\[
  \mathfrak{S}_I/V \xrightarrow{\;\sim\;} \operatorname{Aut}_{\mathrm{gr}}(V)
  = \operatorname{Aut}_{\mathbb{F}_2\text{-vect}}(V)
  \xrightarrow{\;\sim\;} \mathfrak{S}_J
\]

\[
  1 \to V \to \mathfrak{S}_I \to \mathfrak{S}_J \to 1
\]

\[
  1 \to \underset{\text{translations}}{V} \to \operatorname{Aut}_{\mathrm{aff}}(I)
  \to \operatorname{Aut}_{\mathbb{F}_2}(V) \to 1 .
\]

\[
  \begin{aligned}
    \mathfrak{P}_2(I) &\longrightarrow \mathfrak{P}_{2,2}(I) \\
    A &\longmapsto \{A, I \setminus A\}
  \end{aligned}
\]

\[
  \mathrm{Aff}_2(\mathbb{F}_2) \xrightarrow{\;\approx\;} (\mathrm{Ens}_4)
\]

\[
  \begin{aligned}
    (\mathrm{Ens}_4) &\longrightarrow (\mathrm{Gr}) \\
    I &\longmapsto \mathfrak{S}_I
  \end{aligned}
\]

\[
  (\mathrm{Gr})' \longrightarrow (\mathrm{Ens}_4)
\]

\[
  1 \to V \to G^+ \to \mathfrak{S}_J^+ \to 1
\]

\[
  \begin{aligned}
    I &\longrightarrow I_G \\
    i &\longmapsto \mathfrak{S}^+_{I \setminus \{i\}}
      = 3\text{-ss-gr.\ de Sylow de } \mathfrak{S}_I
        \text{ qui fixe } i \,;
  \end{aligned}
\]

\[
  G \xrightarrow{\;\sim\;} \mathfrak{S}_{I_G}
\]

\[
      I \times \underline{\omega}, \quad \text{où } \underline{\omega}
      = \omega(J)\ \bigl(\simeq \omega(I)\bigr)
    \]

\[
  \mathrm{Circ}(I) \longrightarrow V^*
\]

\[
  (**) \qquad \underline{\omega}(I) \simeq \underline{\omega}(J)
\]

\[
  \mathfrak{S}_{I,\mathrm{ab}} \xrightarrow{\;\sim\;} \mathfrak{S}_{J,\mathrm{ab}}
  \simeq \{\pm 1\}
\]

\[
  \underline{\omega}_i(I) \xrightarrow{\;\sim\;}
  \underbrace{\underline{\omega}(I \setminus \{i\})}_{\simeq\, \mathfrak{T}_{\mathrm{tot}}}
\]

\[
  \begin{array}{ccc}
    V^* & \xrightarrow[\;\sim\;]{\;x \,\mapsto\, \mathrm{Cent}_G(x)\;} &
      \text{sous-groupes cycliques d'ordre 4 de } G = \mathfrak{S}_I \\[4pt]
    \text{l'unique élément} \neq 1 \text{ de } \Gamma \cap V &
      \longleftarrow\!\!\!\longrightarrow & \Gamma
  \end{array}
\]

\[
  \begin{array}{c}
    V^* \longleftrightarrow
      \text{ss-groupes cycliques d'ordre 4 de } G = \mathfrak{S}_I \\[4pt]
    x \longmapsto \mathrm{Norm}, \qquad
    D \longmapsto \text{élément non nul du centre de } D \\[4pt]
    \text{ss-groupes de Sylow de } G = \mathfrak{S}_I \ (D)
  \end{array}
\]

\[
  (1) \qquad (S, A, R)
\]

\[
  (2) \qquad
  \begin{array}{c}
    S \\
    \big\downarrow {\scriptstyle \text{degré } 2} \\
    D \\
    (\simeq S/\underline{a})
  \end{array}
  \quad \text{un des diagonales (de card.\ 2)}
\]

\[
  (3) \qquad
  \begin{array}{c}
    A \\
    \big\downarrow {\scriptstyle \text{degré } 2} \\
    C \\
    (\simeq A/\underline{a})
  \end{array}
  \quad \text{un des codiagonales}
\]

\[
  \begin{aligned}
    &(2\ \text{bis}) \qquad S \text{ torseur sous } \Gamma \\
    &(3\ \text{bis}) \qquad A \text{ torseur sous } \Gamma
  \end{aligned}
\]

\[
      D = S/\underline{a}, \quad \text{resp.} \quad C = A/\underline{a}.
    \]

\[
      \mathfrak{D} = \operatorname{Aut} \underline{Q}, \qquad
      \underline{Q} = \{S, A, R\},
    \]

\[
  \mathfrak{D}^+_{\mathrm{ab}} \simeq \mathbb{F}_2 \times \mathbb{F}_2
\]

\[
  \left\{
  \begin{array}{lll}
    \text{image inverse de} & \mathbb{F}_2 \times \{0\} & \leadsto V_S \\
    \text{---\,---} & \{0\} \times \mathbb{F}_2 & \leadsto V_A \\
    \text{---\,---} & \text{diagonale de } \mathbb{F}_2 \times \mathbb{F}_2
      & \leadsto \mathfrak{D}^+
  \end{array}
  \right.
\]

\[
  \left\{
  \begin{array}{lll}
    \text{image inverse de} & (1, 0) & \leadsto C \subset \mathfrak{D}^* \\
    \text{---\,---} & (0, 1) & \leadsto D \subset \mathfrak{D}^* \\
    \text{---\,---} & (1, 1) & \leadsto \underline{\omega}_Q \subset \mathfrak{D}^*
  \end{array}
  \right.
\]

\[
  \mathfrak{D} \simeq
  \underbrace{\{1\} \amalg \{a\} \amalg C \amalg D}_{\text{dic.\ de }
  \mathfrak{D} \text{ en classes de conjugaison}} \amalg \underline{\omega}_Q
  \qquad )
\]

\[
  \underline{\omega}_S \simeq C, \qquad \underline{\omega}_A \simeq D
\]

\[
  \underline{\omega}_S \simeq \underline{\omega}_{V_S^*}
  = \underline{\omega}_{\{\underline{a}\} \amalg C}
  \simeq \underline{\omega}_C \simeq C
\]

\[
  (*) \qquad \underline{\omega}_Q \wedge C \wedge D \simeq \mathbb{1}_{\mathbb{F}_2}
\]

\[
  \varpi \wedge c \wedge d = 0 \iff \varpi c d = 1 \text{ dans } \mathfrak{D}
\]

\[
  \underline{\omega}_{Q'} \wedge C' \wedge D' \simeq \mathbb{1}
  \ (\simeq \underline{\omega}_Q \wedge D \wedge C) \simeq \mathbb{1}
\]

\[
  C \times D
\]

\[
  C \times D \simeq R/\underline{a} \qquad
  R = \operatorname{Rep}(Q) \simeq \operatorname{Rep}(Q^\circ)
\]

\[
  C \wedge D \simeq \underline{\omega}_Q \quad \text{i.e.\ d'une relation}
  \quad \varpi c d = 1
\]

\[
  C \hookrightarrow V_S \setminus \{1, \underline{a}\} \subset \mathfrak{D},
  \qquad D \hookrightarrow V_A \setminus \{1, \underline{a}\} \subset \mathfrak{D},
  \qquad \underline{\omega}_Q \subset \mathfrak{D}^+ \subset \mathfrak{D}
\]

\[
  1 \to \underset{\substack{\text{autom.\ de l'ext.\ } \mathfrak{D} \text{ de}\\
  W \text{ par } \mathbb{F}_2 \text{ qui}\\ \text{induisent l'identité}\\
  \text{sur } W \text{ et } \mathbb{F}_2}}{\check{W}}
  \to \operatorname{Aut} \mathfrak{D} \to
  \underset{\substack{\text{interchange}\\ \text{les deux}\\ \text{éléments de}\\
  W \setminus \{0, \varpi\}}}{\underbrace{\pm 1}} \to 1
\]

\[
  \mathbb{D}_4 = \operatorname{Aut}(Q_0)
\]

\[
  1 \to \mathbb{F}_2 \to \mathbb{D}_4 \to \mathbb{F}_2^{2} \to 1
\]

\[
  (\text{Cubes}) \xrightarrow{\;\approx\;} (\text{groupes isom.\ à }
  \mathbb{D}_4)
\]

\[
  x \longmapsto x \cdot \underline{a} \overset{\mathrm{def}}{=} x'
\]

\[
  (\text{groupes isom.\ à } \mathbb{D}_4) \longrightarrow
  (\text{cubes combinatoires})
\]

\[
  \mathfrak{D} = \Bigl\{ \underbrace{x \in \mathfrak{S},\ \underline{a}}_{
  \text{générateurs}} \Bigm| \underbrace{[x, y]}_{\substack{\text{commutateur}\\
  xyx^{-1}y^{-1}}} = \underline{a} \text{ si } y \neq x, x' \Bigr\}
\]

\[
  A \longrightarrow C,
\]

\[
  S \simeq \prod_{c \in C} A_c = \text{Ens des sections de } A
  \text{ sur } C
\]

\[
  S \hookrightarrow \mathfrak{P}_2(A)
\]

\[
  A \simeq \prod_{d \in D} S_d \qquad (\simeq \text{sections de } S \text{ sur } D)
\]

\[
  S \longrightarrow D
\]

\[
  A \hookrightarrow \mathfrak{P}_2(S)
\]

\[
  \underset{\text{codiagonales}}{C}, \quad
  \underset{\text{diagonales}}{D}, \quad
  \underset{\text{orientations}}{\underline{\omega}}
\]

\[
  S \xrightarrow{\;\deg.\,2\;} D, \qquad A \xrightarrow{\;\deg.\,2\;} C
\]

\[
  (D \simeq S/\underline{a}_S). \qquad (C \simeq A/\underline{a}_A)
\]

\[
  \begin{array}{ll}
    (1) & \underline{\omega}_S \simeq C, \quad \underline{\omega}_A \simeq D \\[6pt]
    (2) & \underline{\omega} \wedge C \wedge D \simeq \mathbb{1}_{\mathbb{F}_2}
      \quad \text{i.e.} \quad
      \left\{ \begin{array}{l}
        \underline{\omega} \simeq C \wedge D \\
        C \simeq D \wedge \underline{\omega} \\
        D \simeq \underline{\omega} \wedge C
      \end{array} \right. \\[18pt]
    (3) & D \simeq S/\underline{a}_S, \quad C \simeq A/\underline{a}_A \\[6pt]
    (4) & S \simeq \prod_{c \in C} A_c, \quad A \simeq \prod_{d \in D} S_d \\[6pt]
    (5) & D \simeq \bigwedge_{c \in C} A_c, \quad C \simeq \bigwedge_{d \in D} S_d
  \end{array}
\]

\[
  \underline{\omega} \simeq \Bigl(\bigwedge_{c \in C} A_c\Bigr) \wedge C
  \quad \text{d'où} \quad
  \underbrace{\underline{\omega} \wedge C}_{D \text{ par } (2)} \simeq
  \bigwedge_{c \in C} A_c .
\]

\[
  (*) \qquad
  \underset{\substack{\text{ens. des}\\ \text{permutations}\\ \text{circulaires}\\ \text{sur } I}}{\mathrm{Circ}(I)} \longrightarrow J, \qquad u \longmapsto u^2 .
\]

\[
  \underset{\substack{\simeq\ \text{ens. des}\\ \text{« transpositions »}\\ \text{de } I\\ \text{(ou « réflexions »)}}}{\mathfrak{P}_2(I)} \longrightarrow J,
  \qquad A \longmapsto \{A, I \setminus A\}
\]

\[
  \text{ou}\quad \forall\, \tau \text{ réflexion}, \ \exists!\, \tau' \text{ réflexion telle que } \tau\tau' \in V^*
\]

\[
  \underset{\underset{\textstyle \underline{\omega}_J\,(\simeq \underline{\omega}_I)}{\| \mathrm{def}}}{\underline{\omega}_Q}
  \ \simeq\
  \bigwedge_{i \in J}
  \underset{\substack{\text{ens. des}\\ \text{deux rotations, d'ordre 4}\\ \text{de la structure carrée}\\ Q_i \text{ définie par } i \in J\\ \text{i.e. ens. des deux orientations}\\ \text{de } Q_i}}{\underline{\omega}_i}
\]

\[
  \underline{\omega}_i \simeq
  \underset{\substack{\text{ens.}\\ \text{des deux}\\ \text{codiagonales}\\ \text{de } Q_i}}{\underline{\omega}} \wedge
  \underset{\substack{\text{ens. des diag.}\\ \text{pour le carré } Q_i}}{D_i}, \quad \text{i.e. } D_i = I/\underset{\substack{\text{translation}\\ \text{indexée par } i}}{a_i}
\]

\[
  \underset{\text{torseur sous } \mathbb{F}_2^J}{\bigwedge_{i \in J} D_i} \simeq \mathbb{1}
\]

\[
  \underset{\text{torseur sous } V_J(\mathbb{F}_2)}{I} \hookrightarrow \underset{\text{torseur sous } \mathbb{F}_2^J}{\prod_{i \in J} D_i}
\]

\[
  8,\ 12,\ 6,\ 4,\ 6,\ 3,\ 24 .
\]

\[
  (1) \qquad M_0 = M/2M = M \otimes_\Lambda \mathbb{F}_2
\]

\[
  (3) \qquad
  \left\{
  \begin{aligned}
    M_i &= M_{x_i} = \ldots \\
    M_J &= \prod_{i \in J} M_i, \quad \text{un torseur sous } M_0^J .
  \end{aligned}
  \right.
\]

\[
  (4) \qquad \sum_{i \in J} x_i = 0
\]

\[
  (6) \qquad \bigwedge_{i \in J} M_i = 0
\]

\[
  (7) \qquad V_J(M_0) = \Bigl\{ (\xi_i)_{i \in J} \in M_0^J \Bigm| \sum \xi_i = 0 \Bigr\}
\]

\[
  (8) \qquad 0 \to V_J(M_0) \to M_0^J \to M_0 \to 0 ,
\]

\[
  (9) \qquad M_i \simeq P_M \wedge^{M_0^J} (M_0, \mathrm{pr}_i) \qquad \text{pour } i \in J
\]

\[
  \mathrm{pr}_i : M_0^J \to M_0 \qquad \text{la projection d'indice } i \in J .
\]

\[
  (10) \qquad \operatorname{End}(M_0) \simeq V_J(M_0) \xrightarrow{\;\sim\;} M_0 \otimes M_0 \simeq M_0 \oplus \mathbb{F}_2^{\underline{\omega}_{M_0}}
\]

\[
  \underline{\omega}_{M_0} \simeq \underline{\omega}(\underset{J}{\underbrace{M_0^*}}) \quad \text{ens. des orientations de } J .
\]

\[
  (12) \qquad
  \left\{
  \begin{aligned}
    \operatorname{End}(M_0) &\xrightarrow{\;\sim\;} V_J(M_0) \\
    u &\longmapsto (u(x_i))_{i \in J}
  \end{aligned}
  \right.
\]

\[
  \operatorname{End}(M_0) \simeq M_0^\vee \otimes M_0 \simeq M_0 \otimes M_0 \quad \text{car } M_0^\vee \simeq M_0
\]

\[
  (13) \qquad
  \begin{array}{ccc}
    M_0 & \hookrightarrow & \operatorname{End}(M_0) \\
    \cup & & \cup \\
    M_0^* & \longleftrightarrow & \operatorname{Aut}(M_0) \simeq \mathfrak{S}_J \\
    x_i & \longmapsto & \sigma_i
  \end{array}
\]

\[
  (14) \qquad
  \begin{array}{ccc}
    \mathbb{F}_2^{\underline{\omega}_{M_0}} & \hookrightarrow & \operatorname{End}(M_0) \\
    \cup & & \cup \\
    \underline{\omega}_{M_0} & \xhookrightarrow{\ \mathrm{can}\ } & \operatorname{Aut}(M_0) \simeq \mathfrak{S}_J \\
    \| & & \\
    \underline{\omega}_J \simeq \mathfrak{S}_J^+ \setminus \{\mathrm{id}_J\} & &
  \end{array}
\]

\[
  (15) \qquad
  \left\{
  \begin{array}{l}
    M_0 \text{ et } \mathbb{F}_2^{\underline{\omega}} \text{ sont orthogonaux l'un de l'autre} \\
    \text{dans } \underline{\operatorname{End}}(M_0) \text{ pour la forme } \operatorname{Tr}(uv)
  \end{array}
  \right.
\]

\[
  (16) \qquad \mathfrak{sl}(M_0) = M_0 \oplus \mathbb{F}_2\, \mathrm{id}_{M_0}
\]

\[
  (17) \qquad \mathbb{F}_2^{\underline{\omega}} \cap \mathfrak{sl}(M_0) \simeq \mathbb{F}_2 \cdot \mathrm{id}_{M_0} \qquad \ldots \qquad ]
\]

\[
  \bigl(\mathbb{Z}/4\mathbb{Z}\text{-modules libres de rang } 2\bigr)
  \longrightarrow
  \text{triplets } (M_0, I, S) \ \text{avec}\
  \left\{
  \begin{array}{l}
    M_0 \text{ vect. de rg } 2 \text{ sur } \mathbb{F}_2 \\
    I \ \ M_0\text{-torseur} \\
    S \ \ \text{torseur sous } \mathbb{F}_2^{\underline{\omega}(M_0)}
  \end{array}
  \right.
\]

\[
  (20) \qquad k : \underset{\substack{\text{ens. des}\\ \text{orientations}\\ \text{de } I}}{\underline{\omega}(I)} \simeq \underset{\substack{\text{ens. des}\\ \text{codiagonales}\\ \text{de } Q}}{C(Q)} .
\]

\[
  (21) \qquad
  \Bigl( \underset{\mathbb{Z}/4\mathbb{Z}}{\underline{\Lambda}}\text{-modules libres de rg } 2 \Bigr)
  \xrightarrow{\;\approx\;}
  \text{triplets } (I, Q, k), \ \text{où}\
  \left\{
  \begin{array}{l}
    I \ \text{ens. à 4 éléments} \\
    Q \ \text{cube combinatoire} \\
    k : \underline{\omega}(I) \simeq C(Q)
  \end{array}
  \right.
\]

\[
  \operatorname{GL}_\Lambda(M) \xrightarrow{\;\det\;} \{\pm 1\}
\]

\[
  (24) \qquad \operatorname{GL}_\Lambda(M) \longrightarrow \mathfrak{D}_Q \underset{\substack{\text{noyau } \mathfrak{D}_Q^+,\\ \text{cyclique d'ordre 4}}}{\underbrace{\xrightarrow{\;\det\;}}} \{\pm 1\} .
\]

\[
  (26) \qquad \operatorname{GL}_\Lambda(M)_{\mathrm{ab}} = \operatorname{GL}_\Lambda(M) / \mathfrak{S}_I^+ \times \underset{\text{centre de } \operatorname{GL}_\Lambda(M)}{\underbrace{\{1, -\mathrm{id}_M\}}} \xrightarrow{\;\sim\;} \underset{\substack{\text{vectoriel de rang 2 sur } \mathbb{F}_2\\ \text{et même \underline{canon.} isom. à}\\ \mathbb{F}_2 \times \mathbb{F}_2}}{(\mathfrak{D}_Q)_{\mathrm{ab}}}
\]

\[
  (27) \qquad \mathfrak{z} = \{1, -\mathrm{id}_M\} \ \bigl(= \operatorname{Cent}(\operatorname{GL}_\Lambda(M)) = \operatorname{Cent}(\operatorname{SL}_\Lambda(M))\bigr)
\]

\[
  \begin{array}{l}
    \underline{\omega} \subset \mathbb{F}_2^{\underline{\omega}} = \mathbb{F}_2^C = V_S \subset \mathfrak{D}_Q \quad (C_Q \simeq \underline{\omega}) \\
    D_Q \simeq \underline{\omega} \wedge \mu \subset \mathbb{F}_2^D = V_A \subset \mathfrak{D}_Q \\
    \underline{\omega}_Q \simeq \mu \subset \mathfrak{D}_Q^+ \subset \mathfrak{D}_Q
  \end{array}
  \qquad
  \begin{array}{l}
    \underline{\omega} \simeq V_S \setminus \mathfrak{z}_Q \longleftrightarrow \mathrm{sg}(g) \\
    \underline{\omega} \wedge \mu \simeq V_A \setminus \mathfrak{z}_Q \longleftrightarrow \mathrm{sg}(g)\det(g) \\
    \mu \simeq \mathfrak{D}_Q^+ \setminus \mathfrak{z}_Q \longleftrightarrow \det(g)
  \end{array}
\]

\[
  \mu \simeq \bigl(\textstyle\bigwedge^2 M\bigr)^* = \operatorname{Or}(M), \qquad \mathfrak{z}_Q = \{1, \underline{a}_Q\} = \operatorname{Cent} \mathfrak{D}_Q
\]

\[
  \begin{aligned}
    \operatorname{GL}_\Lambda(M) &\simeq \mathfrak{S}_I \cdot \mathbb{F}_2^{\underline{\omega}} && (\tfrac12\text{ direct}) \ \text{et} \\
    \operatorname{Ker}(\mathrm{sg} \cdot \det) &\simeq \mathfrak{S}_I \times \mathbb{F}_2 && (\text{prod. direct})
  \end{aligned}
\]

\[
  \begin{aligned}
  \operatorname{GL}_\Lambda(M) &\simeq \underset{\mathfrak{S}_J}{\underbrace{\operatorname{GL}_{\mathbb{F}_2}(M_0)}} \cdot (M_0 \times \mathbb{F}_2^{\underline{\omega}}) \\
  &\simeq \bigl(\underset{\mathfrak{S}_I}{\underbrace{\operatorname{GL}_{\mathbb{F}_2}(M_0) \cdot M_0}}\bigr) \times \mathbb{F}_2^{\underline{\omega}}
  \end{aligned}
\]

\[
  \begin{aligned}
  (29) \qquad M^{\#} &= \{ x \in M \mid x^0 \ (\text{image dans } M_0) \neq 0 \} \\
  &= \text{image inverse de } J = M_0^{\#} \subset M_0 \text{ par } M \to M_0
  \end{aligned}
\]

\[
  (30) \qquad \Delta_M = M^{\#}/(\mathbb{Z}/4\mathbb{Z})^* = M^{\#}/\{\pm 1\} = \mathbb{P}(M)(\mathbb{Z}/4\mathbb{Z})
\]

\[
  (31) \qquad \Delta_M \longrightarrow J
\]

\[
  \mathbb{P}(M)(\mathbb{Z}/4\mathbb{Z}) \longrightarrow \mathbb{P}(M)(\mathbb{F}_2) .
\]

\[
  (32) \qquad f \xrightarrow{\;\sim\;} J \ ;
\]

\[
  \underset{\substack{\wr\\ \operatorname{GL}(M)/\{\pm 1\}}}{\operatorname{GL}(M)'} \longrightarrow \operatorname{Aut}_{\mathrm{oct}}(\Delta)
\]

\[
  (33) \qquad
  \underset{\substack{\|\\ \operatorname{GL}(M)/(\mathfrak{z} = \{\pm 1\})\\ \wr\\ \operatorname{GP}(M)}}{\operatorname{GL}(M)'} \xrightarrow{\;\sim\;} \operatorname{Aut}_{\mathrm{oct}}(\Delta) \simeq \operatorname{Aut}^+(\Delta) \times \underset{\substack{\text{antipodisme}\\ \text{centre de } \operatorname{Aut}_{\mathrm{oct}}(\Delta)}}{\{1, \underline{a}_\Delta\}}
\]

\[
  (34) \qquad \underset{\text{ens. des faces de } \Delta}{\underbrace{F(\Delta_M)}} = P_{M,\Delta} \quad \text{torseur sous } \mathbb{F}_2^J \qquad (J = M_0^{\#})
\]

\[
  (35) \qquad \mathbb{F}_2^J \simeq \underset{\substack{\wr\\ M_0}}{V_J(\mathbb{F}_2)} \times \underset{\text{diag}}{\mathbb{F}_2}
\]

\[
  (36) \qquad P_{M,\Delta} = F(\Delta_M) \xleftarrow{\;\sim\;} P_M/\mathfrak{z} \qquad \underset{\text{centre de } \operatorname{GL}(M)}{\mathfrak{z} = \{\pm \mathrm{id}_M\}}
\]

\[
  (37) \qquad
  \begin{array}{l}
    P_M = \{ (\xi_i)_{i \in J} \in M^{\# J} \mid \xi_i^0 = i_* \ \forall i \in J,\ \sum \xi_i = 0 \} \\
    \quad \big\downarrow \qquad (\xi_i) \longmapsto \{(\dot\xi_i)\} \quad \text{où } \dot\xi = \Lambda^* \xi = \pm \xi \in \Delta \text{ pour } \xi \in M^{\#} \\
    F(\Delta_M) = \{ f \in \mathfrak{P}_3(\Delta_M) \mid \underset{\text{induit par } \Delta \to J}{\underbrace{f \to J}} \text{ bijectif} \}
  \end{array}
\]

\[
  \operatorname{Aut}_{\mathrm{oct}}(\Delta)_{\mathrm{ab}} \simeq \bigl(\operatorname{Aut}^+_{\mathrm{oct}}(\Delta) \times \{1, \underline{a}_\Delta\}\bigr)_{\mathrm{ab}} \simeq \mathbb{F}_2 \times \mathbb{F}_2
\]

\[
  (40) \qquad
  \left\{
  \begin{array}{l}
    C_\Delta \ (\text{« couleurs »}) : \text{ ens.\ des 2 tétraèdres (échangés par l'antipodisme)} \\
    \quad \text{inscrits dans le cube qui correspond à } \Delta, \\
    \quad \text{ou encore des deux couleurs (échiquier\ldots) des faces de l'octaèdre } \Delta \\
    \quad \text{correspond au car.\ \ldots\ de } \operatorname{Aut}^+_{\mathrm{oct}}(\Delta) \times \{1, \underline{a}_\Delta\} \\[4pt]
    \operatorname{Or}(\Delta) : \text{ ens.\ des deux orientations de l'octaèdre } \Delta \\
    \quad \simeq \text{ correspond au car.\ non trivial de } \{1, \underline{a}_\Delta\} \\[4pt]
    \underline{\omega}_\Delta = \underline{\omega}_{I_\Delta} : \text{ ens.\ des deux orientations de } I_\Delta,
      \text{ ou encore de } J_\Delta \\
    \quad \text{correspond au caractère non trivial de } \operatorname{Aut}^+_{\mathrm{oct}}(\Delta)
  \end{array}
  \right.
\]

\[
  (41) \qquad \operatorname{Or}(\Delta) \wedge \underline{\omega}_\Delta \wedge C_\Delta \simeq \mathbb{1}_{\mathbb{F}_2} .
\]

\[
  (42) \qquad
  \left\{
  \begin{array}{l}
    C_\Delta = \bigwedge_{i \in J} \Delta_i \quad \text{quotient de } \overbrace{\prod_{i \in J} \Delta_i}^{\text{torseur sous } \mathbb{F}_2^J} = F(\Delta) \quad \text{par } \underset{\substack{\wr\\ \underbrace{V_J(\mathbb{F}_2)}_{M_0} \times \mathbb{F}_2}}{\mathbb{F}_2^J} \xrightarrow{\text{(somme)}} \mathbb{F}_2 \\[4pt]
    \text{i.e. } C_\Delta = \underbrace{F(\Delta)/M_0}_{(P_M/(M_0 \times \mathfrak{z}) \text{ dans le cas qui nous occupe})}
  \end{array}
  \right.
\]

\[
  (43) \qquad \underline{\omega}_\Delta = \underline{\omega}_{I_\Delta} = \underline{\omega}_J \qquad \text{l'ens. à deux éléments associé à } J
\]

\[
  (44) \qquad \operatorname{Or}(\Delta) \simeq \underset{\substack{\underline{\omega}_J\\ \Delta^+}}{\underbrace{\underline{\omega}_\Delta}} \wedge C_\Delta \qquad \left(\begin{array}{l}\text{correspond au caractère } \det(g) \text{ dans le cas d'un } M) \\ \text{d'où } \boxed{\operatorname{Or}(\Delta) \simeq \mu \overset{\mathrm{def}}{=} (\det M)^*}\end{array}\right.
\]

\[
  (45) \qquad
  \left\{
  \begin{aligned}
    \Delta^+ &= \operatorname{Or}(\Delta) \wedge^{\{1, \underline{a}_\Delta\}} \Delta \\
    \Delta &= \operatorname{Or}(\Delta) \wedge^{\{1, \underline{a}_\Delta\}} \Delta^+
  \end{aligned}
  \right.
\]

\[
  (45) \qquad
  \left\{
  \begin{array}{l}
    \operatorname{Or}(\Delta^+) \simeq \{\pm 1\} = \mathbb{1}_{\mathbb{F}_2} \\
    \underline{\omega}_{\Delta^+} \simeq \underline{\omega}_\Delta \simeq \underline{\omega}_J = \underline{\omega} \quad [\text{car } \underline{a}_\Delta \text{ opère trivialement sur } \underline{\omega}_\Delta] \\
    C_{\Delta^+} \simeq C_\Delta \wedge \mu \simeq \underline{\omega} \qquad \ldots
  \end{array}
  \right.
\]

\[
  (47) \qquad 1 \to \underset{\substack{\wr\\ \{\pm 1\}}}{\mathfrak{z}} \longrightarrow \operatorname{GL}(M) \longrightarrow \underset{\substack{\wr\\ \operatorname{Aut}_{\mathrm{oct}}(\Delta_M)'\\ \simeq \operatorname{Aut}^+_{\mathrm{oct}}(\Delta_M) \times \{1, \underline{a}_\Delta\}}}{\operatorname{GL}(M)'} \longrightarrow 1
\]

\[
  (48) \qquad 1 \to \underset{\substack{\|\\ \{\pm 1\}}}{\mathfrak{z}_0} \longrightarrow \operatorname{GL}(2, \mathbb{Z}/4\mathbb{Z}) \longrightarrow \underset{\substack{\wr\\ \operatorname{Aut}_{\mathrm{oct}}(\Delta_{(\mathbb{Z}/4\mathbb{Z})^2})\\ \text{« octaèdre standard »}}}{\operatorname{GL}(2, \mathbb{Z}/4\mathbb{Z})'} \longrightarrow 1
\]

\[
  K : \underset{\text{codiagonales}}{C_Q} \xrightarrow{\;\sim\;} \underline{\omega}_I
\]

\[
  C_\Delta \simeq F(\Delta)/M_0 \simeq P_{M, \mathbb{F}_2^{\underline{\omega}}}/\mathbb{F}_2 = S_Q/\mathbb{F}_2 \simeq D_Q
\]

\[
  (49) \qquad
  \left|
  \begin{array}{l}
    \text{a) L'octaèdre (non orienté) } \Delta_M = M^{\#}/\pm 1 \ (\text{ens. des sommets}), \\
    \quad \text{l'ens. des trois diagonales s'identifiant à } (M/2M)^* \simeq M_0^{\#} = J \\
    \text{b) L'ensemble à quatre éléments } I, \text{ qu'on peut p.\ ex.\ déduire de } \Delta \\
    \quad \text{comme } I_M \simeq \underset{\text{faces de } \Delta}{\underbrace{F(\Delta)}}/\text{antipodisme} \\
    \text{c) Le cube } Q_M, \ S_M = P_M/M_0
  \end{array}
  \right.
\]

\[
  (50) \qquad
  \left\{
  \begin{array}{llll}
    \underset{\text{codiag.\ de } Q}{C_Q} \simeq \dot{\underline{\omega}}_\Delta & \overset{\mathrm{déf}}{\simeq} \underline{\omega}_I \simeq \underline{\omega}_J & & \text{soit } \underline{\omega} \\
    \underset{\text{diag.\ de } Q}{D_Q} \simeq \underset{\text{« couleurs » des faces}}{C_\Delta} & \simeq \underline{\omega} \wedge \mu & & \text{où } \mu \text{ défini ci-dessous} \\
    \underset{\text{or.\ de } Q}{\underline{\omega}_Q} \simeq \operatorname{Or}(\Delta) & \simeq \det(M)^* & & \text{soit } \mu
  \end{array}
  \right.
\]

\[
  \begin{array}{c}
    \big\updownarrow \\
    \bigl(\underline{I},\ \text{d'où } \underline{J},\ t \in \Gamma(\Sigma_J^*/S)\bigr) \longleftrightarrow \bigl(\underline{J}\ (\text{d'où } \underline{V}_J \text{ et } \Sigma_J),\ \text{section de } \Sigma_J^* \text{ sur } S,\ \text{torseur } I \text{ sous } \underline{V}_J\bigr) \\
    \big\updownarrow \\
    \bigl(\underline{J},\ t \in \Gamma(\Sigma_J^*/S),\ X \text{ rev.\ de } \Sigma_J^*,\ \text{étale sur } \Sigma_J^*,\ \ldots\bigr) \\
    \big\updownarrow \\
    \bigl(\underline{J},\ t \in \Gamma(\Sigma_J^*/S),\ \underline{K} \text{ torseur sous } \underline{J}\bigr)
  \end{array}
\]

\[
  V' \subset \operatorname{End}(V), \qquad
  V' = \operatorname{Aut}^-(V) \cup \{0\}
\]

\[
  W \subset \operatorname{End}(V), \qquad
  W = \operatorname{Aut}^+(V) \cup \{0\}
\]

\[
  \Bigl(\mathrm{Tr}\, xy =
  \begin{cases}
    0 & \text{si } x = y, \text{ ou } x \text{ ou } y = 0\\
    1 & \text{sinon i.e.\ } x, y \in V'^{*},\ x \neq y
  \end{cases}\Bigr),
\]

\[
  0 \to \underset{\lambda \mapsto \lambda \cdot 1}{\mathbb{F}_2} \longrightarrow W
  \xrightarrow{\;\mathrm{Tr}\;} \mathbb{F}_2 \to 0 .
\]

\[
  \mathfrak{sl}(V) \overset{\text{déf}}{=}
  \operatorname{Ker}\bigl(M \xrightarrow{\;\mathrm{Tr}\;} \mathbb{F}_2\bigr)
  = V' + \mathbb{F}_2 \cdot 1_V .
\]

\[
  (1) \qquad V_S \hookrightarrow \underline{\operatorname{Aut}}_S(X) = G
\]

\[
  V_S \longrightarrow \underline{\operatorname{Aut}}_S(X) \quad \text{i.e.}\quad
  V \longrightarrow \operatorname{Aut}_S(X)
\]

\[
  (2) \qquad \Delta_1 \cap \Delta_2 = \emptyset \quad \text{i.e.}\quad
  \Delta_2 \subset X \setminus \Delta_1
\]

\[
  (3) \qquad e_3 = e_1 + e_2
\]

\[
  (4) \qquad
  \begin{cases}
    \Delta_1 = \{0, \infty\}, & \Delta_2 = \{1, -1\} \\
    \sigma_1(z) = -z & \sigma_2(z) = \dfrac{1}{z}
  \end{cases}
\]

\[
  (5) \qquad \sigma_3(z) = -\frac{1}{z} \qquad
  \Delta_3 = \mathbb{V}(z^2 + 1)
\]

\[
  (6) \qquad \Delta = \Delta_1 \amalg \Delta_2 \amalg \Delta_3 \qquad
  \bigl(= \{0, \infty, 1, -1, i, -i\} \text{ dans le cas normalisé}\bigr)
\]

\[
  \Delta \subset X
\]

\[
  (7) \qquad \Gamma = \underline{\operatorname{Aut}}_S(X, \Delta),
\]

\[
  (8) \qquad \Gamma = \underline{\operatorname{Aut}}^{+}_{\mathrm{oct.}}(\Delta)
  \qquad \text{(en tant que sous-schémas de } \underline{\operatorname{Aut}}_S(\Delta)\text{)}
\]

\[
  z \longmapsto az + (a-1)
\]

\[
  a - 1 \in \{1, i, -i\} \quad \text{i.e.}\quad a \in (2, 1+i, 1-i)
\]

\[
  2a - 1 \in \{0, i, -i\} \quad \text{i.e.}\quad
  a \in \Bigl\{\tfrac12, \tfrac12 (i+1), \tfrac12 (1-i)\Bigr\}
\]

\[
  a\underbrace{(i+1)}_{1 : (\frac{1-i}{2})} - 1 \in \{0, 1, -i\}
  \quad \text{i.e.}\quad a \in \Bigl\{\tfrac{1-i}{2}, 1-i, -i\Bigr\}
\]

\[
  \underline{a}_\Delta : \Delta \to \Delta
\]

\[
  (9) \qquad \underline{I} \subset X
\]

\[
  (10) \qquad \underline{\mathfrak{S}}_{\underline{I}} = \underline{\operatorname{Aut}}_S(\underline{I})
\]

\[
  H^0\bigl(S, \mathcal{H}^1(\Sigma^*_{\underline{J}}/S, \underline{V})\bigr)
  \simeq \operatorname{Hom}(\underline{V}_{\underline{J}}, \underline{V})
\]

\[
  (11) \qquad X^* = \underline{K} \wedge_V \Sigma^*_{\underline{J},a},
\]

\[
  (12) \qquad \sum_{i \in \underline{J}} x_i^2 = 0
\]

\[
  \sum x_i = 0 .
\]

\[
  (15) \qquad X/x \xrightarrow{\;\sim\;} I_S \qquad (V_S\text{-isomorphisme})
\]

\[
  (16) \qquad \underline{K} \simeq (I_S) \wedge_{\underline{V}} (\Sigma^*_{J,a}/x)
  \qquad \underline{V} = (V_J)_S .
\]

\[
  (17) \qquad \sigma : E \to E
\]

\[
  (18) \qquad \sigma_{E_0} = -\mathrm{id}_{E_0}
\]

\[
  \sigma x = a - x \qquad a \text{ section convenable}
\]

\[
  \sigma^2 = \mathrm{id}_E
\]

\[
  (19) \qquad
  \begin{cases}
    \underline{I} = E^\sigma \quad \text{sous-schéma des pts fixes de } \sigma \\
    \underline{I} \subset E \\
    \phantom{\underline{I}} = \{x \in E \mid 2x = a\}
  \end{cases}
\]

\[
  (20) \qquad \underline{V} = {}_2E_0 = \operatorname{Ker}(2\,\mathrm{id}_{E_0} : E_0 \to E_0),
\]

\[
  (21) \qquad
  \begin{cases}
    E = \underline{I} \wedge_{\underline{V}} E_0 \\
    \sigma \text{ déduit de la symétrie } x \mapsto -x \text{ de } E_0
    \text{ via la formule précédente}
  \end{cases}
\]

\[
  (22) \qquad
  \begin{array}{c}
    E/\underline{V} \xrightarrow{\;\sim\;} E_0 \\
    \wr\!\wr \\
    E \wedge_{E_0} E \simeq \underline{\operatorname{Pic}}^2_{E/S}
  \end{array}
\]

\[
  (23) \qquad X = E/\{1, \sigma\}, \qquad \Sigma = E_0/\{1, \sigma_0\}
\]

\[
  (24) \qquad X/V \xrightarrow{\;\sim\;} \Sigma ,
\]

\[
  (25) \qquad E \to X \qquad (\text{rev.\ quadratique})
\]

\[
  (26) \qquad \underline{I} \hookrightarrow X
\]

\[
  x + \xi \in \{x, \sigma x = a - x\} \quad \text{i.e.} \quad
  x + \xi = a - x \quad \text{i.e.} \quad 2x + \xi = a
  \quad \text{d'où} \quad 4x = 2a \quad \text{mais } 2x \neq a
\]

\[
  x \text{ pt de } \underline{I} \wedge_{{}_2E_0} ({}_4E_0) \setminus \underline{I}
\]

\[
  (27) \qquad \Delta = \Bigl(\underbrace{\underbrace{\underline{I} \wedge_{\underline{M}_0} \underline{M}}_{\text{degré } 16}
  \setminus \underbrace{\underline{I}}_{\text{degré } 4}}_{\text{degré } 12}\Bigr)
  \Big/ \underbrace{\{1, \sigma\}}_{\text{degré } 2}
\]

\[
  (28) \qquad
  \begin{cases}
    \underline{M} \simeq {}_4E_0 \quad \text{schéma en groupes loc.\ isom.\ à } (\mathbb{Z}/4\mathbb{Z})^2_S \\
    \underline{M}_0 \simeq \underline{M} \otimes_{\mathbb{Z}/4\mathbb{Z}} \mathbb{Z}/2\mathbb{Z}
    \simeq 2\underline{M} = {}_2E_0 = \underline{V}
  \end{cases}
\]

\[
  (30) \qquad \Sigma_{\underline{J}} \xrightarrow{\;f_2^t\;} \Sigma_{\underline{J}}
\]

\[
  (31) \qquad \underline{I}_t = \underline{J} \amalg t(S)
\]

\[
  (32) \quad
  \begin{cases}
    \forall\, i \in J, \text{ si } \sigma_i^t \in \operatorname{Aut}_S(\Sigma_J)
      \text{ est l'automorphisme qui échange } t \text{ et } i, \\
    \qquad \text{et } j \text{ et } k \ (\text{où } \{i,j,k\} = J),
      \text{ et } \Delta_i^t = (\Sigma_J)^{\sigma_i^t}, \\
    f(\Delta_i^t) = \text{section } \xi_i \text{ de } D_{\underline{J}} \simeq J_S
  \end{cases}
\]

\[
  (33) \qquad f_t\bigl(I_S = J_S \cup t(S)\bigr) \subset t(S)
\]

\[
  (34) \quad
  \begin{cases}
    \sigma_0^t(z) = \dfrac{z - t}{z - 1} & (\text{éch.\ } 0 \text{ et } t,\ 1 \text{ et } \infty) \\[1ex]
    \sigma_1^t(z) = t/z & (\text{éch.\ } 1 \text{ et } t,\ 0 \text{ et } \infty) \\[1ex]
    \sigma_\infty^t(z) = t\,\dfrac{z - 1}{z - t} & (\text{éch.\ } \infty \text{ et } t,\ 0 \text{ et } 1)
  \end{cases}
  \qquad \text{NB } \sigma_\infty^t = \sigma_0^t \sigma_1^t
\]

\[
  (35) \quad
  \begin{cases}
    \Delta_0^t : \text{solutions de } z^2 - 2z + t = 0, & \Delta_0^t = \{1 \pm \sqrt{1-t}\} \\
    \Delta_1^t : \text{\phantom{solutions de }} z^2 - t = 0 & \Delta_1^t = \{\pm \sqrt{t}\} \\
    \Delta_\infty^t : \text{\phantom{solutions de }} z^2 - 2tz + t = 0 & \Delta_\infty^t = \{t \pm \sqrt{t(1-t)}\}
  \end{cases}
\]

\[
  (36) \qquad f_2^t : \quad
  \begin{aligned}
    \Delta_0^t &\longrightarrow 0 \\
    \Delta_1^t &\longrightarrow 1 \\
    \Delta_\infty^t &\longrightarrow \infty
  \end{aligned}
  \qquad \underline{\text{avec multiplicité } 2}
\]

\[
  f_2^t(z) = c_t \left( \frac{z^2 - 2z + t}{z^2 - 2tz + t} \right)^2
  \qquad c_t \text{ section de } \underline{\mathcal{O}}_S^*,
\]

\[
  \begin{aligned}
    \zeta^2 - 2\zeta + t &= 2(t - \zeta) \\
    \zeta^2 - 2t\zeta + t &= 2t(1 - \zeta)
  \end{aligned}
\]

\[
  f_2^t(\zeta) = c_t \left( \frac{2(t - \zeta)}{2t(1 - \zeta)} \right)^2
  = c_t/t^2 \left( \frac{t - \zeta}{1 - \zeta} \right)^2
\]

\[
  \frac{t - \zeta}{1 - \zeta}
  = \frac{(t - \zeta)(1 + \zeta)}{(1 - \zeta)(1 + \zeta)}
  = \frac{-\zeta^2 + (t-1)\zeta + t}{1 - \zeta^2}
  = \frac{(t-1)\zeta}{1 - t} = -\zeta
\]

\[
  f_2^t(\zeta) = \frac{c_t}{t^2}\, \zeta^2 = \frac{c_t\, t}{t^2} = \frac{c_t}{t}
  \qquad \text{d'où} \qquad c_t = t
\]

\[
  (37) \qquad f_2^t(z) = t \left( \frac{z^2 - 2z + t}{z^2 - 2tz + t} \right)^2
\]

\[
  (38) \qquad M = {}_4E
\]

\[
  M_0 = M \otimes_{\mathbb{Z}/4\mathbb{Z}} \mathbb{F}_2 \simeq {}_2E .
\]

\[
  (40) \qquad \underline{J} \simeq M_0^*, \quad
  M_0 = V_{\underline{J}}(\mathbb{F}_2) \simeq
  \mathbb{F}_2^{\underline{J}} / \underset{\text{diag}}{\mathbb{F}_2}\ ),
\]

\[
  (41) \qquad \text{« } V_{\underline{J}}(M_0) \text{ »} \simeq
  \underline{\mathrm{End}}_{\mathbb{F}_2, S}(M_0) \simeq
  M_0 \oplus \mathbb{F}_2^{\underline{\omega}}
\]

\[
  (42) \qquad (\mathbb{Z}/4\mathbb{Z})^2_S \simeq M \ (= {}_4E) \ )
\]

\[
  (43) \qquad 0 \to
  \overset{\underline{\mathrm{End}}(M_0)}{\overset{\wr}{V_{\underline{J}}(M_0)}}
  \longrightarrow V_{\underline{J}}(M) \longrightarrow
  \overset{\underline{\mathrm{End}}(M_0)}{\overset{\wr}{V_{\underline{J}}(M_0)}}
  \longrightarrow 1
\]

\[
  (44) \qquad V_{\underline{J}}(A) \simeq
  \operatorname{Ker}\bigl( A^{\underline{J}} \xrightarrow{\ \text{somme}\ } A \bigr)
\]

\[
  (45) \qquad V_{\underline{J}}(A) \simeq \operatorname{Hom}(M_0, A)
  \qquad \text{si } 2\,\mathrm{id}_A = 0
\]

\[
  (46) \qquad M_0 \simeq \mathbb{F}_2^{\underline{J}} / \underset{\text{diag}}{\mathbb{F}_2}
\]

\[
  (47) \qquad \underline{J} \longrightarrow M \,\}
\]

\[
  \underline{J} \hookrightarrow M_0
\]

\[
  \underline{J} \longrightarrow M \setminus M_0
\]

\[
  (\underline{J} \dashrightarrow)\; M \hookrightarrow E \longrightarrow
  \Sigma_{\underline{J}} \simeq E/(1, \sigma_E)
\]

\[
  (48) \qquad \underline{J} \longrightarrow \Delta_t
\]

\[
  f_t : \Delta_t \longrightarrow \underline{J}
\]

\[
  (1) \qquad f : X \longrightarrow S
\]

\[
  X'_0 = \operatorname{Spec} \underline{\mathcal{O}}_X[z]/(z^n - \varphi)
\]

\[
  (2) \qquad T_{X'/S, t'}^{\otimes n} \xrightarrow[\sim]{\ \kappa_{t'}\ } T_{X/S, t}
\]

\[
  (3) \qquad \mu_{n,S} \hookrightarrow \underline{G} .
\]

\[
  f^\circ : X^* \longrightarrow S
\]

\[
  (4) \qquad \xi_{X'/S} \in H^0\bigl(S, R^1 f^\circ_* f^{\circ *}(\underline{G})\bigr) .
\]

\[
  (X', t', G) \longrightarrow (X'_1, t'_1, G_1)
\]

\[
  (\underline{G}, n, i, \xi, T, k)
\]

\[
  T^{\otimes n} \xrightarrow{\ \sim\ } T_{X/S, t}
\]

\[
  \underline{G} = \mu_2 \underset{\text{can}}{\simeq} \{\pm 1\}_S , \quad \text{et où } n = 2
\]

\[
  T^{\otimes 2} \simeq T_{X, t} .
\]

\[
  (1) \qquad J = {}_2E(k)^*
\]

\[
  (2) \qquad E/(\pm \mathrm{id}_E) \simeq \Sigma_J
\]

\[
  (3) \qquad \Sigma_J \simeq \mathbb{P}(V_J(k)) \qquad
  0 \to V_J(k) \to k^J \xrightarrow{\ \text{somme}\ } k \to 0
\]

\[
  (4) \qquad t \in \Sigma_J^* = \Sigma_J \setminus D_J
\]

\[
  (5) \qquad T^{\otimes 2} \xrightarrow{\ \sim\ } T_{\Sigma, t}
\]

\[
  (6) \qquad T_\Sigma \simeq \underline{\mathcal{O}}_\Sigma(2)(\underline{\omega})
\]

\[
  T_0 \simeq \mathcal{O}_t(1)
\]

\[
  T_0^{\otimes 2} = \mathcal{O}_t(2)
  \xrightarrow{\ \sim\ } T_{\Sigma, t} \simeq \mathcal{O}_t(2)(\underline{\omega}) ,
\]

\[
  (7) \qquad \xi \longmapsto \xi \wedge \omega \qquad
  T_0^{\otimes 2} \xrightarrow{\ \sim\ } \mathcal{O}_t(2) \simeq T_{\Sigma, t}
  = \underline{\mathcal{O}}_t(2)(\underline{\omega})
\]

\[
  (8) \qquad T_0 = \underline{\mathcal{O}}_t(1) \wedge_{\mu_4} Q_0
\]

\[
  (9) \qquad T_0^{\otimes 2} \simeq \underline{\mathcal{O}}_t(2) \wedge_{\mu_2} D_{Q_0}
\]

\[
  (10) \qquad D_{Q_0} = Q_0/\{\pm 1\}
\]

\[
  (11) \quad
  \begin{cases}
    \underline{\omega}_{Q_0} \ (\text{ens.\ des orientations de } Q_0)
      \simeq \mu_4^* = \{i, -i\} \\
    D_{Q_0} \ (\text{ens.\ des diagonales de } Q_0) \simeq \underline{\omega}
  \end{cases}
\]

\[
  (12) \qquad T_0^{\otimes 2} \xrightarrow{\ \sim\ } T_{\Sigma, t}
\]

\[
  (13) \qquad T = T_{E, 0} \xleftarrow[\ \sim\ ]{\ \kappa\ } T_0
\]

\[
  (15) \qquad \alpha : \underline{\omega}_J \xrightarrow{\ \sim\ } D_{Q_0} = Q_0/\{\pm 1\}
\]

\[
  (16) \qquad \underline{\omega}_{Q_0} \simeq \mu_4^*
  \qquad \text{i.e.\ d'un élément de } \mu_4^* \wedge \underline{\omega}_{Q_0}
\]

\[
  (17) \qquad \mu_4^* \simeq \operatorname{Aut}^+(Q_0) .
\]

\[
  (18) \qquad s_{i'} = i\, s_i \qquad \text{d'où} \qquad s_i = i' s_{i'}
\]

\[
  (19) \qquad D_{Q_0} \simeq \mu_4^*
\]

\[
  (20) \qquad \alpha : \underline{\omega}_J \simeq \mu_4^*
  \qquad \text{i.e.\ d'un élément de } \underline{\omega}_J \wedge \mu_4^* .
\]

\[
  (21) \quad
  \begin{cases}
    \text{a) d'un $J$ à trois éléments (d'où $\Sigma_J$, $D_J$, $\Sigma_J^*$),} \\
    \text{b) d'un $t \in \Sigma_J^*(k)$} \\
    \text{c) d'un isom $\alpha : \underline{\omega}_J \simeq \mu_4^*$ \ \ $(= \mu_4^*(k))$}
  \end{cases}
\]

\[
  t \in U_{0,3}(k) = \mathbb{P}^1_k(k) \setminus \{0, 1, \infty\}, \quad \text{et } i \in \mu_4^* .
\]

\[
  (22) \qquad M^{\natural} = M \wedge_{\mu_{4S}} Q_{0S}
  \qquad \underline{\text{NB}}\ \ M^{\natural} \simeq M \otimes_{\mathcal{O}_S} \underline{\mathcal{O}}_S^{\natural}
\]

\[
  (23) \qquad M^{\natural} \otimes M^{\natural} \simeq M^{\flat}
  \overset{\text{déf}}{=} M \wedge_{\mu_{2S} = \{\pm 1\}_S} \mu_{4S}^*
\]

\[
  (24) \qquad \mu_4^* = \mu_4 \setminus \mu_2
\]

\[
  (25) \qquad \Sigma_{\underline{J}} = \mathbb{P}(V_{\underline{J}}(\underline{\mathcal{O}}_S))
  \qquad 0 \to V_{\underline{J}}(\mathcal{O}_S) \to
  \underset{\substack{\parallel \\ p_*(\mathcal{O}_{\underline{J}}) \\ p : \underline{J} \to S}}{\underline{\mathcal{O}}_S^{\underline{J}}}
  \xrightarrow{\ \mathrm{tr}\ } \underline{\mathcal{O}}_S
\]

\[
  (26) \qquad T_{0, \Sigma_{\underline{J}}} = \underline{\mathcal{O}}_{\Sigma_{\underline{J}}}(1)^{\natural}
\]

\[
  (27) \qquad T_{0, \Sigma}^{\otimes 2} \simeq \underline{\mathcal{O}}_{\Sigma_J}(2) \wedge_{\{\pm 1\}} \mu_4^*
  \simeq T_{\Sigma_J} \wedge_{\{\pm 1\}} (\mu_4^* \wedge \underline{\omega}_{\underline{J}})
\]

\[
  (28) \qquad \underline{J} = ({}_2E)^* \qquad \text{rev.\ étale cubique de } S
\]

\[
  (29) \qquad E/(\pm \mathrm{id}_E) \xrightarrow{\ \sim\ } \Sigma_{\underline{J}}
\]

\[
  \text{(30)} \qquad T_{E/S,0}^{\otimes 2} \simeq T_{\Sigma_{\underline{J}},t}
\]

\[
  \text{(31)} \qquad \underbrace{\Delta_{E/S}}_{T_{E/S,0}}
  \xrightarrow[\;\sim\;]{k} \underline{O}(1)_t \simeq
\]

\[
  \text{(32)} \qquad T_{\Sigma_{\underline{J}},t} \xrightarrow{\;\sim\;}
  T_{\Sigma_{\underline{J}},t} \wedge^{\{\pm 1\}}
  \bigl(\mu_4^{*} \wedge \underline{\omega}_{\underline{J}}\bigr)
\]

\[
  \text{(33)} \qquad
  \begin{cases}
    \alpha \in \Gamma\bigl(S, \mu_4^{*} \wedge \underline{\omega}_{\underline{J}}\bigr)
      & \text{i.e.\ d'un iso} \\[2pt]
    \alpha : \mu_4^{*} \simeq \underline{\omega}_{\underline{J}}
  \end{cases}
\]

\[
  \begin{cases}
    \text{a) } \underline{J} \text{ rev.\ étale cubique de } S \\
    \text{b) } t \text{ section de } \Sigma_{\underline{J}}^* \\
    \text{c) } \alpha \text{ section de } \mu_4^{*} \wedge \underline{\omega}_{\underline{J}}
      \text{, i.e.\ iso.\ } \mu_4^{*} \simeq \underline{\omega}_{\underline{J}}
  \end{cases}
\]

\[
  \mathfrak{S}_{\underline{J}}^{+} \simeq
  \underbrace{S}_{\substack{\text{correspond}\\ \text{à la section}\\
  \text{unité des}\\ \text{schémas en groupes}}}
  \amalg\ \underline{\omega}_{\underline{J}}
\]

\[
  \underline{P}_{\underline{J}} \subset \Sigma_{\underline{J}}^*,
  \qquad \underline{P}_{\underline{J}} =
  \bigl(\Sigma_{\underline{J}}\bigr)^{\mathfrak{S}_{\underline{J}}^{+}}
\]

\[
  \text{(35)} \qquad u : \underline{J} \xleftarrow{\;\sim\;} \{0, 1, \infty\}_S
\]

\[
  \text{(36)} \qquad \Sigma_{\underline{J}} \xleftrightarrow{\;\sim\;} \mathbb{P}^1_S
\]

\[
  \text{(37)} \qquad U_{03} \times \mu_4^{*}
\]

\[
  \text{(38)} \qquad \Pi_0 = \underline{O}_{U_{0,3} \times \mu_4^{*}}(1)^{\natural}
\]

\[
  \text{(39)} \qquad \Pi_0 \simeq \underline{O}_{U_{0,3}}(1)
\]

\[
  \text{(40)} \qquad k' : \Delta_{E/S} \simeq \underline{O}_{\Sigma}(1)_t
\]

\[
  \text{(41)} \qquad T_{\Sigma,t} \simeq T_{\Sigma,t} \wedge^{\{\pm 1\}}
  \{\underline{\omega}_{\underline{J}}\}
  \quad \text{(évidemment unique)}
\]

\[
  \text{(42)} \qquad \underline{Q}_{E/S}/\mu_2 \simeq \underline{\omega}_{\underline{J}}
\]

\[
  \text{(43)} \qquad Q_{E/S} \wedge^{\mu_4} Q_{0S} \simeq \mathcal{L}_{E/S}
\]

\[
  \mathcal{L}_{E/S} \simeq \underline{\mathrm{Isom}}_{\mu_{4S}\text{-tors.}}
  \bigl(Q_{0S}^{-1}, Q_{E/S}\bigr)
\]

\[
  \text{(44)} \qquad \mathcal{L}_{E/S}/\mu_2 = \mathcal{L}_{E/S}
  \wedge^{\mu_{4S}} \mu_{2S} \simeq
  \bigl(\mu_{4S}^{*} \wedge \underline{\omega}_{\underline{J}}\bigr)
\]

\[
  \text{(45)} \qquad (\mu_4 \times \mathfrak{S}_3)_S
\]

\[
  \text{(46)} \qquad \Pi_0 = \underline{O}_{U_{0,3} \times \mu_4^{*}}(1)
\]

\[
  \text{(47)} \qquad
  \begin{cases}
    \mathfrak{S}_3 \text{ opère sur } U_{03} \times \mu_4^{*} \text{ via }
    \begin{cases}
      \text{action tautol.\ sur } U_{03} \\
      \text{la \ldots\ } \mathrm{sg}(g) \text{ sur } \mu_4^{*}
    \end{cases} \\[10pt]
    \mu_4 \text{ opère sur } U_{03} \times \mu_4^{*} \text{ via }
    \begin{cases}
      \text{action triviale sur } U_{03} \\
      \text{l'h.\ can.\ } \mu_4 \to \mu_2 = \{\pm 1\}_{S_0} \text{ sur } \mu_4^{*}
    \end{cases}
  \end{cases}
\]

\[
  \text{(48)} \qquad \mathbb{D}'_{3S_0} = (\mathfrak{S}_3)_{S_0}
  \times_{\mu_{2S_0}} (\mu_4)_{S_0}
\]

\[
  \begin{cases}
    \mu_{4S}^{*} \text{ muni d'une section canonique } i_S \\
    \mu_{4S} \simeq (\mathbb{Z}/4\mathbb{Z})_S
  \end{cases}
\]

\[
  \text{(49)} \qquad \mathbb{D}'_3 = \mathfrak{S}_3 \times_{\mathbb{Z}/2\mathbb{Z}} (\mathbb{Z}/4\mathbb{Z})
\]

\[
  \text{(50)} \qquad 1 \to \{\pm 1\} \to \mathbb{D}'_3 \to \mathbb{D}_3 \to 1 .
\]

\[
  \begin{aligned}
    &\text{(51)} \qquad T_{g,\lambda}\, \zeta = \lambda\, (g,\lambda) \cdot \zeta
      && \zeta \in \Gamma\underline{O}_{U_{03}}(1)(t) \\
    &\text{(52)} \qquad T_{g,\lambda}\, \eta = \lambda^2\, (g,\lambda) \cdot \eta
      && \eta \in \Gamma\underline{O}_{U_{0,3}}(2)(t) \\
    &\text{(53)} \qquad T_{g,\lambda}(\tau) = \lambda^2\, \mathrm{sg}(g)\, (g,\lambda) \cdot \tau
      && \tau \in \Gamma\,\underline{O}_{U_{03}}(2)(t)
  \end{aligned}
\]

\[
  \begin{aligned}
    &\text{(51 bis)} \qquad T_g\, \zeta = \chi_4(g)\, (g \cdot \zeta) \\
    &\text{(52 bis)} \qquad T_g\, \eta = \mathrm{sg}(g^{0})\, (g \cdot \eta) \\
    &\text{(53 bis)} \qquad T_g(\tau) = g \cdot \tau
  \end{aligned}
\]

\[
  \text{(54)} \qquad \chi_4 : \mathbb{D}'_{3S_0} \to \mu_4
\]

\[
  \text{(*)} \qquad T^{\otimes 2} \simeq T_{\Sigma,t}
\]

\[
  \text{(55)} \qquad \underline{\varepsilon} \longmapsto T = T_0 \wedge^{\mu_{2S}} \underline{\varepsilon}
\]

\[
  \text{(56)} \qquad T_0 = \underline{O}_{\Sigma}(1)(t)^{\natural}
  = \underline{O}_{\Sigma}(1)(t) \wedge^{\mu_4} Q_{0S}
\]

\[
  T_0^{\otimes 2} \simeq \underline{O}_{\Sigma}(2)(t) \wedge^{\mu_2} \mu_4^{*}
  \simeq T_{\Sigma,t} \wedge^{\mu_2} \bigl(\mu_4^{*} \wedge \underline{\omega}_{\underline{J}}\bigr)
\]

\[
  \alpha : \mu_4^{*} \simeq \underline{\omega}_{\underline{J}}
  \quad \text{i.e.\ une section de } \mu_4^{*} \wedge \underline{\omega}_{\underline{J}}
\]

\[
  k_\alpha : T_0^{\otimes 2} \simeq T_{\Sigma,t}
\]

\[
  \text{(57)} \qquad
  \underbrace{\underline{\mathrm{Isom}}_{S,\underline{J}}(E_\alpha, E)}_{
    \substack{\text{l'indice } \underline{J} \text{ signifie qu'on prend les iso.}\\
    \text{compatibles aux } \underline{J} \simeq {}_2E \simeq {}_2E_\alpha}}
  \xrightarrow{\;\sim\;} \mathcal{L}_\alpha(E),
  \qquad u \longmapsto u(k_\alpha)
\]

\[
  \begin{aligned}
    &\text{de } \underline{J} \ (= {}_2E(k)^{*}) \text{ (ens.\ à trois éléments)}, \text{ de la}\\
    &\text{(58)} \quad k\text{-vectoriel } V_{\underline{J}} = \operatorname{Ker}\bigl(k^{\underline{J}} \xrightarrow{\text{somme}} k\bigr) \text{ (d'où} \\
    &\text{(59)} \quad \Sigma_{\underline{J}} = \mathbb{P}^1(V_{\underline{J}}), \text{ avec des } \xi_i \in \Sigma_{\underline{J}}(k) \ (i \in \underline{J}) \text{ distincts,}\\
    &\qquad \Sigma_{\underline{J}}^{*} = \Sigma_{\underline{J}} - \{\xi_i\}_{i \in \underline{J}}\text{)}, \text{ et un } t \in \Sigma_{\underline{J}}^{*} \text{ i.e.\ une droite}\\
    &\text{(60)} \quad F_t \subset V_{\underline{J}},
  \end{aligned}
\]

\[
  \text{(61)} \qquad \nu \ (= \nu_E) : \underline{t}_E^{\otimes 2} \simeq T_{\Sigma_{\underline{J}},t} .
\]

\[
  \text{(62)} \qquad
  \begin{cases}
    T_{\Sigma_{\underline{J}},t} \simeq \underline{O}_t(2)(\underline{\omega}) \\
    \text{avec } \underline{O}_t(2) \overset{\mathrm{def}}{\simeq} \underline{O}_t(1)^{\otimes 2},
    \quad \underline{O}_t(1) \simeq V_{\underline{J}}/F_t \simeq \check{F}_t(\underline{\omega})
  \end{cases}
\]

\[
  \text{(63)} \qquad \nu : \underline{t}_E^{\otimes 2} \simeq F_t^{\otimes(-2)}(\underline{\omega})
  \qquad \text{ou aussi comme} \qquad
  \check{\nu} : \underline{t}^{\otimes -2} \simeq F_t^{\otimes 2}(\underline{\omega})
\]

\[
  \text{(64)} \qquad k : \underline{t} \xrightarrow{\;\sim\;} \underline{O}_t(1)^{\natural}
\]

\[
  \text{(65)} \qquad
  \begin{cases}
    (k_i)_{i \in \mu_4^{*}}, \quad k_i : \underline{t} \xrightarrow{\;\sim\;} \underline{O}_t(1) \simeq \check{F}_t(\underline{\omega}) \\
    k_{i'} = i\, k_i \qquad \text{si } i' = 1/i \text{ est l'autre élément de } \mu_4^{*}
  \end{cases}
\]

\[
  \text{(66)} \qquad
  \begin{cases}
    (\check{k}_i)_{i \in \mu_4^{*}}, \quad \check{k}_i : \check{\underline{t}} \simeq F_t(\underline{\omega}) \\
    \check{k}_{i'} = \tfrac{1}{i}\, \check{k}_i ,
  \end{cases}
\]

\[
  \text{(67)} \qquad
  \begin{cases}
    k_i^{\otimes 2} : \underline{t}^{\otimes 2} \xrightarrow{\;\sim\;} \check{F}_t^{\otimes(-2)} \text{ est de la forme} \\
    k_i^{\otimes 2}(\delta^{\otimes 2}) = \nu(\delta^{\otimes 2}) \wedge \omega_i
  \end{cases}
\]

\[
  \text{(68)} \qquad
  \begin{cases}
    \omega_{i'} = -\omega_i \\
    \text{i.e.\ } i \mapsto \omega_i \text{ est une bijection} \\
    \qquad \alpha = \alpha_k : \mu_4^{*} \xrightarrow{\;\sim\;} \underline{\omega}
  \end{cases}
\]

\[
  \text{(69)} \qquad \check{k}_i(\check{\delta}) = \underbrace{\psi_i(\delta)}_{\in F_t} \wedge \omega_i
  = \underbrace{\psi^{\omega_i}(\delta)} \wedge \omega_i ,
\]

\[
  \text{(70)} \qquad \bigl(\psi^{\omega}(\delta)\bigr)_{\omega \in \underline{\omega}} \in (F_t)^{\underline{\omega}}
\]

\[
  \underbrace{\psi^{\omega_{i'}}(\delta) \wedge \omega_{i'}}_{\substack{\psi^{\omega'}(\delta) \wedge \omega' \\ = -\psi^{\omega'}(\delta) \wedge \omega}}
  = \frac{1}{i}\, \underbrace{\psi^{\omega_i}(\delta) \wedge \omega_i}_{\psi^{\omega}(\delta) \wedge \omega}
  \qquad \text{i.e.\ } \psi^{\omega'}(\delta) = -\frac{1}{i}\, \psi^{\omega}(\delta)
\]

\[
  \text{(71)} \qquad \psi^{\omega'}(\delta) = i_\omega\, \psi^{\omega}(\delta)
  \qquad \text{où } i_\omega \overset{\mathrm{def}}{=} \alpha^{-1}(\omega)
\]

\[
  \text{(72)} \qquad \psi^{\omega}(\lambda\delta) = \frac{1}{\lambda}\, \psi^{\omega}(\delta) ,
\]

\[
  \check{k}_i^{\otimes 2}(\check{\delta}^{\otimes 2}) = \check{\nu}(\check{\delta}^{\otimes 2}) \wedge \omega_i ,
\]

\[
  \text{(73)} \qquad \psi^{\omega}(\delta)^{\otimes 2} = \check{\nu}(\check{\delta}^{\otimes 2}) \wedge \omega .
\]

\[
  F_t \subset V_{\underline{J}} = \operatorname{Ker}\bigl(k^{\underline{J}} \xrightarrow[\text{somme}]{} k\bigr) ,
\]

\[
  \text{(74)} \qquad \bigl(\psi_i^{\omega}(\delta)\bigr)_{\substack{i \in \underline{J} \\ \omega \in \underline{\omega} \\ \delta \in \underline{t}^{*}}},
  \qquad \psi_i^{\omega}(\delta) \in k
\]

\[
  \text{(75)} \qquad \sum_{i \in \underline{J}} \psi_i^{\omega}(\delta) = 0
\]

\[
  \begin{aligned}
    &\text{(76)} \qquad \psi_i^{\omega'}(\delta) = i_\omega\, \psi_i^{\omega}(\delta) \\
    &\text{(77)} \qquad \psi_i^{\omega}(\lambda\delta) = \lambda^{-1}\, \psi_i^{\omega}(\delta) .
  \end{aligned}
\]

\[
  F_t = \Bigl\{ \sum \lambda_i e_i \Bigm| \sum \lambda_i = 0,\
  \sum c_i \lambda_i = 0 \Bigr\}
\]

\[
  \text{(78)} \qquad \sum_{i \in \underline{J}} c_i\, \psi_i^{\omega}(\delta) = 0 .
\]

\[
  (1) \qquad \mathcal{F}_n = \Gamma(E, n\{0_E\})
\]

\[
  (2) \qquad \operatorname{rg} \mathcal{F}_n = n \quad\text{si } n \geqslant 1
\]

\[
  (3) \qquad \mathcal{F}_0 = \mathcal{F}_1 = k \cdot 1_E .
\]

\[
  (4) \qquad \mathcal{F}_\infty \overset{\text{déf}}{=} \varinjlim_n \mathcal{F}_n
\]

\[
  (5) \qquad \mathcal{F}_m \cdot \mathcal{F}_n \subset \mathcal{F}_{n+m} ,
\]

\[
  (6) \qquad 0 \to \mathcal{F}_{n-1} \to \mathcal{F}_n \xrightarrow{\;\varepsilon_n\;}
  t_E^{\otimes n} \to 0 \qquad (n \geqslant 2)
\]

\[
  (7) \qquad t_E \overset{\text{déf}}{=} T_{E, 0_E} .
\]

\[
  (8) \qquad \Theta_\delta : \mathcal{F}_n \longrightarrow \mathcal{F}_{n+1}
  \qquad \text{dérivation de degré $+1$ de l'algèbre graduée } \mathcal{F}_*
\]

\[
  (10) \qquad
  \begin{aligned}
    t_E \otimes \mathcal{F}_n &\xrightarrow{\;\delta_n\;} \mathcal{F}_{n+1} \\
    \delta \otimes \varphi &\longmapsto \Theta_\delta \varphi
  \end{aligned}
\]

\[
  (12) \qquad
  \begin{cases}
    \operatorname{Ker} \delta_n = \operatorname{Ker} \delta_1 =
    t_E \otimes \mathcal{F}_1 = t_E \otimes_k 1_E \\
    \operatorname{Ker} \Theta_\delta = \mathcal{F}_1 = k \cdot 1_E .
  \end{cases}
\]

\[
  (13) \qquad \check f(x) = f(-x)
\]

\[
  (14) \qquad f \mapsto \check f \qquad \mathcal{F}_* \longrightarrow \mathcal{F}_*
  \qquad \text{involution de l'algèbre comm.}
\]

\[
  (15) \qquad (\Theta_\delta f)^\vee = \Theta_{\check\delta} \check f =
  \Theta_{-\delta} \check f = -\Theta_\delta \check f
\]

\[
  \vee \circ \Theta_\delta = -\Theta_\delta \circ \vee .
\]

\[
  (16) \qquad
  \begin{cases}
    \mathcal{F}_n^+ = \{ f \in \mathcal{F}_n \mid \check f = f \} \\
    \mathcal{F}_n^- = \{ f \in \mathcal{F}_n \mid \check f = -f \}
  \end{cases}
\]

\[
  (17) \qquad
  \left\{
  \begin{aligned}
    & \mathcal{F}_n = \mathcal{F}_n^+ \oplus \mathcal{F}_n^- \\
    & \begin{cases}
        \Theta_\delta \mathcal{F}_n^+ \subset \mathcal{F}_{n+1}^- , \quad
        \Theta_\delta \mathcal{F}_n^- \subset \mathcal{F}_{n+1}^+ \\
        t_E \otimes \mathcal{F}_n^+ \xrightarrow{\;\delta_n\;} \mathcal{F}_{n+1}^- , \quad
        t_E \otimes \mathcal{F}_n^- \xrightarrow{\;\delta_n\;} \mathcal{F}_{n+1}^+
      \end{cases} \\
    & \begin{cases}
        \mathcal{F}_n^+ \mathcal{F}_m^+ \subset \mathcal{F}_{n+m}^+ \\
        \mathcal{F}_n^- \mathcal{F}_m^- \subset \mathcal{F}_{n+m}^+
      \end{cases}
      \qquad \mathcal{F}_n^+ \mathcal{F}_m^- \subset \mathcal{F}_{n+m}^-
  \end{aligned}
  \right.
\]

\[
  (18) \qquad \mathcal{F}_2^+ = \mathcal{F}_2 \simeq \Gamma(\Sigma_J, \{t\})
  \qquad \text{où}\quad
  \begin{aligned}
    & J = {}_2E^* \\
    & \Sigma_J \simeq E/(\pm \mathrm{id}_E) \\
    & t \text{ image de } 0_E \\
    & t \in \Sigma_J^* = \Sigma_J \setminus J
  \end{aligned}
\]

\[
  (19) \qquad \mathcal{F}_{2n}^+ \xrightarrow{\;\sim\;} \mathcal{F}_{2n+1}^+
  = \Gamma(\Sigma_J, n\{t\})
\]

\[
  (20) \qquad \mathcal{F}_3^- = \delta_2(t_E \otimes \mathcal{F}_2) \simeq t_E^{\otimes 3}
\]

\[
  (21) \qquad \mathcal{F}_{2n+1}^- \xrightarrow{\;\sim\;} \mathcal{F}_{2n+2}^- \simeq
  \mathcal{F}_3^- \otimes_k \mathcal{F}_{2n-2}^+
\]

\[
  (22) \qquad \delta \in t_E
\]

\[
  (23) \qquad \wp \in \mathcal{F}_2 \qquad \varepsilon_2(\wp) = \delta^{\otimes 2}
\]

\[
  (24) \qquad \wp' = \Theta_\delta(\wp) \qquad \text{donc}\quad
  \varepsilon_3(\wp') = \delta^{\otimes 3}
\]

\[
  (25) \qquad
  \begin{cases}
    1, \wp & \text{base de } \mathcal{F}_2 = \mathcal{F}_2^+ = \mathcal{F}_3^+ \\
    \wp' & \text{base de } \mathcal{F}_3^- \\
    1, \wp, \wp' & \text{base de } \mathcal{F}_3 = \mathcal{F}_3^+ \oplus \mathcal{F}_3^- .
  \end{cases}
\]

\[
  (25') \qquad
  \begin{aligned}
    & \{1, \wp, \ldots, \wp^n\} \text{ base de } \mathcal{F}_{2n}^+ \simeq \mathcal{F}_{2n+1}^+ \\
    & \{\wp', \wp'\wp, \ldots, \wp'\wp^{n-1}\} \text{ base de }
      \mathcal{F}_{2n+1}^- \simeq \mathcal{F}_{2n+2}^-
  \end{aligned}
\]

\[
  (26) \qquad \mathcal{F}_{2n+1}^- = \mathcal{F}_{2n+2}^- = \wp'\, \mathcal{F}_{2n-2}^+
  \qquad \text{pour } n \in \mathbb{N}^* .
\]

\[
  f \longmapsto f \mid J \qquad \mathcal{F}_3 \longrightarrow k^J
\]

\[
  (27) \qquad \operatorname{Ker}(\mathcal{F}_3 \to k^J) = \mathcal{F}_3^- = k \cdot \wp'
\]

\[
  \wp(z) = \frac{1}{z - t}, \qquad \therefore\ t \in k \setminus \{0, 1\} = U_{0,1}(k)
\]

\[
  \wp(0) = \frac{1}{-t}, \qquad \wp(1) = \frac{1}{1 - t}, \qquad \wp(\infty) = 0
\]

\[
  1, \wp, \wp^2, \wp^3
\]

\[
  \varepsilon_6\, \wp'^2 = \varepsilon_6\, \wp^3 = \delta^6 ,
\]

\[
  (28) \qquad \boxed{\;\wp'^2 = \wp^3 + a_1 \wp^2 + a_2 \wp + a_3\;}
  \qquad a_1, a_2, a_3 \in k
\]

\[
  (-\wp^3) \mid J = a_1 (\wp^2 \mid J) + a_2 (\wp \mid J) + a_3 (1_E \mid J)\ )
\]

\[
  (29) \qquad \wp^3 + a_1 \wp^2 + a_2 \wp + a_3 = \prod_{i \in J} \bigl(\wp - \wp(e_i)\bigr) .
\]

\[
  (30) \qquad \mathcal{F}_2 \simeq \underbrace{\mathcal{F}_1}_{k \cdot 1_E} \oplus
  \underbrace{W_E}_{\simeq\, t_E^{\otimes 2}}
\]

\[
  (31) \qquad
  \begin{aligned}
    \mathcal{F}_\infty &\simeq k[P, P'] \big/ P'^2 - (P^3 + a_1 P^2 + a_2 P + a_3) \\
    &\simeq k[P][P'] \big/ \bigl(P'^2 - (P^3 + a_1 P^2 + a_2 P + a_3)\bigr)
  \end{aligned}
\]

\[
  (32) \qquad \Theta_\delta P = P', \qquad \Theta_\delta P' = \tfrac12 (3P^2 + 2a_1 P + a_2)
\]

\[
  2\wp'\, \Theta_\delta(\wp') = (3\wp^2 + 2a_1 \wp + a_2)\, \wp'
\]

\[
  (33) \qquad \wp'' = \tfrac12 (3\wp^2 + 2a_1 \wp + a_2) .
\]

\[
  Z^3 + a_1 Z^2 + a_2 Z + a_3 \in k[Z]
\]

\[
  (34) \qquad \underbrace{\Delta(a_1, a_2, a_3)}_{\text{discriminant}} \neq 0 ,
\]

\[
  (35) \qquad \wp : E \longrightarrow \mathbb{P}^1_k \simeq E/\{\pm \mathrm{id}_E\}
\]

\[
  (36) \qquad Z^3 + a_1 Z^2 + a_2 Z + a_3 = 0 ,
\]

\[
  (37) \qquad \Sigma_J \xrightarrow{\;\sim\;} \mathbb{P}^1_k , \qquad t \mapsto \infty .
\]

\[
  (38) \qquad \wp(x) = x_0 \in k ,
\]

\[
  (39) \qquad X'^2 = x_0^3 + a_1 x_0^2 + a_2 x_0 + a_3 ,
\]

\[
  (40) \qquad X' = \wp'(x) .
\]

\[
  (41) \qquad x + y + z = 0
\]

\[
  (42) \qquad \det \begin{pmatrix}
    1 & \wp(x) & \wp'(x) \\
    1 & \wp(y) & \wp'(y) \\
    1 & \wp(z) & \wp'(z)
  \end{pmatrix} = 0
\]

\[
  x + y + z = 0 \iff (\{x\} - \{0\}) + (\{y\} - \{0\}) + (\{z\} - \{0\})
  \overset{\text{lin.}}{\sim} 0
\]

\[
  \text{i.e.}\quad \{x\} + \{y\} + \{z\} \overset{\text{lin.}}{\sim} 3\{0\}
\]

\[
  (43) \qquad
  \begin{cases}
    f = \wp' + \alpha \wp + \beta , & \alpha, \beta \in k \\
    \operatorname{div} f = \{x\} + \{y\} + \{z\} - 3\{0_E\}
  \end{cases}
\]

\[
  (44) \qquad f(x) = f(y) = f(z) = 0 \quad\text{i.e.}
\]

\[
  (45) \qquad
  \begin{cases}
    \wp'(x) + \alpha \wp(x) + \beta = 0 \\
    \wp'(y) + \alpha \wp(y) + \beta = 0 \\
    \wp'(z) + \alpha \wp(z) + \beta = 0
  \end{cases}
\]

\[
  (46) \qquad z'^2 = z_0^3 + a_1 z_0^2 + a_2 z_0 + a_3 \ ),
\]

\[
  \begin{cases}
    x_0 = \wp(x), & x' = \wp'(x) \\
    y_0 = \wp(y), & y' = \wp'(y)
  \end{cases}
\]

\[
  (*) \qquad z'(x_0 - y_0) + z_0(y' - x') + (x' y_0 - y' x_0) = 0
\]

\[
  (47) \qquad z' = \lambda z_0 + \mu , \quad\text{avec}\quad
  \lambda = \frac{x' - y'}{x_0 - y_0} , \quad
  \mu = \frac{y' x_0 - x' y_0}{x_0 - y_0}
\]

\[
  (48) \qquad z_0^3 + (a_1 - \lambda^2) z_0^2 + (a_2 - 2\lambda\mu) z_0 +
  (a_3 - \mu^2) = 0
\]

\[
  (49) \qquad z_0 \neq x_0 , \quad z_0 \neq y_0 ,
\]

\[
  x_0 + y_0 + z_0 = -(a_1 - \lambda^2) = \lambda^2 - a_1
  = \Bigl(\frac{x' - y'}{x_0 - y_0}\Bigr)^2 - a_1
\]

\[
  (50) \qquad x, y \in E \setminus \{0\}, \quad x \neq y , \quad
  -(x + y) \neq x, y \quad\text{i.e.}\quad
  \begin{aligned}
    & 2x + y \neq 0 , \ 2y + x \neq 0 \\
    & \text{i.e.}\ y \neq -2x , \ x \neq -2y
  \end{aligned}
  \ \Bigr) ,
\]

\[
  (51) \qquad
  \begin{cases}
    z_0 = -x_0 - y_0 + \Bigl(\dfrac{x' - y'}{x_0 - y_0}\Bigr)^2 - a_1 \\[2ex]
    z' = \lambda z_0 + \mu = \dfrac{x' - y'}{x_0 - y_0}
    \Bigl[ -x_0 - y_0 + \Bigl(\dfrac{x' - y'}{x_0 - y_0}\Bigr)^2 - a_1 \Bigr]
    + \dfrac{y' x_0 - x' y_0}{x_0 - y_0} .
  \end{cases}
\]

\[
  (52) \qquad \wp : E \longrightarrow \mathbb{P}^1_k
\]

\[
  (53) \qquad \dot\wp : E/\{\pm \mathrm{id}_E\} \simeq \text{droite projective type } \mathbb{P}^1_k
\]

\[
  (54) \qquad \wp(0_E) = \infty
\]

\[
  (55) \qquad \wp \longmapsto \wp_1 = a\wp + b \qquad a \in k^*,\ b \in k .
\]

\[
  (56) \qquad (\varepsilon_i)_{i \in J} , \qquad \varepsilon_i \in {}_2E(k)^*
\]

\[
  (57) \qquad \zeta_i = \wp(\varepsilon_i) \in \mathbb{P}^1(k) \setminus \{\infty\} = k
\]

\[
  (58) \qquad \underbrace{\textstyle\sum \zeta_i}_{-a_1} = 0
\]

\[
  \wp \longmapsto \wp_1 = a\wp \qquad a \in k^* .
\]

\[
  (59) \qquad
  \begin{cases}
    a_1 = -\sum \zeta_i \\
    a_2 = \sum \zeta_i \zeta_j \\
    a_3 = -\prod \zeta_i
  \end{cases}
\]

\[
  (60) \qquad \underbrace{Z^3 + a_1 Z^2 + a_2 Z + a_3}_{\prod_i (Z - \zeta_i)} = 0
\]

\[
  \delta \in t_E = T_{E,0}
\]

\[
  \varepsilon_2(\wp) = \delta^{\otimes 2}
\]

\[
  \wp' = \Theta_\delta \wp
\]

\[
  \wp'^2 = \wp^3 + a_1 \wp^2 + a_2 \wp + a_3 .
\]

\[
  x \longmapsto x + \varepsilon_i .
\]

\[
  \sigma_i Z = \frac{aZ + b}{cZ + d}
\]

\[
  \frac{a}{c} = \zeta_i , \qquad -\frac{d}{c} = \zeta_i
\]

\[
  \sigma_i Z = \frac{\zeta_i Z + b}{Z - \zeta_i}
\]

\[
  \frac{\zeta_i \zeta_j + b}{\zeta_j - \zeta_i} = \zeta_k
  \quad\text{i.e.}\quad b = \zeta_j \zeta_k - \zeta_i(\zeta_j + \zeta_k)
\]

\[
  (61) \qquad \sigma_i Z = \frac{\zeta_i Z +
  \overbrace{\zeta_j \zeta_k - \zeta_i(\zeta_j + \zeta_k)}^{b_i}}{Z - \zeta_i}
\]

\[
  b_i = 2\underbrace{\zeta_j \zeta_k}_{-a_3/\zeta_i} -
  \underbrace{(\zeta_i \zeta_j + \zeta_j \zeta_k + \zeta_k \zeta_i)}_{a_2}
  = -(a_2 + 2a_3/\zeta_i)
\]

\[
  (62) \qquad \sigma_i Z = \frac{\zeta_i Z - (a_2 + 2a_3/\zeta_i)}{Z - \zeta_i}
  = \frac{\zeta_i^2 Z - (a_2 \zeta_i + 2a_3)}{\zeta_i Z - \zeta_i^2}
\]

\[
  (63) \qquad Z_i^2 - 2\zeta_i Z_i + (a_2 + 2a_3/\zeta_i) = 0
\]

\[
  (64) \qquad Z_i = \zeta_i \pm \sqrt{\zeta_i^2 - a_2 - 2a_3/\zeta_i}
\]

\[
  \frac{1}{\zeta_i}\bigl(\zeta_i^3 - a_2 \zeta_i - 2a_3\bigr)
\]

\[
  \zeta_i^3 = -a_1 \zeta_i^2 - a_2 \zeta_i - a_3
\]

\[
  -\frac{1}{\zeta_i}\bigl(a_1 \zeta_i^2 + 2a_2 \zeta_i + 3a_3\bigr)
\]

\[
  (65) \qquad
  \begin{cases}
    (Z_i - \zeta_i)^2 = -\dfrac{1}{\zeta_i}\bigl(a_1 \zeta_i^2 + 2a_2 \zeta_i + 3a_3\bigr) \\[2ex]
    \text{i.e.}\ Z_i = \zeta_i \pm \sqrt{-\dfrac{1}{\zeta_i}\bigl(a_1 \zeta_i^2 + 2a_2 \zeta_i + 3a_3\bigr)}
  \end{cases}
\]

\[
  (66) \qquad Z_i'^2 = \underbrace{Z_i^3 + a_1 Z_i^2 + a_2 Z_i + a_3}_{(Z_i - \zeta_i)(Z_i - \zeta_j)(Z_i - \zeta_k)} .
\]

\[
  (67) \qquad 2x_i = \varepsilon_i
\]

\[
  (68\,a) \qquad \wp(x_i) = Z_i
\]

\[
  (68\,b) \qquad \wp'(x_i) = Z_i' ,
\]

\[
  (69) \qquad \sum_{i \in J} x_i = 0
\]

\[
  (70) \qquad
  \begin{cases}
    \det \begin{pmatrix} 1 & Z_1 & Z_1' \\ 1 & Z_2 & Z_2' \\ 1 & Z_3 & Z_3' \end{pmatrix} = 0
    \qquad \text{i.e.} \\[3ex]
    (Z_1 Z_2' - Z_2 Z_1') + (Z_2 Z_3' - Z_3 Z_2') + (Z_3 Z_1' - Z_1 Z_3') = 0 .
  \end{cases}
\]

\[
  (71) \qquad
  \begin{cases}
    (Z_i - \zeta_i)^2 = -\dfrac{1}{\zeta_i}\bigl(a_1 \zeta_i^2 + 2a_2 \zeta_i + 3a_3\bigr)
    \quad \bigl[= -(a_1 \zeta_i + 2a_2 + 3\zeta_j \zeta_k)\bigr] \\[2ex]
    Z_i'^2 = Z_i^3 + a_1 Z_i^2 + a_2 Z_i + a_3
    \quad \bigl[= (Z_i - \zeta_i)(Z_i - \zeta_j)(Z_i - \zeta_k)\bigr] \\[2ex]
    \det \begin{pmatrix} 1 & Z_1 & Z_1' \\ 1 & Z_2 & Z_2' \\ 1 & Z_3 & Z_3' \end{pmatrix} = 0
    \\ \quad \text{i.e.}\
    \underbrace{(Z_2 Z_3' - Z_3 Z_2')}_{\varphi_1} +
    \underbrace{(Z_3 Z_1' - Z_1 Z_3')}_{\varphi_2} +
    \underbrace{(Z_1 Z_2' - Z_2 Z_1')}_{\varphi_3} = 0
  \end{cases}
\]

\[
  (72) \qquad (x_i)_{i \in J} \in E(k)^J \quad \text{satisfaisant (67), (69),}
  \qquad 2x_i = \varepsilon_i \ \bigl(\in {}_2E(k)^*\bigr) , \quad \sum x_i = 0
\]

\[
  (73) \qquad Z_i = \wp(x_i) , \quad Z_i' = \wp'(x_i) .
\]

\[
  (74) \qquad \dot\imath \in \mu_4^*(k) ,
\]

\[
  (75) \qquad \omega \in \underline{\omega} = \underline{\omega}_J
  \qquad \text{un ordre circulaire sur } J = {}_2E(k)^* = \{\varepsilon_i\} ,
\]

\[
  (75) \qquad K : \underline{t}_E \simeq \mathcal{O}_{\Sigma_J, t}(1)^{\otimes \,\cdots}
  \xrightarrow[\sim]{\ \text{via choix de } \dot\imath\ } \mathcal{O}_t(1) ,
\]

\[
  (77) \qquad \underline{t}_E^{\otimes 2} \simeq T_{\Sigma_J, t} , \qquad
  \mathcal{O}_t(1)^{\otimes 2} = \mathcal{O}_t(2) \underset{\text{via } \alpha}{\simeq}
  \mathcal{O}_t(2)(\underline{\omega}) \simeq T_{\Sigma_J, t}
\]

\[
  (79) \qquad 0 \to k \cdot 1_{\Sigma_J} \to
  \overbrace{\Gamma(\Sigma_J, \underbrace{\underline{\mathcal{O}}_{\Sigma_J}\{t\}}_{\substack{\simeq\, \underline{\mathcal{O}}_{\Sigma_J}(1) \\ \text{non canoniquement}}})}^{\dot\simeq\, \mathcal{F}_2}
  \to T_{\Sigma_J, t} \to 0
\]

\[
  (80) \qquad \check F_t \simeq \mathcal{O}_t(1)(\underline{\omega})
\]

\[
  \check K : \check{\underline{t}}_E \simeq \underline{\mathcal{O}}_t(1)^\vee \simeq
  \underline{\mathcal{O}}_t(-1) \xrightarrow[\text{via } \omega]{}
  \underline{\mathcal{O}}_t(-1)(\underline{\omega}) \simeq F_t
\]

\[
  \check t_E \hookrightarrow V_J(k) \hookrightarrow k^J
\]

\[
  V_J(\underline{t}_E) = \operatorname{Ker}\bigl(\underline{t}_E^J \xrightarrow[\text{somme}]{} \underline{t}_E\bigr)
\]

\[
  (82) \qquad (\delta_i)_{i \in J} \in \underline{t}_E^J
\]

\[
  (83) \qquad
  \begin{cases}
    \text{a) } \sum \delta_i = 0 \quad \text{exprimant } \delta_i \in V_J(\underline{t}_E) \\
    \text{b) Relation de proportionnalité, exprimant que l'image de } \\
    \qquad \check t_E \hookrightarrow V_J(k) \text{ définie par } (\delta_i) \text{ est } F_t , \\
    \qquad \text{de sorte que } (\delta_i) \text{ définit }
      K : \underline{t}_E \xrightarrow{\sim} \check F_t \xrightarrow[\text{via } \omega]{\sim} \mathcal{O}_t(1) \\
    \text{c) Relation de l'épinglage de Legendre, exprimant que } \\
    \qquad K^{\otimes 2} \text{ est un isom.\ standard déjà connu.}
  \end{cases}
\]

\[
  (83) \qquad \delta_i = \varphi_i^\omega(\delta)\, \delta \qquad
  \text{où}\quad \varphi_i^\omega(\delta) \in k^* \quad
  \bigl(i \in J,\ \omega \in \underline{\omega}(J),\ \delta \in \underline{t}_E^*\bigr)
\]

\[
  (85) \qquad \varphi_i^\omega(\lambda\delta) = \frac{1}{\lambda}\, \varphi_i^\omega(\delta)
\]

\[
  (86) \qquad \sum_i \varphi_i^\omega(\delta) = 0
\]

\[
  (87) \qquad k \cdot \bigl((\varphi_i^\omega(\delta))_i\bigr) = F_t
\]

\[
  (88) \qquad \bigl(\varphi_i^\omega(\delta)\bigr) =
  \underbrace{\check K(\check\delta)}_{\mathcal{O}_t(-1)} \wedge \omega
  \in \mathcal{O}_t(-1)(\underline{\omega}) \simeq F_t \subset V_J(k) \subset k^J
\]

\[
  (89) \qquad \varphi_i^{\omega'}(\delta) = \dot\imath_\omega\, \varphi_i^\omega(\delta)
\]

\[
  \dot\imath_\omega \overset{\text{déf}}{=} \alpha(\omega) , \qquad
  \alpha : \mu_4^* \xrightarrow{\ \sim\ } \underline{\omega} \ \text{étant l'iso.}
\]

\[
  \begin{array}{ccc}
    \xi_1 & \xi_2 & \xi_3 \\
    \downarrow & \downarrow & \downarrow \\
    0 & 1 & \infty
  \end{array}
  \in k , \qquad \infty \mapsto t
\]

\[
  \begin{cases}
    (\xi_1, 1) \\
    (\xi_2, 1) \\
    (\xi_3, 1)
  \end{cases}
  \qquad
  \begin{array}{l}
    \lambda_1 \xi_1, \lambda_1 \\
    \lambda_2 \xi_2, \lambda_2 \\
    \lambda_3 \xi_3, \lambda_3
  \end{array}
\]

\[
  \left[\;
  \begin{aligned}
    &\textstyle\sum \lambda_i = 0 \\
    &\textstyle\sum \lambda_i \xi_i = 0
  \end{aligned}
  \right.
\]

\[
  \begin{aligned}
    \lambda_1 &= \xi_2 - \xi_3 \\
    \lambda_2 &= \xi_3 - \xi_1 \\
    \lambda_3 &= \xi_1 - \xi_2
  \end{aligned}
  \qquad
  \begin{cases}
    e_2 - e_3 = \bigl((\xi_2 - \xi_3)\xi_1,\ \xi_2 - \xi_3\bigr) \\
    e_3 - e_1 = \bigl((\xi_3 - \xi_1)\xi_2,\ \xi_3 - \xi_1\bigr) \\
    e_1 - e_2 = \bigl((\xi_1 - \xi_2)\xi_3,\ \xi_1 - \xi_2\bigr)
  \end{cases}
\]

\[
  \begin{aligned}
    \xi_1 f_1 + f_2 &= \frac{1}{\xi_2 - \xi_3}\,(e_2 - e_3) \\
    \xi_2 f_1 + f_2 &= \frac{1}{\xi_3 - \xi_1}\,(e_3 - e_1) \\
    \xi_3 f_1 + f_2 &= \frac{1}{\xi_1 - \xi_2}\,(e_1 - e_2)
  \end{aligned}
\]

\[
  (\xi_1 - \xi_2) f_1 = \frac{1}{\xi_3 - \xi_1}\, e_1 + \frac{1}{\xi_2 - \xi_3}\, e_2
  - \Bigl(\frac{1}{\xi_3 - \xi_1} + \frac{1}{\xi_2 - \xi_3}\Bigr) e_3
\]

\[
  f_1 = \frac{1}{(\xi_1 - \xi_2)(\xi_3 - \xi_1)}\, e_1
  + \frac{1}{(\xi_1 - \xi_2)(\xi_2 - \xi_3)}\, e_2
  - \frac{1}{\xi_1 - \xi_2}\,(\;\cdot\;)\, e_3
\]

\[
  \begin{aligned}
    f_2 &= \frac{1}{\xi_2 - \xi_3}\,(e_2 - e_3) - \xi_1 f_1 \\
    &= -\frac{\xi_1}{(\xi_1 - \xi_2)(\xi_3 - \xi_1)}\, e_1
      + \Bigl(\frac{1}{\xi_2 - \xi_3} - \frac{\xi_1}{(\xi_1 - \xi_2)(\xi_2 - \xi_3)}\Bigr) e_2 \\
    &\qquad + \Bigl[-\frac{1}{\xi_2 - \xi_3}
      + \frac{\xi_1}{\xi_1 - \xi_2}\Bigl(\frac{1}{\xi_3 - \xi_1} + \frac{1}{\xi_2 - \xi_3}\Bigr)\Bigr] e_3
  \end{aligned}
\]

\[
  \begin{aligned}
    (\xi_1 - \xi_2)(\xi_2 - \xi_3)(\xi_3 - \xi_1)\, f_2
    &= -\xi_1(\xi_2 - \xi_3)\, e_1
      + \bigl[(\xi_1 - \xi_2)(\xi_3 - \xi_1) - \xi_1(\xi_3 - \xi_1)\bigr] e_2 \\
    &\qquad + \bigl[-(\xi_1 - \xi_2)(\xi_3 - \xi_1) + \xi_1(\xi_2 - \xi_3) + \xi_1(\xi_3 - \xi_1)\bigr] e_3 \\
    &= -\xi_1(\xi_2 - \xi_3)\, e_1 - \xi_2(\xi_3 - \xi_1)\, e_2 - \xi_3(\xi_1 - \xi_2)\, e_3
  \end{aligned}
\]

\[
  (\xi_1 - \xi_2)(\xi_2 - \xi_3)(\xi_3 - \xi_1) = \Delta_0
\]

\[
  \boxed{\;
  \begin{aligned}
    -\Delta_0 f_2 &= \xi_1(\xi_2 - \xi_3)\, e_1 + \xi_2(\xi_3 - \xi_1)\, e_2 + \xi_3(\xi_1 - \xi_2)\, e_3 \\
    \Delta_0 f_1 &= (\xi_2 - \xi_3)\, e_1 + (\xi_3 - \xi_1)\, e_2 + (\xi_1 - \xi_2)\, e_3
  \end{aligned}\;}
\]

\[
  = \sum_i \bigl(\wp(\varepsilon_j) - \wp(\varepsilon_k)\bigr)\, e_i
\]

\[
  V = t_E \qquad \text{(espace tangent à l'origine)} \tag{1}
\]

\[
  \Pi \subset V \tag{2}
\]

\[
  V \simeq \Pi \otimes_{\mathbb{Z}} \mathbb{R} \overset{\text{déf}}{=} \Pi_{\mathbb{R}} \tag{3}
\]

\[
  E \simeq V/\Pi \simeq \Pi_{\mathbb{R}}/\Pi , \tag{4}
\]

\[
  V_{\mathbb{C}} = V \otimes_{\mathbb{R}} \mathbb{C} \tag{5}
\]

\[
  \begin{aligned}
    V_{\mathbb{C}} &\longrightarrow V = V_{\mathbb{R}} \\
    z &\longmapsto \tfrac{1}{2}(z + \bar z) = \Re z
  \end{aligned} \tag{6}
\]

\[
  \varphi : L \xrightarrow{\;\sim\;} V \qquad (\mathbb{R}\text{-iso}) \tag{7}
\]

\[
  u x = \varphi(-i\,\varphi^{-1}(x)) = -\varphi(i\,\varphi(x)) \tag{8}
\]

\[
  \psi : V_{\mathbb{C}} \longrightarrow V_c \tag{9}
\]

\[
  \begin{cases}
    \psi(i x) = u x \\
    \psi(x) = x
  \end{cases}
  \qquad \forall\, x \in V = V_{\mathbb{R}}, \tag{10}
\]

\[
  \psi(x + i y) = x + u y . \tag{11}
\]

\[
  \begin{aligned}
    L' = \operatorname{Ker} \psi
    &= \{\, x + i y \mid x, y \in V_{\mathbb{R}},\ x + u y = 0 \ \text{ i.e. } y = u x \,\} \\
    &= \{\, x + i u(x) \mid x \in V_{\mathbb{R}} \,\}
  \end{aligned} \tag{12}
\]

\[
  \varphi : \underset{\substack{\| \\ x + i u(x)}}{z} \longmapsto \tfrac{1}{2}(z + \bar z) = \Re z = x : L \longrightarrow V .
\]

\[
  V_{\mathbb{C}} = \bigoplus_{\omega \in \underline{\omega}} L_\omega \tag{14}
\]

\[
  \varphi_\omega : \mathbb{U} \xrightarrow{\;\sim\;} \mathbb{U} \qquad
  g(z) = \varphi_\omega(g)\cdot z \ \text{ pour } z \in L_\omega \tag{15}
\]

\[
  \Re(\varphi_\omega(g) z) = \Re(g z) = g\, \Re(z) = \text{produit de } \Re(z) \text{ par } \varphi_\omega(g)
\]

\[
  \begin{aligned}
    L_\omega &\xrightarrow{\;\sim\;} V \\
    z &\longmapsto \Re z
  \end{aligned}
\]

\[
  \Sigma = \Sigma_V = P(\mathbb{C}), \qquad \text{où } P = \mathbb{P}(V) \tag{16}
\]

\[
  \begin{aligned}
    L &\longrightarrow V \\
    z &\longmapsto \Re z
  \end{aligned}
\]

\[
  \Sigma_{\mathbb{R}} = P(\mathbb{R})
\]

\[
  z \in \Sigma \setminus \Sigma_{\mathbb{R}} . \tag{17}
\]

\[
  \mathbb{Z}^2 \longrightarrow \Pi \tag{18}
\]

\[
  z_1 \wedge z_2 = \omega \tag{19}
\]

\[
  \mathbb{R}^2 \xrightarrow{\;\sim\;} V \tag{20}
\]

\[
  \begin{cases}
    \mathbb{P}^1_{\mathbb{R}} \simeq \mathbb{P}^1(V), \quad \mathbb{P}^1(\mathbb{C}) \simeq \Sigma_V \\
    \mathfrak{D} \xrightarrow{\;\sim\;} \Sigma^*_{V,\omega}
  \end{cases} \tag{21}
\]

\[
  \mathfrak{D} = \{\, z \in \mathbb{C} \mid \Im z > 0 \,\}, \tag{22}
\]

\[
  L_z = \mathbb{C}\cdot\underset{z e_1 + e_2}{\underbrace{(z, 1)}} . \tag{23}
\]

\[
  \underset{\substack{\| \\ a + ib}}{\lambda} \longmapsto
  \Re\bigl(\underbrace{\lambda(z e_1 + e_2)}_{[(ax - by) + i(ay + bx)] e_1 + (a + ib) e_2}\bigr)
  = (ax - by) e_1 + a e_2
\]

\[
  \lambda_1 = -\frac{i}{y} \qquad \lambda_2 = 1 + i\,\frac{x}{y}
\]

\[
  \lambda_2 / \lambda_1 = i(y + ix) = -x + iy
\]

\[
  \underset{\substack{\text{quotient dans } \mathbb{C}^* \\ \text{au sens de la} \\
  \text{structure compl.\ de } \mathbb{R}^2 \\ \text{définie par } z = x + iy \in \mathfrak{D}}}
  {\underbrace{e_2 / e_1}}
  = (-x + iy)^{-} = -x - iy = -z \ \in \mathfrak{D}^{-}
\]

\[
  z = -(z_2 / z_1) \tag{24}
\]

\[
  g \in \mathrm{SL}(2, \mathbb{Z}) \qquad g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
  \qquad a, b, c, d \in \mathbb{Z},\ ad - bc = 1
\]

\[
  u : \mathbb{Z}^2 \xrightarrow{\;\sim\;} \Pi
\]

\[
  z_1 = u(e_1), \quad z_2 = u(e_2) \in \Pi \subset V
\]

\[
  \begin{aligned}
    z'_1 &= u\,g(e_1) = u(a e_1 + c e_2) = a z_1 + c z_2 \\
    z'_2 &= u\,g(e_2) = u(b e_1 + d e_2) = b z_1 + d z_2 ,
  \end{aligned}
\]

\[
  z' = x' + i y' \in \mathfrak{D}'
\]

\[
  z' = -\frac{z'_2}{z'_1} = -\frac{b z_1 + d z_2}{a z_1 + c z_2}
  = -\frac{d \frac{z_2}{z_1} + b}{c \frac{z_2}{z_1} + a}
  = \frac{d z - b}{-c z + a} \tag{25}
\]

\[
  g^{-1} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

\[
  z' = \frac{a z + b}{c z + d} \tag{25 bis}
\]

\[
  z = z_2 / z_1 ,
\]

\[
  z = -(z_2 / z_1)^{-} = -\bar z_2 / \bar z_1 \in \mathfrak{D}
\]

\[
  E_z = \mathbb{C} / (\mathbb{Z}\cdot 1 + \mathbb{Z}\cdot(-\bar z))
\]

\[
  g z = \frac{a z + b}{c z + d} \qquad \text{pour } g = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}),
\]

\[
  g \cdot u = u \circ g^{-1} \qquad (u : \mathbb{Z}^2 \xrightarrow{\;\sim\;} \Pi_E).
\]

\[
  \mathcal{T}_{1,1} \simeq \mathrm{SL}(2, \mathbb{Z}) \tag{26}
\]

\[
  M_{1,1} \simeq (\mathfrak{D}, \mathrm{SL}(2, \mathbb{Z})) \tag{27}
\]

\[
  \mathcal{T} \longrightarrow \operatorname{Aut}^+_{\mathbb{Z}}(\Pi) \qquad \text{où } \Pi = \pi_1(E, a) \tag{27}
\]

\[
  \mathcal{T} \longrightarrow \operatorname{Aut}^+_{\mathbb{Z}}(H) \tag{28}
\]

\[
  z \longmapsto \exp(2 i \pi z) : \mathbb{C} \longrightarrow \mathbb{C}^*
\]

\[
(1)\quad
\begin{cases}
\text{a) « réseau » } H,\ \mathbb{Z}\text{-module libre de rang } 2 \\
\quad \bigl(H = \pi_1(E, o_E) \simeq H_1(E,\mathbb{Z}) = H^1(E,\mathbb{Z})^{\vee}\bigr) \\
\text{b) Une structure complexe sur } V = H \otimes_{\mathbb{Z}} \mathbb{R}
\ (\simeq t_E, \text{ espace tangent à l'origine de } E).
\end{cases}
\]

\[
(2)\qquad E \simeq H \otimes_{\mathbb{Z}} \mathbb{R} / H
\]

\[
(3)\qquad \tau = z_2 / z_1
\]

\[
(3\,\mathrm{bis})\qquad z_2 = \tau z_1 .
\]

\[
(4)\qquad t : \mathbb{Z}_S^2 \xrightarrow{\;\sim\;} \mathcal{H},
\]

\[
(5)\qquad \tau : S \longrightarrow \mathbb{C} \setminus \mathbb{R} .
\]

\[
(6)\qquad \tau : S \longrightarrow \mathfrak{D}^+ = \{ z \in \mathbb{C} \mid \Im z > 0 \}.
\]

\[
(7)\qquad \mathfrak{X}_{\mathfrak{D}^+} = (\mathfrak{D}^+ \times \mathbb{C}) / \mathbb{Z}^2
\]

\[
(8)\qquad T_{m,n}(\tau, z) = (\tau,\, z + m + n\tau), \qquad
\tau \in \mathfrak{D}^+,\ z \in \mathbb{C} .
\]

\[
(8)\qquad (\mathfrak{X}_{\mathfrak{D}^+})_\tau \simeq \mathbb{C}/(\mathbb{Z} + \mathbb{Z}.\tau)
\simeq \mathbb{C}^* / \text{sous-groupe engendré par } k = \exp(2i\pi\tau)
\]

\[
(9)\qquad \mathrm{SL}(2,\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\;\middle|\; \begin{array}{l} a,b,c,d \in \mathbb{Z} \\ ad - bc = 1 \end{array} \right\}
\]

\[
t \longmapsto t \circ g ,
\]

\[
\begin{aligned}
z'_1 &= a z_1 + c z_2 \\
z'_2 &= b z_1 + d z_2
\end{aligned}
\]

\[
(10)\qquad \tau' = \frac{d\tau + b}{c\tau + a}
\qquad \left(\tau' = \tau \circ g,\ g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\right)
\]

\[
\begin{aligned}
E_\tau &= \mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau), \\
E_{\tau\circ g} &= \mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau')
\qquad \tau' = \tau \circ g = \frac{d\tau + b}{c\tau + a}
\end{aligned}
\]

\[
E_{\tau} \xrightarrow{\;\sim\;} E_{\tau\circ g} .
\]

\[
E_\tau \xrightarrow{\;\sim\;} E_{\tau'}
\qquad \tau' = \tau \circ \tau_\infty(g^{-1}) = \frac{a\tau + b}{c\tau + d}
\]

\[
(14)\qquad T_g.(\tau, z) = \Bigl(g.\tau,\ \frac{z}{c\tau + d}\Bigr)
\qquad \text{où } g = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\
g.\tau = \frac{a\tau + b}{c\tau + d} .
\]

\[
g^{-1} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

\[
(11)\qquad \tau \circ g^{-1} = \frac{a\tau - b}{-c\tau + d} .
\]

\[
(12)\qquad
\begin{cases}
\tau_\infty = \begin{pmatrix} -1 & 0 \\ 0 & +1 \end{pmatrix} \in \mathrm{GL}(2,\mathbb{Z})
\quad \text{d'où} \\[6pt]
\tau_\infty \begin{pmatrix} A & B \\ C & D \end{pmatrix} \tau_\infty^{-1}
= \tau_\infty\Bigl(\begin{pmatrix} A & B \\ C & D \end{pmatrix}\Bigr)
= \begin{pmatrix} A & -B \\ -C & D \end{pmatrix} \\[6pt]
\tau_\infty\Bigl(\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1}\Bigr)
= \begin{pmatrix} d & b \\ c & a \end{pmatrix}
\quad \text{« échange de } a \text{ et de } d \text{ »,}
\end{cases}
\]

\[
(13)\qquad \tau \circ \tau_\infty(g^{-1}) = \frac{a\tau + b}{c\tau + d}
\qquad \text{où } g = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\quad
\tau_\infty(g^{-1}) = \begin{pmatrix} d & b \\ c & a \end{pmatrix}
\]

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix} = g \longmapsto
\tau_\infty(g^{-1}) = \begin{pmatrix} d & b \\ c & a \end{pmatrix} .
\]

\[
(15)\qquad T_{g,m,n}(\tau, z) = \Bigl(g\tau,\ \frac{z}{c\tau + d} + m + n\, g\tau\Bigr)
\]

\[
\begin{array}{c}
\mathfrak{X}_{\mathfrak{D}^+} = \mathfrak{D}^+ \times \mathbb{C} / \mathbb{Z}^2 \\
\Big\downarrow \scriptstyle (\tau, z) \mapsto \tau \\
\mathfrak{D}^+
\end{array}
\qquad
\begin{array}{l}
\mathbb{Z}^2 \text{ opérant par} \\
T_{m,n}(\tau, z) = (\tau, z + m + n\tau)
\end{array}
\]

\[
g\tau = \frac{a\tau + b}{c\tau + d} \qquad
g = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\quad
\begin{array}{l} a,b,c,d \in \mathbb{Z} \\ ad - bc = 1 \end{array}
\]

\[
g \cdot \tau = \tau \circ \tau_\infty(g^{-1})
\qquad \text{où}\quad
\begin{array}{l}
\tau_\infty = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \in \mathrm{GL}(2,\mathbb{Z}) \\[4pt]
\tau_\infty(g^{-1}) = \begin{pmatrix} d & b \\ c & a \end{pmatrix}
\ \text{si } g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\end{array}
\]

\[
(16)\qquad \mathcal{T} \longrightarrow \mathrm{Aut}^+_{\mathbb{Z}}(H)
\]

\[
(17)\qquad z_1, z_2 \in \mathbb{C}^* \quad \text{avec}\quad \tau = z_2/z_1 \in \mathfrak{D}^+
\]

\[
(18)\qquad P\mathfrak{D}^+ = \{ (z_1, z_2) \mid z_1, z_2 \in \mathbb{C}^*,\ z_2/z_1 \in \mathfrak{D}^+ \}
\]

\[
(19)\qquad g(z_1, z_2) = (d z_1 + c z_2,\ b z_1 + a z_2)
\]

\[
(20)\qquad \mathfrak{X}_{P\mathfrak{D}^+} \simeq P\mathfrak{D}^+ \times \mathbb{C} / \mathbb{Z}^2
\]

\[
T_{m,n}(z_1, z_2, z) = (z_1, z_2, z + m z_1 + n z_2)
\]

\[
(21)\qquad T_g(z_1, z_2, z) = (d z_1 + c z_2,\ b z_1 + a z_2,\ z)
\]

\[
(22)\qquad T_\lambda(z_1, z_2, z) = \Bigl(\frac{1}{\lambda} z_1,\ \frac{1}{\lambda} z_2,\ \frac{1}{\lambda} z\Bigr).
\]

\[
1 \longrightarrow N \longrightarrow G \longrightarrow \mathfrak{S}_A \longrightarrow 1
\]

\[
N \simeq \bigl((\mathbb{F}_2)^A\bigr)' = \operatorname{Ker}\bigl(\mathbb{F}_2^A \to \mathbb{F}_2\bigr),
\qquad (x_i) \longmapsto \textstyle\sum x_i
\]

\[
\begin{aligned}
g &= u \cdot x = y u \qquad (u \in \mathfrak{S}_0,\ x \in N), &\quad
ux &= yu,\ \ y = u x u^{-1} = u(x) \\
g' &= u' \cdot x' = y' u' &\quad g'g &= y' \underline{u' y u'^{-1}}\, \underline{u' u} \\
g' g &= u' x' u x = (u' u)\bigl(\underbrace{u^{-1} x' u}_{u^{-1}(x')}\bigr) x = U'' X'' ,
&\quad U'' &= u'u,\ \ X'' = u^{-1}(x')\, x \\
&= \bigl(y' u'(y)\bigr)\, u' u
\end{aligned}
\]

\[
B = \mathbb{Z}[t]\bigl[S\bigr]/\bigl(S^2 - Q(t)\bigr)
\]

\[
S^2 - a = (T + \alpha)^2 - a = T^2 + 2\alpha T + b
\]

\[
B \simeq A[T]/(T^2 + 2\alpha T + b), \qquad
\hat B \simeq \underbrace{A[[T]]}_{C}/(T^2 + 2\alpha T + b)
\]

\[
(1)\qquad P = \underset{\varepsilon}{\pm}\, \underbrace{p_1 p_2 \cdots p_n}_{q}\,
\underbrace{P_1 \cdots P_\ell}_{Q}
\]

\[
P = \pm q\, Q \qquad
\begin{cases}
q \in \mathbb{N}^* \text{ « quadratfrei »} \\
Q \in \mathbb{Z}[t] \text{ à coeff.\ dominant } > 0 \text{, « quadratfrei » dans } \mathbb{Q}[t] \\
\quad \text{à coeff.\ premiers entre eux.}
\end{cases}
\]

\[
P = \varepsilon\, Q \qquad \bigl(\varepsilon \in \{\pm 1\}\bigr)
\]

\[
Q(T) \equiv T^2 \ (2) \quad \text{si } \lambda \equiv 0 \ (4).
\]

\[
\mathbb{E}^1_{\mathbb{Z}} \setminus \{0, \lambda\} = \operatorname{Spec} \mathbb{Z}[T]_{T(T-\lambda)}
\]

\[
\mathbb{P}^1_{\mathbb{Z}} \setminus \{0, \lambda\} = \operatorname{Spec} \mathbb{Z}[T']_{\lambda T' - 1}
\]

\[
T^2 + 2\alpha T + b
\]

\[
x^2 + \alpha x y + \beta y^2, \qquad (x + \beta y)\ \ldots
\]

\[
\mathbb{F}_2[T, \pi] \qquad T^2 + \alpha \pi T + \beta \pi^2, \qquad
\alpha^2 - 4\beta = \alpha^2
\]

\[
4T - 1
\]