\[ \mathbb{S} = \prod_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m,\mathbb{C}} , \qquad \text{donc } \mathbb{S}(\mathbb{R}) \simeq \mathbb{C}^{*} . \]

\[ V_{\mathbb{C}} \simeq \coprod_{p,q} V^{p,q} \]

\[ V^{p,q} = \overline{V^{q,p}} . \]

\[ \operatorname{Rep}(G) \xrightarrow{\ \text{oubli}\ } \operatorname{Modf}(\mathbb{Q}) \xrightarrow{\ \otimes_{\mathbb{Q}} \mathbb{R}\ } \operatorname{Modf}(\mathbb{R}) \xrightarrow{\ \otimes_{\mathbb{R}} \mathbb{C}\ } \operatorname{Modf}(\mathbb{C}) \]

\[ V_{\mathbb{C}} \simeq \coprod_{p,q} V_{\mathbb{C}}^{p,q} \]

\[ \mathbb{S} \xrightarrow{\ j\ } G_{\mathbb{R}} \]

\[ i : \mathbb{G}_{m} \longrightarrow G , \]

\[ V^{n}_{\mathbb{C}} = \coprod_{p+q=n} V^{p,q} \]

\[ (\mathrm{a}) \qquad j \,|\, \mathbb{G}_{m,\mathbb{R}} = i_{\mathbb{R}} \]

\[ \varepsilon : G \longrightarrow \mathbb{G}_{m} \]

\[ (*) \qquad \varepsilon j(\lambda) = \lambda^{2} \ \text{ si } \lambda \in \mathbb{R}^{*} , \qquad \varepsilon(j(u)) = 1 \ \text{ si } u \in \mathbb{U} \]

\[ (\mathrm{b}) \qquad \varepsilon j = \text{Norme} : \mathbb{S} \longrightarrow \mathbb{G}_{m,\mathbb{R}} \]

\[ V \times V \xrightarrow{\ \varphi\ } \mathbb{Q}(-p) \]

\[ (**) \qquad \varphi_{\mathbb{R}}(x, Cy) \]

\[ \varphi : V \otimes V \longrightarrow \mathbb{Q}(-n) \]

\[ V_{\mathbb{C}} \times S = V_{S} \]

\[ s \mapsto j(s) : S \longrightarrow \mathcal{J} . \]

\[ G(\mathbb{R})^{\circ} / H(\mathbb{R}) \cap G(\mathbb{R})^{\circ} , \]

\[ G'(\mathbb{R})^{\circ} / H'(\mathbb{R}) \cap G'(\mathbb{R})^{\circ} , \]

\[ F : C \longrightarrow \text{str. de Hodge} \]

\[ C \longrightarrow \operatorname{Modf}(\mathbb{Q}) , \]

\[ \alpha : \pi = \pi_{1}(S, \xi) \longrightarrow G(\mathbb{Q}) \]

\[ V \longmapsto F(V) \longmapsto F(V)|_{\widetilde{S}} \]

\[ f : \widetilde{S} \longrightarrow \mathcal{J} \]

\[ P_{s} = \operatorname{Isom}_{\otimes}(\omega, \Phi_{s} F) \]

\[ C = \operatorname{Rep}(G) \xrightarrow{\ F\ } \operatorname{Hodge}(S) \xrightarrow{\ V \mapsto V \times_{S} P\ } \operatorname{Hodge}(P) \]

\[ g : P \longrightarrow \mathcal{J} \]

\[ \mathcal{J} / \Gamma_{0} \]

\[ C_{\mathbb{R}} = \operatorname{Centr}_{G_{\mathbb{R}}}(i_{0}) \subset K_{\mathbb{R}} \]

\[ Q = G_{\mathbb{R}} / C_{\mathbb{R}} \qquad \text{(a connected component)} \]

\[ \mathcal{J} = G(\mathbb{R})^{\circ} / G(\mathbb{R})^{\circ} \cap C_{\mathbb{R}}(\mathbb{R}) \subset Q(\mathbb{R}) \]

\[ \mathcal{J} / \Gamma , \]

\[ i_{0} : \mathbb{U}_{\mathbb{R}} \longrightarrow G_{\mathbb{R}} \]

\[ \mathcal{J}/\Gamma = \mathcal{J} = \text{complex projective homogeneous space } X \text{ under } G_{\mathbb{C}} \text{ by Borel,} \]

\[ \check{\Theta}_{X} = \check{\Theta}_{0X} + \check{\omega}(X) \]

\[ \check{\omega} : \text{Dér}(K/k_{0}) \longrightarrow \struck{\operatorname{End}_{K}(V_{K})} \ \ K\text{-dérivations} \]

\[ \check{\Theta}_{X}(\varphi) = \check{\Theta}_{0X}(\varphi) + \check{\omega}(X)(\varphi) \]

\[ \omega : \text{Dér}(K/k_{0}) \longrightarrow \mathfrak{g}_{K} \]

\[ \Theta_{0X}\, \omega(Y) - \Theta_{0Y}\, \omega(X) \ \struck{\ill{}} \ = \ \omega([X,Y]) - [\omega(X), \omega(Y)] \]

\[ \omega \in \Omega^{1}_{K/k_{0}} \otimes_{K} \mathfrak{g}_{K} \quad
\bigl( = \Omega^{1}_{K/k_{0}} \otimes_{k_{0}} \mathfrak{g} \bigr) \]

\[ \omega \struck{_{K}} \in \Omega^{1}_{k/k_{0}} \otimes_{k} \mathfrak{g}_{k} \]

\[ \Omega = \check{\mathfrak{g}} \longrightarrow \Omega^{1}_{k/k_{0}}
\quad \bigl( \longrightarrow \Omega^{1}_{K/k_{0}} \bigr) \]

\[ (1) \qquad T_{DR} : \mathcal{M} \longrightarrow \text{Modf}(k) \]

\[ (2) \qquad T_{B} : \mathcal{M} \longrightarrow \struck{\ill{}}\ \text{Str.\ Hodge} \longrightarrow \text{Modf}(\mathbb{Q}) , \]

\[ (3) \qquad \varphi : T_{B} \otimes_{\mathbb{Q}} \mathbb{C} \ \overset{\sim}{\longrightarrow}\ T_{DR} \otimes_{k} \mathbb{C} \]

\[ (4) \qquad P = \operatorname{Isom}_{\otimes}(T_{B} \otimes_{\mathbb{Q}} k,\ T_{DR}) \]

\[ (5) \qquad T_{DR} = \struck{\ill{}}\ P \times^{G_{k}} (T_{B} \otimes_{\mathbb{Q}} k) . \]

\[ \varphi \in P(\mathbb{C}) \]

\[ (6) \qquad T_{\mathrm{Hdg}} : \mathcal{M} \longrightarrow \text{Mod \struck{bi}gradué f}(k) , \qquad T_{\mathrm{Hdg}} = \operatorname{gr}^{*} T_{DR} . \]

\[ (7) \qquad G'_{k} = P \times^{G_{k}} G_{k} = \operatorname{ad}(P) \simeq \operatorname{Aut}_{\otimes}(T_{DR}) \]

\[ (8) \qquad H' \subset G'_{k} \]

\[ (9) \qquad Q = \operatorname{Isom}_{\otimes,\ \text{ind.\ l'ident.\ sur } \operatorname{gr}^{*}}(T_{DR},\ T_{\mathrm{Hdg}}) , \]

\[ (10) \qquad H'' = Q \times^{H'} H = \operatorname{ad}(H') \simeq \struck{\operatorname{Aut}}\ \text{sous-groupe de} \]

\[ (11) \qquad G'' = Q \times^{H'} G' = \operatorname{Aut}_{\otimes}(T_{\mathrm{Hdg}}) . \]

\[ (12) \qquad i_{1}, i_{2} : \mathbb{G}_{m,k} \longrightarrow G' \]

\[ (13) \qquad \varepsilon\, i_{1}(\lambda) = \varepsilon\, i_{2}(\lambda) = \lambda , \qquad i_{1}(\lambda)\, i_{2}(\lambda) = i(\lambda) . \]

\[ \begin{cases} H' = H'' \times^{H''} Q = \operatorname{ad}_{H''}(Q) \\
\struck{\ill{}} \quad G'' \times^{H''} Q = P \quad (\text{tors.\ à d.\ sous } G'') \\
G' = G'' \times^{G''} Q = \operatorname{ad}_{G''}(Q) \end{cases} \]

\[ \underbrace{\pi(\operatorname{Spec} k, \xi)}_{\text{Gpe fond.\ motivique}} \longrightarrow G \]

\[ \psi \in Q(\mathbb{C}) \]

\[ \overline{\psi} = \psi^{-1} . \]

\[ P \times^{G} \mathbb{G}_{m} \ \simeq\ \mathbb{G}'_{m} \times^{G'} P \]

\[ \dot{\varphi} = \frac{1}{2 i \pi}\, \alpha_{\mathbb{C}} \]

\[ \varphi : T \otimes_{k} L \ \simeq\ T' \otimes_{K} L \qquad \otimes\text{-isom.} \]

\[ \begin{cases} G = \operatorname{Aut}_{\otimes}(T) & \text{gpe affine sur } k \\
P = \operatorname{Isom}_{\otimes}(T_{K}, T') & \text{torseur à droite sous } G_{K} \\
\varphi \in P(L) \end{cases} \]

\[ \dot{g} \in G(L)/G(K) \]

\[ V \cap T'(V) = \{\, x \in V \ \mid\ g^{-1} x \in V \otimes_{k} K \,\} . \]

\[ T_{k} \times T_{k} \longrightarrow G_{K} \qquad \text{par } (s,t) \mapsto t s^{-1} ; \]

\[ R \subset G_{K} \]

\[ R_{K} \subset G_{x,K} . \]

\[ S = p(R_{K}) , \qquad p : G_{K} \longrightarrow G , \]

\[ x \in V \cap T'(V) \quad \Longleftrightarrow \quad x \in V^{H}_{\struck{\ill{}}} \]

\[ V^{H} = V^{G} . \]

\[ P \otimes_{K} P \longrightarrow G_{K} , \qquad (s, u) \mapsto s^{-1} u \]

\[ \operatorname{Spec}(L) \times_{\operatorname{Spec} K} \operatorname{Spec}(L)
\longrightarrow P \times_{\operatorname{Spec} k} P \longrightarrow G_{K} ; \]

\[ x \in V \cap T'(V) = \{\, x \in V \ \mid\ R_{K} \subset G_{x,K} \,\} . \]

\[ V \cap T'(W) = V^{H} , \]

\[ P_{\mathbb{C}} \qquad G_{\mathbb{C}} \qquad\qquad Q_{\mathbb{C}} \]

\[ \pi_{1}(S(\mathbb{C}), \xi) \xrightarrow{\ w\ } G(\mathbb{C}) . \]

\[ G/Q \]

\[ G'_{K} \qquad P \qquad G_{K} \qquad G_{\mathbb{C}}/\mathbb{C} \]

\[ f : \overline{S}_{\mathbb{C}} \longrightarrow \Gamma_{G} \]

\[ f(\overline{s})_{\mathbb{C}} : \struck{\ill{}}_{\mathbb{C}} \longrightarrow (G_{k(t)})_{\mathbb{C}} \simeq G_{\mathbb{C}} \]

\[ \widetilde{u} : \pi_{1}(\operatorname{Spec} k, \xi) \longrightarrow \widetilde{G}(\mathbb{A}) , \qquad \pi_{1}(\operatorname{Spec} k, \xi) = \pi_{1} . \]

\[ (*) \qquad \widetilde{P} = P/R \ \simeq\ \widetilde{P}_{0} \otimes_{k} K \]

\[ \widetilde{G}' = \widetilde{G}'_{0} \otimes_{k} K , \qquad
\widetilde{G}' = \operatorname{Aut}_{(\widetilde{G}_{k})}(\widetilde{P}_{0}) , \]

\[ \widetilde{H}' = H'/(H' \cap R') \ \simeq\ \operatorname{Im}(H' \to \widetilde{G}') , \]

\[ T_{\xi} : \mathcal{M} \longrightarrow T_{B}(M_{\xi}) , \qquad \xi \in S(\mathbb{C}) = \operatorname{Hom}_{k}(\operatorname{Spec} \mathbb{C}, S) \]

\[ P = \operatorname{Isom}_{\otimes}(T_{S}, T_{DR}) \]

\[ M \longmapsto P \times^{G_{S}} M_{S} \qquad \longrightarrow \operatorname{Rep}(G) \]

\[ w : \pi_{1}(S(\mathbb{C}), \xi_{0}) \longrightarrow G(\mathbb{Q}) \subset G(\mathbb{C}) \]

\[ q(\overline{s}\,\sigma) = q(\overline{s}) \cdot w(\sigma) , \quad \overline{s} \in \overline{S}_{\mathbb{C}} , \ \sigma \in \pi_{1}(S(\mathbb{C}), \xi_{0}) , \]

\[ \widetilde{P}_{\mathbb{C}} = \overline{S}_{\mathbb{C}} \times^{\Gamma} G(\mathbb{Q})
\hookrightarrow P_{\mathbb{C}}^{\mathrm{an}} \]

\[ f : \widetilde{P}_{\mathbb{C}} \longrightarrow \Gamma_{G} \]

\[ f : \overline{S}_{\mathbb{C}} \longrightarrow \Gamma_{G} \]

\[ \Gamma \longrightarrow \pi_{1}(S, \xi_{0}) \longrightarrow G(\mathbb{A}) , \]

\[ \widetilde{P}_{\mathbb{C}} / G^{\circ}(\mathbb{Q}) \ \struck{\simeq} \ (P/G^{\circ})_{\mathbb{C}}^{\mathrm{an}} , \]

\[ f : \widetilde{P} \longrightarrow \Gamma_{G} \]

\[ \{\, T_{DR}(M_{y}),\ T_{B}(M_{\xi}),\ \varphi_{M} \,\} \]

\[ \varphi_{M} : T_{B}(M_{\xi}) \otimes_{\mathbb{Q}} \mathbb{C} \ \simeq \ T_{DR}(M_{y}) \otimes_{k(y)} \mathbb{C} . \]

\[ \{\, T_{Hdg}(M),\ T_{B}(M_{\xi}),\ \psi_{M} \,\} \]

\[ \psi_{M} : T_{B}(M_{\xi}) \otimes_{\mathbb{Q}} \mathbb{C} \ \simeq \ T_{Hdg}(M) \otimes_{k(y)} \mathbb{C} . \]

\[ \{\, T_{B}(M_{\xi}),\ \rho_{\ell, M} \,\} \]

\[ \rho_{\ell, M} : \pi \longrightarrow \operatorname{Aut}_{\mathbb{Q}_{\ell}}
\bigl( T_{B}(M_{\xi}) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell} \bigr) . \]

\[ \rho_{M} : \pi' = \pi_{1}(S(\mathbb{C}), \xi) \longrightarrow
\operatorname{Aut}_{\mathbb{Q}} \bigl( T_{D}(M_{\xi}) \bigr) , \]

\[ \bigl( T_{DR}(M),\ \sigma,\ T_{B}(M_{y}),\ \varphi \bigr) \]

\[ T_{DR}(M)(\xi) \ \simeq \ T_{B}(M_{y}) \otimes_{\mathbb{Q}} \mathbb{C} . \]

\[ (4.1) \qquad \{\, T_{DR}(M),\ T_{B}(M),\ \varphi \,\} \]

\[ (4.2) \qquad \{\, T_{B}(M),\ \sigma \,\} \]

\[ G = \underline{\operatorname{Aut}}^{\otimes}\bigl( T_{B}\struck{(M)} \ (M \mapsto \otimes\,\ill{}) \ \text{de la catégorie } \mathrm{Mot}(k) \bigr) \longrightarrow \]

\[ V^{G} \ \subset \ V \cap g(\overline{V}) \]

\[ \struck{\overline{V}^{G} \ \subset \ \overline{V} \cap g(\overline{V})} \]

\[ x \in \overline{V}, \qquad g^{-1} x \in \overline{V} \]

\[ x \in \overline{V} , \quad t\, s^{-1} \in G_{x}(\mathbb{C}) \ \text{ pour } t, s \in \overline{T}(\mathbb{C}) , \]

\[ j_{1}, j_{2} : G_{K} \overset{\rho}{\rightrightarrows} G \]

\[ V_{\mathbb{C}} \ \simeq \ V' \otimes_{K} \mathbb{C} \qquad
[\, V' = P \times^{G_{K}} V, \quad
\rho = \struck{\underline{\operatorname{Isom}}^{\otimes}} \bigl( T_{B} \otimes k,\ T_{DR} \bigr) \,] \]

\[ {V'}^{\,T} \ \bigl\{ = T_{DR} \cap (T_{DR})^{00}_{\mathbb{C}} \bigr\}
\ \ill{} \ \neq \ {V'}^{\,G'} . \]