\[
  \mathrm{Niv}^0(\mathcal{M})^{\circ} \xrightarrow{\qquad\qquad\qquad} \ \ill{}.
\]

\[
  F^{0} \colon \mathrm{Niv}^0(\mathcal{M})^{\circ} \longrightarrow
  \mathrm{Mod}(\mathbb{Q}, \struck{\widehat{\mathbb{Z}}}\,)
\]

\[
  \mathrm{Niv}^0(\mathcal{M})^{\circ} \longrightarrow \mathrm{Mod}(\mathbb{Q}, \pi) \longrightarrow
  \mathrm{Mod}(\mathbb{Q}, \ldots)
\]

\[
  F^{0}(H^{0}(X)) = f_{*}(\mathbb{Q}_X)
\]

\[
  F^{0}(\mathbb{Q}(1)) = \mathbb{Q}.
\]

\[
  \struck{H}\,\mathrm{Ker}\bigl(G^{0} \xrightarrow{\;\varepsilon\;} \mathbb{G}_m\bigr)
\]

\[
  \Gamma \subset \overline{\mathbb{Q}}^{*}
\]

\[
  \begin{cases}
    \lambda \ \text{entier algébrique} \\
    \lambda \ \text{unité } \ell\text{-adique pour } \ell \neq p \\
    |\lambda_i| = p^{\nu/2} \ \text{pour} \ \nu \in \mathbb{Z}^{+} \ \text{convenable,} \\
    \qquad \text{pour tout conjugué } \lambda_i \text{ de } \lambda\,;
  \end{cases}
\]

\[
  \Gamma / \Gamma_{\mathrm{tors}} = \Gamma / (\text{racines de l'unité})
\]

\[
  G \xrightarrow{\;\varepsilon\;} \mathbb{G}_m
\]

\[
  \begin{array}{ccc}
    \mathbb{Z} & \longrightarrow & \Gamma \\
    1 & \longmapsto & p
  \end{array}
\]

\[
  \Gamma / \{\text{racines de l'unité}, p^{\nu}\} = \Gamma'
\]

\[
  0 \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow \Gamma' \longrightarrow \Gamma'/\mathrm{tors}(\Gamma') \longrightarrow 0
\]

\[
  1 \longmapsto \text{classe de } p^{1/2} \text{ dans } \Gamma'
\]

\[
  0 \longrightarrow H^{0} \longrightarrow H \longrightarrow \mu_2 \longrightarrow 0 ,
\]

\[
  H \simeq H^{0} \times \mu_2
\]

\[
  \xi_0 \in H^2(\mathbb{Q}, H) \simeq H^2(\mathbb{Q}, H^{0}) \times H^2(\mathbb{Q}, \mu_2)
\]

\[
  \Gamma \xrightarrow{\;i^{*}\;} \mathbb{Z}
\]

\[
  i^{*}(\lambda) = \nu \iff |\lambda| = p^{\nu/2} \,).
\]

\[
  \xi \in H^2(\mathrm{Spec}\, \mathbb{Q} \bmod T, H)
\]

\[
  T = \coprod_{\ell \neq p} \mathrm{Spec}(\mathbb{Q}_\ell).
\]

\[
  \eta_p \in H^1(\mathbb{Q}_p, G/H)
\]

\[
  \xi_p \ (\text{image de } \xi \text{ dans } H^2(\mathbb{Q}_p, H)) = \partial(\eta_p)
\]

\[
  0 \longrightarrow H \longrightarrow G \longrightarrow G/H \longrightarrow 0 .
\]

\[
  0 \longrightarrow \mathbb{G}_m \longrightarrow G/H \longrightarrow \widehat{\mathbb{Z}} \longrightarrow 0
\]

\[
  H^1(\mathbb{Q}_p, G/H) \xrightarrow{\;\sim\;} H^1(\mathbb{Q}_p, \widehat{\mathbb{Z}})
\]

\[
  \mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p) \simeq \widehat{\mathbb{Z}} \quad \text{donné par Frobenius.}
\]

\[
  \begin{array}{ccccccc}
    \longrightarrow & G^{0} & \longrightarrow & G & \longrightarrow & \widehat{\mathbb{Z}} & \longrightarrow 0 \\
    & \wr & & \wr & & \wr & \\
    & D(\Gamma/\text{racines de } 1) & & D(\Gamma) & \longleftarrow & D(\text{racines de } 1) &
  \end{array}
\]

\[
  \mathbb{G}_m \xrightarrow{\;i\;} G \xrightarrow{\;\varepsilon\;} \mathbb{G}_m
  \qquad\qquad \varepsilon i\,(\chi) = \lambda^{2}
\]

\[
  \mathbb{Z} \xleftarrow{\;i^{*}\;} \Gamma \xleftarrow{\;\varepsilon^{*}\;} \mathbb{Z}
\]

\[
  |\lambda| = p^{\,i^{*}(\lambda)/2} \qquad\qquad \varepsilon^{*}(1) = p
\]

\[
  \begin{array}{ccccc}
    & & (G^{0}_{\mathbb{F}_\ell})_{\mathrm{red}} & \longrightarrow & (G^{0})_{\mathbb{Q}_\ell} \\
    G_{\widehat{\mathbb{F}}_p} & \longrightarrow & (G_{\mathbb{F}_p})_{\mathrm{red}} & \xrightarrow[\text{(épi)}]{\;\alpha\;} & (G)_{\mathbb{Q}_p} \\
    & & & \searrow \quad \widehat{\mathbb{Z}} \quad = & \widehat{\mathbb{Z}}
  \end{array}
\]

\[
  \begin{array}{ccccc}
    \mathbb{Q} & \longleftarrow & \overline{\mathbb{Q}}_p^{*} & \xleftarrow{\;\alpha^{*}\;} & \Gamma \\
    \| & & \downarrow & & \downarrow \\
    \mathbb{Q} & \longleftarrow & \overline{\mathbb{Q}}_p^{*}/\text{racines de } 1 & \longleftarrow & \Gamma/\text{racines de } 1
  \end{array}
\]

\[
  A \otimes_{\mathbb{Q}} \mathbb{Q}_\ell \simeq \mathrm{End}(T_\ell(M)) \quad
  \text{(par Tate--Honda si $M$ une VA)}
\]

\[
  H^2(\mathbb{Q}, H) \simeq \varinjlim H^2(G, K^{*} \otimes \check{M})
\]

\[
  \begin{array}{ccccc}
    H^1(G, I \otimes \check{M}) & \to & H^1(G, C \otimes \check{M}) & \to & H^2(G, K^{*} \otimes \check{M}) \\
    \wr & & \wr & & \wr \\
    \sum_{\mathfrak{p}} \widehat{H}^{-1}(G, D_{\mathfrak{p}} \otimes \check{M}) & \to & \widehat{H}^{-1}(G, \check{M})
    & \to & \widehat{H}^{0}(G, \check{M} \otimes Y)
  \end{array}
\]

\[
  \begin{array}{ccccc}
    \to H^2(G, I \otimes \check{M}) & \to & H^2(G, C \otimes \check{M}) \\
    \wr & & \wr \\
    \to \sum_{\mathfrak{p}} \widehat{H}^{0}(G, D_{\mathfrak{p}} \otimes \check{M}) & \to & \widehat{H}^{0}(G, \check{M})
  \end{array}
\]

\[
  \sum_{\mathfrak{p}} \widehat{H}^{-1}(G, D_{\mathfrak{p}} \otimes \check{M})
  \simeq \sum_{\mathfrak{p}} \widehat{H}^{-1}(G_{\mathfrak{p}}, \check{M}),
  \qquad
  \widehat{H}^{-1}(G_{\mathfrak{p}}, \check{M}) \simeq
  \mathrm{Ker}\bigl(\check{M}_{G_{\mathfrak{p}}} \xrightarrow{N_{G_{\mathfrak{p}}}} \check{M}^{G_{\mathfrak{p}}}\bigr)
\]

\[
  \widehat{H}^{-1}(G, \check{M}) \simeq
  \mathrm{Ker}\bigl(\check{M}_{G} \xrightarrow{N_{G}} \check{M}^{G}\bigr)
\]

\[
  \widehat{H}^{0}(G, D_{\mathfrak{p}} \otimes \check{M}) \simeq \widehat{H}^{0}(G_{\mathfrak{p}}, \check{M})
  \simeq \check{M}^{G_{\mathfrak{p}}} / N_{G_{\mathfrak{p}}} \check{M}
\]

\[
  H^2(\mathbb{Q}_p, H^{0}) \simeq H^2(\mathbb{Q}_p, H/\mu_2)
\]

\[
  \longrightarrow H/\mu_2 \longrightarrow G/\mu_2 \longrightarrow G' \longrightarrow 0,
  \qquad G' \simeq \mathbb{G}_m \times \widehat{\mathbb{Z}}
\]

\[
  \Gamma^{0}/\text{racines de } 1 \simeq M \longrightarrow \mathbb{Z} \quad
  \text{stable par } G_{p}
  \struck{\ill{}}
\]

\[
  \struck{\ill{}}\,, \ \rho_1, \overline{\rho}_1, \ \rho_2, \overline{\rho}_2, \ \ldots,
  \ \rho_\nu, \overline{\rho}_\nu \qquad n_1
\]

\[
  \Gamma^{0} = \mathrm{Ker}\, i^{*} = \text{unités de Weil}
\]

\[
  \mathbb{Z} \xrightarrow{\;\varepsilon^{*}\;} \Gamma \xrightarrow{\;i^{*}\;} \mathbb{Z},
  \qquad i^{*}\varepsilon^{*} = 2\,\mathrm{Id}
  \qquad
  \begin{cases}
    \varepsilon^{*}(1) = p \\
    |\lambda| = p^{\,i^{*}(\lambda)/2}
  \end{cases}
\]

\[
  \struck{\mathbb{Z} \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow \Gamma/\varepsilon^{*}(\mathbb{Z})}
\]

\[
  \begin{array}{ll}
    \mathbb{Z}/2\mathbb{Z} & (\text{engendré par } p^{1/2}) \\
    \downarrow & \\
    \Gamma/(\text{racines de } 1).\,\varepsilon^{*}(\mathbb{Z}) & \longleftrightarrow U \\
    \uparrow & \\
    \Gamma^{0}/\text{racines de } 1 & \longleftrightarrow U^{0}
  \end{array}
\]

\[
  u = \varepsilon\, p^{\nu} \quad
  (\varepsilon \text{ racine de } 1,\ \nu \in \mathbb{Z},\ u \text{ unité de Weil})
  \ \Longrightarrow\ \nu = 0, \text{ donc } u \text{ racine de } 1.
\]

\[
  \mathcal{O} . \mathcal{O}' = \Bigl\{ \sum \lambda_i \mu_i \Bigr\} \subset K, \quad
  \lambda_i \in \mathcal{O}, \ \mu_i \in \mathcal{O}'
\]

\[
  \varphi_{X} \colon \mathcal{O} = \mathrm{End}(X) \xrightarrow{\;\sim\;} \mathcal{O}' = \mathrm{End}(X').
\]

\[
  \mathrm{Brd}(\mathcal{O}) = \pi_{1}\bigl(\underline{\mathrm{End}}(C)\bigr)
  \quad \text{(groupe de Brandt)}
\]

\[
  \mathcal{O}^{\natural *} = \pi_{0}\bigl(\underline{\mathrm{End}}(C)\bigr) = \mathrm{End}(\mathrm{id}_{C})
\]

\[
  \sigma(\lambda)\, x = x \lambda \qquad \text{pour tt } x \in L, \ \lambda \in \mathcal{O}^{\natural}
\]

\[
  \xi \in H^{3}\bigl(\mathrm{Brd}(\mathcal{O}), \mathcal{O}^{\natural *}\bigr)
\]

\[
  \begin{array}{ccc}
    \underline{\underline{\mathrm{Brd}}} & \longrightarrow & \underline{\underline{\mathcal{D}}} \\
    \mathcal{O} & \longmapsto & C_{\mathcal{O}}
  \end{array}
\]

\[
  \varphi \colon C \longrightarrow \mathrm{Mod}(\mathcal{O}_{T})
\]

\[
  \varphi \longmapsto \varphi(X)
\]

\[
  C \longrightarrow \underline{\mathrm{Rep}}(\mathcal{G})
\]

\[
  \det P \simeq A \qquad (\alpha)
\]

\[
  N_{\mathrm{red}}(Q \otimes_{\sigma'} P, \sigma) \simeq
  N_{\mathrm{red}}(Q, \sigma') \otimes N_{\mathrm{red}}(P, \sigma).
\]

\[
  \begin{cases}
    \operatorname{Pic}(A) = 0 \\
    \Phi \cap A^{*} = \{1\}, \quad \Phi \cdot A^{*} = K^{*}
  \end{cases}
\]

\[
  C = \text{catégorie des courbes elliptiques de}
\]

\[
  A' = \prod_{\ell \neq p, \infty} \mathbb{Z}_{\ell},
\]

\[
  \varphi(T) \simeq A'(1) \Bigl(\overset{\mathrm{df}}{=}
  \prod_{\ell \neq p, \infty} \mathbb{Z}_{\ell}(1)\Bigr).
\]

\[
  \mathcal{G}_{A'} \simeq \mathcal{H}_{A'}
\]

\[
  \mathrm{H}^{1}(A', \mu_{2}) \simeq \prod_{\ell \neq p, \infty}
  \mathbb{Z}_{\ell}^{*}/\mathbb{Z}_{\ell}^{*2},
  \qquad
  \mathbb{Z}_{\ell}^{*}/\mathbb{Z}_{\ell}^{*2} \simeq
  \begin{cases}
    \mathbb{Z}/2\mathbb{Z} & \text{si } \ell \neq 2 \\
    \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} & \text{si } \ell = 2
  \end{cases}
\]

\[
  \mathrm{H}^{1}\bigl(\underbrace{\mathbb{Z}[\tfrac{1}{p}]}_{A}, \mu_{2}\bigr)
  \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}
\]

\[
  \Pi/\rho(\mathbb{Z}/2\mathbb{Z}),
\]

\[
  \rho : \mathrm{H}^{1}(\mathbb{Z}[\tfrac{1}{p}], \mu_{2}) \simeq
  (\mathbb{Z}/2\mathbb{Z}) \times \mathbb{Z}/2\mathbb{Z} \hookrightarrow
  \mathrm{H}^{1}(A', \mu_{2})
\]

\[
  \rho(1,0)_{\ell} =
  \begin{cases}
    0 & \text{si } p \text{ [resp.\ } -1\text{] est un carré dans }
        \mathbb{Z}_{\ell}^{*} \\
    1 & \text{sinon}
  \end{cases}
\]

\[
  \exists\ A\text{-algèbre } \sigma, \text{ avec } C \approx
  \mathbb{P}(\sigma/A)
\]

\[
  \Updownarrow
\]

\[
  \Updownarrow
\]

\[
  u : \sigma = \operatorname{End}(X) \longrightarrow \sigma' =
  \operatorname{End}(X')
\]

\[
  \Bigl[\ \operatorname{Hom}_{\mathrm{ad}_{A \to A'}}(C, C') \simeq
  \operatorname{Hom}_{\mathrm{ad}_{A'}}(C_{A'}, C')\ \Bigr]
\]

\[
  \mathbb{P}(\sigma'/A) \approx \mathbb{P}(\sigma/\text{ordre max.\ de }
  \sigma)
\]

\[
  \nu : C \longrightarrow D
\]

\[
  \alpha = (\alpha_{X, P}) : \nu(P \otimes_{\sigma} X)
  \xrightarrow{\ \sim\ } \mathrm{Nr}_{\sigma/A}(P) \otimes_{A} \nu(X)
\]

\[
  \nu(P \otimes_{\sigma} X) \simeq \mathrm{Nr}_{\sigma/A}(P) \otimes T.
\]

\[
  \mathrm{Nr}_{\sigma/A}(P) \simeq A
\]

\[
  \nu(\varphi(X)) \simeq T_{\varphi}.
\]

\[
  \mathrm{N}_{\sigma/A}(P_{\sigma}) \simeq A
  \qquad (\text{i.e.\ } \tau = L \otimes_{A} \nu(X))
\]

\[
  0 \longrightarrow \mathrm{H}^{1}(S, \mu_{n}) \longrightarrow
  \pi_{0}\bigl(\mathrm{Brd}(C, \tau)\bigr) \longrightarrow
  \mathrm{Ram}(C/A) \longrightarrow 0
\]

\[
  \mathrm{Ram}(C/A) \overset{\mathrm{df}}{=}
  \mathrm{Brd}(C)/\operatorname{Pic}(A) \simeq \sum_{p}
  \mathbb{Z}/e_{p}(C)\mathbb{Z}.
\]

\[
  (C_{A'}, T_{A'}) \dashrightarrow (C', T'),
\]

\[
  C_{A'} \xrightarrow{\ \varphi\ } C' \quad A\text{-lin.\ admis.}
\]

\[
  \text{Éq}(C_{A'}, T_{A'}) \ (\text{à droite}) \ \approx\quad
  \text{Éq}(C', T') \ (\text{à gauche}).
\]

\[
  \text{Éq}(C_{A'}, T_{A'}) \approx
  \text{Éq}(C', T')
\]

\[
  \text{Éq}(C, T) \longrightarrow
  \text{Éq}(C)
\]

\[
  (C, T)_{A'} \xrightarrow{\ \sim\ } (C', T')
\]

\[
  \text{Éq}\bigl((C, T)_{A'}\bigr) \simeq
  \text{Éq}(C', T')
\]

\[
  \mathrm{Brd}(C', T') / \operatorname{Im} \mathrm{Brd}(C).
\]

\[
  \begin{array}{ccccc}
    \mathrm{Brd}(C) & \xrightarrow{\ c\ } & \mathrm{Brd}(C, T) &
      \longrightarrow & \mathrm{Brd}(C', T') \\
    \rho & \longmapsto & (\rho, u_{\rho}) & \longmapsto &
      (\rho', u_{\rho'}) \\
    \wr\!\wr \\
    P_{0} & \longmapsto & \mathrm{N}_{\sigma/A}(P_{0}) \otimes u_{\rho} &
      \longmapsto & \mathrm{N}_{\sigma'/A'}(P'_{0}) \otimes (u_{\rho})'
  \end{array}
\]

\[
  E \longmapsto E^{(p)} = E/\operatorname{Ker} F_{E}
\]

\[
  \mathrm{N}_{\sigma/\mathbb{Z}}(\mathfrak{m}) \longrightarrow
  \mathrm{N}_{\sigma/\mathbb{Z}}(\sigma) = \mathbb{Z}, \qquad
  \mathrm{N}_{\sigma/\mathbb{Z}}(\mathfrak{m}) = p\mathbb{Z},
\]

\[
  \mathrm{Brd}(C_{A'}, T_{A'}) \simeq \mathrm{H}^{1}(S', \mu_{2})
\]

\[
  \mathrm{Brd}(C_{p}, T_{p}) \simeq \mathrm{Brd}(C_{A'}, T_{A'})
\]

\[
  \mathrm{H}^{1}(\mathbb{R}, \mu_{2}) = \mathbb{R}^{*}/\mathbb{R}^{*2}.
\]

\[
  A'^{*}/A'^{*2} \times \bigl(\mathbb{Z}_{p}^{*}/\mathbb{Z}_{p}^{*2}
  \times \mathbb{Z}/2\mathbb{Z}\bigr) \times \mathbb{R}^{*}/\mathbb{R}^{*2}
  = \Gamma' .
\]

\[
  \bigl(p, \; f = (0, 1), \; p\bigr)
\]

\[
  \Gamma = \Gamma' / f(\pm 1)
\]

\[
  A'^{*}/A'^{*2} \times \mathbb{Z}_{p}^{*}/\mathbb{Z}_{p}^{*2} \times
  \mathbb{R}^{*}/\mathbb{R}^{*2} = \mathrm{H}^{1}(\mathbb{A}, \mu_{2})
\]

\[
  \mathcal{T}(\overline{\mathbb{F}}_{p}) \ \text{est un torseur canonique
  sous} \ \mathrm{H}^{1}(\mathbb{A}, \mu_{2}) =
  \prod_{\ell} \mathbb{Z}_{\ell}^{*}/\mathbb{Z}_{\ell}^{*2} \quad
  (\text{y compris } p \text{ et } \infty) .
\]

\[
  f \ \text{opère trivialement sur} \ \mathcal{T}(\overline{\mathbb{F}}_{p}) .
\]

\[
  \overline{\mathbb{F}}_{p} = \varinjlim_{n}
  \bigl( \mathbb{F}_{p^{n}} = \mathbb{F}_{p}[T]/\Phi_{p^{n}-1}(T) \bigr) \qquad
  \text{-----}
\]

\[
  \text{Éq}(C_{\pi}) \longrightarrow \text{Éq}(C, T) \longrightarrow
  \text{Éq}(C_{p}, T_{p}) \underset{\mathrm{can}}{\simeq}
  \text{Éq}\bigl(C(\mathbb{Z}_{p}), T(\mathbb{Z}_{p})\bigr)
\]

\[
  \bigl[\,C(\xi), \alpha(\xi)\,\bigr] \quad \text{canoniques.}
\]

\[
  C(\xi_{p}) \xrightarrow{\ \varphi_{C_{\infty}}\ }
  \mathcal{C}\ell(\mathbb{R})
\]

\[
  u_{\xi}(M, o) \in
  \mathrm{Isom}\bigl(\textstyle\bigwedge^{2} M, T_{p}\bigr) /
  \mathbb{Z}_{p}^{*2}
\]

\[
    (\underbrace{P \otimes_{o} M}_{M'}, o')
  \]

\[
    \det M' \simeq (\mathrm{N}_{o/\mathbb{Z}} P) \otimes_{\mathbb{Z}}
    \det M \underset{c}{\simeq} \det M
  \]

\[
  F : C \longrightarrow C'
\]

\[
  P \otimes_{o} F(X) \longrightarrow F(P \otimes_{o} X)
\]

\[
  F \longmapsto F(X) : \mathrm{Hom}_{A\text{-adm}}(C, C')
  \hookrightarrow \mathrm{Mod}(o, C') ,
\]

\[
  C(1) = \mathrm{Hom}_{A\text{-adm}}(C^{\circ}, \mathrm{Mod}(A)) ,
\]

\[
  C \xrightarrow{\ i\ } C(1)
\]

\[
  C \xrightarrow{\ i\ } C_{sp}(1)
\]

\[
  \check{P} \longrightarrow \mathrm{Hom}(P \otimes_{o} X, X) ,
\]

\[
  P \otimes_{o} X \longrightarrow \mathrm{Hom}_{o}(\check{P}, X)
\]

\[
  0 \to o^{Z*} \to o^{*} \to \mathrm{Aut}_{A}\, o \to \mathrm{Brdt}(C)
  \to \mathrm{Is}\, C \to \mathrm{Is}\, \mathcal{O} \to 0
\]

\begin{align*}
  \cdot \to \pi_{1}(\mathcal{B},1) &\to \pi_{1}(C,X) \to
  \pi_{1}(\mathcal{O},o) \to \pi_{0}(\mathcal{B},1) \\
  &\to \pi_{0}(C,X) \to \pi_{0}(\mathcal{O},o) \to 1
\end{align*}

\[
  \to \mathrm{Pic}(A) \to \pi_{0}(\mathcal{B},1) \to \mathrm{Ram}(C/A) \to 0
\]

\[
  \mathrm{Ram}(C/A) \simeq \coprod_{p} \mathbb{Z}/e_{p}\mathbb{Z}
  \quad \text{dans le cas Deuring-Brandt}
\]

\[
  \mathrm{Brd}(C) = \pi_{0}(\mathrm{Eqv}) \quad \text{opère sur}
  \quad \text{Cl. d'isom. de } C
\]

\[
  \mathrm{Pic}(A) \longrightarrow \mathrm{Eqv}(C), \qquad
  L \longmapsto (X \mapsto L \otimes_{A} X)
\]

\[
  \cdot \to \mathrm{Pic}(A) \to \mathrm{Brdt}(C) \to
  \frac{\mathrm{Brdt}(C)}{\mathrm{Pic}(A)} \to 0
\]

\[
  \frac{\mathrm{Brd}\, C}{\mathrm{Pic}(A)} \simeq
  \frac{\mathrm{Div}(C)}{\mathrm{Div}(A)} =
  \coprod_{\substack{p \in \mathrm{Max}\, A \\ p\ \text{ramifié en}\ C}}
  \mathbb{Z}/e_{p}\mathbb{Z}
\]

\[
  \mathrm{Autext}(o) = \mathrm{Brd}\, C = \mathbb{Z}/e\mathbb{Z}
\]

\[
  \cdot \to \mathbb{Z}/2\mathbb{Z} \to o^{*} \to \mathrm{Aut}(o) \to
  \mathbb{Z}/2\mathbb{Z} \to \pi_{0}(C,X) \to \pi_{0}(\mathcal{O},o) \to 1
\]

\[
  ({}_{o}L_{o'}, {}_{o'}M_{o''}) \longmapsto L \otimes_{o'} M ,
\]

\[
  (L, M \mapsto L.M) \quad \text{est commutative :} \quad LM = ML .
\]

\[
  \mathrm{Eqv}(C) = \mathrm{Hom}_{\mathcal{B}_{0}}(C,C)
\]

\[
  \mathrm{Hom}_{\mathcal{B}_{0}}(C,C) \overset{\alpha_{X}}{\simeq}
  \mathrm{Hom}_{\mathcal{B}_{1}}(o,o) = \text{catégorie des }
  (o,o)\text{-bimodules spéciaux}
\]

\[
  c : L \otimes_{o} M \simeq M \otimes_{o} L
\]

\[
  L \simeq L' \subset E , \qquad M \simeq M' \subset E
\]

\[
  L \otimes M \simeq L'M' = M'L' \simeq M \otimes L .
\]

\[
  \begin{array}{ccc}
    LM & \text{transformé par } P \text{ en} &
    P(LM)P^{-1} = (PLP^{-1})(PMP^{-1}) \\
    c_{X} = 1 \ \| & & \| \ P(c_{X}) = 1 \qquad \| \ c_{X'} = 1 \\
    ML & & P(M.L)P^{-1} = (PMP^{-1})(PLP^{-1})
  \end{array}
\]

\[
  \begin{array}{ccc}
    & C' \longmapsto \mathrm{Eqv}(C,C') & \\
    \mathcal{B}_{0} & \longrightarrow &
    \text{la catégorie des torseurs-catégories sous } \mathcal{B}
  \end{array}
\]

\[
  \mathcal{B} \longrightarrow \mathcal{B}'
\]

\[
  C \longrightarrow C_{A'}
\]

\[
  \mathrm{Hom}_{C_{A'}}(X,Y) = \mathrm{Hom}_{C}(X,Y) \otimes_{A} A' .
\]

\[
  \underline{P}(R) \longrightarrow C , \qquad P \longmapsto P \otimes_{R} X
\]

\[
  P \longmapsto P \otimes_{R} X
\]

\[
  G(\mathbf{Q}) \overset{\varphi}{\hookrightarrow} \mathrm{End}_{\mathcal{M}}(E).
\]

\[
  \varphi(u)\,\varphi(u)' = \varepsilon(u)\,1_{E} .
\]

\[
  H = \prod_{i} \Bigl( \prod_{L_i/\mathbf{Q}} \mathbf{G}_{m\,L_i} \Bigr)
\]

\[
  H^{u} = \prod_{i} \Bigl( \prod_{K_i/\mathbf{Q}} \Gamma_i \Bigr)
\]

\[
  \bigl( \mathrm{Ker}(\varepsilon : H \to \mathbf{G}_m) \bigr)^{\circ}
\]

\[ \Updownarrow \]

\[ \Updownarrow \]

\[ \Downarrow \]

\[ \Updownarrow \]

\[
  T_{\mathbf{R}}^{\circ} = U_{\mathbf{R}} \cdot S_{\mathbf{R}} ,
\]

\[
  f^{\nu} = u \cdot j(x) , \qquad u \in U(\mathbf{R}),\ x \in \mathbf{R}^{*} .
\]

\[
  \varepsilon(f)^{\nu} = \varepsilon(u)\,\varepsilon j(x) = \varepsilon j(x) = x^{\alpha} ,
\]

\[ \Updownarrow \]

\[
  x\,\sigma x = 1 ,
\]

\[
  z\,\bar{z} = (y \cdot \sigma y)^{2} = u^{2} ,
\]

\[
  x\,\sigma x = (z\bar{z})/u^{2} = 1 .
\]

\[
  \mu\bar{\mu} = 1 ,
\]

\[
  \lambda\bar{\lambda} = q^{i} \qquad (i \in \mathbf{Z}) .
\]

\[
  \lambda = q^{i/2}u ,
\]

\[
  U \longrightarrow \mathbf{Z}^{P} , \qquad
  u \mapsto (v_{\mathfrak{p}}(u))_{\mathfrak{p} \in P}
\]

\[
  U \simeq \mathbf{Z}^{Q} ,
\]

\[
  M(q^{m}) \subset M(q) \subset M(p), \qquad q = p^{n},
\]

\[
  M(q^{m})^{m} \subset M(q)^{m} \subset M(q^{m}) \subset M(q).
\]

\[
  G(q) \longleftarrow G(q^{m}), \qquad f_{q^{m}} \longmapsto (f_{q})^{m} ;
\]

\[
  M(q) \xrightarrow{\ \text{élévation à la } m\text{-ième puissance}\ } M(q^{m})
\]

\[
  M^{\circ}(q = p^{n}) \longrightarrow P/U
\]

\[
  \lambda \longmapsto \lambda^{1/n} \bmod U
\]

\[
  G \simeq D(M^{\circ} \otimes \mathbf{Q}) \times \mathbf{G}_{m}
\]

\[
  M_{0}(q)/\mathrm{Torsion} \simeq (\mathbf{Z}^{\mathcal{P}})' \qquad (*)
\]

\[
  \mathbf{G}_{m\,L} \xrightarrow{\ v_{\mathfrak{p}}\ } G_{L}.
\]

\[
  v : \mathbf{G}_{m\,L_{0}} \longrightarrow G_{L_{0}}
\]

\[
  \struck{\ill{}}\ \mathbf{Z}/2\mathbf{Z} \to \Gamma \to \Gamma' \to 0,
  \qquad 1 \to \Gamma_{\mathfrak{p}} \subset \Gamma
\]

\[
  G' = \prod_{K_{0}/\mathbf{Q}} U = \operatorname{Ker}\Bigl(\prod_{L/\mathbf{Q}} \mathbf{G}_{m\,L_{0}} \to \prod_{K_{0}/\mathbf{Q}} \mathbf{G}_{m\,K_{0}}\Bigr)
\]

\[
  (**) \qquad \varphi : G' \longrightarrow G
\]

\[
  H^{0}(\mathcal{G}_{\mathbf{R}}) \longrightarrow \mathrm{Pol}(\mathcal{G}_{\mathbf{R}})
\]

\[
  i : \mathbf{G}_{m\,\mathbf{C}} \longrightarrow G_{\mathbf{C}}
\]

\[
  \bar{\imath}\, i = j, \qquad \varepsilon i = \mathrm{id}
\]

\[
  \mathbb{G}_{m,\mathbb{C}} \xrightarrow{\; i \;} G_{\mathbb{C}} ,
  \qquad
  \mathbb{G}_{m,\mathbb{C}} \xrightarrow{\; i' \;} G_{\mathbb{C}}
\]

\[
  \tau\lambda = \overline{\lambda} \qquad \text{pour } \lambda \in
  \overline{\mathbb{Q}}
\]

\[
  \tau^{2} = 1 .
\]

\[
  \lambda^{2} - a\lambda + Q = 0 .
\]

\[
  (T - \lambda)(T - \overline{\lambda}) = 0
  \quad \text{i.e.} \quad
  T^{2} - aT + Q = 0 ,
\]

\[
  \lambda_{\alpha}\overline{\lambda}_{\alpha}
  = N_{\mathbb{L}/\mathbb{K}}(\lambda_{\alpha})
  = N_{\mathbb{L}/\mathbb{K}}(\lambda) = \lambda\overline{\lambda} = Q .
\]

\[
  \mathcal{G} \to \mathcal{G}_{0} ,
  \qquad
  G \xrightarrow{\;\varepsilon\;} \mathbb{G}_{m}
  \qquad \text{épimorphismes}
\]

\[
  \begin{array}{l}
    \mathcal{G} \longrightarrow \mathcal{G}' \longrightarrow 1 \\[2pt]
    1 \longrightarrow G^{0} \longrightarrow G
      \xrightarrow{\ \text{épim.}\ } G' \longrightarrow 1
  \end{array}
\]

\[
  \sum_{i \in \mathbb{Z}} V_{i}(i) , \qquad \text{où les}
\]

\[
  \Bigl(\sum V_{i}(i)\Bigr) \otimes \Bigl(\sum W_{j}(j)\Bigr)
  = \sum_{h} \Bigl(\sum_{i+j=h} V_{i} \otimes W_{j}\Bigr)(h) ,
  \qquad
  \Bigl(\sum V_{i}(i)\Bigr)^{\vee} = \sum \check{V}_{i}(-i) \,\Bigr]
\]

\[
  \sum_{i \in \mathbb{Z}} V_{i}(i) \longmapsto \sum V_{i}(\bar{k}) ,
\]

\[
  \begin{array}{l}
    \mathcal{G} \longrightarrow \mathcal{G}^{(0)} \\[2pt]
    1 \longrightarrow U \longrightarrow G \longrightarrow G^{(0)} =
      \mathbb{G}_{m} \times G' \longrightarrow 1
  \end{array}
\]

\[
  \begin{array}{l}
    \mathcal{M} \longrightarrow \mathcal{M}' \quad \text{catégorie des motifs
      sur } k' \text{ engendrée par les } M \otimes_{k} k',\ M \in \mathcal{M}
      . \\[2pt]
    \mathcal{M}^{(0)} \longrightarrow \mathcal{M}'^{(0)}
  \end{array}
\]

\[
  \mathcal{G}' \longrightarrow \mathcal{G}
\]

\[
  \mathcal{G}'_{\chi'} \longrightarrow \mathcal{G}_{\chi} ,
  \qquad
  \mathcal{G}'_{\psi'} \longrightarrow \mathcal{G}_{\psi} ,
\]

\[
  U' \xrightarrow{\;\sim\;} U ,
  \qquad
  G'^{0} \simeq G^{0} .
\]

\[
  T_{K} = \prod_{K/\mathbb{Q}} \mathbb{G}_{m} ,
  \qquad
  N_{K/\mathbb{Q}} : T_{K} \to T_{\mathbb{Q}} = \mathbb{G}_{m} ,
  \qquad
  V_{K} \, (\subset T_{K}) = \operatorname{Ker} N_{K/\mathbb{Q}} ;
\]

\[
  1 \longrightarrow V_{K} \longrightarrow T_{K} \longrightarrow \mathbb{G}_{m}
  \longrightarrow 1
\]

\[
  V_{K} \xrightarrow{\;N_{K/L}\;} V_{L}
  \xrightarrow{\;\mathrm{can}\;} V_{L}/\operatorname{Im} V_{L_{0}} ,
\]

\[
  V_{\mathbb{R}} \subset (\mathbb{R}^{*r_{1}} \times \mathbb{C}^{*r_{2}})
\]

\[
  \struck{L_{\mathbb{R}} \simeq \mathbb{R}^{r}} \qquad
  L_{0\mathbb{R}} = \ill{}
\]

\[
  V_{K}/V_{K_{0}} \simeq T_{K}/T_{K_{0}} \simeq
  \Bigl[\Bigl(\prod_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}\Bigr) \big/
  \mathbb{G}_{m}\Bigr]^{r} \simeq \mathbb{U}^{r}
\]

\[
  G = \mathrm{Gal}(\widetilde{K}/K)
\]

\[
  V_{K} / (\mathrm{Ker}(N_{K/L}) \cdot V_{L_{0}}) \simeq
  V_{L} / N_{L/K}(V_{L_{0}}) \simeq V_{L}/V_{L_{0}} ,
\]

\[
  1 \longrightarrow V'_{K} \longrightarrow S^{\circ}_{K}
  \xrightarrow{\;\varepsilon\;} \mathbb{G}_{m} \longrightarrow 1
\]

\[
  \mathbb{G}_{m} \xrightarrow{\;i\;} S^{\circ}_{K}
\]

\[
  \mathrm{Spec}\, E_{\overline{K}} \simeq \mathrm{Hom}_{\mathrm{alg}}(E,
  \overline{K})
\]

\[
  (\tau \times 1)\Sigma_{y} = \Sigma'_{y}
\]

\[
  (\tau \times 1)\Sigma = \Sigma' \qquad \text{pour tte conjugaison } \tau
\]

\[
  (1 \times \tau)\Sigma \uncertain{\subset} \Sigma' \qquad
  \text{pour tte conjugaison } \tau
\]

\[
  (\tau \times 1)(\tau \times \tau)\Sigma = \Sigma' \quad \text{i.e.} \quad
  (1 \times \tau)\Sigma = \Sigma' .
\]

\[
  \Psi_{\Sigma} : T_{K} \longrightarrow T_{E}
\]

\[
  \Psi_{\Sigma}(x) = \mathrm{d\acute{e}t}_{t/E}(x_{t}) ,
\]

\[
  \Psi_{\Sigma}(x) = N_{E \otimes K/E}(x^{1_{\Sigma}})
\]

\[
  \mathbb{Z}^{I_{E}} \longrightarrow \mathbb{Z}^{I_{K}}
\]

\[
  \sigma_{1}, \ldots, \sigma_{d'} : K \longrightarrow \overline{E}
\]

\[
  \Psi_{\Sigma}(x) = \sigma_{1}(x)\sigma_{2}(x) \cdots \sigma_{d'}(x)
\]

\[
  \Psi_{\Sigma} | T_{K_{r}} = N_{K_{r}/\mathbb{Q}}
\]

\[
  T_{K} \xrightarrow{\;\mathrm{can}\;} S^{\circ}_{K}
  \xrightarrow{\;\Psi_{\Sigma}\;} T_{E}
\]

\[
  i_{K} : \mathbb{G}_{m,\mathbb{Q}} \longrightarrow S^{\circ}_{K}
\]

\[
  \boxed{\Psi_{\Sigma}\, i_{K} = i_{E/\mathbb{Q}}}
\]

\[
  i_{K}(x)^{d'} = i_{K/\mathbb{Q}}(x) \qquad (x \in \mathbb{Q}^{*})
\]

\[
  \Psi_{\Sigma}(x) = x^{d'} \qquad \text{pour } x \in \mathbb{Q}^{*}
\]

\[
  \Psi_{\Sigma}(S^{\circ}_{K}) = \Psi_{\Sigma}(V'_{K}) \cdot
  i_{E/\mathbb{Q}}(\mathbb{G}_{m})
\]

\[
  \mathrm{Ker}\,\Psi_{\Sigma} \cap \bigcap_{1 \leq i \leq d'}
  \mathrm{Ker}\,\Psi_{\Sigma_{i}} = 1 \qquad \text{dans } S^{\circ}_{K}
\]

\[
  N_{K/L} : S^{\circ}_{K} \longrightarrow S^{\circ}_{L}
\]

\[
  \varphi(\rho\sigma) = -\varphi(\sigma) \qquad \text{pour tt } \sigma \in G
\]

\[
  0 \longrightarrow H^{1}(\mathbb{Q}, T)' \xrightarrow{\;\alpha\;}
  \hat{H}_{0}(G, Y_{\infty} \otimes_{\mathbb{Z}} \check{M})
  \xrightarrow{\;\beta\;} \hat{H}_{0}(G, \check{M}) \longrightarrow 0
\]

\[
  H^{1}(\mathbb{Q}, T)' \simeq \text{gpe des combinaisons linéaires}
\]

\[
  \varinjlim_{T \in \underline{G}/X} T \longrightarrow X ,
\]

\[
  \underline{\underline{A}}^{*}(\hat{\mathbb{Z}})/E^{\circ}
\]

\[
  \underline{\underline{A}}^{*}(\struck{\ill{}}\, \mathbb{Z} \times
  \mathbb{R}^{*}) / \struck{\text{composante}}\,
  \underline{\underline{A}}^{*}(\mathbb{Z})
\]

\[
  G^{\circ} = \underline{\underline{A}}^{*} / \mathfrak{N} ,
\]

\[
  \text{Vectoriels sur } \mathbb{Q} \text{ avec opération de } G
  \xrightarrow{\;\Phi_{\xi}\;} \text{réseaux de Hodge}
\]

\[
  H_{\xi}(V) \subset V_{\mathbb{C}},
\]

\[
  \overline{V}_{\xi} = V \otimes_{\mathbb{Q}} \overline{K}_{\xi} \supset
  H_{\xi}(V)^{(p)}
\]

\[
  k = \mathbb{F}_{q} \;\text{---}\; \bar{k}, \qquad \pi \simeq
  \hat{\mathbb{Z}} \ \text{engendré par } f = \mathrm{frob}^{q}
\]

\[
\begin{aligned}
  H^{0}(\pi, M) &= M^{\pi} = M^{(1-f)} = \operatorname{Ker}(1-f)_{M} \\
  H^{1}(\pi, M) &= M_{\pi} = M_{(1-f)} = \operatorname{Coker}(1-f)_{M} = M/(1-f)M \\
  H^{i}(\pi, M) &= 0 \quad \text{si } i \geq 2
\end{aligned}
\]

\[
  H^{0}(\pi, \mathbb{Z}_{\ell}(n)) = 0 \ \text{si } n \neq 0, \quad =
  \mathbb{Z}_{\ell} \ \text{si } n = 0
\]

\[
\begin{aligned}
  H^{1}(\pi, \mathbb{Z}_{\ell}(n)) &\underset{\text{non canoniquement}}{\simeq}
  \mathbb{Z}_{\ell}/(1-q^{n})\mathbb{Z}_{\ell} \\
  &= \begin{cases}
    \mathbb{Z}_{\ell} & \text{si } n = 0 \\
    0 & \text{si } \ell \nmid (1-q^{n}) \ \text{i.e. } n \not\equiv 0 \ (w_{\ell}(q)) \\
    \mathbb{Z}/\ell^{v_{\ell}(1-q^{n})} & \text{si } \ell \mid 1-q^{n},\ n \neq 0 \ \text{i.e. } n \equiv 0 \ (w_{\ell}(q))
  \end{cases}
\end{aligned}
\]

\[
\left\{
\begin{aligned}
  & H^{i}(\pi, \mathbb{Q}_{\ell}(n)) = 0 \quad \text{si } n \neq 0, \ \text{quel que soit } i \\
  & H^{0}(\pi, \mathbb{Q}_{\ell}) = \mathbb{Q}_{\ell}, \quad H^{1}(\pi, \mathbb{Q}_{\ell}) = \mathbb{Q}_{\ell}
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
  & \mathbb{H}^{i}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Q}}(n)) = 0 \quad \text{si } n \neq 0, \ \text{quel que soit } i \\
  & \mathbb{H}^{0}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Q}}) = \mathbb{Q}, \quad \mathbb{H}^{1}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Q}}) = \underset{\sim}{\mathbb{Q}}.
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
  & \mathbb{H}^{0}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}) = \mathbb{Z} \ (\text{mod } p\text{-groupes ?}), \quad \mathbb{H}^{1}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}) = \mathbb{Z} \ (\text{mod } p\text{-groupes ?}) \\
  & \mathbb{H}^{0}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}(n)) = 0 \ (\text{id}), \quad \mathbb{H}^{1}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}(n)) = \mathbb{Z}/(1-q^{n})\mathbb{Z} \ (\text{mod } p\text{-groupes ?})
\end{aligned}
\right.
\]

\[
\begin{aligned}
  \mathbb{H}^{i}(\operatorname{Spec} \mathbb{Z}, \mathbb{Z}(1)) &=
  \begin{cases} 0 & \text{si } i \neq 3 \\ \mathbb{Z} & \text{si } i = 3 \end{cases} \\
  \mathbb{H}^{i}(\operatorname{Spec} \mathbb{Z}, \mathbb{Z}) &=
  \begin{cases} 0 & \text{si } i \neq 0 \\ \mathbb{Z} & \text{si } i = 0 \end{cases}
\end{aligned}
\]

\[
  \mathbb{H}^{i}(\operatorname{Spec}(\mathbb{Z}), \mathbb{Q}(n))
  \xrightarrow{\ \sim\ } \mathbb{H}^{i}(U, \mathbb{Q}(n))
\]

\[
  i_{x}^{!}(M) = i_{x}^{*}(M)(-1)\,[-2]
\]

\[
  \cdots \to H^{i-2}(x, i_{x}^{*}(M)(-1)) \to H^{i}(X, M) \to H^{i}(U, M) \to
  H^{i-1}(x, i_{x}^{*}(M)(-1)) \to \cdots
\]

\[
  H^{i}(X, M) \xrightarrow{\ \sim\ } H^{i}(U, M)
\]

\[
\begin{aligned}
  0 &\to H^{1}(X, M) \to H^{1}(U, M) \to H^{0}(S, i^{*}(M)(-1)) \to H^{2}(X, M) \\
  &\to H^{2}(U, M) \to H^{1}(S, i^{*}(M)(-1)) \to H^{3}(X, M) \\
  &\to \underset{0\,?}{H^{3}(U, M)} \to 0
\end{aligned}
\]

\[
  H^{i}(X, \mathbb{Q}(n)) \xrightarrow{\ \sim\ } H^{i}(U, \mathbb{Q}(n))
  \qquad \text{tout } i \quad [\text{est-ce vrai ??}]
\]

\[
\begin{aligned}
  0 &\to H^{1}(X, \mathbb{Q}(1)) \to H^{1}(U, \mathbb{Q}(1)) \to \mathbb{Q}^{S} \to H^{2}(X, \mathbb{Q}(1)) \to \\
  &\to H^{2}(U, \mathbb{Q}(1)) \to \mathbb{Q}^{S} \to H^{3}(X, \mathbb{Q}(1)) \to 0
\end{aligned}
\]

\[
  k = \mathbb{F}_{q} \;\text{---}\; \bar{k}, \qquad \pi = \hat{\mathbb{Z}}
  \ \text{engendré par } \mathrm{frob}^{q}_{q}
\]

\[
\begin{aligned}
  H^{0}(\pi, M) &= M^{\pi} \\
  H^{1}(\pi, M) &= \varprojlim_{n} H^{1}(\pi, M/\ell^{n}M) =
\end{aligned}
\]

\[
  0 \to \underset{\substack{\parallel \\ M}}{H^{1}(\pi/\mathfrak{U}, M^{\mathfrak{U}})}
  \to H^{1}(\pi, M) \to
  \underset{\substack{\parallel \\ \mathrm{Hom}(\mathfrak{U}, M^{\pi})}}{H^{0}(\pi/\mathfrak{U}, \mathrm{Hom}(\mathfrak{U}, M))}
  \to \underset{\substack{\parallel \\ M}}{H^{2}(\pi/\mathfrak{U}, M^{\mathfrak{U}})}
\]

\[
\begin{aligned}
  \varphi(xy) &= \varphi(x) + x\varphi(y) \\
  \varphi(x) &= xa - a \\
  \varphi(xy) &= xya - a = xa - a + x(ya - a)
\end{aligned}
\]

\[
\begin{aligned}
  \varphi(x^{2}) &= \varphi(x) + x\varphi(x) \\
  \varphi(x^{3}) &= \varphi(x) + x\varphi(x) + x^{2}\varphi(x) \\
  \varphi(x^{n}) &= (1 + x + \cdots + x^{n-1})\varphi(x) \\
  \text{or } \varphi(x^{-1}x) &= \varphi(x^{-1}) + x^{-1}\varphi(x) \\
  \varphi(x^{-1}) &= -x^{-1}\varphi(x) \\
  \varphi(x^{-n}) &= -(1 + x^{-1} + \cdots + x^{-(n-1)})x^{-1}\varphi(x)
  = -\frac{1 - x^{-n}}{1 - x^{-1}}\, x^{-1}\varphi(x) = \frac{1 - x^{-n}}{1 - x}
\end{aligned}
\]

\[
  \varphi(f^{n}) =
  \begin{cases}
    (1 + x + \cdots + x^{n-1})\varphi(x) = \dfrac{1 - x^{n}}{1 - x}\,\varphi(x) & \text{si } n \geq 1 \\
    0 & \text{si } n = 0
  \end{cases}
\]

\[
  \varphi(f^{n}) = 1 + x + \cdots + x^{n-1} \qquad \frac{1 - f^{n}}{1 - f}\,\varphi(f)
\]

\[
\begin{aligned}
  Z^{1}(\mathbb{Z}, M) &\simeq M \\
  B^{1}(\mathbb{Z}, M) &\simeq (1-f)M \\
  H^{1}(\mathbb{Z}, M) &\simeq M/(1-f)M
\end{aligned}
\]

\[
  H^{1}(X, \mathcal{D}\mathrm{iv}_{X}) = 0
\]

\[
  H^{2}(X, \mathbb{G}_{m,X}) \to H^{2}(X, R^{*}_{X}) \to H^{2}(y,
  \mathbb{G}_{m,y}) = \mathrm{Br}(y)
\]

\[
  \mathrm{Br}(X) \to \mathrm{Br}(y)
\]

\[
  H^{i}_{S}(X, \mathbb{G}_{m}) \to H^{i}(X, \mathbb{G}_{m}) \to H^{i}(U,
  \mathbb{G}_{m}) \to H^{i+1}_{S}(X, \mathbb{G}_{m}) \to
\]

\[
  \mathcal{H}^{i}_{x}(\mathbb{G}_{m}) =
  \begin{cases}
    0 & \text{si } i = 0 \\
    \mathbb{Z}_{x} & \text{si } i = 1 \\
    0 & \text{si } i \geq 2
  \end{cases}
\]

\[
  H^{i}_{S}(X, \mathbb{G}_{m}) = H^{i-1}(S, \mathbb{Z}) =
  \begin{cases}
    0 & \text{si } i = 0 \\
    \mathbb{Z}^{S} & \text{si } i = 1 \\
    0 & \text{si } i = 2 \\
    H^{1}(S, \mathbb{Q}/\mathbb{Z}) & \text{si } i = 3 \\
    0 & \text{si } i \geq 4
  \end{cases}
\]

\[
  0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0
\]

\[
\begin{aligned}
  0 &\to \underset{\substack{\parallel \\ \mathbb{Z}^{*} = \{+1, -1\}}}{H^{0}(\overline{X}, \mathbb{G}_{m})}
  \to H^{0}(\overline{U}, \mathbb{G}_{m}) \to \mathbb{Z}^{S}
  \to \underset{\substack{\parallel \\ 0}}{H^{1}(\overline{X}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0\,!}}{H^{1}(\overline{U}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0}}{H^{2}_{S}(\overline{X}, \mathbb{G}_{m})} \\
  0 &\to \underset{\substack{\parallel \\ 0}}{H^{2}(\overline{X}, \mathbb{G}_{m})}
  \to H^{2}(\overline{U}, \mathbb{G}_{m})
  \to \underset{\substack{\parallel \\ (\mathbb{Q}/\mathbb{Z})^{S}}}{H^{1}(S, \mathbb{Q}/\mathbb{Z})}
  \to \underset{\substack{\parallel \\ \mathbb{Q}/\mathbb{Z}}}{H^{3}(\overline{X}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0\,?}}{H^{3}(\overline{U}, \mathbb{G}_{m})} \to 0
\end{aligned}
\]

\[
\begin{aligned}
  H^{0}(\overline{U}, \mathbb{G}_{m}) &= \mathbb{Z}/2 + \mathbb{Z}^{S} \\
  H^{1}(\overline{U}, \mathbb{G}_{m}) &= 0 \\
  H^{2}(\overline{U}, \mathbb{G}_{m}) &= \operatorname{Ker}\bigl((\mathbb{Q}/\mathbb{Z})^{S} \to \mathbb{Q}/\mathbb{Z}\bigr) \\
  H^{i}(\overline{U}, \mathbb{G}_{m}) &= 0 \quad \text{si } i \geq 3
\end{aligned}
\]

\[
  0 \to \mu_{\ell^{n}} \to \mathbb{G}_{m} \xrightarrow{\ell^{n}}
  \mathbb{G}_{m} \to 0
\]

\[
\begin{aligned}
  H^{1}(\overline{U}, \mu_{\ell^{n}}) &\simeq (\mathbb{Z}/\ell^{n}\mathbb{Z})^{S} \\
  H^{2}(\overline{U}, \mu_{\ell^{n}}) &\simeq \operatorname{Ker}\bigl((\mathbb{Z}/\ell^{n}\mathbb{Z})^{S} \to (\mathbb{Z}/\ell^{n}\mathbb{Z})\bigr) \\
  H^{i}(\overline{U}, \mu_{\ell^{n}}) &= 0 \quad \text{si } i \geq 3
\end{aligned}
\]

\[
  H^{0}(\overline{U}, \mathbb{Z}_{\ell}) \simeq \mathbb{Z}_{\ell}
\]

\[
\left\{
\begin{aligned}
  H^{1}(U, \mathbb{Z}_{\ell}(1)) &\simeq \mathbb{Z}_{\ell}^{S} \\
  H^{2}(\overline{U}, \mathbb{Z}_{\ell}(1)) &\simeq \operatorname{Ker}(\mathbb{Z}_{\ell}^{S} \to \mathbb{Z}_{\ell})
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
  H^{0}(X, \mathbb{Q}(1)) &= 0 \\
  H^{1}(X, \mathbb{Q}(1)) &= 0 \\
  H^{2}(X, \mathbb{Q}(1)) &= 0 \\
  H^{3}(X, \mathbb{Q}(1)) &= \mathbb{Q}
\end{aligned}
\right.
\]

\[
  (5.1.) \qquad 0 \to \mathbb{G}_{m,X} \to R^{*}_{X} \to
  \mathcal{D}\mathrm{iv}_{X} \to 0
\]

\[
  H^{i}(X, \mathcal{D}\mathrm{iv}_{X}) \xrightarrow{\ \partial\ } H^{i+1}(X,
  \mathbb{G}_{m,X})
\]

\[
  i : \mathbb{G}_{m,\mathbb{C}} \to G(s)_{\mathbb{C}}
\]

\[
  \xi(s') \in H^{1}(\mathbb{R}, G^{0u}_{\mathbb{R}})
\]

\[
  u(s') : G(s')_{\mathbb{R}} = G(s)_{\mathbb{R}} \longrightarrow
  G_{\mathbb{R}}^{\xi(s')}
\]

\[
  i : \mu_{2,\mathbb{C}} \longrightarrow G^{0}_{\mathbb{C}}
\]

\[
  \psi(s') = u(s')_{\mathbb{C}} \circ \bigl(i(s') \vert \mu_2\bigr) :
  \mu_{2,\mathbb{C}} \longrightarrow G_{\mathbb{C}}^{\xi(s')}
\]

\[
  f(s') \in G_{\mathbb{R}}^{\xi(s')}(\mathbb{R})
\]

\[
    \xi_{x,\mathbb{Z}} \in \mathbf{T}_\alpha(\mathbb{Z}) \qquad
    \xi_\alpha(\mathbb{Z})(F) = \text{réseau entier dans } F_x ,
  \]

\[
    \begin{cases}
      \xi_{x,\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Z}_\ell \simeq
      \xi_{x,\ell} \\
      \xi_{x,\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{C} \simeq
      \xi_{x,\infty}
    \end{cases}
  \]

\[
  \xi_{x,\mathbb{Z}} \simeq \xi_{\tau x,\mathbb{Z}}
\]

\[
  \begin{array}{l}
    \xi_{a,\infty} \in T_\alpha(\mathbb{R}) \\
    \xi_{\bar{a},\mathbb{Z}} \in T_\alpha(\mathbb{Z})
  \end{array}
  \qquad
  \xi_{\bar{a},\infty} \in T_\alpha(\mathbb{C})
\]

\[
  \xi_{\bar{a},\infty} \overset{\mathrm{can}}{\simeq} \xi_{a,\infty}
  \otimes_{\mathbb{R}} \mathbb{C} \simeq \xi_{\bar{a},\mathbb{Z}}
  \otimes_{\mathbb{Z}} \mathbb{C}
\]

\[
  \xi_{x,\infty} = \xi_{x,\mathbb{R}} \otimes_{\mathbb{R}} \mathbb{C}
  \quad\text{où}\quad
  \xi_{x,\mathbb{R}} \simeq \xi_{x,\mathbb{Z}} \otimes_{\mathbb{Z}}
  \mathbb{R} .
\]

\[
  \xi_{x,\infty} \simeq \xi_{x,\infty\,\mathbb{R}} \otimes_{\mathbb{R}}
  \mathbb{C}
\]

\[
  \tau = \tau'\tau'' = \tau''\tau'
\]

\[
  \boxed{\tau \text{ sur } \xi_{x,\ell}}
\]

\[
  \xi_{x,\mathbb{Q}}(F) \cap \xi_{x,\infty}(F)^{(\rho)} \cap f_\infty\,
  \xi_{x,\infty}(F)^{(\rho)} .
\]

\[
  \pi_1(X, \xi) = \mathrm{Aut}(\xi) = G_\xi(\mathbb{Z}) .
\]

\[
  \xi \in T_{(\alpha)}(k), \quad k \text{ un anneau}
\]

\[
  \pi_1(X,\xi) = \mathrm{Aut}(\xi), \qquad G(X,\xi) =
  \underline{\mathrm{Aut}}(\xi) ;
\]

\[
  \pi_1(X,\xi) \simeq G(X,\xi)(k) .
\]

\[
  M \mapsto \bigl(\xi(M_{\mathbb{C}}),\ \text{structure de Hodge} +
  \text{réalification définie par } f_\infty\bigr)
\]

\[
  G(X,\xi) \to G(Y,\eta),
\]

\[
  G(X,\xi)(k) = \pi_1(X,\xi) \to G(Y,\eta)(k) = \pi_1(Y,\eta) .
\]

\[
  \pi^1(x_j, y_j) \to \pi^1(x_i, y_i)
\]

\[
  \xi_v^{\mathbb{Z}} \mid \mathcal{M}_\alpha(X) \not\simeq
  \xi_w^{\mathbb{Z}} \mid \mathcal{M}_\alpha(X) .
\]

\[
  G_{M,v_{0}}(\mathbb{Z}) = \pi^{1}_{\mathbf{M}}(k, v) \, .
\]

\[
  X = \mathbb{P}^{1}_{\mathbb{C}} - (0) - (1) - (\infty) ,
\]