\[
  (1)\qquad B_{K'} \longrightarrow B_{K^{*}}
\]

\[
  (\uncertain{2})\qquad \Pi_{K'} \longrightarrow \Pi_{K^{*}} .
\]

\[
  (3)\qquad E_{\overline{K}'/K'} \longrightarrow E_{\overline{K}^{*}/K^{*}}
\]

\[
  (\uncertain{4})\qquad
  \begin{cases}
    \Pi_K \simeq \varprojlim \Pi_{K_i} \\
    B_K \simeq \varprojlim B_{K_i} \\
    E_{\overline{K}/K} \xrightarrow{\ \sim\ } \varprojlim E_{\overline{K}_i/K_i}
  \end{cases}
\]

\[
  \operatorname{Ind}(\text{corps t.f.}) \longrightarrow \text{corps}
\]

\[
  (5)\qquad \Gamma_{\mathbb{Q}} = E_{\overline{\mathbb{Q}}_0/\mathbb{Q}}
\]

\[
  (6)\qquad B_K \longrightarrow B_{\mathbb{Q}}, \qquad \Pi_K \longrightarrow \Pi_{\mathbb{Q}}
\]

\[
  (7)\qquad E_{\overline{K}/K} \longrightarrow \Gamma_{\overline{\mathbb{Q}}/\mathbb{Q}} .
\]

\[
  (9)\qquad K = \varinjlim A_i
\]

\[
  \operatorname{Spec} K = \varprojlim U_i
\]

\[
  (10)\qquad E_{\overline{K}/K} = \pi_1(\eta, \bar\eta) \xrightarrow{\ \sim\ } \varprojlim_i E_i
  \qquad (\Gamma_i = \pi_1(U_i, \bar\eta))
\]

\[
  (11)\qquad \overline{U}_i = U_i \otimes_k \overline{\mathbb{Q}}
\]

\[
  (12)\qquad 1 \to \underbrace{\pi_1(\overline{U}_i, \bar\eta)}_{\pi_i}
  \to \underset{\substack{\| \\ E_i}}{\pi_1(U_i, \bar\eta)}
  \to \underset{\substack{\| \\ \Gamma_{\overline{K}/K} = \Gamma_{\overline{\mathbb{Q}}/k}}}{\pi_1(k, \bar\eta)}
  \to 1
\]

\[
  \pi_i = \pi_1(\overline{U}_i, \bar\eta)
\]

\[
  \pi_{\overline{K}/K} \simeq \varprojlim_i \underbrace{\pi_1(\overline{U}_i, \bar\eta)}_{\pi_i}
  \simeq \varprojlim_i \pi_1(U_i(\mathbb{C}), \bar\eta)^{\wedge}
\]

\[
  (\uncertain{14})\qquad \operatorname{dim\,coh} \pi_{\overline{K}/K} \leq n, \qquad
  \operatorname{dim\,coh}_\ell E_{\overline{K}/K} \leq n+2 \ \text{si}\ \ell \neq 2
  \quad (\ell\ \text{nb premier})
\]

\[
  \varphi : \Gamma \longrightarrow \operatorname{Autext}(\pi) .
\]

\[
f^{*}, g^{*} : \mathcal{E}_{\overline{K'}/K'} \rightrightarrows \mathcal{E}_{\overline{K}/K}
\]

\[
\pi_K = \varprojlim_i \pi_1(\overline{U}_i, \overline{\eta}), \qquad \text{où } \overline{U}_i = U_i \otimes_{k} \overline{\mathbb{Q}}
\]

\[
\pi_1(f_i^{*}) = \pi_1(g_i^{*}) : \pi_{K'} \longrightarrow \pi_1(\overline{U}_i)
\]

\[
\pi_{K'} = \varprojlim_j \pi_1(\overline{V}_j, \overline{\eta}'), \qquad \text{avec } \overline{V}_j = \operatorname{Spec}(A_j) \otimes_{k} \overline{\mathbb{Q}} .
\]

\[
X \xrightarrow{\ \mathrm{can}\ } J \xrightarrow{\ \varphi\ } G,
\]

\[
\pi_1(X) \xrightarrow{\ \pi_1(f')\ } \pi_1(U) \xrightarrow{\ \pi_1(j)\ } \pi_1(Y)
\]

\[
\pi_1(X) \xrightarrow{\ \pi_1(f')\ } \pi_1(U) \xrightarrow{\ \pi_1(j')\ } \pi_1(Y)
\]

\[
\pi_1(j')(\gamma) = \operatorname{int}(\alpha)\, \pi_1(j)(\gamma)
\]

\[
(16) \qquad \underset{H^0(K, G)}{G(K)} \longrightarrow H^1(K, H_1(\overline{G}))
\]

\[
0 \to {}_nG \to G \xrightarrow{\ n\,\mathrm{id}_G\ } G \to 1
\]

\[
0 \to H^0(K, G)_n \to H^1(K, {}_nG) \to {}_nH^1(K, G) \to 0
\]

\[
0 \to \varprojlim_n H^0(K, G)_n \to \varprojlim_n H^1(K, {}_nG) \to \varprojlim_n {}_nH^1(K, G)
\]

\[
H^1(K, H_1(\overline{G})) \to \varprojlim_n H^1(K, {}_nG)
\]

\[
H_1(\overline{G}) = \varprojlim_n {}_nG
\]

\[
H^0(K) \longrightarrow \varprojlim H^0(K, G)_n \hookrightarrow \varprojlim H^1(K, {}_nG)
\]

\[
H_1(\alpha j) = H_1(\tau_u)\, H_1(\alpha i)
\]

\[
\widehat{\mathbb{Z}}^J / \widehat{\mathbb{Z}} \ (\text{diagonale})
\]

\[
  E_{K'} \longrightarrow \Gamma_{\overline{\mathbf{Q}}/\mathbf{Q}}
  \xrightarrow{\ \text{caractère cyclotomique}\ } \widehat{\mathbf{Z}}^{*}\ ).
\]

\[
  H_1(B_{\overline{U}}, \mathbf{Z}_\ell) \longrightarrow
  H_1(B_{\overline{V}}, \mathbf{Z}_\ell)
\]

\[
  \pi_U^{\Gamma'^{o}} = \{1\}.
\]

\[
  (1)\qquad 1 \longrightarrow \pi_{U/K} \longrightarrow E_U \longrightarrow
  E_K \longrightarrow 1 .
\]

\[
  U(K) \longrightarrow
  \begin{array}{l}
    \text{classes d'isomorphie de sections de } B_U \text{ sur } B_K,\\
    \text{i.e.\ classes de } \pi_{\overline{U}/K}\text{-conjugaison}\\
    \text{de sections de } E_U \text{ sur } E_K
  \end{array}
\]

\[
  (2)\qquad 1 \longrightarrow L_i \longrightarrow Z(L_i) \longrightarrow
  \Gamma \longrightarrow 1,
  \qquad L_i \simeq T_\infty(\overline{K}^{*}) = T,
\]

\[
  (3)\qquad H^1(\Gamma, T) \simeq \varprojlim_n H^1(K, {}_n\mu) \simeq
  \widehat{K^{*}} = \varprojlim_n K^{*}/K^{*n}
\]

\[
  (3)\qquad \pi^{\Gamma^{o}} = L_i
\]

\[
  L_i = \pi^{\Gamma^{o}}
\]

\[
  \pi^{\Gamma^{o}} \subset \operatorname{Norm}_{\pi}(L_i) = L_i
\]

\[
(4)\qquad \mathcal{E}_{\mathcal{U}_S} \longrightarrow \mathcal{E}_S
\]

\[
(6)\qquad K_1 = \mathbb{Q}(\text{\struck{$X$}}\, T_1)
\]

\[
\pi_1(\mathcal{Y}) \longrightarrow \pi_1(\mathcal{U}).
\]

\[
T_\ell(\widehat{K}^{*}) \simeq H_1(\mathcal{Y}, \mathbb{Z}_\ell) \longrightarrow H_1(\mathcal{U}, \mathbb{Z}_\ell) \;\bigl[\simeq H_1(\mathcal{U}_s, \mathbb{Z}_\ell)\bigr]
\]

\[
(11)\qquad T_\ell\,(\simeq H_1(\mathcal{Y}, \mathbb{Z}_\ell)) \longrightarrow \mathbb{T}_\ell^{I} / T_\ell
\]

\[
\text{\struck{$\operatorname{Ker}\bigl(J^{1}_{\mathcal{U}/S} \to J^{1}_{X/S}\bigr) \simeq \mathbb{G}_m^{I} / \mathbb{G}_m$}}
\]

\[
\text{\struck{$H_1(\mathcal{U}, \mathbb{Z}_\ell) \xrightarrow{\sim} H_1(J^{1}_{\mathcal{U}/S}, \mathbb{Z}_\ell) \simeq \varprojlim {}_{\ell^n}(J^{0}_{\mathcal{U}/S})$}}
\]

\[
V / g_{i_0}^{*}(J_f) \simeq A / (J_{g_{i_0}} + J_f)
\]

\[
T_\ell \longrightarrow T_\ell^{I} / T_\ell
\]

\[
(12)\qquad k_{i_0} : T \;(= T_{\infty}(k)) \longrightarrow \pi_1(\mathcal{U}_k)
\]

\[
1 \longrightarrow \pi \longrightarrow \mathcal{E}_{\mathcal{U}} \longrightarrow \mathcal{E}_S \longrightarrow 1
\]

\[
(13)\qquad 1 \longrightarrow T \longrightarrow N(L_i) \longrightarrow \mathcal{E}_S \longrightarrow 1 .
\]

\[
(14)\qquad c_i \in H^2(\mathcal{E}_S, T), \qquad T = \varprojlim \mu_n ,
\]

\[
(15)\qquad H^2(\mathcal{E}_S, T) \simeq H^2(S, T) \simeq \varprojlim_n H^2(S, \mu_n) ;
\]

\[
(16)\qquad 0 \to \operatorname{Pic}(S)/n \to H^2(S, \mu_n) \to {}_nH^2(S, \mathbb{G}_m) \to 0
\]

\[
(17)\qquad 0 \to \operatorname{Pic}(S)^{\wedge} \to H^2(S, T) \to \varprojlim {}_nH^2(S, \mathbb{G}_m)
\]

\[
(18)\qquad \operatorname{Pic}(S) \longrightarrow \operatorname{Pic}(S)^{\wedge} \hookrightarrow H^2(S, T)
\]

\[
(19)\qquad \xi_i \in \operatorname{Pic}(S)
\]

\[
  (20)\qquad H^1(S, T) \simeq \varprojlim_n H^1(S, \mu_n) .
\]

\[
  0 \to \mathbb{G}_m(S)/\mathbb{G}_m(S)^n \to H^1(S, \mu_n) \to {}_n\mathrm{Pic}(S) \to 0
\]

\[
  (21)\qquad 0 \to \mathbb{G}_m(S)^{\wedge} \to H^1(S, T) \to \varprojlim_n {}_n\mathrm{Pic}\, S .
\]

\[
  (7)\qquad 1 \to I_{\tilde{x}} \to N_{\tilde{x}} \to \operatorname{Gal}(\overline{k(x)}/k(x)) \to 1 ,
\]

\[
  (8)\qquad I_{\tilde{x}} \simeq T_{\infty}\bigl(\overline{k(x)}^{*}\bigr) \simeq T_{\infty}\bigl(\overline{K}^{*}\bigr)
\]

\[
  (22)\qquad \mathbb{G}_m(S) \longrightarrow \mathbb{G}_m(S)^{\wedge} \simeq H^1(S, T)
\]

\[
  g_i(s) = f(s)
\]

\[
  \mathcal{E}_K \longrightarrow \mathcal{E}_{U_K} \longrightarrow \mathcal{E}_U
\]

\[
  (23)\qquad N_{s'} \ \text{(groupe de décomposition de $\mathcal{E}_{K,\overline{K}}$ en $s'$)} \simeq \mathcal{E}_{\tilde{K}}
\]

\[
  (24)\qquad 1 \to \pi \to \mathcal{E}_{U_{\tilde{K}}} \to \mathcal{E}_{\tilde{K}} \to 1 ,
  \qquad \pi = \pi_1(U_{\overline{K}}), \quad \mathcal{E}_{\tilde{K}} = \operatorname{Gal}(\overline{K}/\tilde{K}),
\]

\[
  (25)\qquad 1 \to \pi \to \mathcal{E}_{U} \to \mathcal{E}_{\tilde{\mathcal{O}}} \to 1 ,
  \qquad \mathcal{E}_{\tilde{\mathcal{O}}} \simeq \operatorname{Gal}(\overline{k}/k)
\]

\[
  (27)\qquad 1 \to \pi \to \mathcal{E}_{U_s} \to \mathcal{E}_{k(s)} \to 1
\]

\[
  (28)\qquad \lambda \colon \mathcal{E}_{\tilde{K}} \longrightarrow \mathcal{E}_{U_{\tilde{\mathcal{O}}}} \;\simeq\; \mathcal{E}_{U_s}
\]

\[
  (29)\qquad 1 \to I_{s'} \to N_{s'} \to \mathcal{E}_{k(s)} \to 1
\]

\[
  (29)\qquad \varphi = \mathcal{E}_{f_K} \colon \mathcal{E}_K \longrightarrow \mathcal{E}_{U_K}
\]

\[
  (30)\qquad 1 \to T \to \mathcal{E}_\eta \to \mathcal{E}_k \to 1
\]

\[
  (31)\qquad 1 \to \pi \to \mathcal{E}_{U_\eta} \to \mathcal{E}_\eta \to 1
\]

\[
\text{(1)} \qquad \mathcal{E}_X = \pi_1(X)
\]

\[
\text{(2)} \qquad \mathcal{E}^{(\tilde{X})}_{X} = \pi_1(X ; \tilde{X}) = \operatorname{Aut}_X(\tilde{X})
\]

\[
\text{(3)} \qquad \mathcal{E}^{\xi}_{X} = \pi_1(X, \xi) = \mathcal{E}^{\tilde{X}(\xi)}_{X} ,
\]

\[
\text{(4)} \qquad \mathcal{E}^{\Omega}_{x} \simeq \mathcal{E}^{\overline{k(x)}}_{X} ,
\]

\[
\text{(5)} \qquad \mathcal{E}(f) : \mathcal{E}_X \longrightarrow \mathcal{E}_Y \qquad \text{où } f : X \to Y
\]

\[
\text{(6)} \qquad \mathcal{E}^{\tilde{X}}_{X} \longrightarrow \mathcal{E}^{\tilde{Y}}_{Y}
\]

\[
\text{(7)} \qquad (X, \tilde{X}) \longmapsto \mathcal{E}^{\tilde{X}}_{X}
\]

\[
\text{(8)} \qquad \mathcal{E}^{\xi}_{X} \longrightarrow \mathcal{E}^{\eta}_{\text{\struck{$Y$}}\,Y}
\]

\[
\text{(9)} \qquad \mathcal{E}^{\overline{k(x)}}_{X} \longrightarrow \mathcal{E}^{\overline{k(y)}}_{Y} ,
\]

\[
\text{(10)} \qquad \mathcal{E}^{\tilde{X}}_{X/Y} = \operatorname{Ker}\bigl(\mathcal{E}^{\tilde{X}}_{X} \longrightarrow \mathcal{E}^{\tilde{X}}_{Y} = \mathcal{E}^{\tilde{Y}}_{Y}\bigr)
\]

\[
\text{(11)} \qquad \text{\struck{$\mathcal{E}$}}\,\mathcal{E}^{T}_{X/Y} = \operatorname{Ker}\bigl(\mathcal{E}^{T}_{X} \longrightarrow \mathcal{E}^{T}_{Y}\bigr)
\]

\[ T \longrightarrow Y \]

\[
\text{(12)} \qquad \mathcal{E}^{T}_{X/Y} \simeq \pi_1(X_T ; T) \simeq \mathcal{E}^{T}_{X_T}
\]

\[
\text{(13)} \qquad 1 \longrightarrow \mathcal{E}^{T}_{X/Y} \longrightarrow \mathcal{E}^{T}_{X} \longrightarrow \mathcal{E}^{T}_{Y} \longrightarrow 1
\]

\[
\text{(14)} \qquad X \longrightarrow Y' \longrightarrow Y
\]

\[
\text{(15)} \qquad \mathcal{E}^{T}_{X/Y} \xrightarrow{\ \sim\ } \mathcal{E}^{T}_{X/Y'} ,
\]

\[
\text{(16)} \qquad \mathcal{E}^{T}_{X \times_{Y'} T} \simeq \mathcal{E}^{T}_{X/Y}
\]

\[
\text{(17)} \qquad \mathcal{E}^{T}_{X/Y} \simeq \mathcal{E}^{\tilde{X}}_{X/Y} \simeq \operatorname{Ker}\bigl(\mathcal{E}^{\tilde{X}}_{X} \to \mathcal{E}^{\tilde{Y}}_{Y}\bigr)
\]

\[
\text{(18)} \qquad \mathcal{E}^{T}_{X/Y} \simeq \mathcal{E}^{Z}_{X_{\tilde{Y}}} \simeq \mathcal{E}^{\tilde{X}}_{X_{\tilde{Y}}}
\]

\[
\text{(19)} \qquad \mathcal{E}^{\xi_X}_{X/Y} \simeq \mathcal{E}^{\xi_X}_{X_{(\xi_Y)}} \simeq \pi_1\bigl(X_{(\xi_Y)}, \xi_X\bigr)
\]

\[
\begin{aligned}
\text{(20)} \qquad \mathcal{E}^{\xi_T}_{X/Y} & \simeq \mathcal{E}^{T}_{X/Y} \\
\wr\ \ & \\
\pi_1\bigl(X_{\xi_Y}, \xi_T\bigr) &
\end{aligned}
\]

\[
\text{(21)} \qquad \mathcal{E}^{\overline{L}}_{L} \longrightarrow \mathcal{E}^{\overline{L}}_{U} \longrightarrow \mathcal{E}^{\overline{L}}_{X} = \mathcal{E}^{\mathrm{g}\overline{L}}_{L}
\]

\[
\text{(22)} \qquad U = X \setminus Z
\]

\[
N_{x'_i} \quad \text{et} \quad I_{x'_i}
\]

\[
\text{(23)} \qquad \mathcal{E}^{\overline{L}}_{X} = \mathcal{E}^{\mathrm{g}\overline{L}}_{L} \simeq \mathcal{E}^{\overline{L}}_{U} \big/ \text{ss-groupe fermé invariant engendré}
\]

\[
\text{par les } I_{x'_i} \ (i \in I)
\]

\[
\text{(24)} \qquad \mathcal{E}^{\overline{L}}_{L} \simeq \varprojlim_{U} \mathcal{E}^{\overline{L}}_{U}
\]

\[
\text{(25)} \qquad \pi^{\overline{L}}_{L/K} \simeq \varprojlim_{U} \pi^{\overline{L}}_{U/K} .
\]

\[
\text{(26)} \qquad N_{x'} / I_{x'} \simeq \Gamma_{k(x')}
\]

\[
\text{(27)} \qquad I_{x'} \simeq T_{\circ}(\overline{L}) \simeq T_{\infty}(\overline{K})
\]

\[
\text{(28)} \qquad I_{x'} \subset \pi^{\overline{L}}_{L/K} .
\]

\[
\text{(29)} \quad
\left\{
\begin{aligned}
\tilde{S}_x &= \operatorname{Spec} \tilde{V} , \quad \tilde{V} \text{ hensélisé (pas strict) de } V \\
\tilde{U}_x &= \tilde{S}_x \setminus \{x\} \simeq \operatorname{Spec} \tilde{L}_x \quad (\text{où } \tilde{L}_x \text{ est le corps des fractions de } \tilde{V})
\end{aligned}
\right.
\]

\[
\text{(30)} \quad
\left\{
\begin{aligned}
N_{x'} &\simeq \mathcal{E}^{\overline{L}}_{\tilde{U}_x} \\
I_{x'} &\simeq \operatorname{Noyau} \text{ de l'homom. canonique} \\
& \qquad \mathcal{E}^{\overline{L}}_{\tilde{U}_x} \longrightarrow \mathcal{E}_{\overline{k(x)}}
\end{aligned}
\right.
\]

\[
\text{(31)} \qquad \mathfrak{X}_{L/K} \simeq \varprojlim_{\substack{X \text{ modèle} \\ \text{propre de } L/K}} X
\]

\[
  (32) \qquad (\mathcal{E}_L^{\overline{L}})^{\Gamma^{\circ}} = (1)
\]

\[
  (33) \qquad (\pi_{L/K}^{\overline{L}})^{\Gamma^{\circ}} = (1)
\]

\[
  (34) \qquad 1 \to T^{I(x)} \to \mathcal{E}_{\widetilde{U}_x}^{\overline{L}} \longrightarrow \mathcal{E}_{k(x)}^{\overline{k(x)}} \to 1
\]

\[
  I(x) = \{\, i \in I \mid x \in D_i \,\}, \qquad T = T_{\infty}(\overline{L}^{*}),
\]

\[
  (35) \qquad
  \underset{\substack{\parallel \\ N_V}}{\mathcal{E}_{\widetilde{\eta}}^{\overline{L}}}
  \longrightarrow
  \underset{\substack{\parallel\ \text{dfn} \\ N_x}}{\mathcal{E}_{\widetilde{U}_x}^{\overline{L}}}
  \longrightarrow
  \mathcal{E}_U^{\overline{L}}
\]

\[
  \underset{\substack{\parallel \\ I_x \simeq T^{(I(x))}}}{\mathcal{E}_{\widetilde{U}_x/K}^{\overline{L}}} \longrightarrow \pi_{U/K}^{\overline{L}}
\]

\[
  \mathcal{E}_{\widetilde{\eta}}^{\overline{L}} = N_V \to \mathcal{E}_{\widetilde{U}_x}^{\overline{L}}
\]

\[
  I_V \longrightarrow I_x = T^{I(x)}
\]

\[
  \Delta_x \simeq T^{I(x)} \text{ de } \pi_{L/K}^{\overline{L}},
\]

\[
  D_J^{*} = \{\, x \in X \mid I(x) = J \,\} = \overbrace{\bigcap_{j \in J} D_j}^{D_J} - \bigcup_{i \in I \setminus J} D_J \cap D_i ,
\]

\[
  D_{\{i\}} = D_i \setminus \bigcup_{j \neq i} (D_i \cap D_j).
\]

\[
  (37) \qquad T^{J} \to \mathcal{E}_{U_J}^{\overline{L}} \to \mathcal{E}_{D_J} \to 1
\]

\[
  \pi_2(\overline{D}_J) \to \pi_1(\text{fibre}) \qquad (?)\,!
\]

\[
  \mathcal{E}_{U_J}^{\widetilde{U}_J} \longrightarrow \mathcal{E}_U^{\overline{L}}
\]

\[
  (39) \qquad T^{J} \longrightarrow \pi_{U/K}^{\overline{L}}
\]

\[
  (39) \qquad \mathcal{E}_{D_{J,\alpha}}^{\widetilde{D}_{J,\alpha}} \longrightarrow \mathcal{E}_K^{\overline{K}} .
\]

\[
  (40) \qquad T \xrightarrow{\ \kappa_i\ } \pi_{U/K}^{\overline{L}} ,
\]