\[
\begin{array}{c}
S_{\mathrm{cris}} \\ \downarrow \varphi \\ S_{\mathrm{zar}}
\end{array}
\ \Big)\,\varepsilon
\qquad \varphi\varepsilon = F_S
\]

\[
\varphi^*(\omega) \xrightarrow{\ \varphi^*(C)\ } \varphi^*(\omega^{p}),
\qquad \varphi^*(\omega) = \mathbb{D}^*(G), \quad
\varphi^*(\omega^{p}) = \mathbb{D}^*(G)^{(p)},
\qquad V \downarrow, \quad F = 0 \leftarrow
\]

\[
\mathcal{M} \underset{F}{\overset{V}{\rightleftarrows}} \mathcal{M}^{(p)},
\qquad \mathcal{M} = \mathbb{D}^*(G),
\]

\[
\Delta^*(G) = [\varepsilon^*(\mathcal{M}) \xrightarrow{0} \varepsilon^*(\mathcal{M})]
= \varepsilon^*[\mathcal{M} \to \mathcal{M}],
\]

\[
0 \to {}_F\mathcal{M}^{(p)} \to \mathcal{M} \to {}_V\mathcal{M}^{(p)} \to
\]

\[
\mathcal{M}^{(p)} \underset{V_{\mathcal{M}}}{\overset{F_{\mathcal{M}}}{\rightleftarrows}} \mathcal{M}
\]

\[
\Delta^* = \Delta^*(\mathcal{M}), \qquad
\mathcal{H}^0(\Delta^*) \simeq \mathcal{H}^1(\Delta^*) \simeq \mathcal{M}
\]

\[
\varphi^*(\ell_\bullet[-1]) \overset{\alpha}{\simeq} [\mathcal{M}^{(p)} \xrightarrow{F} \mathcal{M}]
\qquad
\varphi^*(\check{\ell}'_\bullet) \simeq [\mathcal{M} \xrightarrow{V} \mathcal{M}^{(p)}]
\]

\[
\ell_\bullet[-1]^{(p)} \simeq [M^{(p)} \xrightarrow{F_M} M],
\qquad
\check{\ell}'_\bullet{}^{(p)} \simeq [M \xrightarrow{V_M} M^{(p)}] \quad ! \;]
\]

\[
\begin{cases}
\pi_0\, \varepsilon^*(i_{\mathrm{II}}) \text{ ``$=$'' } F_{\check\ell'_\bullet} = {}^tV_{\ell'_\bullet} \\
\varepsilon^*(\pi_{\mathrm{II}})\, i_0 \text{ ``$=$'' } V_{\ell_\bullet}[-1]
\end{cases}
\]

\[
\mathbb{D}^*(G)^{(p)} \underset{V}{\overset{F}{\rightleftarrows}} \mathbb{D}^*(G)
\]

\[
(\ell^{G^*}_\bullet)^{\vee} \xrightarrow{\ \pi\ } \ell^{G}_\bullet
\]

\[
\underline{O}_S \longrightarrow \ell^{G^*}_\bullet \overset{L}{\otimes} \ell^{G}_\bullet
\]

\[
\mathcal{H}^0(\Delta^*_{\mathrm{cris}}(G)) \simeq \mathbb{D}^*(G), \qquad
\mathcal{H}^1(\Delta^*_{\mathrm{cris}}(G)) \simeq \mathbb{D}^*(G) .
\]

\[
\begin{array}{ll}
V_{\ell_\bullet}\colon \ell_\bullet \to \ell^{(p)}_\bullet &
F_{\ell_\bullet} = 0\colon \ell^{(p)}_\bullet \to \ell_\bullet \\
V_{\ell'_\bullet}\colon \ell'_\bullet \to \ell'^{(p)}_\bullet &
F_{\ell'_\bullet} = 0\colon \ell'^{(p)}_\bullet \to \ell'_\bullet
\end{array}
\]

\[
\varepsilon^*(\pi_{\mathrm{II}})\, i_0 \ne V_{\ell_\bullet}[-1]
\]

\[
\begin{cases}
\varphi^*(V_{\ell_\bullet}[-1]) = V_{[\mathcal{M}^{(p)} \xrightarrow{F} \mathcal{M}]} \\
\varphi^*(F_{\check\ell'_\bullet}[-1]) = F_{[\mathcal{M} \xrightarrow{V} \mathcal{M}^{(p)}]}
\end{cases}
\]

\[
\varphi^*(V_{\ell_\bullet}[-1]) = \varphi^*(\varepsilon^*(\pi_{\mathrm{II}})\, i_0)
= \varphi^*\varepsilon^*(\pi_{\mathrm{II}})\, \varphi^*(i_0)
= \bigl[\pi_{\mathrm{II}}^{(p)}\, i_{\mathrm{I}}
= V_{[\mathcal{M}^{(p)} \to \mathcal{M}]}\bigr]
\]

\[
\begin{array}{ll}
\pi_{\mathrm{I}} : & \mathfrak{X}^{p} \underset{V_{\mathfrak{X}}}{\overset{F_{\mathfrak{X}}}{\rightleftarrows}} \mathfrak{X} \\
i_{\mathrm{I}}\,\pi_{\mathrm{II}} : & \mathfrak{X} \xrightarrow{V_{\mathfrak{X}}} \mathfrak{X}^{p}
\end{array}
\]

\[
\left\lbrace
\begin{array}{lll}
\partial_{\mathrm{I}}\,\partial_{\mathrm{II}} : & A^{2} \to A & (\mathrm{nul}\,?) \\
\partial_{\mathrm{II}}\,\partial_{\mathrm{I}} : & B^{2} \to B & (\mathrm{nul}\,?) \\
\pi_{\mathrm{II}}^{(p)}\,\partial_{\mathrm{I}} : & A \xrightarrow{V_A} A^{(p)} & \\
\pi_{\mathrm{I}}\,, i_{\mathrm{II}}^{(p)} : & B^{p} \xrightarrow{F_B} B &
\end{array}
\right.
\]

\[
\begin{array}{ccccccc}
0 & \to & \mathcal{M} & \to & \mathcal{M}^p & \to & \cdot \\
 & & \downarrow & & \downarrow & & \\
 & \mathcal{M}^p & \to & \mathcal{M} & \to & 0 & \to \\
 & & \downarrow & & \downarrow & & \\
 & \mathcal{M} & \to & \mathcal{M}^p & \to & 0 & \to
\end{array}
\]

\[
\begin{array}{ll}
\mathfrak{X}^p \to B \to \mathfrak{X} &
\mathfrak{X} \to A \to \mathfrak{X}^{(p)} \\
X^{(p)} \to b^{(p)} \to X &
X \to a^{(p)} \to X^{(p)}
\end{array}
\]

\[
\left\lbrace
\begin{array}{ll}
\pi_0\, \varepsilon^*(i_{\mathrm{II}}) : & b^{(p)} \xrightarrow{F_b} b \quad ? \\
\varepsilon^*(\pi_{\mathrm{II}})\, i_0 : & a \xrightarrow{V_a} a^{(p)} \quad ?
\end{array}
\right.
\]

\[
\mathcal{M} \overset{\mathbb{L}}{\otimes} \mathcal{O}_{S'} = \Delta^*(S'),
\]

\[
\Bigl[\omega = \underline{H}^1(\ell_\bullet[-1]) = \underline{H}^0(\ell_\bullet),
\quad \nu' = \underline{H}^1(\check\ell'_\bullet)\Bigr]
\]

\[
0 \to \omega \to M \to \nu' \to 0
\]

\[
0 \to \varphi^*(\omega) \to \varphi^*(M) \to \varphi^*(\nu') \to 0, \qquad
\varphi^*(M) = \mathcal{M}^{(p)},
\]

\[
\Bigl[n = \underline{H}^0(\ell_\bullet[-1]) = \underline{H}^{-1}(\ell_\bullet),
\quad t' = \underline{H}^0(\check\ell'_\bullet)\Bigr]
\]

\[
0 \to n \to M \to t' \to 0
\]

\[
0 \to \varphi^*(n) \to \varphi^*(M) \to \varphi^*(t') \to 0
\]

\[
\varphi^*(n) = \operatorname{Ker}(\mathcal{M}^{(p)} \xrightarrow{F_{\mathcal{M}}} \mathcal{M}) .
\]

\[
\varphi^*(n) = \varphi^*(\omega),
\]

\[
\boxed{n = \omega}, \quad \boxed{t' = \nu'} .
\]

\[
\omega \xrightarrow{V_\omega} \omega^{(p)}, \qquad
t'^{(p)} \xrightarrow{F_{t'}} t' \qquad
\bigl(\omega' = \check t', \ \omega' \xrightarrow{V_{\omega'}} \omega'^{(p)},
\ V_{\omega'} = {}^tF_{t'}\bigr)
\]

\[
\omega^{(p)} = \operatorname{Im}(M \xrightarrow{V_M} M^{(p)}),
\]

\[
t'^{(p)} = \operatorname{Coker}(M \xrightarrow{V_M} M^{(p)}),
\]

\[
\mathbf{M}
\]

\[
0 \to G \hookrightarrow G' \twoheadrightarrow G'' \to 0
\]

\[
X_{\mathrm{cris}\,0} \hookrightarrow X_{\mathrm{cris}/n}
\qquad\qquad
\mathcal{M}'' \longrightarrow \mathcal{M}'
\]

\[
\varphi^*(\mathcal{T}) \overset{\alpha}{\simeq} \mathrm{I}(\mathcal{M})
\]

\[
\mathcal{T}_1 \overset{\beta}{\simeq} \varepsilon^*(\Delta^*(\mathcal{M})) \ \struck{\ill{}}
\]

\[
H^1(\mathcal{A}, \mu_2) \simeq \prod_p \mathbb{Q}_p^*/\mathbb{Q}_p^{*2} ,
\]

\[
H^0(\mathcal{A}, \mu_2) = \prod_p (\mathbb{Z}/2\mathbb{Z}) .
\]

\[
(\mathcal{M}, T, T_{\mathcal{A}'}, u_{\mathcal{A}'}, t_\infty, t_p)
\longrightarrow (\mathcal{M}', T' \ \mathrm{etc.})
\]

\[
H^1(\mathbb{Q} \bmod \mathcal{A}, \mu_2) \simeq_2
\Bigl(\prod_p \mathbb{Q}_p\Bigr) \simeq
\Bigl(\prod_p \mathbb{Z}/2\mathbb{Z}\Bigr)/(\pm 1)
\]

\[
\varphi^*(\mathcal{T}(\omega)) \overset{\alpha_\omega}{\simeq}
\mathcal{T}(\varphi^*(\omega))
\]

\[
\mathcal{T}(\omega)_1 \overset{\beta_\omega}{\simeq}
\varepsilon^*\Delta^*(\mathcal{M}) \simeq [\omega \xrightarrow{0} \omega^{p}]
\]

\[
(\omega, F_\omega, V_\omega) \longmapsto
\bigl(\varphi^*(\omega), \varphi^*(F_\omega), \varphi^*(V_\omega),
\mathcal{T}(\omega), \alpha_\omega, \beta_\omega\bigr)
\]

\[
\left\lbrace
\begin{array}{l}
\varphi^*(\omega) \xrightarrow{\ u\ } \varphi^*(\omega') \\
\mathcal{T}(\omega) \xrightarrow{\ v\ } \mathcal{T}(\omega')
\end{array}
\right.
\]

\[
0 \to n \to \mathbf{D}_1 \to t^* \to \omega \to \mathbf{D}_0 \to \nu^* \to 0
\]

\[
0 \to {}_F\tilde{\mathcal{M}} \to {}_{F}\tilde{\mathcal{M}} \xrightarrow{F}
{}_V\mathcal{M} \to \mathcal{M}_F \xrightarrow{V} \tilde{\mathcal{M}}_p \to
\tilde{\mathcal{M}}_V \to 0
\]

\[
X_{\mathrm{cris}} \ \downarrow \varphi \ X_{\mathrm{zar}}
\]

\[
B - A, \qquad B \otimes B \to A
\]

\[
0 \to n \to D_1 \to \tilde{D}_1 \to 0, \qquad
0 \to \tilde{D}_0 \to D_0 \to \nu^* \to 0
\]

\[
(\mathrm{i}) \qquad 0 \to \tilde{D}_1 \to t^* \to \omega \to \tilde{D}_0 \to 0
\]

\[
[t^* \to \omega]\,\struck{\ill{}} \simeq \tilde{\Delta}^*(\mathcal{M})
\]

\[
\operatorname{Ext}^1(S_{\mathrm{zar}} \bmod S_{\mathrm{cris}};
\tilde D_0, \tilde D_1) ;
\]

\[
\operatorname{Ext}^0(S_{\mathrm{zar}} \bmod S_{\mathrm{cris}};
\tilde D_0, \tilde D_1) = 0 \quad !
\]

\[
\operatorname{Ext}^2(S_{\mathrm{zar}} \bmod S_{\mathrm{cris}},
\tilde D_0, \tilde D_1) .
\]

\[
\begin{array}{ccccc}
\tilde{X} & \equiv & \widetilde{X \times X} & \equiv & \ldots \\
| & & & & \\
X & = & X \times X & \equiv & X \times X \times X \\
| & & & & \\
S' & = & S & = & S \\
\downarrow & & \downarrow & & \downarrow \\
\tilde{X} & \Leftarrow & \widetilde{X \times X} & \Lleftarrow & \widetilde{X \times X \times X} \\
\downarrow & & \downarrow & & \downarrow \\
\tilde{X} & \Leftarrow & X \times X & \Lleftarrow & X \times X \times X \\
\downarrow \mathrm{plat} & & \downarrow \mathrm{plat} & & \downarrow \mathrm{plat} \\
S & = & S & = & S
\end{array}
\]

\[
0 \to \ell \to \Delta \to \check{\ell}'_\bullet[1] \to 0
\]

\[
0 \to 0 \to \Theta \xrightarrow{\ \sim\ } \Theta \to 0
\]

\[
\begin{array}{ll}
\operatorname{Fil}^0 \Theta = \Theta & \operatorname{Fil}^0(\Delta) = \Delta \\
\operatorname{Fil}^1 \Theta = 0 & \operatorname{Fil}^1(\Delta) = \ell \\
 & \operatorname{Fil}^2(\Delta) = 0
\end{array}
\qquad
\left.\begin{array}{l} \Delta \to \Theta \\ \ell \to 0 \end{array}\right]
\ \text{i.e.}\ \check{\ell}'_\bullet[1] \to \Theta
\]

\[
\begin{array}{c}
\operatorname{Hom}_{D(X)}(\check{\ell}'_\bullet[1], \Theta) \to \operatorname{Hom}(\Delta, \Theta) \\
\downarrow \\
\operatorname{Hom}_{D(X_{\mathrm{cris}})}(\varphi^{*}(\check{\ell}'_\bullet[1]), \varphi^{*}(\Theta))
\end{array}
\]

\[
\Theta = \underline{O}_X[1]
\]

\[
\begin{array}{c}
\mathbb{H}^0(\ell_\bullet) \to \mathbb{H}^0(\Delta^*) \\
\downarrow \\
\mathbb{H}^0 \varphi^{*}(\ell_\bullet)
\end{array}
\]

\[
0 \to \ell_\bullet \to \check{\Delta}^* \to \check{\ell}_\bullet[1] \to 0
\]

\[
0 \to n_{G^{*}} \to \mathbb{D}^{*}(G)^{\vee} \to t_G \to \omega_{G^{*}} \to
\mathbb{D}^{*}(G^{*}) \to \nu_G \to 0
\]

\[
\begin{gathered}
0 \to \mathbb{H}^1(\ell_\bullet) \to \mathbb{H}^1(\Delta^*) \to
\mathbb{H}^1(\check{\ell}_\bullet[-1]) \to \mathbb{H}^0(\ell_\bullet) \to
\mathbb{H}^0(\Delta^*) \to \mathbb{H}^0(\check{\ell}_\bullet[1]) \to 0 \\
0 \to L_1 \to L_0 \to 0
\end{gathered}
\]

\[
\underbrace{\Gamma(\mathbb{D}^{*}(G)^{\vee}) \to \Gamma(t_G)}\to \mathbb{H}^0(\ell_\bullet)
\qquad
H^1(n_{G^{*}}) \to H^1_{\mathrm{cris}}(n_{G^{*}})
\]

\[
0 \to \underbrace{n_{G^{*}} \to \mathbb{D}^{*}(G)^{\vee}} \to Q \to 0
\]

\[
\to \omega \to \mathbb{D}^{*}(G)
\]

\[
\begin{cases}
H^1(\mathbb{D}^{*}(G)^{\vee}) = 0 \\
\operatorname{Ker}\bigl(H^1(n_{G^{*}}) \to H^1_{\mathrm{cris}}(n_{G^{*}})\bigr) \neq 0
\end{cases}
\]

\[
0 \to Q' \otimes_{\mathbb{Z}_p} \mathcal{O}_S \to M \to
P' \otimes_{\mathbb{Z}_p} \mathcal{O}_S \to 0
\]

\[
\begin{aligned}
F_{Q'} &: F_S^{*}(Q' \otimes_{\mathbb{Z}_p} \mathcal{O}_S) \to
Q' \otimes_{\mathbb{Z}_p} \mathcal{O}_S \\
p \cdot F_{P'} &: F_S^{*}(P' \otimes_{\mathbb{Z}_p} \mathcal{O}_S) \to
P' \otimes_{\mathbb{Z}_p} \mathcal{O}_S ,
\end{aligned}
\]

\[
\mathbb{D} : \mathrm{BTO}(S_0)^{\circ} \longrightarrow C .
\]

\[
\varphi : M \xrightarrow{\ \sim\ } P' \otimes_{\mathbb{Z}_p} \mathcal{O}_S +
Q' \otimes_{\mathbb{Z}_p} \mathcal{O}_S
\]

\[
\varphi' : M \simeq \ldots
\]

\[
L \in \Gamma\bigl(\underbrace{\check{P}' \otimes_{\mathbb{Z}_p} Q'}_{T}
\otimes_{\mathbb{Z}_p} \mathcal{O}_S\bigr)
\]

\[
F_S^{*}(M) \xrightarrow[\ \sim\ ]{F_S^{*}(\varphi)}
P' \otimes \mathcal{O}_S + Q' \otimes \mathcal{O}_S
\]

\[
F_{M,\varphi} = \begin{pmatrix} \mathrm{id} & \Phi \\ 0 & p\,\mathrm{id} \end{pmatrix},
\qquad F_{M,\varphi}(x, y) = (px, y + \Phi x)
\]

\[
\boxed{\ \Phi = \Phi_{M,\varphi} \in \Gamma(T \otimes_{\mathbb{Z}_p} \mathcal{O}_S)\ }
\]

\[
F_{M,\varphi'}\, F_S^{*}(u) = u\, F_{M,\varphi} \qquad \text{i.e.,}
\]

\[
\Phi' + F_S^{*}(L) = \Phi + pL
\]

\[
\boxed{\ \Phi' = \Phi + \bigl(pL - F_S^{*}(L)\bigr)\ }
\]

\[
\nabla_{M,\varphi} = \begin{pmatrix} 0 & \omega_{M,\varphi} \\ 0 & 0 \end{pmatrix}
\qquad
\boxed{\ \omega = \omega_{M,\varphi} \in \Gamma\bigl(\hat{\Omega}^1_{S/W}
\otimes (T \otimes_{\mathbb{Z}_p} \mathcal{O}_S)\bigr)\ }
\]

\[
\boxed{\ \omega' = \omega - dL\ }
\]

\[
\begin{cases}
\boxed{d\omega = 0} & \text{exprime que la connexion } \nabla
\text{ définie par } \omega \text{ est à courbure nulle} \\
\boxed{F_S^{*}(\omega) - p\omega = d\Phi} & \text{exprime que } F_M
\text{ est compatible à la connexion} \\
\boxed{\Phi_0 = 0} & \text{exprimant que } (F_M)_0 \text{ respecte }
\operatorname{Fil}_0 .
\end{cases}
\]

\[
(i+p)(p+1) - (i+p-1)p = \ldots = \underline{2p+i}
\]

\[
E \xrightarrow{\ u\ } F, \quad \omega, \ \omega' \qquad
\boxed{\ v \circ \omega = du + \omega' \circ u\ }
\]

\[
A \in \Gamma\bigl(\hat{\Omega}^1_{S/W} \otimes \operatorname{End}(E_S)\bigr)
\]

\[
A' = F_S^{*}(\nabla) \in \Gamma\bigl(\hat{\Omega}^1_{S/W} \otimes
\underbrace{\operatorname{End}(F_S^{*}(E_S))}_{\simeq\, \operatorname{End}(E_S)}\bigr)
\]

\[
\Phi \circ A^{\sigma} = d\Phi + A \circ \Phi
\]

\[
\boxed{\ d\Phi = \Phi \circ A^{\sigma} - A \circ \Phi\ }
\quad \text{exprime que } F_M \text{ est compatible à la connexion}
\]

\[
\boxed{\ dA + [A, A] = 0\ }
\quad \text{exprime que la connexion } \nabla \text{ définie par } A
\text{ est à courbure nulle}
\]

\[
\boxed{\ \Psi \Phi = p^i\, \mathrm{id}\ }
\]

\[
\sigma(V) = \sigma(U) \cap \mathbb{R}^n
\]

\[
\mathcal{A}_0 = \bigl\lbrace (\sigma_i, (U_i, V_i)) \bigr\rbrace \quad C^r,
\qquad V_i \subset U_i
\]

\[
(\sigma_1, (U_1, V_1)) \qquad (\sigma_2, (U_2, V_2))
\]

\[
U_1 \cap V_2 = V_1 \cap V_2, \qquad U_2 \cap V_1 = V_1 \cap V_2
\]

\[
M = E_S \xrightarrow{\ u\ } E'_S = M' \quad \text{compatible avec}
\quad \nabla_M, \nabla_{M'}, F_M, F_{M'} \ ?
\]

\[
u \circ A = du + A' \circ u
\]

\[
\boxed{\ du = u \circ A - A' \circ u\ }
\]

\[
\boxed{\begin{aligned}
d\varphi &= \varphi(\omega^{\sigma} - \omega) \\
d\omega &= 0 \\
\varphi \psi &= p^{i}
\end{aligned}}
\]

\[
\boxed{\begin{cases}
\omega' - \omega = -\dfrac{du}{u} \\[4pt]
\varphi'/\varphi = u/u^{\sigma}
\end{cases}}
\]

\[
\Gamma(\underline{O}_S^{*}) \to \Gamma(\hat{\Omega}^1_{S/W}) \times
\Gamma(\underline{O}_S)^{!} \to \Gamma(\Omega^2_{S/W}) \times
\Gamma(\Omega^1_S)[1/p]
\]

\[
u \mapsto \Bigl(\frac{du}{u},\ u/u^{\sigma}\Bigr) \qquad
(\varphi, \omega) \mapsto \Bigl(d\omega,\ \frac{d\varphi}{\varphi} -
(\omega^{\sigma} - \omega)\Bigr)
\]

\[
\mathbb{Z}_p \underset{(f)}{\hookrightarrow} W \underset{(e)}{\overset{i}{\hookrightarrow}} A
\]

\[
\pi_{G_0} = F^f \quad (\text{automorphisme de Frobenius de } G_0/\mathbb{F}_q)\ ]
\]

\[
\Bigl[ (*) \quad F_M^f = \pi_M \Bigr]
\qquad (*')\ \exists\, V_M \text{ t.q. } F_M V_M = V_M F_M = p
\]

\[
\sigma_A(\lambda \otimes w) = \lambda \otimes \sigma w
\]

\[
A \otimes_{\mathbb{Z}_p} W \xrightarrow{\ \sim\ } A^{\Sigma}, \qquad
\Sigma \simeq \mathbb{Z}/f\mathbb{Z}
\]

\[
\sigma_A(x_0, x_1, \ldots, x_{f-1}) = (x_1, \ldots, x_{f-1}, x_0)
\]

\[
M_0 \xrightarrow{\varphi_0} M_1 \xrightarrow{\varphi_1} \cdots \to
M_{f-1} \xrightarrow{\varphi_{f-1}} M_0
\]

\[
M_0 \xrightarrow{\ \sim\ } M_\nu
\]

\[
\pi^{-m_\nu}(\varphi_{\nu-1} \varphi_{\nu-2} \cdots \varphi_0), \quad
\text{où } m_\nu \text{ est un entier} \geq 0 \text{ uniquement déterminé}
\]

\[
M \simeq \Omega \otimes_{\mathbb{Z}_p} W = \Omega \otimes_A
(A \otimes_{\mathbb{Z}_p} W)
\]

\[
\varphi_\nu = \pi^{r_\nu}\, \mathrm{id}_{\Omega} \qquad
(r_\nu \geq 0, \ 0 \le \nu \le f-2)
\]

\[
\varphi_{f-1} = \lambda\, \mathrm{id}_{\Omega}, \qquad \lambda \in A, \quad
\text{soit } \lambda = \pi^{r_{f-1}} u,\ u \in A^{*}
\]

\[
F_M^{\,f} = \pi^{r_0 + \cdots + r_{f-2} + r_{f-1}} u = \varpi, \qquad u \in A^{*}
\]

\[
p \ \text{divisible par } \pi^{r_\nu} \text{ pour } 0 \le \nu \le f-2,
\text{ par } u\pi^{r_{f-1}}
\]

\[
\boxed{\ r_\nu \le e\ } \quad \text{pour } 0 \le \nu \le f-1
\]

\[
M/F_M M = A/u\pi^{r_{f-1}}A + A/\pi^{r_0}A + \cdots + A/\pi^{r_{f-2}}A
\]

\[
\boxed{\ \dim G_0 = r_0 + \cdots + r_{f-2} + r_{f-1}\ }
\qquad (\le fe = h)
\]

\[
\boxed{\begin{aligned}
&\dim G_0 = 1 \iff F_M^{\,f} = \varpi_M,\ \varpi \text{ uniformisante de } A \\
&\iff \text{tous les } r_\nu \text{ sauf un seul } r_{\nu_0} \text{ sont nuls},
\ r_{\nu_0} = 1,\ u \text{ une unité}
\end{aligned}}
\]

\[
\begin{cases}
M = \Omega \otimes_{\mathbb{Z}_p} W = \Omega \otimes_A (A \otimes_{\mathbb{Z}_p} W)
\quad (\text{comme } A \otimes_{\mathbb{Z}_p} W\text{-mod.}) \\
F_M = \mathrm{id}_\Omega \otimes_{\mathbb{Z}_p} \sigma + \mathrm{pr}_\nu
\bigl[(\pi - 1) \otimes \sigma\bigr]
\end{cases}
\]

\[
v_0\, \pi^{(\sum_0^{\nu} r_i) - (\sum_0^{\nu} r'_i)} \in
\operatorname{Hom}(\Omega', \Omega), \quad \text{i.e.} \quad
\operatorname{val}(v_0) + \sum_0^{\nu} r_i - \sum_0^{\nu} r'_i \geq 0
\]

\[
\begin{cases}
\text{si } \nu \mathrel{?} \nu' : & \operatorname{Hom}_A(G_0, G'_0) \xrightarrow{\ \sim\ } \operatorname{Hom}(\Omega', \Omega) \\
\text{si } \nu > \nu' : & \operatorname{Hom}_A(G_0, G'_0) \simeq \pi \operatorname{Hom}(\Omega', \Omega)
\end{cases}
\]

\[
M \simeq \Omega \otimes_{\mathbb{Q}_p} W = \Omega \otimes_K (K \otimes_{\mathbb{Q}_p} K_0)
\]

\[
F_M^{\,f} = \varpi \otimes_{\mathbb{Q}_p} \mathrm{id}_{K_0} = \varpi \otimes_K (K \otimes_{\mathbb{Q}_p} K_0)
\]

\[
\begin{cases}
\varpi \text{ laisse invariant un réseau} \\
p^{f} \varpi^{-1} \text{ laisse invariant un réseau}
\end{cases}
\]

\[
\omega_{G_0} = \check{t}_{G_0} \simeq M_\nu / \pi^{r} M_\nu \simeq \Omega / \pi \Omega
\]

\[
G_0 = G_0(A, \pi) \quad \text{correspondant à } (\nu = 0 \text{ et})\ \Omega = A
\]

\[
\Omega \otimes_A G_0(A)
\]

\[
\begin{cases}
M = A \otimes_{\mathbb{Z}_p} W \\
F_M = \mathrm{id}_A \otimes_{\mathbb{Z}_p} \sigma + \mathrm{pr}_0\bigl((\pi - 1) \otimes \sigma\bigr)
\end{cases}
\]

\[
A \otimes_{\mathbb{Z}_p} W \to A
\]

\[
M \otimes_W k = M_A \otimes_A k \ \text{ en une filtration de } M_A \ (\text{sur } A).
\]

\[
0 \to \omega_G \to \Omega \otimes_{\mathbb{Z}_p} A \to t_{G^*} \to 0
\]

\[
(**) \qquad 0 \to \omega_{G^*} \to \check{\Omega} \otimes_{\mathbb{Z}_p} A \xrightarrow{\ p_0\ } t_G \to 0
\qquad\qquad A \otimes_{\mathbb{Z}_p} A \xrightarrow{\ p_0\ } A
\]

\[
p_0 : A \otimes_{\mathbb{Z}_p} A \to A, \qquad p_0(\lambda \otimes \mu) = \lambda\mu
\]

\[
t_G \simeq \check{\Omega} \qquad \text{i.e.} \qquad \omega_G \simeq \Omega
\]

\[
0 \to \omega_G \otimes_A K \to M_K \to t_{G^*} \otimes_A K \to 0,
\qquad M_K = \Omega \otimes_{\mathbb{Z}_p} K
\]

\[
0 \to \omega_{G^*} \otimes_A K \to \check{M}_K \to t_G \otimes_A K \to 0,
\qquad \check{M}_K = \check{\Omega} \otimes_{\mathbb{Z}_p} K
\]

\[
t_G \otimes_A K \simeq \check{M}_K \otimes_{\check{\Omega} \otimes_{\mathbb{Z}_p} K} (K \otimes K, p_0) \simeq \check{\Omega} \otimes_A K
\]

\[
t_G = \pi^{-\alpha} \check{\Omega} \qquad \text{i.e.} \qquad \Omega = \pi^{\alpha} \omega_G,
\]

\[
\mathbb{D}^*(G)_K \simeq M_K \simeq \Omega \otimes_{\mathbb{Z}_p} K
\]

\[
e \in T_p(G)^*(K_\pi)
\]

\[
\overbrace{T_p(G)(K_\pi)}^{T_p} \xrightarrow{\ \sim\ } A, \qquad \lambda \mapsto \lambda e .
\]

\[
(*) \qquad \boxed{\ T_p \otimes_{\mathbb{Z}_p} C \simeq \prod_\sigma C_\sigma\ }
\]

\[
* \qquad \sigma \in \operatorname{Hom}_{\mathbb{Q}_p}(K, C) \simeq \operatorname{Aut}_{\mathbb{Q}_p}(K) \simeq \operatorname{Aut}_{\mathbb{Z}_p}(A) \simeq \operatorname{Hom}_{\mathbb{Z}_p}(A, C).
\]

\[
x \in C_\sigma \Rightarrow \rho_\sigma(g) x = \sigma(\chi_\pi(g)) x
\]

\[
(**) \qquad T_p \otimes C \simeq \mathfrak{t} \otimes_A C(1) + \check{\mathfrak{t}}^{*} \otimes_A C
\]

\[
T_p \otimes C \simeq C_1 \oplus \prod_{\sigma \neq 1} C_\sigma .
\]

\[
\mathfrak{t} \otimes_A C \simeq C_1, \qquad
\check{\mathfrak{t}}^{*} \otimes_A C \simeq \prod_{\sigma \neq 1} C_\sigma,
\]

\[
C_\sigma \to C_\sigma, \quad \text{i.e.} \quad \lambda \mapsto \lambda x_\sigma \quad (x_\sigma \in C)
\]

\[
\boxed{\ \begin{cases}
g(x_\sigma) = \sigma(\chi_\pi(g))\, x_\sigma & \text{si } \sigma \neq 1 \\
g(x_1) = \tau(g)^{-1} \chi_\pi(g)\, x_1 & \text{si } \sigma = 1
\end{cases}
\quad \text{si } g \in \boldsymbol{\pi} = \operatorname{Gal}(\bar{K}/K)\ }
\]

\[
g(x'_1) = \chi_\pi(g)\, x'_1 \qquad \text{pour } g \in \boldsymbol{\pi}
\ (\text{ou} = \text{pour } g \in \text{Inertie de } \boldsymbol{\pi})
\]

\[
g(x) = \underbrace{N_{A^*/\mathbb{Z}_p^*}(\chi_\pi(g))}_{= \tau(g) \text{ si } g \in \text{Inertie}}\, x ,
\]

\[
\begin{gathered}
\ldots\ \tfrac{6}{9} > \tfrac{11}{17}, \qquad \tfrac{2}{3} > \tfrac{11}{17}, \qquad
\tfrac{5}{12} > \tfrac{7}{17}, \qquad \tfrac{1}{4} < \tfrac{?}{11} < \tfrac{2}{7}, \qquad
\tfrac{2}{7} < \tfrac{5}{17}, \\
\tfrac{4}{17} < \tfrac{1}{4}, \qquad \tfrac{1}{9} < \tfrac{2}{17}, \qquad
\tfrac{1}{6} < \tfrac{3}{17}, \qquad \tfrac{7}{16} < \tfrac{12}{17}, \qquad
\tfrac{5}{7} \mathrel{?} \tfrac{13}{18}, \\
\tfrac{4}{13} > \tfrac{7}{23}, \qquad {<}\ \tfrac{51}{13}\,(?), \qquad
\tfrac{3}{10} > \tfrac{7}{24}, \qquad \tfrac{2}{7} < \tfrac{7}{24}, \qquad
\tfrac{1}{3} > \tfrac{7}{23}, \qquad \tfrac{2}{7} < \tfrac{1}{3}, \\
\tfrac{7}{9} < \tfrac{7}{8}\,(?), \qquad \tfrac{4}{5} < \tfrac{14}{17}\,(?), \qquad
\tfrac{11}{17} > \tfrac{13}{23}, \qquad \tfrac{?}{13} > \tfrac{15}{23}, \qquad
\tfrac{10}{14} < \tfrac{13}{18}, \\
\tfrac{13}{17} < \tfrac{10}{13}, \qquad \tfrac{3}{5} < \tfrac{14}{23}, \qquad
\tfrac{3}{4} < \tfrac{13}{17}
\end{gathered}
\]

\[
\tfrac{4}{5} < \tfrac{13}{15}, \qquad \tfrac{12}{14} = \tfrac{6}{7}, \qquad
\tfrac{14}{16} = \tfrac{7}{8} <
\]

\[
\tfrac{12}{14} = \tfrac{6}{7}, \qquad \tfrac{13}{15} < \tfrac{7}{8}, \qquad
\tfrac{3}{4} > \tfrac{11}{15}, \qquad \tfrac{2}{3} < \tfrac{11}{16}
\]

\[
T_p : \mathscr{C} \longrightarrow \operatorname{Mod}_f(\pi, \mathbb{Q}_p)
\qquad \text{où } \pi = \operatorname{Gal}(\bar{L}/L)
\]

\[
\mathscr{C} \cong \operatorname{Rep}_{\mathbb{Q}_p}(G)
\]

\[
\boxed{\ \pi \xrightarrow{\ \alpha\ } G(\mathbb{Q}_\ell)\ }
\]

\[
\mathscr{C} \to \operatorname{Mod}_f(\pi, \mathbb{Q}_p) \xrightarrow{\ \otimes_{\mathbb{Q}_p} C\ } \operatorname{Mod}_f(\pi, C) \to \operatorname{Mod}_f(C)
\]

\[
\boxed{\ \mathbb{G}_{m,C} \overset{i_1}{\underset{i_2}{\rightrightarrows}} G_C\ }
\]

\[
i(\lambda) = i_1(\lambda)\, i_2(\lambda) = i(\lambda) : \mathbb{G}_{m,C} \to G_C
\]

\[
\begin{cases}
\boxed{\ \mathcal{P},\ \text{torseur sous } G_L\ } \\
\bigl(T_{\mathrm{Hdg}} = \mathcal{P} \wedge^{G} (T_p \otimes_{\mathbb{Q}_p} L)\bigr)
\end{cases}
\]

\[
\begin{cases}
\mathbb{G}_{m,L} \overset{j_1}{\underset{j_2}{\rightrightarrows}} G'_L = \operatorname{ad}(\mathcal{P}) \\
j_1(\lambda)\, j_2(\lambda) = i(\lambda)
\end{cases}
\]

\[
T_{p\,C}(M) = T_p(M) \otimes_{\mathbb{Q}_p} C \simeq \sum T_{\mathrm{Hdg}}(M)^{i,j} \otimes C(-i)
\]

\[
C(1) \overset{\rho}{\simeq} C
\]

\[
\tilde\rho : T_{p\,C}(M) \simeq T_{\mathrm{Hdg}\,C}
\]

\[
\tilde\rho \in \mathcal{P}(C) = \operatorname{Hom}_{\operatorname{Spec} \ill{}}(\operatorname{Spec} C, \mathcal{P})
\]

\[
\rho' = u\rho, \qquad u \in C^*
\]

\[
\widetilde{u\rho} = i_1(u^{-1})
\]

\[
\boxed{\ C(1)^* \longrightarrow \mathcal{P}(C)\ }
\]

\[
\check{\mathbb{V}}(C(-1))^* \longrightarrow \mathcal{P}_C,
\qquad
\check{\mathbb{V}}(C(-1))^* = \check{\mathbb{V}}(\mathbb{Z}_p(-1))^*_C
\]

\[
G_L^{\mathcal{P}} \otimes_L C \simeq G_C
\]

\[
\begin{cases}
\boxed{\ Q\ \text{torseur sous } G_K\ } \\
T_{\mathrm{cr}} \simeq Q \wedge^{G_K} (T_p)_K
\end{cases}
\]

\[
\operatorname{gr}(T_{\mathrm{cr}} \otimes_K L) \simeq T_{\mathrm{Hdg}}
\]

\[
U' \subset G' = G_L^{\mathcal{P}}
\]

\[
\boxed{\ R \hookrightarrow \operatorname{Isom}(Q_L, \mathcal{P}) \qquad
(R \text{ torseur sous } U') \qquad G'' \supset U''\ }
\]

\[
F : T_{\mathrm{DR}}^{\sigma} \xrightarrow{\ \sim\ } T_{\mathrm{DR}}
\]

\[
\boxed{\ Q^{\sigma} \simeq Q\ }
\]

\[
\begin{aligned}
&i_1(\infty),\ i_2(\infty) : \mathbb{G}_{m,\mathbb{C}} \rightrightarrows G(\mathbb{C}) \\
&i_1(p),\ i_2(p) : \mathbb{G}_{m,C^{(p)}} \rightrightarrows G(C^{(p)})
\end{aligned}
\]

\[
j_2 : \mathbb{G}_{m,\mathbb{Q}} \to G'
\]

\[
\boxed{\begin{array}{l}
F_p \in G'(\mathbb{Q}_p) \\
F_\infty \in G'(\mathbb{R})
\end{array}}
\quad (\text{Frobenius})
\]

\[
\boxed{\ \rho^{(p)}(j_2, \lambda) \in \mathcal{P}(C^{p})\ }
\]

\[
\begin{cases}
C^{p}(1)^* \times U'(C^{p}) \longrightarrow \mathcal{P}(C^{p}) \\
\bigl[\ \mathbb{G}_{m,C^{p}} \xrightarrow{\ i_2^{(p)}\ } G'_{C^{p}}, \qquad G'_{C^{p}} \supset U'_{C^{p}}\ \bigr]
\end{cases}
\]

\[
(1.1) \qquad 0 \longrightarrow R \xrightarrow{\ v\ } \mathbb{Z}[M] \xrightarrow{\ u\ } M \longrightarrow 0 ,
\]

\[
(1.2) \qquad 0 \to G \to T \to T' \to 0, \qquad T \simeq \mathbb{G}_m^{M}, \quad T' = D(R)
\]

\[
\ell^{G}_\bullet \simeq [\omega_{T'} \to \omega_T] \simeq [R \otimes_{\mathbb{Z}} \mathcal{O}_S \to \mathbb{Z}[M] \otimes_{\mathbb{Z}} \mathcal{O}_S]
\]

\[
(1.3) \qquad \boxed{\ \ell^{G}_\bullet \simeq \underline{M} \overset{L}{\otimes}_{\mathbb{Z}} \mathcal{O}_S\ }
\]

\[
(1.4) \qquad \boxed{\ \ell^{G}_\bullet \simeq \bigl(M \overset{L}{\otimes}_{\mathbb{Z}} \Lambda\bigr) \overset{L}{\otimes}_{\Lambda} \mathcal{O}_S\ }
\]

\[
(1.4') \qquad \ell^{G}_\bullet \simeq \bigl(M \overset{L}{\otimes}_{\mathbb{Z}} \mathbb{F}_p\bigr) \otimes_{\mathbb{F}_p} \mathcal{O}_S .
\]

\[
(1.6) \qquad S \xrightarrow{\ f\ } (B_\pi)
\]

\[
(1.7) \qquad M = f^{-1}(M_0)
\]

\[
M \overset{L}{\otimes}_{\mathbb{Z}} \Lambda = f^*\bigl(M_0 \overset{L}{\otimes}_{\mathbb{Z}} \Lambda\bigr)
\]

\[
(1.8) \qquad \ell^{G}_\bullet \simeq f^*\bigl(M_0 \overset{L}{\otimes}_{\mathbb{Z}} \Lambda\bigr) \overset{L}{\otimes}_{\Lambda} \mathcal{O}_S ,
\]

\[
(2.1) \qquad
\begin{cases}
\omega_G = \underline{M} \otimes_{\mathbb{Z}} \mathcal{O}_S \\
\underline{n}_G = \underline{\operatorname{Tor}}^{\mathbb{Z}}_1(M, \mathcal{O}_S)
\end{cases}
\]

\[
(2.2) \qquad
\begin{cases}
\omega_G = (M \otimes_{\mathbb{Z}} \mathbb{F}_p) \otimes_{\mathbb{F}_p} \mathcal{O}_S = M_p \otimes_{\mathbb{F}_p} \mathcal{O}_S \\
\underline{n}_G \simeq {}_pM \otimes_{\mathbb{F}_p} \mathcal{O}_S
\end{cases}
\]

\[
(2.2') \qquad \omega_G \simeq M \otimes_{\mathbb{F}_p} \mathcal{O}_S, \qquad \underline{n}_G \simeq M \otimes_{\mathbb{F}_p} \mathcal{O}_S
\]

\[
\xi \in \operatorname{Ext}^2_{\mathcal{O}_S}\bigl(M \otimes_{\mathbb{Z}} \mathcal{O}_S, \operatorname{Tor}^{\mathbb{Z}}_1(M, \mathcal{O}_S)\bigr)
\]

\[
\xi_\Lambda \in \operatorname{Ext}^2_\Lambda\bigl(M \otimes_{\mathbb{Z}} \Lambda, \operatorname{Tor}^{\mathbb{Z}}_1(M, \Lambda)\bigr) ;
\]

\[
\xi_\Lambda \in H^2(S, \operatorname{End}(M)) ,
\]

\[
\xi \in H^2(S, \operatorname{End}(E)) \qquad (E = M \otimes_\Lambda \mathcal{O}_S)
\]

\[
H^2(S, \operatorname{End}(M)) \to H^2(S, \operatorname{End}(M) \otimes_\Lambda \mathcal{O}_S) \simeq H^2(S, \operatorname{End}(E))
\]

\[
\xi_0 \in H^2(B_\pi, \operatorname{End}(M_0)) .
\]

\[
H^2(B_\pi, -) \longrightarrow H^2(S, -)
\]