# Local Galois representations associated to
additive polynomials

Takahiro Tsushima

###### Abstract

For an additive polynomial and a positive integer, we define an irreducible
smooth representation of a Weil group of a non-archimedean local field. We
study several invariants of this representation. We deduce a necessary and
sufficient condition for it to be primitive.

††footnotetext: Keywords: Local Galois representations, additive polynomials,
symplectic modules††footnotetext: 2020 Mathematics Subject Classification.
Primary: 11F80; Secondary: 14H37.

##  1 Introduction

Let $p$ be a prime number and $q$ a power of it. An additive polynomial $R(x)$
over $\mathbb{F}_{q}$ is a one-variable polynomial with coefficients in
$\mathbb{F}_{q}$ such that $R(x+y)=R(x)+R(y)$. It is known that $R(x)$ has the
form $\sum_{i=0}^{e}a_{i}x^{p^{i}}\ (a_{e}\neq 0)$ with an integer $e\geq 0$.
Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_{q}$.
We take a separable closure $\overline{F}$ of $F$. Let $W_{F}$ be the Weil
group of $\overline{F}/F$. Let $v_{F}(\cdot)$ denote the normalized valuation
on $F$. We take a prime number $\ell\neq p$. For a non-trivial character
$\psi\colon\mathbb{F}_{p}\to\overline{\mathbb{Q}}_{\ell}^{\times}$, a non-zero
additive polynomial $R(x)$ over $\mathbb{F}_{q}$ and a positive integer $m$
which is prime to $p$, we define an irreducible smooth $W_{F}$-representation
$\tau_{\psi,R,m}$ over $\overline{\mathbb{Q}}_{\ell}$ of degree $p^{e}$ if
$v_{F}(p)$ is sufficiently large. This is unconditional if $F$ has positive
characteristic. The integer $m$ is related to the Swan conductor exponent of
$\tau_{\psi,R,m}$. As $m$ varies, the isomorphism class of $\tau_{\psi,R,m}$
varies.

Let $C_{R}$ denote the algebraic affine curve defined by $a^{p}-a=xR(x)$ in
$\mathbb{A}_{\mathbb{F}_{q}}^{2}=\operatorname{Spec}\mathbb{F}_{q}[a,x]$. This
curve is studied in [6] and [1] in detail. For example, the smooth
compactification of $C_{R}$ is proved to be supersingular if $(p,e)\neq(2,0)$.
The automorphism group of $C_{R}$ contains a semidirect product $Q_{R}$ of a
cyclic group and an extra-special $p$-group ((2.5)). Let $\mathbb{F}$ be an
algebraic closure of $\mathbb{F}_{q}$. Then a semidirect group
$Q_{R}\rtimes\mathbb{Z}$ acts on the base change
$C_{R,\mathbb{F}}:=C_{R}\times_{\mathbb{F}_{q}}\mathbb{F}$ as endomorphisms,
where $1\in\mathbb{Z}$ acts on $C_{R,\mathbb{F}}$ as the Frobenius
endomorphism over $\mathbb{F}_{q}$. The center $Z(Q_{R})$ of $Q_{R}$ is
identified with $\mathbb{F}_{p}$, which acts on $C_{R}$ as $a\mapsto a+\zeta$
for $\zeta\in\mathbb{F}_{p}$. Let $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})$ be the first étale
cohomology group of $C_{R,\mathbb{F}}$ with compact support. Each element of
$Z(Q_{R})$ is fixed by the action of $\mathbb{Z}$ on $Q_{R}$. Thus its
$\psi$-isotypic part $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ is regarded as a
$Q_{R}\rtimes\mathbb{Z}$-representation.

We construct a concrete Galois extension over $F$ whose Weil group is
isomorphic to a subgroup of $Q_{R}\rtimes\mathbb{Z}$ associated to the integer
$m$ (Definition 3.1 and (3.8)). Namely we will define a homomorphism
$\Theta_{R,m}\colon W_{F}\to Q_{R}\rtimes\mathbb{Z}$ in (3.12). As a result,
we define $\tau_{\psi,R,m}$ to be the composite

$W_{F}\xrightarrow{\Theta_{R,m}}Q_{R}\rtimes\mathbb{Z}\to\mathrm{Aut}_{\overline{\mathbb{Q}}_{\ell}}(H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]).$

This is a smooth irreducible representation of $W_{F}$ of degree $p^{e}$.

We state our motivation and reason why we introduce and study
$\tau_{\psi,R,m}$. It is known that the reductions of concentric affinoids in
the Lubin–Tate curve fit into this type of curves $C_{R}$ with special $R$.
For example, see [16] and [17]. When $R$ is a monomial and $m=1$, the
representation $\tau_{\psi,R,m}$ is studied in [9] and [10] in detail. In
these papers, the reduction of a certain affinoid in the Lubin–Tate space is
related to $C_{R}$ in some sense and the supercuspidal representation $\pi$ of
$\mathrm{GL}_{p^{e}}(F)$ which corresponds to $\tau_{\psi,R,m}$ under the
local Langlands correspondence explicitly. The homomorphism $\Theta_{R,1}$
with $R(x)=x^{p^{e}}\ (e\in\mathbb{Z}_{\geq 1})$ does appear in the work [9].
An irreducible representation of a group is said to be primitive if it is not
isomorphic to an induction of any representation of a proper subgroup. The
representation $\tau_{\psi,R,m}$ in [9] and [10] is primitive and this
property makes it complicated to describe $\pi$ in a view point of type
theory. It is an interesting problem to do the same thing for general
$\tau_{\psi,R,m}$. In this direction, it would be valuable to know when
$\tau_{\psi,R,m}$ is primitive. We expect that another $C_{R}$ will be related
to concentric affinoids in the Lubin–Tate spaces as in [9].

We briefly explain the content of each section. In §2, we state several things
on the curves $C_{R}$ and the extra-special $p$-subgroups contained in the
automorphism groups of the curves.

In §3.1 and §3.2, we construct the Galois extension mentioned above and define
$\tau_{\psi,R,m}$. Let $d_{R}:=\mathrm{gcd}\\{p^{i}+1\mid a_{i}\neq 0\\}$. We
show that the Swan conductor exponent of $\tau_{\psi,R,m}$ equals
$m(p^{e}+1)/d_{R}$ (Corollary 3.15). In §3.3, we study primitivity of
$\tau_{\psi,R,m}$. In particular, we write down a necessary and sufficient
condition for $\tau_{\psi,R,m}$ to be primitive. By this, we give examples
such that $\tau_{\psi,R,m}$ is primitive (Example 3.29). The necessary and
sufficient condition is that a symplectic module $(V_{R},\omega_{R})$
associated to $\tau_{\psi,R,m}$ is completely anisotropic (Corollary 3.28). If
$R$ is a monomial, $(V_{R},\omega_{R})$ is studied in §3.4 in more detail. In
Proposition 3.44, a primary module in the sense of [11, §9] is constructed
geometrically by using the Künneth formula.

Our aim in §4 is to show the following theorem.

######  Theorem 1.1.

Assume $p\neq 2$. The following two conditions are equivalent.

  * (1)

There exists a non-trivial finite étale morphism

$C_{R}\to C_{R_{1}};\ (a,x)\mapsto\left(a-\Delta(x),r(x)\right),$

where $\Delta(x)\in\mathbb{F}_{q}[x]$ and $r(x),R_{1}(x)$ are additive
polynomials over $\mathbb{F}_{q}$ such that $d_{R,m}\mid d_{R_{1}}$ and
$r(\alpha x)=\alpha r(x)$ for any $\alpha\in\mu_{d_{R,m}}$.

  * (2)

The $W_{F}$-representation $\tau_{\psi,R,m}$ is imprimitive.

If $\tau_{\psi,R,m}$ is imprimitive, it is written as a form of an induced
representation of a certain explicit $W_{F^{\prime}}$-representation
$\tau^{\prime}_{\psi,R_{1},m}$ associated to a finite extension
$F^{\prime}/F$. The proof of the above theorem depends on several geometric
properties of $C_{R}$ developed in [6] and [1]. See the beginning of §4 for
more details.

### Notation

Let $k$ be a field. Let $\mu(k)$ denote the set of all roots of unity in $k$.
For a positive integer $r$, let $\mu_{r}(k):=\\{x\in k\mid x^{r}=1\\}$.

For a positive integer $i$, let $\mathbb{A}^{i}_{k}$ and $\mathbb{P}^{i}_{k}$
be an $i$-dimensional affine space and a projective space over $k$,
respectively. For a scheme $X$ over $k$ and a field extension $l/k$, let
$X_{l}$ denote the base change of $X$ to $l$. For a closed subset $Z$ of a
variety $X$, we regard $Z$ as a closed subscheme with reduced scheme
structure.

Throughout this paper, we set $q:=p^{f}$ with a positive integer $f$. For a
positive integer $i$, we simply write $\operatorname{Nr}_{q^{i}/q}$ and
$\operatorname{Tr}_{q^{i}/q}$ for the norm map and the trace map from
$\mathbb{F}_{q^{i}}$ to $\mathbb{F}_{q}$, respectively.

Let $X$ be a scheme over $\mathbb{F}_{q}$, let $F_{q}\colon X\to X$ be the
$q$-th power Frobenius endomorphism. Let $\mathbb{F}$ be an algebraic closure
of $\mathbb{F}_{q}$. Let $\mathrm{Fr}_{q}\colon X_{\mathbb{F}}\to
X_{\mathbb{F}}$ be the base change of $F_{q}$. This endomorphism
$\mathrm{Fr}_{q}$ is called the Frobenius endomorphism of $X$ over
$\mathbb{F}_{q}$.

For a Galois extension $l/k$, let $\operatorname{Gal}(l/k)$ denote the Galois
group of the extension.

##  2 Extra-special $p$-groups and affine curves

######  Definition 2.1.

Let $k$ be a field. A polynomial $f(x)\in k[x]$ is called additive if
$f(x+y)=f(x)+f(y)$. Let $\mathscr{A}_{k}$ be the set of all additive
polynomials with coefficients in $k$.

Let $p$ be a prime number. We simply $\mathscr{A}_{q}$ for
$\mathscr{A}_{\mathbb{F}_{q}}$. Let
$R(x):=\sum_{i=0}^{e}a_{i}x^{p^{i}}\in\mathscr{A}_{q}$ with
$e\in\mathbb{Z}_{\geq 0}$ and $a_{e}\neq 0$. Let

$E_{R}(x):=R(x)^{p^{e}}+\sum_{i=0}^{e}(a_{i}x)^{p^{e-i}}\in\mathscr{A}_{q}.$
(2.1)

We always assume

$(p,e)\neq(2,0).$ (2.2)

We simply write $\mu_{r}$ for $\mu_{r}(\mathbb{F})$ for a positive integer
$r$. Let

$d_{R}:=\mathrm{gcd}\\{p^{i}+1\mid a_{i}\neq 0\\}.$

If $a_{i}\neq 0$, we have $\alpha^{p^{i}}=\alpha^{-1}$ and
$\alpha^{p^{e-i}}=\alpha$ for $\alpha\in\mu_{d_{R}}$. Hence we have

$\alpha R(\alpha x)=R(x),\quad E_{R}(\alpha x)=\alpha E_{R}(x)\quad\textrm{for
$\alpha\in\mu_{d_{R}}$}.$ (2.3)

Let

$f_{R}(x,y):=-\sum_{i=0}^{e-1}\left(\sum_{j=0}^{e-i-1}(a_{i}x^{p^{i}}y)^{p^{j}}+(xR(y))^{p^{i}}\right)\in\mathbb{F}_{q}[x,y].$

This is linear with respect to $x$ and $y$. By (2.3), we have

$f_{R}(\alpha x,\alpha y)=f_{R}(x,y)\quad\textrm{for $\alpha\in\mu_{d_{R}}$}.$
(2.4)

######  Lemma 2.2.

We have $f_{R}(x,y)^{p}-f_{R}(x,y)=-x^{p^{e}}E_{R}(y)+xR(y)+yR(x).$ In
particular, if $E_{R}(y)=0$, we have $f_{R}(x,y)^{p}-f_{R}(x,y)=xR(y)+yR(x)$.

###### Proof.

The former equality follows from

$\displaystyle f_{R}(x,y)^{p}-f_{R}(x,y)$
$\displaystyle=xR(y)-(xR(y))^{p^{e}}+\sum_{i=0}^{e-1}(a_{i}x^{p^{i}}y-(a_{i}x^{p^{i}}y)^{p^{e-i}})$
$\displaystyle=-x^{p^{e}}E_{R}(y)+xR(y)+yR(x).$

∎

######  Definition 2.3.

  * (1)

Let $V_{R}:=\\{\beta\in\mathbb{F}\mid E_{R}(\beta)=0\\},$ which is a
$(2e)$-dimensional $\mathbb{F}_{p}$-vector space.

  * (2)

Let

$Q_{R}:=\left\\{(\alpha,\beta,\gamma)\in\mathbb{F}^{3}\mid\alpha\in\mu_{d_{R}},\
\beta\in V_{R},\ \gamma^{p}-\gamma=\beta R(\beta)\right\\}$

be the group whose group law is given by

$(\alpha_{1},\beta_{1},\gamma_{1})\cdot(\alpha_{2},\beta_{2},\gamma_{2}):=\left(\alpha_{1}\alpha_{2},\beta_{1}+\alpha_{1}\beta_{2},\gamma_{1}+\gamma_{2}+f_{R}(\beta_{1},\alpha_{1}\beta_{2})\right).$

This is well-defined according to (2.3) and Lemma 2.2.

  * (3)

Let $H_{R}:=\\{(\alpha,\beta,\gamma)\in Q_{R}\mid\alpha=1\\},$ which is a
normal subgroup of $Q_{R}$.

If $e=0$, we have $p\neq 2$ by (2.2). We have $H_{R}=\mathbb{F}_{p}\subset
Q_{R}=\mu_{2}\times\mathbb{F}_{p}$ if $e=0$.

For a group $G$ and elements $g,g^{\prime}\in G$, let
$[g,g^{\prime}]:=gg^{\prime}g^{-1}g^{\prime-1}$.

######  Lemma 2.4.

For $g=(1,\beta,\gamma),\ g^{\prime}=(1,\beta^{\prime},\gamma^{\prime})\in
H_{R}$, we have
$[g,g^{\prime}]=(1,0,f_{R}(\beta,\beta^{\prime})-f_{R}(\beta^{\prime},\beta))$.
In particular, we have
$f_{R}(\beta,\beta^{\prime})-f_{R}(\beta^{\prime},\beta)\in\mathbb{F}_{p}$.

###### Proof.

This is directly checked. We omit the details. ∎

For a group $G$, let $Z(G)$ denote its center and $[G,G]$ the commutator
subgroup of $G$.

######  Definition 2.5.

A non-abelian $p$-group $G$ is called an extra-special $p$-group if
$[G,G]=Z(G)$ and $|Z(G)|=p$.

######  Lemma 2.6.

Assume $e\geq 1$.

  * (1)

The group $H_{R}$ is non-abelian. We have
$Z(H_{R})=Z(Q_{R})=\\{(1,0,\gamma)\mid\gamma\in\mathbb{F}_{p}\\}$. The
quotient $H_{R}/Z(H_{R})$ is isomorphic to $V_{R}$.

  * (2)

The group $H_{R}$ is an extra-special $p$-group. The pairing $\omega_{R}\colon
V_{R}\times V_{R}\to\mathbb{F}_{p};\ (\beta,\beta^{\prime})\mapsto
f_{R}(\beta,\beta^{\prime})-f_{R}(\beta^{\prime},\beta)$ is a non-degenerate
symplectic form.

###### Proof.

We show (1). Let $X_{\beta}:=\\{x\in\mathbb{F}\mid
f_{R}(\beta,x)=f_{R}(x,\beta)\\}$ for $\beta\in V_{R}$. Then $X_{\beta}$ is an
$\mathbb{F}_{p}$-vector space of dimension $2e-1$ if $\beta\neq 0$. Since
$V_{R}$ has dimension $2e$, we have $V_{R}\nsubseteq X_{\beta}$ for $\beta\in
V_{R}\setminus\\{0\\}$. This implies that $H_{R}$ is non-abelian according to
Lemma 2.4.

Clearly we have $Z:=\\{(1,0,\gamma)\mid\gamma\in\mathbb{F}_{p}\\}\subset
Z(Q_{R})\subset Z(H_{R})$. It suffices to show $Z(H_{R})\subset Z$. Let
$(1,\beta,\gamma)\in Z(H_{R})$. We have $V_{R}\subset X_{\beta}$ by Lemma 2.4.
This implies $\beta=0$. Hence we obtain $Z(H_{R})\subset Z$. The last claim is
easily verified.

We show (2). By Lemma 2.4, we have $[H_{R},H_{R}]\subset Z(H_{R})$. Since
$H_{R}$ is non-abelian, $[H_{R},H_{R}]$ is non-trivial. Hence we have
$[H_{R},H_{R}]=Z(H_{R})$ by $|Z(H_{R})|=p$. Hence $H_{R}$ is extra-special.
Assume $\omega_{R}(\beta,\beta^{\prime})=0$ for any $\beta^{\prime}\in V_{R}$.
We take an element $(1,\beta,\gamma)\in H_{R}$. By Lemma 2.4, we have
$(1,\beta,\gamma)\in Z(H_{R})$. Hence we have $\beta=0$ by (1). ∎

######  Definition 2.7.

  * (1)

Let $C_{R}$ be the affine curve over $\mathbb{F}_{q}$ defined by
$a^{p}-a=xR(x)$.

  * (2)

Let $Q_{R}$ act on $C_{R,\mathbb{F}}$ by

$(a,x)\cdot(\alpha,\beta,\gamma)=\left(a+f_{R}(x,\beta)+\gamma,\alpha^{-1}(x+\beta)\right),$
(2.5)

for $(a,x)\in C_{R,\mathbb{F}}$ and $(\alpha,\beta,\gamma)\in Q_{R}$. This is
well-defined by (2.3) and Lemma 2.2.

The curve $C_{R}$ is studied in [6] and [1].

We take a prime number $\ell\neq p$. For a finite abelian group $A$, let
$A^{\vee}$ denote the character group
$\mathrm{Hom}_{\mathbb{Z}}(A,\overline{\mathbb{Q}}_{\ell}^{\times})$. For a
representation $M$ of $A$ over $\overline{\mathbb{Q}}_{\ell}$ and $\chi\in
A^{\vee}$, let $M[\chi]$ denote the $\chi$-isotypic part of $M$.

According to Lemma 2.6(1), we identify a character
$\psi\in\mathbb{F}_{p}^{\vee}$ with a character of $Z(H_{R})$.

######  Lemma 2.8.

Let $\psi\in\mathbb{F}_{p}^{\vee}\setminus\\{1\\}$.

  * (1)

Let $W\subset V_{R}$ be an $\mathbb{F}_{p}$-subspace of dimension $e$, which
is totally isotropic with respect to $\omega_{R}$. Let $W^{\prime}\subset
H_{R}$ be the inverse image of $W$ by the natural map $H_{R}\to V_{R};\
(1,\beta,\gamma)\mapsto\beta$. Let $\xi\in W^{\prime\vee}$ be an extension of
$\psi\in Z(H_{R})^{\vee}$. Let
$\rho_{\psi}:=\mathrm{Ind}_{W^{\prime}}^{H_{R}}\xi$. Then $\rho_{\psi}$ is a
unique (up to isomorphism) irreducible representation of $H_{R}$ containing
$\psi$. In particular, $\rho_{\psi}|_{Z(H_{R})}$ is a multiple of $\psi$.

  * (2)

The $\psi$-isotypic part $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ is isomorphic to
$\rho_{\psi}$ as $H_{R}$-representations.

###### Proof.

By Lemma 2.6(2) and [8, 16.14(2) Satz], the claim (1) follows. By [16, Remark
3.29], we have $\dim H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]=p^{e}$. Hence the
claim (2) follows from (1). ∎

The representation $\rho_{\psi}$ induces a projective representation
$\bar{\rho}\colon
H_{R}/Z(H_{R})\to\mathrm{PGL}_{p^{e}}(\overline{\mathbb{Q}}_{\ell})$.

######  Lemma 2.9.

The map $\bar{\rho}$ is injective.

###### Proof.

As in the proof of [14, Theorem 6], we have
$\operatorname{Tr}\rho_{\psi}(x)=0$ for $x\in H_{R}\setminus Z(H_{R})$. Assume
$\bar{\rho}(xZ(H_{R}))=1$ for $x\in H_{R}$. Then $\rho_{\psi}(x)$ is a non-
zero scalar matrix. Hence $\operatorname{Tr}\rho_{\psi}(x)\neq 0$. This
implies $x\in Z(H_{R})$. ∎

Let $\mathbb{Z}\ni 1$ act on $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})$ by the pull-back
$\mathrm{Fr}_{q}^{\ast}$. Let $\mathbb{Z}\ni 1$ act on $Q_{R}$ by
$(\alpha,\beta,\gamma)\mapsto(\alpha^{q^{-1}},\beta^{q^{-1}},\gamma^{q^{-1}})$.
The semidirect product $Q_{R}\rtimes\mathbb{Z}$ acts on $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$.

Let $\overline{C}_{R}$ denote the smooth compactification of $C_{R}$.

######  Proposition 2.10.

The projective curve $\overline{C}_{R}$ is supersingular. In particular, this
curve has positive genus. The natural map $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})\to
H^{1}(\overline{C}_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})$ is an
isomorphism.

###### Proof.

The former claim is shown in [6, Theorems (9.4) and (13.7)] ( [1, Proposition
8.5]). The last claim follows from [16, Lemmas 3.27 and 3.28(3)]. ∎

##  3 Local Galois representation

In this section, we define an irreducible smooth $W_{F}$-representation
$\tau_{\psi,R,m}$ and determine several invariants associated to it. In
§3.2.2, we determine the Swan conductor exponent of $\tau_{\psi,R,m}$. In
§3.3, we determine the symplectic module associated to $\tau_{\psi,R,m}$, and
give a necessary and sufficient condition for $\tau_{\psi,R,m}$ to be
primitive. As a result, we obtain several examples such that $\tau_{\psi,R,m}$
is primitive. If $R$ is a monomial, we calculate invariants of the root system
corresponding to $(V_{R},\omega_{R})$ defined in [11] ( Lemma 3.36).

###  3.1 Galois extension

For a valued field $K$, let $\mathcal{O}_{K}$ denote the valuation ring of
$K$.

Let $F$ be a non-archimedean local field. Let $\overline{F}$ be a separable
closure of $F$. Let $\widehat{\overline{F}}$ denote the completion of
$\overline{F}$. Let $v(\cdot)$ denote the unique valuation on
$\widehat{\overline{F}}$ such that $v(\varpi)=1$ for a uniformizer $\varpi$ of
$F$, which we now fix. We simply write $\mathcal{O}$ for
$\mathcal{O}_{\widehat{\overline{F}}}$. Let $\mathfrak{p}$ be the maximal
ideal of $\mathcal{O}$.

Let $q$ be the cardinality of the residue field of $\mathcal{O}_{F}$.

We take $R(x)=\sum_{i=0}^{e}a_{i}x^{p^{i}}\in\mathscr{A}_{q}$. For an element
$a\in\mathbb{F}_{q}$, let $\widetilde{a}\in\mu(F)\cup\\{0\\}$ be its
Teichmüller lift. Let

$\widetilde{R}(x):=\sum_{i=0}^{e}\widetilde{a}_{i}x^{p^{i}},\quad\widetilde{E}_{R}(x):=\widetilde{R}(x)^{p^{e}}+\sum_{i=0}^{e}(\widetilde{a}_{i}x)^{p^{e-i}}\in\mathcal{O}_{F}[x].$

Similarly as (2.3), we have

$\alpha\widetilde{R}(\alpha x)=\widetilde{R}(x),\quad\widetilde{E}_{R}(\alpha
x)=\alpha\widetilde{E}_{R}(x)\quad\textrm{for
$\alpha\in\mu_{d_{R}}(\overline{F})$}.$ (3.1)

######  Definition 3.1.

Let $m$ be a positive integer prime to $p$. Let
$\alpha_{R,\varpi},\beta_{R,m,\varpi},\gamma_{R,m,\varpi}\in\overline{F}$ be
elements such that

$\displaystyle\alpha_{R,\varpi}^{d_{R}}=\varpi,\quad\widetilde{E}_{R}(\beta_{R,m,\varpi})=\alpha_{R,\varpi}^{-m},\quad\gamma_{R,m,\varpi}^{p}-\gamma_{R,m,\varpi}=\beta_{R,m,\varpi}\widetilde{R}(\beta_{R,m,\varpi}).$

For simplicity, we write $\alpha_{R},\beta_{R,m},\gamma_{R,m}$ for
$\alpha_{R,\varpi},\beta_{R,m,\varpi},\gamma_{R,m,\varpi}$, respectively.

######  Remark 3.2.

By $\deg\widetilde{E}_{R}(x)=p^{2e}$ and $\deg\widetilde{R}(x)=p^{e}$, we have

$v(\alpha_{R})=\frac{1}{d_{R}},\quad
v(\beta_{R,m})=-\frac{m}{p^{2e}d_{R}},\quad
v(\gamma_{R,m})=-\frac{m(p^{e}+1)}{p^{2e+1}d_{R}}.$

The integer $m$ controls the depth of ramification of the resulting field
extension $F(\alpha_{R},\beta_{R,m},\gamma_{R,m})/F$. We will understand this
later in §3.2.2.

Let

$\widetilde{f}(x,y):=-\sum_{i=0}^{e-1}\left(\sum_{j=0}^{e-i-1}(\widetilde{a}_{j}x^{p^{i}}y)^{p^{j}}+(x\widetilde{R}(y))^{p^{i}}\right).$

Let $\mathfrak{p}[x]:=\mathfrak{p}\mathcal{O}[x]$ and
$\mathfrak{p}[x,y]:=\mathfrak{p}\mathcal{O}[x,y]$. We assume that

$\displaystyle\begin{aligned}
&\beta_{R,m}^{p^{e}}(\widetilde{E}_{R}(\beta_{R,m}+x)-\widetilde{E}_{R}(\beta_{R,m})-\widetilde{E}_{R}(x)),\quad\beta_{R,m}(\widetilde{R}(\beta_{R,m}+x)-\widetilde{R}(\beta_{R,m})-\widetilde{R}(x)),\\\
&\textrm{$\widetilde{f}(\beta_{R,m},x)^{p}-\widetilde{f}(\beta_{R,m},x)+\beta_{R,m}^{p^{e}}\widetilde{E}_{R}(x)-x\widetilde{R}(\beta_{R,m})-\beta_{R,m}\widetilde{R}(x)$
are contained in $\mathfrak{p}[x]$ and}\\\
&(\gamma_{R,m}+\widetilde{f}(\beta_{R,m},y)+x)^{p}-\gamma_{R,m}^{p}-\widetilde{f}(\beta_{R,m},y)^{p}-x^{p}\in\mathfrak{p}[x,y].\end{aligned}$
(3.2)

For $r\in\mathbb{Q}_{\geq 0}$ and $f,g\in\overline{F}$, we write $f\equiv
g\mod r+$ if $v(f-g)>r$. For a local field $K$ contained in $\overline{F}$,
let $W_{K}$ be the Weil group of $\overline{F}/K$. For $\sigma\in W_{K}$, let
$n_{\sigma}\in\mathbb{Z}$ be the integer such that $\sigma(x)\equiv
x^{q^{-n_{\sigma}}}\mod 0+$ for $x\in\mathcal{O}_{\overline{F}}$. Let
$v_{K}(\cdot)$ denote the normalized valuation on $K$.

######  Definition 3.3.

For $\sigma\in W_{F}$, we set

$\displaystyle\begin{aligned}
a_{R,\sigma}&:=\sigma(\alpha_{R})/\alpha_{R}\in\mu_{d_{R}}(\overline{F}),\quad
b_{R,\sigma}:=a_{R,\sigma}^{m}\sigma(\beta_{R,m})-\beta_{R,m},\\\
c_{R,\sigma}&:=\sigma(\gamma_{R,m})-\gamma_{R,m}-\widetilde{f}(\beta_{R,m},b_{R,\sigma}).\end{aligned}$
(3.3)

In the following, we simply write $a_{\sigma},b_{\sigma},c_{\sigma}$ for
$a_{R,\sigma},b_{R,\sigma},c_{R,\sigma}$, respectively.

For an element $x\in\mathcal{O}$, let $\bar{x}$ denote the image of $x$ by the
map $\mathcal{O}\to\mathcal{O}/\mathfrak{p}$. In the following proofs, for
simplicity, we often write $\alpha$, $\beta$ and $\gamma$ for $\alpha_{R}$,
$\beta_{R,m}$ and $\gamma_{R,m}$, respectively.

######  Lemma 3.4.

We have $b_{\sigma},c_{\sigma}\in\mathcal{O}$, $E_{R}(\bar{b}_{\sigma})=0$ and
$\bar{c}_{\sigma}^{p}-\bar{c}_{\sigma}=\bar{b}_{\sigma}R(\bar{b}_{\sigma})$.

###### Proof.

Using (3.1), the equality $\widetilde{E}_{R}(\beta)=\alpha^{-m}$ in Definition
3.1 and (3.3),

$\widetilde{E}_{R}(\beta+b_{\sigma})=\widetilde{E}_{R}(a_{\sigma}^{m}\sigma(\beta))=a_{\sigma}^{m}\widetilde{E}_{R}(\sigma(\beta))=a_{\sigma}^{m}\sigma(\alpha)^{-m}=\alpha^{-m}=\widetilde{E}_{R}(\beta).$

Using $v(\beta)<0$ in Remark 3.2 and (3.2), we have
$\Delta(x):=\widetilde{E}_{R}(\beta+x)-\widetilde{E}_{R}(\beta)-\widetilde{E}_{R}(x)\in\mathfrak{p}[x]$.
By letting $x=b_{\sigma}$ and applying the previous relationship, we deduce
that $\widetilde{E}_{R}(b_{\sigma})+\Delta(b_{\sigma})=0$. Hence
$b_{\sigma}\in\mathcal{O}$ and $E_{R}(\bar{b}_{\sigma})=0$.

By (3.2), we have

$\beta\widetilde{R}(\beta+b_{\sigma})\equiv\beta\widetilde{R}(\beta)+\beta\widetilde{R}(b_{\sigma}),\quad\widetilde{f}(\beta,b_{\sigma})^{p}-\widetilde{f}(\beta,b_{\sigma})\equiv
b_{\sigma}\widetilde{R}(\beta)+\beta\widetilde{R}(b_{\sigma})\mod 0+.$ (3.4)

Substituting $y=b_{\sigma}\in\mathcal{O}$ to (3.2), we obtain

$\Delta_{1}(x):=(\gamma+\widetilde{f}(\beta,b_{\sigma})+x)^{p}-\gamma^{p}-\widetilde{f}(\beta,b_{\sigma})^{p}-x^{p}\in\mathfrak{p}[x].$

We have
$\sigma(\beta)\widetilde{R}(\sigma(\beta))=(\beta+b_{\sigma})\widetilde{R}(\beta+b_{\sigma})$
by substituting (3.3) and using (3.1). By multiplying the first congruence in
(3.4) by $b_{\sigma}\beta^{-1}$, we obtain
$b_{\sigma}\widetilde{R}(\beta+b_{\sigma})\equiv
b_{\sigma}\widetilde{R}(\beta)+b_{\sigma}\widetilde{R}(b_{\sigma})\mod 0+$.
Hence, we compute

$\displaystyle\sigma(\gamma)^{p}-\sigma(\gamma)$
$\displaystyle=\sigma(\beta)\widetilde{R}(\sigma(\beta))=(\beta+b_{\sigma})\widetilde{R}(\beta+b_{\sigma})$
$\displaystyle\equiv\beta\widetilde{R}(\beta)+b_{\sigma}\widetilde{R}(\beta)+\beta\widetilde{R}(b_{\sigma})+b_{\sigma}\widetilde{R}(b_{\sigma})$
$\displaystyle\equiv\gamma^{p}-\gamma+\widetilde{f}(\beta,b_{\sigma})^{p}-\widetilde{f}(\beta,b_{\sigma})+b_{\sigma}\widetilde{R}(b_{\sigma})$
$\displaystyle\equiv\sigma(\gamma)^{p}-\sigma(\gamma)-(c_{\sigma}^{p}-c_{\sigma}+\Delta_{1}(c_{\sigma}))+b_{\sigma}\widetilde{R}(b_{\sigma})\mod
0+,$

where we have used (3.3) for the last congruence. Hence we obtain
$c_{\sigma}^{p}-c_{\sigma}+\Delta_{1}(c_{\sigma})\equiv
b_{\sigma}\widetilde{R}(b_{\sigma})\mod 0+$. By $b_{\sigma}\in\mathcal{O}$, we
have $c_{\sigma}\in\mathcal{O}$ and
$\bar{c}_{\sigma}^{p}-\bar{c}_{\sigma}=\bar{b}_{\sigma}R(\bar{b}_{\sigma})$. ∎

Assume that

$\displaystyle\begin{aligned}
&(x+\beta_{R,m})^{p^{i}}-x^{p^{i}}-\beta_{R,m}^{p^{i}}\in\mathfrak{p}[x]\quad\textrm{for
$1\leq i\leq e-1$},\\\
&\widetilde{f}(\beta_{R,m},x+y)-\widetilde{f}(\beta_{R,m},x)-\widetilde{f}(\beta_{R,m},y)\in\mathfrak{p}[x,y],\\\
\end{aligned}$ (3.5)

Let

$\Theta_{R,m,\varpi}\colon W_{F}\to Q_{R}\rtimes\mathbb{Z};\
\sigma\mapsto((\bar{a}_{\sigma}^{m},\bar{b}_{\sigma},\bar{c}_{\sigma}),n_{\sigma}).$
(3.6)

######  Lemma 3.5.

The map $\Theta_{R,m,\varpi}$ is a homomorphism.

###### Proof.

Let $\sigma,\sigma^{\prime}\in W_{F}$. Recall that $\sigma(x)\equiv
x^{q^{-n_{\sigma}}}\mod 0+$ for $x\in\mathcal{O}_{\overline{F}}$. Using
Definition 2.3(2), we reduce to checking that

$\bar{a}_{\sigma\sigma^{\prime}}=\bar{a}_{\sigma}\bar{a}_{\sigma^{\prime}}^{q^{-n_{\sigma}}},\quad\bar{b}_{\sigma\sigma^{\prime}}=\bar{a}_{\sigma}^{m}\bar{b}_{\sigma^{\prime}}^{q^{-n_{\sigma}}}+\bar{b}_{\sigma},\quad\bar{c}_{\sigma\sigma^{\prime}}=\bar{c}_{\sigma}+\bar{c}_{\sigma^{\prime}}^{q^{-n_{\sigma}}}+f_{R}(\bar{b}_{\sigma},\bar{a}_{\sigma}^{m}\bar{b}_{\sigma^{\prime}}^{-n_{\sigma}}).$
(3.7)

We easily check that
$a_{\sigma\sigma^{\prime}}=\sigma(a_{\sigma^{\prime}})a_{\sigma}$ and
$b_{\sigma\sigma^{\prime}}=a_{\sigma}^{m}\sigma(b_{\sigma^{\prime}})+b_{\sigma}$.
Hence the first equalities in (3.7) follow. We compute

$\displaystyle c_{\sigma\sigma^{\prime}}$
$\displaystyle=c_{\sigma}+\sigma(c_{\sigma^{\prime}})+\sigma(\widetilde{f}(\beta,b_{\sigma^{\prime}}))+\widetilde{f}(\beta,b_{\sigma})-\widetilde{f}(\beta,b_{\sigma\sigma^{\prime}})$
$\displaystyle\equiv
c_{\sigma}+\sigma(c_{\sigma^{\prime}})+\sigma(\widetilde{f}(\beta,b_{\sigma^{\prime}}))-\widetilde{f}(\beta,a_{\sigma}^{m}\sigma(b_{\sigma^{\prime}}))\mod
0+,$

where we use the second condition in (3.5) for the second congruence. We have

$\displaystyle\sigma(\widetilde{f}(\beta,b_{\sigma^{\prime}}))$
$\displaystyle=-\sum_{i=0}^{e-1}\sum_{j=0}^{e-i-1}(\widetilde{a}_{j}\sigma(b_{\sigma^{\prime}})\sigma(\beta)^{p^{i}})^{p^{j}}-\sum_{i=0}^{e-1}(\sigma(\beta)\widetilde{R}(\sigma(b_{\sigma^{\prime}})))^{p^{i}}$
$\displaystyle\equiv\widetilde{f}(b_{\sigma},a_{\sigma}^{m}\sigma(b_{\sigma^{\prime}}))+\widetilde{f}(\beta,a_{\sigma}^{m}\sigma(b_{\sigma^{\prime}}))\mod
0+,$

where we substitute $\sigma(\beta)=a_{\sigma}^{-m}(\beta+b_{\sigma})$, (3.5)
and (3.1) for the second congruence. The last equality in (3.7) follows from
$\overline{\widetilde{f}(b_{\sigma},a_{\sigma}^{m}\sigma(b_{\sigma^{\prime}}))}=f_{R}(\bar{b}_{\sigma},\bar{a}_{\sigma}^{m}\bar{b}_{\sigma^{\prime}}^{q^{-n_{\sigma}}})$,
since $\widetilde{f}(x,y)$ is a lift of $f_{R}(x,y)$ to
$\mathcal{O}_{F}[x,y]$. ∎

######  Lemma 3.6.

If $v(p)$ is large enough, the conditions (3.2) and (3.5) are satisfied.

###### Proof.

There exists $s\in\mathbb{Z}_{\geq 1}$ such that the coefficients of all
polynomials in (3.2) and (3.5) have the form: $p\cdot\beta_{R,m}^{s}\cdot a$
with $a\in\mathcal{O}_{\overline{F}}$ by Remark 3.2. Since the valuation of
$v(\beta_{R,m})$ is independent of $F$, the claim follows. ∎

In the sequel, we assume that the conditions (3.2) and (3.5) are satisfied.
Let $F^{\rm ur}$ denote the maximal unramified extension of $F$ in
$\overline{F}$.

######  Lemma 3.7.

The extension $F^{\rm ur}(\alpha_{R}^{m},\beta_{R,m},\gamma_{R,m})/F$ is
Galois.

###### Proof.

Let $L_{0}:=F^{\rm ur}(\alpha^{m},\beta,\gamma)$ and $L:=\widehat{L}_{0}$ be
the completion of $L_{0}$. Let $\sigma\in G_{F}$. We note that
$a_{\sigma}\in\mu_{d_{R}}(\overline{F})\subset F^{\rm ur}$ by $p\nmid d_{R}$.
Hence $\sigma(\alpha^{m})=a_{\sigma}^{m}\alpha^{m}\in L_{0}$. We show
$\sigma(\beta),\sigma(\gamma)\in L_{0}$. It suffices to prove

$b_{\sigma},c_{\sigma}\in L_{0},$

since

$\sigma(\beta)=\frac{b_{\sigma}+\beta}{a_{\sigma}^{m}},\quad\sigma(\gamma)=\gamma+c_{\sigma}+\widetilde{f}(\beta,b_{\sigma})$

by (3.3). As in the proof of Lemma 3.4, we have
$(\widetilde{E}_{R}+\Delta)(b_{\sigma})=0$,
$E(x):=(\widetilde{E}_{R}+\Delta)(x)\in\mathcal{O}_{L_{0}}[x]$ and $\deg
E(x)=p^{2e}$. The equation $E(x)\equiv 0\mod 0+$ has $p^{2e}$ different roots.
Hence by Hensel’s lemma, $E(x)=0$ has $p^{2e}$ different roots in
$\mathcal{O}_{L}$. Hence we have $b_{\sigma}\in L\cap\overline{F}=L_{0}$. As
in the proof of Lemma 3.4, we have

$f(c_{\sigma}):=c_{\sigma}^{p}-c_{\sigma}+\Delta_{1}(c_{\sigma})-y=0\
\textrm{with $y\in\mathcal{O}_{L_{0}}$},$

where $f(x)\in\mathcal{O}_{L_{0}}[x]$ with $\deg f(x)=p$. We have $y\equiv
b_{\sigma}\widetilde{R}(b_{\sigma})\mod 0+$. The equation $f(x)\equiv
x^{p}-x-y\equiv x^{p}-x-b_{\sigma}\widetilde{R}(b_{\sigma})\equiv 0\mod 0+$
has $p$ different roots. By Hensel’s lemma, $f(x)=0$ has $p$ different roots
in $\mathcal{O}_{L}$. Hence we have $c_{\sigma}\in L\cap\overline{F}=L_{0}$. ∎

######  Definition 3.8.

Let

$d_{R,m}:=\frac{d_{R}}{\mathrm{gcd}(d_{R},m)},\quad
Q_{R,m}:=\\{(\alpha,\beta,\gamma)\in Q_{R}\mid\alpha\in\mu_{d_{R,m}}\\}.$

######  Lemma 3.9.

We have the isomorphism

$W(F^{\rm
ur}(\alpha_{R}^{m},\beta_{R,m},\gamma_{R,m})/F)\xrightarrow{\sim}Q_{R,m}\rtimes\mathbb{Z};\
\sigma\mapsto((\bar{a}_{\sigma}^{m},\bar{b}_{\sigma},\bar{c}_{\sigma}),n_{\sigma}).$
(3.8)

###### Proof.

We use the notation in the proof of Lemma 3.4. Let

$I:=\operatorname{Gal}(F^{\rm ur}(\alpha^{m},\beta,\gamma)/F^{\rm ur})$

and $\Theta\colon I\to Q_{R,m}$ be the restriction of (3.8). First, we show
that $\Theta$ is injective. Assume $\Theta(\sigma)=1$ for $\sigma\in I$. We
will show $\sigma=1$. By the assumption, $\bar{a}_{\sigma}^{m}=1$,
$\bar{b}_{\sigma}=0$ and $\bar{c}_{\sigma}=0$. We have a natural isomorphism
$\mu_{r}(\overline{F})\xrightarrow{\sim}\mu_{r}$ for an integer $r$ prime to
$p$. By $\bar{a}_{\sigma}^{m}=1$ and $a_{\sigma}\in\mu_{d_{R}}(\overline{F})$
in (3.3), we have $a_{\sigma}^{m}=1$ and $\sigma(\alpha^{m})=\alpha^{m}$. We
recall the equality $\widetilde{E}_{R}(b_{\sigma})+\Delta(b_{\sigma})=0$ in
the proof of Lemma 3.4, where $\Delta(x)\in\mathfrak{p}[x]$ has no constant
coefficient. We have $v(b_{\sigma})>0$ by $\bar{b}_{\sigma}=0$. Assume that
$b_{\sigma}\neq 0$. Then
$v(b_{\sigma})=v(\widetilde{E}_{R}(b_{\sigma})+\Delta(b_{\sigma}))$ by
$E^{\prime}_{R}(0)\neq 0$ and $v(b_{\sigma})>0$. This implies that
$v(b_{\sigma})=\infty$, which is a contradiction. Hence $b_{\sigma}=0$ and
$\sigma(\beta)=\beta$. By the last condition in (3.2) with $y=0$,

$\Lambda(x):=(\gamma+x)^{p}-\gamma^{p}-x^{p}\in\mathfrak{p}[x].$

We have $\sigma(\gamma)^{p}-\sigma(\gamma)=\gamma^{p}-\gamma$ by Definition
3.1. Hence $(\gamma+c_{\sigma})^{p}-\gamma^{p}=c_{\sigma}$ and
$c_{\sigma}^{p}+\Lambda(c_{\sigma})=c_{\sigma}$. Since
$\Lambda(x)\in\mathfrak{p}[x]$ has no constant coefficient, if
$0<v(c_{\sigma})<\infty$, we have
$v(c_{\sigma}^{p}+\Lambda(c_{\sigma}))>v(c_{\sigma})$, which can not occur.
Hence $c_{\sigma}=0$ and $\sigma(\gamma)=\gamma$. We obtain $\sigma=1$. Hence
$\Theta$ is injective. We easily check that $F^{\rm
ur}(\alpha^{m},\beta,\gamma)/F^{\rm ur}$ is a totally ramified extension of
degree $d_{R,m}p^{2e+1}$. Hence $\Theta$ is an isomorphism. By the snake
lemma, (3.8) is an isomorphism. ∎

###  3.2 Galois representations associated to additive polynomials

####  3.2.1 Construction of Galois representation

We assume that (3.2) and (3.5) are satisfied. If the characteristic of $F$ is
positive, these are unconditional. If the characteristic of $F$ is zero, these
conditions are satisfied if the absolute ramification index of $F$ is large
enough as in Lemma 3.6.

######  Definition 3.10.

Let $\psi\in\mathbb{F}_{p}^{\vee}\setminus\\{1\\}$. We define
$\tau_{\psi,R,m,\varpi}$ to be the $W_{F}$-representation which is the
inflation of the irreducible $Q_{R}\rtimes\mathbb{Z}$ -representation $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ by
$\Theta_{R,m,\varpi}$ in (3.6). For simplicity, we write $\tau_{\psi,R,m}$ for
$\tau_{\psi,R,m,\varpi}$.

For a non-archimedean local field $K$, let $I_{K}$ denote the inertia subgroup
of $K$. Then $\operatorname{Ker}\tau_{\psi,R,m}$ contains the open compact
subgroup $I_{F(\alpha_{R}^{m},\beta_{R,m},\gamma_{R,m})}$. Hence the
representation $\tau_{\psi,R,m}$ is a smooth irreducible representation of
$W_{F}$ by Lemma 2.8(1).

Let $G_{F}:=\operatorname{Gal}(\overline{F}/F)$. We consider a general setting
in the following lemma.

######  Lemma 3.11.

Let $\widetilde{\tau}$ be a continuous representation of $G_{F}$ over
$\overline{\mathbb{Q}}_{\ell}$ such that there exists an unramified continuous
character $\phi$ of $G_{F}$ such that $(\widetilde{\tau}\otimes\phi)(G_{F})$
is finite. Assume that $\tau:=\widetilde{\tau}|_{W_{F}}$ is irreducible. Then
$\widetilde{\tau}\otimes\phi$ is primitive if and only if $\tau$ is primitive.

###### Proof.

Let $\widetilde{\tau}^{\prime}:=\widetilde{\tau}\otimes\phi$ and
$\tau^{\prime}:=\tau\otimes\phi|_{W_{F}}$. The subgroup
$\operatorname{Ker}\widetilde{\tau}^{\prime}$ is open by
$|G_{F}/\operatorname{Ker}\widetilde{\tau}^{\prime}|<\infty$. Hence
$\operatorname{Ker}\tau^{\prime}\subset W_{F}$ is open. Therefore
$\tau^{\prime}$ is smooth. Hence so is $\tau$. Since $\tau$ is irreducible and
smooth, we have $\dim\tau<\infty$. We will show that
$\widetilde{\tau}^{\prime}$ is imprimitive if and only if $\tau$ is
imprimitive.

First, assume an isomorphism
$\widetilde{\tau}^{\prime}\simeq\operatorname{Ind}_{H}^{G_{F}}\eta^{\prime}$
with a proper subgroup $H$. We can check
$\operatorname{Ker}\widetilde{\tau}^{\prime}\subset H$. Hence $H$ is open.
Hence we can write $H=G_{F^{\prime}}$ with a finite extension $F^{\prime}/F$.
Hence we obtain an isomorphism
$\tau\simeq\operatorname{Ind}_{W_{F^{\prime}}}^{W_{F}}(\eta^{\prime}|_{W_{F^{\prime}}}\otimes\phi^{-1}|_{W_{F^{\prime}}})$.

To the contrary, assume $\tau\simeq\operatorname{Ind}_{H}^{W_{F}}\sigma$. In
the same manner as above with replacing $G_{F}$ by $W_{F}$, the subgroup $H$
is an open subgroup of $W_{F}$ of finite index by $\dim\tau<\infty$. Hence we
can write $H=W_{F^{\prime}}$ with a finite extension $F^{\prime}/F$. Let
$\sigma^{\prime}:=\sigma\otimes\phi|_{W_{F^{\prime}}}$. We have
$\tau^{\prime}\simeq\operatorname{Ind}_{W_{F^{\prime}}}^{W_{F}}\sigma^{\prime}$.
Frobenius reciprocity implies that
$\sigma^{\prime}(W_{F^{\prime}})\subset\tau^{\prime}(W_{F})$. By the
assumption, the image $\sigma^{\prime}(W_{F^{\prime}})$ is finite. Hence the
smooth $W_{F^{\prime}}$-representation $\sigma^{\prime}$ extends to a smooth
representation of $G_{F^{\prime}}$, for which we write $\widetilde{\sigma}$
([2, Proposition 28.6]). The restriction of
$\operatorname{Ind}_{G_{F^{\prime}}}^{G_{F}}\widetilde{\sigma}$ to $W_{F}$ is
isomorphic to
$\operatorname{Ind}_{W_{F^{\prime}}}^{W_{F}}\sigma^{\prime}\simeq\tau^{\prime}$.
Both of $\operatorname{Ind}_{G_{F^{\prime}}}^{G_{F}}\widetilde{\sigma}$ and
$\widetilde{\tau}^{\prime}$ are smooth irreducible $G_{F}$-representations
whose restrictions to $W_{F}$ are isomorphic to $\tau^{\prime}$. Hence we
obtain an isomorphism
$\widetilde{\tau}^{\prime}\simeq\operatorname{Ind}_{G_{F^{\prime}}}^{G_{F}}\widetilde{\sigma}$
as $G_{F}$-representations by [2, Lemma 28.6.2(2)]. ∎

######  Lemma 3.12.

The eigenvalues of $\mathrm{Fr}_{q}^{\ast}$ on $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ have the forms
$\zeta\sqrt{q}$ with roots of unity $\zeta$. The automorphism
$\mathrm{Fr}_{q}^{\ast}$ is semi-simple.

###### Proof.

The claims follow from Proposition 2.10. ∎

The cohomology group $H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ is regarded as a
representation of $Q_{R,m}\rtimes\widehat{\mathbb{Z}}$. By inflating this by a
natural map $G_{F}\to Q_{R,m}\rtimes\widehat{\mathbb{Z}}$ extending
$\Theta_{R,m,\varpi}$, we obtain a continuous representation of $G_{F}$. We
denote this representation by $\widetilde{\tau}_{\psi,R,m}$. Let $\phi\colon
G_{F}\to\overline{\mathbb{Q}}_{\ell}^{\times}$ be the unramified character
sending a geometric Frobenius to $\sqrt{q}^{-1}$. The image of $G_{F}$ by the
twist $\widetilde{\tau}^{\prime}:=\widetilde{\tau}_{\psi,R,m}\otimes\phi$ is
finite by Lemma 3.12. By fixing an isomorphism
$\overline{\mathbb{Q}}_{\ell}\simeq\mathbb{C}$, we obtain a continuous
representation $\widetilde{\tau}^{\prime}_{\mathbb{C}}$ of $G_{F}$ over
$\mathbb{C}$ by $\widetilde{\tau}^{\prime}$. Then
$\widetilde{\tau}^{\prime}_{\mathbb{C}}$ is primitive if and only if
$\widetilde{\tau}_{\psi,R,m}$ is primitive.

######  Corollary 3.13.

The $W_{F}$-representation $\tau_{\psi,R,m}$ is primitive if and only if the
continuous $G_{F}$-representation $\widetilde{\tau}^{\prime}_{\mathbb{C}}$ is
primitive.

###### Proof.

Clearly $\widetilde{\tau}^{\prime}_{\mathbb{C}}$ is primitive if and only if
$\widetilde{\tau}^{\prime}$ is primitive. We obtain the claim by applying
Lemma 3.11 with $\widetilde{\tau}=\widetilde{\tau}_{\psi,R,m}$ and
$\tau=\tau_{\psi,R,m}$. ∎

####  3.2.2 Swan conductor exponent

In the sequel, we compute the Swan conductor exponent
$\mathrm{Sw}(\tau_{\psi,R,m})$.

We simply write $\alpha,\beta,\gamma$ for
$\alpha_{R},\beta_{R,m},\gamma_{R,m}$ in Definition 3.1, respectively. We
consider the unramified field extension $F_{r}/F$ of degree $r$ such that
$N:=F_{r}(\alpha,\beta,\gamma)$ is Galois over $F$. Let $T:=F_{r}(\alpha)$ and
$M:=T(\beta)$. Then we have

$F\subset F_{r}\subset T\subset M\subset N.$

Let $L/K$ be a Galois extension of non-archimedean local fields with Galois
group $G$. Let $\left\\{G^{i}\right\\}_{i\geq-1}$ denote the upper numbering
ramification groups of $G$ in [15, IV§3]. Let $\psi_{L/K}$ denote the Herbrand
function of $L/K$.

######  Lemma 3.14.

Let $G:=\operatorname{Gal}(N/F)$. Then we have

$\psi_{N/F}(t)=\begin{cases}t&\textrm{if $t\leq 0$},\\\ d_{R}t&\textrm{if
$0<t\leq\frac{m}{d_{R}}$},\\\ p^{2e}d_{R}t-(p^{2e}-1)m&\textrm{if
$\frac{m}{d_{R}}<t\leq\frac{p^{e}+1}{p^{e}}\frac{m}{d_{R}}$},\\\
p^{2e+1}d_{R}t-(p^{e}+1)(p^{e+1}-1)m&\textrm{otherwise}\end{cases}$

and

$G^{i}=\begin{cases}G&\textrm{if $i=-1$},\\\
\operatorname{Gal}(N/F_{r})&\textrm{if $-1<i\leq 0$},\\\
\operatorname{Gal}(N/T)&\textrm{if $0<i\leq\frac{m}{d_{R}}$},\\\
\operatorname{Gal}(N/M)&\textrm{if
$\frac{m}{d_{R}}<i\leq\frac{p^{e}+1}{p^{e}}\frac{m}{d_{R}}$},\\\
\\{1\\}&\textrm{otherwise}.\end{cases}$

###### Proof.

We easily have

$\psi_{T/F}(t)=\begin{cases}t&\textrm{if $t\leq 0$},\\\
d_{R}t&\textrm{otherwise}.\end{cases}$

For a finite Galois extension $L/K$, let
$\\{\operatorname{Gal}(L/K)_{u}\\}_{u\geq-1}$ be the lower numbering
ramification subgroups. Let $1\neq\sigma\in\operatorname{Gal}(M/T)$. Let
$b_{\sigma}=\sigma(\beta)-\beta$ as before. We have
$\widetilde{E}_{R}(\beta+b_{\sigma})=\widetilde{E}_{R}(\beta)$ by the proof of
Lemma 3.4. If $v(b_{\sigma})>0$, we obtain $b_{\sigma}=0$ by the same argument
in the proof of Lemma 3.9. This implies that $\sigma=1$. By (3.2), we have
$v(b_{\sigma})=0$. By $v_{M}(\beta)=-m$, we have
$v_{M}(\sigma(\varpi_{M})-\varpi_{M})=m+1$. Hence

$\operatorname{Gal}(M/T)_{u}=\begin{cases}\operatorname{Gal}(M/T)&\textrm{if
$u\leq m$},\\\
\\{1\\}&\textrm{otherwise},\end{cases}\quad\psi_{M/T}(t)=\begin{cases}t&\textrm{if
$t\leq m$},\\\ p^{2e}t-(p^{2e}-1)m&\textrm{otherwise}.\end{cases}$

Let $1\neq\sigma\in\mathrm{Gal}(N/M)$. If $v_{N}(\sigma(\gamma)-\gamma)>0$, we
obtain $\sigma(\gamma)=\gamma$ in the same way as the proof of Lemma 3.9. This
implies that $\sigma=1$. Hence $v_{N}(\sigma(\gamma)-\gamma)=0$. Let
$\varpi_{N}$ be a uniformizer of $N$. By $v_{N}(\gamma^{-1})=(p^{e}+1)m$, we
have $v_{N}(\sigma(\varpi_{N})-\varpi_{N})=(p^{e}+1)m+1$. Thus

$\displaystyle\operatorname{Gal}(N/M)_{u}$
$\displaystyle=\begin{cases}\operatorname{Gal}(N/M)&\textrm{if
$u\leq(p^{e}+1)m$},\\\ \\{1\\}&\textrm{otherwise},\end{cases}$
$\displaystyle\psi_{N/M}(t)$ $\displaystyle=\begin{cases}t&\textrm{if
$t\leq(p^{e}+1)m$},\\\ pt-(p-1)(p^{e}+1)m&\textrm{otherwise}.\end{cases}$

Hence the former claim follows from
$\psi_{N/F}=\psi_{N/M}\circ\psi_{M/T}\circ\psi_{T/F}$.

We can check

$\displaystyle G_{u}=\begin{cases}G&\textrm{if $u=-1$},\\\
\operatorname{Gal}(N/F_{r})&\textrm{if $-1<u\leq 0$},\\\
\operatorname{Gal}(N/T)&\textrm{if $0<u\leq m$},\\\
\mathrm{Gal}(N/M)&\textrm{if $m<u\leq(p^{e}+1)m$},\\\
\\{1\\}&\textrm{otherwise}\end{cases}$

by using the former claim and [15, Propositions 12(c), 13(c) and 15 in IV§3].
Hence the latter claim follows from $G^{i}=G_{\psi_{N/F}(i)}$. ∎

######  Corollary 3.15.

We have $\mathrm{Sw}(\tau_{\psi,R,m})=m(p^{e}+1)/d_{R}$.

###### Proof.

Recall that the twist $\tau_{\psi,R,m}\otimes\phi$ factors through a finite
group
$Q_{R}\rtimes(\mathbb{Z}/r\mathbb{Z})\simeq\operatorname{Gal}(F_{r}(\alpha,\beta,\gamma)/F)$
with a certain integer $r$. Since $\phi$ is unramified, we have
$\mathrm{Sw}(\tau_{\psi,R,m})=\mathrm{Sw}(\tau_{\psi,R,m}\otimes\phi)$. We
have $\mathrm{Sw}(\tau_{\psi,R,m}\otimes\phi)=m(p^{e}+1)/d_{R}$ by Lemma 3.14
and [7, Théorème 7.7] ([15, Exercise 2 in §2VI]). ∎

###  3.3 Symplectic module associated to Galois representation

Let $\rho\colon W_{F}\to\mathrm{PGL}_{p^{e}}(\overline{\mathbb{Q}}_{\ell})$
denote the composite of $\tau_{\psi,R,m}\colon
W_{F}\to\mathrm{GL}_{p^{e}}(\overline{\mathbb{Q}}_{\ell})$ with the natural
map
$\mathrm{GL}_{p^{e}}(\overline{\mathbb{Q}}_{\ell})\to\mathrm{PGL}_{p^{e}}(\overline{\mathbb{Q}}_{\ell})$.

Let $\rho^{\prime}$ be the projective representation associated to
$\widetilde{\tau}^{\prime}=\widetilde{\tau}_{\psi,R,m}\otimes\phi$.

######  Lemma 3.16.

We have $\rho(W_{F})=\rho^{\prime}(G_{F})$, which is finite.

###### Proof.

Since $\widetilde{\tau}^{\prime}$ is a smooth irreducible
$G_{F}$-representation, we have
$\widetilde{\tau}^{\prime}(G_{F})=(\tau_{\psi,R,m}\otimes\phi)(W_{F})$ ([2,
the proof of Lemma 2 in 28.6]). This implies the claim. ∎

Let $F_{\rho}$ denote the kernel field of $\rho$ and $T_{\rho}$ the maximal
tamely ramified extension of $F$ in $F_{\rho}$. Let

$H:=\mathrm{Gal}(F_{\rho}/T_{\rho})\subset G:=\mathrm{Gal}(F_{\rho}/F).$

The homomorphism $\rho$ induces an injection $\bar{\rho}\colon
G\to\mathrm{PGL}_{p^{e}}(\overline{\mathbb{Q}}_{\ell})$. Let $V_{R}$ be as in
Lemma 2.6.

Let $\tau$ denote the composite

$Q_{R,m}\rtimes\mathbb{Z}\to\mathrm{Aut}_{\overline{\mathbb{Q}}_{\ell}}(H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi])\to\mathrm{Aut}_{\overline{\mathbb{Q}}_{\ell}}(H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi])/\overline{\mathbb{Q}}_{\ell}^{\times}.$

######  Lemma 3.17.

We have an isomorphism $\bar{\rho}(H)\simeq V_{R}$.

###### Proof.

Let $L:=F^{\rm ur}(\alpha_{R}^{m},\beta_{R,m},\gamma_{R,m})$ and $K:=F^{\rm
ur}(\alpha_{R}^{m})$. By (3.8), we have

$W(L/F)\simeq Q_{R,m}\rtimes\mathbb{Z},\quad W(L/K)\simeq H_{R}.$

The subfield $K$ is the maximal tamely ramified extension of $F$ in $L$. We
have $F_{\rho}\subset L$ and $T_{\rho}=F_{\rho}\cap K$. We have isomorphisms
$G=W(F_{\rho}/F)\simeq
W_{F}/\operatorname{Ker}\rho\simeq(Q_{R,m}\rtimes\mathbb{Z})/\operatorname{Ker}\tau$
and $H=\mathrm{Gal}(F_{\rho}/T_{\rho})\simeq
H_{R}/(H_{R}\cap\operatorname{Ker}\tau)$. By Lemma 2.9, we have
$H_{R}\cap\operatorname{Ker}\tau=Z(H_{R})$. Hence the claim follows from the
isomorphism $V_{R}\xrightarrow{\sim}H_{R}/Z(H_{R});\ \beta\mapsto(1,\beta,0)$.
∎

Let

$\mathscr{H}_{0}:=G/H=\mathrm{Gal}(T_{\rho}/F).$

Let $\sigma\in\mathscr{H}_{0}$. We take a lifting $\widetilde{\sigma}\in
G\twoheadrightarrow\mathscr{H}_{0}$ of $\sigma$. Let $\mathscr{H}_{0}$ act on
$H$ by
$\sigma\cdot\sigma^{\prime}:=\widetilde{\sigma}\sigma^{\prime}\widetilde{\sigma}^{-1}$
for $\sigma^{\prime}\in H$. This is well-defined because $H$ is abelian by
Lemma 3.17. We regard $H\simeq V_{R}$ as an
$\mathbb{F}_{p}[\mathscr{H}_{0}]$-module.

By Lemma 3.12, we can take a positive integer $r$ such that
$r\mathbb{Z}\subset\operatorname{Ker}\tau$ and $x^{q^{r}}=x$ for
$x\in\mu_{d_{R,m}}$. Let $\mathbb{Z}/r\mathbb{Z}$ act on $\mu_{d_{R,m}}$ by
$1\cdot x=x^{q^{-1}}$. We take a generator $\alpha\in\mu_{d_{R,m}}$. Let

$\mathscr{H}:=\mu_{d_{R,m}}\rtimes(\mathbb{Z}/r\mathbb{Z})\xrightarrow{\sim}\left\langle\sigma,\tau\mid\sigma^{r}=1,\
\tau^{d_{R,m}}=1,\ \sigma\tau\sigma^{-1}=\tau^{q}\right\rangle,$ (3.9)

where the isomorphism is given by $(\alpha,0)\mapsto\tau$ and
$(1,-1)\mapsto\sigma$. The groups $\mathscr{H}_{0}$ and $\mathscr{H}$ are
supersolvable. We consider the commutative diagram

$\textstyle{Q_{R,m}\rtimes\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Q_{R,m}\rtimes\mathbb{Z})/\operatorname{Ker}\tau\simeq
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{H}\simeq(Q_{R,m}\rtimes\mathbb{Z})/(H_{R}\rtimes
r\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{(Q_{R,m}\rtimes\mathbb{Z})/(\operatorname{Ker}\tau\cdot
H_{R})\simeq\mathscr{H}_{0},}$

where every map is canonical and surjective.

######  Lemma 3.18.

The elements $\varphi(\alpha,0)$ and $\varphi(1,-1)$ in $\mathscr{H}_{0}$ act
on $H\simeq V_{R}$ by $x\mapsto\alpha x$ and $x\mapsto x^{q}$ for $x\in
V_{R}$, respectively.

###### Proof.

These are directly checked. ∎

We can regard $V_{R}$ as an $\mathbb{F}_{p}[\mathscr{H}]$-module via
$\varphi$. Let $\omega_{R}$ be as in Lemma 2.6(2).

######  Lemma 3.19.

We have $\omega_{R}(hx,hx^{\prime})=\omega_{R}(x,x^{\prime})$ for
$h\in\mathscr{H}$.

###### Proof.

The claim for $h=\alpha$ follows from (2.4). For $h=(1,-1)$, the claim follows
from $\omega_{R}(x^{q},x^{\prime
q})=(f_{R}(x,x^{\prime})-f_{R}(x^{\prime},x))^{q}=f_{R}(x,x^{\prime})-f_{R}(x^{\prime},x)=\omega_{R}(x,x^{\prime})$.
∎

######  Definition 3.20.

Let $G$ be a finite group. Let $V$ be an $\mathbb{F}_{p}[G]$-module with
$\dim_{\mathbb{F}_{p}}V<\infty$. Let $\omega\colon V\times V\to\mathbb{F}_{p}$
be a symplectic form. We say that the pair $(V,\omega)$ is symplectic if
$\omega$ is non-degenerate and satisfies
$\omega(gv,gv^{\prime})=\omega(v,v^{\prime})$ for $g\in G$ and
$v,v^{\prime}\in V$.

######  Lemma 3.21.

The $\mathbb{F}_{p}[\mathscr{H}]$-module $(V_{R},\omega_{R})$ is symplectic.

###### Proof.

The claim follows from Lemma 2.6(2) and Lemma 3.19. ∎

######  Definition 3.22.

The $\mathbb{F}_{p}[\mathscr{H}_{0}]$-module $(V_{R},\omega_{R})$ is called a
symplectic module associated to $\tau_{\psi,R,m}$.

######  Definition 3.23.

Let $\sigma\colon\mathbb{F}\to\mathbb{F};\ x\mapsto x^{q}$. For
$f(x)=\sum_{i=0}^{n}a_{i}x^{i}\in\mathbb{F}[x]$, we set
$f^{\sigma}(x):=\sum_{i=0}^{n}\sigma(a_{i})x^{i}$.

Let $k$ be a field. We say that a polynomial $f(x)\in k[x]$ is reduced if the
ring $k[x]/(f(x))$ is reduced.

######  Lemma 3.24.

Let $E(x)\in\mathscr{A}_{\mathbb{F}}$ be a reduced polynomial. Let
$V:=\\{\beta\in\mathbb{F}\mid E(\beta)=0\\}$.

  * (1)

Assume that $E(x)$ is monic and $V$ is stable under $\sigma$. Then we have
$E(x)\in\mathbb{F}_{q}[x]$.

  * (2)

Let $r$ be a positive integer. Assume that $V$ is stable under
$\mu_{r}$-multiplication. Then we have $E(\alpha x)=\alpha E(x)$ for
$\alpha\in\mu_{r}$.

###### Proof.

Recall that $f(x)\in\mathscr{A}_{\mathbb{F}}$ is reduced if and only if
$f^{\prime}(0)\neq 0$.

We show (1). By the assumption, we have
$E^{\sigma}(\beta)=(E(\beta^{q^{-1}}))^{q}=0$ for any $\beta\in V$. Since
$E(x)$ is separable, there exists $\alpha\in\mathbb{F}^{\times}$ such that
$E^{\sigma}(x)=\alpha E(x)$. Since $E(x)$ is monic, we have $\alpha=1$. Hence
we have the claim.

We show (2). Let $\alpha\in\mu_{r}$. By the assumption, $E(\alpha\beta)=0$ for
any $\beta\in V$. Since $E(x)$ is separable, we have $E(\alpha x)=cE(x)$ with
a constant $c\in\mathbb{F}^{\times}$. By considering the derivatives of
$E(\alpha x)$, $cE(x)$ and substituting $x=0$, we have $\alpha=c$ by
$E^{\prime}(0)\neq 0$. Hence the claim follows. ∎

######  Definition 3.25.

Let $f(x)\in\mathscr{A}_{q}$.

  * (1)

A decomposition $f(x)=f_{1}(f_{2}(x))$ with $f_{i}(x)\in\mathscr{A}_{q}$ is
said to be non-trivial if $\deg f_{i}>1$ for $i\in\\{1,2\\}$.

  * (2)

We say that $f(x)\in\mathscr{A}_{q}$ is prime if it does not admit a non-
trivial decomposition $f(x)=f_{1}(f_{2}(x))$ with
$f_{i}(x)\in\mathscr{A}_{q}$.

######  Definition 3.26.

Let $(V,\omega)$ be a symplectic $\mathbb{F}_{p}[\mathscr{H}]$-module. Then
$(V,\omega)$ is said to be completely anisotropic if $V$ does not admit a non-
zero totally isotropic $\mathbb{F}_{p}[\mathscr{H}]$-submodule.

For an $\mathbb{F}_{p}$-subspace $W\subset V$, let $W^{\perp}:=\\{v\in
V\mid\textrm{$\omega(v,w)=0$ for all $w\in W$}\\}$.

######  Proposition 3.27.

The symplectic $\mathbb{F}_{p}[\mathscr{H}]$-module $(V_{R},\omega_{R})$ is
completely anisotropic if and only if there does not exist a non-trivial
decomposition $E_{R}(x)=f_{1}(f_{2}(x))$ with $f_{i}(x)\in\mathscr{A}_{q}$
such that $f_{2}(\alpha x)=\alpha f_{2}(x)$ for $\alpha\in\mu_{d_{R,m}}$ and
$V_{f_{2}}:=\\{\beta\in\mathbb{F}\mid f_{2}(\beta)=0\\}$ satisfies
$V_{f_{2}}\subset V_{f_{2}}^{\perp}$.

###### Proof.

Assume that there exists such a decomposition $E_{R}(x)=f_{1}(f_{2}(x))$.
Since the decomposition is non-trivial, we have $V_{f_{2}}\neq\\{0\\}$. Hence
$V_{f_{2}}$ is a non-zero totally isotropic
$\mathbb{F}_{p}[\mathscr{H}]$-submodule of $V_{R}$. Hence $V_{R}$ is not
completely anisotropic.

Assume that $V_{R}$ is not completely anisotropic. We take a non-zero totally
isotropic $\mathbb{F}_{p}[\mathscr{H}]$-submodule $V^{\prime}\subset V_{R}$.
By [13, 4 in Chapter 1], there exists a monic reduced polynomial
$f(x)\in\mathscr{A}_{\mathbb{F}}$ such that
$V^{\prime}=\\{\beta\in\mathbb{F}\mid f(\beta)=0\\}$. Since $V^{\prime}$ is
stable by $\sigma$, we have $f(x)\in\mathbb{F}_{q}[x]$ by Lemma 3.24(1). Since
$V^{\prime}$ is stable by $\tau$, we have $f(\alpha x)=\alpha f(x)$ for
$\alpha\in\mu_{d_{R,m}}$ by Lemma 3.18 and Lemma 3.24(2). There exist
$f_{1}(x),r(x)\in\mathscr{A}_{q}$ such that $E_{R}(x)=f_{1}(f(x))+r(x)$ and
$\deg r(x)<\deg f(x)$ by [13, Theorem 1]. For any root $\beta\in V^{\prime}$
of $f(x)$, we have $r(\beta)=0$ by $E_{R}(\beta)=0$. Since $f(x)$ is
separable, $r(x)$ is divisible by $f(x)$. Hence $r(x)\equiv 0$ by $\deg
r(x)<\deg f(x)$. By definition, we have $V^{\prime}\subset V^{\prime\perp}$.
Hence the converse is shown. ∎

######  Corollary 3.28.

  * (1)

The $W_{F}$-representation $\tau_{\psi,R,m}$ is primitive if and only if the
symplectic $\mathbb{F}_{p}[\mathscr{H}]$-module $(V_{R},\omega_{R})$ is
completely anisotropic.

  * (2)

The $W_{F}$-representation $\tau_{\psi,R,m}$ is primitive if and only if there
does not exist a non-trivial decomposition $E_{R}(x)=f_{1}(f_{2}(x))$ with
$f_{i}(x)\in\mathscr{A}_{q}$ such that $f_{2}(\alpha x)=\alpha f_{2}(x)$ for
$\alpha\in\mu_{d_{R,m}}$ and $V_{f_{2}}:=\\{\beta\in\mathbb{F}\mid
f_{2}(\beta)=0\\}$ satisfies $V_{f_{2}}\subset V_{f_{2}}^{\perp}$.

  * (3)

If $E_{R}(x)\in\mathscr{A}_{q}$ is prime, the $W_{F}$-representation
$\tau_{\psi,R,m}$ is primitive.

  * (4)

If $R(x)=a_{e}x^{p^{e}}$ and
$\mathbb{F}_{p}(\mu_{d_{R,m}})=\mathbb{F}_{p^{2e}}$, the
$\mathbb{F}_{p}[\mathscr{H}]$-module $V_{R}$ is irreducible. The
$W_{F}$-representation $\tau_{\psi,R,m}$ is primitive. If
$\mathrm{gcd}(p^{e}+1,m)=1$, the condition
$\mathbb{F}_{p}(\mu_{d_{R,m}})=\mathbb{F}_{p^{2e}}$ is satisfied.

###### Proof.

The claim (1) follows from Corollary 3.13, Lemma 3.17, and [11, Theorem 4.1].

The claim (2) follows from the claim (1) and Proposition 3.27. The claim (3)
follows from (2) immediately.

We show (4). We assume that there exists a non-zero
$\mathbb{F}_{p}[\mathscr{H}]$-submodule $W\subset
V_{R}=\\{\beta\in\mathbb{F}\mid(a_{e}x^{p^{e}})^{p^{e}}+a_{e}x=0\\}$. We take
a non-zero element $\beta\in W$. By
$\mathbb{F}_{p}(\mu_{d_{R,m}})=\mathbb{F}_{p^{2e}}$, we have
$\mathbb{F}_{p^{2e}}\beta=\mathbb{F}_{p}(\mu_{d_{R,m}})\beta\subset W$. Since
$V_{R}$ is the set of the roots of a separable polynomial $E_{R}(x)$ of degree
$p^{2e}$, we have $|V_{R}|=p^{2e}$. Hence $W=V_{R}=\mathbb{F}_{p^{2e}}\beta$.
Thus the first claim follows. The second claim follows from the first one and
[11, Theorem 4.1]. If $\mathrm{gcd}(p^{e}+1,m)=1$, we have
$d_{R,m}=d_{R}=p^{e}+1$. Hence the third claim follows from
$\mathbb{F}_{p}(\mu_{p^{e}+1})=\mathbb{F}_{p^{2e}}$. ∎

######  Example 3.29.

For a positive integer $s$, we consider the set

$\mathcal{A}_{q,s}:=\left\\{\varphi(x)\in\mathbb{F}_{q}[x]\ \Big{|}\
\varphi(x)=\sum_{i=0}^{n}c_{i}x^{p^{si}}\right\\},$

which is regarded as a ring with multiplication
$\varphi_{1}\circ\varphi_{2}:=\varphi_{1}(\varphi_{2}(x))$ for
$\varphi_{1},\varphi_{2}\in\mathcal{A}_{q,s}$.

In the following, we give examples such that $E_{R}(x)$ is prime. We write
$d_{R}=p^{t}+1$ with $t\geq 0$. Then we have $E_{R}\in\mathcal{A}_{q,t}$. We
write $q=p^{f}$. Assume $f\mid t$. We have

$E_{R}(x)=\sum_{i=0}^{e}a_{i}x^{p^{e+i}}+\sum_{i=0}^{e}a_{i}x^{p^{e-i}}.$
(3.10)

By $f\mid t$, we have the ring isomorphism
$\Phi\colon\mathcal{A}_{q,t}\xrightarrow{\sim}\mathbb{F}_{q}[y];\
\sum_{i=0}^{r}c_{i}x^{p^{ti}}\mapsto\sum_{i=0}^{r}c_{i}y^{i}$, where
$\mathbb{F}_{q}[y]$ is a usual polynomial ring. The polynomial
$E_{R}(x)\in\mathscr{A}_{q}$ is prime if and only if $\Phi(E_{R}(x))$ is
irreducible in $\mathbb{F}_{q}[y]$ in a usual sense. Recall that a polynomial
$\sum_{i=0}^{r}c_{i}y^{i}\in\mathbb{F}_{q}[y]$ is said to be reciprocal if
$c_{i}=c_{r-i}$ for $0\leq i\leq r$. By (3.10), we know that $\Phi(E_{R}(x))$
is a reciprocal polynomial. The number of the monic irreducible reciprocal
polynomials is calculated in [3, Theorems 2 and 3].

In general, we do not know a necessary and sufficient condition on $R(x)$ for
$E_{R}(x)$ to be prime. The number of prime elements in $\mathcal{A}_{q,s}$ is
calculated in [4] and [12].

######  Proposition 3.30.

Assume $d_{R,m}\in\\{1,2\\}$. There exists an unramified finite extension
$F^{\prime}/F$ such that $\tau_{\psi,R,m}|_{W_{F^{\prime}}}$ is imprimitive.

###### Proof.

For a positive integer $r$, let $F_{r}$ be the unramified extension of $F$ of
degree $r$. We take a non-zero element $\beta\in V_{R}$. Let $t$ be the
positive integer such that $\mathbb{F}_{q^{t}}=\mathbb{F}_{q}(\beta)$. Let
$\mathscr{H}_{t}\subset\mathscr{H}$ be the subgroup generated by
$\sigma^{t},\tau$. The subspace $W_{R}:=\mathbb{F}_{p}\beta\subset V_{R}$ is a
totally isotropic $\mathbb{F}_{p}[\mathscr{H}_{t}]$-submodule because of
$d_{R,m}\leq 2$. Hence $V_{R}$ is not a completely anisotropic
$\mathbb{F}_{p}[\mathscr{H}_{t}]$-module. Thus $\tau_{\psi,R,m}|_{W_{F_{t}}}$
is imprimitive by [11, Theorem 4.1]. ∎

######  Lemma 3.31.

The $W_{T_{\rho}}$-representation $\tau_{\psi,R,m}|_{W_{T_{\rho}}}$ is
imprimitive.

###### Proof.

We take a non-zero element $\beta\in V_{R}$. Then $\mathbb{F}_{p}\beta$ is a
totally isotropic symplectic submodule of the symplectic module $V_{R}$
associated to $\tau_{\psi,R,m}|_{W_{T_{\rho}}}$. Hence
$\tau_{\psi,R,m}|_{W_{T_{\rho}}}$ is imprimitive by Corollary 3.28(1). ∎

###  3.4 Root system associated to irreducible
$\mathbb{F}_{p}[\mathscr{H}]$-module

A root system associated to an irreducible
$\mathbb{F}_{p}[\mathscr{H}]$-module is defined in [11]. We determine the root
system associated to $V_{R}$ in the situation of Corollary 3.28(3).

We recall the definition of a root system.

######  Definition 3.32.

([11, §7])

  * (1)

Let $\Phi$ be the group of the automorphisms of the torus
$(\mathbb{F}^{\times})^{2}$ generated by the automorphisms
$\theta\colon(\alpha,\beta)\mapsto(\alpha^{p},\beta^{p})$ and
$\sigma\colon(\alpha,\beta)\mapsto(\alpha^{q^{-1}},\beta)$. A $\Phi$-orbit of
$(\mathbb{F}^{\times})^{2}$ is called a root system.

  * (2)

Let $W=\Phi(\alpha,\beta)$ be a root system. Let

$\displaystyle\textrm{$a=a(W)$ be the minimal positive integer with
$\alpha^{q^{a}}=\alpha$},$ $b=b(W)$ the minimal positive integer with
$\alpha^{p^{b}}=\alpha^{q^{x}}$, $\beta^{p^{b}}=\beta$ with $x\in\mathbb{Z}$,
and $\displaystyle\textrm{$c=c(W)$ the minimal non-negative integer with
$\alpha^{p^{b}}=\alpha^{q^{c}}$}.$

Let $e^{\prime}=e^{\prime}(W)$ and $f^{\prime}=f^{\prime}(W)$ be the orders of
$\alpha$ and $\beta$, respectively. These integers are independent of
$(\alpha,\beta)$ in $W$.

  * (3)

Let $\mathscr{H}_{d,r}:=\left\langle\sigma,\tau\mid\tau^{d}=1,\ \sigma^{r}=1,\
\sigma\tau\sigma^{-1}=\tau^{q}\right\rangle$ with $q^{r}\equiv 1\pmod{d}$.

  * (4)

We say that a root system $W$ belongs to $\mathscr{H}_{d,r}$ if
$e^{\prime}\mid d$ and $af^{\prime}\mid r$.

  * (5)

Let $W=\Phi(\alpha,\beta)$ be a root system which belongs to
$\mathscr{H}_{d,r}$. Let $\overline{M(W)}$ be the $\mathbb{F}$-module with the
basis

$\\{\theta^{i}\sigma^{j}m\mid 0\leq i\leq b-1,\ 0\leq j\leq a-1\\}$

and with the action of $\mathscr{H}$ by

$\tau m=\alpha m,\quad\sigma^{a}m=\beta m,\quad\theta^{b}m=\sigma^{-c}m.$

######  Theorem 3.33.

([11, Theorems 7.1 and 7.2])

  * (1)

There exists an irreducible $\mathbb{F}_{p}[\mathscr{H}_{d,r}]$-module $M(W)$
such that $M(W)\otimes_{\mathbb{F}_{p}}\mathbb{F}$ is isomorphic to
$\overline{M(W)}$ as $\mathbb{F}[\mathscr{H}_{d,r}]$-modules.

  * (2)

The map $W\mapsto M(W)$ defines a one-to-one correspondence between the set of
root systems belonging to $\mathscr{H}_{d,r}$ and the set of isomorphism
classes of irreducible $\mathbb{F}_{p}[\mathscr{H}_{d,r}]$-modules.

We go back to the original situation. Assume that $R(x)=a_{e}x^{p^{e}}$ and
$\mathbb{F}_{p}(\mu_{d_{R,m}})=\mathbb{F}_{p^{2e}}$. Let $\mathscr{H}$ be as
in (3.9). In the above notation, we have
$\mathscr{H}=\mathscr{H}_{d_{R,m},r}$. As in Corollary 3.28(3), the
$\mathbb{F}_{p}[\mathscr{H}]$-module $V_{R}$ is irreducible.

######  Proposition 3.34.

We write $q=p^{f}$. Let $e_{1}:=\mathrm{gcd}(f,2e)$ and
$\beta:=\operatorname{Nr}_{q/p^{e_{1}}}(-a_{e}^{-(p^{e}-1)})$. Let
$\alpha\in\mu_{d_{R,m}}$ be a primitive $d_{R,m}$-th root of unity. We
consider the root system $W:=\Phi(\alpha,\beta)$.

  * (1)

We have $a(W)=2e/e_{1}$ and $b(W)=e_{1}.$ Further, $c(W)$ is the minimal non-
negative integer such that $fc(W)\equiv e_{1}\pmod{2e}$.

  * (2)

The root system $W$ belongs to $\mathscr{H}$.

  * (3)

We have an isomorphism $V_{R}\simeq M(W)$ as
$\mathbb{F}_{p}[\mathscr{H}]$-modules.

###### Proof.

We show (1). We simply write $a,b,c$ for $a(W),b(W),c(W)$, respectively. By
definition, $a$ is the minimal natural integer such that
$\alpha^{q^{a}}=\alpha$. By $\mathbb{F}_{p}(\alpha)=\mathbb{F}_{p^{2e}}$, $a$
is the minimal positive integer satisfying $fa\equiv 0\pmod{2e}$. Thus we
obtain $a=2e/e_{1}$.

By definition, $b$ is the minimal natural integer such that
$\alpha^{p^{b}}=\alpha^{q^{x}}$ with some integer $x$ and
$\beta^{p^{b}}=\beta$. The first condition implies that $fx\equiv b\pmod{2e}$.
Hence $b$ is divisible by $e_{1}$. By
$\beta\in\mathbb{F}_{p^{e_{1}}}^{\times}$, we have $\beta^{p^{b}}=\beta$ if
$e_{1}\mid b$. Hence $b=e_{1}$.

By definition, $c$ is the minimal non-negative integer such that
$\alpha^{p^{b}}=\alpha^{q^{c}}$. This is equivalent to $e_{1}=b\equiv
fc\pmod{2e}$.

We show (2). The order $e^{\prime}$ of $\alpha$ equals $d_{R,m}$. Let
$f^{\prime}$ be the order of $\beta$. It suffices to show $af^{\prime}\mid r$.
By the choice of $r$, we have $\alpha^{q^{r}}=\alpha$. Hence $2e\mid fr$ by
$\mathbb{F}_{p^{2e}}=\mathbb{F}_{p}(\alpha)$, and $a\mid r$. These imply that
$\mathbb{F}_{p^{2e}}\subset\mathbb{F}_{q^{a}}\subset\mathbb{F}_{q^{r}}$.

Let $\eta\in V_{R}\setminus\\{0\\}$. By
$\eta^{p^{2e}}=-a_{e}^{-(p^{e}-1)}\eta$, $a_{e}\in\mathbb{F}_{q}^{\times}$ and
$2e\mid fr$, we compute

$\eta^{q^{r}}=(\eta^{p^{2e}-1})^{\frac{q^{r}-1}{p^{2e}-1}}\eta=\operatorname{Nr}_{q^{r}/p^{2e}}(-a_{e}^{-(p^{e}-1)})\eta=\operatorname{Nr}_{{q^{a}}/{p^{2e}}}(-a_{e}^{-(p^{e}-1)})^{r/a}\eta.$
(3.11)

The restriction map
$\mathrm{Gal}(\mathbb{F}_{q^{a}}/\mathbb{F}_{p^{2e}})\to\mathrm{Gal}(\mathbb{F}_{q}/\mathbb{F}_{p^{e_{1}}})$
is an isomorphism because of $a=2e/e_{1}$. By
$a_{e}\in\mathbb{F}_{q}^{\times}$, we have
$\operatorname{Nr}_{{q^{a}}/{p^{2e}}}(-a_{e}^{-(p^{e}-1)})=\beta$. Hence
$\eta^{q^{r}}=\beta^{r/a}\eta$ by (3.11). Since $\eta^{q^{r}}=\eta$ by Lemma
3.18, we obtain $\beta^{r/a}=1$. Hence $f^{\prime}\mid(r/a)$.

We show (3). Let $\eta\in V_{R}\setminus\\{0\\}$. Similarly to (3.11), we have
$\sigma^{a}\eta=\eta^{q^{a}}=\beta\eta$. By definition and Lemma 3.18, we have
$\tau\eta=\alpha\eta$. The $\mathbb{F}_{p}[\mathscr{H}]$-module $V_{R}$
satisfies the assumption in [11, Lemma 7.3] by (2). Hence $\\{0\\}\neq
M(W)\subset V_{R}$ by [11, Lemma 7.3]. By the irreducibility of $V_{R}$ in
Corollary 3.28(4), we obtain $M(W)=V_{R}$. ∎

A necessary and sufficient condition for an irreducible
$\mathbb{F}_{p}[\mathscr{H}]$-module to have a symplectic form is determined
in [11, Theorem 8.1]. We recall the result.

######  Theorem 3.35.

([11, Theorem 8.1]) Let $W=\Phi(\alpha,\beta)$ be a root system. The
irreducible $\mathbb{F}_{p}[\mathscr{H}]$-module $M(W)$ has a symplectic form
if and only if

  * (A)

$a(W)\equiv 0\pmod{2}$, $\alpha\in\mu_{q^{a(W)/2}+1}$ and $\beta=-1$,

  * (B)

$b(W),c(W)\equiv 0\pmod{2}$, $\alpha\in\mu_{p^{b(W)/2}+q^{c(W)/2}}$ and
$\beta\in\mu_{p^{b(W)/2}+1}$, or

  * (C)

$b(W)\equiv 0\pmod{2}$, $c(W)\equiv a(W)\pmod{2}$,
$\alpha\in\mu_{p^{b(W)/2}+q^{(a(W)+c(W))/2}}$ and
$\beta\in\mu_{p^{b(W)/2}+1}$.

There are two isomorphism classes of symplectic structures on $M(W)$ in the
case A, $p\neq 2$ and one in all other cases.

######  Lemma 3.36.

Let $W$ be as in Proposition 3.34. Let $v_{2}(\cdot)$ denote the $2$-adic
valuation on $\mathbb{Q}$.

  * (1)

Assume $v_{2}(e)\geq v_{2}(f)$. Then the module $M(W)$ is of type A in Theorem
3.35.

  * (2)

Assume $v_{2}(e)<v_{2}(f)$. Then we have $a(W)\equiv 1\pmod{2}$, $b(W)\equiv
0\pmod{2}$ and $(b(W)/2)\mid e$. Hence we have $\beta\in\mu_{p^{b(W)/2}+1}$.

    * (i)

If $c(W)\equiv 0\pmod{2}$, the module $M(W)$ is of type B in Theorem 3.35.

    * (ii)

If $c(W)\equiv 1\pmod{2}$, the module $M(W)$ is of type C in Theorem 3.35.

###### Proof.

We show (1). Recall that $e_{1}=\mathrm{gcd}(f,2e)$ and
$\beta=\operatorname{Nr}_{q/p_{1}}(-a_{e}^{-(p^{e}-1)})$. We have
$a(W)=2e/e_{1}\equiv 0\pmod{2}$. We have $e_{1}\mid e$ and $f/e_{1}\equiv
1\pmod{2}$. By $(p^{e_{1}}-1)\mid(p^{e}-1)$,

$\beta=(-1)^{\frac{f}{e_{1}}}\bigl{(}a_{e}^{-\frac{p^{e}-1}{p^{e_{1}}-1}}\bigr{)}^{q-1}=-1,$

where we use $a_{e}\in\mathbb{F}_{q}^{\times}$ for the last equality. By
$fa(W)/2=fe/e_{1}$ and $q=p^{f}$, we have
$\alpha^{q^{a(W)/2}+1}=\alpha^{p^{fe/e_{1}}+1}$. Since $fe/e_{1}$ is divisible
by $e$ and $f/e_{1}$ is odd, $d_{R,m}\mid(p^{e}+1)\mid(p^{fe/e_{1}}+1)$. Hence
we obtain $\alpha\in\mu_{q^{a(W)/2}+1}$. Thus the claim follows.

We show (2). Recall $b(W)=e_{1}$. The former claims are clear. By
$(e_{1}/2)\mid e$, we have $(p^{e_{1}/2}-1)\mid(p^{e}-1)$. By definition of
$\beta$ and $a_{e}\in\mathbb{F}_{q}^{\times}$, we obtain

$\beta^{p^{\frac{e_{1}}{2}}+1}=\bigl{(}a_{e}^{-\frac{p^{e}-1}{p^{e_{1}/2}-1}}\bigr{)}^{q-1}=1.$

Hence $\beta\in\mu_{p^{b(W)/2}+1}$. Assume that $c(W)$ is even. We write
$(c(W)/2)f=(e_{1}/2)+le$ with $l\in\mathbb{Z}$ by Proposition 3.34(1). Then
$l$ is odd by $e_{1}=\mathrm{gcd}(f,2e)$. Hence $(p^{e}+1)\mid(p^{le}+1)$.
This implies $\alpha\in\mu_{p^{b(W)/2}+q^{c(W)/2}}$. Hence we obtain (2)(i).
The remaining claim is shown similarly. ∎

####  3.4.1 Künneth formula and primary module

##### Classification results in [11]

We recall classification results on completely anisotropic symplectic modules
given in [11] restricted to the case $p\neq 2$.

######  Theorem 3.37.

([11, Theorem 9.1]) Let $(V,\omega)=\bigoplus_{i=1}^{n}(V_{i},\omega_{i})$ be
a direct sum of irreducible symplectic $\mathbb{F}_{p}[\mathscr{H}]$-modules.
Assume that $p\neq 2$. Then $(V,\omega)$ is completely anisotropic if and only
if, for each isomorphism class, the modules of type B or C occur at most once
and of type A at most twice among $V_{1},\ldots,V_{n}$.

Assume that $p\neq 2$. Let $(M(W),0)$ denote the unique symplectic module on
$M(W)$ which is of type B or C by Theorem 3.35. Let $(M(W),0)$, $(M(W),1)$
denote the two symplectic modules on $M(W)$ in the case where $p\neq 2$ and
$M(W)$ is of type A. We denote by $(M(W),2)$ the completely anisotropic
symplectic module on $M(W)\oplus M(W)$, where $M(W)$ is of type A.

######  Theorem 3.38.

([11, Theorem 8.2]) Each completely anisotropic symplectic
$\mathbb{F}_{p}[\mathscr{H}]$-module is isomorphic to one and only one
symplectic module of the form

$\bigoplus_{i=1}^{n}(M(W_{i}),\nu_{i}),$

where $W_{1},\ldots,W_{n}$ are mutually different root systems belonging to
$\mathscr{H}$.

Let $k$ be a positive integer. Let $R:=\\{R_{i}\\}_{1\leq i\leq k}$ with
$R_{i}\in\mathscr{A}_{q}$. We consider the $k$-dimensional affine smooth
variety $X_{R}$ defined by

$a^{p}-a=\sum_{i=1}^{k}x_{i}R_{i}(x_{i})$

in $\mathbb{A}_{\mathbb{F}_{q}}^{k+1}$. The product group
$Q_{R}:=Q_{R_{1}}\times\cdots\times Q_{R_{k}}$ acts on $X_{R}$ naturally
similarly as (2.5). Let $\mathbb{Z}$ act on $Q_{R}$ naturally. Let
$\psi\in\mathbb{F}_{p}^{\vee}\setminus\\{1\\}$. We regard $H_{\rm
c}^{k}(X_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ as a
$Q_{R}\rtimes\mathbb{Z}$-representation. Let the notation be as in (3.3). Let
$m=\\{m_{i}\\}_{1\leq i\leq k}$, where $m_{i}$ is a positive integer. We have
the homomorphism

$\Theta_{R,m}\colon W_{F}\to Q_{R}\rtimes\mathbb{Z};\
\sigma\mapsto((a_{R_{i},\sigma}^{m_{i}},b_{R_{i},\sigma},c_{R_{i},\sigma})_{1\leq
i\leq k},n_{\sigma}).$ (3.12)

######  Definition 3.39.

We define a smooth $W_{F}$-representation $\tau_{\psi,R,m}$ to be the
inflation of the $Q_{R}\rtimes\mathbb{Z}$-representation $H_{\rm
c}^{k}(X_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ by
$\Theta_{R,m}$.

######  Lemma 3.40.

We have an isomorphism
$\tau_{\psi,R,m}\simeq\bigotimes_{i=1}^{k}\tau_{\psi,R_{i},m_{i}}$ as
$W_{F}$-representations.

###### Proof.

Let $Q_{R_{i},\mathbb{Z}}:=Q_{R_{i}}\rtimes\mathbb{Z}$ and
$\Theta_{R_{i},m_{i}}\colon W_{F}\to Q_{R_{i},\mathbb{Z}}$ be as in (3.6). Let

$\delta^{\prime}\colon Q_{R}\rtimes\mathbb{Z}\to
Q_{R_{1},\mathbb{Z}}\times\cdots\times Q_{R_{k},\mathbb{Z}};\ ((g_{i})_{1\leq
i\leq k},n)\mapsto(g_{i},n)_{1\leq i\leq k}.$

Each $H_{\rm c}^{1}(C_{R_{i},\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$
is regarded as a $Q_{R_{i},\mathbb{Z}}$-representation. By the Künneth
formula, we have an isomorphism $H_{\rm
c}^{k}(X_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]\simeq\bigotimes_{i=1}^{k}(H_{\rm
c}^{1}(C_{R_{i},\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi])$ as
$Q_{R}\rtimes\mathbb{Z}$-representations, where the right hand side is
regarded as a $Q_{R}\rtimes\mathbb{Z}$-representation via $\delta^{\prime}$.
We consider the commutative diagram

$\textstyle{W_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{R,m}}$$\scriptstyle{\delta}$$\textstyle{W_{F}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{R_{1},m_{1}}\times\cdots\times\Theta_{R_{k},m_{k}}}$$\textstyle{Q_{R}\rtimes\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\prime}}$$\textstyle{Q_{R_{1},\mathbb{Z}}\times\cdots\times
Q_{R_{k},\mathbb{Z}},}$

where $\delta$ is the diagonal map. Hence the claim follows. ∎

######  Remark 3.41.

Let $+\colon\prod_{i=1}^{k}Z(Q_{R_{i}})\to\mathbb{F}_{p};\
(1,0,\gamma_{i})_{1\leq i\leq k}\mapsto\sum_{i=1}^{k}\gamma_{i}$ and
$\overline{Q}_{R}:=Q_{R}/\operatorname{Ker}+$. The action of
$Q_{R}\rtimes\mathbb{Z}$ on $H_{\rm
c}^{k}(X_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})$ factors through
$\overline{Q}_{R}\rtimes\mathbb{Z}$. Let $\overline{H}_{R}$ denote the image
of $H_{R_{1}}\times\cdots\times H_{R_{k}}$ under $Q_{R}\to\overline{Q}_{R}$.
The group $\overline{H}_{R}$ is an extra-special $p$-group. The quotient
$\overline{H}_{R}/Z(\overline{H}_{R})$ is isomorphic to
$\bigoplus_{i=1}^{k}V_{R_{i}}$. Moreover, $\overline{Q}_{R}/\overline{H}_{R}$
is supersolvable.

######  Lemma 3.42.

The $W_{F}$-representation $\tau_{\psi,R,m}$ is irreducible.

###### Proof.

The $\overline{H}_{R}$-representation $H_{\rm
c}^{k}(X_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ is irreducible by
[8, 16.14(2) Satz]. The claim follows from this. ∎

Let $\rho_{\psi,R_{i},m_{i}}$ denote the projective representation associated
to $\tau_{\psi,R_{i},m_{i}}$. Let $F_{i}$ denote the kernel field of
$\rho_{\psi,R_{i},m_{i}}$ and $T_{i}$ the maximal tamely ramified extension of
$F$ in $F_{i}$. The field $T_{i}$ is called the tame kernel field of
$\rho_{\psi,R_{i},m_{i}}$. Let $F_{R}:=F_{1}\cdots F_{k}$.

######  Lemma 3.43.

Let $\rho_{\psi,R,m}$ be the projective representation associated to
$\tau_{\psi,R,m}$. The kernel field of $\rho_{\psi,R,m}$ is $F_{R}$.

###### Proof.

By Lemma 3.40, we can check
$\operatorname{Ker}\rho_{\psi,R,m}=\bigcap_{i=1}^{k}\operatorname{Ker}\rho_{\psi,R_{i},m_{i}}$.
The claim follows from this. ∎

Let $T_{R}$ be the maximal tamely ramified extension of $F$ in $F_{R}$. We
have the restriction map
$V_{R}\hookrightarrow\prod_{i=1}^{k}\operatorname{Gal}(F_{i}/T_{i})\simeq\bigoplus_{i=1}^{k}V_{R_{i}}$.
Then $V_{R}:=\operatorname{Gal}(F_{R}/T_{R})$ has a bilinear form stable under
the action of $\mathbb{F}_{p}[\operatorname{Gal}(T_{R}/F)]$ ([11, §4]). The
form on $V_{R}$ is given by $\omega_{R}:=\sum_{i=1}^{k}\omega_{R_{i}}$.

Let $\omega_{R_{i}}$ be the form on $V_{R_{i}}$ in Lemma 2.6(2). We give a
recipe to make an example of $(M(W),2)$ below.

######  Proposition 3.44.

Assume $k=2$. Let $R_{i}(x)=a_{e,i}x^{p^{e}}\neq 0$ for $i\in\\{1,2\\}$.
Assume

$m_{1}\neq m_{2},\quad d:=d_{R_{1},m_{1}}=d_{R_{2},m_{2}}.$

  * (1)

We have an isomorphism $V_{R}\simeq V_{R_{1}}\oplus V_{R_{2}}$.

  * (2)

We have $T_{R}=T_{1}\cdot T_{2}$.

  * (3)

Assume that $p\neq 2$, $v_{2}(e)\geq v_{2}(f)$ and
$\mathbb{F}_{p}(\mu_{d})=\mathbb{F}_{p^{2e}}$. If $(V_{R},\omega_{R})$ is
completely anisotropic as a symplectic
$\mathbb{F}_{p}[\operatorname{Gal}(T_{R}/F)]$-module, $V_{R}$ is isomorphic to
a primary module $(M(W),2)$ with a root system $W$.

###### Proof.

By Lemma 2.9 and Lemma 3.12, there exists an unramified finite extension $E$
of $F$ such that $F_{i}\subset E(\alpha_{R_{i}}^{m_{i}},\beta_{R_{i},m_{i}})$
for $i=1,2$ and $E(\alpha_{R_{i}}^{m_{i}},\beta_{R_{i},m_{i}})/E$ is Galois.
We put $T:=E(\alpha_{R_{i}}^{m_{i}})=E(\varpi^{1/d})$ and
$E_{i}:=T(\beta_{R_{i},m_{i}})$ for $i=1,2$. Let
$n_{i}:=m_{i}d/d_{R}=m_{i}/\mathrm{gcd}(d_{R},m_{i})$. Let
$\\{\operatorname{Gal}(E_{i}/T)^{v}\\}_{v\geq-1}$ be the upper numbering
ramification subgroups of $\operatorname{Gal}(E_{i}/T)$. Similarly as the
proof of Lemma 3.14, we have

$\operatorname{Gal}(E_{i}/T)^{v}=\begin{cases}\operatorname{Gal}(E_{i}/T)&\textrm{if
$v\leq n_{i}$},\\\ \\{1\\}&\textrm{if $v>n_{i}$}.\end{cases}$

Let $H:=E_{1}\cap E_{2}$. Since $E_{i}/T$ is Galois, so is $H/T$. By [15,
Proposition 14 in IV§3], the subgroup $\operatorname{Gal}(H/T)^{v}$ equals
$\operatorname{Gal}(H/T)$ if $v\leq n_{i}$ and $\\{1\\}$ if $v>n_{i}$. Hence
we conclude $\operatorname{Gal}(H/T)=\\{1\\}$ by $n_{1}\neq n_{2}$. We obtain
$H=T$. Hence we have an isomorphism
$\operatorname{Gal}(E_{1}E_{2}/T)\simeq\operatorname{Gal}(E_{1}/T)\times\operatorname{Gal}(E_{2}/T)\simeq
V_{R_{1}}\oplus V_{R_{2}}$. The extension $E_{1}E_{2}/T$ is totally ramified
and the degree is $p$-power. Hence, $T$ is the maximal tamely ramified
extension of $E$ in $E_{1}\cdot E_{2}$. Therefore, $T_{R}=F_{R}\cap T$. We
have the commutative diagram

$\textstyle{\operatorname{Gal}(E_{1}E_{2}/T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Gal}(E_{1}/T)\times\operatorname{Gal}(E_{2}/T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Gal}(F_{R}/T_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\operatorname{Gal}(F_{1}/T_{1})\times\operatorname{Gal}(F_{2}/T_{2}),}$

where every map is the restriction map. The right vertical isomorphism follows
from Lemma 3.17. Clearly $g$ is injective. By the commutative diagram, $g$ is
bijective. Hence we obtain (1).

By the commutative diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Gal}(F_{R}/T_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Gal}(F_{R}/F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{\operatorname{Gal}(T_{R}/F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{R_{1}}\oplus
V_{R_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Gal}(F_{1}/F)\times\operatorname{Gal}(F_{2}/F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Gal}(T_{1}/F)\times\operatorname{Gal}(T_{2}/F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

and injectivity of $g_{1}$, the map $g_{2}$ is injective. Hence
$T_{R}=T_{1}T_{2}$.

We show (3). Let $r:=[E:F]$ and
$\mathscr{H}_{d,r}:=\mu_{d}\rtimes(\mathbb{Z}/r\mathbb{Z})$ as in (3.9). We
identify $\operatorname{Gal}(T/F)$ with $\mathscr{H}_{d,r}$. By $T_{R}\subset
T$, the $V_{R}$, $V_{R_{i}}$ are naturally regarded as
$\mathbb{F}_{p}[\mathscr{H}_{d,r}]$-modules. Let $\alpha$ be a primitive
$d$-th root of unity. Let $W:=\Phi(\alpha,-1)$. Then we have an isomorphism
$V_{R_{i}}\simeq M(W)$ as $\mathbb{F}_{p}[\mathscr{H}_{d,r}]$-modules and know
that $V_{R_{i}}$ is of type A by Proposition 3.34(3), Lemma 3.36(1) and
$d_{R_{1},m_{1}}=d_{R_{2},m_{2}}$. This implies an isomorphism
$V_{R_{1}}\simeq V_{R_{2}}$ as $\mathbb{F}_{p}[\mathscr{H}_{d,r}]$-modules.
Hence the claim follows from the assumption that $(V_{R},\omega_{R})$ is
completely anisotropic and the definition of $(M(W),2)$. ∎

######  Example 3.45.

Assume $p\neq 2$. Let $e=f=1$, $R_{1}(x)=x^{p}$ and
$R_{2}(x)=ax^{p}\in\mathbb{F}_{p}[x]\setminus\\{0\\}$. We assume that
$m_{1}\neq m_{2}$ and $d_{R_{1},m_{1}}=d_{R_{2},m_{2}}=p+1$. We have
$V_{R_{i}}=\\{x\in\mathbb{F}\mid x^{p^{2}}+x=0\\}$ for $i=1,2$.

Let $W\subset V_{R_{1}}\oplus V_{R_{2}}$ be a totally isotropic
$\mathbb{F}_{p}[\operatorname{Gal}(T_{R}/F)]$-subspace. Assume $W\neq\\{0\\}$.
We take a non-zero element $(x_{1},x_{2})\in W$. We have
$f_{R_{1}}(x,y)=-xy^{p}$, $f_{R_{2}}(x,y)=-axy^{p}$ and hence
$\omega_{R}((x_{1},x_{2}),(\xi x_{1},\xi
x_{2}))=(x_{1}^{p+1}+ax_{2}^{p+1})(\xi-\xi^{p})=0$ for any $\xi\in\mu_{p+1}$.
Hence $x_{1}^{p+1}+ax_{2}^{p+1}=0$ and $x_{2}\neq 0$. There exists
$\eta\in\mathbb{F}$ such that $\eta^{p+1}=-a$ and $x_{1}=\eta x_{2}$. By
$\mathbb{F}_{p^{2}}=\mathbb{F}_{p}(\mu_{p+1})$, we have $W_{1}:=\\{(\eta
x,x)\mid x\in V_{R_{2}}\\}\subset W$. We also have
$W_{2}:=\\{(\eta^{p}x,x)\mid x\in V_{R_{2}}\\}\subset W$. Let
$\bigl{(}\frac{\cdot}{p}\bigr{)}$ be the Legendre symbol. If $W_{1}\cap
W_{2}\neq\\{0\\}$, we have $\eta\in\mathbb{F}_{p}$ and $\eta^{2}=-a$. This
implies $\bigl{(}\frac{-a}{p}\bigr{)}=1$.

Assume $\bigl{(}\frac{-a}{p}\bigr{)}=-1$. Then $W=W_{1}\oplus
W_{2}=V_{R_{1}}\oplus V_{R_{2}}$ by $W_{1}\cap W_{2}=\\{0\\}$. This is a
contradiction. Hence $V_{R_{1}}\oplus V_{R_{2}}$ is completely anisotropic if
$\bigl{(}\frac{-a}{p}\bigr{)}=-1$.

If $\bigl{(}\frac{-a}{p}\bigr{)}=1$, we have $W_{1}=W_{2}$, which is the
unique non-zero totally isotropic $\mathbb{F}_{p}[\mathscr{H}]$-subspace.
Hence $V_{R_{1}}\oplus V_{R_{2}}$ is not completely anisotropic.

##  4 Geometric interpretation of imprimitivity

Through this section, we always assume $p\neq 2$. Our aim in this section is
to show Theorem 4.13. To show the theorem, we use the explicit understanding
of the automorphism group of $C_{R}$ and the mechanism of taking quotients of
$C_{R}$ by certain abelian groups, which are developed in [1] and [6].

###  4.1 Quotient of $C_{R}$ and description of $\tau_{\psi,R,m}$

Let $C_{R}$ be as in (2.7). In this subsection, we always assume that there
exists a finite étale morphism

$\phi\colon C_{R}\to C_{R_{1}};\ (a,x)\mapsto(a-\Delta(x),r(x)),$ (4.1)

where $\Delta(x)\in\mathbb{F}_{q}[x]$ and $r(x),R_{1}(x)\in\mathscr{A}_{q}$
satisfy $d_{R,m}\mid d_{R_{1}}$ and $r(\alpha x)=\alpha r(x)$ for
$\alpha\in\mu_{d_{R,m}}$. Since $\phi$ is étale, $r(x)$ is a reduced
polynomial. Hence $r^{\prime}(0)\neq 0$. By the above assumption,

$\displaystyle xR(x)=r(x)R_{1}(r(x))+\Delta(x)^{p}-\Delta(x)$ (4.2)
$\displaystyle r^{\prime}(0)\neq 0,\quad d_{R,m}\mid d_{R_{1}},\quad r(\alpha
x)=\alpha r(x)\quad\textrm{for $\alpha\in\mu_{d_{R,m}}$}.$ (4.3)

Let $e^{\prime}$ be a non-negative integer such that $\deg
R_{1}(x)=p^{e^{\prime}}$ and $e^{\prime}\leq e$. Then $\deg
r(x)=p^{e-e^{\prime}}$ by (4.2).

We have $\alpha R_{1}(\alpha x)=R_{1}(x)$ for $\alpha\in\mu_{d_{R,m}}$ by
$d_{R,m}\mid d_{R_{1}}$ and (2.3). Hence $\Delta(\alpha
x)-\Delta(x)\in\mathbb{F}_{p}$ for $\alpha\in\mu_{d_{R,m}}$ by (4.2). We have
$\Delta(\alpha x)=\Delta(x)$, since the constant coefficient of $\Delta(\alpha
x)-\Delta(x)$ is zero.

######  Lemma 4.1.

Let $\phi\colon C_{R}\to C_{R};\ (x,a)\mapsto(x+c,a+g(x))$ be the automorphism
with $g(x)\in\mathbb{F}_{q}[x]$ and $c\in\mathbb{F}$. Then we have
$E_{R}(c)=0$.

###### Proof.

We have $g(x)^{p}-g(x)=cR(x)+xR(c)+cR(c)$. Let
$\mathcal{P}\colon\mathbb{F}[x]\to\mathbb{F}[x];\ f(x)\mapsto f(x)^{p}-f(x)$.
By the definition of $E_{R}(x)$, we obtain $cR(x)+xR(c)+cR(c)\equiv
E_{R}(c)^{1/p^{e}}x\mod\mathcal{P}(\mathbb{F}[x])$. Thus we must have
$E_{R}(c)=0$. ∎

######  Lemma 4.2.

We have $E_{R_{1}}(r(x))\mid E_{R}(x)$.

###### Proof.

Let $\beta\in\mathbb{F}$ be an element such that $E_{R_{1}}(r(\beta))=0$. We
take an element $\gamma\in\mathbb{F}$ such that
$\gamma^{p}-\gamma=r(\beta)R_{1}(r(\beta))$. The curve $C_{R,\mathbb{F}}$
admits the automorphism $\phi$ defined by

$\phi(a,x)=\left(a+f_{R_{1}}(r(x),r(\beta))+\Delta(x+\beta)-\Delta(x)+\gamma,x+\beta\right).$

This is well-defined by Lemma 2.2 and (4.2). By Lemma 4.1, we have
$E_{R}(\beta)=0$. Since $E_{R_{1}}(r(x))$ is separable, the claim follows. ∎

######  Lemma 4.3.

Let $\alpha,\alpha^{\prime}\in\mu_{d_{R,m}}$. Assume $E_{R_{1}}(r(\alpha
y))=0$ for a certain $y\in\mathbb{F}$. Then we have the equality

$\Delta(\alpha^{\prime}x+\alpha y)+f_{R_{1}}(r(\alpha^{\prime}x),r(\alpha
y))=\Delta(x)+\Delta(y)+f_{R}(\alpha^{\prime}x,\alpha y).$

###### Proof.

By $\Delta(\alpha^{\prime}x+\alpha y)=\Delta(x+(\alpha/\alpha^{\prime})y)$ and
(2.4), we may assume $\alpha^{\prime}=1$ by (4.3). We have $E_{R}(\alpha y)=0$
by Lemma 4.2. Let $\Delta_{1}(x)$ and $\Delta_{2}(x)$ denote the left and
right hand sides of the required equality, respectively. We have
$\Delta_{1}(0)=\Delta(\alpha y)=\Delta(y)=\Delta_{2}(0)$, since
$f_{R}(0,x^{\prime})\equiv 0$ in $\mathbb{F}_{q}[x^{\prime}]$ by definition.
Hence it suffices to show
$\Delta_{1}(x)^{p}-\Delta_{1}(x)=\Delta_{2}(x)^{p}-\Delta_{2}(x)$. By Lemma
4.2 and the assumption, $E_{R_{1}}(r(\alpha y))=E_{R}(\alpha y)=0$. Hence each
$\Delta_{i}(x)^{p}-\Delta_{i}(x)$ for $i=1,2$ equals $(x+\alpha y)R(x+\alpha
y)-r(y)R_{1}(r(y))-r(x)R_{1}(r(x))$ according to Lemma 2.2. Hence the claim
follows. ∎

Let

$U_{R}:=\\{x\in\mathbb{F}\mid r(x)=0\\}\subset
V^{\prime}_{R}:=\\{x\in\mathbb{F}\mid E_{R_{1}}(r(x))=0\\}.$

We have $V^{\prime}_{R}\subset V_{R}$ by Lemma 4.2. Then $U_{R}$ and
$V^{\prime}_{R}$ are regarded as $\mathbb{F}_{p}[\mathscr{H}]$-modules by
$r(x),R_{1}(x)\in\mathbb{F}_{q}[x]$ and (4.3).

######  Lemma 4.4.

We have $V^{\prime}_{R}\subset U^{\perp}_{R}$. In particular, the
$\mathbb{F}_{p}[\mathscr{H}]$-module $U_{R}$ is totally isotropic.

###### Proof.

Let $\beta\in U_{R}$ and $\beta^{\prime}\in V^{\prime}_{R}$. By Lemma 4.3,
$r(\beta)=0$ and $E_{R_{1}}(r(\beta^{\prime}))=0$, we have
$f_{R}(\beta^{\prime},\beta)=f_{R}(\beta,\beta^{\prime})=\Delta(\beta+\beta^{\prime})-\Delta(\beta)-\Delta(\beta^{\prime})$.
Hence $\omega_{R}(\beta,\beta^{\prime})=0$. ∎

Let

$Q^{\prime}_{R,m}:=\\{(\alpha,\beta,\gamma)\in Q_{R,m}\mid\beta\in
V^{\prime}_{R}\\}.$

Then $Q^{\prime}_{R,m}$ is a subgroup of $Q_{R,m}$ of index
$p^{e-e^{\prime}}$, because of (4.3) and
$[V_{R}:V^{\prime}_{R}]=p^{e-e^{\prime}}$. We have the map

$\pi\colon Q^{\prime}_{R,m}\to Q_{R_{1},m};\
(\alpha,\beta,\gamma)\mapsto(\alpha,r(\beta),\gamma-\Delta(\beta)).$

######  Corollary 4.5.

The map $\pi$ is a homomorphism.

###### Proof.

The claim follows from Lemma 4.3 and (4.3). ∎

We have

$U^{\prime}_{R}:=\\{(1,\beta,\Delta(\beta))\in Q^{\prime}_{R,m}\mid\beta\in
U_{R}\\}=\operatorname{Ker}\pi.$ (4.4)

The space $V^{\prime}_{R}$ is stable by the $q$-th power map. Hence we can
consider the semidirect product $Q^{\prime}_{R,m}\rtimes\mathbb{Z}$. The map
$\pi$ induces $\pi^{\prime}\colon Q^{\prime}_{R,m}\rtimes\mathbb{Z}\to
Q_{R_{1},m}\rtimes\mathbb{Z}$.

##### Quotient of $C_{R}$

Let $\phi$ be as in (4.1). We can check that $\phi$ factors through
$C_{R,\mathbb{F}}\to
C_{R,\mathbb{F}}/U^{\prime}_{R}\xrightarrow{\bar{\phi}}C_{R_{1},\mathbb{F}}$
by (2.5). We obtain an isomorphism $\bar{\phi}\colon
C_{R,\mathbb{F}}/U^{\prime}_{R}\xrightarrow{\sim}C_{R_{1},\mathbb{F}}$.

######  Lemma 4.6.

We have $\phi((a,x)g)=\phi(a,x)\pi^{\prime}(g)$ for $g\in
Q^{\prime}_{R,m}\rtimes\mathbb{Z}$.

###### Proof.

The claim follows from Lemma 4.3. ∎

Let $\tau^{\prime}_{\psi,R_{1},m}$ denote the
$Q^{\prime}_{R,m}\rtimes\mathbb{Z}$-representation which is the inflation of
the $Q_{R_{1},m}\rtimes\mathbb{Z}$-representation $H_{\rm
c}^{1}(C_{R_{1},\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ by
$\pi^{\prime}$. By (3.6), we have the homomorphism $\Theta_{R,m}\colon
W_{F}\to Q_{R,m}\rtimes\mathbb{Z}$. We define a $W_{F}$-representation
$\tau^{\prime\prime}_{\psi,R_{1},m}$ to be the inflation of
$\mathrm{Ind}_{Q^{\prime}_{R,m}\rtimes\mathbb{Z}}^{Q_{R,m}\rtimes\mathbb{Z}}\tau^{\prime}_{\psi,R_{1},m}$
via $\Theta_{R,m}$. We have $\dim\tau^{\prime\prime}_{\psi,R_{1},m}=p^{e}$ by
$[Q_{R,m}:Q^{\prime}_{R,m}]=p^{e-e^{\prime}}$ and
$\dim\tau^{\prime}_{\psi,R_{1},m}=p^{e^{\prime}}$.

######  Proposition 4.7.

We have an isomorphism
$\tau_{\psi,R,m}\simeq\tau^{\prime\prime}_{\psi,R_{1},m}$ as
$W_{F}$-representations.

###### Proof.

By Lemma 4.6, we have the injection

$\tau^{\prime}_{\psi,R_{1},m}=H_{\rm
c}^{1}(C_{R_{1},\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]\xrightarrow{\phi^{\ast}}H_{\rm
c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$

as $Q^{\prime}_{R,m}\rtimes\mathbb{Z}$-representations. Hence we have a non-
zero homomorphism

$\mathrm{Ind}_{Q^{\prime}_{R,m}\rtimes\mathbb{Z}}^{Q_{R,m}\rtimes\mathbb{Z}}\tau^{\prime}_{\psi,R_{1},m}\to
H_{\rm c}^{1}(C_{R,\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi]$ (4.5)

as $Q_{R,m}\rtimes\mathbb{Z}$-representations by Frobenius reciprocity. Since
the target is irreducible by Lemma 2.8(2), the map (4.5) is surjective. By
comparing the dimensions, (4.5) is an isomorphism. By inflating it by
$\Theta_{R,m}$, we obtain the claim. ∎

We consider the open subgroup
$W^{\prime}:=\Theta_{R,m}^{-1}(Q^{\prime}_{R,m}\rtimes\mathbb{Z})\subset
W_{F}$ of index $p^{e-e^{\prime}}$. We can write $W^{\prime}=W_{F^{\prime}}$
with a finite field extension $F^{\prime}/F$ of degree $p^{e-e^{\prime}}$. Let

$\tau^{\prime}_{\psi,R_{1},m}\colon
W_{F^{\prime}}\xrightarrow{\Theta_{R,m}}Q^{\prime}_{R,m}\rtimes\mathbb{Z}\xrightarrow{\pi^{\prime}}Q_{R_{1},m}\rtimes\mathbb{Z}\to\mathrm{Aut}_{\overline{\mathbb{Q}}_{\ell}}(H_{\rm
c}^{1}(C_{R_{1},\mathbb{F}},\overline{\mathbb{Q}}_{\ell})[\psi])$ (4.6)

be the composite.

######  Corollary 4.8.

We have an isomorphism
$\tau_{\psi,R,m}\simeq\operatorname{Ind}_{W_{F^{\prime}}}^{W_{F}}\tau^{\prime}_{\psi,R_{1},m}$
as $W_{F}$-representations. If $e^{\prime}<e$, the $W_{F}$-representation
$\tau_{\psi,R,m}$ is imprimitive.

###### Proof.

The assertion follows from Proposition 4.7. ∎

###  4.2 Totally isotropic subspace and geometry of $C_{R}$

Let $(1,\beta,\gamma)\in H_{R}$ so, as in Definition 2.3(2), we know that
$\gamma^{p}-\gamma=\beta R(\beta)$. We obtain
$(f_{R}(\beta,\beta)-2\gamma)^{p}=f_{R}(\beta,\beta)-2\gamma$ by definition of
the pairing $\omega_{R}$ (Lemma 2.6(2)). Assume

$\beta\neq 0,\quad\gamma=\frac{f_{R}(\beta,\beta)}{2}.$ (4.7)

The following lemma is given in [6, Propositions (9.1) and (13.5)] and [1,
Proposition 7.2]. This lemma gives an algorithm of taking quotients of $C_{R}$
by certain abelian groups.

######  Lemma 4.9.

Let $C_{R}$ be as in Definition 2.7. Assume $e\geq 1$.

  * (1)

Let

$u:=x^{p}-\beta^{p-1}x,\quad v:=a+(x/\beta)(\gamma(x/\beta)-f_{R}(x,\beta)).$
(4.8)

Then there exists $P_{1}(u)\in\mathscr{A}_{\mathbb{F}}$ of degree $p^{e-1}$
such that $v^{p}-v=uP_{1}(u)$.

  * (2)

Let $U:=\\{(1,\xi\beta,\xi^{2}\gamma)\in
H_{R}\mid\xi\in\mathbb{F}_{p}\\}=\langle(1,\beta,\gamma)\rangle$. Then the
quotient $C_{R,\mathbb{F}}/U$ is isomorphic to $C_{P_{1},\mathbb{F}}$.

###### Proof.

We show (1). Let $x_{1}:=x/\beta$ and $u_{1}:=u/\beta^{p}$. Then
$u_{1}=x_{1}^{p}-x_{1}$. We compute

$\displaystyle v^{p}-v$ $\displaystyle=xR(x)+\gamma^{p}x_{1}^{2p}-\gamma
x_{1}^{2}-x_{1}^{p}{f_{R}(x,\beta)}^{p}+x_{1}f_{R}(x,\beta)$
$\displaystyle=xR(x)+\gamma(x_{1}^{2p}-x_{1}^{2})+\beta^{-2p+1}R(\beta)x^{2p}$
$\displaystyle\ \ -u_{1}f_{R}(x,\beta)-(x/\beta)^{p}(\beta R(x)+xR(\beta))$
$\displaystyle=u\beta^{-p}(-\beta
R(x)+\beta^{-p+1}R(\beta)x^{p}+\gamma(x_{1}^{p}+x_{1})-f_{R}(x,\beta)),$

where we use $\gamma^{p}-\gamma=\beta R(\beta)$ and Lemma 2.2 for the second
equality. Let $P(x):=\beta^{-p}(-\beta
R(x)+\beta^{-p+1}R(\beta)x^{p}+\gamma(x_{1}^{p}+x_{1})-f_{R}(x,\beta))$. Since
$P(x)$ is additive, there exists $P_{1}(u)\in\mathscr{A}_{\mathbb{F}}$ such
that $P(x)=P_{1}(u)+\alpha x$ with a constant $\alpha$. By (4.7), we have
$P(\beta)=\beta^{-p}(2\gamma-f_{R}(\beta,\beta))=0$. Hence $\alpha=0$. By
$\deg P(x)=p^{e}$, we have $\deg P_{1}(u)=p^{e-1}$. Hence we obtain (1).

We show (2). We easily check that the finite étale morphism of degree $p$:
$C_{R,\mathbb{F}}\to C_{P_{1},\mathbb{F}};\ (a,x)\mapsto(v,u)$ factors through
$C_{R,\mathbb{F}}\to C_{R,\mathbb{F}}/U\to C_{P_{1},\mathbb{F}}$. Hence we
obtain the claim. Since $C_{R,\mathbb{F}}\to C_{R,\mathbb{F}}/U$ is a finite
étale morphism of degree $p$, the claim follows. ∎

Let

$\Delta_{0}(x):=-(x/\beta)(\gamma(x/\beta)-f_{R}(x,\beta)).$

We have

$xR(x)=uP_{1}(u)+\Delta_{0}(x)^{p}-\Delta_{0}(x)$ (4.9)

by Lemma 4.9(1). We write $u(x)$ for $u$.

Let $(1,\beta^{\prime},\gamma^{\prime})\in H_{R}$ be an element satisfying
(4.7). Assume $\omega_{R}(\beta,\beta^{\prime})=0$. Then $(1,\beta,\gamma)$
commutes with $(1,\beta^{\prime},\gamma^{\prime})$. Hence the action of
$(1,\beta^{\prime},\gamma^{\prime})$ induces the automorphism of
$C_{P_{1},\mathbb{F}}\simeq C_{R,\mathbb{F}}/U$ ((2.5)).

######  Lemma 4.10.

Let
$\pi(\beta^{\prime},\gamma^{\prime}):=(1,u(\beta^{\prime}),\gamma^{\prime}-\Delta_{0}(\beta^{\prime}))$.

  * (1)

We have $\pi(\beta^{\prime},\gamma^{\prime})\in H_{P_{1}}$ and
$f_{P_{1}}(u(\beta^{\prime}),u(\beta^{\prime}))=2(\gamma^{\prime}-\Delta_{0}(\beta^{\prime}))$.

  * (2)

The action of $(1,\beta^{\prime},\gamma^{\prime})$ on $C_{R,\mathbb{F}}$
induces $\pi(\beta^{\prime},\gamma^{\prime})$ on $C_{P_{1},\mathbb{F}}$.

###### Proof.

Let
$\Delta_{1}(x):=f_{R}(x,\beta^{\prime})-\Delta_{0}(x+\beta^{\prime})+\Delta_{0}(x)$.
By (4.8), the action of $(1,\beta^{\prime},\gamma^{\prime})$ on
$C_{R,\mathbb{F}}$ induces the automorphism of $C_{P_{1},\mathbb{F}}$ given by
$u\mapsto u+u(\beta^{\prime})$ and $v\mapsto v+\Delta_{1}(x)+\gamma^{\prime}$
on $C_{P_{1},\mathbb{F}}$. We can easily check that
$\Delta_{1}(x)-\Delta_{1}(0)$ is an additive polynomial such that
$\Delta_{1}(\beta)-\Delta_{1}(0)=\omega_{R}(\beta,\beta^{\prime})=0$. Hence
there exists $g(u)\in\mathbb{F}_{q}[u]$ such that
$\Delta_{1}(x)=g(u(x))+\Delta_{1}(0)$. Lemma 4.1 implies that
$E_{P_{1}}(u(\beta^{\prime}))=0$. Hence $u(\beta^{\prime})\in V_{P_{1}}$. We
show (1). The former claim follows from (4.9). By using
$\Delta_{0}(0)=E_{P_{1}}(u(\beta^{\prime}))=E_{R}(\beta^{\prime})=0$ in the
same way as Lemma 4.3, we have

$\Delta_{0}(x+\beta^{\prime})+f_{P_{1}}(u(x),u(\beta^{\prime}))=\Delta_{0}(x)+\Delta_{0}(\beta^{\prime})+f_{R}(x,\beta^{\prime}).$
(4.10)

Substituting $x=\beta^{\prime}$, and using
$\Delta_{0}(2\beta^{\prime})=4\Delta_{0}(\beta^{\prime})$ and (4.7) for
$(\beta^{\prime},\gamma^{\prime})$, we obtain the latter claim in (1).

By (4.10), we have

$v+f_{R}(x,\beta^{\prime})-\Delta_{0}(x+\beta^{\prime})+\Delta_{0}(x)+\gamma^{\prime}=v+f_{P_{1}}(u(x),u(\beta^{\prime}))+\gamma^{\prime}-\Delta_{0}(\beta^{\prime}).$

Hence the claim (2) follows from (2.5). ∎

Assume that $V_{R}$ is not completely anisotropic. Let $U_{R}$ be a non-zero
totally isotropic $\mathbb{F}_{p}[\mathscr{H}]$-submodule in $V_{R}$. There
exists a monic reduced polynomial $r(x)\in\mathscr{A}_{\mathbb{F}}$ such that
$U_{R}=\\{x\in\mathbb{F}\mid r(x)=0\\}$ by [13, Theorem 7]. Since $U_{R}$ is
an $\mathbb{F}_{p}[\mathscr{H}]$-module, we have

$r(\alpha x)=\alpha r(x)$ for $\alpha\in\mu_{d_{R,m}}$ and
$r(x)\in\mathbb{F}_{q}[x]$ (4.11)

by Lemma 3.24. We write $\deg r(x)=p^{e-e^{\prime}}$ with a non-negative
integer $0\leq e^{\prime}<e$.

We take a basis $\beta_{1},\ldots,\beta_{e-e^{\prime}}$ of $U_{R}$ over
$\mathbb{F}_{p}$. Let $(1,\beta_{i},\gamma_{i})\in H_{R}$ be an element which
satisfies (4.7). Let
$U_{i}:=\\{(1,\xi\beta_{i},\xi^{2}\gamma_{i})\mid\xi\in\mathbb{F}_{p}\\}\subset
H_{R}$, which is a subgroup. Since $U_{R}$ is totally isotropic, we have
$\omega_{R}(\beta_{i},\beta_{j})=0$. Thus $g_{i}g_{j}=g_{j}g_{i}$ for any
$g_{i}\in U_{i}$ and $g_{j}\in U_{j}$ by Lemma 2.6(2). Let

$U^{\prime}_{R}:=U_{1}\cdots U_{e-e^{\prime}}\subset H_{R},$ (4.12)

which is an abelian subgroup.

######  Proposition 4.11.

Assume that $V_{R}$ is not completely anisotropic. Then there exist
$R_{1}(x)\in\mathscr{A}_{\mathbb{F}}$ of degree $p^{e^{\prime}}$ and a
polynomial $\Delta(x)\in\mathbb{F}[x]$ such that $\Delta(0)=0$ and the
quotient $C_{R,\mathbb{F}}/U^{\prime}_{R}$ is isomorphic to the affine curve
$C_{R_{1},\mathbb{F}}$ and the isomorphism is induced by $\pi\colon
C_{R,\mathbb{F}}\to C_{R_{1},\mathbb{F}};\ (a,x)\mapsto(a-\Delta(x),r(x))$. In
particular, we have $xR(x)=r(x)R_{1}(r(x))+\Delta(x)^{p}-\Delta(x)$.
Furthermore, we have $d_{R,m}\mid d_{R_{1}}$.

###### Proof.

By applying Lemmas 4.9 and 4.10 successively, the quotient
$C_{R,\mathbb{F}}/U^{\prime}_{R}$ is isomorphic to the curve
$C_{R_{1},\mathbb{F}}$ with some $R_{1}(x)\in\mathscr{A}_{\mathbb{F}}$, and we
obtain $\pi\colon C_{R,\mathbb{F}}\to C_{R_{1},\mathbb{F}};\
(a,x)\mapsto(a-\Delta(x),r(x))$. By (4.8), we have $\Delta(0)=0$. Since
$U_{R}$ is an $\mathbb{F}_{p}[\mathscr{H}]$-module, the subgroup
$A:=\\{(\alpha,0,0)\in Q_{R,m}\mid\alpha\in\mu_{d_{R,m}}\\}$ normalizes
$U^{\prime}_{R}$. Hence $A$ acts on the quotient $C_{R_{1},\mathbb{F}}$. We
recall that $b^{p}-b=yR_{1}(y)$ is the defining equation of
$C_{R_{1},\mathbb{F}}$. Then $A\ni(\alpha,0,0)$ acts on $C_{R_{1},\mathbb{F}}$
is given by $b\mapsto b+\Delta(x)-\Delta(\alpha^{-1}x),\ y=r(x)\mapsto
r(\alpha^{-1}x)=\alpha^{-1}y$ through the morphism $\pi$ by (4.11). By [6,
Theorems (4.1) and (13.3)] or [1, Theorem 4.3.2], we must have
$\alpha\in\mu_{d_{R_{1}}}$. Hence the last claim follows. ∎

######  Corollary 4.12.

Let the assumption be as in Proposition 4.11. We have
$\Delta(x),R_{1}(x)\in\mathbb{F}_{q}[x]$.

###### Proof.

We use the same notation in Definition 3.23. We consider the equality
$xR(x)=r(x)R_{1}(r(x))+\Delta(x)^{p}-\Delta(x)$ in Proposition 4.11. Let
$S(x):=-R_{1}^{\sigma}(x)+R_{1}(x)$ and
$\Pi(x):=\Delta^{\sigma}(x)-\Delta(x)$. We have
$S(x)\in\mathscr{A}_{\mathbb{F}}$. By $r(x),R(x)\in\mathbb{F}_{q}[x]$,

$\Pi(x)^{p}-\Pi(x)=r(x)S(r(x)).$ (4.13)

Assume $S(x)\neq 0$. We have the non-constant morphism
$f\colon\mathbb{A}_{\mathbb{F}}^{1}\to C_{S,\mathbb{F}};\
x\mapsto(\Pi(x),r(x))$ by $\deg r(x)>0$. Let $\overline{C}_{S,\mathbb{F}}$ be
the smooth compactification of $C_{S,\mathbb{F}}$. The morphism $f$ extends to
a non-constant morphism
$\mathbb{P}_{\mathbb{F}}^{1}\to\overline{C}_{S,\mathbb{F}}$. Hence this is a
finite morphism. By the Riemann–Hurwitz formula, we know that the genus of
$\overline{C}_{S,\mathbb{F}}$ equals zero. This is a contradiction by Lemma
2.10. Hence $S(x)\equiv 0$ and $R_{1}(x)\in\mathbb{F}_{q}[x]$. We have
$\Pi(x)\in\mathbb{F}_{p}$ by (4.13). We have $\Pi(0)=0$ by $\Delta(0)=0$ as in
Proposition 4.11. Hence $\Pi(x)\equiv 0$. Thus the claim follows. ∎

###  4.3 Theorem

Finally, we summarize the contents of §4.1 and §4.2 as a theorem.

######  Theorem 4.13.

Assume $p\neq 2$. The following conditions are equivalent.

  * (1)

There exists a non-trivial finite étale morphism

$C_{R}\to C_{R_{1}};\ (a,x)\mapsto(a-\Delta(x),r(x)),$

where $\Delta(x)\in\mathbb{F}_{q}[x]$ and $r(x),R_{1}(x)\in\mathscr{A}_{q}$
satisfy $d_{R,m}\mid d_{R_{1}}$ and $r(\alpha x)=\alpha r(x)$ for
$\alpha\in\mu_{d_{R,m}}$.

  * (2)

The $\mathbb{F}_{p}[\mathscr{H}]$-module $(V_{R},\omega_{R})$ is not
completely anisotropic.

  * (3)

The $W_{F}$-representation $\tau_{\psi,R,m}$ is imprimitive.

If the above equivalent conditions are satisfied, the $W_{F}$-representation
$\tau_{\psi,R,m}$ is isomorphic to
$\operatorname{Ind}_{W_{F^{\prime}}}^{W_{F}}\tau^{\prime}_{\psi,R_{1},m}$,
where $\tau^{\prime}_{\psi,R_{1},m}$ is given in (4.6).

###### Proof.

Assume (1). Since $C_{R}\to C_{R_{1}}$ is non-trivial, we have $e^{\prime}<e$,
where $\deg r(x)=p^{e-e^{\prime}}$. Then we have (2) by Lemma 4.4. Assume (2).
We obtain (1) by (4.11), Proposition 4.11 and Corollary 4.12.

The equivalence of (2) and (3) follows from Corollary 3.28(1).

The last claim follows from Corollary 4.8. ∎

#### Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers 20K03529/21H00973.

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Takahiro Tsushima
Department of Mathematics and Informatics, Faculty of Science, Chiba
University 1-33 Yayoi-cho, Inage, Chiba, 263-8522, Japan
<EMAIL_ADDRESS>