\begin{align*} [a]_{f,f}(\alpha) \begin{cases} =0, &a\in \mfr^n_K;\\ \in \Lambda_{f,n-i}\setminus\Lambda_{f,n-i-1}, &a\in \mfr^i_K\setminus \mfr^{i+1}_K, 0\le i\le n-1, \end{cases}\\ \forall a\in \ocal_K,\alpha\in \Lambda_{f,n}\setminus \Lambda_{f,n-1}. \end{align*}

\begin{equation*} \begin{cases} \Phi^{L'}_{\pi,n}(r+\mfr^n_K) &= \Phi^L_{\pi,n}(r+\mfr^n_K)|_{L_{\pi,n}}, \quad\forall r\in \ocal_K; \\ \sigma^{L'}_{\pi,\pi_0,n} &= \sigma^L_{\pi,\pi_0,n}|_{L_{\pi,n}}; \\ \vp^{L'}_{\pi,n}(x) &= \vp^L_{\pi,n}(x)|_{L_{\pi,n}}, \quad\forall x\in K^\times, \end{cases} \quad\forall n\in\mathbb{N}^* \end{equation*} 和 \begin{equation*} \begin{cases} \Phi^{L'}_\pi(r+\mfr^n_K) &= \Phi^L_\pi(r+\mfr^n_K)|_{L_\pi}, \quad\forall r\in \ocal_K; \\ \sigma^{L'}_{\pi,\pi_0} &= \sigma^L_{\pi,\pi_0}|_{L_\pi}; \\ \vp^{L'}_\pi(x) &= \vp^L_\pi(x)|_{L_\pi}, \quad\forall x\in K^\times. \end{cases} \end{equation*}

\begin{equation*} \begin{cases} f'(H_{k+1}) &= u\pi H_{k+1} + O(2(k+1)), \\ H^\sigma_{k+1} (f(X_1),\cdots,f(X_n)) &= H^\sigma_{k+1} (\pi X_1,\cdots,\pi X_n) + O(2(k+1)) \\ &= \pi^{k+1} H^\sigma_{k+1} + O(2(k+1)), \end{cases} \end{equation*}

\begin{equation*} \begin{cases} \Phi_L(\vp)\overset{\operatorname{def}}{=}\vp(1+I): \operatorname{End}_A((A/I)_L)\to (A/I)^\mathrm{rev},\\ \Phi_R(\vp)\overset{\operatorname{def}}{=}\vp(1+I): \operatorname{End}_A((A/I)_R)\to A/I, \end{cases} \end{equation*}

\begin{equation*} \begin{cases} \operatorname{Im} \vp_{\pi,n} &= \{\tau\in \operatorname{Aut}(L_{\pi,n}/K): \tau|_L\in \sigma^\mathbb{Z}\},\\ \ker \vp_{\pi,n} &=\begin{cases} (1+\mfr^n_K)\times \operatorname{Nm}_{K_m/K}(\pi)^\mathbb{Z}, &L = K_m;\\ 1+\mfr^n_K, &L = \Khat. \end{cases} \end{cases} \end{equation*}

\begin{equation*} \alpha(e_i\alpha^j) = \begin{cases} e_i\alpha^{j+1}, &0\le j\le n-2;\\ -(a_0e_i+a_1e_i\alpha+\cdots+a_{n-1}e_i\alpha^{n-1}), &j=n-1. \end{cases} \end{equation*}

\begin{equation*} \Gal(L_{\pi,n}/L)_j\cong \begin{cases} \Gal(L_{\pi,n}/L), &j=0; \\ (1+\mfr^l_K)/(1+\mfr^n_K), &q^{l-1}\le j\le q^l-1, 1\le l\le n-1; \\ \operatorname{id}_{L_{\pi,n}}, &j\ge q^{n-1}, \end{cases} \end{equation*}

\begin{equation*} G_n\cap \sigma^\mathbb{Z} = \begin{cases} \sigma^\mathbb{Z}, &1\le n \le n_1,\\ \sigma^{p^l \mathbb{Z}}, &n_{p^{l-1}}+1 \le n \le n_{p^l}, 1\le l\le k-1,\\ \{\operatorname{id}_L\}, &n \ge n_{p^{k-1}}+1; \end{cases} \end{equation*}

\begin{equation*} G_i = \begin{cases} \sigma^\mathbb{Z}, &1\le i \le i_1;\\ \sigma^{p^l \mathbb{Z}}, &i_{p^{l-1}}+1 \le i \le i_{p^l}, 1\le l\le k-1;\\ \{\operatorname{id}_L\}, &i \ge i_{p^{k-1}}+1, \end{cases} \end{equation*}

\begin{equation*} \#\Gal(L_{\pi,n}/L)_j = \begin{cases} q^{n-1} (q-1), &j=0; \\ q^{n-l}, &q^{l-1}\le j\le q^l-1, 1\le l\le n-1; \\ 1, &j\ge q^{n-1}. \end{cases} \end{equation*}

\label{prod}为方便起见, 即使$A$无单位元, 对$a\in A$和$\bm{k}\in\mathbb{N}^n$, 我们同样使用记号$a \displaystyle{\prod^{n}_{i=1}} X^{k_i}_i$, 此时它只是形式记号.