\[
  \begin{cases}
    \gamma \mapsto \widetilde{\sigma}_{2}(\gamma) = \sigma_{2}(\gamma) \cdot j_{2}(\chi(\gamma)) \\
    \pi \longrightarrow E
  \end{cases}
\]

\[
  \gamma \mapsto \sigma_{1}(\gamma) u(\gamma) = \widetilde{\sigma}_{1}(\gamma) .
\]

\[
  \sigma_{2}(\gamma)\, j_{2}(\chi(\gamma)) = \sigma_{1}(\gamma)\, u(\gamma) ,
\]

\[
  \sigma_{1}(\gamma)^{-1} \sigma_{2}(\gamma) = \struck{\ill{}}\; u(\gamma) \cdot j_{2}(\chi(\gamma^{-1})) .
\]

\[
  \begin{cases}
    W_{i}(\mathcal{E}(-1)) = W_{i-2}(\mathcal{E}) \\
    F^{i}(\mathcal{E}(-1)) = F^{i-1}(\mathcal{E})
  \end{cases}
\]

\[
  \begin{cases}
    W_{i}(\mathcal{E}(-n)) = W_{i+2n}(\mathcal{E}) \\
    F^{i}(\mathcal{E}(-n)) = F^{i+n}(\mathcal{E})
  \end{cases}
\]

\[
  \begin{cases}
    P = \operatorname{Isom}(T_{B} \otimes k,\ T_{DR}) \\
    Q \subset P' = \operatorname{Isom}(T_{DR},\ T_{Hdg}) \simeq Q \times^{H'} G' \\
    P'' = \operatorname{Isom}(T_{Hdg},\ T_{B} \otimes k)
  \end{cases}
  \qquad \uncertain{\text{déf.}} / k
\]

\[
  \begin{cases}
    \mathrm{Fil} = \operatorname{Filt}_{\otimes,\ \ill{}}(T_{B}) \\
    \mathrm{Big} = \operatorname{Bigr}_{\otimes,\ \ill{}}(T_{B})
  \end{cases}
  \qquad \uncertain{\text{déf.}} / \mathbb{Q}
\]

\[
  P \times P' \to P'', \qquad P' \times P'' \to P, \qquad P'' \times P \to P',
  \qquad Q \subset P' .
\]

\[
  \beta\alpha = \gamma^{-1}, \qquad \gamma\beta = \alpha^{-1}, \qquad \alpha\gamma = \beta^{-1} .
\]

\[
  \operatorname{deg\,tr} k(g_{0}) = \dim G'' \quad (= \dim G - 1) .
\]

\[
  \lambda = \int_{C} P_{\mathrm{an}}(1_{Y}) = \int_{C} \frac{dz}{z} = 2i\pi .
\]

\[
  \boxed{\ P_{\mathrm{an}}\!\left(\tfrac{1}{2i\pi}\, 1_{Y}\right) = P_{\mathrm{top}}(1_{Y})\ }
\]

\[
  \begin{cases}
    P_{\mathrm{top}}(1_{Y}) = \dfrac{1}{2i\pi}\, P_{\mathrm{an}}(1_{Y}) \\[4pt]
    P_{\mathrm{an}}(1_{Y}) = 2i\pi\, P_{\mathrm{top}}(1_{Y})
  \end{cases}
\]

\[
  a^{p}(X) \xrightarrow{\ P_{\mathrm{alg}}\ } H^{2p}(X, \Omega^{\bullet}_{X})
\]

\[
  P_{\sigma}^{(p)} \colon a^{p}(X) \xrightarrow{\ \mathrm{can} \circ P_{\mathrm{top}}\ } H^{2p}(X, \Omega^{\bullet}_{X})
\]

\[
  P_{\sigma}^{(p)} = \left(\frac{1}{2i\pi}\right)^{p} P_{\mathrm{alg}} ,
\]

\[
  f : X \longrightarrow Y
\]

\[
  H^{i}(X, \mathbb{Q}(0)) \longrightarrow H^{i}(X_{\mathrm{top}}, \mathbb{Z})
\]

\[
  \boxed{\ \xi(y)(x) = \mathrm{pr}^{\mathrm{top}}_{2*}\left( \mathrm{pr}^{*}_{1}(x)\, P_{\mathrm{top}}(\xi) \right)\ }
\]

\[
  \xi_{\mathbb{C}}\left(H^{i}(X, \mathbb{Q}(0))\right) = \xi\left(H^{i}(X, \mathbb{Q}(0))\right) \otimes_{\mathbb{Z}} \mathbb{C} \ \xrightarrow{\ \sim\ }\ H^{*i}(X, \Omega^{*}_{X}) = \xi_{\mathbb{C}, \infty}\left(H^{i}(X, \mathbb{Q}(0))\right)
\]

\[
  x \longmapsto \mathrm{pr}^{\mathrm{top}}_{2*}\left( \mathrm{pr}^{*}_{1}(x)\, P_{\mathrm{top}}(\xi) \right) = \mathrm{pr}^{\mathrm{top}}_{2*}\left( \mathrm{pr}^{\mathrm{an}}_{1}(x) \left(\tfrac{1}{2i\pi}\right)^{n} P_{\mathrm{alg}}(\xi) \right)
\]

\[
  = \left(\tfrac{1}{2i\pi}\right)^{-n} \mathrm{pr}^{\mathrm{alg}}_{2*}\left( \mathrm{pr}^{*}_{1}(x) \left(\tfrac{1}{2i\pi}\right)^{n} P_{\mathrm{alg}}(\xi) \right)
\]

\[
  \xi \otimes_{\mathbb{Z}} \mathbb{Z}_{\ell} \ \xrightarrow{\ \alpha_{1}\ }\ \xi_{\ell}
  \qquad
  \xi \otimes_{\mathbb{Z}} K \ \xrightarrow{\ \alpha_{2}\ }\ \xi_{K}
\]

\[
  \xi\left(H^{2}(\mathbb{P}^{1}, \mathbb{Z}(0))\right) = \xi\left(\mathbb{Z}(-1)\right)
\]

\[
  \alpha_{\infty}(u) = \frac{1}{2\sqrt{-1}\,\pi}\, v
\]

\[
  \prod_{\ell} M_{\ell} = M_{\ell} \times M_{\infty \mathbb{C}}
\]

\[
  \mathrm{Betti}(M)_{K} \simeq DR\struck{(\subseteq DR(M)} \qquad \text{($\otimes$-isom.\ can.)}
\]

\[
  DR(M)_{K} = DR(M_{K})
\]

\[
  DR(M_{K})^{\tau}\ \left(= H^{*}(X(K), \mathbb{R}) \text{ si } M = M^{*}(X)\right)
\]

\[
  f_{\mathbb{R}} \in G(\mathbb{Q}) \ \struck{G(\mathbb{Q})}
\]

\[
  f_{\mathbb{R}}\, i(\lambda)\, f_{\mathbb{R}}^{-1} = i\left(\struck{\ill{}}\, i(\struck{\ill{}}\, \lambda')\right)
\]

\[
  (*) \qquad f_{\mathbb{R}}\, i(\lambda)\, f_{\mathbb{R}}^{-1} = i(\lambda^{-1}) \qquad (\lambda \in U_{\mathbb{R}}\ \struck{\ill{}})
\]

\[
  V \simeq V^{+} + V^{-}, \qquad W = W^{+} + W^{-+} + W^{-},
\]

\[
  \overline{\alpha} = f_{\infty}\, \alpha
\]

\[
  f_{\infty} = 1 \iff \struck{\iff} \alpha \in P(\mathbb{R}) \ \text{ i.e. } \ \mathbb{R}(\alpha) = \mathbb{R}
\]

\[
  \boxed{\ \varepsilon(f_{\infty})\, \varepsilon(f_{\infty}) = -1\ }
\]

\[
  \boxed{\ f_{\infty} \neq 1 \neq 1 \ \text{ i.e. } \ \mathbb{R}(\alpha) = \mathbb{C}\ }
\]

\begin{align}
  \left(f_{\infty} = 1\right)
    &\iff \left(\mathbb{R}(\alpha) = \mathbb{R} \ \struck{\mathbb{R}(\alpha) = \mathbb{R}}\right) \notag \\
    &\iff M \text{ est engendré par } \mathbb{Q}(-2n), \text{ pour quelques } n \geqslant 0 \notag
\end{align}

\[
  \Updownarrow
\]

\[
  G \simeq G_{m} \quad \text{ou} \quad G \simeq \{e\}
\]

\[
  \boxed{\ f_{\mathbb{R}\infty} \in G(\mathbb{Q})\ \struck{G(\mathbb{Q})}\ } \qquad (f_{\infty}^{2} = 1)
\]

\[
  \tau_{M} = (f_{\infty})_{M}\ \struck{(f_{\infty})_{M}}\ \sigma_{M} = \sigma_{M}\, (f_{\infty})_{M}
\]

\[
  P = \mathrm{Isom}_{\otimes}\ \struck{\mathrm{Isom}_{\otimes}}\ \left(T_{B} \otimes_{\mathbb{Q}} \mathbb{R},\ T_{DR}\right)
\]

\[
  j : S_{\mathbb{R}}\ \struck{\to S_{\mathbb{R}}}\ \longrightarrow G_{\mathbb{R}}
\]

\[
  \boxed{\ f_{\infty}\, j(\lambda)\, f_{\infty}^{-1} = \struck{j(\ill{})} = j(\uncertain{\lambda'})\ }
\]

\[
  \beta \in Q(\mathbb{R}) \subset P'(\ \struck{)} \subset P'(\mathbb{R})
\]

\[
  \mathbb{R}(\gamma) = \mathbb{R}(\beta, \gamma) = \mathbb{R}(\beta, \alpha) = \mathbb{R}(\beta, \alpha) = \mathbb{R}(\alpha)
\]

\[
  \mathbb{R}(\gamma) = \mathbb{R}(\alpha)
\]

\[
  f_{\infty}\, C\, f_{\infty}^{-1} = C' = C^{-1} = \eta\, C \qquad (\eta = i(-1))
\]

\[
  f_{\infty} \in \mathrm{Norm}_{G_{\mathbb{R}}}\left(\mathrm{im}_{G_{\mathbb{R}}}(C, C^{-1})\right)
\]

\[
  \dot{f}_{\infty} \in \mathrm{Norm}\ \struck{\mathrm{Norm}}_{\dot{G}_{\mathbb{R}}}\left(\dot{C}, \dot{C}^{-1}\right) \overset{\mathrm{df}}{=} K'
\]

\[
  \dot{K} = \mathrm{Norm}\ \struck{\mathrm{Norm}}_{\dot{G}_{\mathbb{R}}}\left(\dot{C}\right) \subset K'
\]

\[
  H^{*}(X_{\mathrm{top}}, K) \ \xrightarrow{\ \sim\ }\ \struck{(K)} \ \xrightarrow{\ \sim\ }\ H^{*}(X)
\]

\[
  \varphi : \coprod_{p,q} H^{q}(X, \struck{H^{q}(X,}\ \Omega^{p}_{X}) \ \xrightarrow{\ \sim\ }\ H^{*}(X)
\]

\[
  \left[\ \varphi_{K'} : \coprod_{p,q} H^{q}(X, \struck{\Omega\,}(X, \Omega^{p}_{X}) \otimes_{K} K' \ \xrightarrow{\ \sim\ }\ H^{*}(X \struck{\ }) \otimes_{K} K'\ \right]
\]

\[
  \varphi : \coprod_{p,q} H^{q}(\struck{;}\ H^{q}(X, \Omega^{p}_{X}) \ \xrightarrow{\ \sim\ }\ H^{*}(X)
\]

\[
  H^{p,q}(X) = \varphi\left(H^{q}(X, \Omega^{p}_{X})\right) \subset F^{p} H^{q}(X)
\]

\[
  H^{*}(X)_{T} = \struck{H^{*}(X} = H^{*}(X) \otimes_{K} \widehat{K}_{T}.
\]

\[
  \boxed{\ \varphi_{T} : \coprod_{p,q} H^{q}(X, \struck{\Omega\,}(X, \Omega^{p}_{X})_{T} \ \longrightarrow\ H^{*}(X)_{T}\ }
\]

\[
  H^{p,q}_{T}(X) = \varphi_{T} \struck{=} \varphi_{T}\left(H^{q}(X, \Omega^{p}_{X})_{T}\right) \cap H^{*}(X) \ \subset\ F^{p}\!\left(H^{q}(X)\right)
\]

\[
  H^{p,q}_{T}(X, \mathbb{R}) = \varphi_{T} \struck{=} \varphi_{T}\left(H^{q}(X, \Omega^{p}_{X})_{T}\right) \cap H^{*}(X_{T}, \mathbb{R}) \cap H^{*}(X)
\]

\[
  \mathbb{R}_{T} = \struck{\mathbb{R}_{T}} = \mathbb{R} \cap K, \qquad \text{donc } \widehat{K}_{T} \simeq \mathbb{C}
\]

\[
  K_{r} \simeq \bigcap_{T} \mathbb{R}_{T} \ \struck{= \mathbb{R}_{T}} \ \simeq\ \text{plus grand sous-corps tot.\ réel de } K
\]

\[
  \xi : M \rightsquigarrow M_{\xi_{0}} \otimes_{\xi_{0}} \quad (= T(\mathbb{Q}))
\]

\[
  P_{i} = \underline{\mathrm{Isom}}(\xi, \xi_{v_{i}})
\]

\[
  GL(n, \widehat{\mathbb{Z}}) \times \cdots \times GL(n, \widehat{\mathbb{Z}})
  \,\backslash\, GL(n, \widehat{\mathbb{Q}}) \times \cdots \times GL(n, \widehat{\mathbb{Q}})
  \,/\, GL(n, \widehat{\mathbb{Q}})
\]

\[
  M^{\mathbb{Z}}_{v} \subset M^{\mathbb{A}_{\mathbb{C}}}
  \quad\text{où}\quad
  M^{\mathbb{A}_{\mathbb{C}}} = \struck{\ill{}} \prod M(\ell)
\]

\[
  M^{\mathbb{Z}}_{\sigma v} = \dot\sigma M^{\mathbb{Z}}_{v} ,
\]

\[
  \dot\sigma\bigl((x_{\ell})_{\ell\ \mathrm{fini}},\ x_{\infty}\bigr)
  = \bigl(\tau_{\ell} x_{\ell},\ \dot\sigma_{\infty} x_{\infty}\bigr)
\]

\[
  g \in G(\mathbb{A}_{\mathbb{C}}) = \prod_{\ell\ \mathrm{fini}} G(\mathbb{Z}_{\ell}) \times G(\mathbb{C})
\]

\[
  \underline{M}^{\mathbb{Z}}_{w} = g\, M^{\mathbb{Z}}_{v}
\]

\[
  M^{\mathbb{Z}}_{\sigma v} = \struck{g_{\sigma}}\ g_{\sigma} M^{\mathbb{Z}}_{v}
  = \dot\sigma_{\infty} M^{\mathbb{Z}}_{v}
\]

\[
  \mathrm{Gal}(k/k_{0}) \xrightarrow{\ \varphi_{v}\ } G(\mathbb{A}_{\mathbb{C}})/G(\widehat{\mathbb{Z}})
\]

\[
  M^{\mathbb{Z}}_{\sigma v} = g_{\sigma} \varphi_{v}(\sigma) M^{\mathbb{Z}}_{v}
  \quad\text{i.e.}\quad
  \dot\sigma_{\infty}(M^{\mathbb{Z}}_{v}) = \varphi_{v}(\sigma) M^{\mathbb{Z}}_{v}
\]

\[
  g_{\sigma} = \bigl((\sigma_{\ell})_{\ell\ \mathrm{fini}},\ \dot\sigma_{\infty}\bigr)
\]

\[
  \mathrm{Gal}(k/k_{0}) \xrightarrow{\ \varphi_{v,\infty}\ } G_{\struck{\ill{}}}(k)
  \subset GL\bigl(M_{k}(\infty)\bigr)
\]

\[
  \sigma_{*} = \bigl((\sigma_{\ell})_{\ell\ \mathrm{fini}},\ \varphi_{v}(\sigma)\bigr) .
\]

\[
  \varphi_{v}(\tau\sigma) = {}^{\tau}\varphi_{v}(\sigma)\, \varphi_{v}(\tau) ,
  \qquad
  \psi_{v}(\tau\sigma) = \psi_{v}(\tau)\, {}^{\tau}\psi_{v}(\sigma) .
\]

\[
  \mathrm{Gal}(k/k_{0}) \xrightarrow{\ \varphi_{v}\ } G(k)
  \quad\bigl[\subset GL(M_{k}(\infty))\bigr]
\]

\[
  \mathrm{Gal}(k/k_{0}) \longrightarrow G(\mathbb{A}_{\mathbb{C}}) .
\]

\[
  M_{w}^{\mathbb{Z}} = g\,M_{v}^{\mathbb{Z}}
\]

\begin{align*}
  M_{\sigma w}^{\mathbb{Z}} &= \dot{\sigma}(M_{w}^{\mathbb{Z}})
    = \dot{\sigma}\,g\,M_{v}^{\mathbb{Z}}
    = \dot{\sigma}\,g\,\dot{\sigma}^{-1}\,
      (\dot{\sigma} \cdot M_{v}^{\mathbb{Z}}) \\
  &= (\dot{\sigma}\,g\,\dot{\sigma}^{-1})\,\struck{\ill{}}\;\sigma_{*}\,
    M_{\sigma\ill{}}^{\mathbb{Z}} \\
  &= \dot{\sigma}\,g\,\dot{\sigma}^{-1}\,\struck{\ill{}}\;\sigma_{*}\,g^{-1}
    \;\; g\,M_{v}^{\mathbb{Z}}
\end{align*}

\[
  M_{\sigma w}^{\mathbb{Z}} = \struck{\ill{}}\;\sigma'_{*}\,M_{w}^{\mathbb{Z}}
\]

\[
  \sigma'_{*} = \dot{\sigma}\,g\,\dot{\sigma}^{-1}\,\sigma_{*}\,g^{-1}
\]

\[
  \sigma'_{\ell} = \sigma_{\ell}\,g_{\ell}\,\sigma_{\ell}^{-1}\,
    \sigma_{\ell}\,g_{\ell}^{-1} = \sigma_{\ell}
\]

\begin{align*}
  \sigma'_{\infty} &= \dot{\sigma}_{\infty}\,g_{\infty}\,
     \dot{\sigma}_{\infty}^{-1}\,\sigma_{\infty}\,g_{\infty}^{-1} \\
  &= g_{\infty}^{\dot{\sigma}_{\infty}}\,\sigma_{\infty}\,g_{\infty}^{-1}
\end{align*}

\[
  \struck{\varphi_{w}(\sigma) = g_{\infty}^{\dot{\sigma}_{\infty}}\,
    \varphi_{v}(\sigma)\,g_{\infty}}
\]

\[
  \boxed{\;\varphi_{w}(\sigma) = g_{\infty}\,\varphi_{v}(\sigma)\,
    g_{\infty}^{-1}\;}
\]