Local Galois representations associated to additive polynomials

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 obtain a necessary and sufficient condition for it to be primitive.

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 $\sum _{i=0}^e a_i x^{p^i}\ (a_e \ne 0)$ with an integer $e \ge 0$. 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 $\ell \ne p$. For a non-trivial character $\psi :\mathbb {F}_p \rightarrow \overline{\mathbb {Q}}_{\ell }^{\times }$, 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 $a^p-a=x R(x)$ in $\mathbb {A}_{\mathbb {F}_q}^2 ={{\,\textrm{Spec}\,}}\mathbb {F}_q[a,x]$. This curve is studied in [6] and [1] in detail. For example, the smooth compactification of $C_R$ is proved to be supersingular if $(p,e) \ne (2,0)$. The automorphism group of $C_R$ contains a semidirect product $Q_R$ of a cyclic group and an extra-special p-group (Definition 2.7). 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 $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })$ 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 $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ 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 Lemma 3.9(1)). Namely we will define a homomorphism $\Theta _{R,m,\varpi } :W_F \rightarrow Q_R \rtimes \mathbb {Z}$ in (3.17). As a result, we define $\tau _{\psi ,R,m}$ to be the composite

$$\begin{aligned} W_F \xrightarrow {\Theta _{R,m,\varpi }} Q_R \rtimes \mathbb {Z} \rightarrow \hbox {Aut}_{\overline{\mathbb {Q}}_{\ell }}(H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]). \end{aligned}$$

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 [18] and [19]. 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 $\hbox {GL}_{p^e}(F)$ which corresponds to $\tau _{\psi ,R,m}$ under the local Langlands correspondence explicitly. The homomorphism $\Theta _{R,1}$ with $R(x)=x^{p^e}\ (e \in \mathbb {Z}_{\ge 1})$ 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. For example, see [2]. 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 $d_R:=\hbox {gcd}\{p^i+1 \mid a_i \ne 0\}$. 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. Using this criterion, we give examples 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 [9, 11] is constructed geometrically via the Künneth formula.

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

Theorem 1.1

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

• (1) There exists a non-trivial finite étale morphism

  $$\begin{aligned} C_R \rightarrow C_{R_1};\ (a,x) \mapsto \left( a-\Delta (x),r(x)\right) , \end{aligned}$$

  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 $W_{F'}$-representation $\tau '_{\psi ,R_1,m}$ associated to a finite extension $F'/F$. 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.

1.1 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 the reduced scheme structure.

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

Let X be a scheme over $\mathbb {F}_q$ and let $F_q :X \rightarrow X$ be the q-th power Frobenius endomorphism. Let $\mathbb {F}$ be an algebraic closure of $\mathbb {F}_q$. Let $\hbox {Fr}_q :X_{\mathbb {F}} \rightarrow X_{\mathbb {F}}$ be the base change of $F_q$ to $X_{\mathbb {F}}$. This endomorphism $\hbox {Fr}_q$ is called the Frobenius endomorphism of X over $\mathbb {F}_q$.

For a Galois extension l/k, let ${{\,\textrm{Gal}\,}}(l/k)$ 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 write $\mathscr {A}_q$ for $\mathscr {A}_{\mathbb {F}_q}$. Let $R(x):=\sum _{i=0}^e a_i x^{p^i} \in \mathscr {A}_q$ with $e \in \mathbb {Z}_{\ge 0}$ and $a_e \ne 0$. Let

$$\begin{aligned} E_R(x):=R(x)^{p^e}+\sum _{i=0}^e (a_i x)^{p^{e-i}} \in \mathscr {A}_q. \end{aligned}$$

(2.1)

We always assume

$$\begin{aligned} (p,e) \ne (2,0). \end{aligned}$$

(2.2)

This condition and $a_e \ne 0$ guarantee that $E_R(x)$ is a separable polynomial of degree $p^{2e}$. We simply write $\mu _r$ for $\mu _r(\mathbb {F})$ for a positive integer r. Let

$$\begin{aligned} d_R:=\hbox {gcd}\{p^i+1 \mid a_i \ne 0 \}. \end{aligned}$$

If $a_i \ne 0$, we have $\alpha ^{p^i}=\alpha ^{-1}$ and $\alpha ^{p^{e-i}}=\alpha$ for $\alpha \in \mu _{d_R}$. Hence

$$\begin{aligned} \alpha R(\alpha x)= R(x), \quad E_R(\alpha x)=\alpha E_R(x) \quad \hbox {for }\alpha \in \mu _{d_R}. \end{aligned}$$

(2.3)

We consider the polynomial

$$\begin{aligned} f_R(x,y):=-\sum _{i=0}^{e-1}\left( \sum _{j=0}^{e-i-1} (a_i x^{p^i} y)^{p^j} +(x R(y))^{p^i}\right) \in \mathbb {F}_q[x,y]. \end{aligned}$$

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

$$\begin{aligned} f_R(\alpha x,\alpha y) =f_R(x,y) \quad \hbox {for }\alpha \in \mu _{d_R}. \end{aligned}$$

(2.4)

Lemma 2.2

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

Proof

The former equality follows from

$$\begin{aligned} f_R(x,y)^{p}-f_R(x,y)&=xR(y)-(xR(y))^{p^e} +\sum _{i=0}^{e-1} ( a_i x^{p^i}y-(a_i x^{p^i} y)^{p^{e-i}}) \\&\quad =-x^{p^e} E_R(y)+xR(y)+y R(x). \end{aligned}$$

$\square$

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

  $$\begin{aligned} Q_R:=\left\{ (\alpha ,\beta ,\gamma ) \in \mathbb {F}^3 \mid \alpha \in \mu _{d_R}, \ \beta \in V_R,\ \gamma ^{p}-\gamma =\beta R(\beta ) \right\} \end{aligned}$$

  be the group whose group law is given by

  $$\begin{aligned} (\alpha _1,\beta _1,\gamma _1) \cdot (\alpha _2,\beta _2,\gamma _2):=\left( \alpha _1\alpha _2,\beta _1+\alpha _1 \beta _2,\gamma _1+\gamma _2+f_R(\beta _1,\alpha _1 \beta _2)\right) . \end{aligned}$$

  (2.5)

  We check that this is well-defined and $Q_R$ is a group. From (2.3), it follows that $E_R(\alpha _1 \beta _2)=\alpha _1 E_R(\beta _2)=0$. Furthermore, letting $\gamma :=\gamma _1+\gamma _2+f_R(\beta _1,\alpha _1 \beta _2)$, we compute

  $$\begin{aligned} \gamma ^p-\gamma= & {} \beta _1 R(\beta _1)+\beta _2 R(\beta _2)+\beta _1 R(\alpha _1 \beta _2) +\alpha _1 \beta _2R(\beta _1)\\= & {} (\beta _1+\alpha _1 \beta _2) R(\beta _1+\alpha _1 \beta _2), \end{aligned}$$

  where we use Lemma 2.2 for the first equality and use (2.3) for the last one. Hence the right hand side of (2.5) is in $Q_R$. Via (2.4), both of $((\alpha _1,\beta _1,\gamma _1) \cdot (\alpha _2,\beta _2,\gamma _2))\cdot (\alpha _3,\beta _3,\gamma _3)$ and $(\alpha _1,\beta _1,\gamma _1) \cdot ((\alpha _2,\beta _2,\gamma _2)\cdot (\alpha _3,\beta _3,\gamma _3))$ equal

  $$\begin{aligned}{} & {} (\alpha _1\alpha _2\alpha _3,\beta _1+\alpha _1(\beta _2+\alpha _2 \beta _3),\gamma _1+\gamma _2+\gamma _3 + f_R(\beta _1,\alpha _1(\beta _2+\alpha _2\beta _3))\\{} & {} \qquad +f_R(\beta _2,\alpha _2 \beta _3)). \end{aligned}$$

  Finally, (1, 0, 0) is the identity element of $Q_R$ and the inverse element of $(\alpha ,\beta ,\gamma ) \in Q_R$ is given by

  $$\begin{aligned} (\alpha ,\beta ,\gamma )^{-1}=(\alpha ^{-1},-\alpha ^{-1} \beta ,-\gamma +f_R(\beta ,\beta )), \end{aligned}$$

  (2.6)

  where the right hand side is in $Q_R$ due to Lemma 2.2 and (2.3).

• (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 $p \ne 2$ 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 $g,g' \in G$, let $[g,g']:=gg'g^{-1}g'^{-1}$.

Lemma 2.4

For $g=(1,\beta ,\gamma ),\ g'=(1,\beta ',\gamma ') \in H_R$, we have $[g,g']=(1,0,f_R(\beta ,\beta ')-f_R(\beta ',\beta ))$. In particular, we have $f_R(\beta ,\beta ')-f_R(\beta ',\beta ) \in \mathbb {F}_p$.

Proof

Using (2.6) and letting $\gamma _1:=-\gamma -\gamma '+f_R(\beta ,\beta )+f_R(\beta ',\beta ')+f_R(\beta ,\beta ')$, we compute

$$\begin{aligned}{}[g,g']&=(1,\beta ,\gamma ) (1,\beta ',\gamma ') (1,-\beta ,-\gamma +f_R(\beta ,\beta )) (1,-\beta ',-\gamma '+f_R(\beta ',\beta ')) \\&=(1,\beta +\beta ',\gamma +\gamma '+f_R(\beta ,\beta ')) (1,-\beta -\beta ',\gamma _1) \\&=(1,0,f_R(\beta ,\beta )+f_R(\beta ',\beta ')+2f_R(\beta ,\beta ')-f_R(\beta +\beta ',\beta +\beta ')) \\&=(1,0,f_R(\beta ,\beta ')-f_R(\beta ',\beta )). \\ \end{aligned}$$

$\square$

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 $e \ge 1$.

• (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 $\mathbb {F}_p$-bilinear form $\omega _R :V_R \times V_R \rightarrow \mathbb {F}_p;\ (\beta ,\beta ') \mapsto f_R(\beta ,\beta ')-f_R(\beta ',\beta )$ 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 $\beta \ne 0$. Since $V_R$ has dimension 2e, we have $V_R \nsubseteq X_{\beta }$ for $\beta \in V_R {\setminus } \{0\}$. We take $\beta ' \in V_R {\setminus } X_{\beta }$ and $g=(1,\beta ,\gamma ), g'=(1,\beta ',\gamma ') \in H_R$. Then $[g,g']=(1,0,f_R(\beta ,\beta ')-f_R(\beta ',\beta )) \ne 1$ in $H_R$ according to Lemma 2.4. Hence $H_R$ is non-abelian.

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 $\beta =0$. Thus 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 $|Z(H_R)|=p$. Thus $H_R$ is extra-special. Assume $\omega _R(\beta ,\beta ')=0$ for any $\beta ' \in V_R$. We take an element $(1,\beta ,\gamma ) \in H_R$. By Lemma 2.4, we have $(1,\beta ,\gamma ) \in Z(H_R)$. Thus $\beta =0$ by (1). $\square$

Definition 2.7

• (1) Let $C_R$ be the affine curve over $\mathbb {F}_q$ defined by $a^{p}-a=x R(x)$.

• (2) Let $Q_R$ act on $C_{R,\mathbb {F}}$ by

  $$\begin{aligned} (a,x) \cdot (\alpha ,\beta ,\gamma )= \left( a+f_R(x,\beta )+\gamma , \alpha ^{-1} (x+\beta )\right) , \end{aligned}$$

  (2.7)

  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 $\ell \ne p$. For a finite abelian group A, let $A^{\vee }$ denote the character group $\hbox {Hom}_{\mathbb {Z}}(A,\overline{\mathbb {Q}}_{\ell }^{\times })$. 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 $W' \subset H_R$ be the inverse image of W by the natural map $H_R \rightarrow V_R;\ (1,\beta ,\gamma ) \mapsto \beta$. Let $\xi \in W'^{\vee }$ be an extension of $\psi \in Z(H_R)^{\vee }$. Let $\rho _{\psi }:=\hbox {Ind}_{W'}^{H_R} \xi$. 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 $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ is isomorphic to $\rho _{\psi }$ as $H_R$-representations.

Proof

From Lemma 2.4, it follows that the subgroup $W' \subset H_R$ is abelian, since W is totally isotropic via $\omega _R$. Hence an extension $\xi \in W'^{\vee }$ of $\psi$ always exists. From Lemma 2.6(2) and [8, 16.14(2) Satz], the claim (1) follows. By [18, Remark 3.29], we have $\dim H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]=p^e$. Hence the claim (2) follows from (1).

$\square$

The representation $\rho _{\psi }$ induces a projective representation

$$\begin{aligned} \bar{\rho }_{\psi } :H_R/Z(H_R) \rightarrow \hbox {PGL}_{p^e}(\overline{\mathbb {Q}}_{\ell }). \end{aligned}$$

Lemma 2.9

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

Proof

As in the proof of [15, Theorem 6], we have ${{\,\textrm{Tr}\,}}\rho _{\psi }(x)=0$ for $x \in H_R {\setminus } Z(H_R)$. Assume $\bar{\rho }_{\psi } (x Z(H_R))=1$ for $x \in H_R$. Then $\rho _{\psi }(x)$ is a non-zero scalar matrix. Hence ${{\,\textrm{Tr}\,}}\rho _{\psi }(x) \ne 0$. This implies $x \in Z(H_R)$. $\square$

Let $\mathbb {Z} \ni 1$ act on $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })$ by the pull-back $\hbox {Fr}_q^*$. 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 $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$.

A smooth projective geometrically connected curve X over $\mathbb {F}_q$ is said to be supersingular when the Jacobian of $X_{\mathbb {F}}$ is isogenous to a power of a supersingular elliptic curve.

Proposition 2.10

Let $\overline{C}_R$ denote the smooth compactification of $C_R$. The projective curve $\overline{C}_R$ is supersingular. In particular, this curve has positive genus. The natural map $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell }) \rightarrow H^1(\overline{C}_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })$ is an isomorphism.

Proof

The former claim is shown in [6, Theorems (9.4) and (13.7)] ( [1, Proposition 8.5], [17, Theorem 1.1]). The last claim follows from [18, Lemmas 3.27 and 3.28(3)]. $\square$

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. In Lemma 3.36, if R is a monomial, we calculate invariants of the root system corresponding to $(V_R,\omega _R)$ defined in [11].

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. We denote the characteristic of F by ${{\,\mathrm{\hbox {char}}\,}}F$. 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 $\mathcal {O}$.

For an element $x \in \mathcal {O}$, let $\bar{x}$ denote the image of x by the reduction map $\mathcal {O} \rightarrow \mathcal {O}/\mathfrak {p}$. For a positive integer r prime to p, we have the bijection

$$\begin{aligned} \mu _r(\overline{F}) \cup \{0\} \xrightarrow {\sim } \mu _r(\mathbb {F}) \cup \{0\};\ x \mapsto \bar{x}. \end{aligned}$$

(3.1)

The inverse of this map is given by Teichmüller lift. Let q be the cardinality of the residue field of $\mathcal {O}_F$. For an element $a \in \mathbb {F}_q$, let $\widetilde{a} \in \mu _{q-1}(F) \cup \{0\}$ denote its lift via (3.1).

We take $R(x)=\sum _{i=0}^e a_i x^{p^i} \in \mathscr {A}_q$. Let

$$\begin{aligned} \widetilde{R}(x):=\sum _{i=0}^e \widetilde{a}_i x^{p^i}, \quad \widetilde{E}_R(x):=\widetilde{R}(x)^{p^e}+\sum _{i=0}^e (\widetilde{a}_i x)^{p^{e-i}} \in \mathcal {O}_F[x]. \end{aligned}$$

Similarly as in (2.3),

$$\begin{aligned} \alpha \widetilde{R}(\alpha x)=\widetilde{R}(x), \quad \widetilde{E}_R(\alpha x)=\alpha \widetilde{E}_R(x) \quad \hbox { for}\ \alpha \in \mu _{d_R}(\overline{F}). \end{aligned}$$

(3.2)

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

$$\begin{aligned} \alpha _{R,\varpi }^{d_R}=\varpi , \quad \widetilde{E}_R(\beta _{R,m,\varpi })=\alpha _{R,\varpi }^{-m}, \quad \gamma _{R,m,\varpi }^{p}-\gamma _{R,m,\varpi }= \beta _{R,m,\varpi } \widetilde{R}(\beta _{R,m,\varpi }). \end{aligned}$$

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

$$\begin{aligned} v(\alpha _R)=\frac{1}{d_R}, \quad v(\beta _{R,m})=-\frac{m}{p^{2e}d_R}, \quad v(\gamma _{R,m})=-\frac{m(p^e+1)}{p^{2e+1}d_R}. \end{aligned}$$

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

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

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

$$\begin{aligned} \begin{aligned}&\beta _{R,m}^{p^e}(\widetilde{E}_R(\beta _{R,m}+x)-\widetilde{E}_R(\beta _{R,m})-\widetilde{E}_R(x)),\\&\beta _{R,m} (\widetilde{R}(\beta _{R,m}+x) -\widetilde{R}(\beta _{R,m})-\widetilde{R}(x)), \\&\widetilde{f}(\beta _{R,m},x)^{p}-\widetilde{f}(\beta _{R,m},x)+\beta _{R,m}^{p^e} \widetilde{E}_R(x)-x \widetilde{R}(\beta _{R,m})-\beta _{R,m}\\&\widetilde{R}(x)\hbox { are contained in }\mathfrak {p}[x]\hbox { and}\\&(\gamma _{R,m}+\widetilde{f}(\beta _{R,m},y)+x)^{p}-\gamma _{R,m}^{p}-\widetilde{f}(\beta _{R,m},y)^{p}-x^{p} \in \mathfrak {p}[x,y]. \end{aligned} \end{aligned}$$

(3.3)

If ${{\,\mathrm{\hbox {char}}\,}}F=p$, these differences are zero by $(x+y)^p=x^p+y^p$ and Lemma 2.2. Thus (3.3) is always satisfied in this case.

For $r \in \mathbb {Q}_{\ge 0}$ 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 $\overline{F}/K$. Let

$$\begin{aligned} n :W_K \twoheadrightarrow \mathbb {Z};\ \sigma \mapsto n_{\sigma } \end{aligned}$$

(3.4)

denote the homomorphism defined by $\sigma (x)\equiv x^{q^{-n_{\sigma }}} \mod 0+$ 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

$$\begin{aligned} \begin{aligned} a_{R,\sigma }&:=\sigma (\alpha _R)/\alpha _R \in \mu _{d_R}(\overline{F}), \quad b_{R,\sigma }:= a_{R,\sigma }^m \sigma (\beta _{R,m})-\beta _{R,m}, \\ c_{R,\sigma }&:=\sigma (\gamma _{R,m})-\gamma _{R,m} -\widetilde{f}(\beta _{R,m},b_{R,\sigma }). \end{aligned} \end{aligned}$$

(3.5)

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

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 (3.2), the equality $\widetilde{E}_R(\beta )=\alpha ^{-m}$ in Definition 3.1 and (3.5),

$$\begin{aligned} \widetilde{E}_R(\beta +b_{\sigma }) =\widetilde{E}_R(a_{\sigma }^m \sigma (\beta )) =a_{\sigma }^m \widetilde{E}_R(\sigma (\beta )) =a_{\sigma }^m \sigma (\alpha )^{-m}=\alpha ^{-m} =\widetilde{E}_R(\beta ). \end{aligned}$$

Using $v(\beta )<0$ in Remark 3.2 and (3.3), 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 obtain that $\widetilde{E}_R(b_{\sigma })+\Delta (b_{\sigma })=0$. Hence $b_{\sigma } \in \mathcal {O}$ and $E_R(\bar{b}_{\sigma })=0$.

By (3.3), we have

$$\begin{aligned}{} & {} \beta \widetilde{R}(\beta +b_{\sigma }) \equiv \beta \widetilde{R}(\beta ) +\beta \widetilde{R}(b_{\sigma }), \nonumber \\{} & {} \quad \widetilde{f}(\beta ,b_{\sigma })^{p}-\widetilde{f}(\beta ,b_{\sigma }) \equiv b_{\sigma }\widetilde{R}(\beta )+\beta \widetilde{R}(b_{\sigma }) \mod 0+. \end{aligned}$$

(3.6)

Substituting $y=b_{\sigma } \in \mathcal {O}$ to (3.3), we obtain

$$\begin{aligned} \Delta _1(x):=(\gamma +\widetilde{f}(\beta , b_{\sigma })+x)^{p}-\gamma ^{p}-\widetilde{f}(\beta , b_{\sigma })^{p}-x^{p} \in \mathfrak {p}[x]. \end{aligned}$$

We have $\sigma (\beta )\widetilde{R}(\sigma (\beta ))=(\beta +b_{\sigma }) \widetilde{R}(\beta +b_{\sigma })$ by substituting (3.5) and using (3.2). By multiplying the first congruence in (3.6) 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

$$\begin{aligned} \sigma (\gamma )^{p}-\sigma (\gamma )&=\sigma (\beta )\widetilde{R}(\sigma (\beta ))=(\beta +b_{\sigma }) \widetilde{R}(\beta +b_{\sigma }) \\&\equiv \beta \widetilde{R}(\beta )+b_{\sigma }\widetilde{R}(\beta )+\beta \widetilde{R}(b_{\sigma })+b_{\sigma }\widetilde{R}(b_{\sigma }) \\&\equiv \gamma ^{p}-\gamma +\widetilde{f}(\beta , b_{\sigma })^{p}-\widetilde{f}(\beta ,b_{\sigma })+b_{\sigma }\widetilde{R}(b_{\sigma }) \\&\equiv \sigma (\gamma )^{p}-\sigma (\gamma ) -(c_{\sigma }^{p}-c_{\sigma }+\Delta _1(c_{\sigma })) +b_{\sigma }\widetilde{R}(b_{\sigma }) \mod 0+, \end{aligned}$$

where we have used (3.5) for the last congruence. Therefore, 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 $\bar{c}_{\sigma }^p-\bar{c}_{\sigma }=\bar{b}_{\sigma }R(\bar{b}_{\sigma })$. $\square$

Assume that

$$\begin{aligned} \begin{aligned}&(x+\beta _{R,m})^{p^i}-x^{p^i}-\beta _{R,m}^{p^i} \in \mathfrak {p}[x] \quad \hbox {for }1 \le i \le e-1, \\&\widetilde{f}(\beta _{R,m},x+y)-\widetilde{f}(\beta _{R,m},x)-\widetilde{f}(\beta _{R,m},y) \in \mathfrak {p}[x,y], \end{aligned} \end{aligned}$$

(3.7)

which are satisfied if ${{\,\mathrm{\hbox {char}}\,}}F=p$, because these differences are zero. Let

$$\begin{aligned} \Theta _{R,m,\varpi } :W_F \rightarrow Q_R \rtimes \mathbb {Z};\ \sigma \mapsto ((\bar{a}_{\sigma }^m,\bar{b}_{\sigma },\bar{c}_{\sigma }), n_{\sigma }). \end{aligned}$$

(3.8)

Lemma 3.5

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

Proof

Let $\sigma , \sigma ' \in W_F$. 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 the claim to checking that

$$\begin{aligned} \bar{a}_{\sigma \sigma '}= & {} \bar{a}_{\sigma } \bar{a}_{\sigma '}^{q^{-n_{\sigma }}}, \quad \nonumber \\ \bar{b}_{\sigma \sigma '}= & {} \bar{a}_{\sigma }^m \bar{b}_{\sigma '}^{q^{-n_{\sigma }}}+\bar{b}_{\sigma }, \nonumber \\ \bar{c}_{\sigma \sigma '}= & {} \bar{c}_{\sigma }+\bar{c}_{\sigma '}^{q^{-n_{\sigma }}} +f_R(\bar{b}_{\sigma },\bar{a}_{\sigma }^m \bar{b}_{\sigma '}^{-n_{\sigma }}). \end{aligned}$$

(3.9)

We easily check that $a_{\sigma \sigma '}=\sigma (a_{\sigma '}) a_{\sigma }$ and $b_{\sigma \sigma '}= a_{\sigma }^m \sigma (b_{\sigma '})+b_{\sigma }$. Hence the first two equalities in (3.9) follow. We compute

$$\begin{aligned} c_{\sigma \sigma '}&=c_{\sigma }+\sigma (c_{\sigma '}) +\sigma (\widetilde{f}(\beta ,b_{\sigma '})) +\widetilde{f}(\beta ,b_{\sigma })- \widetilde{f} (\beta ,b_{\sigma \sigma '}) \\&\equiv c_{\sigma }+\sigma (c_{\sigma '}) +\sigma (\widetilde{f}(\beta ,b_{\sigma '})) -\widetilde{f} (\beta ,a_{\sigma }^m \sigma (b_{\sigma '})) \mod 0+, \end{aligned}$$

where we use the second condition in (3.7) for the second congruence. We have

$$\begin{aligned} \sigma (\widetilde{f}(\beta ,b_{\sigma '}))&= -\sum _{i=0}^{e-1} \sum _{j=0}^{e-i-1} (\widetilde{a}_j \sigma (b_{\sigma '}) \sigma (\beta )^{p^i})^{p^j} -\sum _{i=0}^{e-1} (\sigma (\beta ) \widetilde{R}(\sigma (b_{\sigma '})))^{p^i} \\&\equiv \widetilde{f}(b_{\sigma }, a_{\sigma }^m\sigma (b_{\sigma '}))+\widetilde{f} (\beta , a_{\sigma }^m \sigma (b_{\sigma '})) \mod 0+, \end{aligned}$$

where we substitute $\sigma (\beta )=a_{\sigma }^{-m}(\beta +b_{\sigma })$, (3.7) and (3.2) for the second congruence. The last equality in (3.9) follows from $\overline{\widetilde{f}(b_{\sigma }, a_{\sigma }^m\sigma (b_{\sigma '}))}=f_R (\bar{b}_{\sigma },\bar{a}_{\sigma }^m \bar{b}_{\sigma '}^{q^{-n_{\sigma }}})$, since $\widetilde{f}(x,y)$ is a lift of $f_R(x,y)$ to $\mathcal {O}_F[x,y]$. $\square$

Lemma 3.6

If v(p) is large enough, the conditions (3.3) and (3.7) are satisfied.

Proof

There exists $s \in \mathbb {Z}_{\ge 1}$ such that the coefficients of all polynomials in (3.3) and (3.7) 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. $\square$

In the sequel, we assume that the conditions (3.3) and (3.7) are satisfied. Let $F^\textrm{ur}$ denote the maximal unramified extension of F in $\overline{F}$.

Lemma 3.7

The extension $F^\textrm{ur}(\alpha _R^m,\beta _{R,m},\gamma _{R,m})/F$ is Galois.

Proof

Let $L_0:=F^\textrm{ur}(\alpha ^m,\beta ,\gamma )$ and $L:=\widehat{L}_0$ be the completion of $L_0$. Let $\sigma \in G_F$. We note that $a_{\sigma } \in \mu _{d_R}(\overline{F}) \subset F^\textrm{ur}$ by $p \not \mid d_R$. 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

$$\begin{aligned} b_{\sigma }, c_{\sigma } \in L_0, \end{aligned}$$

since

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

by (3.5). 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. Thus by Hensel’s lemma, $E(x)=0$ has $p^{2e}$ different roots in $\mathcal {O}_L$. Hence $b_{\sigma } \in L \cap \overline{F} =L_0$. As in the proof of Lemma 3.4, we have

$$\begin{aligned} f(c_{\sigma }):=c_{\sigma }^p-c_{\sigma }+ \Delta _1(c_{\sigma })-y=0\ \hbox {with }y \in \mathcal {O}_{L_0}, \end{aligned}$$

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 $c_{\sigma } \in L \cap \overline{F}=L_0$. $\square$

Definition 3.8

Let

$$\begin{aligned} d_{R,m}:=\frac{d_R}{\hbox {gcd}(d_R,m)}, \quad Q_{R,m}:=\{(\alpha ,\beta ,\gamma ) \in Q_R \mid \alpha \in \mu _{d_{R,m}}\}. \end{aligned}$$

We have

$$\begin{aligned} F^\textrm{ur} \subset F^\textrm{ur}(\alpha _R^m) \subset F^\textrm{ur}(\alpha _R^m,\beta _{R,m}) \subset F^\textrm{ur}(\alpha _R^m,\beta _{R,m},\gamma _{R,m}). \end{aligned}$$

(3.10)

Using Definition 3.1 and $p \not \mid d_{R,m}$, the first extension is a tamely ramified extension of degree $d_{R,m}$. According to Remark 3.2, the second and last extensions are totally ramified extensions of degree $p^{2e}$ and p, respectively. Thus

$$\begin{aligned}{}[F^\textrm{ur}(\alpha _R^m,\beta _{R,m},\gamma _{R,m}):F^\textrm{ur}]=d_{R,m} p^{2e+1}. \end{aligned}$$

(3.11)

Lemma 3.9

• (1) The homomorphism $\Theta _{R,m,\varpi }$ in (3.8) induces the isomorphism

  $$\begin{aligned} W(F^\textrm{ur}(\alpha _R^m,\beta _{R,m},\gamma _{R,m})/F) \xrightarrow {\sim } Q_{R,m} \rtimes \mathbb {Z};\ \sigma \mapsto ((\bar{a}_{\sigma }^m,\bar{b}_{\sigma },\bar{c}_{\sigma }),n_{\sigma }).\nonumber \\ \end{aligned}$$

  (3.12)

• (2) The homomorphism $\Theta _{R,m,\varpi }$ induces

  $$\begin{aligned} {{\,\textrm{Gal}\,}}(F^\textrm{ur}(\alpha _R^m,\beta _{R,m},\gamma _{R,m})/F^\textrm{ur})&\xrightarrow {\sim } Q_{R,m}, \\ {{\,\textrm{Gal}\,}}(F^\textrm{ur}(\alpha _R^m,\beta _{R,m},\gamma _{R,m})/F^\textrm{ur}(\alpha _R^m))&\xrightarrow {\sim } H_R. \end{aligned}$$

Proof

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

$$\begin{aligned} I:={{\,\textrm{Gal}\,}}(F^\textrm{ur}(\alpha ^m,\beta ,\gamma )/F^\textrm{ur}) \supset P:={{\,\textrm{Gal}\,}}(F^\textrm{ur}(\alpha ^m,\beta ,\gamma )/F^\textrm{ur}(\alpha ^m)) \end{aligned}$$

and $\Theta :I \rightarrow Q_{R, m}$ be the restriction of (3.12). By the snake lemma, the assertion (1) is reduced to showing that $\Theta$ is an isomorphism. In order to show that $\Theta$ is an isomorphism, it suffices to show that $\Theta$ is injective according to (3.11) and $|Q_{R,m}|=d_{R,m} p^{2e+1}$. If $\Theta$ is injective, it follows that $\Theta |_P :P \rightarrow H_R$ is an isomorphism from $|P|=|H_R|=p^{2e+1}$.

We will now 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 $\bar{c}_{\sigma }=0$. Recall (3.1). By $\bar{a}_{\sigma }^m=1$ and $a_{\sigma } \in \mu _{d_R}(\overline{F})$ in (3.5), 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 write $\widetilde{E}_R(x)+\Delta (x)=\sum _{i=1}^r c_i x^i \in \mathcal {O}[x]$. From $E'_R(0) \ne 0$ and $\Delta (x) \in \mathfrak {p}[x]$, it follows that $v(c_1)=0$. We have $v(b_{\sigma })>0$ by $\bar{b}_{\sigma }=0$. Thus, for an integer $2 \le i \le r$, we obtain $v(c_1b_{\sigma })=v(b_{\sigma })<v(b_{\sigma }^i)\le v(c_i b_{\sigma }^i)$. Hence $v(b_{\sigma })=v(c_1 b_{\sigma })=v(\widetilde{E}_R(b_{\sigma })+\Delta (b_{\sigma }))=\infty$. Hence $b_{\sigma }=0$ and $\sigma (\beta )=\beta$.

By the last condition in (3.3) with $y=0$,

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

Definition 3.1 induces $\sigma (\gamma )^p-\sigma (\gamma )=\gamma ^p-\gamma$. Thus $(\gamma +c_{\sigma })^p-\gamma ^p=c_{\sigma }$ 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 $\sigma (\gamma )=\gamma$. As a result, we obtain $\sigma = 1$. Thus $\Theta$ is injective. $\square$

3.2 Galois representations associated to additive polynomials

3.2.1 Construction of Galois representation

We assume that (3.3) and (3.7) are satisfied. If ${{\,\mathrm{\hbox {char}}\,}}F$ is positive, these are unconditional. If ${{\,\mathrm{\hbox {char}}\,}}F$ 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 $Q_R \rtimes \mathbb {Z}$ -representation $H_\textrm{c}^1 (C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ by $\Theta _{R,m,\varpi }$ in (3.8). 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 ${{\,\textrm{Ker}\,}}\tau _{\psi ,R,m}$ contains the open compact subgroup $I_{F(\alpha _R^m,\beta _{R,m},\gamma _{R,m})}$ by Lemma 3.9(1). According to Lemma 3.9(2) and Lemma 2.8(1), $\tau _{\psi ,R,m}|_{I_{F(\alpha _R^m)}}$ is irreducible. Hence the representation $\tau _{\psi ,R,m}$ is a smooth irreducible representation of $W_F$.

Let $G_F:={{\,\textrm{Gal}\,}}(\overline{F}/F)$. 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 $\widetilde{\tau }':=\widetilde{\tau } \otimes \phi$ and $\tau ':=\tau \otimes \phi |_{W_F}$. The subgroup ${{\,\textrm{Ker}\,}}\widetilde{\tau }'$ is open by $|G_F/{{\,\textrm{Ker}\,}}\widetilde{\tau }'|<\infty$. Hence ${{\,\textrm{Ker}\,}}\tau ' \subset W_F$ is open. Therefore $\tau '$ is smooth. Hence so is $\tau$. Since $\tau$ is irreducible and smooth, we have $\dim \tau <\infty$. We will show that $\widetilde{\tau }'$ is imprimitive if and only if $\tau$ is imprimitive.

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

To the contrary, assume $\tau \simeq {{\,\textrm{Ind}\,}}_H^{W_F} \sigma$. 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 $H=W_{F'}$ with a finite extension $F'/F$. Let $\sigma ':=\sigma \otimes \phi |_{W_{F'}}$. We have $\tau ' \simeq {{\,\textrm{Ind}\,}}_{W_{F'}}^{W_F} \sigma '$. From Frobenius reciprocity, we have that $\sigma '(W_{F'}) \subset \tau '(W_F)$. By the assumption, the image $\sigma '(W_{F'})$ is finite. Hence the smooth $W_{F'}$-representation $\sigma '$ extends to a smooth representation of $G_{F'}$, for which we write $\widetilde{\sigma }$ ( [2, Proposition 28.6]). The restriction of ${{\,\textrm{Ind}\,}}_{G_{F'}}^{G_F} \widetilde{\sigma }$ to $W_F$ is isomorphic to ${{\,\textrm{Ind}\,}}_{W_{F'}}^{W_F} \sigma ' \simeq \tau '$. Both of ${{\,\textrm{Ind}\,}}_{G_{F'}}^{G_F} \widetilde{\sigma }$ and $\widetilde{\tau }'$ are smooth irreducible $G_F$-representations whose restrictions to $W_F$ are isomorphic to $\tau '$. Hence we obtain an isomorphism $\widetilde{\tau }' \simeq {{\,\textrm{Ind}\,}}_{G_{F'}}^{G_F} \widetilde{\sigma }$ as $G_F$-representations by [2, Lemma 28.6.2(2)]. $\square$

We identify as ${{\,\textrm{Gal}\,}}(\mathbb {F}/\mathbb {F}_q)\xrightarrow {\sim } \widehat{\mathbb {Z}}$, which sends the geometric Frobenius to 1. The group ${{\,\textrm{Gal}\,}}(\mathbb {F}/\mathbb {F}_q)$ acts on $Q_{R,m}$ naturally. Then $H_\textrm{c}^1 (C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ is regarded as a representation of $Q_{R,m} \rtimes \widehat{\mathbb {Z}}$. We identify as ${{\,\textrm{Gal}\,}}(F^\textrm{ur}/F) \xrightarrow {\sim } {{\,\textrm{Gal}\,}}(\mathbb {F}/\mathbb {F}_q) \xrightarrow {\sim } \widehat{\mathbb {Z}}$, where the first isomorphism is the natural map. Let $\widehat{n} :G_F \xrightarrow {\mathrm{rest.}} {{\,\textrm{Gal}\,}}(F^{\textrm{ur}}/F) \xrightarrow {\sim } \widehat{\mathbb {Z}}$ be the composite, which is an extension of $n :W_F \rightarrow \mathbb {Z};\ \sigma \mapsto n_{\sigma }$ in (3.4). Then we have

$$\begin{aligned}{} & {} \widehat{\Theta }_{R,m,\varpi } :G_F \rightarrow Q_{R,m} \rtimes \widehat{\mathbb {Z}};\ \sigma \mapsto ((\bar{a}_{\sigma }^m,\bar{b}_{\sigma },\bar{c}_{\sigma }),\widehat{n}_{\sigma }), \end{aligned}$$

which is defined by the same formulas as in (3.5). Then $\widehat{\Theta }_{R,m,\varpi }$ extends $\Theta _{R,m,\varpi }$. By inflating $H_\textrm{c}^1 (C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ via $\widehat{\Theta }_{R,m,\varpi }$, we obtain a continuous representation of $G_F$, which we denote by $\widetilde{\tau }_{\psi ,R,m}$.

We fix an isomorphism $\iota :\mathbb {C} \xrightarrow {\sim } \overline{\mathbb {Q}}_{\ell }$, and work with the choice of square root q in $\overline{\mathbb {Q}}_{\ell }$ given by $\iota (\sqrt{q})$.

Lemma 3.12

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

Proof

As in [14, §2.3], it is well-known that a smooth projective geometrically connected curve X over $\mathbb {F}_q$ is supersingular if and only if all the eigenvalues of $\hbox {Fr}_q^*$ on $H^1(X_{\mathbb {F}},\overline{\mathbb {Q}}_{\ell })$ have the form $\zeta \sqrt{q}$ with $\zeta \in \mu (\overline{\mathbb {Q}}_{\ell })$. Hence the claim follows from Proposition 2.10. $\square$

Let $\phi :G_F \rightarrow \overline{\mathbb {Q}}_{\ell }^{\times }$ be the unramified character sending a geometric Frobenius to $\sqrt{q}^{-1}$. The image of $G_F$ by the twist $\widetilde{\tau }':=\widetilde{\tau }_{\psi ,R,m} \otimes \phi$ is finite by Lemma 3.12.

By the isomorphism $\iota :\overline{\mathbb {Q}}_{\ell } \simeq \mathbb {C}$, we obtain a continuous representation $\widetilde{\tau }'_{\mathbb {C}}$ of $G_F$ over $\mathbb {C}$ by $\widetilde{\tau }'$. Then $\widetilde{\tau }'_{\mathbb {C}}$ 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 $\widetilde{\tau }'_{\mathbb {C}}$ is primitive.

Proof

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

3.2.2 Swan conductor exponent

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

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

$$\begin{aligned} F \subset F_r \subset T \subset M \subset N. \end{aligned}$$

Let L/K be a Galois extension of non-archimedean local fields with Galois group G. Let $\left\{ G^i\right\} _{i \ge -1}$ denote the upper numbering ramification groups of G in [16, IV §3]. Let $\psi _{L/K}$ denote the Herbrand function of L/K.

Lemma 3.14

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

$$\begin{aligned} \psi _{N/F}(t)= \left\{ \begin{array}{ll} {t} &{} \text {if }t \le 0, \\ {d}_R t &{} \text {if }0<t \le \frac{m}{d_R}, \\ {p}^{2e} d_R t-(p^{2e}-1)m &{} \text {if } \frac{m}{d_R}<t \le \frac{p^e+1}{p^e} \frac{m}{d_R}, \\ {p}^{2e+1} d_R t-(p^e+1)(p^{e+1}-1)m &{} \text {otherwise} \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} G^i=\left\{ \begin{array}{ll} G &{} \hbox {if }i =-1, \\ {{\,\textrm{Gal}\,}}(N/F_r) &{} \hbox {if }-1<i \le 0, \\ {{\,\textrm{Gal}\,}}(N/T) &{} \hbox {if }0<i \le \frac{m}{d_R}, \\ {{\,\textrm{Gal}\,}}(N/M) &{} \hbox {if }\frac{m}{d_R} <i \le \frac{p^e+1}{p^e }\frac{ m}{d_R}, \\ \{1\} &{} \hbox {otherwise}. \end{array}\right. \end{aligned}$$

Proof

Similarly as in (3.10), T/F is a totally ramified extension of degree $d_R$. We easily have

$$\begin{aligned} \psi _{T/F}(t)=\left\{ \begin{array}{ll} t &{} \hbox {if }t \le 0, \\ d_R t &{} \hbox {otherwise}. \end{array}\right. \end{aligned}$$

For a finite Galois extension L/K, let $\{{{\,\textrm{Gal}\,}}(L/K)_u\}_{u \ge -1}$ be the lower numbering ramification subgroups. Let $1 \ne \sigma \in {{\,\textrm{Gal}\,}}(M/T)$. 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 arguments as in the proof of Lemma 3.9. This induces $\sigma =1$. Hence $v(b_{\sigma })=0$. From $v_M(\beta )=-m$, we obtain $v_M(\sigma (\varpi _M)-\varpi _M)=m+1$. Thus

$$\begin{aligned}{} & {} {{\,\textrm{Gal}\,}}(M/T)_u=\left\{ \begin{array}{ll} {{\,\textrm{Gal}\,}}(M/T) &{}\hbox {if }u \le m, \\ \{1\} &{} \hbox {otherwise}, \end{array}\right. \\{} & {} \psi _{M/T}(t)=\left\{ \begin{array}{ll} t &{} \hbox {if }t \le m, \\ p^{2e}t-(p^{2e}-1)m &{} \hbox {otherwise}. \end{array}\right. \end{aligned}$$

Let $1 \ne \sigma \in \hbox {Gal}(N/M)$. 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. From $v_N(\gamma ^{-1})=(p^e+1)m$, it follows that $v_N(\sigma (\varpi _N)-\varpi _N)=(p^e+1)m+1$. Thus

$$\begin{aligned} {{\,\textrm{Gal}\,}}(N/M)_u&=\left\{ \begin{array}{ll} {{\,\textrm{Gal}\,}}(N/M) &{}\hbox {if }u \le (p^e+1)m, \\ \{1\} &{} \hbox {otherwise}, \end{array}\right. \\ \psi _{N/M}(t)&= \left\{ \begin{array}{ll} t &{} \hbox { if}\ t \le (p^e+1)m, \\ p t-(p-1)(p^e+1)m &{} \hbox {otherwise}. \end{array}\right. \end{aligned}$$

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

We can check

$$\begin{aligned} G_u =\left\{ \begin{array}{ll} G &{} \hbox {if }u =-1, \\ {{\,\textrm{Gal}\,}}(N/F_r) &{} \hbox {if }-1<u \le 0, \\ {{\,\textrm{Gal}\,}}(N/T) &{} \hbox {if }0<u \le m, \\ \hbox {Gal}(N/M) &{} \hbox {if }m<u \le (p^e+1)m, \\ \{1\} &{} \hbox {otherwise} \end{array}\right. \end{aligned}$$

by using the former claim and [16, Propositions 12(c), 13(c) and 15 in IV§3]. Hence the latter claim follows from $G^i=G_{\psi _{N/F}(i)}$. $\square$

Corollary 3.15

We have $\hbox {Sw}(\tau _{\psi ,R,m})= m(p^e+1)/d_R$.

Proof

Before Corollary 3.13, it is stated that the twist $\tau _{\psi ,R,m} \otimes \phi$ factors through a finite group $Q_R \rtimes (\mathbb {Z}/r \mathbb {Z}) \simeq {{\,\textrm{Gal}\,}}(F_r(\alpha ,\beta ,\gamma )/F)$ with a certain integer r. Since $\phi$ is unramified, $\hbox {Sw}(\tau _{\psi ,R,m})=\hbox {Sw}(\tau _{\psi ,R,m} \otimes \phi )$. It follows that $\hbox {Sw}(\tau _{\psi ,R,m} \otimes \phi ) =m(p^e+1)/d_R$ from Lemma 3.14 and [7, Théorème 7.7] ( [16, Exercise 2 in §2VI]). $\square$

3.3 Symplectic module associated to Galois representation

We simply write $\hbox {PGL}(\overline{\mathbb {Q}}_{\ell })$ for $\hbox {Aut}_{\overline{\mathbb {Q}}_{\ell }}(H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ])/\overline{\mathbb {Q}}_{\ell }^{\times }$. Let $\rho$ denote the composite

$$\begin{aligned}{} & {} W_F \xrightarrow {\tau _{\psi ,R,m}} \hbox {Aut}_{\overline{\mathbb {Q}}_{\ell }}(H_{\textrm{c}}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]) \xrightarrow {\mathrm{can.}} \hbox {PGL}(\overline{\mathbb {Q}}_{\ell }). \end{aligned}$$

Namely, $\rho$ is the projective representation associated to $\tau _{\psi ,R,m}$. Similarly, let $\rho '$ be the projective representation associated to $\widetilde{\tau }'=\widetilde{\tau }_{\psi ,R,m} \otimes \phi$.

Lemma 3.16

We have $\rho (W_F)=\rho '(G_F)$, which is finite.

Proof

Since $\widetilde{\tau }'$ is a smooth irreducible $G_F$-representation, we have that $\widetilde{\tau }'(G_F) =(\tau _{\psi ,R,m} \otimes \phi )(W_F)$ ( [2, the proof of Lemma 2 in 28.6]). Thus the claim follows. $\square$

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

$$\begin{aligned} H:=\hbox {Gal}(F_{\rho }/T_{\rho }) \subset G:=\hbox {Gal}(F_{\rho }/F). \end{aligned}$$

The homomorphism $\rho$ induces an injection $\bar{\rho } :G \rightarrow \hbox {PGL}(\overline{\mathbb {Q}}_{\ell })$.

Recall that $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ is regarded as a $Q_R \rtimes \mathbb {Z}$-representation as in §2. We have the subgroup $Q_{R,m} \rtimes \mathbb {Z} \subset Q_R \rtimes \mathbb {Z}$. Now we regard $H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ as a $Q_{R,m} \rtimes \mathbb {Z}$-representation. Let $\tau$ denote the composite

$$\begin{aligned} Q_{R,m} \rtimes \mathbb {Z} \rightarrow \hbox {Aut}_{\overline{\mathbb {Q}}_{\ell }}(H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]) \rightarrow \hbox {PGL}(\overline{\mathbb {Q}}_{\ell }). \end{aligned}$$

From Definition 3.10 and Lemma 3.9(1), it follows that $\tau \circ \Theta _{R,m,\varpi }=\rho$. Let $i :H_R \hookrightarrow Q_{R,m} \rtimes \mathbb {Z}$ be the natural inclusion. Since $\tau \circ i$ equals $\bar{\rho }_{\psi }$ in Lemma 2.9, we obtain ${{\,\textrm{Ker}\,}}(\tau \circ i)=Z(H_R)$ according to the lemma. Let $V_R$ be as in Lemma 2.6. Then we have $\hbox {pr} :H_R/Z(H_R) \xrightarrow {\sim } V_R;\ (1,\beta ,\gamma ) Z(H_R) \mapsto \beta$. Thus we have a commutative diagram

(3.13)

Lemma 3.17

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

Proof

Let $L:=F^\textrm{ur}(\alpha _R^m,\beta _{R,m},\gamma _{R,m})$ and $K:=F^\textrm{ur}(\alpha _R^m)$. From Lemma 3.9, we recall

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

The subfield K is the maximal tamely ramified extension of F in L. In the sequel, we freely use (3.13). From Lemma 3.9(1), it follows that $F_{\rho } \subset L$ and $T_{\rho }=F_{\rho } \cap K$. The homomorphism $\Theta _{R,m,\varpi }$ induces $G=W(F_{\rho }/F) \simeq W_F/{{\,\textrm{Ker}\,}}\rho \xrightarrow {\sim } (Q_{R,m} \rtimes \mathbb {Z})/{{\,\textrm{Ker}\,}}\tau$. Thus we have a commutative diagram

where the two horizontal sequences are exact. This induces that

$$\begin{aligned} H=\hbox {Gal}(F_{\rho }/T_{\rho }) \simeq H_R/{{\,\textrm{Ker}\,}}(\tau \circ i)=H_R/Z(H_R) \xrightarrow {\sim } V_R. \end{aligned}$$

Since $\overline{\rho }$ is injective, the claim follows. $\square$

Let

$$\begin{aligned} \mathscr {H}_0:=G/H=\hbox {Gal}(T_{\rho }/F). \end{aligned}$$

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 $\sigma \cdot \sigma ':=\widetilde{\sigma } \sigma '\widetilde{\sigma }^{-1}$ for $\sigma ' \in H$. This is well-defined because H is abelian according to 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 $r \mathbb {Z} \subset {{\,\textrm{Ker}\,}}\tau$ 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

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

(3.14)

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

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. $\square$

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 $\omega _R(hx,hx')=\omega _R(x,x')$ for $h \in \mathscr {H}$.

Proof

The claim for $h=\alpha$ follows from (2.4). For $h=(1,-1)$, the claim follows from $\omega _R(x^q,x'^q)=(f_R(x,x')-f_R(x',x))^q =f_R(x,x')-f_R(x',x)=\omega _R(x,x')$. $\square$

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 $\omega :V \times V \rightarrow \mathbb {F}_p$ be a symplectic form. We say that the pair $(V,\omega )$ is symplectic if $\omega$ is non-degenerate and satisfies $\omega (gv,gv')=\omega (v,v')$ for $g \in G$ and $v,v' \in V$.

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. $\square$

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 $\sigma :\mathbb {F} \rightarrow \mathbb {F};\ x \mapsto x^q$. 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. An additive polynomial $f(x) \in \mathscr {A}_{\mathbb {F}} \setminus \{0\}$ is reduced if and only if $f'(0) \ne 0$.

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

We show (1). The assumption implies that $E^{\sigma }(\beta )=(E(\beta ^{q^{-1}}))^q=0$ for any $\beta \in V$. Since E(x) is separable, there exists $\alpha \in \mathbb {F}^{\times }$ such that $E^{\sigma }(x)=\alpha E(x)$. Then $\alpha =1$, because E(x) is monic. 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, there exists a constant $c \in \mathbb {F}^{\times }$ such that $E(\alpha x)=c E(x)$. By considering the derivatives of $E(\alpha x)$, cE(x) and substituting $x=0$, we obtain $\alpha =c$ by $E'(0) \ne 0$. Hence the claim follows. $\square$

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 proper $\mathbb {F}_p[\mathscr {H}]$-submodule.

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

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 $V_{f_2} \ne \{0\}$. Hence $V_{f_2}$ is a non-zero totally isotropic proper $\mathbb {F}_p[\mathscr {H}]$-submodule of $V_R$. Thus $V_R$ 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 $V' \subset V_R$. According to [13, 4 in Chap. 1], there exists a monic reduced polynomial $f(x) \in \mathscr {A}_{\mathbb {F}}$ such that $V'=\{\beta \in \mathbb {F}\mid f(\beta )=0\}$. Since $V'$ is stable by $\sigma$, we have $f(x) \in \mathbb {F}_q[x]$ by Lemma 3.24(1). Since $V'$ is stable by $\tau$, we have $f(\alpha x)=\alpha f(x)$ for $\alpha \in \mu _{d_{R,m}}$ from 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 $\deg r(x)<\deg f(x)$ according to [13, Theorem 1]. For any root $\beta \in V'$ of f(x), we have $r(\beta )=0$ from $E_R(\beta )=0$. Since f(x) is separable, r(x) is divisible by f(x). Hence $\deg r(x)<\deg f(x)$ induces $r(x) \equiv 0$. From definition, we obtain $V' \subset V'^{\perp }$. Thus the converse is shown. $\square$

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 $\hbox {gcd}(p^e+1,m)=1$, 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 (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 $W \subset V_R=\{\beta \in \mathbb {F} \mid (a_e x^{p^e})^{p^e}+a_ex=0 \}$. We take a non-zero element $\beta \in W$. Then $\mathbb {F}_p(\mu _{d_{R,m}})=\mathbb {F}_{p^{2e}}$ implies $\mathbb {F}_{p^{2e}} \beta =\mathbb {F}_p(\mu _{d_{R,m}})\beta \subset W$. 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 $\hbox {gcd}(p^e+1,m)=1$, we have $d_{R,m}=d_R=p^e+1$. Hence the third claim follows from $\mathbb {F}_p(\mu _{p^e+1})= \mathbb {F}_{p^{2e}}$. $\square$

Example 3.29

For a positive integer s, we consider the set

$$\begin{aligned} \mathcal {A}_{q,s}:= \left\{ \varphi (x) \in \mathbb {F}_q[x]\ \Big |\ \varphi (x)=\sum _{i=0}^n c_i x^{p^{si}}\right\} , \end{aligned}$$

which is regarded as a ring with multiplication $\varphi _1 \circ \varphi _2:=\varphi _1(\varphi _2(x))$ for $\varphi _1,\varphi _2 \in \mathcal {A}_{q,s}$. The number of prime elements in $\mathcal {A}_{q,s}$ in the sense of Definition 3.25(2) is calculated in [4] and [12]. We will review this now.

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

$$\begin{aligned} E_R(x)=\sum _{i=0}^e a_i x^{p^{e+i}} +\sum _{i=0}^e a_i x^{p^{e-i}}. \end{aligned}$$

(3.15)

Because of $f \mid t$, we have the ring isomorphism $\Phi :\mathcal {A}_{q,t} \xrightarrow {\sim } \mathbb {F}_q[y];\ \sum _{i=0}^r c_i x^{p^{ti}} \mapsto \sum _{i=0}^r c_i y^i$, 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 $0 \le i \le r$. Via (3.15), 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.

Proposition 3.30

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

Proof

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 $\sigma ^t,\tau$. Since $d_{R,m} \le 2$, according to Lemma 3.18, $\tau$ acts on $V_R$ as multiplication by sign. Thus the subspace $W_R:= \mathbb {F}_p \beta \subset V_R$ is an $\mathbb {F}_p[\mathscr {H}_t]$-submodule, since $\sigma ^t$ acts on $W_R$ trivially. From Lemma 2.6(2), it follows that $\omega _R(\zeta \beta ,\zeta ' \beta )= \zeta \zeta '\omega _R(\beta ,\beta )=0$ for any $\zeta ,\zeta ' \in \mathbb {F}_p$. Thus $W_R$ is a totally isotropic proper $\mathbb {F}_p[\mathscr {H}_t]$-submodule of $V_R$. Hence $V_R$ is not a completely anisotropic $\mathbb {F}_p[\mathscr {H}_t]$-module. Let $F_t/F$ be the unramified extension of degree t in $\overline{F}$. Then $\tau _{\psi ,R,m}|_{W_{F_t}}$ is imprimitive by [11, Theorem 4.1]. $\square$

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). $\square$

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(4).

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 $\theta :(\alpha ,\beta ) \mapsto (\alpha ^p,\beta ^p)$ and $\sigma :(\alpha ,\beta )\mapsto (\alpha ^{q^{-1}},\beta )$. A $\Phi$-orbit of $(\mathbb {F}^{\times })^2$ is called a root system.

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

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

  Let $e'=e'(W)$ and $f'=f'(W)$ 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 $e' \mid d$ and $a f' \mid r$.

• (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

  $$\begin{aligned} \{\theta ^i \sigma ^j m \mid 0 \le i \le b-1,\ 0 \le j \le a-1\} \end{aligned}$$

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

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

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 (3.14). In the above notation, we have $\mathscr {H}=\mathscr {H}_{d_{R,m},r}$. As in Corollary 3.28(4), the $\mathbb {F}_p[\mathscr {H}]$-module $V_R$ is irreducible.

Proposition 3.34

We write $q=p^f$. Let $e_1:=\hbox {gcd}(f,2e)$ and $\beta :={{\,\textrm{Nr}\,}}_{q/p^{e_1}}(-a_e^{-(p^e-1)})$. 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 $f c(W) \equiv e_1 \pmod {2e}$.

• (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 $\alpha ^{q^{a}}=\alpha$. Because of $\mathbb {F}_p(\alpha )=\mathbb {F}_{p^{2e}}$, a is the minimal positive integer satisfying $f a \equiv 0 \pmod {2e}$. Thus we obtain $a =2e/e_1$.

From 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 $e_1$. The congruence $fx \equiv e_1 \pmod {2e}$ has a solution x and $\beta \in \mathbb {F}_{p^{e_1}}$ means $\beta ^{p^{e_1}}=\beta$. Thus $b=e_1$.

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 have shown (1).

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

Let $\eta \in V_R \setminus \{0\}$. Using $\eta ^{p^{2e}}=-a_e^{-(p^e-1)} \eta$, $a_e \in \mathbb {F}_q^{\times }$ and $2e \mid fr$, we compute

$$\begin{aligned} \eta ^{q^r}=(\eta ^{p^{2e}-1})^{\frac{q^r-1}{p^{2e}-1}}\eta ={{\,\textrm{Nr}\,}}_{q^r/p^{2e}}(-a_e^{-(p^e-1)}) \eta ={{\,\textrm{Nr}\,}}_{{q^a}/{p^{2e}}}(-a_e^{-(p^e-1)})^{r/a} \eta . \nonumber \\ \end{aligned}$$

(3.16)

The restriction map $\hbox {Gal}(\mathbb {F}_{q^a}/\mathbb {F}_{p^{2e}})\rightarrow \hbox {Gal}(\mathbb {F}_{q}/\mathbb {F}_{p^{e_1}})$ is an isomorphism because of $a=2e/e_1$. Since $a_e \in \mathbb {F}_q^{\times }$, we have ${{\,\textrm{Nr}\,}}_{{q^a}/{p^{2e}}} (-a_e^{-(p^e-1)})=\beta$. Hence $\eta ^{q^r}=\beta ^{r/a} \eta$ by (3.16). Since $\eta ^{q^r}=\eta$ by Lemma 3.18 and Definition 3.32(3), we obtain $\beta ^{r/a}=1$. Thus $f' \mid (r/a)$.

We show (3). Let $\eta \in V_R \setminus \{0\}$. Similarly to (3.16), we have $\sigma ^a \eta =\eta ^{q^a}=\beta \eta$. From 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 [11, Lemma 7.3] induces $\{0\} \ne M(W) \subset V_R$. Since $V_R$ is irreducible as in Corollary 3.28(4), we obtain $M(W)=V_R$. $\square$

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, $p \ne 2$ 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 $v_2(e) \ge v_2(f)$. 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 $e_1=\hbox {gcd}(f,2e)$ and $\beta ={{\,\textrm{Nr}\,}}_{q/p_1}(-a_e^{-(p^e-1)})$. We have $e_1 \mid e$, $a(W)=2e/e_1 \equiv 0 \pmod {2}$ and $f/e_1 \equiv 1 \pmod {2}$. From $(p^{e_1}-1) \mid (p^e-1)$, it follows that

$$\begin{aligned} \beta =(-1)^{\frac{f}{e_1}}\bigl (a_e^{-\frac{p^e-1}{p^{e_1}-1}}\bigr )^{q-1}=-1, \end{aligned}$$

where we use $a_e \in \mathbb {F}_q^{\times }$ for the last equality. By $f a(W)/2=fe/e_1$ and $q=p^f$, we have $q^{a(W)/2}=p^{fe/e_1}$. Since $fe/e_1$ is divisible by e and $f/e_1$ is odd, $d_{R,m} \mid (p^e+1) \mid p^{fe/e_1}+1 =q^{a(W)}+1$. 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. Since $(e_1/2) \mid e$, we have $(p^{e_1/2}-1) \mid (p^e-1)$. From the definition of $\beta$ and $a_e \in \mathbb {F}_q^{\times }$, it follows that

$$\begin{aligned} \beta ^{p^{\frac{e_1}{2}}+1}= \bigl (a_e^{-\frac{p^e-1}{p^{e_1/2}-1}}\bigr )^{q-1}=1. \end{aligned}$$

Hence $\beta \in \mu _{p^{b(W)/2}+1}$. Assume that c(W) is even. We write $(c(W)/2) f=(e_1/2)+l e$ with $l \in \mathbb {Z}$ by Proposition 3.34(1). Then l is odd by $e_1=\hbox {gcd}(f,2e)$. Thus $(p^e+1) \mid (p^{le}+1)$. This induces that $\alpha \in \mu _{p^{b(W)/2}+q^{c(W)/2}}$. Hence (2)(i) follows. The remaining claim is shown similarly. $\square$

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 $p \ne 2$.

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 $p \ne 2$. 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 $p \ne 2$. 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 $p \ne 2$ 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

$$\begin{aligned} \bigoplus _{i=1}^n(M(W_i),\nu _i), \end{aligned}$$

where $W_1,\ldots , W_n$ are mutually different root systems belonging to $\mathscr {H}$.

Let k be a positive integer. Let $R:=\{R_i\}_{1 \le i \le k}$ with $R_i \in \mathscr {A}_q$. We consider the k-dimensional affine smooth variety $X_R$ defined by

$$\begin{aligned} a^p-a=\sum _{i=1}^k x_iR_i(x_i) \end{aligned}$$

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 (2.7). Let $\mathbb {Z}$ act on $Q_R$ naturally. Let $\psi \in \mathbb {F}_p^{\vee } \setminus \{1\}$. We regard $H_\textrm{c}^k(X_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ as a $Q_R \rtimes \mathbb {Z}$-representation. Let the notation be as in (3.5). Let $m=\{m_i\}_{1 \le i \le k}$, where $m_i$ is a positive integer. We have the homomorphism

$$\begin{aligned} \Theta _{R,m,\varpi } :W_F \rightarrow Q_R \rtimes \mathbb {Z};\ \sigma \mapsto ((a_{R_i,\sigma }^{m_i},b_{R_i,\sigma },c_{R_i,\sigma })_{1 \le i \le k}, n_{\sigma }). \end{aligned}$$

(3.17)

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 $H_\textrm{c}^k(X_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ by $\Theta _{R,m,\varpi }$.

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 $\Theta _{R_i,m_i,\varpi } :W_F \rightarrow Q_{R_i,\mathbb {Z}}$ be as in (3.8). Let

$$\begin{aligned} \delta ' :Q_R \rtimes \mathbb {Z} \rightarrow Q_{R_1,\mathbb {Z}} \times \cdots \times Q_{R_k,\mathbb {Z}};\ ((g_i)_{1 \le i \le k},n) \mapsto (g_i,n)_{1 \le i \le k}. \end{aligned}$$

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

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

Remark 3.41

Let $+ :\prod _{i=1}^k Z(Q_{R_i}) \rightarrow \mathbb {F}_p;\ (1,0,\gamma _i)_{1 \le i \le k} \mapsto \sum _{i=1}^k\gamma _i$ and $\overline{Q}_R:=Q_R/{{\,\textrm{Ker}\,}}+$. The action of $Q_R \rtimes \mathbb {Z}$ on $H_\textrm{c}^k(X_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })$ 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 $Q_R \rightarrow \overline{Q}_R$. 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 $H_\textrm{c}^k(X_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ is irreducible by [8, 16.14(2) Satz]. The claim follows from this. $\square$

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 ${{\,\textrm{Ker}\,}}\rho _{\psi ,R,m}=\bigcap _{i=1}^k {{\,\textrm{Ker}\,}}\rho _{\psi ,R_i,m_i}$. The claim follows from this. $\square$

Let $T_R$ be the maximal tamely ramified extension of F in $F_R$. We have the restriction map $V_R \hookrightarrow \prod _{i=1}^k {{\,\textrm{Gal}\,}}(F_i/T_i) \simeq \bigoplus _{i=1}^k V_{R_i}$. Then $V_R:={{\,\textrm{Gal}\,}}(F_R/T_R)$ has a bilinear form stable under the action of $\mathbb {F}_p[{{\,\textrm{Gal}\,}}(T_R/F)]$ ( [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 $R_i(x)=a_{e,i}x^{p^e} \ne 0$ for $i \in \{1,2\}$. Assume

$$\begin{aligned} m_1 \ne m_2, \quad d:=d_{R_1,m_1}=d_{R_2,m_2}. \end{aligned}$$

• (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 $p \ne 2$, $v_2(e) \ge v_2(f)$ and $\mathbb {F}_p(\mu _d)=\mathbb {F}_{p^{2e}}$. If $(V_R,\omega _R)$ is completely anisotropic as a symplectic $\mathbb {F}_p [{{\,\textrm{Gal}\,}}(T_R/F)]$-module, $V_R$ is isomorphic to a primary module (M(W), 2) with a root system W.

Proof

Via 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 $n_i:=m_i d/d_R=m_i/\hbox {gcd}(d_{R},m_i)$. Let $\{{{\,\textrm{Gal}\,}}(E_i/T)^v\}_{v \ge -1}$ be the upper numbering ramification subgroups of ${{\,\textrm{Gal}\,}}(E_i/T)$. Similarly as the proof of Lemma 3.14, we have

$$\begin{aligned} {{\,\textrm{Gal}\,}}(E_i/T)^v=\left\{ \begin{array}{ll} {{\,\textrm{Gal}\,}}(E_i/T) &{} \hbox {if }v \le n_i, \\ \{1\} &{} \hbox {if }v > n_i. \end{array}\right. \end{aligned}$$

Let $H:=E_1\cap E_2$. Since $E_i/T$ is Galois, so is H/T. By [16, Proposition 14 in IV§3], the subgroup ${{\,\textrm{Gal}\,}}(H/T)^v$ equals ${{\,\textrm{Gal}\,}}(H/T)$ if $v \le n_i$ and $\{1\}$ if $v>n_i$. Hence we conclude ${{\,\textrm{Gal}\,}}(H/T)=\{1\}$ by $n_1 \ne n_2$. We obtain $H=T$. Thus we have an isomorphism ${{\,\textrm{Gal}\,}}(E_1E_2/T) \simeq {{\,\textrm{Gal}\,}}(E_1/T) \times {{\,\textrm{Gal}\,}}(E_2/T) \simeq V_{R_1} \oplus V_{R_2}$. The extension $E_1E_2/T$ 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

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

We have the commutative diagram

where the two horizontal sequences are exact. Since $g_1$ is injective, so is $g_2$. 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 (3.14). We identify ${{\,\textrm{Gal}\,}}(T/F)$ with $\mathscr {H}_{d,r}$. Since $T_R \subset T$, 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 induces 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). $\square$

Example 3.45

Assume $p \ne 2$. Let $e=f=1$, $R_1(x)=x^p$ and $R_2(x)=a x^p \in \mathbb {F}_p[x] {\setminus } \{0\}$. We assume that $m_1 \ne m_2$ 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 $\mathbb {F}_p[{{\,\textrm{Gal}\,}}(T_R/F)]$-subspace. Assume $W \ne \{0\}$. We take a non-zero element $(x_1,x_2) \in W$. We have $f_{R_1}(x,y)=-xy^p$, $f_{R_2}(x,y)=-a xy^p$ and hence $\omega _R((x_1,x_2),(\xi x_1,\xi x_2))= (x_1^{p+1}+a x_2^{p+1})(\xi -\xi ^p)=0$ for any $\xi \in \mu _{p+1}$. Thus $x_1^{p+1}+a x_2^{p+1}=0$ and $x_2 \ne 0$. There exists $\eta \in \mathbb {F}$ such that $\eta ^{p+1}=-a$ and $x_1=\eta x_2$. Since $\mathbb {F}_{p^2}=\mathbb {F}_p(\mu _{p+1})$, we have $W_1:=\{(\eta x,x) \mid x \in V_{R_2}\} \subset W$ and $W_2:=\{(\eta ^p x, x) \mid x \in V_{R_2}\} \subset W$. Let $\bigl (\frac{\cdot }{p}\bigr )$ be the Legendre symbol. If $W_1 \cap W_2 \ne \{0\}$, 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 $p \ne 2$. 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

$$\begin{aligned} \phi :C_R \rightarrow C_{R_1};\ (a,x) \mapsto (a-\Delta (x),r(x)), \end{aligned}$$

(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 $r'(0)\ne 0$. The above assumption implies that

$$\begin{aligned}&xR(x)=r(x)R_1(r(x)) +\Delta (x)^p-\Delta (x) \end{aligned}$$

(4.2)

$$\begin{aligned}&r'(0)\ne 0, \quad d_{R,m} \mid d_{R_1},\quad r(\alpha x)=\alpha r(x) \quad \hbox { for}\ \alpha \in \mu _{d_{R,m}}. \end{aligned}$$

(4.3)

Let $e'$ be a non-negative integer such that $\deg R_1(x)=p^{e'}$ and $e' \le e$. Then $\deg r(x)=p^{e-e'}$ 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 $\varphi :C_{R,\mathbb {F}} \rightarrow C_{R,\mathbb {F}};\ (x,a) \mapsto (x+c,a+g(x))$ be the automorphism with $g(x) \in \mathbb {F}[x]$ and $c \in \mathbb {F}$. Then we have $E_R(c)=0$.

Proof

From the definition of $\varphi$, we have that

$$\begin{aligned} g(x)^p-g(x)=cR(x)+xR(c)+cR(c). \end{aligned}$$

(4.4)

Let $\mathcal {P} :\mathbb {F}[x] \rightarrow \mathbb {F}[x];\ f(x) \mapsto f(x)^p-f(x)$. Since $\mathbb {F}$ is algebraically closed, $cR(c) \equiv 0 \mod \mathcal {P}(\mathbb {F})$. From (4.4) and the definition of $E_R(x)$, it follows that

$$\begin{aligned} 0 \equiv cR(x)+xR(c)+cR(c) \equiv E_R(c)^{1/p^e}x \mod \mathcal {P}(\mathbb {F}[x]). \end{aligned}$$

Hence there exists $h(x) \in \mathbb {F}[x]$ such that $h(x)^p-h(x)=E_R(c)^{1/p^e} x$ in $\mathbb {F}[x]$. By considering degrees, we obtain $h(x)=0$ and $E_R(c)=0$. $\square$

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 $\gamma ^p-\gamma =r(\beta ) R_1(r(\beta ))$. Let $\varphi :C_{R,\mathbb {F}} \rightarrow C_{R,\mathbb {F}}$ be the automorphism defined by

$$\begin{aligned} \varphi (a, x)=\left( a+f_{R_1}(r(x),r(\beta ))+\Delta (x+\beta )-\Delta (x)+\gamma , x+\beta \right) . \end{aligned}$$

This is well-defined by Lemma 2.2 and (4.2). From Lemma 4.1 it follows that $E_R(\beta )=0$. Since $E_{R_1}(r(x))$ is separable, the claim follows. $\square$

Lemma 4.3

Let $\alpha ,\alpha ' \in \mu _{d_{R,m}}$. Assume $E_{R_1}(r(\alpha y))=0$ for a certain $y \in \mathbb {F}$. Then we have the equality

$$\begin{aligned} \Delta (\alpha ' x+\alpha y)+f_{R_1}(r(\alpha ' x),r(\alpha y))= \Delta (x) +\Delta (y)+f_R(\alpha ' x,\alpha y). \end{aligned}$$

Proof

By $\Delta (\alpha ' x+\alpha y) =\Delta (x+(\alpha /\alpha ') y)$ and (2.4), we may assume $\alpha '=1$ by (4.3). Lemma 4.2 induces that $E_R(\alpha y)=0$. 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 $f_R(0,x') \equiv 0$ in $\mathbb {F}_q[x']$ by definition. Hence it suffices to show $\Delta _1(x)^p-\Delta _1(x)=\Delta _2(x)^p-\Delta _2(x)$. Lemma 4.2 and the assumption imply $E_{R_1}(r(\alpha y))=E_R(\alpha y)=0$. Therefore, for each $i=1,2$, we have $\Delta _i(x)^p-\Delta _i(x)= (x+\alpha y)R(x+\alpha y)-r(y) R_1(r(y))-r(x) R_1(r(x))$ according to Lemma 2.2. Hence the claim follows. $\square$

Let

$$\begin{aligned} U_R:=\{x \in \mathbb {F} \mid r(x)=0\} \subset V'_R:=\{x \in \mathbb {F} \mid E_{R_1}(r(x))=0\}. \end{aligned}$$

We obtain $V'_R \subset V_R$ via Lemma 4.2. Then $U_R$ and $V'_R$ are regarded as $\mathbb {F}_p[\mathscr {H}]$-modules according to $r(x), R_1(x) \in \mathbb {F}_q[x]$ and (4.3).

Lemma 4.4

We have $V'_R \subset U^{\perp }_R$. In particular, the $\mathbb {F}_p[\mathscr {H}]$-module $U_R$ is totally isotropic.

Proof

Let $\beta$ be in $U_R$ and $\beta '$ be in $V'_R$ so that $r(\beta )=0$ and $E_{R_1}(r(\beta '))=0$. From Lemma 4.3, it follows that $f_R(\beta ',\beta )=f_R(\beta ,\beta ') =\Delta (\beta +\beta ')-\Delta (\beta )-\Delta (\beta ')$. Hence $\omega _R(\beta ,\beta ')=0$. $\square$

Let

$$\begin{aligned} Q'_{R,m}:=\{(\alpha ,\beta ,\gamma ) \in Q_{R,m} \mid \beta \in V'_R\}. \end{aligned}$$

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

$$\begin{aligned} \pi :Q'_{R,m} \rightarrow Q_{R_1,m};\ (\alpha ,\beta ,\gamma ) \mapsto (\alpha ,r(\beta ),\gamma -\Delta (\beta )). \end{aligned}$$

Corollary 4.5

The map $\pi$ is a homomorphism.

Proof

The claim follows from Lemma 4.3 and (4.3). $\square$

We have

$$\begin{aligned} U'_R:=\{(1,\beta ,\Delta (\beta ))\in Q'_{R,m} \mid \beta \in U_R\}={{\,\textrm{Ker}\,}}\pi . \end{aligned}$$

(4.5)

The space $V'_R$ is stable by the q-th power map. Hence we can consider the semidirect product $Q'_{R,m} \rtimes \mathbb {Z}$. The map $\pi$ induces $\pi ' :Q'_{R,m} \rtimes \mathbb {Z} \rightarrow Q_{R_1,m} \rtimes \mathbb {Z}$.

Quotient of $C_R$ Let $\phi$ be as in (4.1). We can check that $\phi$ factors through $C_{R,\mathbb {F}} \rightarrow C_{R,\mathbb {F}}/U'_R \xrightarrow {\bar{\phi }} C_{R_1,\mathbb {F}}$ by (2.7). We obtain an isomorphism $\bar{\phi } :C_{R,\mathbb {F}}/U'_R \xrightarrow {\sim } C_{R_1,\mathbb {F}}$.

Lemma 4.6

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

Proof

The claim follows from Lemma 4.3. $\square$

Let $\tau '_{\psi ,R_1,m}$ denote the $Q'_{R,m} \rtimes \mathbb {Z}$-representation which is the inflation of the $Q_{R_1,m} \rtimes \mathbb {Z}$-representation $H_\textrm{c}^1(C_{R_1,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]$ by $\pi '$. We have the homomorphism $\Theta _{R,m,\varpi } :W_F \rightarrow Q_{R,m} \rtimes \mathbb {Z}$ as in (3.8). We define the $W_F$-representation $\tau ''_{\psi ,R_1,m}$ to be the inflation of $\hbox {Ind}_{Q'_{R,m} \rtimes \mathbb {Z}} ^{Q_{R,m} \rtimes \mathbb {Z}} \tau '_{\psi ,R_1,m}$ via $\Theta _{R,m,\varpi }$. We have $\dim \tau ''_{\psi ,R_1,m} =p^e$, since $[Q_{R,m}:Q'_{R,m}]=p^{e-e'}$ and $\dim \tau '_{\psi ,R_1,m}=p^{e'}$.

Proposition 4.7

We have an isomorphism $\tau _{\psi ,R,m} \simeq \tau ''_{\psi ,R_1,m}$ as $W_F$-representations.

Proof

Lemma 4.6 induces an injection

$$\begin{aligned} \tau '_{\psi ,R_1,m}=H_\textrm{c}^1(C_{R_1,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ] \xrightarrow {\phi ^*} H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ] \end{aligned}$$

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

$$\begin{aligned} \hbox {Ind}_{Q'_{R,m} \rtimes \mathbb {Z}} ^{Q_{R,m} \rtimes \mathbb {Z}} \tau '_{\psi ,R_1,m} \rightarrow H_\textrm{c}^1(C_{R,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ] \end{aligned}$$

(4.6)

as $Q_{R,m} \rtimes \mathbb {Z}$-representations via Frobenius reciprocity. Since the target is irreducible by Lemma 2.8, the map (4.6) is surjective. We know that (4.6) is an isomorphism by comparing the dimensions. By inflating (4.6) via $\Theta _{R,m,\varpi }$, we obtain the claim. $\square$

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

$$\begin{aligned} \tau '_{\psi ,R_1,m} :W_{F'} \xrightarrow {\Theta _{R,m,\varpi }} Q'_{R,m} \rtimes \mathbb {Z} \xrightarrow {\pi '} Q_{R_1,m} \rtimes \mathbb {Z} \rightarrow \hbox {Aut}_{\overline{\mathbb {Q}}_{\ell }}(H_\textrm{c}^1(C_{R_1,\mathbb {F}},\overline{\mathbb {Q}}_{\ell })[\psi ]) \nonumber \\ \end{aligned}$$

(4.7)

be the composite.

Corollary 4.8

We have an isomorphism $\tau _{\psi ,R,m} \simeq {{\,\textrm{Ind}\,}}_{W_{F'}}^{W_F}\tau '_{\psi ,R_1,m}$ as $W_F$-representations. If $e'<e$, the $W_F$-representation $\tau _{\psi ,R,m}$ is imprimitive.

Proof

The assertion follows from Proposition 4.7. $\square$

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 the definition of the pairing $\omega _R$ in Lemma 2.6(2). Assume we have that

$$\begin{aligned} \beta \ne 0, \quad \gamma =\frac{f_R(\beta ,\beta )}{2}. \end{aligned}$$

(4.8)

The following lemma is given in [6, Proposition (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 $e \ge 1$.

• (1) Let

  $$\begin{aligned} u:=x^p-\beta ^{p-1}x, \quad v:=a+(x/\beta )(\gamma (x/\beta )-f_R(x,\beta )). \end{aligned}$$

  (4.9)

  Then there exists $P_1(u) \in \mathscr {A}_{\mathbb {F}}$ of degree $p^{e-1}$ such that $v^p-v=u P_1(u)$.

• (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

$$\begin{aligned} v^p-v&= xR(x) +\gamma ^p x_1^{2p}-\gamma x_1^2-x_1^p {f_R(x,\beta )}^p+x_1 f_R(x,\beta ) \\&=xR(x)+\gamma (x_1^{2p}-x_1^2)+ \beta ^{-2p+1} R(\beta ) x^{2p}\\&\ \ -u_1 f_R(x,\beta )-(x/\beta )^p(\beta R(x)+x R(\beta )) \\&=u \beta ^{-p} (-\beta R(x)+\beta ^{-p+1} R(\beta ) x^p+\gamma (x_1^p+x_1)-f_R(x,\beta )), \end{aligned}$$

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 $P(x)=P_1(u)+\alpha x$ for a constant $\alpha$. By (4.8), we have $P(\beta )=\beta ^{-p}(2\gamma -f_R(\beta ,\beta ))=0$. Thus $\alpha =0$. From $\deg P(x)=p^e$, it follows that $\deg P_1(u)=p^{e-1}$. Hence we obtain (1).

We show (2). We easily check that the finite étale morphism of degree p: $C_{R,\mathbb {F}} \rightarrow C_{P_1,\mathbb {F}};\ (a,x) \mapsto (v,u)$ factors through $C_{R,\mathbb {F}} \rightarrow C_{R,\mathbb {F}}/U \rightarrow C_{P_1,\mathbb {F}}$. Since $C_{R,\mathbb {F}} \rightarrow C_{R,\mathbb {F}}/U$ is a finite étale morphism of degree p, the claim follows. $\square$

Let

$$\begin{aligned} \Delta _0(x):=-(x/\beta )(\gamma (x/\beta )- f_R(x,\beta )). \end{aligned}$$

From Lemma 4.9(1), it follows that

$$\begin{aligned} xR(x)=u P_1(u)+\Delta _0(x)^p-\Delta _0(x). \end{aligned}$$

(4.10)

We write u(x) for u.

Let $(1,\beta ',\gamma ') \in H_R$ be an element satisfying (4.8). Assume $\omega _R (\beta ,\beta ')=0$. Then $(1,\beta ,\gamma )$ commutes with $(1,\beta ',\gamma ')$. Hence the action of $(1,\beta ',\gamma ')$ on $C_{R,\mathbb {F}}$ in (2.7) induces the automorphism of $C_{P_1,\mathbb {F}} \simeq C_{R,\mathbb {F}}/U$.

Lemma 4.10

Let $\pi (\beta ',\gamma '):=(1,u(\beta '),\gamma '-\Delta _0(\beta '))$.

• (1) We have $\pi (\beta ',\gamma ') \in H_{P_1}$ and $f_{P_1}(u(\beta '),u(\beta '))=2(\gamma '-\Delta _0(\beta '))$.

• (2) The action of $(1,\beta ',\gamma ')$ on $C_{R,\mathbb {F}}$ induces $\pi (\beta ',\gamma ')$ on $C_{P_1,\mathbb {F}}$.

Proof

Let $\Delta _1(x):=f_R(x,\beta ')-\Delta _0(x+\beta ')+\Delta _0(x)$. By (4.9), the action of $(1,\beta ',\gamma ')$ on $C_{R,\mathbb {F}}$ induces the automorphism of $C_{P_1,\mathbb {F}}$ given by $u \mapsto u+u(\beta ')$ and $v \mapsto v+\Delta _1(x)+\gamma '$ on $C_{P_1,\mathbb {F}}$. Using (4.8), we can easily check that $\Delta _1(x)-\Delta _1(0)$ is an additive polynomial such that $\Delta _1(\beta )-\Delta _1(0)=\omega _R(\beta ,\beta ')=0$. Hence there exists $g(u) \in \mathbb {F}_q[u]$ such that $\Delta _1(x)=g(u(x))+\Delta _1(0)$. Lemma 4.1 induces that $E_{P_1}(u(\beta '))=0$. Thus $u(\beta ') \in V_{P_1}$. We show (1). The former claim follows from (4.10). Using $\Delta _0(0)=E_{P_1}(u(\beta '))=E_R(\beta ')=0$ in the same way as Lemma 4.3, we have

$$\begin{aligned} \Delta _0(x+\beta ')+f_{P_1}(u(x),u(\beta ')) =\Delta _0(x)+\Delta _0(\beta ')+f_R(x,\beta '). \end{aligned}$$

(4.11)

Substituting $x=\beta '$, and using $\Delta _0(2 \beta ')=4 \Delta _0(\beta ')$ and (4.8) for $(\beta ',\gamma ')$, we obtain the latter claim in (1).

By (4.11),

$$\begin{aligned} v+f_R(x,\beta ')-\Delta _0(x+\beta ')+\Delta _0(x)+\gamma '=v+f_{P_1}(u(x),u(\beta '))+\gamma '-\Delta _0(\beta '). \end{aligned}$$

Hence the claim (2) follows from (2.7). $\square$

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

$$\begin{aligned} r(\alpha x)=\alpha r(x)\hbox { for }\alpha \in \mu _{d_{R,m}}\hbox { and }r(x) \in \mathbb {F}_q[x] \end{aligned}$$

(4.12)

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

We take a basis $\{\beta _1,\ldots ,\beta _{e-e'}\}$ of $U_R$ over $\mathbb {F}_p$. Let $(1,\beta _i,\gamma _i) \in H_R$ be an element satisfying (4.8). 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 $g_j \in U_j$ via Lemma 2.6(2). We consider the abelian subgroup

$$\begin{aligned} U'_R:=U_1\cdots U_{e-e'} \subset H_R. \end{aligned}$$

(4.13)

Proposition 4.11

Assume that $V_R$ is not completely anisotropic. Then there exist $R_1(x) \in \mathscr {A}_{\mathbb {F}}$ of degree $p^{e'}$ and a polynomial $\Delta (x) \in \mathbb {F}[x]$ such that $\Delta (0)=0$ and the quotient $C_{R,\mathbb {F}}/U'_R$ is isomorphic to the affine curve $C_{R_1,\mathbb {F}}$ and the isomorphism is induced by $\pi :C_{R,\mathbb {F}} \rightarrow C_{R_1,\mathbb {F}};\ (a,x) \mapsto (a-\Delta (x),r(x))$. 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, we know that the quotient $C_{R,\mathbb {F}}/U'_R$ is isomorphic to the curve $C_{R_1,\mathbb {F}}$ with some $R_1(x) \in \mathscr {A}_{\mathbb {F}}$, and we obtain $\pi :C_{R,\mathbb {F}} \rightarrow C_{R_1,\mathbb {F}};\ (a,x) \mapsto (a-\Delta (x),r(x))$. By (4.9), 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 $U'_R$. Hence A acts on the quotient $C_{R_1,\mathbb {F}}$. We recall that $b^p-b=y R_1(y)$ is the defining equation of $C_{R_1,\mathbb {F}}$. Through the morphism $\pi$, the action of $A \ni (\alpha ,0,0)$ on $C_{R_1,\mathbb {F}}$ is given by $b \mapsto b+\Delta (x)-\Delta (\alpha ^{-1} x),\ y=r(x) \mapsto r(\alpha ^{-1}x)=\alpha ^{-1} y$ by (4.12). From [6, Theorem (13.3)] or [1, Theorem 4.3.2], it follows that $\alpha \in \mu _{d_{R_1}}$. Hence the last claim follows. $\square$

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 as 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 $\Pi (x):=\Delta ^{\sigma }(x)-\Delta (x)$. Then $S(x) \in \mathscr {A}_{\mathbb {F}}$. Since $r(x), R(x) \in \mathbb {F}_q[x]$,

$$\begin{aligned} \Pi (x)^p-\Pi (x)=r(x) S(r(x)). \end{aligned}$$

(4.14)

Assume $S(x) \ne 0$. We have the non-constant morphism $f :\mathbb {A}_{\mathbb {F}}^1 \rightarrow C_{S,\mathbb {F}};\ x \mapsto (\Pi (x),r(x))$, since r(x) is non-constant. Let $\overline{C}_{S,\mathbb {F}}$ be the smooth compactification of $C_{S,\mathbb {F}}$. The morphism f extends to a non-constant morphism $\mathbb {P}_{\mathbb {F}}^1 \rightarrow \overline{C}_{S,\mathbb {F}}$. Hence this is a finite morphism. From the Riemann–Hurwitz formula, it follows that the genus of $\overline{C}_{S,\mathbb {F}}$ equals zero. This contradicts to 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 (4.14). As in Proposition 4.11, $\Delta (0)=0$ induces $\Pi (0)=0$. Hence $\Pi (x) \equiv 0$. Thus the claim follows. $\square$

4.3 Theorem

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

Theorem 4.13

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

• (1) There exists a non-trivial finite étale morphism

  $$\begin{aligned} C_R \rightarrow C_{R_1};\ (a,x) \mapsto (a-\Delta (x),r(x)), \end{aligned}$$

  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 ${{\,\textrm{Ind}\,}}_{W_{F'}}^{W_F} \tau '_{\psi , R_1, m}$, where $\tau '_{\psi , R_1, m}$ is given in (4.7).

Proof

Assume (1). The degree of the finite covering $C_R \rightarrow C_{R_1}$ equals $\deg r(x)$. Since $C_R \rightarrow C_{R_1}$ is not an isomorphism, we have $\deg r(x)>1$. Thus (2) follows from Lemma 4.4 and $U_R \ne \{0\}$. Assume (2). Then (1) follows from (4.12), 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. $\square$