\[
  (1) \qquad E \times E' \longrightarrow \mathcal{O}_S \qquad (x, x') \longmapsto \langle x, x' \rangle
\]

\[
  (2) \qquad \underline{\det}(E) \otimes \underline{\det}(E') \simeq \mathcal{O}_S
\]

\[
  \underline{\det} E \simeq \Lambda^3 E \quad )
\]

\[
  \begin{array}{ll}
    E \times \Lambda^2 E \longrightarrow \Lambda^3 E = \underline{\det} E & (x, \omega) \longmapsto x \wedge \varphi = \varphi \wedge x \\
    E' \times \Lambda^2 E' \longrightarrow \Lambda^3 E' = \underline{\det} E' & (x', \varphi') \longmapsto x' \wedge \varphi' \; (= \varphi' \wedge x')
  \end{array}
\]

\[
  (3) \qquad
  \begin{cases}
    \Lambda^2 E \overset{\alpha_0}{\simeq} \check{E} \otimes \underline{\det} E \simeq E' \otimes \underline{\det}(E) \\
    \Lambda^2 E' \overset{\alpha'_0}{\simeq} \check{E}' \otimes \underline{\det} E' \simeq E \otimes \underline{\det}(E')
  \end{cases}
\]

\[
  (4) \qquad \omega \in \Gamma(S, \underline{\det}(E))
\]

\[
  (5) \qquad \mathcal{O}_S \xrightarrow{\ \sim\ } \underline{\det}(E) \qquad \lambda \longmapsto \lambda\omega
\]

\[
  (4') \qquad \omega' = \omega^{-1} \in \Gamma(S, \underline{\det}(E'))
\]

\[
  (5') \qquad \mathcal{O}_S \xrightarrow{\ \sim\ } \underline{\det}(E') \qquad \lambda \longmapsto \lambda\omega'
\]

\[
  (6) \qquad
  \begin{cases}
    \Lambda^2 E \xrightarrow[\sim]{\ \alpha\ } E' \\
    \Lambda^2 E' \xrightarrow[\sim]{\ \alpha'\ } E
  \end{cases}
\]

\[
  (7) \qquad \langle x, \alpha(\varphi) \rangle\, \omega = x \wedge \varphi , \qquad
  \langle \alpha'(\varphi'), x' \rangle\, \omega' = \varphi' \wedge x'
\]

\[
  (8) \qquad (e_1, e_2, e_3) \qquad (e_i \in \Gamma(S, E))
\]

\[
  (8') \qquad (e'_1, e'_2, e'_3) \qquad (e'_i \in \Gamma(S, E'))
\]

\[
  \langle e_i, e'_j \rangle = \delta_{ij} \qquad (\text{symb.\ de Kronecker})
\]

\[
  (9) \qquad
  \begin{cases}
    \bar{e}_1 = e_2 \wedge e_3 ,\ \bar{e}_2 = e_3 \wedge e_1 ,\ \bar{e}_3 = e_1 \wedge e_2 & (\text{base de } \Lambda^2 E) \\
    \bar{e}'_1 = e'_2 \wedge e'_3 ,\ \bar{e}'_2 = e'_3 \wedge e'_1 ,\ \bar{e}'_3 = e'_1 \wedge e'_2 & (\text{--- } \Lambda^2 E')
  \end{cases}
\]

\[
  (10) \qquad
  \begin{cases}
    \alpha(\bar{e}_i) = e'_i \\
    \alpha'(\bar{e}'_i) = e_i
  \end{cases}
\]

\[
  f(x)(y) \; \bigl(= \langle y, f(x) \rangle\bigr) = \varphi(x, y)
\]

\[
  (12) \qquad (x', y') \longmapsto [x', y']_{\varphi} = f\alpha'(x' \wedge y')
\]

\[
  (16) \qquad
  \begin{cases}
    [[e'_1 \wedge e'_2], e'_3] = (ap' - qr')e'_1 + (p'r - bq)e'_2 + (p'q' - pq)e'_3 \\
    [[e'_2 \wedge e'_3], e'_1] = (q'r' - qr)e'_1 + (bq' - rp')e'_2 + (q'p - cr)e'_3 \\
    [[e'_3, e'_1], e'_2] = (r'q - ap)e'_1 + (r'p' - rp)e'_2 + (cr' - pq')e'_3
  \end{cases}
\]

\[
  (17) \qquad \mathrm{Jac}(e_1, e_2, e_3) = [[e_1, e_2], e_3] + [[e_2, e_3], e_1] + [[e_3, e_1], e_2]
\]

\[
  = [a(p' - p) + (q'r' - qr)]e'_1 + [b(q' - q) + (r'p' - rp)]e'_2 + [c(r' - r) + (p'q' - pq)]e'_3
\]

\[
  (18) \qquad
  \begin{cases}
    a(p' - p) + (q'r' - qr) = 0 & \text{ou } (ap' - qr) - (ap - q'r') = 0 \\
    b(q' - q) + (r'p' - rp) = 0 & \text{ou } (bq' - rp) - (bq - r'p') = 0 \\
    c(r' - r) + (p'q' - pq) = 0 & \text{ou } (cr' - pq) - (cr - p'q') = 0
  \end{cases}
\]

\[
  (19) \qquad
  \begin{cases}
    \mathrm{Min}^{12}_{13}(\varphi) - \mathrm{Min}^{12}_{13}({}^{t}\varphi) \\
    \mathrm{Min}^{23}_{21}(\varphi) - \mathrm{Min}^{23}_{21}({}^{t}\varphi) \\
    \mathrm{Min}^{31}_{32}(\varphi) - \mathrm{Min}^{31}_{32}({}^{t}\varphi)
  \end{cases}
\]

\[
  (11) \qquad \varphi = (\varphi_{ij})_{1 \leq i, j \leq 3} \qquad \varphi_{ij} = \varphi(e_i, e_j)
\]

\[
  [e'_2, e'_3]_{\varphi} = \varphi\alpha'(e'_2 \wedge e'_3) = \varphi\alpha'(\bar{e}'_1) \underset{(10)}{=} \varphi(e_1) = \sum_j \varphi_{1j}\, e'_j
\]

\[
  (12) \qquad
  \begin{cases}
    [e'_2, e'_3]_{\varphi} = \sum_j \varphi_{1j}\, e'_j \\
    [e'_3, e'_1]_{\varphi} = \sum_j \varphi_{2j}\, e'_j \\
    [e'_1, e'_2]_{\varphi} = \sum_j \varphi_{3j}\, e'_j
  \end{cases}
\]

\[
  (13) \qquad [[e'_1 \wedge e'_2], e'_3] = \sum_j (\varphi_{32}\varphi_{1j} - \varphi_{31}\varphi_{2j})\, e'_j
\]

\[
  (13\ \text{bis}) \qquad
  \begin{cases}
    [[e'_2 \wedge e'_3], e'_1] = \sum_j (\varphi_{13}\varphi_{2j} - \varphi_{12}\varphi_{3j})\, e'_j \\
    [[e'_3 \wedge e'_1], e'_2] = \sum_j (\varphi_{21}\varphi_{3j} - \varphi_{23}\varphi_{1j})\, e'_j
  \end{cases}
\]

\[
  (14) \qquad
  \begin{cases}
    \varphi_{11} = a ,\ \varphi_{22} = b ,\ \varphi_{33} = c \\
    \varphi_{23} = p ,\ \varphi_{31} = q ,\ \varphi_{12} = r \\
    \varphi_{32} = p' ,\ \varphi_{13} = q' ,\ \varphi_{21} = r'
  \end{cases}
\]

\[
  (15) \qquad \varphi = \begin{pmatrix} a & r & q' \\ r' & b & p \\ q & p' & c \end{pmatrix}
\]

\[
  (21) \qquad q(x, y, z) = xy + z^2
\]

\[
  (22) \qquad \varphi\bigl((x, y, z), (x', y', z')\bigr) = (xy' + yx') + 2zz'
\]

\[
  (23) \qquad
  \begin{cases}
    q(e_1) = q(e_2) = 0 ,\ q(e_3) = 1 \\
    \varphi(e_1, e_2) = 1 ,\ \varphi(e_2, e_3) = \varphi(e_3, e_1) = 0 \\
    \varphi(e_1, e_1) = \varphi(e_2, e_2) = 0 \qquad \varphi(e_3, e_3) = 2
  \end{cases}
\]

\[
  (23\ \text{bis}) \qquad
  \begin{cases}
    a = b = 0 ,\ c = 1 \\
    p = p' = q = q' = 0 \quad r = r' = 1
  \end{cases} ,
  \quad \varphi = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix}
\]

\[
  (24) \qquad
  \begin{cases}
    [e'_2, e'_3] = e'_2 \\
    [e_3, e'_1] = e'_1 \\
    [e'_1, e'_2] = 2e'_3
  \end{cases}
\]

\[
  (25) \qquad X = e'_1 ,\ Y = e'_2 ,\ H = e'_3
\]

\[
  (26) \qquad [H, X] = X ,\ [H, Y] = -Y ,\quad [X, Y] = 2H
\]

\[
  \varphi(x, y) = \sum_{1 \leq i, j \leq 3} \varphi_{ij}\, x_i y_j
\]

\[
  (27) \qquad \delta(\varphi) = \det(\varphi_{ij})\, \omega'^{\otimes 2} \qquad (\text{où } \omega' = e'_1 \wedge e'_2 \wedge e'_3)
\]

\[
  (28) \qquad q(x, y, z) = ax^2 + by^2 + cz^2 + pyz + qzx + rxy
\]

\[
  (29) \qquad \varphi = \begin{pmatrix} 2a & r & q \\ r & 2b & p \\ q & p & 2c \end{pmatrix}
\]

\[
  (30) \qquad \delta(q) = \delta(\varphi) = 2(4abc + pqr - ap^2 - bq^2 - cr^2)\, \omega'^{\otimes 2}
\]

\[
  (31) \qquad \delta'(q) = (4abc + pqr - ap^2 - bq^2 - cr^2)\, \omega'^{\otimes 2} ,
\]

\[
  (32) \qquad \delta'(q) = -\omega'^{\otimes 2} \qquad (\omega' = e'_1 \wedge e'_2 \wedge e'_3)
\]

\[
  (33) \qquad \delta'(q) = -\omega^{\otimes(-2)} .
\]

\[
  (34) \qquad \delta'(q) = (-1)^n \omega^{\otimes(-2)}
\]

\[
  (35) \qquad q(x) = \sum_1^n x_{2i} x_{2i+1} \quad \text{resp.} \quad q(x) = \sum_1^n x_{2i} x_{2i+1} + x_{2n+1}^2
\]

\[
  (36) \qquad \delta'(q) = (-1)^n \omega^{\otimes(-2)} \qquad \omega = e_1 \wedge \cdots \wedge e_N
\]

\[
  (36) \qquad f : E \longrightarrow E'
\]

\[
  (37) \qquad \Lambda^2 f : \Lambda^2 E \longrightarrow \Lambda^2 E'
\]

\[
  (38) \qquad f' = \alpha \circ \Lambda^2 f \circ \alpha^{-1} : E' \longrightarrow E
\]

\[
  (39) \qquad f'\bigl(\alpha(x \wedge y)\bigr) = \alpha'\bigl(f(x) \wedge f(y)\bigr) \qquad x, y \in \Gamma(S, E)
\]

\[
  (40) \qquad \bigl\langle f'\bigl(\alpha(x \wedge y)\bigr), x' \bigr\rangle\, \omega' = f(x) \wedge f(y) \wedge x' \qquad (x' \in \Gamma(S, E'))
\]

\[
  (40\ \text{bis}) \qquad [x, y]_{\varphi'} = \alpha'\Lambda^2 f(x \wedge y) = \alpha'\bigl(f(x) \wedge f(y)\bigr)
\]

\[
  (40\ \text{ter}) \qquad \bigl\langle [x, y]_{\varphi'}, x' \bigr\rangle\, \omega' = f(x) \wedge f(y) \wedge x' \quad )
\]

\[
  (41) \qquad \underset{(\text{discriminant})}{\delta(\varphi)} = \Delta\, \omega^{\otimes -2} , \qquad \Delta \in \Gamma(S, \mathcal{O}_S)
\]

\[
  (42) \qquad
  \begin{cases}
    f'f = \Delta\, \mathrm{id}_E \\
    ff' = \Delta\, \mathrm{id}_{E'}
  \end{cases}
\]

\[
  (43) \qquad \delta(\varphi') = \Delta' . \omega'^{\otimes 2} \qquad \Delta' \in \Gamma(S, \mathcal{O}_S)
\]

\[
  (44) \qquad
  \begin{cases}
    f'f'' = \Delta'\, \mathrm{id}_E \\
    f''f' = \Delta'\, \mathrm{id}_{E'}
  \end{cases}
\]

\[
  (45) \qquad f' = \Delta f^{-1} , \qquad f'' = \Delta' f'^{-1} = (\Delta'\Delta^{-1}) f
\]

\[
  (45\ \text{bis}) \qquad \Delta f'' = \Delta' f
\]

\[
  \det(f') = \Delta^3 \det(f)^{-1} = \Delta^3 (\Delta\omega^{\otimes -2})^{-1}
\]

\[
  (47) \qquad \det f' = \Delta^2 \omega^{\otimes 2}
\]

\[
  \Delta' = \Delta^2 \quad \text{i.e.}
\]

\[
  (48) \qquad f'' = \Delta f
\]

\[
  (49) \qquad f[x, y]_{\varphi'} = [f(x), f(y)]_{\varphi} ;
\]

\[
  f\alpha'\bigl(f(x) \wedge f(y)\bigr) ,
\]

\[
  (50) \qquad [x, y]_{\varphi'} = \alpha'\bigl(f(x) \wedge f(y)\bigr)
\]

\[
  f^{-1}\bigl([x', y']\bigr) = [f^{-1}(x'), f^{-1}(y')]
\]

\[
  (51) \qquad \rho_E(x') : E \longrightarrow E
\]

\[
  (52) \qquad \varphi\bigl(\rho_E(x')x, y\bigr) + \varphi\bigl(x, \rho_E(x')y\bigr) = 0 \qquad \text{ce qui implique}
\]

\[
  (53) \qquad \rho_E(x')[x, y]_{\varphi'} = [\rho_E(x')x, y]_{\varphi'} + [x, \rho_E(x')y]_{\varphi'}
\]

\[
  (54) \qquad E' \xrightarrow{\ \rho_E\ } \underline{\mathrm{Hom}}^{\varphi}(E, E)
\]

\[
  (55) \qquad \rho_E\bigl([x', y']_{\varphi'}\bigr) = [\rho_E(x'), \rho_E(y')] .
\]

\[
  (56) \qquad f\bigl(\rho_E(x')x\bigr) = \mathrm{ad}_{[\ ,\ ]_{\varphi}}(x')\, f(x) \quad
  \Bigl(\overset{\text{déf}}{=} [x', f(x)]_{\varphi} \underset{(12)}{=} f\alpha'\bigl(x' \wedge f(x)\bigr)\Bigr)
\]

\[
  (56\ \text{bis}) \qquad \rho_E(\xi').x = \alpha'\bigl(\xi' \wedge f(x)\bigr)
\]

\[
  -\bigl\langle x, [\xi', x']_{\varphi'} \bigr\rangle = \bigl\langle \alpha'(\xi' \wedge f(x)), x' \bigr\rangle
\]

\[
  -\bigl\langle x, f\alpha'(\xi' \wedge x') \bigr\rangle
\]

\[
  = -\bigl\langle \alpha'(\xi' \wedge x'), {}^{t}f(x) \bigr\rangle
\]

\[
  -\xi' \wedge x' \wedge f(x) = \xi' \wedge f(x) \wedge x'
\]

\[
  \bigl\langle y, f(\rho_E(\xi').x) \bigr\rangle + \bigl\langle \rho_E(\xi').y, f(x) \bigr\rangle = 0
\]

\[
  \bigl\langle y, \rho_{E'}(\xi')f(x) \bigr\rangle + \bigl\langle \rho_E(\xi')y, f(x) \bigr\rangle = 0
\]

\[
  (57) \qquad \rho_{E'}(\xi')x' \overset{\text{déf}}{=} [\xi', x']_{\varphi'} ,
\]

\[
  (58) \qquad \rho_E(\xi') = -{}^{t}\rho_{E'}(\xi')
\]

\[
  (59) \qquad \bigl\langle \rho_E(\xi')x, x' \bigr\rangle + \bigl\langle x, \rho_{E'}(\xi')x' \bigr\rangle = 0
\]

\[
  (59\ \text{bis}) \qquad \bigl\langle \rho_E(\xi')x, x' \bigr\rangle = -\bigl\langle x, \underbrace{[\xi', x']_{\varphi'}}_{f\alpha'(\xi' \wedge x')} \bigr\rangle
  = -\varphi\bigl(\alpha'(\xi' \wedge x'), x\bigr)
\]

\[
  X = e'_1 ,\ Y = e'_2 ,\ H = e'_3 , \quad \text{et}
\]

\[
  (61) \qquad \omega' = -X \wedge H \wedge Y
\]

\[
  (62) \qquad \tilde{X} = e_2 ,\ \tilde{Y} = e_1 ,\ \tilde{H} = e_3
\]

\[
  (63) \qquad \tilde{X}, \tilde{H}, \tilde{Y}
\]

\[
  (64) \qquad \omega = \tilde{X} \wedge \tilde{H} \wedge \tilde{Y}
\]

\[
  (65) \qquad f(\tilde{X}) = X ,\ f(\tilde{Y}) = Y ,\ f(\tilde{H}) = 2H
\]

\[
  \Delta = -2
\]

\[
  (66) \qquad f'(X) = 2\tilde{X} ,\ f'(Y) = 2(\tilde{Y}) ,\ f'(H) = \tilde{H}
\]

\[
  (67) \qquad
  \begin{cases}
    \varphi'(X, X) = \varphi'(Y, Y) = 0 & \varphi'(H, H) = 1 \\
    \varphi'(X, H) = \varphi'(Y, H) = 0 & \varphi'(X, Y) = 2
  \end{cases}
\]

\[
  \xi = xX + hH + yY , \qquad \xi' = x'X + h'H + y'Y
\]

\[
  (68) \qquad \varphi'(\xi, \xi') = \varphi\bigl((x, h, y); (x', h', y')\bigr) = hh' + 2(xy' + yx')
\]

\[
  (69) \qquad q'(\xi) = \varphi'(\xi, \xi) = h^2 + 4xy
\]

\[
  (70) \qquad [\tilde{H}, \tilde{X}] = 2\tilde{X} ,\ [\tilde{H}, \tilde{Y}] = -2\tilde{Y} ,\ [\tilde{X}, \tilde{Y}] = \tilde{H}
\]

\[
  (71) \qquad \mathrm{SL}(2)_S \longrightarrow \mathrm{GP}(1)_S
\]

\[
  (72) \qquad
  \begin{cases}
    \rho_E(X) = \tilde{\rho}_E(\tilde{X}) : & \tilde{X} \mapsto 0 ,\ \tilde{Y} \mapsto \tilde{H} ,\ \tilde{H} \mapsto -2\tilde{X} \\
    \rho_E(Y) = \tilde{\rho}_E(\tilde{Y}) : & \tilde{X} \mapsto -\tilde{H} ,\ \tilde{Y} \mapsto 0 ,\ \tilde{H} \mapsto 2\tilde{Y} \\
    \rho_E(H)\ (= \tfrac{1}{2}\tilde{\rho}_{\tilde{H}}\ \text{sic}) : & \tilde{X} \mapsto \tilde{X} ,\ \tilde{Y} \mapsto -\tilde{Y} ,\ \tilde{H} \mapsto 0
  \end{cases}
\]

\[
  \begin{cases}
    \mathrm{Cas}_{E'}(x') = \mathrm{Tr}\, \rho_{E'}(x')^2 \\
    \mathrm{Cas}_{E}(x) = \mathrm{Tr}\, \tilde{\rho}_{E}(x)^2
  \end{cases}
\]

\[
  \mathrm{Cas}_{E'}(x') = \Delta\, \varphi(x, x)
\]

\[
  (73) \qquad
  \begin{cases}
    \mathrm{Cas}_{E'}(x', y') = \mathrm{Tr}\bigl(\rho_{E'}(x') \circ \rho_{E'}(y')\bigr) \\
    \mathrm{Cas}_{E}(x, y) = \mathrm{Tr}\bigl(\tilde{\rho}_{E}(x) \circ \tilde{\rho}_{E}(y)\bigr)
  \end{cases}
\]

\[
  (74) \qquad
  \begin{cases}
    \mathrm{Cas}_{E'}(x', y') = 2\varphi'(x', y') \\
    \mathrm{Cas}_{E}(x, y) = -2\Delta\, \varphi(x, y)
  \end{cases}
\]

\[
  \begin{aligned}
    &\mathrm{Cas}_{E'}(H, H) = 1 + 1 + 0 = 2 \\
    &\varphi'(H, H) = 1 \\
    &\mathrm{Cas}_{E}(\tilde{H}, \tilde{H}) = \mathrm{Cas}_{E'}(2H, 2H) = 4\, \mathrm{Cas}_{E'}(H, H) = 8 \\
    &\qquad (\ldots = 4 + 4 + 0) \\
    &\varphi(\tilde{H}, \tilde{H}) = 2 \\
    &\qquad \Delta = -2
  \end{aligned}
\]

\[
  (75) \qquad \mathrm{Cas}_E(x, x) = -2\Delta\, \varphi(x, x) = -4\Delta\, q(x)
\]

\[
  (76) \qquad \mathrm{Cas}_E(x, x) = 8q(x)
\]

\[
  (77) \qquad \mathrm{Cas}_{E'}(x', x') = 2\varphi'(x', x') = 2q'(x')
\]

\[
  (78) \qquad
  \begin{cases}
    [\ ,\ ]_{\lambda\omega, \mu\varphi} = \lambda\mu\, [\ ,\ ]_{\omega, \varphi} \\
    [\ ,\ ]_{\lambda\omega, (\mu\varphi)'} = \lambda\mu^2\, [\ ,\ ]_{\omega, \varphi'}
  \end{cases}
\]

\[
  (79) \qquad [\ ,\ ]_{\omega, \varphi} = [\ ,\ ]_{\omega_1, \varphi_1}
  \iff \exists \lambda \in \Gamma(S, \mathcal{O}_S^{*}),\ \omega_1 = \lambda\omega,\ \varphi_1 = \lambda^{-1}\varphi
\]

\[
  (80) \qquad f' = g' \iff g = \varepsilon f , \quad \varepsilon \in \Gamma(S, \mu_2) \ \text{i.e.}\ \varepsilon^2 = 1
\]

\[
  (81) \qquad [\ ,\ ]_{\omega_1, (\varphi_1)'} = [\ ,\ ]_{\omega, (\varphi)'}
  \iff \omega_1 = \lambda\omega ,\ \varphi_1 = \mu\varphi
\]

\[
  (82) \qquad \delta'(q) = -\omega^{\otimes(-2)}
\]

\[
  (83) \qquad G = \underline{\mathrm{SO}}(q) = \underline{\mathrm{Aut}}(E, q, \omega)
\]

\[
  (84) \qquad 1 \to \mu_2 \to \widetilde{G} \xrightarrow{F} G \to 1
\]

\[
  (85) \qquad
  \begin{cases}
    \widetilde{\mathfrak{g}} = \underline{\mathrm{Lie}}(\widetilde{G}) \\
    \mathfrak{g} = \underline{\mathrm{Lie}}(G)
  \end{cases}
\]

\[
  (86) \qquad F : \widetilde{G} \to G
\]

\[
  (87) \qquad G \xrightarrow{\ \rho\ } \underline{\mathrm{Aut}}_{S\text{-}\mathrm{gr}}(\widetilde{G})
\]

\[
  (90) \qquad \widetilde{\mathfrak{g}} \xrightarrow{\ \mathfrak{f}\ } \mathfrak{g} \qquad \text{hom.\ d'alg.\ de Lie}
\]

\[
  (91) \qquad \mathfrak{g} \xrightarrow{\ \rho_{\mathfrak{g}}\ } \underline{\mathrm{D\acute{e}r}}(\widetilde{\mathfrak{g}})
\]

\[
  (93) \qquad
  \begin{cases}
    E \xrightarrow[\sim]{\ i\ } \widetilde{\mathfrak{g}} \\
    E' \xrightarrow[\sim]{\ j\ } \mathfrak{g}
  \end{cases}
\]

\[
  \mathfrak{g} \xrightarrow{\ \rho_{\mathfrak{g}}\ } \underline{\mathrm{D\acute{e}r}}(\widetilde{\mathfrak{g}})
  \xrightarrow[(i)]{\ \sim\ } \underline{\mathrm{D\acute{e}r}}(E) \to \underline{\mathrm{End}}(E)
\]

\[
  E' \longrightarrow \underline{\mathrm{D\acute{e}r}}(E, [\ ,\ ]_{\varphi'})
\]

\[
  E' \xrightarrow{\ j\ } \mathfrak{g}
\]

\[
  \rho_E(xX + hH + yY) :
  \begin{cases}
    \tilde{X} \mapsto h\tilde{X} - y\tilde{H} \\
    \tilde{Y} \mapsto x\tilde{H} - h\tilde{Y} \\
    \tilde{H} \mapsto -2x\tilde{X} + 2y\tilde{Y}
  \end{cases}
\]

\[
  \rho_E(xX + hH + yY) = 0 \Longrightarrow x = y = h = 0 \qquad \text{o.k.}
\]

\[
  E \xrightarrow{\ \sim\ } \widetilde{\mathfrak{g}}
\]

\[
  \begin{aligned}
    F_3 &= T \\
    F_4 &= T - 1 \\
    F_5 &= T^2 - T - 1 \\
    F_6 &= T - 2
  \end{aligned}
\]

\[
  \zeta_0 \xrightarrow[e_1]{} \zeta_1 \quad \zeta_2 \quad \zeta_3
\]

\[
  \zeta_1 - \zeta_0 = e_1 , \qquad \zeta_2 - \zeta_1 = e_2
\]

\[
  \begin{cases}
    u(\zeta_0) = \zeta_0 + e_1 \\
    u(e_1) = \zeta_2 - \zeta_1 = e_2 \\
    u(e_2) = \zeta_3 - \zeta_2 = -e_1 + (\alpha - 1)e_2
  \end{cases}
\]

\[
  \mathrm{Alg\,Sp\acute{e}c}(\mathcal{M}) \simeq \mathbb{P}(\mathfrak{g}) \simeq \check{\mathbb{P}}(\widetilde{\mathfrak{g}}) \simeq \bigl[\check{P} = \mathrm{Dr}(P)\bigr] \simeq \mathrm{Div}_2^{+}(X)
\]

\[
  (X - Y) \longrightarrow \overline{\mathcal{T}}
\]

\[
  \mathcal{T} \xrightarrow{\ 2\,\mathrm{id}_{\mathcal{T}}\ } \mathcal{T}
\]

\[
  (1) \qquad \forall s \in S , \quad u_s \notin k(s) \cdot 1
\]

\[
  (2) \qquad \det(\alpha\,\mathrm{id} + \beta u) = \alpha^2 + \alpha\beta\, \mathrm{Tr}\, u + \beta^2 \det u \quad \text{inv.}
\]

\[
  C_{\underline{\mathrm{GP}}(V)}(u) \simeq \mathbb{P}^1 - Q_{\mathrm{Tr}\,u,\, \det u}
\]

\[
  \alpha^2 + \tau\,\alpha\beta + \delta\,\beta^2
\]

\[
  \tau^2 - 4\delta \quad \text{inv.}
\]

\[
  g h g^{-1} = \lambda h \qquad \text{OPS} \quad \det u = -1
\]

\[
  v = \begin{pmatrix} x & y \\ z & t \end{pmatrix} \qquad
  v \longmapsto [\underline{u}, v] \qquad M_2 \xrightarrow{\ \mathrm{fu}\ } M_2
\]

\[
  \det(\alpha + \beta u) = \prod (\alpha + \beta\mu_i) = \alpha^2 + \alpha\beta\,\mathrm{Tr}\,u + \beta^2 \det u
\]

\[
  g h^2 g^{-1} = (\lambda h)^2 = \lambda^2 h^2 = h^2
\]

\[
  g u g^{-1} = \lambda u^{-1} \qquad \forall \ \ldots
\]

\[
  \underbrace{g(\alpha + \beta u_0)g^{-1}}_{\alpha + \beta\, g u_0 g^{-1} = \alpha + \beta\lambda_0 u_0} = \lambda(\alpha, \beta)\,(\alpha + \beta u_0)
\]

\[
  \begin{aligned}
    &g u g^{-1} = u + \lambda \\
    &\mathrm{Tr}\,u = \mathrm{Tr}\,u + 2\lambda
  \end{aligned}
\]

\[
  [H, X] = X , \quad [H, Y] = -Y , \quad [X, Y] = 2H = 0 \ \text{(en car.~2)}
\]

\[
  N/H \simeq \lbrace \pm 1 \rbrace_{S} \subset (\mathbf{Z}/p\mathbf{Z})^{\times}_{S} .
\]

\[
  u = \mathrm{id}_V + v
\]

\[
  v^2 = 0, \quad v \neq 0 ,
\]

\[
  \mathcal{M} \mapsto G, \quad \mathcal{M} \mapsto \widetilde{G}, \quad
  \mathcal{M} \mapsto (E, q, \omega), \quad
  \mathcal{M} \mapsto (P \supset X), \quad \mathcal{M} \mapsto X
\]

\[
  (1) \qquad \mathcal{M} = \check{V} \otimes V \simeq \underline{\mathrm{End}}(V)
\]

\[
  (2) \qquad X \overset{\varphi}{\simeq} \mathbb{P}(V) \;\bigl(\simeq \mathbb{P}(\check{V})\bigr)
\]

\[
  (3) \qquad \varphi(L) = L' \otimes L = L^{\perp} \otimes L \subset \check{V} \otimes V = \mathcal{M}
\]

\[
  (4) \qquad Q = \mathrm{Var}(\Delta) \subset \check{\mathbb{P}}(\mathcal{M})
\]

\[
  (5) \qquad \Delta(u) = -\det u
\]

\[
  (6) \qquad Q \xrightarrow[\sim]{\ \psi\ } \mathbb{P}(V) \times \mathbb{P}(V) \quad (\simeq X \times X)
\]

\[
  V \twoheadrightarrow I \hookrightarrow V
\]

\[
  (7) \qquad u \longmapsto (\underbrace{\mathrm{Ker}\, u}_{L}, \underbrace{\mathrm{Im}\, u}_{M})
\]

\[
  \underset{L^{\perp} \otimes L}{\underline{L}}, \ \underset{M^{\perp} \otimes M}{\underline{M}} \subset \mathcal{M} = \underline{\mathrm{End}}(V)
\]

\[
  (8) \qquad
  \begin{cases}
    \underline{L} \subset \bar{\mathfrak{g}}, \ q \mid \underline{L} = 0 \\
    \underline{M} \subset \mathfrak{g}, \ q \mid \underline{M} = 0
  \end{cases}
\]

\[
  (9) \qquad
  \begin{cases}
    J = \underline{O}_S \cdot u \subset \mathcal{M} & (J \in \Gamma(Q/S)) \\
    J \circ L = 0 \\
    \underline{M} \circ J = 0
  \end{cases}
\]

\[
  (10) \qquad \mathcal{A} = \underline{O}_S \oplus J \hookrightarrow \mathcal{M} = \underline{\mathrm{End}}(V)
\]

\[
  (11) \qquad \mathcal{A} \xrightarrow{\ \rho\ } \underline{O}_S
\]

\[
  (13) \qquad V \xrightarrow[\sim]{\ \alpha\ } \check{V} \quad \bigl(V \to \check{V} \otimes \det V \xrightarrow{\sim} \check{V}\bigr)
\]

\[
  (14) \qquad u \longmapsto u' = \alpha^{-1}\, ({}^{t}u)\, \alpha
\]

\[
  (15) \qquad (uv)' = v' u'
\]

\[
  (16) \qquad u' = \mathrm{Tr}\, u \cdot 1 - u
\]

\[
  (17) \qquad
  \begin{cases}
    u + u' = (\mathrm{Tr}\, u)\, 1 \\
    u u' = (\det u)\, 1
  \end{cases}
\]

\[
  (18) \qquad \rho'(u) = \rho(u')
\]

\[
  (19) \qquad Q \longrightarrow \check{\mathbb{P}}(\mathfrak{g}) = \mathbb{P}(\widetilde{\mathfrak{g}}) \quad \bigl(= \Sigma\bigr)
\]

\[
  (20) \qquad Q \simeq \operatorname{Spec}(\mathcal{A}_{\Sigma})
\]

\[
  \mathrm{Tr}\, u \, x' \otimes x = \langle ux, x' \rangle
\]

\[
  (21) \qquad \mathrm{AlgSpec}(\mathcal{M}) \simeq \underbrace{\mathbb{P}(\widetilde{\mathfrak{g}})}_{P} \ \bigl(\simeq \check{\mathbb{P}}(\mathfrak{g})\bigr) \simeq \lbrace \ldots \rbrace
\]

\[
  D \longmapsto D \cap X = Y
\]

\[
  (22) \qquad \check{P} = \mathrm{Dr}(P) \xrightarrow{\ \sim\ } \mathrm{Div}_2^{+}(X)
\]

\[
  (23) \qquad Y \simeq \operatorname{Spec} \mathcal{A}
\]

\[
  (24) \qquad Y = X^{u}
\]

\[
  \operatorname{int}(u)\, \mathcal{A} \ \bigl(\overset{\mathrm{def}}{=} u \mathcal{A} u^{-1}\bigr) = \mathcal{A}
\]

\[
  (25) \qquad \mathcal{T} = \check{\mathbb{V}}(\mathcal{A})^{*}/\mathbb{G}_m \subset G
\]

\[
  \mathrm{GP}(1) \simeq \underline{\mathrm{Aut}}(\mathfrak{X}_0)
\]

\[
  (1) \qquad \widetilde{G}_0/\mathfrak{z}_0 = \mathrm{SL}(2)/\mu_2 \simeq \mathrm{GP}(1) = G_0
\]

\[
  (3) \qquad \xi = \begin{pmatrix} z & x \\ y & -z \end{pmatrix}
\]

\[
  (4) \qquad q_0(\xi) \overset{\mathrm{déf}}{=} -\det \xi = xy + z^2
\]

\[
  (5) \qquad \det(g \xi g^{-1}) = \det \xi
\]

\[
  (6) \qquad \left\lbrace
  \begin{array}{l}
    (x, y, z) \longmapsto \begin{pmatrix} z & x \\ y & -z \end{pmatrix} \\[4pt]
    E_0 = \underline{O}^3_{S_0} \xrightarrow{\ \sim\ } \widetilde{\mathfrak{g}}_0 = \mathfrak{sl}(2)_{S_0}
  \end{array}\right.
\]

\[
  (7) \qquad G_0 = \mathrm{GP}(1) \longrightarrow \underline{\mathrm{Aut}}(\mathfrak{X}_0)
\]

\[
  (8) \qquad R \longmapsto R \overset{\mathrm{GP}(1)_S}{\wedge} \mathfrak{X}_{0_S}
\]

\[
  \begin{array}{ccc}
    C_{0_S} & \longrightarrow & C_{5_S} \\
    \shortparallel & & \shortparallel \\
    \mathrm{Tors}\bigl(\mathrm{GP}(1)_S\bigr) & & \mathrm{Drpr}(S)
  \end{array}
\]

\[
  (9) \qquad R \longmapsto R/B_{0_S}
\]

\[
  (10) \qquad B_0 \subset \mathrm{GP}(1) = G_0
\]

\[
  (10) \qquad \widetilde{B}_0 = \left\lbrace \begin{pmatrix} \lambda & \mu \\ 0 & \lambda^{-1} \end{pmatrix}
  \;\middle|\; \lambda \text{ dans } \mathbb{G}_m,\ \mu \text{ dans } \mathbb{G}_a \right\rbrace
\]

\[
  (11) \qquad \widetilde{B}'_0 = \left\lbrace \begin{pmatrix} \lambda & 0 \\ \mu & \lambda^{-1} \end{pmatrix}
  \;\middle|\; \lambda \text{ dans } \mathbb{G}_m,\ \mu \text{ dans } \mathbb{G}_a \right\rbrace
\]

\[
  (12) \qquad \begin{array}{c} g \longmapsto g \cdot s_0 \\ G_0 \longrightarrow X_0 \end{array}
\]

\[
  (13) \qquad G_0/B_0 \longrightarrow X_0 ,
\]

\[
  (14) \qquad R \wedge X_{0_S} \simeq R/B_{0_S}
\]

\[
  (15) \qquad \mathfrak{X} \longmapsto \underline{\mathrm{Isom}}(\mathfrak{X}_0, \mathfrak{X})
\]

\[
  (16) \qquad \mathrm{Tors}\bigl(\underline{\mathrm{Aut}}(\mathfrak{X}_{0_S})\bigr) \simeq
  \mathrm{Tors}\bigl(\mathrm{GP}(1)_S\bigr)
\]

\[
  (17) \qquad \mathrm{GP}(1)_S \xrightarrow{\ \sim\ } \underline{\mathrm{Aut}}(\mathfrak{X}_{0_S})
  \ \bigl(\simeq \underline{\mathrm{Aut}}(\mathfrak{X}_0)_S\bigr)
\]

\[
  (18) \qquad \mathfrak{X} \longmapsto \underline{\mathrm{Aut}}(\mathfrak{X})
\]

\[
  (19) \qquad \widetilde{G} \longmapsto \widetilde{G}/\mathrm{Cent}(\widetilde{G})
\]

\[
  (20) \qquad \mathcal{M} \longmapsto \check{\mathbb{V}}(\mathcal{M})^{*}/\mathbb{G}_{m_S}
\]

\[
  (21) \qquad G \longmapsto \underline{\mathrm{Isom}}(G_{0_S}, G) \overset{G_{0_S}}{\wedge} \mathfrak{X}_{0_S} ,
\]

\[
  (22) \qquad G \longmapsto \widetilde{G} = \text{Rev.\ simplement connexe de } G
\]

\[
  (23) \qquad \begin{array}{c}
    G \longmapsto \mathfrak{g} = \underline{\mathrm{Lie}}(G) \\
    \mathrm{Fo}\bigl(\mathrm{GP}(1)_S\bigr) \xrightarrow{\ \sim\ } \mathrm{Fo}\bigl(\mathfrak{gp}(1)_S\bigr)
  \end{array}
\]

\[
  (24) \qquad \begin{array}{c}
    G \longmapsto \widetilde{\mathfrak{g}} = \underline{\mathrm{Lie}}(\widetilde{G}) \\
    \mathrm{Fo}\bigl(\mathrm{GP}(1)_S\bigr) \xrightarrow{\ \sim\ } \mathrm{Fo}\bigl(\mathfrak{sl}(2)_S\bigr)
  \end{array}
\]

\[
  G \longmapsto (E, q, \omega)
\]

\[
  G \longmapsto X
\]

\[
  (25) \qquad G \longmapsto \underline{\mathrm{Bor}}(G)
\]

\[
  (26) \qquad X \xrightarrow{\ \sim\ } \underline{\mathrm{Bor}}(G)
\]

\[
  (27) \qquad \mathfrak{X} \longmapsto R = \underline{\mathrm{Isom}}(\mathfrak{X}_{0_S}, \mathfrak{X})
  \longmapsto R \overset{\mathrm{GP}(1)_S}{\wedge} \widetilde{G}_{0_S} ,
\]

\[
  (28) \qquad \mathfrak{X} \longmapsto G = \underline{\mathrm{Aut}}_S(\mathfrak{X})
  \longmapsto \widetilde{G} = \text{Rev.\ simpl.\ conn.\ } G
\]

\[
  \mathrm{Quat}(S) \longrightarrow \mathrm{Fo}\bigl(\mathrm{SL}(2)_S\bigr)
\]

\[
  (29) \qquad \mathcal{M} \longmapsto \widetilde{G} \simeq
  \mathrm{Ker}\Bigl(\check{\mathbb{V}}(\mathcal{M})^{*} \xrightarrow{\ \det\ } \mathbb{G}_m\Bigr)
\]

\[
  \mathrm{Fo}\bigl(\mathrm{SL}(2)_S\bigr) \longrightarrow \mathrm{Fo}\bigl(\mathfrak{sl}(2)_S\bigr)
\]

\[
  (30) \qquad \widetilde{G} \longmapsto \widetilde{\mathfrak{g}} = \underline{\mathrm{Lie}}(\widetilde{G})
\]

\[
  (31) \qquad \widetilde{G} \longmapsto X \simeq \underline{\mathrm{Bor}}(\widetilde{G})
\]

\[
  (32) \qquad \left\lbrace
  \begin{array}{l}
    \text{Centre de } \mathcal{M} = \underline{O}_S \cdot 1 \\
    \bigl(\text{Centre de } \mathcal{M}^{*} = \underline{O}_S^{*} \cdot 1 \text{ ou encore }
    \bigl(\text{Centre de } \mathcal{L} = \mathbb{V}(\mathcal{M})^{*}\bigr) = \mathbb{G}_m \subset \mathcal{L}(\mathcal{M})\bigr)
  \end{array}\right.
\]

\[
  (33) \qquad \mathbb{V}(\mathcal{M}) \overset{\det}{\underset{\mathrm{tr}}{\rightrightarrows}} \mathbb{E}^{1}
\]

\[
  (34) \qquad 1 \longrightarrow \mathbb{G}_m \xrightarrow{\text{incl.\ du centre}}
  \mathcal{L}(\mathcal{M}) \longrightarrow G \longrightarrow 1
\]

\[
  (35) \qquad 1 \longrightarrow \widetilde{G} \longrightarrow \mathcal{L}(\mathcal{M})
  \xrightarrow{\ \det\ } \mathbb{G}_m \longrightarrow 1
\]

\[
  (34') \qquad 0 \longrightarrow \underline{O}_S \xrightarrow{\lambda \mapsto \lambda \cdot 1}
  \mathcal{M} \longrightarrow \mathfrak{g} \longrightarrow 0
\]

\[
  (35') \qquad 0 \longrightarrow \widetilde{\mathfrak{g}} \longrightarrow \mathcal{M}
  \xrightarrow{\ \mathrm{tr}\ } \underline{O}_S \longrightarrow 0 .
\]

\[
  (36) \qquad \left\lbrace
  \begin{array}{l}
    \widetilde{G} \longrightarrow G \\
    \widetilde{\mathfrak{g}} \xrightarrow{\ f\ } \mathfrak{g}
  \end{array}\right.
\]

\[
  (37) \qquad \widetilde{G} \xhookrightarrow{\ \mathrm{incl.}\ } \mathcal{L}(\mathcal{M})
  \xrightarrow{\ \mathrm{proj}\ } G
\]

\[
  (38) \qquad \widetilde{\mathfrak{g}} \xhookrightarrow{\ \mathrm{incl}\ } \mathcal{M}
  \xrightarrow{\ \mathrm{proj}\ } \mathfrak{g}
\]

\[
  (39) \qquad \Phi(u, v) = \mathrm{Tr}(uv)
\]

\[
  (40) \qquad \mathcal{M} \simeq \check{\mathcal{M}}
\]

\[
  (41) \qquad \widetilde{\mathfrak{g}} \times \mathfrak{g} \longrightarrow \underline{O}_S
\]

\[
  (42) \qquad \widetilde{\mathfrak{g}} \simeq \check{\mathfrak{g}}, \qquad
  \mathfrak{g} \simeq \check{\widetilde{\mathfrak{g}}}
\]

\[
  (43) \qquad \left\lbrace
  \begin{array}{l}
    \Phi_1(u, v) = -\mathrm{Tr}\, u \,\mathrm{Tr}\, v + \mathrm{Tr}(uv) = -\mathrm{Tr}\, u \,\mathrm{Tr}\, v + \Phi(u, v) \\
    \text{donc } -\det(u + v) + \det u + \det v = \Phi_1(u, v) \\
    \qquad = -\mathrm{Tr}\, u \,\mathrm{Tr}\, v + \mathrm{Tr}(uv)
  \end{array}\right.
\]

\[
  \Phi_1(1, u) = -\Phi(1, u) = -\mathrm{Tr}\, u
\]

\[
  (42) \qquad \left\lbrace
  \begin{array}{l}
    \varphi = \Phi \,|\, \widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}}
    = \Phi_1 \,|\, \widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}} :
    \widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}} \to \underline{O}_S \\
    \varphi(u, v) = \mathrm{Tr}\, uv
  \end{array}\right.
\]

\[
  (43) \qquad \left\lbrace
  \begin{array}{l}
    q = Q \,|\, \widetilde{\mathfrak{g}} : \widetilde{\mathfrak{g}} \longrightarrow \underline{O}_S \\
    q(u) = -\det u
  \end{array}\right.
\]

\[
  (44) \qquad \left\lbrace
  \begin{array}{l}
    q(x + y) = q(x) + q(y) + \varphi(x, y) \\
    \varphi(x, x) = 2 q(x)
  \end{array}\right.
\]

\[
  (45) \qquad \left\lbrace
  \begin{array}{l}
    \mathrm{Tr}\bigl(\mathrm{ad}_{\widetilde{\mathfrak{g}}}(u)\, \mathrm{ad}_{\widetilde{\mathfrak{g}}}(v)\bigr) = 4 \varphi(u, v) \\
    \mathrm{Tr}\bigl(\mathrm{ad}_{\widetilde{\mathfrak{g}}}(u)^2\bigr) = 8 q(u)
  \end{array}\right.
\]

\[
  (46) \qquad \omega \in \Gamma\, \underline{\det}\, \widetilde{\mathfrak{g}} \simeq \Gamma\, \underline{\det}\, \mathcal{M}
\]

\[
  \mathcal{M} \simeq \underline{\mathrm{End}}(V) \simeq \check{V} \otimes V
\]

\[
  (47) \qquad \det(\check{V} \otimes V) \simeq \det \check{V} \otimes \det V
  \simeq (\det V)^{-1} \otimes \det V \simeq \underline{O}_S
\]

\[
  (48) \qquad \underbrace{\delta'(q)}_{\text{discriminant divisé}} = -\,\omega^{\otimes 2}
\]

\[
  (49) \qquad \delta(\varphi) = \delta(q) = -2\,\omega^{\otimes 2}
\]

\[
  (50) \qquad (E, q, \omega) = (\widetilde{\mathfrak{g}}, q, \omega) \qquad \text{i.e. } E = \widetilde{\mathfrak{g}}
\]

\[
  (51) \qquad f : \widetilde{\mathfrak{g}} \longrightarrow \mathfrak{g}
\]

\[
  \underbrace{\langle \bar{u}, f(v) \rangle}_{\mathrm{Tr}\, uv}
  = \underbrace{\varphi(u, v)}_{\mathrm{Tr}\, uv}
  \qquad u, v \in \Gamma(S, \widetilde{\mathfrak{g}})
\]

\[
  (52) \qquad \boxed{\ \omega = \widetilde{X} \wedge \widetilde{H} \wedge \widetilde{Y}
  = -\,\underset{e_1}{\widetilde{X}} \wedge \underset{e_2}{\widetilde{Y}} \wedge \underset{e_3}{\widetilde{H}}\ }
\]

\[
  (53) \qquad f' : \mathfrak{g} \longrightarrow \widetilde{\mathfrak{g}}
\]

\[
  (54) \qquad \left\lbrace
  \begin{array}{l}
    f f' = -2\, \mathrm{id}_{\mathfrak{g}} \\
    f' f = -2\, \mathrm{id}_{\widetilde{\mathfrak{g}}}
  \end{array}\right.
\]

\[
  (55) \qquad f'(\bar{u}) = (\mathrm{Tr}\, u)\, 1 - 2u
\]

\[
  (56) \qquad \left\lbrace
  \begin{array}{l}
    P = \check{\mathbb{P}}(\widetilde{\mathfrak{g}}) \simeq \mathbb{P}(\mathfrak{g}) \\
    X = \mathbb{V}(q) \subset P
  \end{array}\right.
\]

\[
  (57) \qquad \widetilde{B} = \check{\mathbb{V}}(\widetilde{\mathfrak{L}}) \cap
  \underbrace{\mathbb{V}(\mathcal{M})^{*}}_{\mathcal{L}}
  \qquad (B = \widetilde{B}/\mathbb{G}_m)
\]

\[
  (q \,|\, \mathfrak{L}^{\perp}) = 0
\]

\[
  (58) \qquad \underline{O}_S 1 \subset \widetilde{\mathfrak{g}}
\]

\[
  (59) \qquad X \longrightarrow \mathbb{P}(\widetilde{\mathfrak{g}})
\]

\[
  (60) \qquad X \longrightarrow \mathbb{P}(\widetilde{\mathfrak{g}}/\underline{O}_S)
\]

\[
  (61) \qquad \mathbb{P}(\widetilde{\mathfrak{g}}/\underline{O}_S) \simeq X^{(2)}
  \quad (\text{frobénisé})
\]

\[
  X \longrightarrow X^{(2)}
\]