\le i\le l.\end{aligned}$$
(2.7)

In particular, as the \(e_i\) are decreasing,

$$\begin{aligned} (k')^{p^{e_i}}\subseteq k(\alpha _1^{p^{e_i}},\ldots ,\alpha _{i-1}^{p^{e_i}}). \end{aligned}$$
(2.8)

Combining this with the main result of [10], we get that the sequence of exponents is an invariant of the field extension \(k'/k\). More precisely:

Lemma 2.9

Let \(k'/k\) be a finite purely inseparable field extension. Suppose that \(\alpha _{1},\ldots ,\alpha _l\) and \(\beta _1,\ldots ,\beta _{l'}\) are normal generating sequences of \(k'/k\) with sequences of exponents \(e_1,\ldots , e_l\) and \(e'_1,\ldots , e'_{l'}\). Then \(l=l'\) and \(e_i=e'_i\) for \(1\le i< l\).

By the lemma, the numbers \({{\,\mathrm{m}\,}}(k'/k)\), \({{\,\mathrm{mr}\,}}(k'/k,r)\) defined in the introduction are both invariants of \(k'/k\) and \((k'/k,r)\) respectively. To relate these with \({\mathfrak {m}}\), we first prove:

Lemma 2.10

For a normal generating sequence \(\alpha _1,\ldots , \alpha _l\) of \(k'/k\), define \(a_i:=1\otimes \alpha _i-\alpha _i\otimes 1\in {\mathfrak {m}}\). Then:

  1. (i)

    the ideal \({\mathfrak {m}}\) is generated by the \(a_i\) over \(k'\);

  2. (ii)

    the \(p^{e_i}\)-power map takes \({\mathfrak {m}}\) into the \(k'\)-subspace \(k'[a_1^{p^{e_i}},\ldots ,a_{i-1}^{p^{e_i}}]\cap {\mathfrak {m}}\).

Proof

(i) Since \({\mathfrak {m}}\) is a codimension 1 (left) \(k'\)-subspace of \({\mathfrak {B}}\), it is a codimension \([k':k]\) k-subspace of \({\mathfrak {B}}\). Define \(J=\{ (b_1,b_2,\ldots ,b_l)\in {\mathbb {Z}}^l \mid 0\le b_i\le p^{e_i}-1 \}\). For \(b=(b_1,...,b_l)\in J\), we set \(x^b=\prod _{i=1}^{l}\alpha _{i}^{b_i}\). Then \(\{x^b\}_{b\in J}\) is a k-basis of \(k'\) and \(\{1\otimes x^b\}\) is a \(k'\)-basis of \({\mathfrak {B}}\). For \(b\ne (0,\ldots , 0)\), the elements \(a_b:=1\otimes x^b-x^b\otimes 1=1\otimes x^b-x^b (1\otimes 1)\) are therefore linearly independent, generating a codimension 1 subspace of \({\mathfrak {B}}\), which is contained in the codimension 1 kernel \({\mathfrak {m}}\) of the multiplication map \(x\otimes y\mapsto xy\); hence they form a \(k'\)-basis of \({\mathfrak {m}}\). We wish to show \(a_b\in {\mathfrak {m}}':={\mathfrak {m}}\cap k'[a_1,\ldots ,a_l]\) by induction on \(\sum b_i\), the case \(\sum b_i=1\) being trivial. So let \(\sum b_i>1\). Without loss of generality, suppose \(b_1>0\) and let \(b'=b-(1,0,\ldots ,0)\). Then inductively \(a_{b'}=1\otimes x^{b'}-x^{b'}\otimes 1\in {\mathfrak {m}}'\). Now one checks directly that \(a_b=x^{b'}a_1+\alpha _1x^{b'}-a_{b'}\cdot a_1\in {\mathfrak {m}}'\) as required. (And it follows that \({\mathfrak {m}}={\mathfrak {m}}'\).)

(ii) Define \({\mathfrak {m}}':=k'[a_1^{p^{e_i}},\ldots ,a_{i-1}^{p^{e_i}}]\cap {\mathfrak {m}}\). From (i), we can express any element of \({\mathfrak {m}}\) as a polynomial in \(a_1,\ldots , a_l\) over \(k'\). Since \(k'\) has characteristic p, the \(p^{e_i}\)-power map distributes over addition and so it suffices to check that \((a_1^{r_1}\cdots a_l^{r_l})^{p^{e_i}}\in {\mathfrak {m}}'\) for \(r_1,\ldots , r_l\in {\mathbb {Z}}_{\ge 0}\). For this, it suffices to check that \(a_j^{p^{e_i}}\in {\mathfrak {m}}'\) for \(j\ge i\). Now \(a_j^{p^{e_i}}=(1\otimes \alpha _j -\alpha _j\otimes 1)^{p^{e_i}}=1\otimes \alpha _j^{p^{e_i}}-\alpha _j^{p^{e_i}}\otimes 1\) and by (2.8), we can write \(\alpha _j^{p^{e_i}}\) as a k-polynomial P in \(k[\alpha _1^{p^{e_i}},\ldots ,\alpha _{i-1}^{p^{e_i}}]\) with no constant term. But applying part (i) with \(l=i-1\), \(a_j^{p^{e_i}}\) in place of \(a_j\) for \(1\le j\le i-1\), and \(k[a_1^{p^{e_i}},\dots , a_{i-1}^{p^{e_i}}]\) in place of \(k'\), we deduce that \(1\otimes P-P\otimes 1\) is indeed a polynomial in the \(a_j^{p^{e_i}}\) with \(1\le j\le i-1\) as required.\(\square \)

We may now identify the minimal integer n such that \({\mathfrak {m}}^n=0\) with the integer \({{\,\mathrm{m}\,}}(k'/k)\). In fact, we also prove something stronger:

Lemma 2.11

With notation as above, let \(d\le l\) and suppose \(m_1,\ldots ,m_d\in {\mathfrak {m}}\) and \(f_1,\ldots , f_d\in {\mathbb {Z}}_{\ge 0}\).

If \(\sum f_i\ge {{\,\mathrm{mr}\,}}(k'/k,d)-d+1=\sum p^{e^i}-d+1\), then \(m_1^{f_1}\cdots m_d^{f_d}=0\).

In particular, the minimum n such that \({\mathfrak {m}}^{n}=0\) is \({{\,\mathrm{m}\,}}(k'/k)\).

Proof

Assume for a contradiction that \(m:=m_1^{f_1}\cdots m_d^{f_d}\ne 0\). We proceed by induction on d, observing that the product of any factors of m is also non-zero. It does no harm to assume that \(f_1\ge \cdots \ge f_d\). If \(d=1\), then \({{\,\mathrm{mr}\,}}(k'/k,1)=p^{e_1}=p^e\) and indeed \(m=0\), a contradiction. Otherwise, let c be maximal such that \(f_c\ge p^{e_c}\). This implies \(\sum _{i=1}^c f_i\ge {{\,\mathrm{mr}\,}}(k'/k,c)-c+1\) so if \(c<d\), then \(m_1^{f_1}\cdots m_c^{f_c}\) is zero by induction, a contradiction. Therefore \(c=d\). Let \((q_i,r_i)\) be the quotient and remainder when \(p^{e_d}\) is divided into \(f_i\). As \(f_i\ge f_d\ge p^{e_d}\), we have each \(q_i\ge 1\). Now

$$\begin{aligned} f_1+\cdots +f_d=p^{e_d}(q_1+\cdots +q_d)+(r_1+\cdots +r_d)\ge p^{e_1}+\cdots +p^{e_d}-d+1 \end{aligned}$$

and so

$$\begin{aligned} p^{e_d}(q_1+\cdots +q_d)&\ge p^{e_d}(q_1+\cdots +q_{d-1}+1)\\&\ge p^{e_1}+\cdots +p^{e_d}-d+1-d(p^{e_d}-1)\\&=p^{e_d}(p^{e_1-e_d}+\cdots +p^{e_{d-1}-e_d}-d+1)+1. \end{aligned}$$

Since the second expression above is a multiple of \(p^{e_d}\), we have

$$\begin{aligned}q_1+\cdots +q_{d}\ge p^{e_1-e_d}+\cdots +p^{e_{d-1}-e_d}-(d-2). \end{aligned}$$

Now as \(q_ip^{e_d}\le f_i\) for each i, we have \(m'=m_1^{q_1p^{e_d}}\cdots m_d^{q_dp^{e_d}}\ne 0\). By Lemma 2.10(ii), we may write each \(m_i^{p^{e_d}}\) as a polynomial in \(k'[a_1^{p^{e_d}},\ldots ,a_{d-1}^{p^{e_d}}]\). After expanding \(m'\), any constituent monomial will be of the form \(a_1^{\nu _1}\cdots a_{d-1}^{\nu _{d-1}}\) where \(\sum _{i=1}^{d-1}\nu _i=p^{e_d}\sum _{i=1}^d q_i\ge p^{e_1}+\cdots +p^{e_{d-1}}-p^{e_d}(d-2)\) and each \(\nu _i\) is divisible by \(p^{e_d}\). Furthermore, as \(a_{d-1}^{p^{e_{d-1}}}\) can be re-expressed in terms of the \(a_i\) with \(i<d-1\), we may assume \(\nu _{d-1}\le p^{e_{d-1}}-p^{e_d}\). It follows that

$$\begin{aligned} \sum _{i=1}^{d-2}\nu _i\ge p^{e_1}+\cdots +p^{e_{d-2}}-p^{e_d}(d-3). \end{aligned}$$

Continuing inductively, \(\nu _1\ge p^{e_1}-p^{e_d}+1\). Thus \(\nu _1\ge p^{e_1}\) and hence all monomials in \(m'\) are zero, a contradiction.

We now tackle the second statement. To see that \(n\ge {{\,\mathrm{m}\,}}(k'/k)\), observe that \(a:=\prod _{i=1}^{l}a_i^{p^{e_i}-1}\) is a product of \({{\,\mathrm{m}\,}}(k'/k)-1\) elements from \(\{a_1,\ldots , a_l\}\). Expanding this product in terms of the k-basis \(\{x^b\otimes x^c\mid b,c\in J\}\) of \({\mathfrak {B}}\), the coefficient of \(1\otimes x^b\) where \(b=(p^{e_1}-1,\ldots ,p^{e_l}-1)\) is easily seen to be 1. It follows that \(a\ne 0\).

For the upper bound, take \(d=l\) in the first part of the lemma, noting that \({{\,\mathrm{m}\,}}(k'/k)={{\,\mathrm{mr}\,}}(k'/k,l)-l+1\). \(\square \)

3 Bounds in the case of the general linear group

In this section, \(G'={{\,\mathrm{GL}\,}}_r\) and \(G=\mathrm {R}_{k'/k}(G')\), where \(k'/k\) is purely inseparable of exponent e. In [1, Lem. 4.4], it was shown that the exponent e(R) of \(R:={\mathscr {R}}_u(G_{{\bar{k}}})\) is at most s, where s is chosen such that \(p^s\ge r^2(p^e-1)\). Using the Cayley–Hamilton theorem in its full generality (see [2, III.8.11]) yields an improvement.

Proof of Theorem 1.1(i)

Let \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\), as in Lemma 2.5. By the Cayley–Hamilton theorem, we have \(\chi _M(M)=0\), where \(\chi _M(\lambda )=\lambda ^r+\sum _{t=0}^{r-1}f_t \lambda ^t\) is the characteristic polynomial of M with degree r. Observe that the coefficient \(f_t\) is either zero or a homogeneous polynomial in the entries of \({\mathfrak {m}}\) with degree \(r-t\). Since the matrices \(M^{r-1},\ldots ,M,I_r\) all commute and \(k'\otimes _{k} k'\) has characteristic p, we get

$$\begin{aligned} M^{rp^e}=-(f_{r-1}^{p^e}M^{(r-1)p^e}+\cdots +f_1^{p^e}M^{p^e}+f_0^{p^e}I_r)=0, \end{aligned}$$

since \(f_t^{p^e}=0\) for all \(f_t\in {\mathfrak {m}}\). Choosing s such that \(p^s\ge p^er\), we get \(M^{p^s}=M^{p^er}M^{p^s-p^er}=0\). \(\square \)

Let \(B'\subseteq G'\) be the upper Borel subgroup, whose points are upper triangular matrices, and \(B=\mathrm {R}_{k'/k}(B')\subseteq G\) its Weil restriction—a pseudo-Borel subgroup in the terminology of [5, C.2]. We now specify certain elements \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\) with \(M\in B(k')=B'({\mathfrak {B}})\) to construct lower bounds on e(R) which depend on r and the exponents \(e_i\) of a normal generating sequence. Moreover, we calculate exactly the exponent of the intersection \(R\cap B_{{\bar{k}}}\).

Before we give the proof, we make some elementary remarks about powers of matrices. If A is a commutative ring and \(E_{i,j}\in {{\,\mathrm{Mat}\,}}(r,A)\) is the \(r\times r\) elementry matrix with \(e_{i,j}=1\) and zeros everywhere else, then we have \(E_{i,j}E_{l,s}=E_{i,s}\) if and only if \(j=l\) and 0 else. Write \(M=\sum a_{ij}E_{i,j}\) for a general element of \({{\,\mathrm{Mat}\,}}(r,A)\) . Then after expansion, any non-zero term in the power \(M^n\) is of the form \(a_w\cdot E_{i_1,i_{n+1}}\) where \(a_w=a_{i_1,i_2}\cdot a_{i_2,i_3}\cdots a_{i_l,i_{n+1}}\), and \(w\in (i_1,\ldots ,i_{n+1})\in \{1,\ldots ,r\}^{n+1}\). Furthermore, the (ij)-th entry of \(M^n\) is the sum over all \(a_w\) such that \(i_1=i\) and \(i_{n+1}=j\). For \(w\in \{1,\ldots ,r\}^{n+1}\), say n is the length of w. Now, if M is an upper triangular matrix, then \(a_w\) will be zero unless \(i_j\le i_{j+1}\) for each \(1\le j\le n+1\). Hence the only w yielding non-zero \(a_w\) are non-decreasing sequences of integers from \(\{1,\ldots ,r\}\), whose length is bounded above by \({{\,\mathrm{m}\,}}(k'/k)\) (Lemma 2.11).

Proof of Theorem 1.1

(ii)-(iv). We work inside \(B'({\mathfrak {m}})\). There are two cases. Suppose we have \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\). Then there is an integer \(q<r\) with q maximal such that \((p^{e_{q+1}}-1)+\cdots +(p^{e_l}-1)\ge q-1\). Define the matrix \(M\in B'({\mathfrak {m}})\) as follows.

  1. (i)

    Start with \(M=0\);

  2. (ii)

    fill the first q spaces on the diagonal with \(a_1,\ldots , a_q\) respectively;

  3. (iii)

    Fill up the first \(q-1\) elements of the superdiagonal with \(p^{e_l}-1\) entries of \(a_l\), then \(p^{e_{l-1}}-1\) entries of \(a_{l-1}\) and so on.

  4. (iv)

    If one has now written down \(p^{e_{q+1}}-1\) entries \(a_q\), then stop. Otherwise put \(a_q\) in the q-th position of the superdiagonal. If, after this, there are only \(p^{e_{q+1}}-1-\tau \) entries \(a_q\) with \(\tau \ge 1\) on the superdiagonal, we set also the \((q+1)\)st diagonal element equal to \(a_q\).

We claim \(M^{{{\,\mathrm{m}\,}}(k'/k)-1}\ne 0\). Take the sequence

$$\begin{aligned}w=(\underbrace{1,\ldots ,1}_{p^{e_1}\text { times}},\underbrace{2,\ldots ,2}_{p^{e_2}\text { times}},\ldots , \underbrace{q,\ldots ,q}_{p^{e_q}\text { times}},\underbrace{q+1,\ldots ,q+1}_{\tau +1\text { times}})\end{aligned}$$

and observe that \(a_w\ne 0\). A little thought shows that this is the unique sequence w of this length with \(a_w\ne 0\) starting at 1 and ending at \(q+1\). This proves the claim. Hence we conclude \(e(R\cap B_{{\bar{k}}})\ge {{\,\mathrm{em}\,}}(k'/k)={{\,\mathrm{E}\,}}(k'/k,r)\). Since \(M^{{{\,\mathrm{m}\,}}(k'/k)}=0\) by Lemmas 2.11 and 2.5, we have \(e(R\cap B_{{\bar{k}}})\le {{\,\mathrm{em}\,}}(k'/k)\) and so this case is done.

Now suppose \(\sum _{i=r+1}^l(p^{e_i}-1)\ge r-1\). One constructs a similar M. Start as before with \(M=0\). One fills the diagonal with \(a_1,\ldots , a_r\) and the superdiagonal with any \(r-1\) elements from the list \((\underbrace{a_{r+1},\ldots ,a_{r+1}}_{p^{e_{r+1}}-1\text { times}},\ldots ,\underbrace{a_l,\ldots ,a_l}_{p^{e_l}-1 \text { times}})\). This gives a matrix M such that \(M^m\ne 0\) where \(m=\sum _{i=1}^r(p^{e_i}-1)+r-1={{\,\mathrm{mr}\,}}(k'/k,r)-1\). Hence \(e(R\cap B_{{\bar{k}}})\ge {{\,\mathrm{emr}\,}}(k'/k,r)\).

We need to demonstrate the upper bound for this case. Take a matrix \(M\in B'({\mathfrak {m}})\) and any sequence w of length \({{\,\mathrm{mr}\,}}(k'/k)\) such that \(a_{w}\ne 0\) in a product of \({{\,\mathrm{mr}\,}}(k'/k,r)\) copies of M. Suppose the entries on the diagonal of M are \(m_1,\ldots ,m_r\in {\mathfrak {m}}\). Then \(a_w=m_1^{f_1}\cdots m_r^{f_r}\cdot \mu \) for non-negative integers \(f_1,\ldots ,f_r\) and \(\mu \) a product of s entries from above the diagonal; we have \(s\le r-1\). Since \({{\,\mathrm{mr}\,}}(k'/k)=\sum f_i+s\), we have \(\sum f_i\ge {{\,\mathrm{mr}\,}}(k'/k,r)-r+1\). We are now done by Lemma 2.11.

For part (iii) of the theorem, we note that any lower bound for \(e(R\cap B_{{\bar{k}}})\) is automatically one for e(R). For part (iv), we just observe that \({{\,\mathrm{E}\,}}(k'/k,r)={{\,\mathrm{em}\,}}(k'/k)\) is also an upper bound for e(R) by Lemmas 2.11 and 2.5.\(\square \)

4 Extended example of

\({{\,\mathrm{SL}\,}}_2\) and \({{\,\mathrm{PGL}\,}}_2\). In this section, we give a complete description of e(R) in case \(G'={{\,\mathrm{SL}\,}}_2,{{\,\mathrm{PGL}\,}}_2\), or \({{\,\mathrm{GL}\,}}_2\).

Proposition 4.1

Let \(k'/k\) be purely inseparable of characteristic p and exponent e. Let G be \(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_2)\), \(\mathrm {R}_{k'/k}({{\,\mathrm{PGL}\,}}_2)\), or \(\mathrm {R}_{k'/k}({{\,\mathrm{SL}\,}}_2)\), the last of these only if \(p\ne 2\). Then e(R) is e if \(k'/k\) is simple and \(e+1\) otherwise.

Proof

If \(G'\) is \({{\,\mathrm{GL}\,}}_2\), then Theorem 1.1(iv) gives the exponent \(e(R)=e\) for \(k'/k\) simple of exponent e, and at least \(e+1\) when \(k'/k\) is non-simple. However, by Theorem 1.1(i), we have e(R) is at most \(e+1\). As the map \({{\,\mathrm{GL}\,}}_2\rightarrow {{\,\mathrm{PGL}\,}}_2\) is smooth, and \({{\,\mathrm{SL}\,}}_2\subset {{\,\mathrm{GL}\,}}_2\), we immediately obtain e or \(e+1\) respectively as upper bounds for both the cases \(G'={{\,\mathrm{SL}\,}}_2\) and \(G'={{\,\mathrm{PGL}\,}}_2\). In fact, the matrices constructed in the proof of Theorem 1.1(ii)-(iv) and their relevant powers are clearly non-zero in their image in \({{\,\mathrm{PGL}\,}}_2({\mathfrak {B}})\) and so the exponents of the radicals in case \(G'={{\,\mathrm{GL}\,}}_2\) and \(G'={{\,\mathrm{PGL}\,}}_2\) coincide. Now by Proposition 2.3(iv), these also coincide for \({{\,\mathrm{SL}\,}}_2\) outside of characteristic 2, since the isogeny \(f:{{\,\mathrm{SL}\,}}_2\rightarrow {{\,\mathrm{PGL}\,}}_2\) is smooth. \(\square \)

We see some different behaviour for \({{\,\mathrm{SL}\,}}_2\) in characteristic 2; in particular, the exponents of \(R_u(G_{\bar{k}})\cap R_u(B_{\bar{k}})\) and \(R_u(G_{\bar{k}})\) do not coincide.

Recall that a unital ring R has characteristic 2 if the equation \(1+1=0\) holds in R. We prove a general formula for \(2^s\)-th powers of matrices over rings of characteristic 2.

Lemma 4.2

Let R be a commutative ring of characteristic 2 and \(M=\begin{pmatrix} a &{}\quad b \\ c &{}\quad d \end{pmatrix}\) with \(a,b,c,d\in R\). Then for all \(s\in {\mathbb {N}}_0\), we have

$$\begin{aligned} M^{2^s}=\begin{pmatrix} a^{2^s}+\sum _{i=0}^{s-1} b^{2^i}c^{2^i}t^{2^s-2^{i+1}} &{} bt^{2^s-1} \\ ct^{2^s-1} &{} d^{2^s}+\sum _{i=0}^{s-1} b^{2^i}c^{2^i}t^{2^s-2^{i+1}} \end{pmatrix}, \end{aligned}$$

where \(t=a+d\).

Proof

We argue by induction on s. For \(s=0\), we just get M on both sides. So assume the result for s and set \(l_s=\sum _{i=0}^{s-1} b^{2^i}c^{2^i}t^{2^s-2^{i+1}}\). We note that

$$\begin{aligned} l_s^2+bct^{2^{s+1}-2}=\sum _{i=1}^{s} b^{2^{i}}c^{2^{i}}t^{2^{s+1}-2^{i+1}}+bct^{2^{s+1}-2}=l_{s+1} \end{aligned}$$

holds. And so we get

$$\begin{aligned} M^{2^{s+1}} =(M^{2^s})^2&=\begin{pmatrix} a^{2^{s+1}}+l_s^2+bct^{2^{s+1}-2} &{} bt^{2^s-1}(a^{2^s}+d^{2^s}) \\ ct^{2^s-1}(a^{2^s}+d^{2^s}) &{} d^{2^{s+1}}+l_s^2+bct^{2^{s+1}-2} \end{pmatrix}\\&=\begin{pmatrix} a^{2^{s+1}}+l_{s+1} &{} bt^{2^{s+1}-1} \\ ct^{2^{s+1}-1} &{} d^{2^{s+1}}+l_{s+1} \end{pmatrix}. \end{aligned}$$

\(\square \)

Proposition 4.3

Let \(G'={{\,\mathrm{SL}\,}}_2\), \(p=2\), and suppose \(k'/k\) has normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) with exponents \(e_1\ge e_2\ge \cdots \ge e_l\). Let \(B'\) be an upper Borel subgroup of \(G'\) and \(B=\mathrm {R}_{k'/k}(B')\) a pseudo-Borel subgroup.

  1. (i)

    We have \(e(R\cap B_{{\bar{k}}})=e\).

  2. (ii)

    We have \(e(R)={\left\{ \begin{array}{ll}e+1\text { if }e_1=e_2,\\ e\text { otherwise.}\end{array}\right. }\)

Proof

We first calculate the exponent of \(B'({\mathfrak {m}})=\ker B'({\mathfrak {B}})\rightarrow B'(k')\). A general element is \(I+M\) with \(M=\begin{pmatrix} a &{}\quad b\\ 0 &{}\quad d\\ \end{pmatrix}\), where \(a,b,d\in {\mathfrak {m}}=\langle a_1,\ldots ,a_l\rangle \), satisfying \(a+d+ad=0\) from the condition \(\det (I+M)=1\). As \(a\in {\mathfrak {m}}\), \(1+a\) is invertible in \({\mathfrak {B}}\), and so it follows that \(d=a(1+a)^{-1}\) and \(a+d=ad=a^2(1+a)^{-1}\). By Lemma 4.2, the \(2^s\)-th power of this matrix is

$$\begin{aligned}\begin{pmatrix} a^{2^s} &{}b(a^2(1+a)^{-1})^{2^s-1}\\ 0 &{} d^{2^s}\\ \end{pmatrix}.\end{aligned}$$

Choosing for example \(a=a_1=1\otimes \alpha _1-\alpha _1\otimes 1\), one sees the minimal s for which \(M^{2^s}=0\) is at least \(e_1=e\) and also that if \(s\ge e\), then the diagonal entries vanish. For \(s=e\), the off-diagonal entry is a multiple of \((a^2)^{2^e-1}\), which has at least \(2^e\) factors of a, hence is also 0. So \(e(R\cap B_{{\bar{k}}})=e\), proving (i).

For (ii), a general element of \(G'({\mathfrak {m}})\) is a matrix \(I+M\) where \(M=\begin{pmatrix} a &{}\quad b\\ c &{}\quad d\\ \end{pmatrix}\), satisfying \(a+d+ad+bc=0\), since \(\det (I+M)=1\). If \(l=1\) (i.e. \(k'/k\) is simple of exponent \(e=e_1\)), then it is clear that \(e(R)=e\). Suppose \(e_2\le e_1-1=e-1\). We may write \(d=(1+a)^{-1}(a+bc)\), hence \(t=a+d=(1+a)^{-1}(a^2+bc)\). Therefore \(t^{2^{e-1}}=(1+a)^{-2^{e-1}}(a^{2^e}+b^{2^{e-1}}c^{2^{e-1}})=(1+a)^{-2^{e-1}}(b^{2^{e-1}}c^{2^{e-1}})\). Now by Lemma 2.10, we have \(m^{2^{e-1}}\in \langle a_1^{2^{e-1}}\rangle \subseteq {\mathfrak {m}}\) for any \(m\in {\mathfrak {m}}\). It follows that \(b^{2^{e-1}}c^{2^{e-1}}\) has a factor of \(a_1^{2^{e-1}}\cdot a_1^{2^{e-1}}=a_1^{2^e}=0\). By the same token, \(t^{2^{e-1}}=0\) and hence, using the lemma, \(M^{2^e}=0\). Therefore \(e(R)=e\) as required.

Now suppose \(e_1=e_2\). Since \(G\subseteq \mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_2)\), we have \(e(R)\le e+1\). Set \(b=a_1\), \(c=a_2\), \(d=0\), and \(a=a_1a_2\). Then

$$\begin{aligned}M^{2^e}=\begin{pmatrix} (a_1a_2)^{2^e}+\sum _{i=1}^{e-1} (a_1a_2)^{2^i+2^e-2^{i+1}} &{} a_1(a_1a_2)^{2^e-1}\\ a_2(a_1a_2)^{2^e-1} &{} \sum _{i=1}^{e-1} (a_1a_2)^{2^i+2^e-2^{i+1}}\\ \end{pmatrix}.\end{aligned}$$

The off-diagonal entries are zero, but \(\sum _{i=1}^{e-1} (a_1a_2)^{2^i+2^e-2^{i+1}}=\sum _{i=1}^e (a_1a_2)^{2^e-2^{i}}\) and the \(i=e-1\) entry is \((a_1a_2)^{2^{e-1}}\). Provided \(e_2=e_1\), this element is non-zero by the same argument as in the proof of Lemma 2.11—in a k-basis of \({\mathfrak {B}}\), the coefficient of \(1\otimes \alpha _1^{2^{e-1}}\alpha _2^{2^{e-1}}\) in \((a_1a_2)^{2^{e-1}}\) is evidently 1. It follows that \(M^{2^e}\ne 0\) and \(e(R)=e+1\). \(\square \)

5 Simple groups of rank at least 2

For a positive integer r, recall that the algebraic k-group \(\mu _r\) is the functor returning the r-th roots of unity in any k-algebra A; it is the kernel of the r-power map \(\phi _r:{{\,\mathrm{{\mathbb {G}}_m}\,}}\rightarrow {{\,\mathrm{{\mathbb {G}}_m}\,}},x\mapsto x^r\). (We have \(\mu _r\) étale if and only if \(p\not \mid r\).) Say a \(k'\)-group L is a skewed \({{\,\mathrm{GL}\,}}_n\) if \(L\cong {{\,\mathrm{GL}\,}}_n/K\), where \(K\cong \mu _r\), with \(r\not \mid n\); the quotient by \(\mu _r\) induces the r-power map on the central torus of \({{\,\mathrm{GL}\,}}_n\). Since \(K\cap {{\,\mathrm{SL}\,}}_n=1\), the derived subgroup \({\mathscr {D}}(L)\) of L is still isomorphic to \({{\,\mathrm{SL}\,}}_n\) and notably there is still a central torus \(C\subset L\) which contains \(Z({{\,\mathrm{SL}\,}}_n)\). In [12, Ex. 5.17], it is explained that if \(\mu _n\subset T\) is a finite subgroup of a split k-torus of rank at least 2, then any automorphism of \(\mu _n\) (which is some r-power map on \(\mu _n\) with \((r,n)=1\)) lifts to an automorphism of T. In our case, the quotient map by \(K\times 1\) on \({{\,\mathrm{GL}\,}}_n\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) lifts to an automorphism of \({{\,\mathrm{GL}\,}}_n\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) defined on the central torus \({{\,\mathrm{{\mathbb {G}}_m}\,}}\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) by \((x,y)\mapsto (x^ry^{-n},x^by^a)\) where a and b are integers such that \(ar+bn=1\). It follows that if L is a skewed \({{\,\mathrm{GL}\,}}_n\)-group, then \(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\cong {{\,\mathrm{GL}\,}}_n\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\).

The group \({{\,\mathrm{SL}\,}}_{n+1}\) contains a (Levi) subgroup \(L:={{\,\mathrm{GL}\,}}_n\) embedded block diagonally via \(g\mapsto {{\,\mathrm{diag}\,}}(g,\det {g}^{-1})\). If \(G'\) is a split simple k-group of type \(A_n\), then \(G'\cong {{\,\mathrm{SL}\,}}_{n+1}/\mu _r\) with \(\mu _r\subseteq Z({{\,\mathrm{SL}\,}}_{n+1})\cong \mu _{n+1}\), so \(r\mid n+1\). But then \(r\not \mid n\), hence L maps under the quotient by \(\mu _r\) to a skewed \({{\,\mathrm{GL}\,}}_n\)-subgroup of \(G'\). This observation establishes the type A case of the following lemma.

Lemma 5.1

Let \(G'\) be a split simple \(k'\)-group of rank r. Then one of the following holds:

  1. (i)

    \(G'\) contains a skewed \({{\,\mathrm{GL}\,}}_r\) subgroup L.

  2. (ii)

    There are étale isogenies \(\pi :L\rightarrow L/Z\), and \(\phi :{{\,\mathrm{GL}\,}}_r\rightarrow L/Z\) or \(\phi :L\rightarrow {{\,\mathrm{GL}\,}}_r\).

  3. (iii)

    \(p=2\) and either \(G'\) is of type \(F_4\) or \(G'\) is of type B, C, or D, but not isomorphic to \({{\,\mathrm{SO}\,}}_{2r}\), \({{\,\mathrm{SO}\,}}_{2r+1}\), or \({{\,\mathrm{Sp}\,}}_{2r}\).

  4. (iv)

    \(p=3\) and G is of type \(E_6\).

In cases (i) and (ii), if we let \(R={\mathscr {R}}_u(\mathrm {R}_{k'/k}(L)_{{\bar{k}}})\), then we have \(e(R)=e({\mathscr {R}}_u(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r)_{{\bar{k}}}))\).

Proof

If \(G'={{\,\mathrm{SO}\,}}_{2r}\), \({{\,\mathrm{SO}\,}}_{2r+1}\), or \({{\,\mathrm{Sp}\,}}_{2r}\), then [7, 4.5] promises a Levi subgroup \(L\cong {{\,\mathrm{GL}\,}}_r\) as required so (i) holds in all these cases. The central isogenies \({{\,\mathrm{Sp}\,}}_{2r}\rightarrow {{\,\mathrm{PSp}\,}}_{2r}\), \({{\,\mathrm{Spin}\,}}_n\rightarrow {{\,\mathrm{SO}\,}}_n\), \({{\,\mathrm{SO}\,}}_n\rightarrow {{\,\mathrm{PSO}\,}}_n\), \({{\,\mathrm{Spin}\,}}_{2r}\rightarrow {{\,\mathrm{HSpin}\,}}_{2r}\), or \({{\,\mathrm{HSpin}\,}}_{2r}\rightarrow {{\,\mathrm{PSO}\,}}_{2r}\) (where \({{\,\mathrm{HSpin}\,}}_{2r}\) occurs only if n even) are all quotients by central subgroups isomorphic to \(\mu _2\), hence are étale maps when \(p>2\) and hence (ii) holds for all classical groups when \(p>2\). Otherwise (iii) holds.

Let now \(G'\) be exceptional. Note that groups of type \(E_7\) and \(E_8\) contain simple \(A_7\) and \(A_8\) maximal rank subgroups, evident from the Borel–de-Siebenthal algorithm. Hence each contains a skewed \({{\,\mathrm{GL}\,}}_7\) and \({{\,\mathrm{GL}\,}}_8\) subgroup L respectively, satisfying (i) as required. By [8, Thm. A], any Levi subgroup L of \(G_2\) is isomorphic to \({{\,\mathrm{GL}\,}}_2\) (rather than \({{\,\mathrm{SL}\,}}_2\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) for example), hence satisfies (i). If \(G'\) is of type \(F_4\), then we take L to be an \(A_3\) Levi subgroup of the \({{\,\mathrm{Spin}\,}}_9\) subgroup of \(G'\). Now L admits an étale isogeny with the \({{\,\mathrm{GL}\,}}_4\) Levi subgroup of \({{\,\mathrm{SO}\,}}_9\cong {{\,\mathrm{Spin}\,}}_9/Z({{\,\mathrm{Spin}\,}}_9)\) if \(p\ne 2\); so L satisfies (ii) in that case, and if \(p=2\), then L is listed in (iii). It remains to deal with the case \(G'=E_6\). Using Bourbaki notation [3], we take L to be the \(A_5\) Levi subgroup corresponding to the nodes \(\{1,3,4,5,6\}\) of the Dynkin diagram explicitly. Assume first \(G'\) is simply connected. Then \({\mathscr {D}}(L)\) is simply connected. Since L centralises the \({{\,\mathrm{SL}\,}}_2\) subgroup generated by the \(\pm {{\tilde{\alpha }}}\) root groups, where \({{\tilde{\alpha }}}\) is the longest root, the central torus C of L is the image of the cocharacter \(h_{\tilde{\alpha }}:{{\,\mathrm{{\mathbb {G}}_m}\,}}\rightarrow G', t\mapsto h_{\tilde{\alpha }}(t)\). One calculates \(h_{\tilde{\alpha }}(t)=h_{\alpha _1}(t)h_{\alpha _2}(t^2)h_{\alpha _3}(t^2)h_{\alpha _4}(t^3)h_{\alpha _5}(t^2)h_{\alpha _6}(t)\). This is inside \({\mathscr {D}}(L)\cong {{\,\mathrm{SL}\,}}_6\) iff \(t^2=1\); therefore \({\mathscr {D}}(L)\cap C\cong \mu _2\)—so L is not a skewed \({{\,\mathrm{GL}\,}}_r\). However, if \(p\ne 3\), then there is an étale isogeny from L to \(L/Z\cong {{\,\mathrm{GL}\,}}_6\) where Z is a diagonal copy of \(\mu _3\) in both factors \({\mathscr {D}}(L)\) and C of L, so that (ii) holds. Besides which, since \(Z(G')\cong \mu _3\), then if \(p\ne 3\), the quotient by \(Z(G')\) is étale and \(L/Z(G')\) satisfies (ii) for the adjoint group \(G/Z(G')\). Otherwise \(p=3\) and (iv) holds.

We prove the last statement. Suppose (ii) holds. Then we note that if \(\pi :A\rightarrow B\) is an étale central isogeny of reductive groups, then \(\mathrm {R}_{k'/k}(A)\rightarrow \mathrm {R}_{k'/k}(B)\) is an isogeny whose kernel has trivial intersection with the unipotent radical. As in Proposition 2.3(iv), it follows that the induced map \({\mathscr {R}}_u(\mathrm {R}_{k'/k}(A)_{{\bar{k}}})\rightarrow {\mathscr {R}}_u(\mathrm {R}_{k'/k}(B)_{{\bar{k}}})\) is an isomorphism as required. If (i) holds, then L is a skewed \({{\,\mathrm{GL}\,}}_r\)-subgroup and by the remarks preceding the lemma, \(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\cong {{\,\mathrm{GL}\,}}_r\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\). Observe \({\mathscr {R}}_u(\mathrm {R}_{k'/k}(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}})_{{\bar{k}}})\cong {\mathscr {R}}_u(\mathrm {R}_{k'/k}(L)_{{\bar{k}}})\times {\mathscr {R}}_u(\mathrm {R}_{k'/k}({{\,\mathrm{{\mathbb {G}}_m}\,}})_{{\bar{k}}})\); since L already contains \(C\cong {{\,\mathrm{{\mathbb {G}}_m}\,}}\), the exponent of the product is equal to that of the first factor, e(R). The same argument with \({{\,\mathrm{GL}\,}}_r\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) in place of \(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) implies the claim. \(\square \)

Proof of Corollary 1.3

With the hypotheses on p, by Lemma 5.1, we get a reductive subgroup L of \(G'\), with

$$\begin{aligned} e({\mathscr {R}}_u(\mathrm {R}_{k'/k}(L))_{{\bar{k}}})=e({\mathscr {R}}_u(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r))_{{\bar{k}}})\ge {{\,\mathrm{E}\,}}(k'/k,r). \end{aligned}$$

By Proposition 2.3(iii), it follows that \(e(R)\ge {{\,\mathrm{E}\,}}(k'/k,r)\). If \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\), then Weil restricting a faithful representation \(G'\rightarrow {{\,\mathrm{GL}\,}}_n\) from \(k'\) to k gives an embedding of G in \({{\,\mathrm{GL}\,}}_m\) with \(m=[k':k]\cdot n\ge r\). Now by Proposition 2.3(iii) again, \(e(R)\le e({\mathscr {R}}_u(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_m))_{{\bar{k}}})\) which is equal to \({{\,\mathrm{E}\,}}(k'/k,m)={{\,\mathrm{em}\,}}(k'/k)\) by Theorem 1.1.\(\square \)