# On the quaternionic Serre Weights

Yang CHEN and Haoran WANG

###### Abstract.

We give a representation theoretic description of the set of quaternionic
Serre weights for generic two-dimensional mod $p$ representations of
$\operatorname{Gal}(\overline{{\mathbb{Q}}}_{p}/K)$, where $K$ is a finite
unramified extension of ${\mathbb{Q}}_{p}$.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Preliminaries
    1. 2.1 Irreducible ${\mathbb{F}}$-representations of $\operatorname{GL_{2}}(k)$
    2. 2.2 Cuspidal types
    3. 2.3 Automorphic forms on quaternion algebras over totally real field
    4. 2.4 Serre weights for $\mathrm{GL}_{2}$
      1. 2.4.1 The reducible case
      2. 2.4.2 The irreducible case
  3. 3 Explicit description of the set $W_{D}(\overline{\rho})$
    1. 3.1 Preparation for the calculation of $W_{D}(\overline{\rho})$ in semisimple case
    2. 3.2 Explicit description of $W_{D}(\overline{\rho})$ for generic $\overline{\rho}$

##  1\. Introduction

Let $p$ be a prime number. Serre [Ser87] conjectured that every odd
irreducible continuous representation $\overline{r}:{\rm
Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}\left(\overline{\mathbb{F}}_{p}\right)$
is modular in the sense that it arises from some modular eigenform. Serre also
formulated a refined conjecture which predicts the minimal weight and level of
the modular eigenforms. The original Serre’s conjecture has been proved by
Khare and Wintenberger [KW09]. In fact Serre [Ser87] asked whether a “mod $p$
Langlands philosophy” exists, which is now known to be true for ${\rm GL}_{2}$
over $\mathbb{Q}$ [Eme11] [Bre10].

In [BDJ10] Buzzard, Diamond and Jarvis generalized Serre’s refined conjecture
to Hilbert modular forms over totally real field $F$ in which $p$ is
unramified. As there is no obvious notion of minimal weight, [BDJ10] used
irreducible $\overline{\mathbb{F}}_{p}$-representations of
$\operatorname{GL}_{2}(\mathcal{O}_{F}/p)$ as a generalization of the notion
of weight, and predicted all the possible weights for totally odd irreducible
continuous modular representation $\overline{r}:{\rm
Gal}(\overline{F}/F)\to\mathrm{GL}_{2}\left(\overline{\mathbb{F}}_{p}\right).$
Let’s call the set of all possible weights the set of ${\rm GL}_{2}$-Serre
weights for $\overline{r}.$ The weight part of Serre’s conjecture formulated
in [BDJ10] and its natural generalizations to definite or indefinite
quaternion algebra $B$ over $F$ which splits at all places above $p$ are
proved in [GLS14]. There are important generalizations of weight part of
Serre’s conjecture formulated in [Sch08] [Her09] [GHS18]. We refer to the
introduction of [GHS18] for a historical account.

In this paper we consider the weight part of Serre’s conjecture for
$\overline{r}:{\rm
Gal}(\overline{F}/F)\to\mathrm{GL}_{2}\left(\overline{\mathbb{F}}_{p}\right)$
when $B$ is a definite or indefinite quaternion algebra over totally real
field $F$ ramified at all places above $p.$ The possible weights are
irreducible $\overline{\mathbb{F}}_{p}$-representations of
$({\mathcal{O}}_{B}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_{p})^{\times},$ where
${\mathcal{O}}_{B}$ is a fixed maximal order. Let’s call the set of all
possible weights the set of quaternionic Serre weights for $\overline{r}.$
Recently, Scholze [Sch18] proposed a mod $p$ Jacquet-Langlands correspondence
which satisfies some local-global compatibility. Recent works like [Lud17]
[Paš22] [HW22] give some study on this mod $p$ Jacquet-Langlands
correspondence. We would like to view the question of determining the set of
quaternionic Serre weights as part of the mod $p$ Langlands program. Up to
multiplicity, it gives the
$({\mathcal{O}}_{B}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_{p})^{\times}$-socle of
some interesting mod $p$ representation of
$(B\otimes_{\mathbb{Q}}\mathbb{A}_{f})^{\times}.$

When $F={\mathbb{Q}},$ the set of quaternionic Serre weights is completely
determined in [Kha01]. Under some Taylor-Wiles type assumption, [GS11]
determined, in terms of irreducible
$\overline{\mathbb{F}}_{p}$-representations of
$({\mathcal{O}}_{B}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_{p})^{\times},$ the set
of quaternionic Serre weights in most cases if $\overline{r}|_{{\rm
Gal}(\overline{F}_{v}/F_{v})}$ is semisimple for all $v|p.$ In the non-
semisimple case, they gave an explicit description in terms of $p$-adic Hodge
theory. However, in order to understand the mod $p$ Jacquet-Langlands
correspondence, one still wants to have an explicit description in terms of
representations of
$({\mathcal{O}}_{B}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_{p})^{\times}.$ Note
that [GS11] had no assumption on the ramification of $p$ in the totally real
field $F.$

In this paper, we assume $p$ is unramified in $F$ and we give a representation
theoretic description of the set of quaternionic Serre weights for generic
2-dimensional modular $\overline{r}$ under the Taylor-Wiles type assumption.
The generic condition is the one introduced in [BP12] for $\overline{r}|_{{\rm
Gal}(\overline{F}_{v}/F_{v})}$ for all $v|p.$ It follows from [GS11] that
every quaternionic Serre weight is of the form $\otimes_{v|p}\sigma_{v}$, for
$\sigma_{v}$ lies in a set $W_{B_{v}}(\overline{r}|_{{\rm
Gal}(\overline{F}_{v}/F_{v})})$ which depends only on $\overline{r}|_{{\rm
Gal}(\overline{F}_{v}/F_{v})}.$ The question is then reduced to determine the
set $W_{B_{v}}(\overline{r}|_{{\rm Gal}(\overline{F}_{v}/F_{v})})$ for each
$v|p.$ We let $K:=F_{v}$ which is unramified of degree $f$ over
${\mathbb{Q}}_{p}$ with residue field $k.$ Let
$\overline{\rho}:=\overline{r}|_{{\rm Gal}(\overline{F}_{v}/F_{v})}(1),$
$D:=B_{v}$ and $W_{D}(\overline{\rho}):=W_{B_{v}}(\overline{r}|_{{\rm
Gal}(\overline{F}_{v}/F_{v})}).$ We deduce our results from the existing
knowledge of ${\rm GL}_{2}$-Serre weights ([Dia07] [BP12] [Bre14]). Under the
generic assumption, it is easy to see that an irreducible mod $p$
representation $\overline{\psi}$ of $\mathcal{O}_{D}^{\times}$ is in
$W_{D}(\overline{\rho})$ if and only if
$W_{\operatorname{GL}_{2}}(\overline{\rho})$ intersects with the set of
Jordan-Hölder factors of the reduction mod $p$ of the tame cuspidal type
$\Theta([\overline{\psi}]),$ where $[\overline{\psi}]$ is the Techmüller lift
of $\overline{\psi},$ see Proposition 2.4111This has already been indicated in
Remark 5.10 of [GG15] for a precise statement. This gives a way to compute
$\overline{\psi}.$ We then prove that such $\overline{\psi}$’s can be
parameterised by $(\vec{w},\vec{d})$ with
$\vec{w}=(w_{0},...,w_{f-1})\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ and
$\vec{d}=(d_{0},...,d_{f-1})\in\\{-1,0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$
subject to some relations. Our main results in the semisimple case are given
in Theorems 3.9, 3.14 respectively. We use Theorem 3.9 as an example to
illustrate our explicit description.

######  Theorem 1.1 (Theorem 3.9).

Let $q=p^{f}$ and let $l$ be the quadratic extension of $k.$ We fix an
embedding $\overline{\iota}:l\hookrightarrow\mathbb{F}$, where $\mathbb{F}$ is
a sufficiently large finite extension of $\mathbb{F}_{p}$. Let
$\overline{\kappa}_{0}:k\hookrightarrow\mathbb{F}$ be given by
$\overline{\iota}^{q+1}=\overline{\kappa}_{0}\circ\operatorname{Nm}_{l/k}$.
Let
$\overline{\rho}:\operatorname{Gal}\left(\overline{{\mathbb{Q}}}_{p}/K\right)\rightarrow\mathrm{GL}_{2}\left(\mathbb{F}\right)$
be a continuous representation such that its restriction to inertia is:

$\left(\begin{array}[]{cc}\omega_{f}^{r_{0}+1+p\left(r_{1}+1\right)+\cdots+p^{f-1}\left(r_{f-1}+1\right)}&0\\\
0&1\end{array}\right)\otimes(\overline{\kappa}_{0}\circ\operatorname{res}\circ\operatorname{Art}_{K}^{-1})$

with $-1\leq r_{i}\leq p-2$. Assume $\overline{\rho}$ is generic in the sense
that $0\leq r_{i}\leq p-3$ for all $i$ and not all $r_{i}$ equal to $0$ or
equal to $p-3$. Then $W_{D}(\overline{\rho})$ consists of
$\overline{\psi}=\overline{\iota}^{\sum_{i=0}^{f-1}q^{w_{i}}p^{i}r_{i}+(1-q)\sum_{i=0}^{f-1}d_{i}p^{i}},$
where $w_{i}\in\\{0,1\\}$ and $d_{i}\in\\{-1,0,1\\}$ satisfying:

  * •

For $i>0$, $w_{i}=\left\\{\begin{array}[]{cl}1&\text{ if }d_{i}=-1\\\ 0&\text{
if }d_{i}=1\end{array}\right.$, and if $d_{i}=0$, then $(w_{i-1},w_{i})=(0,0)$
or $(1,1)$.

  * •

$w_{0}=\left\\{\begin{array}[]{cl}1&\text{ if }d_{0}=-1\\\ 0&\text{ if
}d_{0}=1\end{array}\right.$, and if $d_{0}=0$, then $(w_{f-1},w_{0})=(0,1)$ or
$(1,0)$.

We write $\overline{\psi}_{\vec{w},\vec{d}}$ for such $\overline{\psi}$ and we
have
$\overline{\psi}_{\vec{w},\vec{d}}=\overline{\psi}_{\vec{w}^{\prime},\vec{d}^{\prime}}$
if and only if $(\vec{w},\vec{d})=(\vec{w}^{\prime},\vec{d}^{\prime}).$

To determine the set $W_{D}(\overline{\rho})$ for non-semisimple
$\overline{\rho},$ we first give a partition of $W_{D}(\overline{\rho}^{{\rm
ss}})$ where $\overline{\rho}^{{\rm ss}}$ denotes the semisimplification of
$\overline{\rho}.$ We identify the set of ${\rm GL}_{2}$-Serre weights for
$\overline{\rho}^{{\rm ss}}$ with the set of $f$-tuples
$\vec{v}\in\\{0,1\\}^{\mathbb{Z}/f\mathbb{Z}}.$ We have

$W_{D}(\overline{\rho}^{{\rm
ss}})=\bigsqcup\limits_{{\vec{v}}\in\\{0,1\\}^{\mathbb{Z}/f\mathbb{Z}}}W_{D}^{\vec{v}}(\overline{\rho}^{{\rm
ss}}).$

Each subset $W_{D}^{\vec{v}}(\overline{\rho}^{{\rm ss}})$ can be explicited
described.

######  Theorem 1.2 (Theorem 3.12).

Let $\overline{\rho}$ be the generic reducible nonsplit Galois representation
as in Definition 2.7. Then

$W_{D}(\overline{\rho})=\bigsqcup_{{\vec{v}}\leq\vec{v}(\overline{\rho})}W_{D}^{{\vec{v}}}(\overline{\rho}^{{\rm
ss}}),$

where $\leq$ is the partial order which is translated from the partial order
defined in [BP12, §11] and
$\vec{v}(\overline{\rho})\in\\{0,\ldots,f-1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$
is the $f$-tuple associated to $\overline{\rho}$ which is defined in
Definition 2.7.

###  1.1. Acknowledgements

We thank Prof. Yongquan Hu for several interesting discussions during the
preparation of the paper. We thank Prof. Christophe Breuil and Prof. Florian
Herzig for their comments on an earlier draft. The work is supported by
National Natural Science Foundation of China Grants 11971028.

##  2\. Preliminaries

We keep the notations as in the introduction. Let $K$ be unramified of degree
$f$ over $\mathbb{Q}_{p}$ with residue field $k.$ Let $l$ be the quadratic
extension of $k$ in $\overline{k}.$ Let $E$ be a finite extension of
$\mathbb{Q}_{p}$ with integer ring ${\mathcal{O}}$ uniformizer $\varpi_{E}$
and residue field $\mathbb{F}.$ We assume $\mathbb{F}$ is sufficiently large.
We fix an embedding $\overline{\iota}:l\hookrightarrow\mathbb{F}$. Let
$\overline{\kappa}_{0}:k\hookrightarrow\mathbb{F}$ be given by
$\overline{\iota}^{p^{f}+1}=\overline{\kappa}_{0}\circ\operatorname{Nm}_{l/k}$.
We define embeddings $\overline{\kappa}_{i}:k\hookrightarrow\mathbb{F}$ by
$\overline{\kappa}_{i+1}^{p}=\overline{\kappa}_{i}$ for $i\in{\mathbb{Z}}$.

###  2.1. Irreducible ${\mathbb{F}}$-representations of
$\operatorname{GL_{2}}(k)$

Irreducible ${\mathbb{F}}$-representations of $\mathrm{GL}_{2}(k)$ are of the
form:

(1)
$\overline{\sigma}_{\vec{t},\vec{s}}:=\otimes_{j=0}^{f-1}\left(\operatorname{det}^{t_{j}}\operatorname{Sym}^{s_{j}}k^{2}\right)\otimes_{k,\overline{\kappa}_{-j}}\mathbb{F},$

where $0\leq s_{j},$ $t_{j}\leq p-1$ and not all $t_{j}$ are equal to $p-1$.
An irreducible ${\mathbb{F}}$-representations of $\mathrm{GL}_{2}(k)$ can also
be written in the form:

(2)
$\operatorname{Sym}^{r_{0}}{\mathbb{F}}^{2}\otimes_{{\mathbb{F}}}\left(\operatorname{Sym}^{r_{1}}{\mathbb{F}}^{2}\right)^{\mathrm{Fr}}\otimes\cdots\otimes_{{\mathbb{F}}}\left(\mathrm{Sym}^{r_{f-1}}{\mathbb{F}}^{2}\right)^{\mathrm{Fr}^{f-1}}\otimes_{{\mathbb{F}}}(\overline{\eta}\circ\operatorname{det})$

where the $r_{i}$ are integers between $0$ and $p-1$, $\overline{\eta}$ is a
smooth character $k^{\times}\rightarrow{\mathbb{F}}^{\times}$,
$\mathrm{GL}_{2}(k)$ acts on the first $\operatorname{Sym}$ via the fixed
embedding $\overline{\kappa}_{0}:k\hookrightarrow{\mathbb{F}}$ and on the
others via twists by powers of the Frobenius $\mathrm{Fr}$ where
$\operatorname{Fr}(x):=x^{p}\left(x\in\mathbb{F}_{p^{f}}\right)$. Such
representation is denoted by
$\left(r_{0},\ldots,r_{f-1}\right)\otimes(\overline{\eta}\circ\operatorname{det})$.
The relation between the above two descriptions of irreducible
${\mathbb{F}}$-representations of $\mathrm{GL}_{2}(k)$ is given by the
following lemma.

######  Lemma 2.1.

We have
$\overline{\sigma}_{\vec{t},\vec{s}}=(s_{0},...,s_{f-1})\otimes(\overline{\kappa}_{0}\circ\operatorname{det}^{\sum_{i=0}^{f-1}p^{i}t_{i}}).$

###### Proof.

This follows from
$\operatorname{Sym}^{s_{j}}k^{2}\otimes_{k,\overline{\kappa}_{-j}}\mathbb{F}=(\operatorname{Sym}^{s_{j}}{\mathbb{F}}^{2})^{\mathrm{Fr}^{j}}$
and
$\operatorname{det}^{t_{j}}\otimes_{k,\overline{\kappa}_{-j}}\mathbb{F}=\overline{\kappa}_{0}\circ\operatorname{det}^{p^{j}t_{j}}$
for $0\leq j\leq f-1$. ∎

###  2.2. Cuspidal types

Let $\psi:l^{\times}\rightarrow\mathcal{O}^{\times}$ be a multiplicative
character which does not factor through the norm
$\operatorname{Nm}_{l/k}:l^{\times}\rightarrow k^{\times}$. Let $\Theta(\psi)$
be the cuspidal type of irreducible $E$-representations of
$\operatorname{GL}_{2}(k)$ as in [Dia07, §1]. Let
$\overline{\Theta}(\psi)^{{\rm ss}}$ be the semisimplification of the
reduction mod $\varpi_{E}$ of any $\operatorname{GL}_{2}(k)$-stable
${\mathcal{O}}$-lattice in $\Theta(\psi).$ Then the Jordan-Hölder factors of
$\overline{\Theta}(\psi)^{{\rm ss}}$ are described in [Dia07, Prop. 1.3]. We
follow the presentation of [EGS15, §3.3].

We can write $\psi$ in the form $[\overline{\iota}]^{(q+1)b+1+c}$, where
$0\leq b\leq q-2$, $0\leq c\leq q-1$ and $[\overline{\iota}]$ is the
Teichmüller lift of $\overline{\iota}$. Write $c=\sum_{i=0}^{f-1}c_{i}p^{i}$,
where $0\leq c_{i}\leq p-1$. If $J\subseteq\mathcal{S}:=\\{0,...,f-1\\}$, we
set $J_{0}=J\triangle\\{f-1\\}$, i.e. $J_{0}=J\cup\\{f-1\\}$ if $f-1\notin J$,
and $J_{0}=J\backslash\\{f-1\\}$ if $f-1\in J$. We define
$\mathcal{P}_{\Theta(\psi)}$ to be the collection of subsets of $\SS$
consisting of those $J$ satisfying the conditions:

  * •

if $j\in J$ and $j-1\notin J_{0}$ then $c_{j}\neq p-1$, and

  * •

if $j\notin J$ and $j-1\in J_{0}$ then $c_{j}\neq 0$.

For any $J\in\mathcal{P}_{\Theta(\psi)}$ we define $s_{J,i}$ and $t_{J,i}$ by

(3)
$s_{J,i}=\left\\{\begin{array}[]{cl}p-1-c_{i}-\delta_{\left(J_{0}\right)^{c}}(i-1)&\text{
if }i\in J\\\ c_{i}-\delta_{J_{0}}(i-1)&\text{ if }i\notin
J\end{array}\right.,$ (4)
$t_{J,i}=\left\\{\begin{array}[]{cl}c_{i}+\delta_{J^{c}}(i-1)&\text{ if }i\in
J\\\ 0&\text{ if }i\notin J\end{array}\right.,$

where, for $J\subseteq\SS$, $\delta_{J}(i)=1$ if $i\in J$, and 0 otherwise.

######  Lemma 2.2.

The Jordan-Hölder factors of $\overline{\Theta}(\psi)^{{\rm ss}}$ are
parameterised by $\mathcal{P}_{\Theta(\psi)}$ as follows: for
$J\in\mathcal{P}_{\Theta(\psi)}$, let

(5)
$\overline{\Theta}(\psi)_{J}:=\overline{\sigma}_{\vec{t}_{J},\vec{s}_{J}}\otimes(\overline{\kappa}_{0}\circ\mathrm{det}^{b+\delta_{J}(0)\delta_{J}(f-1)+\delta_{J^{c}}(0)\delta_{J^{c}}(f-1)}).$

###### Proof.

See [Dia07, Prop. 1.3] or [EGS15, §3.3]. ∎

A subset $J$ of $\SS$ can be identified with an $f$-tuple
$\vec{u}:=(u_{0},...,u_{f-1})\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$, where
$u_{i}=0$ if $i\notin J$ and $u_{i}=1$ if $i\in J$. For
$\vec{u}:=(u_{0},...,u_{f-1})\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$,
define $\vec{u}_{0}=(u_{0,0},...,u_{0,f-1})$ by letting $u_{0,i}=u_{i}$ for
$0\leq i<f-1$ and $u_{0,f-1}=1-u_{f-1}$. If $\vec{u}$ corresponds to $J$, then
$\vec{u}_{0}$ corresponds to $J_{0}$ under the above identification. If
$\vec{u}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ corresponds to
$J\in\mathcal{P}_{\Theta(\psi)},$ then $\vec{u}$ satisfies

  * •

if $u_{i}=1$ and $u_{0,j-1}=0$ then $c_{j}\neq p-1$, and

  * •

if $u_{i}=0$ and $u_{0,j-1}=1$ then $c_{j}\neq 0$.

We abuse notation by letting $\mathcal{P}_{\Theta(\psi)}$ denote the set of
$\vec{u}$’s satisfying above conditions.

######  Definition 2.3.

For $\vec{u}:=(u_{0},...,u_{f-1})\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$,
we define:

(6) $s_{\vec{u},i}=\left\\{\begin{array}[]{cl}p-2+u_{0,i-1}-c_{i}&\text{ if
}u_{i}=1\\\ c_{i}-u_{0,i-1}&\text{ if }u_{i}=0\end{array}\right.,$ (7)
$t_{\vec{u},i}=\left\\{\begin{array}[]{cl}c_{i}+(1-u_{i-1})&\text{ if
}u_{i}=1\\\ 0&\text{ if }u_{i}=0\end{array}\right.,$ (8)
$\overline{\Theta}(\psi)_{\vec{u}}:=\overline{\sigma}_{\vec{t}_{\vec{u}},\vec{s}_{\vec{u}}}\otimes(\overline{\kappa}_{0}\circ\mathrm{det}^{b+u_{0}u_{f-1}+(1-u_{0})(1-u_{f-1})}).$

Then if $\vec{u}\in\mathcal{P}_{\Theta(\psi)}$ corresponds to
$J\in\mathcal{P}_{\Theta(\psi)}$, we have $s_{\vec{u},i}=s_{J,i}$,
$t_{\vec{u},i}=t_{J,i}$, and
$\overline{\Theta}(\psi)_{\vec{u}}=\overline{\Theta}(\psi)_{J}$.

###  2.3. Automorphic forms on quaternion algebras over totally real field

Let $F$ be a totally real field and $B$ be a quaternion algebra over $F.$ We
assume $B$ is either _definite_ (i.e. it is ramified at all infinite places),
or $B$ is _indefinite_ (i.e. it splits at exactly one infinite place of $F$).
Let ${\mathcal{O}}_{B}$ be a fixed maximal order of $B.$ Since our main result
is on the local side, we simplify the global setup by assuming that $p$ is
inert in $F.$ Let $v$ denote the unique place of $B$ over $p.$

We define the space of automorphic forms associated to $B$ as in [HW22, §5.2].
Up to normalization, it is identical to the space used in [GS11] (resp.
[BD14]) when $B$ is definite (resp. indefinite). Let $\overline{r}:{\rm
Gal}(\overline{F}/F)\to{\rm GL}_{2}(\mathbb{F})$ be an absolutely irreducible
totally odd representation, which is modular in the sense of [BD14, §3.1].
Assume $\overline{r}|_{{\rm Gal}(\overline{F}/F(\sqrt[p]{1}))}$ is absolutely
irreducible and, if $p=5,$ the image of $\overline{r}({\rm
Gal}(\overline{F}/F(\sqrt[p]{1})))$ in ${\rm PGL}_{2}(\mathbb{F})$ is not
isomorphic to ${\rm PSL}_{2}(\mathbb{F}_{5}).$ Then $\overline{r}$ gives a
maximal ideal of the abstract Hecke algebra which acts on the space of modular
forms in the usual way, see [HW22, §5.2] [GS11] [BD14]. We denote by
$\pi^{B}(\overline{r})$ the admissible smooth representation of
$(B\otimes_{\mathbb{Q}}\mathbb{A}_{f})^{\times}$ over $\mathbb{F}$ given by
[HW22, (5.4)]. When $B$ is indefinite, $\pi^{B}(\overline{r})$ is the
representation denoted by $\pi_{D}(\overline{\rho})$ in [BD14, §3.1] up to
twist. We assume the hypothesis (H0) of [BD14] is satisfied. Then by [BD14,
Cor. 3.2.3], $\pi^{B}(\overline{r})$ is non-zero.

Let $\sigma$ be an irreducible representation of
${\mathcal{O}}_{B_{v}}^{\times}$ over $\mathbb{F}.$ We say that $\sigma$ is a
Serre weight for $\overline{r}$ with respect to $B$ (at $v$) if

(9) ${\rm
Hom}_{{\mathcal{O}}_{B_{v}}^{\times}}(\sigma,\pi^{B}(\overline{r}))\neq 0.$

Let $W_{B}(\overline{r})$ denote the set of $\sigma$’s satisfying (9). If $B$
splits at $v,$ [GLS14] proves that the set $W_{B}(\overline{r})$ depends only
on $\overline{r}|_{G_{I_{v}}},$333In fact [GLS14] establishes this also in
non-generic case. where $I_{F_{v}}$ is the inertia subgroup of
$G_{F_{v}}:={\rm Gal}(\overline{F}_{v}/F_{v})$. If moreover
$\overline{r}|_{G_{F_{v}}}$ is generic in the sense of [BP12], then the set
$W_{B}(\overline{r})$ is equal to the set $W_{\rm GL_{2}}(\overline{\rho})$
with $\overline{\rho}:=\overline{r}|_{G_{F_{v}}}(1).$ The generic condition of
[BP12] and the set $W_{\rm GL_{2}}(\overline{\rho})$ are recalled in §2.4 with
$K=F_{v}$.

Assume $B$ is ramified at $v.$ Then irreducible representations of
${\mathcal{O}}_{B_{v}}^{\times}$ over $\mathbb{F}$ are the same as
multiplicative characters $\overline{\psi}:l^{\times}\to\mathbb{F}^{\times},$
where $l$ is the quadratic extension of the residue field $k$ of $F_{v}.$ If
moreover $\overline{r}|_{G_{F_{v}}}$ is generic, then [GS11, Thm. 8.3] shows
that $W_{B}(\overline{r})$ depends only on $\overline{r}|_{I_{F_{v}}}$ and it
only consists of characters of type I defined in [GS11, Def. 3.1]. Recall
$\overline{\psi}:l^{\times}\rightarrow\mathbb{F}^{\times}$ is a type I
character if it does not factor through the norm
$\operatorname{Nm}_{l/k}:l^{\times}\rightarrow k^{\times}$. In the following,
we will write $D$ for the quaternion algebra $B_{v}$ over $F_{v}.$ So we
denote $W_{D}(\overline{\rho}):=W_{B}(\overline{r})$ with
$\overline{\rho}:=\overline{r}|_{G_{F_{v}}}(1)$ as above.

######  Proposition 2.4.

Let $\overline{\psi}:l^{\times}\rightarrow\mathbb{F}^{\times}$ be a type I
character. Then $\overline{\psi}\in W_{D}(\overline{\rho})$ if and only if
${\rm JH}\left(\overline{\Theta}(\psi)^{{\rm ss}}\right)\cap
W_{\mathrm{GL}_{2}}(\overline{\rho})\neq\emptyset,$ where $\psi$ denotes the
Techmüller lift of $\overline{\psi}.$

###### Proof.

Let $B$ be a definite quaternion algebra over $F$ which ramified at $v.$ Via
the natural projection map ${\mathcal{O}}_{B_{v}}^{\times}\to l^{\times},$
$\psi$ can be viewed as a character of ${\mathcal{O}}_{B_{v}}^{\times}.$ By
the classical local Langlands and Jacquet-Langlands correspondence, $\psi$ is
a $K$-type for the Weil-Deligne type $[\psi\oplus\psi^{q_{v}},N=0]$ in the
sense of [BD14, §3.2], where $q_{v}$ is the cardinality of $k$. By [BD14, Lem.
3.2.1], there is an automorphic representation $\pi^{B}$ of
$(B\otimes_{\mathbb{Q}}\mathbb{A})^{\times}$ whose weight is the trivial
representation, such that $\overline{r}\cong r_{\pi^{B}}\mod\varpi_{E}$ and
$r_{\pi^{B}}|_{G_{F_{v}}}$ is of inertial type $[\psi\oplus\psi^{q_{v}},N=0].$
Here $r_{\pi^{B}}$ is the Galois representation associated to $\pi^{B}.$

By global Jacquet-Langlands correspondence, there is an automorphic
representation $\pi$ of ${\rm GL}_{2}(\mathbb{A}_{F})$ with the same
infinitesimal character as the trivial representation such that
$\overline{r}\cong r_{\pi}\mod\varpi_{E},$ where $r_{\pi}$ is the Galois
representation associated to $\pi,$ and $r_{\pi}|_{G_{F_{v}}}$ is of inertial
type $[\psi\oplus\psi^{q_{v}},N=0].$ Then $\Theta(\psi)$ is a $K$-type (in the
sense of [BD14, §3.2]) of the supercuspidal representation $\pi_{v}$, see the
proof of [GS11, Lem. 3.3]. By [BD14, Lem. 3.2.1] again, $\overline{r}$ is
modular of weight $\sigma,$ with respect to ${\rm GL}_{2}$ at $v,$ for some
$\sigma\in{\rm JH}\left(\overline{\Theta}(\psi)^{{\rm ss}}\right).$

For the converse, we may reverse the above argument. This finishes the proof
of the proposition. ∎

###  2.4. Serre weights for $\mathrm{GL}_{2}$

We recall the explicit description of $W_{\mathrm{GL}_{2}}(\overline{\rho})$
associated to generic $\overline{\rho}$ in [BP12, §11]. For our latter use, we
slightly generalize the definition of [BP12]. Let $K$ be the unramified
extension of degree $f$ over $\mathbb{Q}_{p}.$

####  2.4.1. The reducible case

Let $\left(x_{0},\ldots,x_{f-1},x\right)$ be $f+1$ variables. We define a set
$\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$ of $f$-tuples
$\lambda(x_{0},\ldots,x_{f-1},x):=\left(\lambda_{0}\left(x_{0},x\right),\ldots,\lambda_{f-1}\left(x_{f-1},x\right)\right)$
where $\lambda_{i}\left(x_{i},x\right)\in\mathbb{Z}[x]\pm x_{i}$ as below.
Note that our $\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},p\right)$ is
the same as $\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1}\right)$ defined
in [BP12, §11].

If $f=1$,
$\lambda_{0}\left(x_{0},x\right)\in\left\\{x_{0},x-3-x_{0}\right\\}$. If
$f>1$, then:

  1. (i)

$\lambda_{i}\left(x_{i},x\right)\in\left\\{x_{i},x_{i}+1,x-2-x_{i},x-3-x_{i}\right\\}$
for $i\in\\{0,\ldots,f-1\\}$

  2. (ii)

if $\lambda_{i}\left(x_{i},x\right)\in\left\\{x_{i},x_{i}+1\right\\}$, then
$\lambda_{i+1}\left(x_{i+1},x\right)\in\left\\{x_{i+1},x-2-x_{i+1}\right\\}$

  3. (iii)

if $\lambda_{i}\left(x_{i},x\right)\in\left\\{x-2-x_{i},x-3-x_{i}\right\\}$,
then
$\lambda_{i+1}\left(x_{i+1},x\right)\in\left\\{x-3-x_{i+1},x_{i+1}+1\right\\}$

with the conventions that $x_{f}=x_{0}$ and
$\lambda_{f}\left(x_{f},x\right)=\lambda_{0}\left(x_{0},x\right).$

We use a different parameterisation of the set
$\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$ compared to [BP12,
§11], see Remark 2.5 below. An $f$-tuple $\lambda$ of the set
$\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$ can be naturally
identified with an $f$-tuple
${\vec{v}}:=(v_{0},...,v_{f-1})\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$,
where $v_{i}=0$ if
$\lambda_{i}\left(x_{i},x\right)\in\left\\{x_{i},x_{i}+1\right\\}$ and
$v_{i}=1$ if
$\lambda_{i}\left(x_{i},x\right)\in\left\\{x-2-x_{i},x-3-x_{i}\right\\}$. We
set $\ell(\lambda):=\\#\\{0\leq i\leq f-1\mid v_{i}=1\\}$. Let
$\lambda_{\vec{v}}$ denote the $f$-tuple $\lambda$ parameterised by
${\vec{v}}$.

For $\lambda\in\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$, we
define:

(10)
$e(\lambda):=\left\\{\begin{array}[]{cl}\frac{1}{2}\left(\sum_{i=0}^{f-1}x^{i}\left(x_{i}-\lambda_{i}\left(x_{i},x\right)\right)\right)&\text{if
}\lambda_{f-1}\left(x_{f-1},x\right)\in\left\\{x_{f-1},x_{f-1}+1\right\\}\\\
\frac{1}{2}\left(x^{f}-1+\sum_{i=0}^{f-1}x^{i}\left(x_{i}-\lambda_{i}\left(x_{i},x\right)\right)\right)&\text{otherwise}\end{array}\right.$

######  Remark 2.5.

As explained in [BP12, §11], the set
$\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1}\right)$ can be naturally
identified with the set of subsets $J$ of $\\{0,\ldots,f-1\\}$ as follows: set
$i\in J$ if and only if $\lambda_{i}(x_{i})\in\\{p-3-x_{i},x_{i}+1\\}$. Since
$v_{i}=1$ in our parameterisation if and only if $i+1\in J$, our
$\ell(\lambda)(x_{0},\ldots,x_{f-1},x)$ is the same as the
$\ell(\lambda)(x_{0},\ldots,x_{f-1},p)$ in [BP12, §11] which is defined as the
cardinality of $J$.

We first consider the case where $\overline{\rho}$ is reducible split. Let
$\omega_{f}$ denote the fundamental character of inertia group $I_{K}$ of
niveau $f$ associated to $\overline{\kappa}_{0}$; that is, $\omega_{f}$ is the
$reciprocal$ of the character defined by the composite

$I_{K}\stackrel{{\scriptstyle\operatorname{Art}_{K}^{-1}}}{{\longrightarrow}}\mathcal{O}_{K}^{\times}\stackrel{{\scriptstyle\operatorname{res}}}{{\longrightarrow}}k^{\times}\stackrel{{\scriptstyle\overline{\kappa}_{0}}}{{\longrightarrow}}\mathbb{F}^{\times}.$

Let $\omega_{2f}$ be the fundamental character of niveau $2f$ associated to
the embedding $\overline{\iota}$.

######  Lemma 2.6.

Let
$\overline{\rho}:\operatorname{Gal}\left(\overline{{\mathbb{Q}}}_{p}/K\right)\rightarrow\mathrm{GL}_{2}\left({\mathbb{F}}\right)$
be a continuous representation such that its restriction to inertia is:

$\left(\begin{array}[]{cc}\omega_{f}^{r_{0}+1+p\left(r_{1}+1\right)+\cdots+p^{f-1}\left(r_{f-1}+1\right)}&0\\\
0&1\end{array}\right)\otimes(\overline{\eta}\circ\operatorname{res}\circ\operatorname{Art}_{K}^{-1})$

with $-1\leq r_{i}\leq p-2$, where $\overline{\eta}$ is a smooth character
$k^{\times}\rightarrow{\mathbb{F}}^{\times}$. Assume $0\leq r_{i}\leq p-3$ for
all $i$ and not all $r_{i}$ equal to $0$ or equal to $p-3$. Then
$W_{\operatorname{GL}_{2}}(\overline{\rho})$, the set of
$\operatorname{GL}_{2}$-Serre weights associated to $\overline{\rho}$,
consists of:

$\sigma_{\vec{v}}(\overline{\rho}):=\lambda_{\vec{v}}(r_{0},...,r_{f-1},p)\otimes(\overline{\eta}\circ\operatorname{det}^{e(\lambda_{\vec{v}})\left(r_{0},\ldots,r_{f-1},p\right)})$

for $\vec{v}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ and
$\lambda_{\vec{v}}\in\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$.

###### Proof.

See [Dia07, Prop. 1.1 and Prop. 1.3] and [Dia07, §3] ∎

We now consider the case where $\overline{\rho}$ is reducible nonsplit. Let
$\leq$ be the partial order on $\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$
defined by $\vec{v}\leq\vec{v}^{\prime}$ if $v_{i}=1$, then $v_{i}^{\prime}=1$
for every $0\leq i\leq f-1$. Then
$\lambda_{{\vec{v}}}\leq\lambda_{{\vec{v}}^{\prime}}$ in the sense of [BP12,
§11] if and only if $\vec{v}\leq\vec{v}^{\prime}$.

######  Definition 2.7.

Let
$\overline{\rho}:\operatorname{Gal}\left(\overline{{\mathbb{Q}}}_{p}/K\right)\rightarrow\mathrm{GL}_{2}\left({\mathbb{F}}\right)$
be a continuous representation such that its restriction to inertia is:

$\left(\begin{array}[]{cc}\omega_{f}^{r_{0}+1+p\left(r_{1}+1\right)+\cdots+p^{f-1}\left(r_{f-1}+1\right)}&\ast\\\
0&1\end{array}\right)\otimes(\overline{\eta}\circ\operatorname{res}\circ\operatorname{Art}_{K}^{-1})$

with $\ast\neq 0$ and $0\leq r_{i}\leq p-3$ for all $i$ and not all $r_{i}$
equal to 0 or equal to $p-3$. As explained in [Bre14, §4], there exists an
object $M$ of the Fontaine-Laffaille category such that
$\overline{\rho}\simeq\operatorname{Hom}_{\operatorname{Fil}^{\cdot},\varphi.}\left(M,A_{\operatorname{cris}}\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}\right)$.
Explicitly, $M$ is a free $k\otimes_{{\mathbb{F}}_{p}}{\mathbb{F}}$-module of
rank $2$ which can be written as $M=M^{0}\times\cdots\times M^{f-1}$ where
$M^{j}={\mathbb{F}}e^{j}\oplus{\mathbb{F}}f^{j}$, $s_{j}:=r_{f-j}$,
$\operatorname{Fil}^{0}M^{j}=M^{j}$,
$\operatorname{Fil}^{1}M^{j}=\operatorname{Fil}^{s_{j}+1}M^{j}={\mathbb{F}}f^{j}$,
$\mathrm{Fil}^{s_{j}+2}M^{j}=0$ and

$\left\\{\begin{array}[]{ccc}\varphi\left(e^{j}\right)&=&\alpha_{j+1}e^{j+1}\\\
\varphi_{s_{j}+1}\left(f^{j}\right)&=&\beta_{j+1}\left(f^{j+1}+\mu_{j+1}e^{j+1}\right)\end{array}\right.$

for
$j\in\\{0,\cdots,f-1\\},\alpha_{j},\beta_{j}\in{\mathbb{F}}^{\times},\mu_{j}\in{\mathbb{F}}$.
Note that the nullity of $\mu_{j}$ is independent of the choice of
$\left(e^{j},f^{j}\right)_{j}$. We define an $f$-tuple
$\vec{v}(\overline{\rho})\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ associated
to $\overline{\rho}$ by setting $\vec{v}(\overline{\rho})_{j}=1$ if and only
if $\mu_{f-j}=0$. We also associate a non-empty subset
$\mathcal{D}(\overline{\rho})\subseteq\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$
to $\overline{\rho}$:

(11)
$\mathcal{D}(\overline{\rho}):=\\{\lambda_{\vec{v}}\in\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)\mid\vec{v}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}},\vec{v}\leq\vec{v}(\overline{\rho})\\}.$

######  Lemma 2.8.

Let $\overline{\rho}$ be generic reducible nonsplit as in Definition 2.7. Then
$W_{\operatorname{GL}_{2}}(\overline{\rho})$, the set of
$\operatorname{GL}_{2}$-Serre weights for $\overline{\rho}$, consists of:

$\sigma_{\vec{v}}(\overline{\rho}):=\lambda_{\vec{v}}(r_{0},...,r_{f-1},p)\otimes(\overline{\eta}\circ\operatorname{det}^{e(\lambda_{\vec{v}})\left(r_{0},\ldots,r_{f-1},p\right)})$

for $\vec{v}\leq\vec{v}(\overline{\rho})$ in
$\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ and
$\lambda_{\vec{v}}\in\mathcal{R}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$.

###### Proof.

See [Bre14, Prop. A.3]. ∎

####  2.4.2. The irreducible case

Let $\left(x_{0},\ldots,x_{f-1},x\right)$ be $f+1$ variables. We define a set
$\mathcal{ID}\left(x_{0},\ldots,x_{f-1},x\right)$ of $f$-tuples
$\lambda\left(x_{0},\ldots,x_{f-1},x\right):=\left(\lambda_{0}\left(x_{0},x\right),\ldots,\lambda_{f-1}\left(x_{f-1},x\right)\right)$
where $\lambda_{i}\left(x_{i},x\right)\in\mathbb{Z}[x]\pm x_{i}$ as below.
Note that our $\mathcal{I}\mathcal{D}\left(x_{0},\ldots,x_{f-1},p\right)$ is
the same as $\mathcal{I}\mathcal{D}\left(x_{0},\ldots,x_{f-1}\right)$ defined
in [BP12, §11].

If $f=1$,
$\lambda_{0}\left(x_{0},x\right)\in\left\\{x_{0},x-1-x_{0}\right\\}$. If
$f>1$, then:

  1. (i)

$\lambda_{0}\left(x_{0},x\right)\in\left\\{x_{0},x_{0}-1,x-2-x_{0},x-1-x_{0}\right\\}$
and $\lambda_{i}\left(x_{i},x\right)\in$
$\left\\{x_{i},x_{i}+1,x-2-x_{i},x-3-x_{i}\right\\}$ if $i>0$

  2. (ii)

if $i>0$ and
$\lambda_{i}\left(x_{i},x\right)\in\left\\{x_{i},x_{i}+1\right\\}$, or $i=0$
and $\lambda_{0}\left(x_{0},x\right)\in\left\\{x_{0},x_{0}-1\right\\}$, then
$\lambda_{i+1}\left(x_{i+1},x\right)\in\left\\{x_{i+1},x-2-x_{i+1}\right\\}$

  3. (iii)

if $0<i<f-1$ and
$\lambda_{i}\left(x_{i},x\right)\in\left\\{x-2-x_{i},x-3-x_{i}\right\\}$, or
$i=0$ and
$\lambda_{0}\left(x_{0},x\right)\in\left\\{x-2-x_{0},x-1-x_{0}\right\\}$, then
$\lambda_{i+1}\left(x_{i+1},x\right)\in$
$\left\\{x_{i+1}+1,x-3-x_{i+1}\right\\}$

  4. (iv)

if
$\lambda_{f-1}\left(x_{f-1},x\right)\in\left\\{x-2-x_{f-1},x-3-x_{f-1}\right\\}$,
then $\lambda_{0}\left(x_{0},x\right)\in\\{x-1-x_{0},x_{0}-1\\}$

with the conventions that $x_{f}=x_{0}$ and
$\lambda_{f}\left(x_{f},x\right)=\lambda_{0}\left(x_{0},x\right)$.

We use a different parameterisation of the set
$\mathcal{I}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$ compared to [BP12,
§11], see Remark 2.9 below. An $f$-tuple $\lambda$ of the set
$\mathcal{ID}\left(x_{0},\ldots,x_{f-1},x\right)$ can be naturally identified
with an $f$-tuple
${\vec{v}}:=(v_{0},...,v_{f-1})\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$,
where $v_{i}=0$ if $i>0$ and
$\lambda_{i}\left(x_{i},x\right)\in\left\\{x_{i},x_{i}+1\right\\}$ (resp.
$\lambda_{0}\left(x_{0},x\right)\in\left\\{x_{0},x_{0}-1\right\\}$), and
$v_{i}=1$ if $i>0$ and
$\lambda_{i}\left(x_{i},x\right)\in\left\\{x-2-x_{i},x-3-x_{i}\right\\}$
(resp.
$\lambda_{0}\left(x_{0},x\right)\in\left\\{x-2-x_{0},x-1-x_{0}\right\\}$). Let
$\lambda_{\vec{v}}$ denote the $f$-tuple $\lambda$ parameterised by
${\vec{v}}$.

For $\lambda\in\mathcal{I}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$, we
define :

(12)
$e(\lambda):=\left\\{\begin{array}[]{cl}\frac{1}{2}\left(\sum_{i=0}^{f-1}x^{i}\left(x_{i}-\lambda_{i}\left(x_{i},x\right)\right)\right)&\begin{subarray}{c}\text{if
}f>1\text{ and
}\lambda_{f-1}\left(x_{f-1},x\right)\in\left\\{x_{f-1},x_{f-1}+1\right\\}\\\
\text{ (resp. $\lambda_{0}(x_{0},x)=x_{0}$)}\end{subarray}\\\
\frac{1}{2}\left(x^{f}-1+\sum_{i=0}^{f-1}x^{i}\left(x_{i}-\lambda_{i}\left(x_{i},x\right)\right)\right)&\text{otherwise}\end{array}\right.$

######  Remark 2.9.

As explained in [BP12, §11], the set
$\mathcal{I}\mathcal{D}\left(x_{0},\ldots,x_{f-1}\right)$ can be naturally
identified with the set of subsets $J$ of $\\{0,\ldots,f-1\\}$ as follows: for
$i>0$, set $i\in J$ if and only if
$\lambda_{i}(x_{i})\in\\{p-3-x_{i},x_{i}+1\\}$; set $0\in J$ if and only if
$\lambda_{0}(x_{0})\in\\{p-1-x_{0},x_{0}-1\\}$. Then $v_{i}=1$ if and only if
$i+1\in J$.

######  Lemma 2.10.

Let
$\overline{\rho}:\operatorname{Gal}\left(\overline{\mathbb{Q}}_{p}/K\right)\rightarrow\mathrm{GL}_{2}\left({\mathbb{F}}\right)$
be a continuous representation such that its restriction to inertia is:

$\left(\begin{array}[]{cc}\omega_{2f}^{r_{0}+1+p\left(r_{1}+1\right)+\cdots+p^{f-1}\left(r_{f-1}+1\right)}&0\\\
0&\omega_{2f}^{p^{f}(r_{0}+1)+p^{f+1}\left(r_{1}+1\right)+\cdots+p^{2f-1}\left(r_{f-1}+1\right)}\end{array}\right)\otimes(\overline{\eta}\circ\operatorname{res}\circ\operatorname{Art}_{K}^{-1})$

with $0\leq r_{0}\leq p-1$ and $-1\leq r_{i}\leq p-2$ for $i>0$. Assume $1\leq
r_{0}\leq p-2$ and $0\leq r_{i}\leq p-3$ for $i>0$. Then
$W_{\operatorname{GL}_{2}}(\overline{\rho})$, the set of
$\operatorname{GL}_{2}$-Serre weights associated to $\overline{\rho}$,
consists of:

$\sigma_{\vec{v}}(\overline{\rho}):=\lambda_{\vec{v}}(r_{0},...,r_{f-1},p)\otimes(\overline{\eta}\circ\operatorname{det}^{e(\lambda_{\vec{v}})\left(r_{0},\ldots,r_{f-1},p\right)})$

for $\vec{v}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ and
$\lambda_{\vec{v}}\in\mathcal{I}\mathcal{D}\left(x_{0},\ldots,x_{f-1},x\right)$.

###### Proof.

See [Dia07, Prop. 1.1 and Prop. 1.3] and [Dia07, §3] ∎

##  3\. Explicit description of the set $W_{D}(\overline{\rho})$

###  3.1. Preparation for the calculation of $W_{D}(\overline{\rho})$ in
semisimple case

We assume $\overline{\rho}$ is semisimple and generic in this section. Recall
that we have descriptions of $\operatorname{JH}(\overline{\Theta}(\psi)^{{\rm
ss}})$ and $W_{\mathrm{GL}_{2}}(\overline{\rho})$ in Lemmas 2.2, 2.6, 2.8 and
2.10. We assume $\overline{\eta}=\overline{\kappa}_{0}$ in
$\overline{\rho}|_{I_{K}}$. Our goal is to determine all the
$\overline{\psi}$’s such that there exist
$\vec{u}\in\mathcal{P}_{\Theta([\overline{\psi}])}$,
$\vec{v}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ satisfying
$\overline{\Theta}([\overline{\psi}])_{\vec{u}}=\sigma_{\vec{v}}(\overline{\rho})$.

Let $q=p^{f}.$ Let $\overline{\psi}=\overline{\iota}^{(q+1)b+1+c}$ with $0\leq
b\leq q-2,$ $0\leq c\leq q-1.$ By (8) and Lemmas 2.6, 2.10, we have:

$\overline{\sigma}_{\vec{t}_{\vec{u}},\vec{s}_{\vec{u}}}\otimes\bar{\kappa}_{0}\circ\mathrm{det}^{b+u_{0}u_{f-1}+(1-u_{0})(1-u_{f-1})}=\lambda_{\vec{v}}(r_{0},\ldots,r_{f-1},p)\otimes\overline{\kappa}_{0}\circ\operatorname{det}^{e(\lambda_{\vec{v}})\left(r_{0},\ldots,r_{f-1},p\right)}.$

By Lemma 2.1 we get the following equations:

(13) $s_{\vec{u},i}=\lambda_{\vec{v},i}\left(r_{i},p\right),$ (14)
$b+u_{0}u_{f-1}+(1-u_{0})(1-u_{f-1})+\sum_{i=0}^{f-1}p^{i}t_{\vec{u},i}\equiv
e(\lambda_{\vec{v}})(r_{0},\ldots,r_{f-1},p)(\text{mod }q-1).$

######  Lemma 3.1.

Let $\vec{u},\vec{v}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ satisfy
$\overline{\Theta}([\overline{\psi}])_{\vec{u}}=\sigma_{\vec{v}}(\overline{\rho})$.
Then $\overline{\psi}$ is uniquely determined by $\vec{u},\vec{v}$, and
$\vec{u}\in\mathcal{P}_{\Theta([\overline{\psi}])}$. Write
$\overline{\psi}_{\vec{u},{\vec{v}}}$ for such $\overline{\psi}$.

###### Proof.

If $\overline{\psi}=\overline{\iota}^{(q+1)b+1+c}$ satisfies
$\overline{\Theta}([\overline{\psi}])_{\vec{u}}=\sigma_{\vec{v}}(\overline{\rho})$,
then $(c_{i})_{i}$ is determined by (6) and (13), and $b$ is determined by
(14). By (6), (13) and the definition of $\mathcal{P}_{\Theta(\psi)}$, it
suffices to verify $\lambda_{\vec{v},i}(r_{i},p)\neq-1$ which follows from the
generic condition. ∎

######  Definition 3.2.

For
$z=z(x_{0},\ldots,x_{f-1},x)\in\oplus_{i=0}^{f-1}{\mathbb{Z}}[x]x_{i}\oplus{\mathbb{Z}}[x]\subset{\mathbb{Z}}[x_{0},\ldots,x_{f-1},x]$,
we can write $z$ uniquely as $z=L(z)+C(z)$ with

$L(z):=\Sigma_{i=0}^{f-1}a_{i}x_{i},\ C(z):=\Sigma_{i}a^{\prime}_{i}x^{i}.$

Let $C_{i}(z):=a^{\prime}_{i}$. Define $S(z)=S(z)(x_{0},\ldots,x_{f-1},x)$ to
be the unique element in ${\mathbb{Z}}[x_{0},\ldots,x_{f-1},x]$ such that
$\operatorname{deg}_{x}(S(z))<f$ and $z\equiv S(z)\
(\operatorname{mod}x^{f}-1)$.

######  Lemma 3.3.

We have
$\frac{1}{2}S(\Sigma_{i=0}^{f-1}(x^{i}(x_{i}-\lambda_{\vec{v},i})))\in\oplus_{i=0}^{f-1}{\mathbb{Z}}[x]x_{i}\oplus{\mathbb{Z}}[x]$
and

$e(\lambda_{\vec{v}})(r_{0},...,r_{f-1},p)=\frac{1}{2}S(\Sigma_{i=0}^{f-1}(x^{i}(x_{i}-\lambda_{\vec{v},i})))(r_{0},...,r_{f-1},p).$

###### Proof.

Since $L(x_{i}-\lambda_{\vec{v},i})\in\\{0,2x_{i}\\}$, we have
$\frac{1}{2}L(S(\Sigma_{i=0}^{f-1}(x^{i}(x_{i}-\lambda_{\vec{v},i}))))\in\oplus_{i=0}^{f-1}{\mathbb{Z}}[x]x_{i}$.
Since $\operatorname{deg}_{x}C(x_{i}-\lambda_{\vec{v},i})\leq 1$ and
$C_{1}(x_{i}-\lambda_{\vec{v},i})\in\\{0,-1\\}$, we see that
$\frac{1}{2}C(S(\Sigma_{i=0}^{f-1}(x^{i}(x_{i}-\lambda_{\vec{v},i}))))\in{\mathbb{Z}}[x]$
follows from the fact $C_{1}(x_{i}-\lambda_{\vec{v},i})=-1$ if and only if
$C_{0}(x_{i+1}-\lambda_{\vec{v},i+1})$ is odd. The first claim follows. Since
$\operatorname{deg}_{x}(\Sigma_{i=0}^{f-1}(x^{i}(x_{i}-\lambda_{\vec{v},i})))\leq
f$, we obtain $S(\Sigma_{i=0}^{f-1}(x^{i}(x_{i}-\lambda_{\vec{v},i})))$ by
substituting $x^{f}$ by $1$ in $x^{f-1}\lambda_{\vec{v},f-1}$ if
$\operatorname{deg}_{x}(\lambda_{\vec{v},f-1})=1$. This coincides with the
definition of $e(\lambda_{\vec{v}})$ and we get the second claim. ∎

######  Definition 3.4.

Motivated by (6), (7) and (13), we define:

(15)
$c_{\vec{u},\vec{v},i}(x_{i},x):=\left\\{\begin{array}[]{cl}x-2+u_{0,i-1}-\lambda_{\vec{v},i}(x_{i},x)&\text{
if }u_{i}=1\\\ u_{0,i-1}+\lambda_{\vec{v},i}(x_{i},x)&\text{ if
}u_{i}=0\end{array}\right.,$ (16)
$t_{\vec{u},\vec{v},i}(x_{i},x):=\left\\{\begin{array}[]{cl}c_{\vec{u},\vec{v},i}(x_{i},x)+(1-u_{i-1})&\text{
if }u_{i}=1\\\ 0&\text{ if }u_{i}=0\end{array}\right.,$ (17)
$b_{\vec{u},\vec{v}}(x_{0},...,x_{f-1},x):=\frac{1}{2}S(\Sigma_{i=0}^{f-1}x^{i}(x_{i}-\lambda_{\vec{v},i}))-S(u_{0}u_{f-1}+(1-u_{0})(1-u_{f-1})+\Sigma_{i=0}^{f-1}x^{i}t_{\vec{u},\vec{v},i}).$

We write $c_{\vec{u},\vec{v},i}$, $t_{\vec{u},\vec{v},i}$,
$b_{\vec{u},\vec{v}}$ for $c_{\vec{u},\vec{v},i}(x_{i},x)$,
$t_{\vec{u},\vec{v},i}(x_{i},x)$, $b_{\vec{u},\vec{v}}(x_{0},...,x_{f-1},x)$
in the following.

######  Lemma 3.5.

We have

(18)
$\begin{split}b_{\vec{u},\vec{v}}(x_{0},...,x_{f-1},x)&=\frac{1}{2}(\sum_{0\leq
i<f-1}x^{i}(x_{i}-\lambda_{\vec{v},i})+S(x^{f-1}(x_{f-1}-\lambda_{\vec{v},f-1})))\\\
&-(u_{0}u_{f-1}+(1-u_{0})(1-u_{f-1})+\sum_{0\leq
i<f-1}x^{i}t_{\vec{u},\vec{v},i}+S(x^{f-1}t_{\vec{u},\vec{v},f-1})),\end{split}$

and

(19)
$1+\Sigma_{i=0}^{f-1}p^{i}c_{i}+(q+1)b\equiv(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})(r_{0},...,r_{f-1},p)~{}(\operatorname{mod}q^{2}-1).$

where $c_{0},...,c_{f-1}$ and $b$ satisfy (6), (13) and (14).

###### Proof.

The first claim follows from
$\operatorname{deg}_{x}(x_{i}-\lambda_{\vec{v},i})\leq 1$ and
$\operatorname{deg}_{x}(t_{\vec{u},\vec{v},i})\leq 1$. Since we compute
$1+c+(q+1)b$ $\operatorname{mod}q^{2}-1$, it suffices to determine $b$
$\operatorname{mod}q-1$ and the second claim follows. ∎

Next we calculate
$L(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})$
and
$C(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})=\Sigma_{i=0}^{2f-1}d_{i}x^{i}$
in Proposition 3.7 and Proposition 3.8.

######  Lemma 3.6.

$L(c_{\vec{u},\vec{v},i})$, $L(t_{\vec{u},\vec{v},i})$,
$L(\lambda_{\vec{v},i})$ are determined by $u_{i}$ and $v_{i}$ as follows.

$(L(c_{\vec{u},\vec{v},i}),L(t_{\vec{u},\vec{v},i}),L(\lambda_{\vec{v},i}))$ | $u_{i}=0$ | $u_{i}=1$
---|---|---
$v_{i}=0$ | $(x_{i},0,x_{i})$ | $(-x_{i},-x_{i},x_{i})$
$v_{i}=1$ | $(-x_{i},0,-x_{i})$ | $(x_{i},x_{i},-x_{i})$

###### Proof.

This follows from (15), (16) and the definition of $\lambda_{\vec{v},i}$. ∎

######  Proposition 3.7.

$L(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})$
are determined by $\vec{u}$ and $\vec{v}$ as follows:

$L(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})=\Sigma_{i=0}^{f-1}x^{i+w_{i}f}x_{i},$

where $w_{i}\in\\{0,1\\}$ and $w_{i}\equiv u_{i}+v_{i}\
(\operatorname{mod}2)$.

###### Proof.

Since $\operatorname{deg}_{x}L(x_{f-1}-\lambda_{\vec{v},f-1})\leq 1$ and
$\operatorname{deg}_{x}L(t_{\vec{u},\vec{v},f-1})\leq 1$, we have

$L(S(x^{f-1}(x_{f-1}-\lambda_{\vec{v},f-1})))=L(x^{f-1}(x_{f-1}-\lambda_{\vec{v},f-1})),$
$L(S(x^{f-1}t_{\vec{u},\vec{v},f-1}))=L(x^{f-1}t_{\vec{u},\vec{v},f-1}),\
\text{and}$
$L(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})=\sum_{i=0}^{f-1}L(x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})(-x^{i}t_{\vec{u},\vec{v},i}+\frac{1}{2}x^{i}(x_{i}-\lambda_{\vec{v},i}))).$

It follows from the table in Lemma 3.6 that

$\begin{split}L(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})&=\sum_{i=0}^{f-1}L(x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})(-x^{i}t_{\vec{u},\vec{v},i}+\frac{1}{2}x^{i}(x_{i}-\lambda_{\vec{v},i})))\\\
&=\sum_{\begin{subarray}{c}0\leq i\leq f-1\\\
u_{i}=0,v_{i}=0\end{subarray}}x^{i}x_{i}+\sum_{\begin{subarray}{c}0\leq i\leq
f-1\\\ u_{i}=0,v_{i}=1\end{subarray}}x^{i+f}x_{i}\\\
&+\sum_{\begin{subarray}{c}0\leq i\leq f-1\\\
u_{i}=1,v_{i}=0\end{subarray}}x^{i+f}x_{i}+\sum_{\begin{subarray}{c}0\leq
i\leq f-1\\\ u_{i}=1,v_{i}=1\end{subarray}}x^{i}x_{i}.\end{split}$

∎

Since $\operatorname{deg}_{x}(C(c_{\vec{u},\vec{v},i}))\leq 1$,
$\operatorname{deg}_{x}(C(b_{\vec{u},\vec{v}}))<f$, let $d_{i}\in{\mathbb{Z}}$
for $0\leq i\leq 2f-1$ be given by:

(20)
$C(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})=\Sigma_{i=0}^{2f-1}d_{i}x^{i}.$

######  Proposition 3.8.

We have

(21) $d_{i}+d_{i+f}=0$

for $0\leq i\leq f-1$, and $d_{i}$ is determined by
$u_{i-1},u_{i},v_{i-1},v_{i}$ as follows.

For $i>0$, we have

Table 1.  $d_{i}(i>0)$ | $(u_{i-1},u_{i})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{i-1},v_{i})$ | $(0,0)$ | $0$ | $-1$ | $1$ | $0$
$(0,1)$ | $-1$ | $0$ | $0$ | $1$
$(1,0)$ | $1$ | $-1$ | $1$ | $-1$
$(1,1)$ | $-1$ | $1$ | $-1$ | $1$

For $i=0$ and $\overline{\rho}$ is reducible split, we have

Table 2.  $d_{0}$ | $(u_{f-1},u_{0})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{f-1},v_{0})$ | $(0,0)$ | $1$ | $0$ | $0$ | $-1$
$(0,1)$ | $0$ | $1$ | $-1$ | $0$
$(1,0)$ | $1$ | $-1$ | $1$ | $-1$
$(1,1)$ | $-1$ | $1$ | $-1$ | $1$

For $i=0$ and $\overline{\rho}$ is irreducible, we have

Table 3.  $d_{0}$ | $(u_{f-1},u_{0})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{f-1},v_{0})$ | $(0,0)$ | $1$ | $0$ | $0$ | $-1$
$(0,1)$ | $0$ | $1$ | $-1$ | $0$
$(1,0)$ | $0$ | $0$ | $0$ | $0$
$(1,1)$ | $0$ | $0$ | $0$ | $0$

###### Proof.

Note first $\operatorname{deg}_{x}(C(c_{\vec{u},\vec{v},i}))\leq 1$,
$\operatorname{deg}_{x}(C(\lambda_{\vec{v},i}))\leq 1$,
$\operatorname{deg}_{x}(C(t_{\vec{u},\vec{v},i}))\leq 1$, and
$\operatorname{deg}_{x}(C(b_{\vec{u},\vec{v}}))<f$. By (17) and (20), if
$0<i\leq f-1$, then we have

(22)
$d_{i}=C_{i}(x^{i-1}c_{\vec{u},\vec{v},i-1}+x^{i}c_{\vec{u},\vec{v},i}-x^{i-1}t_{\vec{u},\vec{v},i-1}-x^{i}t_{\vec{u},\vec{v},i}-\frac{1}{2}(x^{i-1}\lambda_{\vec{v},i-1}+x^{i}\lambda_{\vec{v},i})),$
$d_{i+f}=C_{i}(-x^{i-1}t_{\vec{u},\vec{v},i-1}-x^{i}t_{\vec{u},\vec{v},i}-\frac{1}{2}(x^{i-1}\lambda_{\vec{v},i-1}+x^{i}\lambda_{\vec{v},i})),$

and

(23)
$\begin{split}d_{i}+d_{i+f}=&C_{i}(x^{i-1}c_{\vec{u},\vec{v},i-1}+x^{i}c_{\vec{u},\vec{v},i}-2(x^{i-1}t_{\vec{u},\vec{v},i-1}+x^{i}t_{\vec{u},\vec{v},i})-(x^{i-1}\lambda_{\vec{v},i-1}+x^{i}\lambda_{\vec{v},i}))\end{split}$

The above sum is determined by $u_{i-1},u_{i},v_{i-1},v_{i}$ as follows.

By definition of $\lambda_{\vec{v}}$, we have

$C_{i}(x^{i-1}\lambda_{\vec{v},i-1})=\left\\{\begin{aligned} 0\text{, if
}v_{i-1}=0\\\ 1\text{, if }v_{i-1}=1\end{aligned}\right.,\quad
C_{i}(x^{i}\lambda_{\vec{v},i})=\left\\{\begin{aligned} 0\text{, if
}(v_{i-1},v_{i})=(0,0)\\\ -2\text{, if }(v_{i-1},v_{i})=(0,1)\\\ 1\text{, if
}(v_{i-1},v_{i})=(1,0)\\\ -3\text{, if }(v_{i-1},v_{i})=(1,1)\\\
\end{aligned}\right..$

By (15), we have

$C_{i}(x^{i-1}c_{\vec{u},\vec{v},i-1})$ | $u_{i-1}=0$ | $u_{i-1}=1$
---|---|---
$v_{i-1}=0$ | $0$ | $1$
$v_{i-1}=1$ | $1$ | $0$
$C_{i}(x^{i}c_{\vec{u},\vec{v},i})$ | $(u_{i-1},u_{i})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{i-1},v_{i})$ | $(0,0)$ | $0$ | $-2$ | $1$ | $-1$
$(0,1)$ | $-2$ | $0$ | $-1$ | $1$
$(1,0)$ | $1$ | $-3$ | $2$ | $-2$
$(1,1)$ | $-3$ | $1$ | $-2$ | $2$

By (16), we have

$C_{i}(x^{i-1}t_{\vec{u},\vec{v},i-1})$ | $u_{i-1}=0$ | $u_{i-1}=1$
---|---|---
$v_{i-1}=0$ | $0$ | $1$
$v_{i-1}=1$ | $0$ | $0$
$C_{i}(x^{i}t_{\vec{u},\vec{v},i})$ | $(u_{i-1},u_{i})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{i-1},v_{i})$ | $(0,0)$ | $0$ | $-1$ | $0$ | $-1$
$(0,1)$ | $0$ | $1$ | $0$ | $1$
$(1,0)$ | $0$ | $-2$ | $0$ | $-2$
$(1,1)$ | $0$ | $2$ | $0$ | $2$

Combining (23) and tables above, we have $d_{i}+d_{i+f}=0$ for $i>0$.
Combining (22) and tables above, we have Table 1.

If $i=0$, then we have

(24)
$\begin{split}d_{0}&=C_{0}\Big{(}1+c_{\vec{u},\vec{v},0}-t_{\vec{u},\vec{v},0}-S\left(x^{f-1}t_{\vec{u},\vec{v},f-1}\right)-\left(u_{0}u_{f-1}+\left(1-u_{0}\right)\left(1-u_{f-1}\right)\right)\\\
&-\frac{1}{2}\left(x^{0}\lambda_{\vec{v},0}+S\left(x^{f-1}\lambda_{\vec{v},f-1}\right)\right)\Big{)},\end{split}$

$\begin{split}d_{f}=&C_{f}\left(x^{f-1}c_{\vec{u},\vec{v},f-1}\right)+C_{0}\Big{(}-t_{\vec{u},\vec{v},0}-S\left(x^{f-1}t_{\vec{u},\vec{v},f-1}\right)-\left(u_{0}u_{f-1}+\left(1-u_{0}\right)\left(1-u_{f-1}\right)\right)\\\
&-\frac{1}{2}\left(x^{0}\lambda_{\vec{v},0}+S\left(x^{f-1}\lambda_{\vec{v},f-1}\right)\right)\Big{)},\end{split}$

and

(25)
$\begin{split}d_{0}+d_{f}=&C_{0}\left(1+c_{\vec{u},\vec{v},0}-2\left(t_{\vec{u},\vec{v},0}-S\left(x^{f-1}t_{\vec{u},\vec{v},f-1}\right)\right)-2\left(u_{0}u_{f-1}+\left(1-u_{0}\right)\left(1-u_{f-1}\right)\right)\right.\\\
&\left.-\left(x^{0}\lambda_{\vec{v},0}+S\left(x^{f-1}\lambda_{\vec{v},f-1}\right)\right)\right)+C_{f}\left(x^{f-1}c_{\vec{u},\vec{v},f-1}\right).\end{split}$

The above sum is determined by $u_{f-1},u_{0},v_{f-1},v_{0}$ as follows.

$u_{0}u_{f-1}+(1-u_{0})(1-u_{f-1})=\left\\{\begin{aligned} 1\text{, if
}(u_{f-1},u_{0})=(0,0)\\\ 0\text{, if }(u_{f-1},u_{0})=(0,1)\\\ 0\text{, if
}(u_{f-1},u_{0})=(1,0)\\\ 1\text{, if }(u_{f-1},u_{0})=(1,1)\\\
\end{aligned}\right.$

If $\overline{\rho}$ is reducible split, by definition of $\lambda_{\vec{v}}$,
we have

$C_{0}(S(x^{f-1}\lambda_{\vec{v},f-1}))=\left\\{\begin{aligned} 0\text{, if
}v_{f-1}=0\\\ 1\text{, if }v_{f-1}=1\end{aligned}\right.,\quad
C_{0}(x^{0}\lambda_{\vec{v},0})=\left\\{\begin{aligned} 0\text{, if
}(v_{f-1},v_{0})=(0,0)\\\ -2\text{, if }(v_{f-1},v_{0})=(0,1)\\\ 1\text{, if
}(v_{f-1},v_{0})=(1,0)\\\ -3\text{, if }(v_{f-1},v_{0})=(1,1)\\\
\end{aligned}\right..$

By (15), we have

$C_{f}(x^{f-1}c_{\vec{u},\vec{v},f-1})$ | $u_{f-1}=0$ | $u_{f-1}=1$
---|---|---
$v_{f-1}=0$ | $0$ | $1$
$v_{f-1}=1$ | $1$ | $0$
$C_{0}(c_{\vec{u},\vec{v},0})$ | $(u_{f-1},u_{0})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{f-1},v_{0})$ | $(0,0)$ | $1$ | $-1$ | $0$ | $-2$
$(0,1)$ | $-1$ | $1$ | $-2$ | $0$
$(1,0)$ | $2$ | $-2$ | $1$ | $-3$
$(1,1)$ | $-2$ | $2$ | $-3$ | $1$

By (16), we have

$C_{0}(S(x^{f-1}t_{\vec{u},\vec{v},f-1}))$ | $u_{f-1}=0$ | $u_{f-1}=1$
---|---|---
$v_{f-1}=0$ | $0$ | $1$
$v_{f-1}=1$ | $0$ | $0$
$C_{0}(t_{\vec{u},\vec{v},0})$ | $(u_{f-1},u_{0})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{f-1},v_{0})$ | $(0,0)$ | $0$ | $0$ | $0$ | $-2$
$(0,1)$ | $0$ | $2$ | $0$ | $0$
$(1,0)$ | $0$ | $-1$ | $0$ | $-3$
$(1,1)$ | $0$ | $3$ | $0$ | $1$

Combining (24), (25) and tables above, we get Table 2 and $d_{0}+d_{f}=0$ for
reducible split $\overline{\rho}$.

If $\overline{\rho}$ is irreducible, by definition of $\lambda_{\vec{v}}$, we
have

$C_{0}(S(x^{f-1}\lambda_{\vec{v},f-1}))=\left\\{\begin{aligned} 0\text{, if
}v_{f-1}=0\\\ 1\text{, if }v_{f-1}=1\end{aligned}\right.,\quad
C_{0}(x^{0}\lambda_{\vec{v},0})=\left\\{\begin{aligned} 0\text{, if
}(v_{f-1},v_{0})=(0,0)\\\ -2\text{, if }(v_{f-1},v_{0})=(0,1)\\\ -1\text{, if
}(v_{f-1},v_{0})=(1,0)\\\ -1\text{, if }(v_{f-1},v_{0})=(1,1)\\\
\end{aligned}\right..$

By (15), we have

$C_{f}(x^{f-1}c_{\vec{u},\vec{v},f-1})$ | $u_{f-1}=0$ | $u_{f-1}=1$
---|---|---
$v_{f-1}=0$ | $0$ | $1$
$v_{f-1}=1$ | $1$ | $0$
$C_{0}(c_{\vec{u},\vec{v},0})$ | $(u_{f-1},u_{0})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{f-1},v_{0})$ | $(0,0)$ | $1$ | $-1$ | $0$ | $-2$
$(0,1)$ | $-1$ | $1$ | $-2$ | $0$
$(1,0)$ | $0$ | $0$ | $-1$ | $-1$
$(1,1)$ | $0$ | $0$ | $-1$ | $-1$

By (16), we have

$C_{0}(S(x^{f-1}t_{\vec{u},\vec{v},f-1}))$ | $u_{f-1}=0$ | $u_{f-1}=1$
---|---|---
$v_{f-1}=0$ | $0$ | $1$
$v_{f-1}=1$ | $0$ | $0$
$C_{0}(t_{\vec{u},\vec{v},0})$ | $(u_{f-1},u_{0})$
---|---
$(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
$(v_{f-1},v_{0})$ | $(0,0)$ | $0$ | $0$ | $0$ | $-2$
$(0,1)$ | $0$ | $2$ | $0$ | $0$
$(1,0)$ | $0$ | $1$ | $0$ | $-1$
$(1,1)$ | $0$ | $1$ | $0$ | $-1$

Combining (24), (25) and tables above, we get Table 3 and $d_{0}+d_{f}=0$ for
irreducible $\overline{\rho}$. ∎

###  3.2. Explicit description of $W_{D}(\overline{\rho})$ for generic
$\overline{\rho}$

We now give explicit description of $W_{D}(\overline{\rho}).$ Let
$\overline{\psi}=\overline{\iota}^{(q+1)b+1+c}\in W_{D}(\overline{\rho})$ with
$0\leq b\leq q-2,$ $0\leq c\leq q-1$. By (19) and (21), we have

$\begin{split}(q+1)b+1+c\equiv&(L(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})(r_{0},...,r_{f-1},p)\\\
&+C(1+\Sigma_{i=0}^{f-1}x^{i}c_{\vec{u},\vec{v},i}+(1+x^{f})b_{\vec{u},\vec{v}})(r_{0},...,r_{f-1},p))~{}(\operatorname{mod}q^{2}-1)\\\
\equiv&\sum_{i=0}^{f-1}q^{w_{i}}p^{i}r_{i}+\sum_{i=0}^{2f-1}d_{i}p^{i}~{}(\operatorname{mod}q^{2}-1)\\\
\equiv&\sum_{i=0}^{f-1}q^{w_{i}}p^{i}r_{i}+\sum_{i=0}^{f-1}(d_{i}+qd_{i+f})p^{i}~{}(\operatorname{mod}q^{2}-1)\\\
\equiv&\sum_{i=0}^{f-1}q^{w_{i}}p^{i}r_{i}+(1-q)\sum_{i=0}^{f-1}d_{i}p^{i}~{}(\operatorname{mod}q^{2}-1).\end{split}$

######  Theorem 3.9.

Let $\overline{\rho}$ be generic reducible split as in Lemma 2.6, then
$\overline{\psi}=\overline{\iota}^{(q+1)b+1+c}\in W_{D}(\overline{\rho})$, if
and only if

$(q+1)b+1+c\equiv\sum_{i=0}^{f-1}q^{w_{i}}p^{i}r_{i}+(1-q)\sum_{i=0}^{f-1}d_{i}p^{i}~{}(\operatorname{mod}q^{2}-1)$

where $w_{i}\in\\{0,1\\}$ and $d_{i}\in\\{-1,0,1\\}$ satisfy the following
relations.

  * •

For $i>0$, $w_{i}=\left\\{\begin{array}[]{cl}1&\text{ if }d_{i}=-1\\\ 0&\text{
if }d_{i}=1\end{array}\right.$, and if $d_{i}=0$, then $(w_{i-1},w_{i})=(0,0)$
or $(1,1)$.

  * •

$w_{0}=\left\\{\begin{array}[]{cl}1&\text{ if }d_{0}=-1\\\ 0&\text{ if
}d_{0}=1\end{array}\right.$, and if $d_{0}=0$, then $(w_{f-1},w_{0})=(0,1)$ or
$(1,0)$.

Note that, from the above relations, if $d_{i}=0$ for all $0\leq i\leq f-1$,
$\vec{w}$ does not exist, and if $(d_{0},\ldots,d_{f-1})\neq(0,\ldots,0)$,
there is exactly one $\vec{w}$. We write $\overline{\psi}_{\vec{w},\vec{d}}$
for such $\overline{\psi}$ and we have

(26)
$\overline{\psi}_{\vec{w},\vec{d}}=\overline{\psi}_{\vec{w}^{\prime},\vec{d}^{\prime}}\Longleftrightarrow\vec{w}=\vec{w}^{\prime},\
\vec{d}=\vec{d}^{\prime}.$

As a consequence,

$W_{D}(\overline{\rho})=\\{\overline{\psi}_{\vec{w},\vec{d}}\mid\vec{d}\in\\{-1,0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}},\
\vec{d}\neq(0,...,0)\\},$

where $\vec{w}$ is uniquely determined by $\vec{d}$. We have
$\\#W_{D}(\overline{\rho})=3^{f}-1$.

###### Proof.

The relations of $\vec{w}$ and $\vec{d}$ follow from Proposition 3.7, Tables 1
and 2 in Proposition 3.8. We need to prove $\vec{w}$ is uniquely determined by
$\vec{d}\neq(0,...,0)$ and construct $\vec{u}$ and $\vec{v}$ for each
$\vec{d}\neq(0,...,0)$.

If $d_{i}=0$ for all $0\leq i\leq f-1$, we deduce from the first relation that
all $w_{i}$ should be the same, while the second relation tells us that
$w_{0}$ and $w_{f-1}$ are different. This is a contradiction.

If $\vec{d}\neq 0$, we will show the uniqueness of $\vec{w}$. By the two
relations, we have if $d_{i}=-1$ or $1$ , $w_{i}$ is determined by $d_{i}$; if
$d_{i}=0$, $w_{i}$ is determined by the nearest $d_{j}\neq 0$ $(j<i)$ if there
exists $j<i$, $d_{j}\neq 0$, and by the farthest $d_{j}\neq 0$ $(j>i)$ if
$d_{j}=0$ for all $j<i$.

For $\vec{d}\neq(0,...,0)$, to show $\overline{\psi}_{\vec{w},\vec{d}}\in
W_{D}(\overline{\rho})$, by Lemma 3.1, it suffices to find
$\vec{u},\vec{v}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ such that
$\overline{\psi}_{\vec{w},\vec{d}}=\overline{\psi}_{\vec{u},\vec{v}}$. Let
$v_{i-1}\equiv d_{i}(\text{mod }2)$ for $0\leq i\leq f-1$, we fix the choice
of $\vec{v}$, and construct $\vec{u}$. By Table 1 and Table 2, if $d_{i}=-1$
or $1$, $u_{i}$ is determined by $(v_{i-1},v_{i},d_{i})$, and if $d_{i}=0$,
$u_{i}$ is determined by $(v_{i-1},v_{i},u_{i-1})$.

If
$\overline{\psi}_{\vec{w},\vec{d}}=\overline{\psi}_{\vec{w}^{\prime},\vec{d}^{\prime}}$,
then

$0\equiv(q-1)\sum_{i=0}^{f-1}p^{i}((w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i}))\
(\operatorname{mod}q^{2}-1).$

Note that $0\leq r_{i}\leq p-3$ for all $i$ since $\overline{\rho}$ is
generic. Since
$\sum_{i=0}^{f-1}p^{i}\left|(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})\right|\leq\sum_{i=0}^{f-1}p^{i}(r_{i}+2)<q+1$,
we have
$\sum_{i=0}^{f-1}p^{i}((w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i}))=0$.
Since $\left|(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})\right|\leq
p-1$ for all $i$, we have
$(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})=0$ for all $i$. Since
$w_{i}\leq w^{\prime}_{i}$ if $d_{i}>d^{\prime}_{i}$ by the relations above,
we have $d_{i}=d^{\prime}_{i}$ for all $i$. Then $\vec{w}=\vec{w}^{\prime}$
since $\vec{w}$ is uniquely determined by $\vec{d}$. ∎

######  Definition 3.10.

Let $\overline{\rho}$ be generic reducible split as in Lemma 2.6. For each
$f$-tuple ${\vec{v}}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$, we define a
set $U_{\vec{v}}$ consisting of $f$-tuples
${\vec{u}}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$ such that:

  * •

If $v_{f-1}=1$, then $u_{f-1}-u_{0}\equiv v_{f-1}-v_{0}+1\
(\operatorname{mod}2)$;

  * •

if $0<i\leq f-1$ and $v_{i-1}=1$, then $u_{i-1}-u_{i}\equiv v_{i-1}-v_{i}\
(\operatorname{mod}2)$.

For ${\vec{v}}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$, we define:

(27)
$W_{D}^{\vec{v}}(\overline{\rho}):=\\{\overline{\psi}:l^{\times}\rightarrow{\mathbb{F}}^{\times}\mid\exists\vec{u}\in
U_{\vec{v}},\text{ s.t.
}\sigma_{\vec{v}}(\overline{\rho})=\overline{\Theta}([\overline{\psi}])_{\vec{u}}\\},$

where
$\sigma_{\vec{v}}(\overline{\rho})=\lambda_{\vec{v}}\left(r_{0},\ldots,r_{f-1},p\right)\otimes\left(\overline{\kappa}_{0}\circ\operatorname{det}^{e(\lambda_{\vec{v}})\left(r_{0},\ldots,r_{f-1},p\right)}\right)$,
and define:

$W_{D,\vec{v}}(\overline{\rho}):=\\{\overline{\psi}:l^{\times}\rightarrow{\mathbb{F}}^{\times}\mid\exists\vec{u}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}},\text{
s.t. }\overline{\psi}=\overline{\psi}_{\vec{u},\vec{v}}\\}.$

######  Proposition 3.11.

We have the following properties:

  1. (i)

If $\ell(\lambda_{\vec{v}})=f$, $W_{D}^{\vec{v}}(\overline{\rho})=\emptyset$.

  2. (ii)

If $\ell(\lambda_{\vec{v}})=d<f$,
$\\#W_{D}^{\vec{v}}(\overline{\rho})=2^{f-d}$.

  3. (iii)

Assume $\overline{\psi}=\overline{\psi}_{\vec{u},\vec{v}}\in
W_{D}^{\vec{v}}(\overline{\rho})$, $\vec{u}\in U_{\vec{v}}$. If there exist
$\vec{u}^{\prime}$ and ${\vec{v}}^{\prime}$, such that
$\overline{\Theta}([\overline{\psi}])_{\vec{u}^{\prime}}=\sigma_{\vec{v}^{\prime}}(\overline{\rho})$,
then ${\vec{v}}\leq{{\vec{v}}^{\prime}}$. In particular, if
$W_{D}^{\vec{v}}(\overline{\rho})\cap
W_{D}^{\vec{v}^{\prime}}(\overline{\rho})\neq\emptyset$, then
${\vec{v}}={{\vec{v}}^{\prime}}$.

  4. (iv)

$W_{D}(\overline{\rho})=\bigsqcup\limits_{{\vec{v}}}W_{D}^{\vec{v}}(\overline{\rho})$.

  5. (v)

$W_{D}^{\vec{v}}(\overline{\rho})=W_{D,\vec{v}}(\overline{\rho})\backslash\bigcup_{\vec{v}^{\prime}<\vec{v}}W_{D,\vec{v}^{\prime}}(\overline{\rho})$.

###### Proof.

For (i), since $\ell(\lambda_{\vec{v}})=f$, we have $v_{i}=1$ for $0\leq i\leq
f-1$, then $u_{f-1}-u_{0}\equiv 1\ (\operatorname{mod}2)$ and
$u_{i-1}-u_{i}\equiv 0\ (\operatorname{mod}2)$ for $0<i\leq f-1$. It is easy
to see that $U_{\vec{v}}=\emptyset$. For (ii), there exists $i$, such that
$v_{i}=0$. Let $u_{i+1}=0$ or $1$. Then we determine $u_{j}$ for
$j=i+2,...,f-1,0,...,i$ in turn. If $v_{j}=0$, there are two choices for
$u_{j+1}$, and if $v_{j}=1$, $u_{j+1}$ is uniquely determined by $u_{j}$, so
there are $2^{f-d}$ choices for $\vec{u}$. Since
$\overline{\psi}_{\vec{u},\vec{v}}=\overline{\psi}_{\vec{u}^{\prime},\vec{v}}$
if and only if $\vec{u}=\vec{u}^{\prime}$ by $w_{i}\equiv u_{i}+v_{i}\
(\operatorname{mod}2)$ and (26), we have
$\\#W_{D}^{\vec{v}}(\overline{\rho})=2^{f-d}$.

Next we prove (iii). We need to prove: for $0\leq i\leq f-1$, if $v_{i}=1$,
then $v^{\prime}_{i}=1$. Since $\overline{\psi}\in W_{D}(\overline{\rho})$, we
may suppose $\overline{\psi}=\overline{\psi}_{\vec{w},\vec{d}}$ by Theorem 3.9
where $\vec{w},\vec{d}$ are determined by $\overline{\psi}$. Since $v_{i}=1$,
$d_{i+1}$ can be $-1$ or $1$ by Table 1 and Table 2. We will show that if
$v_{i}^{\prime}=0$, then $w_{i}\neq w^{\prime}_{i}$, and we deduce a
contradiction. We take $d_{i+1}=-1$ and $i\neq f-1$ as an example. The other
cases can be treated similarly. By $u_{i}-u_{i+1}\equiv v_{i}-v_{i+1}\
(\operatorname{mod}2)$, Table 1 and Table 2, we have $u_{i}=0$. If
$v^{\prime}_{i}=0$, we have $u^{\prime}_{i}=0$ by $d_{i+1}=-1$ and Table 1.
Then $w_{i}\equiv u_{i}+v_{i}\equiv 1\ (\operatorname{mod}2)$ and
$w^{\prime}_{i}\equiv u^{\prime}_{i}+v^{\prime}_{i}\equiv 0\
(\operatorname{mod}2)$, which is a contradiction.

If there exists $\overline{\psi}\in W_{D}^{\vec{v}}(\overline{\rho})\cap
W_{D}^{\vec{v}^{\prime}}(\overline{\rho})$, then there exist $\vec{u}$ and
$\vec{v}$ such that $\overline{\psi}=\overline{\psi}_{\vec{u},\vec{v}}$ since
$\overline{\psi}\in W_{D}^{\vec{v}}(\overline{\rho})$. Since
$\overline{\psi}_{\vec{u},\vec{v}}\in
W_{D}^{\vec{v}^{\prime}}(\overline{\rho})$, we have
$\vec{v}^{\prime}\leq\vec{v}$. Similarly we also have
$\vec{v}\leq\vec{v}^{\prime}$, then $\vec{v}=\vec{v}^{\prime}$. We deduce that
$W_{D}^{\vec{v}}(\overline{\rho})\cap
W_{D}^{\vec{v}^{\prime}}(\overline{\rho})=\emptyset$ if
$\vec{v}\neq\vec{v}^{\prime}$. By (ii) and Theorem 3.9, we have
$\\#\bigsqcup\limits_{{\vec{v}}}W_{D}^{\vec{v}}(\overline{\rho})=\sum_{d=0}^{f-1}\tbinom{f}{d}2^{f-d}=3^{f}-1=\\#W_{D}(\overline{\rho})$
which implies (iv).

By (iii) we have $W_{D}^{\vec{v}}(\overline{\rho})\subseteq
W_{D,\vec{v}}(\overline{\rho})\backslash\bigcup\limits_{\vec{v}^{\prime}<\vec{v}}W_{D,\vec{v}^{\prime}}(\overline{\rho})$.
For $\overline{\psi}\in
W_{D,\vec{v}}(\overline{\rho})\backslash\bigcup\limits_{\vec{v}^{\prime}<\vec{v}}W_{D,\vec{v}^{\prime}}(\overline{\rho})$,
by (iv) there exists $\vec{v}^{\prime\prime}$ such that $\overline{\psi}\in
W_{D}^{\vec{v}^{\prime\prime}}(\overline{\rho})$. By (iii), we have
$\vec{v}^{\prime\prime}\leq\vec{v}$. Since
$\overline{\psi}\notin\bigcup_{\vec{v}^{\prime}<\vec{v}}W_{D,\vec{v}^{\prime}}(\overline{\rho})$,
we have $\vec{v}^{\prime\prime}=\vec{v}$, i.e. $\overline{\psi}\in
W_{D}^{\vec{v}}(\overline{\rho})$. ∎

The following theorem determines $W_{D}(\overline{\rho})$ for non-semisimple
generic $\overline{\rho}$.

######  Theorem 3.12.

Let $\overline{\rho}$ be generic reducible nonsplit as in Definition 2.7 and
let
$\vec{v}(\overline{\rho})\in\\{0,\ldots,f-1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$
be the $f$-tuple associated to $\overline{\rho}.$ Then

$W_{D}(\overline{\rho})=\bigsqcup_{{\vec{v}}\leq\vec{v}(\overline{\rho})}W_{D}^{{\vec{v}}}(\overline{\rho}^{{\rm
ss}}).$

Note that each $W_{D}^{{\vec{v}}}(\overline{\rho}^{{\rm ss}})$ is explicit.

###### Proof.

By Lemma 2.8, we have
$W_{D}(\overline{\rho})=\bigcup\limits_{\vec{v}\leq\vec{v}(\overline{\rho})}W_{D,\vec{v}}(\overline{\rho}^{{\rm
ss}})$. By (iv) and (v) of Proposition 3.11, we have
$\bigcup\limits_{\vec{v}\leq\vec{v}(\overline{\rho})}W_{D,\vec{v}}(\overline{\rho}^{{\rm
ss}})=\bigsqcup\limits_{{\vec{v}}\leq\vec{v}(\overline{\rho})}W_{D}^{{\vec{v}}}(\overline{\rho}^{{\rm
ss}})$. ∎

######  Remark 3.13.

If $\overline{\rho}$ is reducible nonsplit, i.e.
$\ell(\lambda_{\vec{v}(\overline{\rho})})=d<f$, by Proposition 3.11(ii),
$\\#W_{D}(\overline{\rho})=\sum_{k=0}^{d}\tbinom{d}{k}2^{f-k}=3^{d}2^{f-d}.$
Note that this number also appears in [BP12, Prop. 14.7].

######  Theorem 3.14.

Let $\overline{\rho}$ be generic irreducible as in Lemma 2.10, then
$\overline{\psi}=\overline{\iota}^{(q+1)b+1+c}\in W_{D}(\overline{\rho})$, if
and only if

$(q+1)b+1+c\equiv\sum_{i=0}^{f-1}q^{w_{i}}p^{i}r_{i}+(1-q)\sum_{i=0}^{f-1}d_{i}p^{i}~{}(\operatorname{mod}q^{2}-1)$

where $w_{i}\in\\{0,1\\}$ and $d_{i}\in\\{-1,0,1\\}$ satisfy the following
relations.

  * •

For $i>0$, $w_{i}=\left\\{\begin{array}[]{cl}1&\text{ if }d_{i}=-1\\\ 0&\text{
if }d_{i}=1\end{array}\right.$, and if $d_{i}=0$, then $(w_{i-1},w_{i})=(0,0)$
or $(1,1)$.

  * •

$(w_{f-1},w_{0})=\left\\{\begin{array}[]{cl}(1,1)&\text{ if }d_{0}=-1\\\
(0,0)&\text{ if }d_{0}=1\end{array}\right.$.

Note that, from the above relations, we have:

  * •

For every $0<i\leq f-1$, if $d_{0}=1$, $d_{i}=-1$, and $d_{j}=0$ for every
$i<j\leq f-1$, or $d_{0}=-1$, $d_{i}=1$, and $d_{j}=0$ for every $i<j\leq
f-1$, $\vec{w}$ does not exist;

  * •

For every $0\leq i\leq f-1$, if $d_{0}=1$, $d_{i}=1$, and $d_{j}=0$ for every
$i<j\leq f-1$, or $d_{0}=-1$, $d_{i}=-1$, and $d_{j}=0$ for every $i<j\leq
f-1$, there is exactly one $\vec{w}$;

  * •

If $d_{0}=0$, there are exactly two choices for $\vec{w}$.

We write $\overline{\psi}_{\vec{w},\vec{d}}$ for such $\overline{\psi}$, and
we have
$\overline{\psi}_{\vec{w},\vec{d}}=\overline{\psi}_{\vec{w}^{\prime},\vec{d}^{\prime}}$
if and only if $(\vec{w},\vec{d})=(\vec{w}^{\prime},\vec{d}^{\prime}).$

###### Proof.

The two relations of $\vec{w}$ and $\vec{d}$ follow from Proposition 3.7,
Table 1 and Table 3 in Proposition 3.8.

For every $0<i\leq f-1$, if $d_{0}=1$, $d_{i}=-1$, and $d_{j}=0$ for every
$i<j\leq f-1$, we deduce from the second relation that $w_{f-1}=w_{0}=0$, and
from the first relation that $w_{i}=1$ and $w_{i}=w_{i+1}=\cdots=w_{f-1}$,
which is a contradiction; if $d_{0}=-1$, $d_{i}=1$, and $d_{j}=0$ for every
$i<j\leq f-1$, then $\vec{w}$ does not exist for similar reason.

For every $0\leq i\leq f-1$, if $d_{0}=1$, $d_{i}=1$, and $d_{j}=0$ for every
$i<j\leq f-1$, we will show the uniqueness of $\vec{w}$. By the relations
above, we have $w_{0}=0$, $w_{j}=0$ for $i\leq j\leq f-1$. For $1<j<i$, we
have if $d_{j}=-1$ or $1$ , $w_{j}$ is determined by $d_{j}$; if $d_{j}=0$,
$w_{j}$ is determined by the nearest $d_{k}\neq 0$ $(k<j)$. If $d_{0}=-1$,
$d_{i}=-1$, and $d_{j}=0$ for every $i<j\leq f-1$, $\vec{w}$ is uniquely
determined by $\vec{d}$ for similar reason.

Next for each choice of $\vec{d}$ in the second case, to show
$\overline{\psi}_{\vec{w},\vec{d}}\in W_{D}(\overline{\rho})$, by Lemma 3.1,
it suffices to find $\vec{u},\vec{v}\in\\{0,1\\}^{{\mathbb{Z}}/f{\mathbb{Z}}}$
such that
$\overline{\psi}_{\vec{w},\vec{d}}=\overline{\psi}_{\vec{u},\vec{v}}$. Let
$v_{f-1}=0$, $v_{k-1}\equiv d_{k}\ (\operatorname{mod}2)$ for $0\leq k\leq
f-1$, and let $u_{0}=v_{0}$, $u_{k}=0$ for $i\leq k\leq f-1$; for $0<k<i$,
$u_{k}$ is uniquely determined in the sense that, from Table 1, if $d_{k}=-1$
or $1$, $u_{k}$ is determined by $(v_{k-1},v_{k},d_{k})$, and if $d_{k}=0$,
$u_{k}$ is determined by $(v_{k-1},v_{k},u_{k-1})$. For every $0\leq i\leq
f-1$, if $d_{0}=-1$, $d_{i}=-1$, and $d_{j}=0$ for every $i<j\leq f-1$, let
$v_{f-1}=0$, $v_{k-1}\equiv d_{k}\ (\operatorname{mod}2)$ for $0\leq k\leq
f-1$, and let $u_{0}\equiv 1+v_{0}\ (\operatorname{mod}2)$, $u_{k}=1$ for
$i\leq k\leq f-1$; for $0<k<i$, $u_{k}$ is uniquely determined for similar
reason.

If $d_{0}=0$, then $w_{0}$ can be $0$ or $1$, and other $w_{i}$’s are uniquely
determined by $w_{0}$ and the two relations above. Let $v_{f-1}=1$, and
$v_{k-1}\equiv d_{k}\ (\operatorname{mod}2)$ otherwise, then let $u_{0}\equiv
w_{0}-v_{0}\ (\operatorname{mod}2)$, and other $u_{i}$’s are uniquely
determined.

If
$\overline{\psi}_{\vec{w},\vec{d}}=\overline{\psi}_{\vec{w}^{\prime},\vec{d}^{\prime}}$,
then

$\begin{split}0&\equiv(\sum_{i=0}^{f-1}q^{w_{i}}p^{i}r_{i}+(1-q)\sum_{i=0}^{f-1}d_{i}p^{i})-(\sum_{i=0}^{f-1}q^{w^{\prime}_{i}}p^{i}r_{i}+(1-q)\sum_{i=0}^{f-1}d^{\prime}_{i}p^{i})\
(\operatorname{mod}q^{2}-1)\\\
&\equiv(q-1)\sum_{i=0}^{f-1}p^{i}((w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i}))\
(\operatorname{mod}q^{2}-1).\end{split}$

Note that $1\leq r_{0}\leq p-2$ and $0\leq r_{i}\leq p-3$ for $i>0$ since
$\overline{\rho}$ is generic. Since
$\sum_{i=0}^{f-1}p^{i}\left|(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})\right|\leq\sum_{i=0}^{f-1}p^{i}(r_{i}+2)<q+1$,
we have
$\sum_{i=0}^{f-1}p^{i}((w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i}))=0$.
Since $\left|(w_{0}-w^{\prime}_{0})r_{0}-(d_{0}-d^{\prime}_{0})\right|\leq p$
and $\left|(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})\right|\leq p-1$
for $i>0$, we have $(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})=0$ for
all $i$, or for some $0<i\leq f-1$,
$(w_{0}-w^{\prime}_{0})r_{0}-(d_{0}-d^{\prime}_{0})=p$,
$(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})=-1$,
$(w_{j}-w^{\prime}_{j})r_{i}-(d_{j}-d^{\prime}_{j})=p-1$ for $1<j<i$, and
$(w_{j}-w^{\prime}_{j})r_{i}-(d_{j}-d^{\prime}_{j})=0$ for $i<j\leq f-1$. For
the first case, since $w_{i}\leq w^{\prime}_{i}$ if $d_{i}>d^{\prime}_{i}$ by
relations above, we have $d_{i}=d^{\prime}_{i}$ and
$(w_{i}-w^{\prime}_{i})r_{i}=0$ for all $i$. Since $1\leq r_{0}\leq p-2$, we
have $w_{0}=w^{\prime}_{0}$. Then $\vec{w}$ is uniquely determined by
$(b_{i})_{i}$ once $w_{0}$ is fixed. For the second case, we have $w_{0}=1$,
$w^{\prime}_{0}=0$, $d_{0}=-1$, $d_{0}-1$. Then $w_{f-1}=1$,
$w^{\prime}_{f-1}=0$ by the second relation which implies
$d_{f-1}\in\\{-1,0\\}$, $d^{\prime}_{f-1}\in\\{0,1\\}$ by the first relation.
Since $(w_{f-1}-w^{\prime}_{f-1})r_{f-1}-(d_{f-1}-d^{\prime}_{{f-1}})=-1$ or
$0$, we have this should be $0$ and $d_{f-1}=d^{\prime}_{{f-1}}=0$. Then we
have $w_{f-2}=1$, $w^{\prime}_{f-2}=0$ by the first relation which implies
$d_{f-2}\in\\{-1,0\\}$, $d^{\prime}_{f-2}\in\\{0,1\\}$. We repeat this
procedure and deduce that
$(w_{i}-w^{\prime}_{i})r_{i}-(d_{i}-d^{\prime}_{i})=0$ for all $i\leq f-1$
which is a contradiction.

∎

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Department of Mathematical Sciences, Tsinghua University, Beijing, 100084
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Academy for Multidisciplinary Studies, Capital Normal University, Beijing,
100048
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