2. Preliminaries

We write $\underline{i}$ for an element $(i_0,\ldots,i_{f-1})\in\mathbb{Z}^f$ . For $a\in\mathbb{Z}$ , we write $\underline{a}\overset{\mathrm{def}}{=}(a,\ldots,a)\in\mathbb{Z}^f$ . For $J\subseteq\mathcal{J}$ , we define $\underline{e}^J\in\mathbb{Z}^f$ by $e^J_j\overset{\mathrm{def}}{=}\delta_{j\in J}$ . We say that $\underline{i}\leq\underline{i}'$ if $i_j\leq i'_j$ for all j. We define the left shift $\delta:\mathbb{Z}^f\to\mathbb{Z}^f$ by $\delta(\underline{i})_j\overset{\mathrm{def}}{=} i_{j+1}$ . Abusing notation, we still denote by $\underline{i}$ the integer $\sum_{j=0}^{f-1}i_jp^j\in\mathbb{Z}$ .

A Serre weight is an isomorphism class of an absolutely irreducible representation of $\operatorname{GL}_2(\mathbb{F}_q)$ over $\mathbb{F}$ . For $\lambda=(\underline{\lambda}_1,\underline{\lambda}_2)\in\mathbb{Z}^{2f}$ such that $\underline{0}\leq\underline{\lambda}_1-\underline{\lambda}_2\leq\underline{p}-\underline{1}$ , we define

\begin{equation*} F(\lambda)\overset{\mathrm{def}}{=}\bigotimes_{j=0}^{f-1}\,((\operatorname{Sym}^{\lambda_{1,j}-\lambda_{2,j}}\mathbb{F}_q^2\otimes_{\mathbb{F}_q}\operatorname{det}^{\lambda_{2,j}})\otimes_{\mathbb{F}_q,\sigma_j}\mathbb{F}).\end{equation*}

We also denote it by $(\underline{\lambda}_1-\underline{\lambda}_2)\otimes\operatorname{det}^{\underline{\lambda}_2}$ .

For $\lambda=(\underline{\lambda}_1,\underline{\lambda}_2)\in\mathbb{Z}^{2f}$ , we define the character $\chi_{\lambda}:I\to\mathbb{F}^{\times}$ by $({\kern-1pt}\begin{smallmatrix}a&b\\pc&d\end{smallmatrix}{\kern-1pt})\mapsto(\overline{a})^{\underline{\lambda}_1}(\overline{d})^{\underline{\lambda}_2}$ , where $a,d\in\mathcal{O}_K^{\times}$ and $b,c\in\mathcal{O}_K$ . In particular, if $\underline{0}\leq\underline{\lambda}_1-\underline{\lambda}_2\leq\underline{p}-\underline{1}$ , then $\chi_{\lambda}$ is the I-character acting on $F(\lambda)^{I_1}$ . We still denote by $\chi_{\lambda}$ its restriction to H.

We write $\underline{r}=(r_0,\ldots,r_{f-1})$ with $r_j$ as in (1.2). For $\underline{b}\in\mathbb{Z}^f$ such that $-\underline{r}\leq\underline{b}\leq\underline{p}-\underline{2}-\underline{r}$ , we denote by $\sigma_{\underline{b}}$ the Serre weight $F(\mathfrak{t}_{(\underline{r},\underline{0})}(\underline{b}))$ (see [Reference Breuil, Herzig, Hu, Morra and SchraenBHH⁺23, § 2.4] for $\mathfrak{t}_{(\underline{r},\underline{0})}(\underline{b})\in\mathbb{Z}^{2f}$ ).

For $\overline{\rho}$ as in (1.3), we let $W(\overline{\rho})$ be the set of Serre weights of $\overline{\rho}$ defined in [Reference Buzzard, Diamond and JarvisBDJ10, § 3] and $J_{\overline{\rho}}\subseteq\mathcal{J}$ be the subset as in [Reference BreuilBre14, (17)]. Then by [Reference BreuilBre14, Proposition A.3] and [Reference Breuil, Herzig, Hu, Morra and SchraenBHH⁺23, (14)], the subset $J_{\overline{\rho}}\subseteq\mathcal{J}$ is characterized by

\begin{equation*} W(\overline{\rho})=\bigg\{ \sigma_{\underline{b}}:~\begin{array}{ll}b_j=0&\text{if}~j\notin J_{\overline{\rho}}\\ b_j\in\{ 0,1 \}&\text{if}~j\in J_{\overline{\rho}}\end{array}\!\! \bigg\}.\end{equation*}

In particular, $\overline{\rho}$ is semisimple if and only if $J_{\overline{\rho}}=\mathcal{J}$ .

For each $J\subseteq\mathcal{J}$ , as in [Reference WangWan24, § 2] we define the (distinct) Serre weight $\sigma_J\overset{\mathrm{def}}{=} F(\lambda_J)$ with $\lambda_J\overset{\mathrm{def}}{=}(\underline{s}^J+\underline{t}^J,\underline{t}^J)$ , where

(2.1) \begin{align} s^J_j&\overset{\mathrm{def}}{=} \begin{cases} r_j&\text{if}~j\notin J,~j+1\notin J,\\ r_j+1&\text{if}~j\in J,~j+1\notin J,\\ p-2-r_j&\text{if}~j\notin J,~j+1\in J,\\ p-1-r_j&\text{if}~j\in J,~j+1\in J,~j\notin J_{\overline{\rho}},\\ p-3-r_j&\text{if}~j\in J,~j+1\in J,~j\in J_{\overline{\rho}}; \end{cases} \end{align}

(2.2) \begin{align} t^J_j&\overset{\mathrm{def}}{=} \begin{cases} 0&\text{if}~j\notin J,~j+1\notin J,\\ -1&\text{if}~j\in J,~j+1\notin J,\\ r_j+1&\text{if}~j\notin J,~j+1\in J~\text{or}~j\in J,~j+1\in J,~j\in J_{\overline{\rho}},\\ r_j&\text{if}~j\in J,~j+1\in J,~j\notin J_{\overline{\rho}}. \end{cases}\end{align}

In particular, for $J\subseteq J_{\overline{\rho}}$ we have $\sigma_J=\sigma_{\underline{e}^J}$ (see [Reference WangWan24, § 2]). We also define the Serre weight $\sigma_{J}^s\overset{\mathrm{def}}{=}F(\lambda_{J^s})$ with $\lambda_{J^s}\overset{\mathrm{def}}{=}(\underline{s}^{J^s}+\underline{t}^{J^s},\underline{t}^{J^s})$ , where $s^{J^s}_j\overset{\mathrm{def}}{=} p-1-s^J_j$ and $t^{J^s}_j\overset{\mathrm{def}}{=} r_j-t^J_j$ . We write $\chi_J\overset{\mathrm{def}}{=}\chi_{\lambda_J}$ and $\chi_{J^s}\overset{\mathrm{def}}{=}\chi_{\lambda_{J^s}}$ so that $\chi_J$ (respectively, $\chi_{J^s}$ ) is the I-character acting on $\sigma_J^{I_1}$ (respectively, $(\sigma_J^s)^{I_1}$ ). For each I-character $\chi$ , we denote by $\chi^s$ its conjugation by the matrix $({\kern-1pt}\begin{smallmatrix}0&1\\[1pt]p&0\end{smallmatrix}{\kern-1pt})$ . Then we have $\chi_J^s=\chi_{J^s}$ and $\chi_{J^s}^s=\chi_J$ for all $J\subseteq\mathcal{J}$ .

For $J\subseteq\mathcal{J}$ and $k\in\mathbb{Z}$ , we write $J+k\overset{\mathrm{def}}{=}\{ j+k:j\in J \}\subseteq\mathcal{J}$ . Then we define

(2.3) \begin{equation}\begin{array}{lll} J^{{\rm ss}}\overset{\mathrm{def}}{=} J\cap J_{\overline{\rho}},&J^{{\rm nss}}\overset{\mathrm{def}}{=} J\setminus J_{\overline{\rho}},&J^c\overset{\mathrm{def}}{=}\mathcal{J}\setminus J,\\ \partial J\overset{\mathrm{def}}{=} J\setminus(J-1),&\delta_{{\rm ss}}(J)\overset{\mathrm{def}}{=}(J-1)^{{\rm ss}},&J^{\delta}\overset{\mathrm{def}}{=} J\Delta(J-1)^{{\rm ss}}.\end{array}\end{equation}

Here we recall that $J\Delta J'\overset{\mathrm{def}}{=} (J\setminus J')\sqcup(J'\setminus J)$ .

Conversely, suppose that $\sigma_{J^s}=\sigma_{(J-1)^{{\rm ss}}}$ . Comparing the $I_1$ -invariants, we get $\chi_{J^s}=\chi_{(J-1)^{{\rm ss}}}$ . In particular, we have $\sum_{j=0}^{f-1}p^jt^{J^s}_j\equiv\sum_{j=0}^{f-1}p^jt^{(J-1)^{{\rm ss}}}_j\,\operatorname{mod}\,(q-1)$ , which implies $\underline{t}^{J^s}=\underline{t}^{(J-1)^{{\rm ss}}}$ since $-1\leq t^{J}_j\leq p-3$ for all $J\subseteq\mathcal{J}$ and $j\in\mathcal{J}$ by (1.2) and (2.2). By (2.2) and using $r_j\geq1$ for all j, we have $\{ j:t^{J}_j\in\{ r_j,r_j+1 \} \}+1=J$ for all $J\subseteq\mathcal{J}$ , hence we conclude that $J^c=(J-1)^{{\rm ss}}$ .

If, moreover, $J_{\overline{\rho}}\neq\mathcal{J}$ , then $J^*\subseteq\mathcal{J}$ is uniquely characterized by the property that $j\in J^*$ if and only if $j\notin J_{\overline{\rho}}$ or ( $j\in J_{\overline{\rho}}$ and $j+1\notin J$ ).

3. The relation between $I_1$ -invariants

Let $\pi$ be as in (1.1). We study the relation between $I_1$ -invariants of $\pi$ , which generalizes some results of [Reference WangWan24, § 5]. The main results are Propositions 3.1 and 3.4.

From now on, we identify $\pi^{K_1}$ with $D_0(\overline{\rho})$ (see the introduction), which is the (finite-dimensional) representation of $\operatorname{GL}_2(\mathcal{O}_K)$ defined in [Reference Breuil and PaškūnasBP12, § 13]. For each $J\subseteq\mathcal{J}$ , the character $\chi_J$ appears in $\pi^{I_1}=D_0(\overline{\rho})^{I_1}$ with multiplicity one by [Reference WangWan24, Lemma 4.1(ii)]. We fix a choice of $0\neq v_J\in\pi^{I_1}$ with I-character $\chi_J$ . For each $j\in\mathcal{J}$ we define

\begin{equation*} Y_j\overset{\mathrm{def}}{=}\sum\limits_{a\in\mathbb{F}_q^{\times}}a^{-p^j}\bigg(\begin{matrix}1&[a]\\0&1\end{matrix}\bigg)\in\mathbb{F}[\![N_0]\!].\end{equation*}

For $\underline{i}=(i_0,\ldots,i_{f-1})\in\mathbb{Z}^f$ , we write $\underline{Y}^{\underline{i}}$ for $\prod_{j=0}^{f-1}Y_j^{i_j}$ .

For $J,J'\subseteq\mathcal{J}$ such that $(J-1)^{{\rm ss}}=(J')^{{\rm ss}}$ , we define $\mu_{J,J'}\in\mathbb{F}^{\times}$ as in [Reference WangWan24, (47)]. In particular, in the case $(J')^{{\rm nss}}\neq\mathcal{J}$ such that

(3.1) \begin{equation} (J')^{{\rm nss}}\subseteq(J-1)^{{\rm nss}}\Delta(J'-1)^{{\rm nss}},\end{equation}

the element $\mu_{J,J'}\in\mathbb{F}^{\times}$ is defined by the formula (see [Reference WangWan24, Proposition 5.10])

(3.2) \begin{equation} \bigg[\prod_{j+1\in J\Delta J'}Y_j^{s^{J'}_j}\prod_{j+1\notin J\Delta J'}Y_j^{p-1}\bigg]\bigg(\begin{matrix}p&0\\0&1\end{matrix}\bigg)(\underline{Y}^{-\underline{e}^{(J\cap J')^{{\rm nss}}}}v_J)=\mu_{J,J'}v_{J'},\end{equation}

where $\underline{Y}^{-\underline{e}^{(J\cap J')^{{\rm nss}}}}v_J\in D_0(\overline{\rho})$ is a suitable shift of $v_J$ defined in [Reference WangWan24, Proposition 4.2]. As in the discussion after [Reference WangWan24, (47)], for $J_1,J_2,J_3,J_4\subseteq\mathcal{J}$ such that $(J_1-1)^{{\rm ss}}=(J_2-1)^{{\rm ss}}=J_3^{{\rm ss}}=J_4^{{\rm ss}}$ we have $\mu_{J_1,J_3}/\mu_{J_1,J_4}=\mu_{J_2,J_3}/\mu_{J_2,J_4}$ . In particular, for J,J’ such that $J^{{\rm ss}}=(J')^{{\rm ss}}$ , the quantity $\mu_{J'',J}/\mu_{J'',J'}$ does not depend on J” such that $(J''-1)^{{\rm ss}}=J^{{\rm ss}}$ , and we denote it by $\mu_{*,J}/\mu_{*,J'}$ .

For each $J\subseteq\mathcal{J}$ , the character $\chi_J^s$ also appears in $\pi^{I_1}=D_0(\overline{\rho})^{I_1}$ with multiplicity one by [Reference WangWan24, Lemma 4.1(ii)]. We fix a choice of $0\neq v_{J^s}\in\pi^{I_1}$ with I-character $\chi_J^s$ . Since p acts trivially on $\pi$ , by rescaling the vectors $v_J$ and $v_{J^s}$ we assume from now on that $v_{J^s}=({\kern-1pt}\begin{smallmatrix}0&1\\p&0\end{smallmatrix}{\kern-1pt})v_J$ for all $J\subseteq\mathcal{J}$ . We prove some analogous results for the vectors $v_{J^s}$ .

Proof. (i) Since $v_J\in\pi^{I_1}$ has I-character $\chi_J$ , by Frobenius reciprocity there is a $\operatorname{GL}_2(\mathcal{O}_K)$ -equivariant map

(3.4) \begin{equation} \begin{aligned} \operatorname{Ind}_{I}^{\operatorname{GL}_2(\mathcal{O}_K)}(\chi^s_{J^s})=\operatorname{Ind}_{I}^{\operatorname{GL}_2(\mathcal{O}_K)}(\chi_{J})&\stackrel{\alpha}{\twoheadrightarrow}\langle\operatorname{GL}_2(\mathcal{O}_K)v_J\rangle=\bigg\langle\!\!\operatorname{GL}_2(\mathcal{O}_K)\bigg(\begin{matrix}p&0\\0&1\end{matrix}\bigg)v_{J^s}\!\!\bigg\rangle\hookrightarrow\pi,\\ \phi&\mapsto v_J=\bigg(\,\begin{matrix}0&1\\p&0\end{matrix}\bigg)v_{J^s}, \end{aligned} \end{equation}

where $\phi\in\operatorname{Ind}_{I}^{\operatorname{GL}_2(\mathcal{O}_K)}(\chi_{J})$ is supported on I such that $\phi(\operatorname{id})=1$ . By [Reference WangWan24, Proposition 4.2], the $\operatorname{GL}_2(\mathcal{O}_K)$ -subrepresentation $V\overset{\mathrm{def}}{=}\big\langle\!\operatorname{GL}_2(\mathcal{O}_K)v_J\big\rangle\subseteq D_0(\overline{\rho})$ lies in $D_{0,J^{{\rm ss}}}(\overline{\rho})$ (as defined in [Reference WangWan24, Lemma 4.1(i)]), hence has socle $\sigma_{J^{{\rm ss}}}$ . As a quotient of $\operatorname{Ind}_{I}^{\operatorname{GL}_2(\mathcal{O}_K)}(\chi_{J})$ , V has cosocle $\sigma_J$ , which equals $\sigma_{\underline{a}^J}$ with $\underline{a}^J$ as in [Reference WangWan24, (5)]. In particular, we have $V=I(\sigma_{J^{{\rm ss}}},\sigma_{\underline{a}^J})$ in the notation of [Reference WangWan24, Proposition 5.1(iii)]. From the explicit formula of $\underline{a}^J$ , one checks that $a^J_j=\delta_{j\in J^{{\rm ss}}}$ if and only if $j\notin J^{{\rm nss}}$ . Then we deduce from [Reference WangWan24, Lemma 3.2(i)] (applied to $\lambda=\lambda_{J^s}$ ) that $\sigma_{J^{{\rm ss}}}$ corresponds to the subset $(J^{{\rm nss}})^c-1$ (in the sense of [Reference WangWan24, Lemma 3.2(i)]), hence V is isomorphic to the quotient $Q(\chi^s_{J^s},(J^{{\rm nss}})^c-1)$ of $\operatorname{Ind}_{I}^{\operatorname{GL}_2(\mathcal{O}_K)}(\chi^s_{J^s})$ in the notation of [Reference WangWan24, Lemma 3.2(iii)]. By Lemma 2.1(ii), for $j+1\notin J^{{\rm nss}}$ we have

\begin{equation*} p-2-s^{J^s}_j+\delta_{j\in(J^{{\rm nss}})^c}=s^J_j-\delta_{j\in J^{{\rm nss}}}=s^{J^{{\rm ss}}}_j. \end{equation*}

Then by [Reference WangWan24, Lemma 3.2(iii)] (applied to $\lambda=\lambda_{J^s}$ ), the left-hand side of (3.3) is non-zero in $\sigma_{J^{{\rm ss}}}$ and is the unique (up to scalar) H-eigencharacter in $\sigma_{J^{{\rm ss}}}$ killed by all $Y_j$ . It follows that the left-hand side of (3.3) is a non-zero $I_1$ -invariant of $\sigma_{J^{{\rm ss}}}$ , hence is a scalar multiple of $v_{J^{{\rm ss}}}$ .

(ii) By the proof of (i) and using $J^{{\rm nss}}\neq\mathcal{J}$ , the $\operatorname{GL}_2(\mathcal{O}_K)$ -equivariant surjection $\alpha$ in (3.4) is not an isomorphism, hence it maps $\sigma_{J}^s=\operatorname{soc}(\operatorname{Ind}_{I}^{\operatorname{GL}_2(\mathcal{O}_K)}(\chi_{J}))$ to zero. By [Reference WangWan24, Lemma 3.2(iii)(a)] (applied to $\lambda=\lambda_{J^s}$ ) we have

\begin{equation*} \underline{Y}^{\underline{p}-\underline{1}-\underline{s}^{J^s}}\bigg(\begin{matrix}p&0\\0&1\end{matrix}\bigg)v_{J^s}+(-1)^{f-1+\underline{s}^{J^s}+\underline{t}^{J^s}} \bigg(\prod_{j=0}^{f-1}(s^{J^s}_j)!{\kern-1pt}\bigg)v_{J^s}=0. \end{equation*}

Since $\underline{s}^{J^s}=\underline{p}-\underline{1}-\underline{s}^J$ , this proves (ii) with

(3.5) \begin{equation} \mu_{J^s,J}=(-1)^{\underline{1}+\underline{s}^{J^s}+\underline{t}^{J^s}} \bigg(\prod_{j=0}^{f-1}(p-1-s^{J}_j)!{\kern-1pt}\bigg)^{\!{-}1}=(-1)^{\underline{t}^{J^s}}\bigg(\prod_{j=0}^{f-1}(s^{J}_j)!{\kern-1pt}\bigg), \end{equation}

where the second equality uses

(3.6) \begin{equation} ((p-1-r)!)^{-1}\equiv(-1)^{r+1}r!~\operatorname{mod} p~\forall\,0\leq r\leq p-1, \end{equation}

and the uniqueness is clear.

Corollary 3.2. Suppose that $J_{\overline{\rho}}\neq\mathcal{J}$ and $J_{\overline{\rho}}\neq\emptyset$ . Let $J^*$ be as in Lemma 2.2. Then we have

\begin{equation*} (-1)^{|J^*\cap(J^*-1)^{{\rm nss}}|}\mu_{(J^*-1)^{{\rm ss}},J^*}\mu_{J^*,(J^*-1)^{{\rm ss}}}=1. \end{equation*}

Proof. Note that $J_{\overline{\rho}}\neq\emptyset$ implies $(J^*)^{{\rm nss}}\neq\mathcal{J}$ . Since $\sigma_{(J^*-1)^{{\rm ss}}}=\sigma_{J^*}^s$ by Lemma 2.2, we have $\underline{s}^{(J-1)^{{\rm ss}}}=\underline{s}^{J^s}$ and $\underline{t}^{((J-1)^{{\rm ss}})^s}=\underline{t}^J$ . Then by (3.5) we have

\begin{equation*} \begin{aligned} &\mu_{(J^*-1)^{{\rm ss}},J^*}=\mu_{(J^*)^s,J^*}=(-1)^{\underline{t}^{J^s}}\bigg(\prod_{j=0}^{f-1}(s^{J}_j)!{\kern-1pt}\bigg),\\ &\mu_{J^*,(J^*-1)^{{\rm ss}}}=\mu_{((J^*-1)^{{\rm ss}})^s,(J^*-1)^{{\rm ss}}}=(-1)^{\underline{t}^{J}}\bigg(\prod_{j=0}^{f-1}(s^{J^s}_j)!{\kern-1pt}\bigg). \end{aligned} \end{equation*}

Hence, we have

\begin{equation*} \mu_{(J^*-1)^{{\rm ss}},J^*}\mu_{J^*,(J^*-1)^{{\rm ss}}}=(-1)^{\underline{t}^{J^s}+\underline{t}^J}\bigg(\prod_{j=0}^{f-1}(s^{J}_j)!(s^{J^s}_j)!{\kern-1pt}\bigg) =(-1)^{\underline{t}^{J^s}+\underline{t}^J+\underline{s}^J+\underline{1}}=(-1)^f, \end{equation*}

where the second equality uses (3.6) and the last equality follows from Lemma 2.3. Moreover, from the structure of $J^*$ one can check that $|J^*\cap(J^*-1)^{{\rm nss}}|\equiv f~\operatorname{mod}2$ , which completes the proof.

Proposition 3.4. For $J\subseteq\mathcal{J}$ such that $J^{{\rm nss}}\neq\mathcal{J}$ , we have

\begin{equation*} \frac{\mu_{J^s,J}}{\mu_{J^s,J^{{\rm ss}}}}=\frac{\mu_{*,J}}{\mu_{*,J^{{\rm ss}}}}. \end{equation*}

Proof. Let $J_0\overset{\mathrm{def}}{=}(J^{{\rm ss}}\sqcup(J^c-1)^{{\rm nss}})+1$ . Then both the pairs $(J_0,J)$ and $(J_0,J^{{\rm ss}})$ satisfy (3.1). We consider the elements $B_1\overset{\mathrm{def}}{=}({\kern-1pt}\begin{smallmatrix}p&0\\0&1\end{smallmatrix}{\kern-1pt})v_{J^s}\in\pi$ and

\begin{equation*} B_2\overset{\mathrm{def}}{=}\bigg[\prod_{j+1\notin J_0\Delta J}Y_j^{2\delta_{j\in J_0\cap J^{{\rm nss}}}+p-1-s^{J_0}_j+\delta_{j\in J_0\Delta J}}\bigg]\bigg(\begin{matrix}p&0\\0&1\end{matrix}\bigg)(\underline{Y}^{-\underline{i}}v_{J_0})\in\pi. \end{equation*}

By Proposition 3.1(ii), we have $\underline{Y}^{\underline{s}^J}B_1=\mu_{J^s,J}v_J$ . By Lemma 3.3(ii) and (3.2) applied to $(J_0,J)$ , we also have

\begin{equation*} \underline{Y}^{\underline{s}^J}B_2=\bigg[\prod_{j+1\in J_0\Delta J}Y_j^{s^{J'}_j}\prod_{j+1\notin J_0\Delta J}Y_j^{p-1}\bigg]\bigg(\begin{matrix}p&0\\0&1\end{matrix}\bigg)(\underline{Y}^{-\underline{e}^{J_0\cap J^{{\rm nss}}}}v_J)=\mu_{J_0,J}v_J. \end{equation*}

In particular, we deduce from [Reference WangWan24, Lemma 3.1(ii)] that $B_1$ and $B_2$ are H-eigenvectors with the same H-eigencharacter.

Moreover, by the proof and the notation of Proposition 3.1(i), we have $B_1\in I(\sigma_{J^{{\rm ss}}},\sigma_{\underline{a}^J})\cong Q(\chi^s_{J^s},(J^{{\rm nss}})^c-1)$ . By [Reference WangWan24, Proposition 5.7] applied to $(J,J',\underline{i})=(J_0,(J_0\Delta J)-1,\underline{e}^{J_0\cap J^{{\rm nss}}})$ and using Lemma 3.3(i), we have $Y_{j'}B_2\in I(\sigma_{J^{{\rm ss}}},\sigma_{\underline{a}^J})$ for all $j'\in\mathcal{J}$ . Since $I(\sigma_{J^{{\rm ss}}},\sigma_{\underline{a}^J})$ is multiplicity-free as an H-representation by [Reference WangWan24, Proposition Lemma 3.2(ii),(iii)] and using $J^{{\rm nss}}\neq\mathcal{J}$ , combining with the first paragraph we deduce that $Y_{j'}B_1=(\mu_{J^s,J}/\mu_{J_0,J})Y_{j'}B_2$ for all $j'\in\mathcal{J}$ .

Then we have

\begin{equation*} \begin{aligned} \mu_{J^s,J^{{\rm ss}}}v_{J^{{\rm ss}}}&=\bigg[\prod_{j+1\notin J^{{\rm nss}}}Y_j^{s^{J^{{\rm ss}}}_j}\prod_{j+1\in J^{{\rm nss}}}Y_j^{p-1}\bigg]B_1\\[4pt] &=\frac{\mu_{J^s,J}}{\mu_{J_0,J}}\bigg[\prod_{j+1\notin J^{{\rm nss}}}Y_j^{s^{J^{{\rm ss}}}_j}\prod_{j+1\in J^{{\rm nss}}}Y_j^{p-1}\bigg]B_2\\[4pt] &=\frac{\mu_{J^s,J}}{\mu_{J_0,J}}\bigg[\prod_{j+1\notin J_0\Delta J^{{\rm ss}}}Y_j^{s^{J^{{\rm ss}}}_j}\prod_{j+1\in J_0\Delta J^{{\rm ss}}}Y_j^{p-1}\bigg]\underline{Y}^{p\delta(\underline{e}^{J_0\cap J^{{\rm nss}}})}\bigg(\begin{matrix}p&0\\0&1\end{matrix}\bigg)(\underline{Y}^{-\underline{e}^{J_0\cap J^{{\rm nss}}}}v_{J_0})\\[4pt] &=\frac{\mu_{J^s,J}}{\mu_{J_0,J}}\bigg[\prod_{j+1\notin J_0\Delta J^{{\rm ss}}}Y_j^{s^{J^{{\rm ss}}}_j}\prod_{j+1\in J_0\Delta J^{{\rm ss}}}Y_j^{p-1}\bigg]\bigg(\begin{matrix}p&0\\0&1\end{matrix}\bigg)v_{J_0}\\[4pt] &=\frac{\mu_{J^s,J}}{\mu_{J_0,J}}\mu_{J_0,J^{{\rm ss}}}v_{J^{{\rm ss}}}, \end{aligned} \end{equation*}

where the first equality follows from Proposition 3.1(i), the third equality follows from Lemma 3.3(iii), the fourth equality follows from [Reference WangWan24, Lemma 3.1(i)], and the last equality follows from (3.2) applied to $(J_0,J^{{\rm ss}})$ . Therefore, we have $\mu_{J^s,J^{{\rm ss}}}=(\mu_{J^s,J}/\mu_{J_0,J})\mu_{J_0,J^{{\rm ss}}}$ , which completes the proof.

4. Kisin modules

Let $\overline{\rho}$ be as in (1.3). In particular, $\overline{\rho}$ is generic in the sense of [Reference Dotto and LeDL21, § 3.2.2]. We study the Kisin modules following [Reference Dotto and LeDL21] that can be used to describe the tamely potentially Barsotti–Tate deformation rings of $\overline{\rho}$ . When $\overline{\rho}$ is non-semisimple, we define and compute the elements in the Kisin modules that form one part of the computation of the constants in the diagram $(\pi^{I_1}\hookrightarrow\pi^{K_1})$ . The main result is Proposition 4.3.

Up to enlarging $\mathbb{F}$ , we fix an fth root $\beta\overset{\mathrm{def}}{=}\sqrt[f]{\xi}\in\mathbb{F}^{\times}$ of $\xi$ (see (1.3)). We let $T_K$ be the Lubin–Tate variable as in [Reference WangWan26, § 2]. By [Reference WangWan26, (44)], the Lubin–Tate $(\varphi,\mathcal{O}_K^{\times})$ -module $D_K(\overline{\rho})$ associated to $\overline{\rho}$ has the following form ( $a\in\mathcal{O}_K^{\times}$ ):

\begin{equation*}\left\{\!\!\!\begin{array}{r@l@l} D_K(\overline{\rho})&=&\displaystyle \prod_{j=0}^{f-1}D_{K,\sigma_j}(\overline{\rho})=\prod_{j=0}^{f-1}(\mathbb{F}(\!(T_K)\!)e_0^{(j)}\oplus\mathbb{F}(\!(T_K)\!)e_1^{(j)})\\[16pt] \varphi(e_0^{(j+1)}\ e_1^{(j+1)})&=&(e_0^{(j)}\ e_1^{(j)})\operatorname{Mat}(\varphi^{(j)})\\[6pt] a(e_0^{(j)}\ e_1^{(j)})&=&(e_0^{(j)}\ e_1^{(j)})\operatorname{Mat}(a^{(j)}),\end{array}\right.\end{equation*}

where

(4.1) \begin{equation} \operatorname{Mat}(\varphi^{(j)})=\bigg(\begin{matrix}\beta\,T_K^{-(q-1)(r_j+1)}&\beta^{-1} d_j\\[2pt]0&\beta^{-1}\end{matrix}\bigg)\end{equation}

for some $d_j\in\mathbb{F}$ , and $\operatorname{Mat}(a^{(j)})\in I_2+\operatorname{M}_2(T_K^{q-1}\mathbb{F}[\![T_K^{q-1}]\!])$ which uniquely determines $\operatorname{Mat}(a^{(j)})$ .

By [Reference WangWan26, Lemma 5.1], the Fontaine–Laffaille module $\operatorname{FL}(\overline{\rho})$ associated to $\overline{\rho}$ (see [Reference Fontaine and LaffailleFL82]) has the following form:

(4.2) \begin{equation}\left\{\!\!\!\begin{array}{r@l@l} \operatorname{FL}(\overline{\rho})&=&\displaystyle \prod_{j=0}^{f-1}\operatorname{FL}_{\sigma_j}(\overline{\rho})=\prod_{j=0}^{f-1}(\mathbb{F} e_0^{(j)}\oplus\mathbb{F} e_1^{(j)})\\[5pt] \operatorname{Fil}^{k}\operatorname{FL}_{\sigma_j}(\overline{\rho})&=&\mathbb{F} e_0^{(j)}~\text{exactly for}~1\leq k\leq r_j+1\\[5pt] \varphi_{r_{j+1}+1}(e_0^{(j+1)})&=&\beta^{-1}(e_0^{(j)}-d_{j+1}e_1^{(j)})\\[5pt] \varphi(e_1^{(j+1)})&=&\beta\,e_1^{(j)},\end{array}\right.\end{equation}

where $d_j\in\mathbb{F}$ is as in (4.1). In particular, by [Reference BreuilBre14, (18)] with $e^j=e_1^{(f-j)}$ , $f^j=e_0^{(f-j)}$ , $\alpha_j=\beta$ , $\beta_j=\beta^{-1}$ and $\mu_j=d_{f+1-j}$ for all $j\in\mathcal{J}$ in [Reference BreuilBre14, (16)], we deduce that $d_j=0$ if and only if $j\in J_{\overline{\rho}}$ .

We fix a compatible system $(p_n)_n$ of p-power roots of $(-p)$ in $\overline{\mathbb{Q}}_p$ and define $K_{\infty}\overset{\mathrm{def}}{=}\bigcup_{n\geq0}K(p_n)$ . Let $\overline{\mathcal{M}}$ be the étale $\varphi$ -module over $k(\!(v)\!)\otimes_{\mathbb{F}_p}\mathbb{F}$ in the sense of [Reference Dotto and LeDL21, § 3.2.1] such that $\mathbb{V}^*(\overline{\mathcal{M}})\cong\overline{\rho}|_{G_{K_{\infty}}}$ , where $\mathbb{V}^*$ is Fontaine’s anti-equivalence of categories (see [Reference FontaineFon90]). Then as in the proof of [Reference Dotto and LeDL21, Proposition 3.3] and using (4.2), we can take

\begin{equation*}\left\{\!\!\!\begin{array}{r@l@l} \overline{\mathcal{M}}&=&\displaystyle\prod_{j=0}^{f-1}\overline{\mathcal{M}}^{(j)}=\prod_{j=0}^{f-1}(\mathbb{F}(\!(v)\!)e_0^{(j)}\oplus\mathbb{F}(\!(v)\!)e_1^{(j)})\\[15pt] \varphi(e_0^{(j)}\ e_1^{(j)})&=&(e_0^{(j+1)}\ e_1^{(j+1)})\operatorname{Mat}(\varphi_{\overline{\mathcal{M}}}^{(j)})\end{array}\right.\end{equation*}

with

(4.3) \begin{equation} \operatorname{Mat}(\varphi_{\overline{\mathcal{M}}}^{(f-1-j)})=\begin{pmatrix}\beta^{-1}&0\\0&\beta\end{pmatrix}\begin{pmatrix}v^{r_j+1}&0\\-\beta^{-2}d_jv^{r_j+1}&1\end{pmatrix},\end{equation}

where $d_j\in\mathbb{F}$ is as in (4.1).

We write $\underline{W}\overset{\mathrm{def}}{=}(S_2)^f$ and $X^*(\underline{T})\overset{\mathrm{def}}{=}(\mathbb{Z}^2)^f\cong\mathbb{Z}^{2f}$ . For $w\in\underline{W}$ and $\lambda\in X^*(\underline{T})$ , we write the corresponding jth components $w_j\in S_2$ and $\lambda_j\in\mathbb{Z}^2$ and define $w\lambda\in X^*(\underline{T})$ with $(w\lambda)_j=w_j(\lambda_j)$ . We denote by $\tau(w,\lambda):I_K\to\operatorname{GL}_2(\mathcal{O})\subseteq\operatorname{GL}_2(E)$ the associated tame inertial type defined in [Reference Dotto and LeDL21, § 2.3.2], which is a two-dimensional representation of $I_K$ over E that factors through the tame inertial quotient and extends to $G_K$ . If, moreover, $(w,\mu)$ is a good pair (see [Reference Dotto and LeDL21, § 2.3.2]), which will always be the case in this article, we write $R_w(\lambda)$ for the associated tame type of K defined in [Reference Dotto and LeDL21, § 2.3.1], which is a smooth irreducible representation of $\operatorname{GL}_2(\mathbb{F}_q)$ over E.

For $\chi:I\to\mathbb{F}^{\times}$ a character, we write $\theta^{\circ}(\chi)\overset{\mathrm{def}}{=}\operatorname{Ind}_{I}^{\operatorname{GL}_2(\mathcal{O}_K)}([\chi])$ , which is an $\mathcal{O}$ -lattice in the principal series type $\theta(\chi)\overset{\mathrm{def}}{=}\theta^{\circ}(\chi)[1/p]$ , where $[\chi]:I\to\mathcal{O}^{\times}$ is the Techmüller lift of $\chi$ . We let $\varphi^{\chi}\in\theta^{\circ}(\chi)$ be the unique element supported on I such that $\varphi^{\chi}(\operatorname{id})=1$ . If, moreover, $\chi\neq\chi^s$ , we write $\sigma(\chi)$ for the unique Serre weight such that $\chi$ is the I-character acting on $\sigma(\chi)^{I_1}$ . Then the cosocle of $\theta^{\circ}(\chi)$ is isomorphic to $\sigma(\chi)$ , and we denote the image of $\varphi^{\chi}$ in $\sigma(\chi)$ by $\varphi^{\chi}$ as well.

For R an $\mathcal{O}$ -algebra and $\tau$ a two-dimensional tame inertial type, we define a Kisin module over R of type $\tau$ and its eigenbasis as in [Reference Dotto and LeDL21, § 3.1], with the caveat that we only consider modules of rank two. For each Kisin module $\mathfrak{M}$ with a fixed eigenbasis, we define the matrices $A^{(j)}\in\operatorname{M}_2(R[\![v]\!])$ for $0\leq j\leq f-1$ as in [Reference Dotto and LeDL21, § 3.4].

From now on, we write $\mu\overset{\mathrm{def}}{=}(\underline{r}+\underline{1},\underline{0})\in X^*(\underline{T})$ and $\eta\overset{\mathrm{def}}{=}(\underline{1},\underline{0})\in X^*(\underline{T})$ . We let $w,w'\in\underline{W}$ such that (see [Reference Dotto and LeDL21, Proposition 3.11]):

1. (i) $(w_j,w_j')\neq(\mathfrak{w},\operatorname{id})$ for all $j\in\mathcal{J}$ ;

2. (ii) if $(w_j,w_j')=(\operatorname{id},\mathfrak{w})$ , then $j\in J_{\overline{\rho}}$ (or, equivalently, $d_j=0$ );

where $\mathfrak{w}$ is the unique non-trivial element in $S_2$ . As in [Reference Dotto and LeDL21, § 3.5] with $s_{\overline{\rho}}=1$ , we consider the tame inertial type $\tau=\tau(w,\mu-w'\eta)$ . We let $\overline{\mathfrak{M}}^{\tau}$ be the Kisin module over $\mathbb{F}$ of type $\tau$ given by the matrices $\overline{A}^{(f-1-j)}=\operatorname{Mat}(\phi_{\overline{\mathcal{M}}}^{(f-1-j)})v^{-(\mu_j-w'_j\eta_j)}\dot{w}_j\in\operatorname{M}_2(\mathbb{F}[\![v]\!])$ for $j\in\mathcal{J}$ , where $v^{(a,b)}\in\operatorname{M}_2(\mathbb{F}(\!(v)\!))$ denotes the diagonal matrix $({\kern-1pt}\begin{smallmatrix}v^a&0\\0&v^b\end{smallmatrix}{\kern-1pt})$ for $(a,b)\in\mathbb{Z}^2$ , and $\dot{w}_j\in\operatorname{M}_2(\mathbb{F})$ denotes the corresponding permutation matrix associated to $w_j$ . Then by [Reference Le, Le Hung, Levin and MorraLLL⁺20, Proposition 3.2.1], the étale $\varphi$ -module over $k(\!(v)\!)\otimes_{\mathbb{F}_p}\mathbb{F}$ associated to $\overline{\mathfrak{M}}^{\tau}$ in the sense of [Reference Le, Le Hung, Levin and MorraLLL⁺20, § 3.2] is isomorphic to $\overline{\mathcal{M}}$ . Concretely, from (4.3) we have (cf. [Reference Dotto and LeDL21, (14)])

\begin{equation*} \overline{A}^{(f-1-j)}=\begin{cases} \bigg(\begin{matrix}\beta^{-1}&0\\0&\beta\end{matrix}\bigg)\bigg(\begin{matrix}v&0\\-\beta^{-2}d_jv&1\end{matrix}\bigg)&\text{if}~(w_j,w'_j)=(\operatorname{id},\operatorname{id}),\\[10pt] \bigg(\begin{matrix}\beta^{-1}&0\\0&\beta\end{matrix}\bigg)\bigg(\begin{matrix}1&0\\0&v\end{matrix}\bigg)&\text{if}~(w_j,w'_j)=(\operatorname{id},\mathfrak{w}),\\[10pt] \bigg(\begin{matrix}\beta^{-1}&0\\0&\beta\end{matrix}\bigg)\bigg(\begin{matrix}0&1\\v&-\beta^{-2}d_j\end{matrix}\bigg)&\text{if}~(w_j,w'_j)=(\mathfrak{w},\mathfrak{w}).\end{cases}\end{equation*}

Let $R_0\overset{\mathrm{def}}{=}\mathcal{O}[\![X_j,Y_j,Z_j,Z'_j]\!]_{j=0}^{f-1}/(f_j)_{j=0}^{f-1}$ with $f_j=Y_j$ if $j\notin J_{\overline{\rho}}$ and $f_j=X_jY_j-p$ if $j\in J_{\overline{\rho}}$ . For $\tau=\tau(w,\mu-w'\eta)$ as above, we let $\mathfrak{M}^{\tau}$ be the Kisin module over $R_0$ given by the matrices $A^{(f-1-j)}\in\operatorname{M}_2(R_0[\![v]\!])$ for $j\in\mathcal{J}$ such that (cf. [Reference Dotto and LeDL21, (15)])

(4.4) \begin{equation} A^{(f-1-j)}=D^{(f-1-j)}A^{\prime(f-1-j)}\end{equation}

with

(4.5) \begin{equation}\begin{aligned} D^{(f-1-j)}&\overset{\mathrm{def}}{=}\bigg(\begin{matrix}Z'_j+[\beta]^{-1}&0\\0&Z_j+[\beta]\end{matrix}\bigg),\\ A^{\prime(f-1-j)}&\overset{\mathrm{def}}{=}\begin{cases} \bigg(\begin{matrix}v+p&0\\ (X_j-[\beta^{-2}d_j])v&1\end{matrix}\bigg)&\text{if}~(w_j,w'_j)=(\operatorname{id},\operatorname{id}),\\[10pt] \bigg(\begin{matrix}1&-Y_j\\0&v+p\end{matrix}\bigg)&\text{if}~(w_j,w'_j)=(\operatorname{id},\mathfrak{w}),\\[10pt] \bigg(\begin{matrix}-Y_j&1\\v&X_j\end{matrix}\bigg)&\text{if}~(w_j,w'_j)=(\mathfrak{w},\mathfrak{w})~\text{and}~j\in J_{\overline{\rho}},\\[10pt] \bigg(\begin{matrix}-p(X_j-[\beta^{-2}d_j])^{-1}&1\\v&X_j-[\beta^{-2}d_j]\end{matrix}\bigg)&\text{if}~(w_j,w'_j)=(\mathfrak{w},\mathfrak{w})~\text{and}~j\notin J_{\overline{\rho}}.\end{cases}\end{aligned}\end{equation}

In particular, $\mathfrak{M}^{\tau}$ is a deformation of $\overline{\mathfrak{M}}^{\tau}$ to $R_0$ .

For each $J\subseteq\mathcal{J}$ , we define $s^*,w,w'\in\underline{W}$ by the formula

(4.6) \begin{equation} \begin{aligned} s^*_{j-1}=\operatorname{id}&\Leftrightarrow j\notin J,\\ w_j=\operatorname{id}&\Leftrightarrow j\notin J,~j+1\notin J~\text{or}~j\in J,~j+1\in J,\\ w'_j=\operatorname{id}&\Leftrightarrow j\notin J,~j+1\notin J~\text{or}~j\in J,~j+1\in J,~j\notin J_{\overline{\rho}}. \end{aligned}\end{equation}

In particular, we have $w_j=s^*_js^*_{j-1}$ for all j by (4.6) and $\chi_J=\chi_{(s^*)^{-1}(\mu-w'\eta)}$ by Lemma 4.1 below, hence an isomorphism of tame types $\theta(\chi_J)\cong R_w(\mu-w'\eta)$ by [Reference Dotto and LeDL21, § 2.3.1] (cf. [Reference Dotto and LeDL21, Lemma 3.13]). Then we define $U_p(\chi_J)\overset{\mathrm{def}}{=}\prod_{j=0}^{f-1}U_p(\chi_J)_j\in R_0$ with (cf. [Reference Dotto and LeDL21, Proposition 3.22])

(4.7) \begin{equation} U_p(\chi_J)_j\overset{\mathrm{def}}{=}(p(A^{(f-1-j)}_{s^*_{j-1}(1)s^*_{j-1}(1)})^{{-}1}\operatorname{mod} v)\in R_0,\end{equation}

where $A^{(f-1-j)}_{s^*_{j-1}(1)s^*_{j-1}(1)}$ is the $(s^*_{j-1}(1),s^*_{j-1}(1))$ th entry of the matrix $A^{(f-1-j)}$ in (4.4) for the Kisin module $\mathfrak{M}^{\tau}$ over $R_0$ of type $\tau=\tau(w,\mu-w'\eta)$ .

Proof. Recall from § 2 that $\chi_J=\chi_{\lambda_J}$ with $\lambda_J=(\underline{s}^J+\underline{t}^J,\underline{t}^J)\in X^*(\underline{T})$ . Concretely, by (2.1) and (2.2) we have

(4.8) \begin{equation} (\lambda_J)_j= \begin{cases} (r_j,0)&\text{if}~j\notin J,~j+1\notin J,\\ (r_j,-1)&\text{if}~j\in J,~j+1\notin J,\\ (p-1,r_j+1)&\text{if}~j\notin J,~j+1\in J,\\ (p-2,r_j+1)&\text{if}~j\in J,~j+1\in J,~j\in J_{\overline{\rho}},\\ (p-1,r_j)&\text{if}~j\in J,~j+1\in J,~j\notin J_{\overline{\rho}}. \end{cases} \end{equation}

By (4.6) we also have

(4.9) \begin{equation} \begin{aligned} &((s^*)^{-1}(\mu-w'\eta))_j\\ &\quad=\left\{\!\!\!\begin{array}{ll@l} (r_j,0) & \text{if}~s^*_j=\operatorname{id},~w'_j=\operatorname{id}, & \text{equivalently},~j\notin J,~j+1\notin J,\\ (r_j+1,-1) & \text{if}~s^*_j=\operatorname{id},~w'_j=\mathfrak{w}, & \text{equivalently},~j\in J,~j+1\notin J,\\ (-1,r_j+1) & \text{if}~s^*_j=\mathfrak{w},~w'_j=\mathfrak{w}, & \text{equivalently}, \begin{aligned} &j\notin J,~j+1\notin J,\\ &\text{or}~j\in J,~j+1\in J,~j\in J_{\overline{\rho}}, \end{aligned}\\ (0,r_j) & \text{if}~s^*_j=\mathfrak{w},~w'_j=\operatorname{id}, & \text{equivalently},~j\in J,~j+1\in J,~j\notin J_{\overline{\rho}}. \end{array}\right. \end{aligned} \end{equation}

Combining (4.8) and (4.9), we have

\begin{equation*} \lambda_J-(s^*)^{-1}(\mu-w'\eta)=(p\underline{e}^{J-1}-\underline{e}^{J},\underline{0})\quad\text{in}~X^*(\underline{T}). \end{equation*}

Since $x^{p\underline{e}^{J-1}-\underline{e}^{J}}=1$ for all $x\in\mathbb{F}_q^{\times}$ , we deduce the equality $\chi_J=\chi_{(s^*)^{-1}(\mu-w'\eta)}$ .

Lemma 4.2. Let $J\subseteq\mathcal{J}$ . Then $U_p(\chi_J)$ is a product of a 1-unit of $R_0$ , an integer power of p, the scalar $[\beta]^{|J^c|-|J|}$ , and the quantity $U_p(\chi_J)'\overset{\mathrm{def}}{=}\prod_{i=0}^{f-1}U_p(\chi_J)'_j\in R_0$ with

(4.10) \begin{equation} U_p(\chi_J)'_j= \begin{cases} 1&\text{if}~j\notin J,~j+1\notin J~\text{or}~j\in J,~j+1\in J,\\ Y_j&\text{if}~j\in J,~j+1\notin J,~j\in J_{\overline{\rho}},\\ -X_j&\text{if}~j\notin J,~j+1\in J,~j\in J_{\overline{\rho}},\\ -[\beta^{-2}d_j]^{-1}&\text{if}~j\in J,~j+1\notin J,~j\notin J_{\overline{\rho}},\\ [\beta^{-2}d_j]&\text{if}~j\notin J,~j+1\in J,~j\notin J_{\overline{\rho}}. \end{cases} \end{equation}

In particular, we have $U_p(\chi_J)\in R_0[1/p]^{\times}$ . Here, a 1-unit means an element of $1+\mathfrak{m}_0$ , where $\mathfrak{m}_0$ is the maximal ideal of $R_0$ .

Proof. Let $s^*,w,w'\in\underline{W}$ be as in Lemma 4.1. By (4.5) and (4.6) we have

\begin{equation*} D^{(f-1-j)}_{s^*_{j-1}(1)s^*_{j-1}(1)}= \begin{cases} Z_j'+[\beta]^{-1}\in[\beta]^{-1}(1+\mathfrak{m}_0)&\text{if}~s^*_{j-1}=\operatorname{id},~\text{equivalently},~j\notin J,\\ Z_j+[\beta]\in[\beta](1+\mathfrak{m}_0)&\text{if}~s^*_{j-1}=\mathfrak{w},~\text{equivalently},~j\in J. \end{cases} \end{equation*}

Hence, we have

(4.11) \begin{equation} \bigg(\prod_{j=0}^{f-1}D^{(f-1-j)}_{s^*_{j-1}(1)s^*_{j-1}(1)}\!\bigg)^{\!{-}1}\in[\beta]^{|J^c|-|J|}(1+\mathfrak{m}_0). \end{equation}

By (4.5) we also have

(4.12) \begin{equation} (A^{\prime(f-1-j)}_{s^*_{j-1}(1)s^*_{j-1}(1)}\operatorname{mod} v)= \begin{cases} 1~\text{or}~p&\text{if}~w_j=\operatorname{id},\\ X_j&\text{if}~(w_j,w'_j,s^*_{j-1})\!=\!(\mathfrak{w},\mathfrak{w},\mathfrak{w})~\text{and}~j\in J_{\overline{\rho}},\\ -Y_j&\text{if}~(w_j,w'_j,s^*_{j-1})\!=\!(\mathfrak{w},\mathfrak{w},\operatorname{id})~\text{and}~j\in J_{\overline{\rho}},\\ X_j-[\beta^{-2}d_j]&\text{if}~(w_j,w'_j,s^*_{j-1})\!=\!(\mathfrak{w},\mathfrak{w},\mathfrak{w})~\text{and}~j\notin J_{\overline{\rho}},\\ -p(\!X_j-[\beta^{-2}d_j])^{-1}&\text{if}~(w_j,w'_j,s^*_{j-1})\!=\!(\mathfrak{w},\mathfrak{w},\operatorname{id})~\text{and}~j\notin J_{\overline{\rho}}. \end{cases} \end{equation}

Since $X_j-[\beta^{-2}d_j]\in-[\beta^{-2}d_j](1+\mathfrak{m}_0)$ for all $j\in\mathcal{J}$ and $X_jY_j=p$ in $R_0$ for $j\in J_{\overline{\rho}}$ , from (4.12) and (4.6) we deduce that

(4.13) \begin{equation} (p(A^{\prime(f-1-j)}_{s^*_{j-1}(1)s^*_{j-1}(1)})^{{-}1}\operatorname{mod} v)\in p^{\mathbb{Z}_{\geq0}}U_p(\chi_J)'_j(1+\mathfrak{m}_0)\quad\text{in}~R_0 \end{equation}

for $U_p(\chi_J)'_j$ as in (4.10). The result is then a combination of (4.11) and (4.13).

In the rest of this section we suppose that $J_{\overline{\rho}}\neq\mathcal{J}$ . Then for each $J\subseteq\mathcal{J}$ , there exists $i\geq0$ such that $\delta_{{\rm ss}}^i(J)=\emptyset$ (see (2.3) for $\delta_{{\rm ss}}$ ) and we define $\ell(J)\overset{\mathrm{def}}{=}\operatorname{min}\{ i\geq0:\delta_{{\rm ss}}^i(J)=\emptyset \}$ . We then define

(4.14) \begin{equation} \widetilde{U}_p(J)\overset{\mathrm{def}}{=}\frac{\prod_{i=0}^{\ell(J)-1}U_p(\chi_{\delta_{{\rm ss}}^i(J)})}{\prod_{i=0}^{\ell(J^{{\rm ss}})-1}U_p(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})}\in R_0[1/p].\end{equation}

Since $J_{\overline{\rho}}\neq\mathcal{J}$ , there is a unique decomposition of $J_{\overline{\rho}}$ into a disjoint union of intervals (in $\mathbb{Z}/f\mathbb{Z}$ ) not adjacent to each other $J_{\overline{\rho}}=J_1\sqcup\cdots\sqcup J_t$ . For each $1\leq i\leq t$ , we write $J_i=\{ j_i,j_i+1,\ldots,j_i+k_i \}$ with $j_i\in\mathcal{J}$ and $k_i\geq0$ . Then we define

(4.15) \begin{equation} A^{{\rm ss}}(J)\overset{\mathrm{def}}{=}\sum_{i=1}^{t}(\delta_{j_i+k_i\in\partial(J^c)}(k_i+1))\in\mathbb{Z}.\end{equation}

Proposition 4.3. Suppose that $J_{\overline{\rho}}\neq\mathcal{J}$ . Let $J\subseteq\mathcal{J}$ . Then $\widetilde{U}_p(J)$ is a product of a 1-unit of $R_0$ , an integer power of p, and the scalar $[U_p(J)]$ with

(4.16) \begin{equation} U_p(J)=(-1)^{A(J)}\beta^{B(J)}d(J)\in\mathbb{F}^{\times}, \end{equation}

where

(4.17) \begin{equation} \begin{aligned} A(J)&\overset{\mathrm{def}}{=} A^{{\rm ss}}(J)+\sum_{j\notin J_{\overline{\rho}}}\delta_{j\in\partial J}\in\mathbb{Z},\\ B(J)&\overset{\mathrm{def}}{=}\sum_{i=0}^{\ell(J)-1}(|\delta_{{\rm ss}}^i(J)^c|-|\delta_{{\rm ss}}^i(J)|)-\sum_{i=0}^{\ell(J^{{\rm ss}})-1}(|\delta_{{\rm ss}}^i(J^{{\rm ss}})^c|-|\delta_{{\rm ss}}^i(J^{{\rm ss}})|)\in\mathbb{Z},\\ d(J)&\overset{\mathrm{def}}{=}\bigg(\prod_{j\in J^{{\rm nss}}}d_j\!\bigg)^{\!{-}1}\bigg[\frac{\prod_{i=0}^{\ell(J)-1}\prod_{j\in(\delta_{{\rm ss}}^i(J)-1)^{{\rm nss}}}d_j}{\prod_{i=0}^{\ell(J^{{\rm ss}})-1}\prod_{j\in(\delta_{{\rm ss}}^i(J^{{\rm ss}})-1)^{{\rm nss}}}d_j}\bigg]\in\mathbb{F}^{\times}. \end{aligned} \end{equation}

Proof. We write $\widetilde{U}_p(J)'\overset{\mathrm{def}}{=}\prod_{j=0}^{f-1}\widetilde{U}_p(J)'_j\in R_0[1/p]$ with

\begin{equation*} \widetilde{U}_p(J)'_j\overset{\mathrm{def}}{=}\frac{\prod_{i=0}^{\ell(J)-1}U_p(\chi_{\delta_{{\rm ss}}^i(J)})'_j}{\prod_{i=0}^{\ell(J^{{\rm ss}})-1}U_p(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})'_j}=\frac{\prod_{i\geq0}U_p(\chi_{\delta_{{\rm ss}}^i(J)})'_j}{\prod_{i\geq0}U_p(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})'_j}, \end{equation*}

where each $U_p(\chi_{J'})'_j$ is defined in (4.10) and the equality uses $U_p(\chi_{\emptyset} )'_j=1$ for all $j\in\mathcal{J}$ . By Lemma 4.2, it suffices to show that $\widetilde{U}_p(J)'\in p^{\mathbb{Z}}(-1)^{A(J)}[d(J)]$ for $A(J)\in\mathbb{Z}$ and $d(J)\in\mathbb{F}^{\times}$ as in (4.17).

We fix $j\in\mathcal{J}$ and compute $\widetilde{U}_p(J)'_j$ . By definition, for $i\geq0$ we have

(4.18) \begin{equation} \begin{aligned} j\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(j+i\in J{,}~\text{and}~j+i'\in J_{\overline{\rho}}~\text{for}~0\leq i'\leq i-1),\\ j\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(j+i\in J{,}~\text{and}~j+i'\in J_{\overline{\rho}}~\text{for}~0\leq i'\leq i). \end{aligned} \end{equation}

We let $0\leq k\leq f-1$ be the unique integer such that $j+i\in J_{\overline{\rho}}$ for $1\leq i\leq k$ , and $j+k+1\notin J_{\overline{\rho}}$ . We also write it as k(j) to emphasize its dependence on j. We separate the following two cases.

Case 1. Assume that $j\in J_{\overline{\rho}}$ . Then we write

\begin{align*} &\begin{aligned} S_1(J)_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J)})'_j=Y_j\}\\ &=\#\{i\geq0:j\in\delta_{{\rm ss}}^i(J),~j+1\notin\delta_{{\rm ss}}^i(J)\}=\bigg(\sum_{i'=0}^{k}\delta_{j+i'\in\partial J}\!\bigg)+\delta_{j+k+1\in J}, \end{aligned}\\ &\begin{aligned} S_2(J)_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J)})'_j=-X_j\}\\&=\#\{i\geq0:j\notin\delta_{{\rm ss}}^i(J),~j+1\in\delta_{{\rm ss}}^i(J)\}=\sum_{i'=0}^{k}\delta_{j+i'\in\partial(J^c)}, \end{aligned}\\ &\begin{aligned} S_1(J^{{\rm ss}})_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})'_j=Y_j\}\\ &=\#\{i\geq0:j\in\delta_{{\rm ss}}^i(J^{{\rm ss}}),~j+1\notin\delta_{{\rm ss}}^i(J^{{\rm ss}})\}=\bigg(\sum_{i'=0}^{k-1}\delta_{j+i'\in\partial J}\!\bigg)+\delta_{j+k\in J}, \end{aligned}\\ &\begin{aligned} S_2(J^{{\rm ss}})_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})'_j=-X_j\}\\&=\#\{i\geq0:j\notin\delta_{{\rm ss}}^i(J^{{\rm ss}}),~j+1\in\delta_{{\rm ss}}^i(J^{{\rm ss}})\}=\sum_{i'=0}^{k-1}\delta_{j+i'\in\partial(J^c)}, \end{aligned} \end{align*}

where in each formula the first equality follows from (4.10) and the second equality follows from (4.18). In particular, we have

\begin{align*}& (S_1(J)_j-S_1(J^{{\rm ss}})_j)-(S_2(J)_j-S_2(J^{{\rm ss}})_j)=(\delta_{j+k\in\partial J}+\delta_{j+k+1\in J}-\delta_{j+k\in J})-\delta_{j+k\in\partial(J^c)}\\&\quad =(\delta_{j+k\in J}(1-\delta_{j+k+1\in J})+\delta_{j+k+1\in J}-\delta_{j+k\in J})-(1-\delta_{j+k\in J})\delta_{j+k+1\in J}=0. \end{align*}

Then using $X_jY_j=p$ in $R_0$ since $j\in J_{\overline{\rho}}$ , we deduce that

(4.19) \begin{equation} \widetilde{U}_p(J)'_j=Y_j^{S_1(J)_j-S_1(J^{{\rm ss}})_j}(-X_j)^{S_2(J)_j-S_2(J^{{\rm ss}})_j}\in(-1)^{\delta_{j+k\in\partial(J^c)}}p^{\mathbb{Z}}. \end{equation}

Case 2. Assume that $j\notin J_{\overline{\rho}}$ . Then we write

\begin{align*} &\begin{aligned} N_1(J)_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J)})'_j=-[\beta^{-2}d_j]^{-1}\}\\ &=\#\{i\geq0:j\in\delta_{{\rm ss}}^i(J),~j+1\notin\delta_{{\rm ss}}^i(J)\}=\delta_{j\in\partial J}, \end{aligned}\\ &\begin{aligned} N_2(J)_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J)})'_j=[\beta^{-2}d_j]\}\\&=\#\{i\geq0:j\notin\delta_{{\rm ss}}^i(J),~j+1\in\delta_{{\rm ss}}^i(J)\}=\delta_{j\in\partial(J^c)}+\sum_{i'=2}^{k+1}\delta_{j+i'\in J}, \end{aligned}\\ &\begin{aligned} N_1(J^{{\rm ss}})_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})'_j=-[\beta^{-2}d_j]^{-1}\}\\ &=\#\{i\geq0:j\in\delta_{{\rm ss}}^i(J^{{\rm ss}}),~j+1\notin\delta_{{\rm ss}}^i(J^{{\rm ss}})\}=0, \end{aligned}\\ &\begin{aligned} N_2(J^{{\rm ss}})_j&\overset{\mathrm{def}}{=}\#\{i\geq0:U_p(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})'_j=[\beta^{-2}d_j]\}\\&=\#\{i\geq0:j\notin\delta_{{\rm ss}}^i(J^{{\rm ss}}),~j+1\in\delta_{{\rm ss}}^i(J^{{\rm ss}})\}=\sum_{i'=1}^{k}\delta_{j+i'\in J}, \end{aligned} \end{align*}

where in each formula the first equality follows from (4.10) and the second equality follows from (4.18). In particular, we have

\begin{equation*} \begin{aligned} N(J)_j&\overset{\mathrm{def}}{=}(N_2(J)_j-N_2(J^{{\rm ss}})_j)-(N_1(J)_j-N_1(J^{{\rm ss}})_j)\\ &=(\delta_{j\in\partial(J^c)}+\delta_{j+k+1\in J}-\delta_{j+1\in J})-\delta_{j\in\partial J}\\ &=(1-\delta_{j\in J})\delta_{j+1\in J}+\delta_{j+k+1\in J}-\delta_{j+1\in J}+\delta_{j\in J}(1-\delta_{j+1\in J})=\delta_{j+k+1\in J}-\delta_{j\in J}. \end{aligned} \end{equation*}

Hence, we have

(4.20) \begin{equation} \widetilde{U}_p(J)'_j=(-1)^{N_1(J)_j+N_1(J^{{\rm ss}})_j}[\beta^{-2}d_j]^{N(J)_j}=(-1)^{\delta_{j\in\partial J}}[\beta^{-2}d_j]^{N(J)_j}. \end{equation}

By the definition of d(J), we have $d(J)=\prod_{j\notin J_{\overline{\rho}}}d_j^{M(J)_j}$ with (for each $j\notin J_{\overline{\rho}}$ )

(4.21) \begin{align} M(J)_j&=-\delta_{j\in J}+\bigg(\sum_{i=0}^{\ell(J)-1}\delta_{j+1\in\delta_{{\rm ss}}^i(J)}\!\bigg)-\bigg(\sum_{i=0}^{\ell(J^{{\rm ss}})-1}\delta_{j+1\in\delta_{{\rm ss}}^i(J^{{\rm ss}})}\!\bigg)\nonumber\\ &=-\delta_{j\in J}+\bigg(\sum_{i=0}^{k(j)}\delta_{j+i+1\in J}\!\bigg)-\bigg(\sum_{i=0}^{k(j)-1}\delta_{j+i+1\in J}\!\bigg)\nonumber\\ &=\delta_{j+k(j)+1\in J}-\delta_{j\in J}=N(J)_j. \end{align}

By the definition of k(j), we have that $j+k(j)+1$ is the first place after j that is not in $J_{\overline{\rho}}$ , hence we have

(4.22) \begin{equation} \sum_{j\notin J_{\overline{\rho}}}N(J)_j=\sum_{j\notin J_{\overline{\rho}}}(\delta_{j+k(j)+1\in J}-\delta_{j\in J})=0. \end{equation}

Combining (4.19)–(4.21) and (4.22), we deduce that

\begin{equation*} \begin{aligned} \widetilde{U}_p(J)'&=\prod_{j=0}^{f-1}\widetilde{U}_p(J)'_j\\ &\in p^{\mathbb{Z}}(-1)^{\sum_{j\in J_{\overline{\rho}}}\delta_{j+k(j)\in\partial(J^c)}+\sum_{j\notin J_{\overline{\rho}}}\delta_{j\in\partial J}}[\beta]^{-2\sum_{j\notin J_{\overline{\rho}}}N(J)_j}\bigg(\prod_{j\notin J_{\overline{\rho}}}[d_j]^{N(J)_j}\bigg)\\ &=p^{\mathbb{Z}}(-1)^{A(J)}[d(J)] \end{aligned} \end{equation*}

for $A(J)\in\mathbb{Z}$ and $d(J)\in\mathbb{F}^{\times}$ as in (4.17), which completes the proof.

5. Constants in the Diamond diagrams

We review the strategy of [Reference Dotto and LeDL21] to compute the constants in the diagram $(\pi^{I_1}\hookrightarrow\pi^{K_1})$ . When $\overline{\rho}$ is non-semisimple, we specialize the general formula to some particular constants that are needed to prove the main result, Theorem 1.1 (see Example 5.3).

For $0\leq i\leq q-1$ , we define (with the convention that $0^0\overset{\mathrm{def}}{=}1$ )

(5.1) \begin{align}\begin{aligned} S_i&\overset{\mathrm{def}}{=}\sum\limits_{\lambda\in\mathbb{F}_q^{\times}}[\lambda]^i\bigg(\begin{matrix}\lambda&1\\1&0\end{matrix}\bigg)\in\mathcal{O}[\operatorname{GL}_2(\mathbb{F}_q)],\\ S^+_i&\overset{\mathrm{def}}{=}\sum\limits_{\lambda\in\mathbb{F}_q^{\times}}[\lambda]^i\bigg(\begin{matrix}1&0\\\lambda&1\end{matrix}\bigg)\in\mathcal{O}[\operatorname{GL}_2(\mathbb{F}_q)].\end{aligned}\end{align}

Recall that we identify $\pi^{K_1}$ with $D_0(\overline{\rho})$ . As in [Reference Dotto and LeDL21, Definition 4.1], we let $R:D_0(\overline{\rho})^{I_1}\to(\operatorname{soc}_{\operatorname{GL}_2(\mathcal{O}_K)}D_0(\overline{\rho}))^{I_1}$ be the unique map defined as follows: if $\chi:I\to\mathbb{F}^{\times}$ is an I-character such that $D_0(\overline{\rho})^{I_1}[\chi]\neq0$ , then $R|_{D_0(\overline{\rho})^{I_1}[\chi]}$ is given by $S_{i(\chi)}$ for some unique $0\leq i(\chi)\leq q-1$ , except when $\chi$ appears in $(\operatorname{soc}_{\operatorname{GL}_2(\mathcal{O}_K)}D_0(\overline{\rho}))^{I_1}$ , in which case $R|_{D_0(\overline{\rho})^{I_1}[\chi]}$ is the identity. Given $\chi$ , we write $R\chi$ for the I-character such that $R(D_0(\overline{\rho})^{I_1}[\chi])=D_0(\overline{\rho})^{I_1}[R\chi]$ . Then we define $g_{\chi}:D_0(\overline{\rho})^{I_1}[R\chi]\to D_0(\overline{\rho})^{I_1}[R\chi^s]$ by the formula $g_{\chi}(R(v))=R(\!\!({\kern-1pt}\begin{smallmatrix}0&1\\p&0\end{smallmatrix}{\kern-1pt})v)$ for $v\in D_0(\overline{\rho})^{I_1}[\chi]$ . In particular, we have $g_{\chi^s}=g_{\chi}^{-1}$ for all $\chi$ .

Example 5.1. Suppose that $J_{\overline{\rho}}\neq\mathcal{J}$ . We let $J\subseteq\mathcal{J}$ such that $J\neq J^*$ (see Lemma 2.2 for $J^*$ , which implies $J^{{\rm nss}}\neq\mathcal{J}$ ). By [Reference WangWan24, Lemma 4.1(iii)] we have

(5.2) \begin{equation} R\chi_J=\chi_{J^{{\rm ss}}}xs,\quad R\chi_J^s=\chi_{(J-1)^{{\rm ss}}}. \end{equation}

In particular, we have $R\chi_J=\chi_J$ if and only if $J=J^{{\rm ss}}$ and we have $R\chi_J^s\neq\chi_J^s$ by Lemma 2.2.

Recall that $0\neq v_J\in D_0(\overline{\rho})^{I_1}[\chi_J]$ and $0\neq v_{J^s}=({\kern-1pt}\begin{smallmatrix}0&1\\p&0\end{smallmatrix}{\kern-1pt})v_J\in D_0(\overline{\rho})^{I_1}[\chi_{J^s}]$ as defined in § 3. By (3.2) with $J'=(J-1)^{{\rm ss}}$ and [Reference Breuil, Herzig, Hu, Morra and SchraenBHH⁺25a, Lemma 3.2.2.5(i)], we have

(5.3) \begin{equation} R(v_{J^s})=S_{i(\chi_J^s)}v_{J^s}=(-1)^{f-1}P_1(\chi_J)\mu_{J,(J-1)^{{\rm ss}}}v_{(J-1)^{{\rm ss}}} \end{equation}

with

(5.4) \begin{align}\begin{aligned} i(\chi_J^s)&=\sum_{j+1\in J\Delta(J-1)^{{\rm ss}}}(p-1-s^{(J-1)^{{\rm ss}}}_j)p^j,\\ P_1(\chi_J)&\overset{\mathrm{def}}{=}\prod_{j+1\in J\Delta(J-1)^{{\rm ss}}}(p-1-s^{(J-1)^{{\rm ss}}}_j)!\in\mathbb{F}^{\times}.\end{aligned} \end{align}

If $J\subseteq J_{\overline{\rho}}$ , then we have $R(v_J)=v_J=v_{J^{{\rm ss}}}$ by definition. If $J\nsubseteq J_{\overline{\rho}}$ , then by Proposition 3.1(i) and [Reference Breuil, Herzig, Hu, Morra and SchraenBHH⁺25a, Lemma 3.2.2.5(i)] we have

(5.5) \begin{equation} R(v_J)=S_{i(\chi_J)}v_J=(-1)^{f-1}P_2(J)\mu_{J^s,J^{{\rm ss}}}v_{J^{{\rm ss}}} \end{equation}

with

(5.6) \begin{align}\begin{aligned} i(\chi_J)&=\sum_{j+1\notin J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j)p^j,\\ P_2(J)&\overset{\mathrm{def}}{=}\prod_{j+1\notin J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j)!\in\mathbb{F}^{\times}.\end{aligned} \end{align}

By the definition of $g_{\chi_J}$ and combining (5.3) and (5.5), we get

(5.7) \begin{equation} g_{\chi_J}(v_{J^{{\rm ss}}})= \left\{\begin{aligned} (-1)^{f-1}P_1(\chi_J)\mu_{J,(J-1)^{{\rm ss}}}v_{(J-1)^{{\rm ss}}}&\quad\text{if}~J\subseteq J_{\overline{\rho}},\\[2pt] \frac{(-1)^{f-1}P_1(\chi_J)\mu_{J,(J-1)^{{\rm ss}}}}{(-1)^{f-1}P_2(J)\mu_{J^s,J^{{\rm ss}}}}v_{(J-1)^{{\rm ss}}}&\quad\text{if}~J\nsubseteq J_{\overline{\rho}}. \end{aligned}\right. \end{equation}

For $J\nsubseteq J_{\overline{\rho}}$ we define (see above (4.14) for $\ell(J)$ )

(5.8) \begin{equation} \gamma(J)\overset{\mathrm{def}}{=}(-1)^{|J\cap(J-1)^{{\rm nss}}|}\frac{\mu_{*,J}}{\mu_{*,J^{{\rm ss}}}}\Bigg[\frac{\prod_{i=0}^{\ell(J)-1}(-1)^{f-1}\mu_{\delta_{{\rm ss}}^i(J),\delta_{{\rm ss}}^{i+1}(J)}}{\prod_{i=0}^{\ell(J^{{\rm ss}})-1}(-1)^{f-1}\mu_{\delta_{{\rm ss}}^i(J^{{\rm ss}}),\delta_{{\rm ss}}^{i+1}(J^{{\rm ss}})}}\Bigg]. \end{equation}

Consider the following two maps:

\begin{equation*} \begin{aligned} &D_0(\overline{\rho})^{I_1}[R\chi_J]\xrightarrow{g_{\chi_J}} D_0(\overline{\rho})^{I_1}[R\chi_{\delta_{{\rm ss}}(J)}]\xrightarrow{g_{\chi_{\delta_{{\rm ss}}(J)}}}\cdots\rightarrow D_0(\overline{\rho})^{I_1}[R\chi_{\emptyset}],\\ &D_0(\overline{\rho})^{I_1}[R\chi_J]=D_0(\overline{\rho})^{I_1}[R\chi_{J^{{\rm ss}}}]\xrightarrow{g_{\chi_{J^{{\rm ss}}}}} D_0(\overline{\rho})^{I_1}[R\chi_{\delta_{{\rm ss}}(J^{{\rm ss}})}]\xrightarrow{g_{\chi_{\delta_{{\rm ss}}(J^{{\rm ss}})}}}\cdots\rightarrow D_0(\overline{\rho})^{I_1}[R\chi_{\emptyset}]. \end{aligned} \end{equation*}

Suppose that the composition

(5.9) \begin{equation} \bigg(\prod^{0}_{i=\ell(J^{{\rm ss}})-1}g_{\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})}}\!\bigg)^{\!{-}1}\circ\bigg(\prod^{0}_{i=\ell(J)-1}g_{\chi_{\delta_{{\rm ss}}^i(J)}}\!\bigg):D_0(\overline{\rho})^{I_1}[R\chi_J]\to D_0(\overline{\rho})^{I_1}[R\chi_J] \end{equation}

is given by the scalar $g(J)\in\mathbb{F}^{\times}$ . Applying the first formula of (5.7) to $\delta_{{\rm ss}}^i(J^{{\rm ss}})$ for $0\leq i\leq\ell(J^{{\rm ss}})-1$ and $\delta_{{\rm ss}}^i(J)$ for $1\leq i\leq\ell(J)-1$ , and applying the second formula of (5.7) to J, we get

(5.10) \begin{align} \gamma(J)&=(-1)^{|J\cap(J-1)^{{\rm nss}}|}\frac{\mu_{*,J}}{\mu_{*,J^{{\rm ss}}}}(-1)^{f-1}\mu_{J^s,J^{{\rm ss}}}(P_2(J)/P_1(J))g(J)\nonumber\\ &=(-1)^{f-1+|J\cap(J-1)^{{\rm nss}}|}\mu_{J^s,J}(P_2(J)/P_1(J))g(J), \end{align}

where $P_1(J)\overset{\mathrm{def}}{=}(\prod_{i=0}^{\ell(J)-1}P_1(\chi_{\delta_{{\rm ss}}^i(J)}))/(\prod_{i=0}^{\ell(J^{{\rm ss}})-1}P_1(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})}))$ and the last equality follows from Proposition 3.4.

We let $M_{\infty}$ be the $\mathcal{O}[\operatorname{GL}_2(K)]$ -module as in [Reference Dotto and LeDL21, § 6.2]. Then $M_{\infty}$ has a minimal arithmetic action of $R_{\infty}$ in the sense of [Reference Dotto and LeDL21, § 4.2], where $R_{\infty}$ is a suitable power series ring over $R_{\overline{\rho}}$ , the universal framed $\mathcal{O}$ -deformation ring for $\overline{\rho}$ . In particular, we have $M_{\infty}^{\vee}[\mathfrak{m}_{\infty}]\cong\pi$ for $\pi$ as in (1.1), where $(-)^{\vee}$ is the Pontrjagin dual and $\mathfrak{m}_{\infty}$ is the maximal ideal of $R_{\infty}$ . We let $M_{\infty}(-)$ be the corresponding patching functor.

Let $\theta$ be a non-scalar tame type. For each Serre weight $\sigma\in\operatorname{JH}(\overline{\theta})$ where $\overline{\theta}$ is the reduction modulo $\varpi$ of any $\mathcal{O}$ -lattice in $\theta$ , we fix a lattice $\theta^{\sigma}$ in $\theta$ with irreducible cosocle $\sigma$ . Such a lattice is unique up to homothety and we rescale it when necessary. For $\chi:I\to\mathbb{F}^{\times}$ a character such that $\chi\neq\chi^s$ and $\sigma(\chi)\in\operatorname{JH}(\overline{\theta})$ , we write $\theta^{\chi}$ for $\theta^{\sigma(\chi)}$ . We let $\operatorname{pr}_{\chi^s}:\theta(\chi^s)^{R\chi^s}\to\sigma(R\chi^s)$ and $\operatorname{pr}_{\chi}:\theta(\chi^s)^{R\chi}\to\sigma(R\chi)$ be the normalized surjections as in [Reference Dotto and LeDL21, (23), (24)].

We fix a tame inertial type $\tau_0\overset{\mathrm{def}}{=}\tau(w,\mu-w\eta)$ with associated tame type $\theta_0\overset{\mathrm{def}}{=} R_w(\mu-w\eta)$ for some fixed $w\in\underline{W}$ satisfying $w_j=\mathfrak{w}$ for $j\in J_{\overline{\rho}}$ and $w_0w_1\cdots w_{f-1}=\mathfrak{w}$ if $J_{\overline{\rho}}\neq\mathcal{J}$ . Then we have $W(\overline{\rho})\subseteq\operatorname{JH}(\overline{\theta}_0)$ by [Reference Dotto and LeDL21, Proposition 3.11] and $\theta_0$ is a cuspidal type if $J_{\overline{\rho}}\neq\mathcal{J}$ . Moreover, the ring $R_0$ defined above (4.4) is a power series ring over $R_{\overline{\rho}}^{\tau_0}$ (see [Reference Dotto and LeDL21, § 3.5.1]), where $R_{\overline{\rho}}^{\tau_0}$ is the quotient of $R_{\overline{\rho}}$ parametrizing potentially crystalline lifts of $\overline{\rho}$ with Hodge–Tate weights (1,0) in each embedding and inertial type $\tau_0$ . In particular, all the arguments of [Reference Dotto and LeDL21, § 4] still hold, replacing the so-called central type $\theta=R_{\underline{\mathfrak{w}}}(\mu-\underline{\mathfrak{w}}\eta)$ with the type $\theta_0$ .

For any $\mathcal{O}$ -lattice $\theta_0^{\circ}$ in $\theta_0$ , the patched module $M_{\infty}(\theta_0^{\circ})$ is supported on $R_{\infty}(\tau_0)\overset{\mathrm{def}}{=}R_{\infty}\otimes_{R_{\overline{\rho}}}R_{\overline{\rho}}^{\tau_0}$ . We let $Q(\chi^s)^{R\chi^s}$ (respectively, $Q(\chi^s)^{R\chi}$ ) be the quotient of $\theta(\chi^s)^{R\chi^s}/\varpi$ (respectively, $\theta(\chi^s)^{R\chi}/\varpi$ ) as in [Reference Dotto and LeDL21, Proposition 4.18]. Then the surjection $\operatorname{pr}_{\chi^s}$ (respectively, $\operatorname{pr}_{\chi}$ ) factors through $Q(\chi^s)^{R\chi^s}$ (respectively, $Q(\chi^s)^{R\chi}$ ). As in the discussion after [Reference Dotto and LeDL21, (27)], we fix a surjection $\alpha:\theta_0^{R\chi^s}\to Q(\chi^s)^{R\chi^s}$ which induces a surjection $\alpha:\theta_0^{R\chi}\to Q(\chi^s)^{R\chi}$ . Then as in [Reference Dotto and LeDL21, (29)] there is a commutative diagram

where we refer to [Reference Dotto and LeDL21, § 4.4] for the maps $\iota$ , $\widetilde{h}_{\chi}$ , $\overline{h}_{\chi}$ and the element $\widetilde{U}_p(\chi)\in R_{\infty}(\tau_0)$ . Moreover, by [Reference Dotto and LeDL21, Lemma 5.5] and the definition of $\widetilde{U}'_p(\chi)$ in [Reference Dotto and LeDL21, Definition 3.22], we deduce that $\widetilde{U}_p(\chi)\widetilde{U}_p(\chi^s)$ is a product of an integer power of p and a 1-unit of $R_{\infty}(\tau_0)$ (i.e. an element of $1+\mathfrak{m}_{\infty}(\tau_0)$ , where $\mathfrak{m}_{\infty}(\tau_0)$ is the maximal ideal of $R_{\infty}(\tau_0)$ ). In particular, $\widetilde{U}_p(\chi)\in R_{\infty}(\tau_0)[1/p]^{\times}$ .

For our purposes, we consider a cycle of characters but in a different order from that in [Reference Dotto and LeDL21, § 4.5]. Namely, we consider the I-characters $\psi_0,\psi_1,\ldots,\psi_n$ and $\psi'_0,\psi'_1,\ldots,\psi'_m$ appearing in $D_0(\overline{\rho})^{I_1}$ such that $R\psi_0=R\psi'_0$ , $R\psi_n^s=R\psi_m^{\prime s}$ , $R\psi_i^s=R\psi_{i+1}$ for $0\leq i\leq n-1$ and $R\psi_i^{\prime s}=R\psi'_{i+1}$ for $0\leq i\leq m-1$ . Here we allow $m=-1$ , in which case we are reduced to the situation considered in [Reference Dotto and LeDL21, § 4.5]. We fix a surjection $\alpha_0:\theta_0^{R\psi_0^s}\to Q(\psi_0^s)^{R\psi_0^s}$ . Similarly to [Reference Dotto and LeDL21, (32)], we define the surjection $\alpha'_0:\theta_0^{R\psi_0^{\prime s}}\to Q(\psi_0^{\prime s})^{R\psi_0^{\prime s}}$ by the following commutative diagram (if $m\geq0$ ).

Then as in [Reference Dotto and LeDL21, (32)] we define the surjections $\alpha_i:\theta_0^{R\psi_i^s}\to Q(\psi_i^s)^{R\psi_i^s}$ for $1\leq i\leq n$ inductively by the commutative diagram

and we define the surjections $\alpha_i':\theta_0^{R\psi_i^{\prime s}}\to Q(\psi_i^{\prime s})^{R\psi_i^{\prime s}}$ for $1\leq i\leq m$ inductively in a similar way.

Analogously to the picture in [Reference Dotto and LeDL21, § 4.5], we give a picture for $n=1$ and $m=0$ , where all the patched modules are cyclic $R_{\infty}$ -modules, as follows.

Suppose that $\operatorname{pr}_{\psi_n^s}\circ\alpha_n=c(\psi,\psi')\operatorname{pr}_{\psi_m^{\prime s}}\circ\alpha_m'$ for $c(\psi,\psi')\in\mathbb{F}^{\times}$ . Suppose that the composition

(5.11) \begin{equation} \overline{h}_{\psi_n}^{-1}\!\circ\!\cdot\!\cdot\!\cdot\!\circ\!\overline{h}_{\psi_0}^{-1}\!\circ\!\overline{h}_{\psi_0'}\!\circ\!\cdot\!\cdot\!\cdot\!\circ\!\overline{h}_{\psi_m'}\!:\!M_{\infty}(\sigma(R\psi_n^s))/\mathfrak{m}_{\infty}\!=\!M_{\infty}(\sigma(R\psi_m^{\prime s}))/\mathfrak{m}_{\infty}\!\to\!M_{\infty}(\sigma(R\psi_n^s))/\mathfrak{m}_{\infty}\end{equation}

is given by multiplication by $h(\psi,\psi')\in\mathbb{F}^{\times}$ . Analogously to [Reference Dotto and LeDL21, (34)], there exists $\nu\in\mathbb{Z}$ such that the element

(5.12) \begin{equation} p^{-\nu}\bigg(\prod_{i=0}^n\widetilde{U}_p(\psi_i)\!\bigg)^{\!{-}1}\bigg(\prod_{i=0}^m\widetilde{U}_p(\psi_i')\!\bigg)\in R_{\infty}(\tau_0)[1/p]^{\times}\end{equation}

lies in $R_{\infty}(\tau_0)$ and reduces to $h(\psi,\psi')c(\psi,\psi')^{-1}$ modulo $\mathfrak{m}_{\infty}(\tau_0)$ .

Example 5.3. Suppose that $J_{\overline{\rho}}\neq\mathcal{J}$ . We let $J\subseteq\mathcal{J}$ such that $J\nsubseteq J_{\overline{\rho}}$ and $J\neq J^*$ (see Lemma 2.2 for $J^*$ ). Then we take $n=\ell(J)-1$ (see above (4.14) for $\ell(J)$ ), $m=\ell(J^{{\rm ss}})-1$ , $\psi_i=\chi_{\delta_{{\rm ss}}^{n-i}(J)}^s$ for $0\leq i\leq n$ and $\psi_i'=\chi_{\delta_{{\rm ss}}^{m-i}(J^{{\rm ss}})}^s$ for $0\leq i\leq m$ (using (5.2)), and write c(J) for $c(\psi,\psi')\in\mathbb{F}^{\times}$ . Let $g(J),U_p(J)\in\mathbb{F}^{\times}$ be as in (5.9) and (4.16). Then by (5.11) and [Reference Dotto and LeDL21, Proposition 4.16] we have $g(J)=h(\psi,\psi')$ . Moreover, using [Reference Dotto and LeDL21, Lemma 5.5] together with the definition of $\widetilde{U}_p(J)$ in (4.14), Proposition 4.3 and (5.12), we deduce that $U_p(J)=h(\psi,\psi')c(J)^{-1}$ , hence $g(J)=U_p(J)c(J)$ . Combining with (5.10), we conclude that (see (5.8) for $\gamma(J)$ )

(5.13) \begin{equation} \gamma(J)=U_p(J)c'(J) \end{equation}

with $c'(J)\overset{\mathrm{def}}{=}(-1)^{f-1+|J\cap(J-1)^{{\rm nss}}|}\mu_{J^s,J}(P_2(J)/P_1(J))c(J)$ .

6. Computation of constants

Throughout this section, we suppose that $J_{\overline{\rho}}\neq\mathcal{J}$ , or equivalently that $\overline{\rho}$ is non-semisimple. We compute c’(J) defined in (5.13) for $J\nsubseteq J_{\overline{\rho}}$ and $J\neq J^*$ (see Lemma 2.2 for $J^*$ ). The main results are Propositions 6.7 and 6.12. Together with the results of § 4, we finish the computation of all the constants in the diagram $(\pi^{I_1}\hookrightarrow\pi^{K_1})$ that we need.

6.1 Relation between S-operators

For $a\in\mathbb{Z}$ , we define $a_j\in\{ 0,1,\ldots,p-1 \}$ for $0\leq j\leq f-1$ by writing $a=\sum_{j=0}^{f-1}a_jp^j+Q(q-1)$ for some $Q\in\mathbb{Z}$ and we define $a_{q}\overset{\mathrm{def}}{=}\sum_{j=0}^{f-1}a_jp^j\in\{ 0,1,\ldots,q-2 \}$ . If $(q-1)\nmid a$ , we write $S_a$ (respectively, $S^+_a$ ) for the operators $S_{a_q}$ (respectively, $S^+_{a_q}$ ) defined in (5.1). For $0\neq b\in K$ , we let $u\in\mathbb{Z}$ be such that $b\in p^u\mathcal{O}_K^{\times}$ , then we define the leading term of b to be the element of $\mathbb{F}_q^{\times}$ that is the reduction modulo p of $p^{-u}b\in\mathcal{O}_K^{\times}$ .

For $a,b\in\mathbb{Z}$ , we define

(6.1) \begin{align}\begin{aligned} u(a,b)&\overset{\mathrm{def}}{=}(p-1)^{-1}\sum_{j=0}^{f-1}(p-1-(a_j+b_j-(a+b)_j))\in\mathbb{Z},\\ J(a,b)&\overset{\mathrm{def}}{=}(-1)^{f-1+u(a,b)}\prod_{j=0}^{f-1}(a_j!b_j!((a+b)_j!)^{-1})\in\mathbb{F}_q^{\times}.\end{aligned}\end{align}

More generally, for $a_1,\ldots,a_n\in\mathbb{Z}$ , we define

(6.2) \begin{align}\begin{aligned} u(a_1,\ldots,a_n)&\overset{\mathrm{def}}{=}(p-1)^{-1}\sum_{j=0}^{f-1} \bigg(\sum_{i=1}^n(p-1-(a_i)_j)-\bigg(p-1-\bigg(\sum_{i=1}^na_i{\kern-1pt}\bigg)_{\!j}\bigg)\!\bigg)\in\mathbb{Z},\\ J(a_1,\ldots,a_n)&\overset{\mathrm{def}}{=}(-1)^{(n-1)(f-1)+u(a_1,\ldots,a_n)}\prod_{j=0}^{f-1} \bigg(\!\bigg(\prod_{i=1}^n(a_i)_j!\bigg)\bigg(\!\bigg(\sum_{i=1}^na_i{\kern-1pt}\bigg)_{\!j}!\bigg)^{\!{-}1}\bigg)\in\mathbb{F}_q^{\times}.\end{aligned}\end{align}

Lemma 6.1.

1. (i) Let $0\lt a,b\lt q-1$ such that $a+b\neq q-1$ . Then we have

   \begin{equation*} S^+_{a}S^+_{b}=\widetilde{J}(a,b)S^+_{a+b},\quad S_{a}S^+_{b}=\widetilde{J}(a,b)S_{a+b}, \end{equation*}
   where $0\neq\widetilde{J}(a,b)\in\mathcal{O}_K$ has leading term J(a,b).

2. (ii) Let $0\lt a_1,\ldots,a_n\lt q-1$ such that $(q-1)$ does not divide $\sum_{i=1}^ka_i$ for all $1\leq k\leq n-1$ . Then we have

   \begin{equation*} \begin{aligned} S^+_{a_1}\ldots S^+_{a_n}&=\widetilde{J}(a_1,\ldots,a_n)S^+_{a_1+\cdots+a_n}&\!\!\!\!\!\quad\text{if}~(q-1)\nmid\sum_{i=1}^na_i,\\ S^+_{a_1}\ldots S^+_{a_n}&=J'(a_1,\ldots,a_n)S^+_0+\widetilde{J}(a_1,\ldots,a_n)\bigg(\begin{matrix}1&0\\0&1\end{matrix}\bigg)&\!\!\!\!\!{\kern-.5pt}\quad\text{if}~(q-1)\mid\sum_{i=1}^na_i, \end{aligned} \end{equation*}
   where $0\neq\widetilde{J}(a_1,\ldots,a_n)\in\mathcal{O}_K$ has leading term $J(a_1,\ldots,a_n)$ , and $J'(a_1,\ldots,a_n)\in\mathcal{O}_K$ .

Proof. Part (i) is [Reference Dotto and LeDL21, Lemmas 2.3 and 2.4]. The first formula of (ii) follows from (i) by induction, and the second formula of (ii) is [Reference Dotto and LeDL21, Proposition 2.5].

Lemma 6.2. Let v be an H-eigenvector in an $\mathcal{O}[\operatorname{GL}_2(\mathbb{F}_q)]$ -module with H-eigencharacter $\chi$ . Then $S_iv$ has H-eigencharacter $\chi^s\alpha^{-i}$ and $S^+_iv$ has H-eigencharacter $\chi\alpha^i$ , where $\alpha$ is the H-character $({\kern-1pt}\begin{smallmatrix}a&0\\0&d\end{smallmatrix}{\kern-1pt})\mapsto ad^{-1}$ .

Proof. This is [Reference Dotto and LeDL21, Lemma 2.7].

Lemma 6.3. Let $J\subseteq\mathcal{J}$ and v be an I-eigenvector in an $\mathcal{O}[\operatorname{GL}_2(\mathbb{F}_q)]$ -module with I-character $[\chi_J]$ . Let $a,b\in\mathbb{Z}$ such that $(q-1)$ does not divide $a,a-b,a-b-\underline{s}^J$ (see (2.1) for $\underline{s}^J$ , which also denotes the integer $\sum_{j=0}^{f-1}s^J_jp^j\in\mathbb{Z}$ ). Then we have (see (2.2) for $\underline{t}^J$ )

\begin{equation*} S_aS_bv=(-1)^{\underline{t}^J}\widetilde{J}(a,-b-\underline{s}^J)S_{a-b-\underline{s}^J}v, \end{equation*}

where $0\neq\widetilde{J}(a,-b-\underline{s}^J)\in\mathcal{O}_K$ has leading term $J(a,-b-\underline{s}^J)$ .

Proof. This is [Reference Dotto and LeDL21, Proposition 2.8] with $\chi=\chi_J$ .

Lemma 6.4. Suppose that $J\neq J^*$ . Then we have (see (5.4) for $i(\chi_J^s)$ and see (2.3) for $J^{\delta}$ )

\begin{equation*} J(i(\chi_J^s),-\underline{s}^J)=(-1)^{1+\sum_{j+1\in J^{\delta}}s^{(J-1)^{{\rm ss}}}_j}\Bigg[\frac{\prod_{j+1\in J^{\delta},j\notin J^{\delta}}(-1)(s^{(J-1)^{{\rm ss}}}_j+1)}{\prod_{j+1\notin J^{\delta},j\in J^{\delta}}(s^{(J-1)^{{\rm ss}}}_j+1)}\Bigg]. \end{equation*}

Proof. We write $a\overset{\mathrm{def}}{=} i(\chi_J^s)\in\mathbb{Z}$ and $b\overset{\mathrm{def}}{=}-\underline{s}^J\in\mathbb{Z}$ so that $a_j=\delta_{j+1\in J^{\delta}}(p-1-s^{(J-1)^{{\rm ss}}}_j)$ and $b_j=p-1-s^J_j$ for each $j\in\mathcal{J}$ . By Lemma 2.1(i) we have

(6.3) \begin{equation} a_j+b_j= \begin{cases} (p-1-s^{(J-1)^{{\rm ss}}}_j)+(p-1-s^J_j)=p-\delta_{j\in J^{\delta}}&\text{if}~j+1\in J^{\delta},\\[3pt] p-1-s^J_j=p-1-s^{(J-1)^{{\rm ss}}}_j-\delta_{j\in J^{\delta}}&\text{if}~j+1\notin J^{\delta}, \end{cases} \end{equation}

hence we have

(6.4) \begin{align} a+b&\equiv\sum_{j=0}^{f-1}(a_j+b_j)p^j\nonumber\\ &=\sum_{j+1\in J^{\delta}}(p-\delta_{j\in J^{\delta}})p^j+\sum_{j+1\notin J^{\delta}}(p-1-s^{(J-1)^{{\rm ss}}}_j-\delta_{j\in J^{\delta}})p^j\nonumber\\ &=\bigg(\sum_{j+1\notin J^{\delta}}(p-1-s^{(J-1)^{{\rm ss}}}_j)p^j\!\bigg)+\bigg(\sum_{j+1\in J^{\delta}}p^{j+1}-\sum_{j\in J^{\delta}}p^j\!\bigg)\nonumber\\ &\equiv\sum_{j+1\notin J^{\delta}}(p-1-s^{(J-1)^{{\rm ss}}}_j)p^j\quad{\operatorname{mod}}\,(q-1), \end{align}

which implies $(a+b)_j=0$ if $j+1\in J^{\delta}$ and $(a+b)_j=p-1-s^{(J-1)^{{\rm ss}}}_j$ if $j+1\notin J^{\delta}$ .

Then by (6.1) using (6.3) and (6.4), we obtain both

\begin{align*} u(a,b)&=(p-1)^{-1}\sum_{j=0}^{f-1}(p-1-(a_j+b_j-(a+b)_j))\\ &=(p-1)^{-1}\bigg(\sum_{j+1\notin J^{\delta}}(p-1+\delta_{j\in J^{\delta}})+\sum_{j+1\in J^{\delta}}(-1+\delta_{j\in J^{\delta}})\!\bigg)\\ &=(p-1)^{-1}((p-1)\#\{ j:j+1\notin J^{\delta} \}+\#\{ j:j\in J^{\delta} \}-\#\{ j:j+1\in J^{\delta} \})\\ &=f-|J^{\delta}| \end{align*}

and

\begin{align*} J(i(\chi_J^s),-\underline{s}^J)&=(-1)^{f-1+u(a,b)}\prod_{j=0}^{f-1}(a_j!b_j!((a+b)_j!)^{-1})\\ &=(-1)^{f-1+u(a,b)}\Bigg[\frac{\prod_{j+1\in J^{\delta},j\in J^{\delta}}(a_j!(p-1-a_j)!)\prod_{j+1\in J^{\delta},j\notin J^{\delta}}(a_j!(p-a_j)!)}{\prod_{j+1\notin J^{\delta},j\in J^{\delta}}(a+b)_j}\Bigg]\\ &=(-1)^{f-1+f-|J^{\delta}|+\sum_{j+1\in J^{\delta}}(s^{(J-1)^{{\rm ss}}}_j+1)}\Bigg[\frac{\prod_{j+1\in J^{\delta},j\notin J^{\delta}}(s^{(J-1)^{{\rm ss}}}_j+1)}{\prod_{j+1\notin J^{\delta},j\in J^{\delta}}(p-1-s^{(J-1)^{{\rm ss}}}_j)}\Bigg]\\ &=(-1)^{1+\sum_{j+1\in J^{\delta}}s^{(J-1)^{{\rm ss}}}_j}\Bigg[\frac{\prod_{j+1\in J^{\delta},j\notin J^{\delta}}(-1)(s^{(J-1)^{{\rm ss}}}_j+1)}{\prod_{j+1\notin J^{\delta},j\in J^{\delta}}(s^{(J-1)^{{\rm ss}}}_j+1)}\Bigg], \end{align*}

where the third equality follows from (3.6) and the last equality uses $\#\{ j:j+1\in J^{\delta},j\notin J^{\delta} \}=\#\{ j:j+1\notin J^{\delta},j\in J^{\delta} \}$ .

Lemma 6.5. Let $J\nsubseteq J_{\overline{\rho}}$ . Then for each $j\in\mathcal{J}$ we have (see (5.6) for $i(\chi_J)$ )

\begin{equation*} (i(\chi_J)+\underline{s}^J)_j=\delta_{j+1\in J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j). \end{equation*}

Proof. This follows from the computation

\begin{align*} i(\chi_J)+\underline{s}^J&\equiv\sum_{j=0}^{f-1}(i(\chi_J)_j+s^J_j)p^j\\ &=\sum_{j+1\notin J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j+s^J_j)p^j+\sum_{j+1\in J^{{\rm nss}}}s^J_jp^j\\ &=\sum_{j+1\notin J^{{\rm nss}}}(p-\delta_{j\notin J^{{\rm nss}}})p^j+\sum_{j+1\in J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j-\delta_{j\notin J^{{\rm nss}}})p^j\\ &=\bigg(\sum_{j+1\in J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j)p^j\!\bigg)+\bigg(\sum_{j+1\notin J^{{\rm nss}}}p^{j+1}-\sum_{j\notin J^{{\rm nss}}}p^j\!\bigg)\\ &\equiv\sum_{j+1\in J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j)p^j\quad{\operatorname{mod}}\,(q-1), \end{align*}

where the second equality follows from Lemma 2.1(ii).

6.2 The case $(J-1)^{{\rm ss}}=J^{{\rm ss}}$

Lemma 6.6. Let $\emptyset\neq J\subseteq\mathcal{J}$ such that $J\neq J^*$ and $(J-1)^{{\rm ss}}=J^{{\rm ss}}$ (which implies $J\nsubseteq J_{\overline{\rho}}$ and $J^{{\rm nss}}\neq\mathcal{J}$ ). Then we have (see §4 for $\theta^{\circ}(\chi_J)$ and $\varphi^{\chi_J}$ )

\begin{equation*} S_{i(\chi_J^s)}S_0\varphi^{\chi_J}=\widetilde{c}(\chi_J)S_{i(\chi_J)}\varphi^{\chi_J}\quad\text{in}~\theta^{\circ}(\chi_J), \end{equation*}

where $0\neq\widetilde{c}(\chi_J)\in\mathcal{O}_K$ has leading term $c(\chi_J)\overset{\mathrm{def}}{=}(-1)^{\underline{t}^J}J(i(\chi_J^s),-\underline{s}^J)$ .

Proof. Since $(J-1)^{{\rm ss}}=J^{{\rm ss}}$ , we deduce from (5.6) and (6.4) that $i(\chi_J^s)-\underline{s}^J\equiv i(\chi_J)~\operatorname{mod}\,(q-1).$ Moreover, our assumption implies $J^{{\rm nss}}\notin\{ \emptyset,\mathcal{J} \}$ , hence $(q-1)\nmid i(\chi_J^s)$ and $(q-1)\nmid i(\chi_J)$ . Then we conclude using Lemma 6.3.

Recall that $c'(J)=(-1)^{f-1+|J\cap(J-1)^{{\rm nss}}|}\mu_{J^s,J}(P_2(J)/P_1(J))c(J)$ for $J\nsubseteq J_{\overline{\rho}}$ and $J\neq J^*$ with $\mu_{J^s,J}$ defined in Proposition 3.1(ii), $P_i(J)$ defined in Example 5.1 for $i\in\{ 1,2 \}$ and c(J) defined in Example 5.3.

Proposition 6.7. Let $\emptyset\neq J\subseteq\mathcal{J}$ such that $J\neq J^*$ and $(J-1)^{{\rm ss}}=J^{{\rm ss}}$ . Then we have (see (5.13) for c’(J) and see (4.17) for A(J))

\begin{equation*} c'(J)=(-1)^{A(J)}. \end{equation*}

Proof. By (3.5), using (3.6) and Lemma 2.1(ii), we have

(6.5) \begin{equation} \mu_{J^s,J}=(-1)^{\underline{t}^{J^s}+\sum_{j+1\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+\delta_{j\in J^{{\rm nss}}})}\Bigg[\frac{\prod_{j+1\notin J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+\delta_{j\in J^{{\rm nss}}})!}{\prod_{j+1\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+\delta_{j\notin J^{{\rm nss}}})!}\Bigg]. \end{equation}

Since $(J-1)^{{\rm ss}}=J^{{\rm ss}}$ , we have $\delta_{{\rm ss}}^{i+1}(J)=\delta_{{\rm ss}}^{i}(J^{{\rm ss}})$ for all $i\geq0$ , hence we have

(6.6) \begin{align} P_2(J)/P_1(J)&=P_2(J)/P_1(\chi_J)\nonumber\\ &=\frac{\prod_{j+1\notin J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j)!}{\prod_{j+1\in J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j)!} =(-1)^{\underline{s}^{J^{{\rm ss}}}+\underline{1}}\Bigg[\frac{\prod_{j+1\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j)!}{\prod_{j+1\notin J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j)!}\Bigg]. \end{align}

Recall from Example 5.3 that $c(J)=c(\psi,\psi')$ with $\psi_i=\chi_{\delta_{{\rm ss}}^{n-i}(J)}^s$ for $0\leq i\leq n\overset{\mathrm{def}}{=}\ell(J)-1$ and $\psi_i'=\chi_{\delta_{{\rm ss}}^{m-i}(J^{{\rm ss}})}^s$ for $0\leq i\leq m\overset{\mathrm{def}}{=}\ell(J^{{\rm ss}})-1$ . Since $(J-1)^{{\rm ss}}=J^{{\rm ss}}$ , we have $n=m+1$ and $\psi_{i+1}=\psi'_i$ (hence $\overline{h}_{\psi_{i+1}}=\overline{h}_{\psi'_{i}}$ ) for $0\leq i\leq m$ , hence c(J) is also equal to $c(\psi,\psi')$ with $n=0$ , $m=-1$ and $\psi_0=\chi_J^s$ by definition. Then we deduce from Lemma 6.4 to Lemma 6.6 that

(6.7) \begin{equation} c(J)=c(\chi_J)=(-1)^{1+\underline{t}^J+\sum_{j+1\in J^{{\rm nss}}}s^{J^{{\rm ss}}}_j}\Bigg[\frac{\prod_{j+1\in J^{{\rm nss}},j\notin J^{{\rm nss}}}(-1)(s^{J^{{\rm ss}}}_j+1)}{\prod_{j+1\notin J^{{\rm nss}},j\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+1)}\Bigg]. \end{equation}

Moreover, we have

(6.8) \begin{align} &|J\cap(J-1)^{{\rm nss}}|+\#\{ j:j+1\in J^{{\rm nss}},j\in J^{{\rm nss}} \}+\#\{ j:j+1\in J^{{\rm nss}},j\notin J^{{\rm nss}} \}\nonumber\\ &\quad=|J^{{\rm nss}}\cap(J-1)|+\#\{ j:j+1\in J^{{\rm nss}},j\in J^{{\rm nss}} \}+\#\{ j:j+1\notin J^{{\rm nss}},j\in J^{{\rm nss}} \}\nonumber\\ &\quad=|J^{{\rm nss}}\cap(J-1)|+|J^{{\rm nss}}|\equiv\#\{ j\in J^{{\rm nss}},j+1\notin J \}=A(J)\quad{\operatorname{mod}}\,2, \end{align}

where the last equality follows from the definition of A(J) using $(J-1)^{{\rm ss}}=J^{{\rm ss}}$ .

By the definition of c’(J) and combining (6.5), (6.6) and (6.7), we deduce that $c'(J)=(-1)^{d}$ with

\begin{align*} d&=(f-1+|J\cap(J-1)^{{\rm nss}}|)+\bigg(\underline{t}^{J^s}+\sum_{j+1\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+\delta_{j\in J^{{\rm nss}}})\!\bigg)\\ &\quad+(\underline{s}^{J^{{\rm ss}}}+\underline{1})+\bigg(1+\underline{t}^J+\sum_{j+1\in J^{{\rm nss}}}s^{J^{{\rm ss}}}_j+\#\{ j:j+1\in J^{{\rm nss}},j\notin J^{{\rm nss}} \}\!\bigg)\\ &\equiv\underline{t}^J+\underline{t}^{J^s}+\underline{s}^{J^{{\rm ss}}}+A(J)\equiv A(J)\quad{\operatorname{mod}}\,2, \end{align*}

where the first congruence follows from (6.8) and the second congruence follows from Lemma 2.3. This completes the proof.

Remark 6.8. When $(J-1)^{{\rm ss}}=J^{{\rm ss}}$ , we are in the situation of [Reference HuHu16], and the constant $g(J)=U_p(J)c(J)$ (hence the constant $\gamma(J)=U_p(J)c'(J)$ ) can be computed by [Reference HuHu16, Theorem 4.7]. Here we remark that the term $(-1)^{e(\tau)(r_0,\ldots,r_{f-1})}$ is missing in the formula of [Reference HuHu16, Theorems 4.5 and 4.7].

6.3 The case $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$

Lemma 6.9. Let $\emptyset\neq J\subseteq J_{\overline{\rho}}$ (which implies $J\neq J^*$ and $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$ ). Then we have

\begin{equation*} S_{i(\chi_J^s)}S_0\varphi^{\chi_J}=\widetilde{c}(\chi_J)S^+_{i^+(\chi_J)}\varphi^{\chi_J}\quad\text{in}~\theta^{\circ}(\chi_J), \end{equation*}

where $i^+(\chi_J)\overset{\mathrm{def}}{=} q-1-i(\chi_J^s)$ and $0\neq\widetilde{c}(\chi_J)\in\mathcal{O}_K$ has leading term $c(\chi_J)\overset{\mathrm{def}}{=} J(i(\chi_J^s),-\underline{s}^J)$ .

Proof. The proof is the same as [Reference Dotto and LeDL21, Lemmas 5.11 and 5.28].

Lemma 6.10. Let $J\subseteq\mathcal{J}$ such that $J\nsubseteq J_{\overline{\rho}}$ , $J\neq J^*$ and $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$ . Then we have

(6.9) \begin{equation} S_{i(\chi_J^s)}S_0\varphi^{\chi_J}=\widetilde{c}(\chi_J)S^+_{i^+(\chi_J)}S_{i(\chi_J)}\varphi^{\chi_J}\quad\text{in}~\theta(\chi_J), \end{equation}

where $i^+(\chi_J)\overset{\mathrm{def}}{=} i(\chi_J)-i(\chi_J^s)+\underline{s}^J$ , and $0\neq\widetilde{c}(\chi_J)\in K$ has leading term

\begin{equation*} c(\chi_J)\overset{\mathrm{def}}{=}(-1)^{\underline{t}^J}\frac{J(i(\chi_J^s),-\underline{s}^J)}{J(i^+(\chi_J),-i(\chi_J)-\underline{s}^J)}. \end{equation*}

Proof. Recall from Example 5.1 that $R\chi_J=\chi_{J^{{\rm ss}}}$ , $R\chi_J^s=\chi_{(J-1)^{{\rm ss}}}$ and $R\chi_J^s\neq\chi_J^s$ . Then the assumption $J\nsubseteq J_{\overline{\rho}}$ implies $J\neq(J-1)^{{\rm ss}}$ , hence $R\chi_J^s\neq\chi_J$ , and the assumption $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$ implies $R\chi_J\neq R\chi_{J}^s$ . As in the proof of [Reference Dotto and LeDL21, Lemma 5.11] (using Lemma 6.2 and [Reference Dotto and LeDL21, Lemma 2.11]), equality (6.9) holds for some $0\neq\widetilde{c}(\chi_J)\in K$ .

Next we compute $\widetilde{c}(\chi_J)\in K$ . The assumption $J\nsubseteq J_{\overline{\rho}}$ implies $J^{\delta}\neq\emptyset$ , hence $(q-1)\nmid i(\chi_J^s)$ . The assumption $J\neq J^*$ implies $J^{\delta}\neq\mathcal{J}$ , hence $(q-1)\nmid(i(\chi_J^s)-\underline{s}^J)$ by (6.4). Then by Lemma 6.3 we have

(6.10) \begin{equation} S_{i(\chi_J^s)}S_0\varphi^{\chi_J}=\widetilde{J}_1S_{i(\chi_J^s)-\underline{s}^J}\varphi^{\chi_J}, \end{equation}

where $0\neq\widetilde{J}_1\in\mathcal{O}_K$ has leading term $J_1\overset{\mathrm{def}}{=}(-1)^{\underline{t}^J}J(i(\chi_J^s),-\underline{s}^J)$ .

We choose $0\lt z\lt q-1$ such that none of the numbers

\begin{equation*} z-i(\chi_J^s)+\underline{s}^J,~z-i(\chi_J^s),~z+i^+(\chi_J),~z+i^+(\chi_J)-i(\chi_J) \end{equation*}

are multiples of $(q-1)$ , which is possible since $q\geq7$ . On the one hand, by Lemma 6.3 we have

(6.11) \begin{equation} S_zS_{i(\chi_J^s)-\underline{s}^J}\varphi^{\chi_J}=\widetilde{J}_2S_{z-i(\chi_J^s)}\varphi^{\chi_J}, \end{equation}

where $0\neq\widetilde{J}_2\in\mathcal{O}_K$ has leading term $J_2\overset{\mathrm{def}}{=}(-1)^{\underline{t}^J}J(z,-i(\chi_J^s))$ . On the other hand, by Lemmas 6.1(i) and 6.3 we have

(6.12) \begin{equation} S_zS^+_{i^+(\chi_J)}S_{i(\chi_J)}\varphi^{\chi_J}=\widetilde{J}_3S_{z+i^+(\chi_J)}S_{i(\chi_J)}\varphi^{\chi_J}=\widetilde{J}_3\widetilde{J}_4S_{z-i(\chi_J^s)}\varphi^{\chi_J}, \end{equation}

where $0\neq\widetilde{J}_3\in\mathcal{O}_K$ has leading term $J_3\overset{\mathrm{def}}{=} J(z,i^+(\chi_J))$ and $0\neq\widetilde{J}_4\in\mathcal{O}_K$ has leading term $J_4\overset{\mathrm{def}}{=}(-1)^{\underline{t}^J}J(z+i^+(\chi_J),-i(\chi_J)-\underline{s}^J)$ . Since $S_{z-i(\chi_J^s)}\varphi^{\chi_J}\neq0$ as explained in the proof of [Reference Dotto and LeDL21, Lemma 5.28], we deduce. combining (6.9)–(6.11) and (6.12), that $0\neq\widetilde{c}(\chi_J)=(\widetilde{J}_1\widetilde{J}_2)/(\widetilde{J}_3\widetilde{J}_4)\in K$ has leading term

\begin{equation*} c(\chi_J)=\frac{J_1J_2}{J_3J_4}=\frac{(-1)^{\underline{t}^J}J(i(\chi_J^s),-\underline{s}^J)J(z,-i(\chi_J^s))}{J(z,i^+(\chi_J))J(z+i^+(\chi_J),-i(\chi_J)-\underline{s}^J)}=(-1)^{\underline{t}^J}\frac{J(i(\chi_J^s),-\underline{s}^J)}{J(i^+(\chi_J),-i(\chi_J)-\underline{s}^J)}, \end{equation*}

where the last equality follows from the formula (applied with $a=z,~b=i^+(\chi_J), c=-i(\chi_J)-\underline{s}^J$ )

\begin{equation*} J(a,b)J(a+b,c)=J(a,b+c)J(b,c)\quad\text{for}~a,b,c\in\mathbb{Z}, \end{equation*}

which can be deduced from the explicit formula (6.1).

For $J\subseteq\mathcal{J}$ such that $J\neq J^*$ and $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$ , we write

\begin{equation*}\begin{aligned} i^+(J)&\overset{\mathrm{def}}{=}\sum_{i=0}^{\ell(J)-1}i^+(\chi_{\delta_{{\rm ss}}^i(J)})\in\mathbb{Z},\\ \beta(J)&\overset{\mathrm{def}}{=} J(i^+(\chi_J),i^+(\chi_{\delta_{{\rm ss}}(J)}),\ldots,i^+(\chi_{\delta_{{\rm ss}}^{\ell(J)-1}(J)}))\in\mathbb{F}^{\times},\end{aligned}\end{equation*}

where each $i^+(\chi_{J'})$ is defined in either Lemma 6.9 or Lemma 6.10.

Proposition 6.11. Let $J\subseteq\mathcal{J}$ such that $J\nsubseteq J_{\overline{\rho}}$ , $J\neq J^*$ and $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$ . Then we have

\begin{equation*} c(J)=\frac{\beta(J)}{\beta(J^{{\rm ss}})}\frac{\prod_{i=0}^{\ell(J)-1}c(\chi_{\delta_{{\rm ss}}^i(J)})}{\prod_{i=0}^{\ell(J^{{\rm ss}})-1}c(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})}, \end{equation*}

where c(J) is defined in Example 5.3 and each $c(\chi_{J'})$ is defined in either Lemma 6.9 or Lemma 6.10.

Proof. Recall from Example 5.3 that $c(J)=c(\psi,\psi')$ with $\psi_i=\chi_{\delta_{{\rm ss}}^{n-i}(J)}^s$ for $0\leq i\leq n\overset{\mathrm{def}}{=}\ell(J)-1$ and $\psi_i'=\chi_{\delta_{{\rm ss}}^{m-i}(J^{{\rm ss}})}^s$ for $0\leq i\leq m\overset{\mathrm{def}}{=}\ell(J^{{\rm ss}})-1$ . Recall from § 5 the maps $\operatorname{pr}_{\chi}$ , $\alpha_i$ and $\alpha'_i$ . If $x\in\theta_0^{R\psi_n^s}$ is an H-eigenvector such that $\operatorname{pr}_{\psi_n^s}\alpha_n(x)=\varphi^{R\psi_n^s}\in\sigma(R\psi_n^s)$ , then the proof of [Reference Dotto and LeDL21, Proposition 5.14] (using Lemmas 6.9, 6.10 and the cuspidality of $\theta_0$ ) and the argument that follows show that

(6.13) \begin{equation} \operatorname{pr}_{\psi_0}\alpha_0\bigg(\prod_{i=n}^0[c(\chi_{\delta_{{\rm ss}}^{i}(J)})]S^+_{i^+(\chi_{\delta_{{\rm ss}}^{i}(J)})}(x)\!\bigg)=\varphi^{R\psi_0}=\varphi^{\chi_{\emptyset}}\in\sigma(\chi_{\emptyset}). \end{equation}

Similarly, if $x'\in\theta_0^{R\psi_m^{\prime s}}$ is an H-eigenvector such that $\operatorname{pr}_{\psi_m^{\prime s}}\alpha_m'(x')=\varphi^{R\psi_m^{\prime s}}\in\sigma(R\psi_m^{\prime s})$ , then we have

(6.14) \begin{equation} \operatorname{pr}_{\psi_0'}\alpha_0'\bigg(\prod_{i=m}^0[c(\chi_{\delta_{{\rm ss}}^{i}(J^{{\rm ss}})})]S^+_{i^+(\chi_{\delta_{{\rm ss}}^{i}(J^{{\rm ss}})})}(x')\!\bigg)=\varphi^{R\psi_0'}=\varphi^{\chi_{\emptyset}}\in\sigma(\chi_{\emptyset}). \end{equation}

We compare H-eigencharacters of (6.13) and (6.14) using Lemma 6.2. Since both x and x’ have H-eigencharacters $R\psi_n^s=R\psi_m^{\prime s}=\chi_{J^{{\rm ss}}}$ by (5.2), we deduce that $i^+(J)\equiv i^+(J^{{\rm ss}})~{\operatorname{mod}}\,(q-1)$ . Since $R\chi_{\delta_{{\rm ss}}^i(J)}\neq\chi_{\emptyset}$ for $0\leq i\leq n-1$ and $R\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})}\neq\chi_{\emptyset}$ for $0\leq i\leq m-1$ by the choice of n and m, we deduce that $(q-1)\nmid i^+(\delta_{{\rm ss}}^i(J))$ for $1\leq i\leq n$ and $(q-1)\nmid i^+(\delta_{{\rm ss}}^i(J^{{\rm ss}}))$ for $1\leq i\leq m$ . Moreover, since $\theta_0$ is a cuspidal type, we have $S^+_0(x)=S^+_0(x')=0$ by [Reference Dotto and LeDL21, Lemma 2.9]. Hence, by Lemma 6.1(ii) we have

(6.15) \begin{align}\begin{aligned} \bigg(\prod_{i=n}^0S^+_{i^+(\chi_{\delta_{{\rm ss}}^{i}(J)})}\!\bigg)(x)&=\widetilde{\beta}(J)S^+_{i^+(J)}(x),\\ \bigg(\prod_{i=m}^0S^+_{i^+(\chi_{\delta_{{\rm ss}}^{i}(J^{{\rm ss}})})}\!\bigg)(x)&=\widetilde{\beta}(J^{{\rm ss}})S^+_{i^+(J^{{\rm ss}})}(x),\end{aligned} \end{align}

where $0\neq\widetilde{\beta}(J)\in\mathcal{O}_K$ (respectively, $0\neq\widetilde{\beta}(J^{{\rm ss}})\in\mathcal{O}_K$ ) has leading term $\beta(J)$ (respectively, $\beta(J^{{\rm ss}})$ ). Combining (6.13)–(6.15) and the argument following the proof of [Reference Dotto and LeDL21, Proposition 5.14], we deduce that

\begin{equation*} c(J)=\frac{\beta(J)}{\beta(J^{{\rm ss}})}\frac{\prod_{i=0}^{n}c(\chi_{\delta_{{\rm ss}}^i(J)})}{\prod_{i=0}^{m}c(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})}, \end{equation*}

which completes the proof.

Proposition 6.12. Let $J\subseteq\mathcal{J}$ such that $J\nsubseteq J_{\overline{\rho}}$ , $J\neq J^*$ and $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$ . Then we have

\begin{equation*} c'(J)=(-1)^{A(J)}. \end{equation*}

Proof. By the definition c′(J) and Proposition 6.11 we have

(6.16) \begin{equation} c'(J)=(-1)^{f-1+|J\cap(J-1)^{{\rm nss}}|}\mu_{J^s,J}\frac{P_2(J)}{P_1(J)}\frac{\beta(J)}{\beta(J^{{\rm ss}})}\frac{\prod_{i=0}^{\ell(J)-1}c(\chi_{\delta_{{\rm ss}}^i(J)})}{\prod_{i=0}^{\ell(J^{{\rm ss}})-1}c(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})}.\end{equation}

Recall from the proof of Proposition 6.11 that $i^+(J)\equiv i^+(J^{{\rm ss}})~{\operatorname{mod}}\,(q-1)$ , hence by (6.2) we have

(6.17) \begin{equation} \frac{\beta(J)}{\beta(J^{{\rm ss}})}=\frac{(-1)^{u(J)}}{(-1)^{u(J^{{\rm ss}})}}\frac{((-1)^{f-1}\prod_{j=0}^{f-1}(i^+(\chi_J)_j)!)\prod_{i=1}^{\ell(J)-1}((-1)^{f-1}\prod_{j=0}^{f-1}(i^+(\chi_{\delta_{{\rm ss}}^i(J)})_j)!)}{\prod_{i=0}^{\ell(J^{{\rm ss}})-1}((-1)^{f-1}\prod_{j=0}^{f-1}\!(i^+(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})_j)!)},\end{equation}

where $u(J)\overset{\mathrm{def}}{=} u(i^+(\chi_J),i^+(\chi_{\delta_{{\rm ss}}(J)}),\ldots,i^+(\chi_{\delta_{{\rm ss}}^{\ell(J)-1}(J)}))\in\mathbb{Z}$ , and similarly for $u(J^{{\rm ss}})$ . Moveover, by Lemmas 6.9, 6.10 and (6.1) we have

(6.18) \begin{align}\begin{aligned} \prod_{j=0}^{f-1}(i^+(\chi_{J'})_j)!&=\prod_{j=0}^{f-1}(p-1-i(\chi_{J'}^s)_j)!\quad\text{for}~J'\subseteq J_{\overline{\rho}},\\ \frac{\prod_{j=0}^{f-1}(i^+(\chi_J)_j)!}{J(i^+(\chi_J),-i(\chi_J)-\underline{s}^J)}&=(-1)^{f-1+u'(J)}\frac{\prod_{j=0}^{f-1}(p-1-i(\chi_J^s)_j)!}{\prod_{j=0}^{f-1}((-i(\chi_J)-\underline{s}^J)_j)!},\end{aligned}\end{align}

where $u'(J)\overset{\mathrm{def}}{=} u(i^+(\chi_J),-i(\chi_J)-\underline{s}^J)$ . Combining (6.16)–(6.18), the definition of $c(\chi_J)$ in Lemma 6.10 and the definition of each $c(\chi_J')$ for $J'\subseteq J_{\overline{\rho}}$ in Lemma 6.9, we deduce that

\begin{equation*} c'(J)=(-1)^{U(J)}\alpha'(J)\alpha(J),\end{equation*}

where

(6.19) \begin{align}\begin{aligned} U(J)&\overset{\mathrm{def}}{=} u(J)-u(J^{{\rm ss}})-u'(J)\in\mathbb{Z},\\ \alpha'(J)&\overset{\mathrm{def}}{=}(-1)^{|J\cap(J-1)^{{\rm nss}}|+\underline{t}^J}\mu_{J^s,J}P_2(J)\bigg/\bigg(\prod_{j=0}^{f-1}((-i(\chi_J)-\underline{s}^J)_j)!\!\bigg)\in\mathbb{F}^{\times},\\ \alpha(J)&\overset{\mathrm{def}}{=}\bigg(\prod_{i=0}^{\ell(J)-1}\alpha(\chi_{\delta_{{\rm ss}}^i(J)})\!\bigg)\bigg/\bigg(\prod_{i=0}^{\ell(J^{{\rm ss}})-1}\alpha(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})\!\bigg)\in\mathbb{F}^{\times},\end{aligned}\end{align}

with (for each $J'\neq J^*$ )

\begin{equation*} \alpha(\chi_{J'})\overset{\mathrm{def}}{=} J(i(\chi_{J'}^s),-\underline{s}^{J'})\bigg({\kern-1pt}(-1)^{f-1}\prod_{j=0}^{f-1}(p-1-i(\chi_{J'}^s))!\!\bigg)\bigg/P_1(\chi_{J'})\in\mathbb{F}^{\times}.\end{equation*}

Then the proposition follows from an explicit computation of the constants U(J), $\alpha'(J)$ and $\alpha(J)$ , which is given in Lemma 6.13.

6.4 Explicit computations

Here we prove Lemma 6.13, which will finish the proof of Proposition 6.12. To state the result, for $J\subseteq\mathcal{J}$ and $j\in\mathcal{J}$ we define (in $\mathbb{F}^{\times}$ )

(6.20) \begin{equation} \alpha'(J)_j\overset{\mathrm{def}}{=}\begin{cases} 1&\text{if}~j\notin J^{{\rm nss}},~j+1\notin J^{{\rm nss}},\\ r_j+1&\text{if}~j\in J^{{\rm nss}},~j+1\notin J,\\ p-1-r_j&\text{if}~j\in J^{{\rm nss}},~j+1\in J^{{\rm ss}},\\ -(r_j!(r_j+1)!)^{-1}&\text{if}~j\notin J,~j+1\in J^{{\rm nss}},\\ -((r_j+1)!(r_j+2)!)^{-1}&\text{if}~j\in J^{{\rm ss}},~j+1\in J^{{\rm nss}},\\ (r_j!)^{-2}&\text{if}~j\in J^{{\rm nss}},~j+1\in J^{{\rm nss}}.\end{cases}\end{equation}

Lemma 6.13. Let $J\subseteq\mathcal{J}$ such that $J\nsubseteq J_{\overline{\rho}}$ , $J\neq J^*$ and $(J-1)^{{\rm ss}}\neq J^{{\rm ss}}$ . Let $U(J)\in\mathbb{Z}$ , $\alpha'(J),\alpha(J)\in\mathbb{F}^{\times}$ be as in (6.19). Then we have:

1. (i) $U(J)=A^{{\rm ss}}(J)$ (see (4.15) for $A^{{\rm ss}}(J)$ );

2. (ii) $\alpha'(J)=(-1)^{|(\partial J)^{{\rm nss}}|}\prod_{j=0}^{f-1}\alpha'(J)_j$ ;

3. (iii) $\alpha(J){\prod_{j=0}^{f-1}\alpha'(J)_j}=1$ .

In the rest of this subsection, we prove Lemma 6.13. We start with some more notation that is needed in the proof. For $J\subseteq\mathcal{J}$ and $j\in\mathcal{J}$ , we define

\begin{equation*}\begin{aligned} I(J)^{1}_j&\overset{\mathrm{def}}{=}\{ i\geq0:j+1\in\delta_{{\rm ss}}^i(J),~j+1\notin\delta_{{\rm ss}}^{i+1}(J) \},\\ I(J)^{2}_j&\overset{\mathrm{def}}{=}\{ i\geq0:j+1\notin\delta_{{\rm ss}}^i(J),~j+1\in\delta_{{\rm ss}}^{i+1}(J) \},\\ I(J)^{3}_j&\overset{\mathrm{def}}{=}\{ i\geq0:j+1\in\delta_{{\rm ss}}^i(J),~j+1\in\delta_{{\rm ss}}^{i+1}(J) \},\\ I(J)^{4}_j&\overset{\mathrm{def}}{=}\{ i\geq0:j+1\notin\delta_{{\rm ss}}^i(J),~j+1\notin\delta_{{\rm ss}}^{i+1}(J) \}.\end{aligned}\end{equation*}

Since $j\in\delta_{{\rm ss}}^{i+1}(J)$ if and only if $(j+1\in\delta_{{\rm ss}}^{i}(J)~\text{and}~j\in J_{\overline{\rho}})$ , by (2.1) we have

(6.21) \begin{equation} s^{\delta_{{\rm ss}}^{i+1}(J)}_j=\begin{cases} r_j+\delta_{j\in J_{\overline{\rho}}}&\text{if}~i\in I(J)^{1}_j,\\ p-2-r_j&\text{if}~i\in I(J)^{2}_j,\\ p-2-r_j-\delta_{j\in J_{\overline{\rho}}}&\text{if}~i\in I(J)^{3}_j,\\ r_j&\text{if}~i\in I(J)^{4}_j.\end{cases}\end{equation}

Then we define

\begin{equation*}\begin{aligned} I(J)^{t+4}_j&\overset{\mathrm{def}}{=} I(J)^t_j\cap\{i\geq0:j\notin(\delta_{{\rm ss}}^i(J))^{\delta}\}&\text{for}~t\in\{ 1,2 \},\\ I(J)^{t+4}_j&\overset{\mathrm{def}}{=} I(J)^t_j\cap\{i\geq0:j\in(\delta_{{\rm ss}}^i(J))^{\delta}\}&\text{for}~t\in\{ 3,4 \}.\end{aligned}\end{equation*}

We also define

\begin{equation*}\begin{aligned} i(J)^{t}_j&\overset{\mathrm{def}}{=}|I(J)^{t}_j|-|I(J^{{\rm ss}})^{t}_j|&\text{for}~t\in\{ 1,2 \},\\ i(J)^{t}_j&\overset{\mathrm{def}}{=}(|I(J)^{t+2}_j|-|I(J^{{\rm ss}})^{t+2}_j|)-(|I(J)^{t+4}_j|-|I(J^{{\rm ss}})^{t+4}_j|)&\text{for}~t\in\{ 3,4 \}.\end{aligned}\end{equation*}

Finally, for $t\in\{ 1,2,5,6,7,8 \}$ we define

\begin{equation*} I^{\circ}(J)^t_j\overset{\mathrm{def}}{=} I(J)^t_j\cap\{ 0 \}.\end{equation*}

If, moreover, $j+1\in J_{\overline{\rho}}$ , we let $k\geq0$ be such that $j+i+1\in J_{\overline{\rho}}$ for $0\leq i\leq k$ and $j+k+2\notin J_{\overline{\rho}}$ . Then we define (for $t\in\{ 1,2,5,6,7,8 \}$ )

\begin{equation*} I^{\geq}(J)^t_j\overset{\mathrm{def}}{=}\{ i\geq0:i+k\in I(J)^t_j \}.\end{equation*}

Proof of Lemma 6.13(i). Recall that $U(J)=u(J)-u(J^{{\rm ss}})-u'(J)$ with u(J) and $u(J^{{\rm ss}})$ defined in (6.17) and u’(J) defined in (6.18). By (6.1) we have

(6.22) \begin{align} u'(J)&=(p-1)^{-1}\sum_{j=0}^{f-1}(p-1-(i^+(\chi_J)_j+(-i(\chi_J)-\underline{s}^J)_j-(-i(\chi_J^s))_j))\nonumber\\ &=(p-1)^{-1}\sum_{j=0}^{f-1}((p-1-i^+(\chi_J)_j)-(p-1-(i(\chi_J)+\underline{s}^J)_j)+(p-1-i(\chi_J^s)_j)). \end{align}

By Lemma 6.9 we have $p-1-i^+(\chi_{J'})_j=i(\chi_{J'}^s)_j$ for $\emptyset\neq J'\subseteq J_{\overline{\rho}}$ and $j\in\mathcal{J}$ . Hence, by (6.2) we have

(6.23) \begin{align}\begin{aligned} u(J)&=(p-1)^{-1}\sum_{j=0}^{f-1}\bigg({\kern-1pt}(p-1-i^+(\chi_J)_j)+\bigg(\sum_{i=1}^{\ell(J)-1}i(\chi_{\delta_{{\rm ss}}^i(J)}^s)_j\!\bigg)-(p-1-i^+(J)_j)\!\bigg),\\ u(J^{{\rm ss}})&=(p-1)^{-1}\sum_{j=0}^{f-1}\bigg(\!\bigg(\sum_{i=0}^{\ell(J^{{\rm ss}})-1}i(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})}^s)_j\!\bigg)-(p-1-i^+(J^{{\rm ss}})_j)\!\bigg).\end{aligned}\end{align}

Moreover, from the proof of Proposition 6.11 we have $i^+(J)\equiv i^+(J^{{\rm ss}})~{\operatorname{mod}}\,(q-1)$ . Combining (6.22) and (6.23), we deduce that

\begin{equation*} U(J)=u(J)-u(J^{{\rm ss}})-u'(J)=(p-1)^{-1}\sum_{j=0}^{f-1}U(J)_j,\end{equation*}

where (for each $j\in\mathcal{J}$ )

(6.24) \begin{equation} U(J)_j\overset{\mathrm{def}}{=}\sum_{i=0}^{\ell(J)-1}i(\chi_{\delta_{{\rm ss}}^i(J)}^s)_j-\sum_{i=0}^{\ell(J^{{\rm ss}})-1}i(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})}^s)_j-(i(\chi_J)+\underline{s}^J)_j\in\mathbb{Z}.\end{equation}

By (5.4), we have

(6.25) \begin{align} \sum_{i=0}^{\ell(J)-1}i(\chi_{\delta_{{\rm ss}}^i(J)}^s)_j&=\sum_{i\geq0}\delta_{j\in(\delta_{{\rm ss}}^i(J))^{\delta}}(p-1-s^{\delta_{{\rm ss}}^{i+1}(J)}_j)=\sum_{i\in I(J)^1_j\sqcup I(J)^2_j}(p-1-s^{\delta_{{\rm ss}}^{i+1}(J)}_j)\nonumber\\ &=|I(J)^1_j|(p-1-r_j-\delta_{j\in J_{\overline{\rho}}})+|I(J)^2_j|(r_j+1),\end{align}

where the last equality follows from (6.21). Similarly, we have

(6.26) \begin{equation} \sum_{i=0}^{\ell(J^{{\rm ss}})-1}i(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})}^s)_j=|I(J^{{\rm ss}})^1_j|(p-1-r_j-\delta_{j\in J_{\overline{\rho}}})+|I(J^{{\rm ss}})^2_j|(r_j+1).\end{equation}

Moreover, by Lemma 6.5 and (2.1) we have

(6.27) \begin{equation} (i(\chi_J)+\underline{s}^J)_j=\delta_{j+1\in J^{{\rm nss}}}(p-1-s^{J^{{\rm ss}}}_j)=\delta_{j+1\in J^{{\rm nss}}}(p-1-r_j-\delta_{j\in J^{{\rm ss}}}).\end{equation}

Combining (6.24)–(6.26) and (6.27), we deduce that

(6.28) \begin{equation} U(J)_j=i(J)^1_j(p-1-r_j-\delta_{j\in J_{\overline{\rho}}})+i(J)^2_j(r_j+1)-\delta_{j+1\in J^{{\rm nss}}}(p-1-r_j-\delta_{j\in J^{{\rm ss}}}).\end{equation}

To compute each $U(J)_j$ explicitly, we separate the following cases.

Case 1. $j+1\in J_{\overline{\rho}}$ . Let $k\geq0$ such that $j+i+1\in J_{\overline{\rho}}$ for $0\leq i\leq k$ and $j+k+2\notin J_{\overline{\rho}}$ . By (4.18) we have

(6.29) \begin{align}\begin{aligned} j+1\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(0\leq i\leq k+1~\text{and}~j+i+1\in J),\\ j+1\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(0\leq i\leq k~\text{and}~j+i+1\in J).\end{aligned}\end{align}

In particular, for $t\in\{ 1,2 \}$ we have $i\in I(J)^{t}_j\Leftrightarrow i\in I(J^{{\rm ss}})^{t}_j$ for $0\leq i\leq k-1$ , hence

(6.30) \begin{equation} |I(J)^t_j|-|I(J^{{\rm ss}})^t_j|=|I^{\geq}(J)^t_j|-|I^{\geq}(J^{{\rm ss}})^t_j|.\end{equation}

We denote $\operatorname{ch}^1_J\overset{\mathrm{def}}{=}(\delta_{j+k+1\in J},\delta_{j+k+2\in J})\in\{ 0,1 \}^2$ . Then from the definition with a case-by-case examination we obtain the following table, where the second and third columns use (6.29), the fourth column uses (6.30), and the last column uses (6.28).

$\operatorname{ch}^1_J$	$I^{\geq}(J)^{1,2^{\vphantom{(}}}_j$	$I^{\geq}(J^{{\rm ss}})^{1,2}_j$	$i(J)^{1,2}_j$	$U(J)_j$
(1,1)	$\{ 1 \},\emptyset$	$\{ 0 \},\emptyset$	0,0	0
(1,0)	$\{ 0 \},\emptyset$	$\{ 0 \},\emptyset$	0,0	0
(0,1)	$\{ 1 \},\{ 0 \}$	$\emptyset,\emptyset$	1,1	$p-\delta_{j\in J_{\overline{\rho}}}$
(0,0)	$\emptyset,\emptyset$	$\emptyset,\emptyset$	0,0	0

We provide a detailed argument for one case. The other cases, as well as the tables in the proof of Lemma 6.13(iii), are similar and straightforward. Suppose that $j+k+1\in J$ and $j+k+2\in J$ . For $i\geq0$ , we deduce from (6.29) that $j+1\in\delta_{{\rm ss}}^{k+i}(J)$ if and only if $i\in\{ 0,1 \}$ . It follows that $k+i\in I(J)^1_j$ if and only if $i=1$ , and that $k+i\notin I(J)^2_j$ for all $i\geq0$ . Hence, $I^{\geq}(J)^1_j=\{ 1 \}$ and $I^{\geq}(J)^2_j=\emptyset$ . Similarly, we deduce from (6.29) that $j+1\in\delta_{{\rm ss}}^{k+i}(J^{{\rm ss}})$ if and only if $i=0$ . It follows that $k+i\in I(J^{{\rm ss}})^1_j$ if and only if $i=1$ , and that $k+i\notin I(J^{{\rm ss}})^2_j$ for all $i\geq0$ . Hence, $I^{\geq}(J^{{\rm ss}})^1_j=\{ 0 \}$ and $I^{\geq}(J^{{\rm ss}})^2_j=\emptyset$ . Then by (6.30) we have $i(J)^1_j=1-1=0$ and $i(J)^2_j=0-0=0$ . Finally, by (6.28) we have $U(J)_j=0+0-0=0$ .

Case 2. $j+1\notin J_{\overline{\rho}}$ . In this case, we have $j+1\notin\delta_{{\rm ss}}^i(J)$ for $i\geq1$ and $j+1\notin\delta_{{\rm ss}}^i(J^{{\rm ss}})$ for $i\geq0$ . Hence, $I(J)^1_j=\{ 0 \}$ if $j+1\in J$ , $I(J)^1_j=\emptyset$ if $j+1\notin J$ , and $I(J)^2_j=I(J^{{\rm ss}})^1_j=I(J^{{\rm ss}})^2_j=\emptyset$ . By (6.28) we have

\begin{equation*}\begin{aligned} U(J)_j&=\delta_{j+1\in J}(p-1-r_j-\delta_{j\in J_{\overline{\rho}}})-\delta_{j+1\in J^{{\rm nss}}}(p-1-r_j-\delta_{j\in J^{{\rm ss}}})\\ &=-\delta_{j+1\in J}(\delta_{j\in J_{\overline{\rho}}}-\delta_{j\in J^{{\rm ss}}})=-\delta_{j+1\in J,\,j\in J_{\overline{\rho}}\setminus J}.\end{aligned}\end{equation*}

As in § 4, we decompose $J_{\overline{\rho}}$ into a disjoint union of intervals (in $\mathbb{Z}/f\mathbb{Z}$ ) not adjacent to each other, $J_{\overline{\rho}}=J_1\sqcup\cdots\sqcup J_t$ , and for each $1\leq i\leq t$ we write $J_i=\{ j_i,j_i+1,\ldots,j_i+k_i \}$ with $j_i\in\mathcal{J}$ and $k_i\geq0$ . Combining Case 1 and Case 2, we get (for each $i\in\mathbb{Z}/t\mathbb{Z}$ )

\begin{equation*} U(J)_j=\begin{cases} p\delta_{j_i+k_i\in\partial(J^c)}&\text{if}~j=j_i-1,\\ (p-1)\delta_{j_i+k_i\in\partial(J^c)}&\text{if}~j=j_i,j_i+1,\ldots,j_i+k_i-1,\\ -\delta_{j_i+k_i\in\partial(J^c)}&\text{if}~j=j_i+k_i,\\ 0&\text{if}~j=j_i+k_i+1,j_i+k_i+2,\ldots,j_{i+1}-2.\end{cases}\end{equation*}

Hence, we conclude that

\begin{equation*}\begin{aligned} U(J)&=(p-1)^{-1}\sum_{j=0}^{f-1}U(J)_j=(p-1)^{-1}\sum_{i=1}^{t}\bigg(\sum_{j=j_i-1}^{j_{i+1}-2}U(J)_j\!\bigg)\\ &=(p-1)^{-1}\sum_{i=1}^{t}(\delta_{j_i+k_i\in\partial(J^c)}(p+k_i(p-1)-1))=\sum_{i=1}^{t}(\delta_{j_i+k_i\in\partial(J^c)}(k_i+1)),\end{aligned}\end{equation*}

which equals $A^{{\rm ss}}(J)$ by (4.15).

Proof of Lemma 6.13(ii). As in (6.5) we have

(6.31) \begin{equation} \mu_{J^s,J}=(-1)^{\underline{t}^{J^s}+\sum_{j+1\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+\delta_{j\in J^{{\rm nss}}})}\Bigg[\frac{\prod_{j+1\notin J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+\delta_{j\in J^{{\rm nss}}})!}{\prod_{j+1\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+\delta_{j\notin J^{{\rm nss}}})!}\Bigg].\end{equation}

By (5.6) and using (3.6), we have

(6.32) \begin{equation} P_2(J)=(-1)^{\sum_{j+1\notin J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+1)}\bigg(\prod_{j+1\notin J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j)!{\kern-1pt}\bigg)^{\!{-}1}.\end{equation}

By Lemma 6.5 and using $(p-1)!\equiv-1~{\operatorname{mod}}\,p$ , we have

(6.33) \begin{equation} \prod_{j=0}^{f-1}((-i(\chi_J)-\underline{s}^J)_j)!=(-1)^{f-|J^{{\rm nss}}|}\prod_{j+1\in J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j)!\end{equation}

Combining (6.31), (6.32) and (6.33), we deduce that

(6.34) \begin{align} \alpha'(J)&=(-1)^{|J\cap(J-1)^{{\rm nss}}|+\underline{t}^J}\mu_{J^s,J}P_2(J)\bigg/\bigg(\prod_{j=0}^{f-1}((-i(\chi_J)-\underline{s}^J)_j)!{\kern-1pt}\bigg)\nonumber\\ &=(-1)^{d}\Bigg[\frac{\prod_{j\notin J^{{\rm nss}},j+1\notin J^{{\rm nss}}}(1)\prod_{j\in J^{{\rm nss}},j+1\notin J^{{\rm nss}}}(s^{J^{{\rm ss}}}_j+1)}{\prod_{j\notin J^{{\rm nss}},j+1\in J^{{\rm nss}}}(-(s^{J^{{\rm ss}}}_j)!(s^{J^{{\rm ss}}}_j+1)!)\prod_{j\in J^{{\rm nss}},j+1\in J^{{\rm nss}}}((s^{J^{{\rm ss}}}_j)!)^2}\Bigg],\end{align}

where (using Lemma 2.3)

\begin{equation*}\begin{aligned} d&=|J\cap(J-1)^{{\rm nss}}|+\underline{t}^J+\underline{t}^{J^s}+\underline{s}^{J^{{\rm ss}}}+\underline{1}+f-|J^{{\rm nss}}|\\ &\equiv|J^{{\rm nss}}|-|J\cap(J-1)^{{\rm nss}}|=|(\partial J)^{{\rm nss}}|\quad{\operatorname{mod}}\,2.\end{aligned}\end{equation*}

Moreover, a case-by-case examination using (2.1) shows that each term on the right-hand side of (6.34) agrees with the corresponding $\alpha'(J)_j$ defined in (6.20). For example, in the case $j\in J^{{\rm nss}}$ and $j+1\notin J^{{\rm nss}}$ , we deduce from (2.1) that $s^{J^{{\rm ss}}}_j+1=r_j+1$ if $j+1\notin J$ and $s^{J^{{\rm ss}}}_j+1=p-1-r_j$ if $j+1\in J^{{\rm ss}}$ . Hence, we conclude that $\alpha'(J)=(-1)^{|(\partial J)^{{\rm nss}}|}\prod_{j=0}^{f-1}\alpha'(J)_j$ .

Proof of Lemma 6.13(iii). For $J'\neq J^*$ , by (5.4) and using (3.6), we have

(6.35) \begin{align}\begin{aligned} \prod_{j=0}^{f-1}(p-1-i(\chi_{J'}^s)_j)!&=(-1)^{f-|J^{\prime\delta}|}\bigg(\prod_{j+1\in J^{\prime\delta}}(s^{(J'-1)^{{\rm ss}}}_j)!{\kern-1pt}\bigg),\\ P_1(\chi_{J'})&=(-1)^{\sum_{j+1\in J^{\prime\delta}}(s^{(J'-1)^{{\rm ss}}}_j+1)}\bigg(\prod_{j+1\in J^{\prime\delta}}(s^{(J'-1)^{{\rm ss}}}_j)!{\kern-1pt}\bigg)^{\!{-}1}.\end{aligned}\end{align}

Combining (6.35) with Lemma 6.4, we deduce that

\begin{equation*}\begin{aligned} \alpha(\chi_{J'})&=J(i(\chi_{J'}^s),-\underline{s}^J)((-1)^{f-1}\prod_{j=0}^{f-1}(p-1-i(\chi_{J'}^s))!)/P_1(\chi_{J'})\\ &=\Bigg[\frac{\prod_{j+1\in J^{\prime\delta},j\notin J^{\prime\delta}}(-1)(s^{(J'-1)^{{\rm ss}}}_j\!+1)}{\prod_{j+1\notin J^{\prime\delta},j\in J^{\prime\delta}}(s^{(J'-1)^{{\rm ss}}}_j\!+1)}\Bigg]\bigg(\prod_{j+1\in J^{\prime\delta}}(s^{(J'-1)^{{\rm ss}}}_j)!{\kern-1pt}\bigg)^{\!2}.\end{aligned}\end{equation*}

Then we have (using (6.21))

\begin{align*}&\prod_{i=0}^{\ell(J)-1}\alpha(\chi_{\delta_{{\rm ss}}^i(J)})=\Bigg[\frac{\prod_{i\in I(J)^{5}_j\sqcup I(J)^{6}_j}(-1)(s^{\delta_{{\rm ss}}^{i+1}(J)}_j+1)}{\prod_{i\in I(J)^{7}_j\sqcup I(J)^{8}_j}(s^{\delta_{{\rm ss}}^{i+1}(J)}_j+1)}\Bigg]\bigg(\prod_{i\in I(J)^{1}_j\sqcup I(J)^{2}_j}(s^{\delta_{{\rm ss}}^{i+1}(J)}_j)!{\kern-1pt}\bigg)^{\!2}\\&\quad =\bigg[\frac{(p-1-r_j-\delta_{j\in J_{\overline{\rho}}})^{|I(J)^{5}_j|}(r_j+1)^{|I(J)^{6}_j|}}{(p-1-r_j-\delta_{j\in J_{\overline{\rho}}})^{|I(J)^{7}_j|}(r_j+1)^{|I(J)^{8}_j|}}\bigg]((r_j+\delta_{j\in J_{\overline{\rho}}})!)^{2|I(J)^{1}_j|}((p-2-r_j)!)^{2|I(J)^{2}_j|}.\end{align*}

A similar formula holds with each J replaced with $J^{{\rm ss}}$ . Hence, we have

(6.36) \begin{equation} \alpha(J)=\bigg(\prod_{i=0}^{\ell(J)-1}\alpha(\chi_{\delta_{{\rm ss}}^i(J)})\!\bigg)\!\bigg/\!\bigg(\prod_{i=0}^{\ell(J^{{\rm ss}})-1}\alpha(\chi_{\delta_{{\rm ss}}^i(J^{{\rm ss}})})\!\bigg)=\prod_{j=0}^{f-1}\alpha(J)_j,\end{equation}

where (for each $j\in\mathcal{J}$ )

(6.37) \begin{equation} \alpha(J)_j\overset{\mathrm{def}}{=}((r_j+\delta_{j\in J_{\overline{\rho}}})!)^{2i(J)^{1}_j}((p-2-r_j)!)^{2i(J)^{2}_j}(p-1-r_j-\delta_{j\in J_{\overline{\rho}}})^{i(J)^{3}_j}(r_j+1)^{i(J)^{4}_j}\in\mathbb{F}^{\times}.\end{equation}

To compute each $\alpha(J)_j$ explicitly, we separate the following cases.

Case 1. $j+1\in J_{\overline{\rho}},~j\in J_{\overline{\rho}}$ . Let $k\geq0$ such that $j+i+1\in J_{\overline{\rho}}$ for $0\leq i\leq k$ and $j+k+2\notin J_{\overline{\rho}}$ . By (4.18) we have

(6.38) \begin{align}\begin{aligned} j+1\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(0\leq i\leq k+1~\text{and}~j+i+1\in J),\\ j\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(0\leq i\leq k+2~\text{and}~j+i\in J),\\ j+1\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(0\leq i\leq k~\text{and}~j+i+1\in J),\\ j\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(0\leq i\leq k+1~\text{and}~j+i\in J).\end{aligned}\end{align}

In particular, for $t\in\{ 1,2,5,6,7,8 \}$ we have $i\in I(J)^{t}_j\Leftrightarrow i\in I(J^{{\rm ss}})^{t}_j$ for $0\leq i\leq k-1$ , hence

(6.39) \begin{equation} |I(J)^t_j|-|I(J^{{\rm ss}})^t_j|=|I^{\geq}(J)^t_j|-|I^{\geq}(J^{{\rm ss}})^t_j|.\end{equation}

We denote $\operatorname{ch}^2_J\overset{\mathrm{def}}{=}(\delta_{j+k\in J},\delta_{j+k+1\in J},\delta_{j+k+2\in J})\in\{ 0,1 \}^3$ . Then from the definition with a case-by-case examination we obtain the following table, where the second and third columns use (6.38), the fourth column uses (6.39), and the last column follows from (6.37) using the equality $((r_j+1)!(p-2-r_j)!)^2=1$ in $\mathbb{F}$ .

$\operatorname{ch}^{2}_J$	$I^{\geq}(J)^{1,2,5,6,7,8^{\vphantom{(}}}_j$	$I^{\geq}(J^{{\rm ss}})^{1,2,5,6,7,8}_j$	$i(J)^{1,2,3,4}_j$	$\alpha(J)_j$
(1,1,1)	$\{ 1 \},\emptyset,\{ 1 \},\emptyset,\emptyset,\{ 2 \}$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\{ 1 \}$	0,0,0,0	1
(0,1,1)	$\{ 1 \},\emptyset,\{ 1 \},\emptyset,\{ 0 \},\{ 2 \}$	$\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset,\{ 1 \}$	0,0,0,0	1
(1,1,0)	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\{ 1 \}$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\{ 1 \}$	0,0,0,0	1
(0,1,0)	$\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset,\{ 1 \}$	$\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset,\{ 1 \}$	0,0,0,0	1
(1,0,1)	$\{ 1 \},\{ 0 \},\emptyset,\emptyset,\emptyset,\{ 2 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	1,1,0,0	1
(0,0,1)	$\{ 1 \},\{ 0 \},\emptyset,\{ 0 \},\emptyset,\{ 2 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	1,1,0,0	1
(1,0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	0,0,0,0	1
(0,0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	0,0,0,0	1

In particular, since $\alpha'(J)_j=1$ by (6.20), we deduce that $\alpha(J)_j\alpha'(J)_j=1$ .

Case 2. $j+1\in J_{\overline{\rho}},~j\notin J_{\overline{\rho}}$ . Let $k\geq0$ such that $j+i+1\in J_{\overline{\rho}}$ for $0\leq i\leq k$ and $j+k+2\notin J_{\overline{\rho}}$ . By (4.18) we have

(6.40) \begin{align}\begin{aligned} j+1\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(0\leq i\leq k+1~\text{and}~j+i+1\in J),\\ j\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(i=0~\text{and}~j\in J),\\ j+1\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(0\leq i\leq k~\text{and}~j+i+1\in J),\\ j\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(\text{impossible}).\end{aligned}\end{align}

We denote $\operatorname{ch}^3_J\overset{\mathrm{def}}{=}(\delta_{j\in J},\delta_{j+1\in J},\delta_{j+2\in J})\in\{ 0,1 \}^3$ .

If $k=0$ , then from the definition with a case-by-case examination we obtain the following table, where the second and third columns use (6.40), and the last column follows from (6.37) using the equality $(r_j!(p-2-r_j)!)^2=(r_j+1)^{-2}$ in $\mathbb{F}$ .

$\operatorname{ch}^{3}_J$	$I(J)^{1,2,5,6,7,8^{\vphantom{(}}}_j$	$I(J^{{\rm ss}})^{1,2,5,6,7,8}_j$	$i(J)^{1,2,3,4}_j$	$\alpha(J)_j$
(1,1,1)	$\{ 1 \},\emptyset,\{ 1 \},\emptyset,\{ 0 \},\emptyset$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	$0,0,-1,0$	$(p-1-r_j)^{-1}$
(0,1,1)	$\{ 1 \},\emptyset,\{ 1 \},\emptyset,\emptyset,\emptyset$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	0,0,0,0	1
(1,1,0)	$\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	$0,0,-1,0$	$(p-1-r_j)^{-1}$
(0,1,0)	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	0,0,0,0	1
(1,0,1)	$\{ 1 \},\{ 0 \},\{ 1 \},\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	1,1,1,0	$(p-1-r_j)^{-1}$
(0,0,1)	$\{ 1 \},\{ 0 \},\{ 1 \},\{ 0 \},\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	1,1,1,1	$-1$
(1,0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$0,0,0,-1$	$(r_j+1)^{-1}$
(0,0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	0,0,0,0	1

If $k\geq1$ , then for $t\in\{ 1,2,5,6,7,8 \}$ we have $i\in I(J)^{t}_j\Leftrightarrow i\in I(J^{{\rm ss}})^{t}_j$ for $1\leq i\leq k-1$ , hence

(6.41) \begin{equation} |I(J)^t_j|-|I(J^{{\rm ss}})^t_j|=(|I^{\circ}(J)^t_j|-|I^{\circ}(J^{{\rm ss}})^t_j|)+(|I^{\geq}(J)^t_j|-|I^{\geq}(J^{{\rm ss}})^t_j|).\end{equation}

By (6.40) and a case-by-case examination, we have

(6.42) \begin{align}\begin{aligned} I^{\geq}(J)^{1,2,5,6,7,8}_j&=\begin{cases} \{ 1 \},\{ 0 \},\{ 1 \},\{ 0 \},\emptyset,\emptyset&\text{if}~j+k+1\notin J,~j+k+2\in J\\ \emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset&\text{otherwise},\end{cases}\\ I^{\geq}(J^{{\rm ss}})^{1,2,5,6,7,8}_j&=\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset.\end{aligned}\end{align}

If $(\delta_{j+k+1\in J},\delta_{j+k+2\in J})\neq(0,1)$ , then from the definition with a case-by-case examination we obtain the following table, where the second and third columns use (6.40), the fourth column uses (6.41) and (6.42), and the last column follows from (6.37).

$\operatorname{ch}^{3}_J$	$I^{\circ}(J)^{1,2,5,6,7,8^{\vphantom{(}}}_j$	$I^{\circ}(J^{{\rm ss}})^{1,2,5,6,7,8}_j$	$i(J)^{1,2,3,4}_j$	$\alpha(J)_j$
(1,1,1)	$\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \},\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$0,0,-1,0$	$(p-1-r_j)^{-1}$
(0,1,1)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	0,0,0,0	1
(1,1,0)	$\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	$0,0,-1,0$	$(p-1-r_j)^{-1}$
(0,1,0)	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	0,0,0,0	1
(1,0,1)	$\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset$	$0,0,0,-1$	$(r_j+1)^{-1}$
(0,0,1)	$\emptyset,\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset$	$\emptyset,\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset$	0,0,0,0	1
(1,0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$0,0,0,-1$	$(r_j+1)^{-1}$
(0,0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	0,0,0,0	1

If $(\delta_{j+k+1\in J},\delta_{j+k+2\in J})=(0,1)$ , then we have a similar table with the first three columns unchanged. By (6.41) and (6.42), each number of the fourth column should be added by 1. Then by (6.37) and using the equality $(r_j!(p-2-r_j)!)^2(r_j+1)(p-1-r_j)=-1$ in $\mathbb{F}$ , each $\alpha(J)_j$ of the last column should be multiplied by $(-1)$ .

Combining the cases $k=0$ and $k\geq1$ and using the definition of $\alpha'(J)_j$ in (6.20), we conclude that $\alpha(J)_j\alpha'(J)_j=-1$ if $j+k+1\notin J$ and $j+k+2\in J$ , and $\alpha(J)_j\alpha'(J)_j=1$ otherwise.

Case 3. $j+1\notin J_{\overline{\rho}},~j\notin J_{\overline{\rho}}$ . In this case, by (4.18) we have $j,j+1\notin\delta_{{\rm ss}}^i(J)$ for all $i\geq1$ and $j,j+1\notin\delta_{{\rm ss}}^i(J^{{\rm ss}})$ for all $i\geq0$ . We denote $\operatorname{ch}^4_J\overset{\mathrm{def}}{=}(\delta_{j\in J},\delta_{j+1\in J})\in\{ 0,1 \}^2$ . Then from the definition with a case-by-case examination we obtain the following table.

$\operatorname{ch}^{4}_J$	$I(J)^{1,2,5,6,7,8^{\vphantom{(}}}_j$	$I(J^{{\rm ss}})^{1,2,5,6,7,8}_j$	$i(J)^{1,2,3,4}_j$	$\alpha(J)_j$
(1,1)	$\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	1,0,0,0	$(r_j!)^2$
(1,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$0,0,0,-1$	$(r_j+1)^{-1}$
(0,1)	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	1,0,1,0	$(r_j!)^2(p-1-r_j)$
(0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	0,0,0,0	1

Together with (6.20), we conclude that $\alpha(J)_j\alpha'(J)_j=1$ .

Case 4. $j+1\notin J_{\overline{\rho}},~j\in J_{\overline{\rho}}$ . In this case, by (4.18) we have

(6.43) \begin{align}\begin{aligned} j+1\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(i=0~\text{and}~j+1\in J),\\ j\in\delta_{{\rm ss}}^i(J)&\Leftrightarrow(i\in\{ 0,1 \}~\text{and}~j+i\in J),\\ j+1\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(\text{impossible}),\\ j\in\delta_{{\rm ss}}^i(J^{{\rm ss}})&\Leftrightarrow(i=0~\text{and}~j\in J).\end{aligned}\end{align}

Then from the definition with a case-by-case examination we obtain the following table, where the second and third columns use (6.43) and the last column follows from (6.37).

$\operatorname{ch}^{4}_J$	$I(J)^{1,2,5,6,7,8^{\vphantom{(}}}_j$	$I(J^{{\rm ss}})^{1,2,5,6,7,8}_j$	$i(J)^{1,2,3,4}_j$	$\alpha(J)_j$
(1,1)	$\{ 0 \},\emptyset,\{ 0 \},\emptyset,\emptyset,\{ 1 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	1,0,1,0	$((r_j+1)!)^2(p-2-r_j)$
(1,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{ 0 \}$	0,0,0,0	1
(0,1)	$\{ 0 \},\emptyset,\emptyset,\emptyset,\emptyset,\{ 1 \}$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$1,0,0,-1$	$((r_j+1)!)^2(r_j+1)^{-1}$
(0,0)	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	$\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset$	0,0,0,0	1

Together with (6.20), we conclude that $\alpha(J)_j\alpha'(J)_j=-1$ if $j\notin J,~j+1\in J$ , and $\alpha(J)_j\alpha'(J)_j=1$ otherwise.

To give a summary of Cases 1–4, we write $J_{\overline{\rho}}=J_1\sqcup\cdots\sqcup J_t$ as in § 4 where each $J_i=\{ j_i,j_i+1,\ldots,j_i+k_i \}$ with $j_i\in\mathcal{J}$ and $k_i\geq0$ . Then we have (for each $i\in\mathbb{Z}/t\mathbb{Z}$ )

\begin{equation*} \alpha(J)_j\alpha'(J)_j=\begin{cases} (-1)^{\delta_{j_i+k_i\in\partial(J^c)}}&\text{if}~j=j_i-1,\\ 1&\text{if}~j=j_i,j_i+1,\ldots,j_i+k_i-1,\\ (-1)^{\delta_{j_i+k_i\in\partial(J^c)}}&\text{if}~j=j_i+k_i,\\ 1&\text{if}~j=j_i+k_i+1,j_i+k_i+2,\ldots,j_{i+1}-2.\end{cases}\end{equation*}

Therefore, we conclude from (6.36) that

\begin{equation*} \alpha(J){\prod_{j=0}^{f-1}\alpha'(J)_j}=\prod_{j=0}^{f-1}(\alpha(J)_j\alpha'(J)_j)=(-1)^{2\#\{ j:j+1\notin J_{\overline{\rho}},~j+1\in J,~j\in J_{\overline{\rho}},~j\notin J \}}=1,\end{equation*}

which completes the proof.