\[
-v d^2 + w cd + c^2
\]

\[
\det(d\rho + c) = d^2 \det\rho + cd\, \mathrm{Tr}\,\rho + c^2
\]

\[
\rho = \begin{pmatrix} 0 & v_1 \\ v & w \end{pmatrix}
\qquad
d_0 \rho + c_0 = \begin{pmatrix} c_0 & d_0 v_1 \\ d_0 v & d_0 w + c_0 \end{pmatrix}
\]

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\begin{pmatrix} c & d \\ dv & dw + \uncertain{c} \end{pmatrix}
= \begin{pmatrix} ac + bdv & \\ & \end{pmatrix}
\]

\[
\underline{a} = \alpha^{-1} u = \frac{1}{\delta_\alpha}
\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\begin{pmatrix} \mu & \lambda \\ 0 & 1 \end{pmatrix}
= \frac{1}{\delta_\alpha}
\begin{pmatrix} d\mu & d\lambda - b \\ -c\mu & -c\lambda + a \end{pmatrix}
= c_0 \rho + d_0
\]

\[
c_0 = d\mu \struck{/} \delta_\alpha = \frac{-dv}{\uncertain{-v^2} + wcd + c^2}
\]

\[
c_0 = \frac{d\mu}{\delta_\alpha} \qquad d_0 = \frac{-c\mu}{\delta_\alpha v}
\]

\[
\boxed{\underline{a} = \frac{\mu}{\delta_\alpha}\Bigl(-\frac{c}{v}\rho + d\Bigr)}
\]

\[
\det \underline{a} = \Bigl(\frac{\mu}{\delta_\alpha}\Bigr)^2
\Bigl[\frac{c^2}{v^2}(-v v_1) + \frac{dc}{v} w + d^2\Bigr] \ldots
\]

\[
\widetilde{\Pi}{}^{\circ D}_{n} \struck{\overset{\text{déf}}{=}}
\{\tilde\rho, \tilde\sigma \mid \tilde\rho^{\,n} = \tilde\sigma^{2}\}
\qquad
\boxed{\tilde\rho^{\,n} = \tilde\sigma^{2} = \tilde\omega_0} \ \text{élément central}
\]

\[
1 \to \mathbb{Z} \to \widetilde{\Pi^{\circ D}_{n}} \to \Pi^{D}_{n} \to 1
\qquad \text{ext. centrale}
\]

\[
\boxed{\tilde\rho_0 = \tilde\sigma\tilde\rho\tilde\sigma^{-1}\tilde\rho}
= \tilde\sigma(\tilde\rho)\,\tilde\rho
= (\tilde\sigma\tilde\rho)^2\,\tilde\omega_0^{-1}
\qquad
\boxed{\tilde\rho_i = \tilde\rho^{\,i}(\tilde\rho_0)}
\qquad \tilde\rho_i \text{ relève } \rho_i
\]

\begin{align*}
\tilde\rho_{n-1}\tilde\rho_{n-2}\cdots\tilde\rho_0
&= \underbrace{\tilde\rho^{\,n-1}\tilde\rho_0\tilde\rho^{-(n-1)}}\,
\underbrace{\tilde\rho^{\,n-2}\tilde\rho_0\tilde\rho^{-(n-2)}} \cdots
(\tilde\rho\tilde\rho_0\tilde\rho^{-1})\,\tilde\rho_0 \\
&= \tilde\rho^{\,n-1}\tilde\rho_0\tilde\rho^{-1}\tilde\rho_0\tilde\rho^{-1} \cdots
\tilde\rho^{-1}\tilde\rho_0\tilde\rho^{-1}\tilde\rho_0
\qquad (n \text{ facteurs } \tilde\rho_0) \\
&= \tilde\rho^{\,n}\,\underbrace{(\tilde\rho^{-1}\tilde\rho_0)\cdots(\tilde\rho^{-1}\tilde\rho_0)}_{n \text{ facteurs}}
= \underbrace{\tilde\rho^{\,n}}_{\tilde\omega_0}(\tilde\rho^{-1}\tilde\rho_0)^n
\end{align*}

\[
\tilde\rho^{-1}\tilde\rho_0 = \tilde\rho^{-1}\tilde\sigma\tilde\rho\tilde\sigma^{-1}\tilde\rho
= (\tilde\rho^{-1}\tilde\sigma)(\tilde\rho)
\]

\[
(\tilde\rho^{-1}\tilde\rho_0)^n = (\tilde\rho^{-1}\tilde\sigma)(\tilde\rho^{\,n})
= (\tilde\rho^{-1}\tilde\sigma)(\tilde\omega_0) = \tilde\omega_0
\]

\[
\boxed{\tilde\rho_{n-1}\tilde\rho_{n-2}\cdots\tilde\rho_0 = \tilde\omega_0^{2}}
\]

\[
1 \to \mathbb{Z} \to \widetilde{\Pi}_n \to \Pi_n \to 1
\]

\[
\widetilde{\Pi}_n = \{\tilde\rho_0, \ldots, \tilde\rho_{n-1}, \tilde\omega_0 \mid
\tilde\omega_0 \text{ central},\ \tilde\rho_{n-1}\tilde\rho_{n-2}\cdots\tilde\rho_0 = \tilde\omega_0^{2}\}
\]

\[
\widetilde{\Pi}{}^{\circ}_n = \{\tilde\rho_0, \ldots, \tilde\rho_{n-1} \mid
\tilde\rho_{n-1}\tilde\rho_{n-2}\cdots\tilde\rho_0 \text{ central i.e. }
\tilde\rho_{n-1}\tilde\rho_{n-2}\cdots\tilde\rho_0
= \tilde\rho_{n-2}\tilde\rho_{n-3}\cdots\tilde\rho_0\tilde\rho_{n-1} = \cdots\}
\]

\[
1 \to \mathbb{Z} \to \widetilde{\Pi}{}^{\circ}_n \to \Pi_n \to 1
\]

\[
\widetilde{\Pi}_n \simeq \widetilde{\Pi}{}^{\circ}_n \wedge_{\mathbb{Z}\varpi} \mathbb{Z}\omega
\qquad \varpi_0 \longmapsto \omega_0^{2}
\]

\[
\sigma^{-1}(\rho_0) = \rho_1,\quad \sigma^{-1}(\rho_1) = \rho_0,\quad
\sigma^{-1}(\rho_2) = \rho_0(\rho_{n-1}),\ \ldots\
\sigma^{-1}(\rho_i) = \rho_n\cdots\rho_{n+2-i}(\rho_{n+1-i})\ \ldots
\]

\[
\tilde\sigma^{-1}(\tilde\rho_0) = \tilde\rho_1\tilde z_0,\quad
\tilde\sigma^{-1}(\tilde\rho_1) = \tilde\rho_0\tilde z_1\ \cdots\
\tilde\sigma^{-1}(\tilde\rho_i) = (\tilde\rho_n\cdots\tilde\rho_{n+2-i})(\tilde\rho_{n+1-i})\,\tilde z_i
\]

\[
z_{n-1}z_{n-2}\cdots z_0 = 1
\]

\[
\tilde\sigma^{-1}(\tilde\rho_0) = \rho_1 + \struck{\ill{}}\, z_0,\quad
\tilde\sigma^{-1}(\rho_1) = \rho_0 + z_1,\ \ldots\
\tilde\sigma^{-1}(\rho_i) = \rho_{n+1-i} + z_i
\]

\[
\tilde\sigma(\tilde\rho_0) = \tilde\sigma^{-1}(\tilde\rho_0)
= \tilde\rho\,\tilde\sigma^{-1}\tilde\rho\,\tilde\sigma
\qquad
\bigl(\tilde\rho_0 = \tilde\sigma\tilde\rho\tilde\sigma^{-1}\tilde\rho\bigr)
\]

\[
\tilde\rho_1 = \tilde\rho\tilde\sigma\tilde\rho\tilde\sigma^{-1}
\]

\[
\tilde\rho_1^{-1}\,\tilde\sigma^{-1}(\tilde\rho_0)
= \tilde\sigma\tilde\rho^{-1}\tilde\sigma^{-1}\tilde\rho^{-1}\tilde\rho\,\tilde\sigma^{-1}\tilde\rho\tilde\sigma
= \tilde\sigma\underbrace{\tilde\rho^{-n}}_{\tilde\omega_0^{-1}}\tilde\sigma
= \struck{z_0}\ \tilde\omega_0^{-1}\underbrace{\tilde\sigma^{2}}_{\tilde\omega_0} = 1
\]

\[
\tilde\sigma^{-1}(\tilde\rho_1) = \tilde\sigma^{-1}(\tilde\rho_1)
= \tilde\sigma^{-1}\tilde\rho\tilde\sigma\tilde\rho
\]

\[
\tilde\rho_0^{-1}\,\tilde\sigma^{-1}(\tilde\rho_1)
= \tilde\rho^{-1}\tilde\sigma\,\tilde\rho^{-1}\tilde\sigma^{-1}\tilde\sigma^{-1}\tilde\rho\tilde\sigma\tilde\rho = 1
\]

\[
\struck{\ill{}}\ \tilde\rho_i = \tilde\rho^{\,i}\tilde\sigma\tilde\rho\tilde\sigma^{-1}\tilde\rho^{\,1-i}
\]

\[
\tilde\sigma(\tilde\rho_i) = \tilde\sigma\tilde\rho^{\,i}\tilde\sigma\tilde\rho\tilde\sigma^{-1}\tilde\rho^{\,1-i}\tilde\sigma^{-1}
\]

\[
\tilde\rho_n\cdots\tilde\rho_{n+2-i}\,\tilde\rho_{n+1-i}\,\tilde\rho_{n+2-i}^{-1}
\]

\[
\mathbb{D}'_3 = \{\rho, \sigma \mid \rho^3 = \sigma^2,\ \sigma^4 = 1\ (= \rho^6),\
\sigma(\rho) = \rho^{-1}\}
\]

\[
1 \to \mathbb{Z}/2\mathbb{Z} \to \mathbb{D}'_3 \to \mathbb{D}_3 \to 1
\]

\[
\Pi^{\mathbb{D}}_{0,3} \longrightarrow \Pi^{\mathbb{D}}_{03}/\Pi'_{0,3}
\xrightarrow{\ \sim\ } \mathbb{Z}/6\mathbb{Z} \times_{\mathbb{Z}/2\mathbb{Z}} \mathbb{D}_3
\simeq (\mathbb{F}_3 \times \mathbb{F}_3) \rtimes \{1, \sigma\}
\longrightarrow \mathbb{Z}/6\mathbb{Z} \longrightarrow \mathbb{Z}/2\mathbb{Z}
\]

\[
\Pi^{\mathbb{D}}_{0,3} = \{\rho, \sigma \mid \rho^3 = \sigma^2 = 1\}
\qquad
\mathrm{SL}(2,\mathbb{Z}) = \{\rho, \sigma \mid
\underbrace{\rho^3 = \sigma^2}_{\omega},\ \rho^6 = 1\ (= \sigma^4)\}
\]

\[
\Pi^{\mathbb{D}}_{0,3}/\Pi'_{0,3} \simeq \{\rho, \sigma \mid
\rho^3 = \sigma^2 = (\sigma\rho)^6 = 1,\ (\sigma\rho)^2 = (\rho\sigma)^2
\ \struck{\text{central}}\}
\qquad 18
\]

\[
\Pi^{\mathbb{D}}_{0,3}/\Pi'_{0,3} \xrightarrow{\substack{\rho \mapsto \rho \\ \sigma \mapsto \sigma}}
\mathbb{D}_3 = \{\rho, \sigma \mid \rho^3 = \sigma^2 = 1,\ \sigma(\rho) = \rho^{-1}\}
\quad \text{i.e. } (\sigma\rho)^2 = 1
\]

\[
\Pi^{\mathbb{D}}_{0,3}/\Pi'_{0,3} \longrightarrow \mathbb{Z}/6\mathbb{Z}
\qquad \rho \longmapsto 4 \bmod 6, \quad \sigma \longmapsto 3 \bmod 6
\]

\[
\mathcal{E} = \{\rho, \sigma \mid \underbrace{\rho^3 = \sigma^2}_{\omega},\
\rho^6 = 1\ (= \sigma^4),\ \underbrace{(\sigma\rho)^2\omega}_{\rho_0 = \sigma\rho\sigma^{-1}\rho}
\ \text{invariant par } \rho \text{ et d'ordre } 3\}
\qquad 36
\]

\[
\text{i.e. } (\sigma\rho)^6 = \omega, \quad (\sigma\rho)^2 = (\rho\sigma)^2
\]

\[
\mathcal{E} \longrightarrow \mathbb{Z}/12\mathbb{Z} \simeq (\mathrm{SL}(2,\mathbb{Z}))_{\mathrm{ab}}
\qquad
\begin{aligned}
\rho &\longmapsto 10 \bmod 12 & \sigma\rho = \varepsilon_0 &\longmapsto 1 \bmod 12 \\
\sigma &\longmapsto 3 \bmod 12 & (\sigma\rho)^2\omega = \rho_0 &\longmapsto 8 \bmod 12 \\
\omega &\longmapsto 6 \bmod 12 & &
\end{aligned}
\]

\[
\mathbb{D}'_3 = \{\rho, \sigma \mid \underbrace{\rho^3 = \sigma^2}_{\omega},\
\rho^6 = 1\ (= \sigma^4)\}\ \struck{\ill{}}
\qquad \rho_0 \longmapsto (2 \bmod 6, 1)
\qquad \rho_0 = (\sigma\rho)^2\omega
\]

\[
\widetilde{\Pi}{}^{D}_n = \{\tilde\rho, \tilde\sigma, \tilde\omega \mid
\tilde\omega \text{ central},\ \tilde\rho^{\,n} = \tilde\sigma^{2} = \tilde\omega^{\nu}\}
\]

\[
\mathbb{Z}\tilde\omega^{2} \longrightarrow \widetilde{\Pi}{}^{D}_n
\longrightarrow \Pi^{D}_n = \{\rho, \sigma \mid \rho^n = \sigma^2 = 1\}
\longrightarrow 1
\]

\[
\rho \longmapsto (n+1 \bmod 2n) = (1 - n \bmod 2n)
\qquad
\sigma \longmapsto n \bmod 2n
\]

\[
\tilde\rho_0 = (\rho, n+1) \qquad \tilde\sigma_0 = (\sigma, n)
\]

\[
\tilde\rho_0^{\,n} = (\rho^n, n(n+1)) = \struck{\tilde\omega^{\ill{}}}
\qquad
\tilde\sigma_0^{\,2} = (\underbrace{\sigma^2}_{1}, 2n)
\qquad
\tilde\rho_0^{\,n}/\tilde\sigma_0^{\,2} = (1, n(n-1)) = \tilde\omega^{\alpha}
\]

\[
\alpha = \frac{n-1}{2} \qquad (\ill{}\ \alpha + 1)
\]

\[
(\tilde\rho_0\tilde\omega^{\beta})^{n} = (\tilde\sigma_0\tilde\omega^{\gamma})^{2}
\quad \text{i.e.} \quad
\tilde\rho_0^{\,n}\tilde\sigma_0^{-2} = \tilde\omega^{\struck{\ill{}}\,2\gamma - n\beta}
\]

\[
\gamma - n\beta = \alpha = \frac{n-1}{2}
\qquad \struck{\gamma - 2n\beta =}
\qquad \gamma =
\]

\[
(\alpha+1)\cdot 2 - 1 \cdot n = 1
\qquad
\underbrace{\alpha(\alpha+1)\cdot 2}_{\gamma} - \underbrace{\alpha}_{\beta}\cdot n = \alpha
\]

\[
\tilde\rho_0\tilde\omega^{\beta} = \Bigl(\rho,\ n + 1 + 2n\overbrace{\tfrac{n-1}{2}}^{\beta = \alpha}\Bigr)
= (\rho, n^2 + 1)
\]

\[
\tilde\sigma_0\tilde\omega^{\gamma} = \bigl(\sigma,\ n + 2n\underbrace{\gamma}_{\frac{(n-1)(n+1)}{4} = \frac{n^2-1}{4}}\bigr)
= \Bigl(\sigma, \frac{n^3+n}{2}\Bigr)
\]

\[
\tilde\rho_0^{\,n} = (1, n^3 + n) = \tilde\omega^{\overbrace{\scriptstyle\frac{n^2+1}{2}}^{x}}
\qquad
\tilde\sigma_0^{\,2} = (1, n^3 + n) = \tilde\omega^{\frac{n^2+1}{2}}
\]

\[
\tilde\rho = \tilde\rho_0\,\tilde\omega^{2x} \qquad
\tilde\sigma = \tilde\sigma_0\,\tilde\omega^{nx} \qquad x =
\]

\[
\tilde\rho^{\,*} = (\rho, n^2 + 1) \qquad
\tilde\sigma^{*} = \Bigl(\sigma, \frac{n^3+n}{2}\Bigr)
\qquad
\tilde\rho^{\,n} \simeq \tilde\omega^{\,?} = \tilde\omega^{\frac{n^2+1}{2}}
\]

\[
(\tilde\rho\,\tilde\omega^{\beta})^n = (\tilde\sigma\,\tilde\omega^{\gamma})^2
\quad \Leftrightarrow \quad
\tilde\omega^{\nu}\,\tilde\omega^{\beta n} = \tilde\omega^{\nu}\,\tilde\omega^{\gamma 2}
\]

\[
\beta n = 2\gamma \qquad
\mathbb{Z}^2 \longrightarrow \mathbb{Z}, \ (\beta, \gamma)
\qquad
\beta = 2x,\ \gamma = nx \qquad \beta n = 2\gamma = (2n)x
\]

\[
\frac{n^2+1}{2} \bmod 2n \qquad \frac{n^2+1}{2} \bmod 2n
\qquad \struck{n^2}\ 5 \bmod 6 =
\qquad 13 \bmod 10 \,/\, 3
\]

\[
n = 2\alpha + 1 \qquad
\frac{n^2+1}{2} = \frac{4\alpha^2 + 4\alpha + 2}{2} = \underbrace{2\alpha^2 + 2\alpha + 1}
\ \bmod 4\alpha + 2
\]

\begin{align*}
&(4\alpha+2)\frac{\alpha}{2} + \alpha + 1 && \text{si } \alpha \text{ pair} \\
&\underbrace{(4\alpha+2)\,\frac{\alpha+1}{2}}_{2\alpha^2 + 3\alpha + 1} - \alpha && \text{si } \alpha \text{ impair}
\end{align*}

\[
\frac{n^2+1}{2} \equiv \alpha + 1 = \frac{n+1}{2} \quad (2n) \quad \text{si } \alpha \text{ pair}
\qquad
\frac{n^2+1}{2} \equiv -\alpha = \frac{1-n}{2} \quad (2n) \quad \text{si } \alpha \text{ impair}
\]

\[
\tilde\rho^{\,n} = \tilde\sigma^{2} = \tilde\omega^{\frac{n+1}{2}}
\ \text{si } \frac{n+1}{2} \text{ est impair i.e. } \alpha = \frac{n-1}{2} \text{ pair}
\]

\[
\tilde\rho^{\,n} = \tilde\sigma^{2} = \tilde\omega^{-\frac{n-1}{2}}
\ \text{si } \frac{n-1}{2} \text{ est impair}
\]

\[
\Pi^{D}_{03} \longrightarrow
\underbrace{\mathbb{Z}/6\mathbb{Z} \times_{\mathbb{Z}/2\mathbb{Z}} \mathbb{D}_3}_{18}
= \{\nu, g \in \mathbb{Z}/6\mathbb{Z} \times \mathbb{D}_3 \mid \mathrm{sg}^{+}(g) \equiv \nu\ (2)\}
\]

\[
(\Pi_{0,3})_{\mathrm{ab}} \otimes_{\mathbb{Z}} \mathbb{F}_3
= \mathbb{F}_3^3/\mathbb{F}_3 \supset \ill{}(\mathbb{F}_3^{3\,\uncertain{*}})/\mathbb{F}_3
\]

\[
\rho_0\ \rho_1\ \rho_\infty \quad
\begin{cases}
\rho_0\rho_1^{-1},\ \rho_1\rho_\infty^{-1},\ \rho_\infty\rho_0^{-1},\ \rho_0^3 \\
\struck{[\rho_0, \rho_1]\,[\rho_1, \rho_\infty]\,[\rho_\infty, \rho_0]}
\end{cases}
\]

\[
\mathrm{SL}(2, \mathbb{Z}/6\mathbb{Z})/\{\pm 1\}
\simeq \bigl(\underbrace{\mathrm{SL}(2, \struck{\ill{}}\,\mathbb{Z}/2\mathbb{Z})}_{6} \times
\underbrace{\mathrm{SL}(2, \mathbb{Z}/3\mathbb{Z})}_{24}\bigr)/\pm 1
\qquad \text{d'ordre } 72
\]

\[
\{\rho, \sigma \mid \rho^3 = \sigma^2 = \struck{\ill{}}\,(\sigma\rho)^6 = 1,\
\sigma\rho\sigma\rho = \rho\sigma\rho\sigma\}
\]

\[
\rho_0 = \sigma\rho\sigma\rho \qquad \rho_1 = \rho\sigma\rho\sigma
\qquad
\struck{[\rho_0, \rho_1] = \ldots = [\rho_1, \rho_0]}
\]

\[
\rho_0 \quad \rho_1 \quad \rho_\infty \quad 5
\qquad
\rho_0 = \struck{\ill{}}\,\sigma\rho\sigma^{-1}\rho = (\sigma\rho)^2\omega
\]

\[
\mathcal{T}_{11} \simeq \pi_1(\overbrace{U_{0,3}, \mathbb{D}'_3}^{\Pi_{0,3}^{\mathbb{D}'_3}}; P_+)
\quad \text{et ainsi}
\qquad
\mathcal{T}_{11} \simeq \mathrm{SL}(2,\mathbb{Z})
\]

\[
\rho' = (\rho, 2), \quad \sigma' = (\sigma, 3) \qquad
\text{alors } \rho'^{6} = \sigma'^{4} = 1, \quad
\rho'^{3} = \sigma'^{2} = (\underbrace{1, 2 \bmod 4}_{\omega})
\]

\[
\vec{S}(X_\bullet) \simeq G_0 \qquad
S(X_\bullet) \simeq G_0/H_S, \quad A(X_\bullet) \simeq G_0/H_A, \quad
F(X_\bullet) \simeq G_0/H_F
\]

\[
H_S \cap H_A = e, \quad H_A \cap H_F = e, \quad H_S \cap H_F = e
\]

\[
G_0 \hookrightarrow G_0/H_S \times G_0/H_A, \quad
G_0 \hookrightarrow G_0/H_A \times G_0/H_F, \quad
G_0 \hookrightarrow \struck{\ill{}}\,G_0/H_A \times G_0/H_F
\]

\[
S \quad \vec{A} \quad F \quad \struck{\ill{}}
\]

\[
A_1 = \mathrm{Im}(R \to S \times A) \qquad
A_2 = \mathrm{Im}(R \to A \times F) \qquad
A_3 = \mathrm{Im}(R \to S \times F)
\]

\[
A_{1\,\Omega} \simeq A_{2\,\Omega} \simeq A_{3\,\Omega} \quad (
\]

\[
\mathrm{Pol}_\tau \qquad \mathrm{Pol}_\tau^{\Omega} \simeq \mathrm{Pol}_{\tau\,/\Omega}
\]

\[
S\mathcal{T}_{1,1} \simeq \text{image inverse de } \mathrm{SL}(2,\mathbb{Z})
\text{ dans } \widetilde{\mathrm{SL}(2,\mathbb{R})}
\simeq \{\rho, \sigma \mid \rho^3 = \sigma^2\ (= \omega)\}
\quad (\omega \text{ élément central})
\]

\[
\mathcal{T}_{11} = S\mathcal{T}_{11}/\langle\omega^2\rangle
\]

\[
(\uncertain{S}\mathcal{T}_{11})_{\mathrm{ab}} \simeq \mathbb{Z}
\qquad
\struck{(\rho, \sigma)}\quad
\rho \longmapsto 2, \quad \sigma \longmapsto 3, \quad \omega \longmapsto 6
\]

\[
(\mathcal{T}_{11})_{\mathrm{ab}} \simeq \mathbb{Z}/12\mathbb{Z}
\qquad
\rho \longmapsto 2 \bmod 12, \quad \sigma \longmapsto 3 \bmod 12
\]

\[
\tilde\rho_0^{\,n} = \tilde\sigma_0^{\,n} =
\qquad\qquad
\tilde\sigma^{2} = \tilde\rho^{\,n} = \tilde\omega
\]

\[
\struck{\ill{}}\quad
\tilde\rho \longmapsto \tilde\rho\,\tilde\omega^{-1} = \tilde\rho_0
\]

\[
\tilde\rho_0^{\,n} \longmapsto \struck{\ill{}} = \tilde\rho^{\,n}\,\tilde\omega^{-n}
= \tilde\omega^{\,1-n} = \omega^{\frac{1-n}{2}}
\]

\[
\tilde\sigma_0 = \tilde\sigma_{\uncertain{a}}\,\tilde\omega^{s}
\qquad
\tilde\sigma_0^{\,2} = \tilde\sigma^{2}\,\tilde\omega^{2s} = \tilde\omega^{\,1+2s}
\]

\[
1 + 2s = \frac{1-n}{2}
\qquad \text{OK si } \frac{1-n}{2} \text{ impair}
\]

\[
\tilde\rho = (\rho_0, 2) = (\rho, 2)
\qquad
\tilde\sigma = (\sigma_0, 3) = (\sigma, 3)
\qquad
\rho_0 = (\rho, \dot 2)
\qquad
\sigma_0 = (\sigma, \dot 3) = (\sigma, -1)
\]

\[
PM_{11} \simeq (T^{*}U_{03}, \mathfrak{S}'_3)
\qquad
\text{(fibré \struck{tangent} \uncertain{cotangent} de \struck{\ill{}}\,$U_{03}$ !!)}
\]

\[
\mathcal{T}_{11} \simeq \pi_1(T^{*}U_{03}, \mathfrak{S}'_3) \ldots
\]

\[
\pi \qquad \ell_1 \ \cdots\ \ell_n
\]

\[
u \in \mathrm{Aut}\,\pi, \quad \sigma \in \mathfrak{S}_n, \quad
(g_i)_{i \in n},\ g_i \in \pi
\qquad
u(\ell_i) = g_i\,\ell_{\sigma(i)}\,g_i^{-1}
\]

\[
1 \to \pi_1(\mathrm{Aut}(X)) \to \mathbb{Z}^n \to S\mathcal{T} \to \mathcal{T}
\to \mathfrak{S}_n \to 1
\]

\[
1 \to \underbrace{\pi_1\mathrm{Aut}^{+}(X)}_{0 \text{ ou } \mathbb{Z}}
\to \mathbb{Z}^n \to S\mathcal{T} \to \mathcal{T} \to \mathfrak{S}_n \to 1
\]

\[
\pi_1(G) = \pi_1 \longrightarrow F
\]

\[
\pi_0 \quad \pi_1 \quad \text{op.}
\]

\[
1 \to F \to E \to \pi_0 \to 1
\qquad E = \pi_0(\widetilde{A})
\]

\[
1 \to \pi_1(A) \to \pi_1 \to F \to \pi_0(A) \to \pi_0 \to 1
\]

\[
\struck{\ill{}}\ \to \struck{\ill{}} \to F \to \pi_0(\widetilde{A}) \to \pi_0 \to 1
\qquad \pi_0(\widetilde{A}) = E
\]

\[
H^1(U_{0,3}, \mathfrak{S}_3; \mathbb{G}_m) \simeq \mathbb{Z}/6\mathbb{Z}
\]

\[
0 \to V_J \to \mathcal{O}^J \to \mathcal{O} \to 0 ,
\]

\[
P\Sigma_J = \mathbb{V}(\mathcal{O}(1))^{*}_{\Sigma_J}
\]

\[
\mathcal{O}(1)^{\otimes 6} = \mathcal{O}(6) \simeq V_J^{\otimes 3}(\omega),
\]

\[
\Gamma(\Sigma_J, \mathcal{O}(6)) \simeq \mathrm{Sym}^6 V_J
\]

\[
\varphi \in \Gamma(\Sigma_J, \mathcal{O}(6)), \qquad
\varphi = \prod_{\substack{i, j \in J \\ i \neq j}} (e_i - e_j)
= \prod_{i \in J} \underbrace{(e_j - e_k)^2}_{J = \{i, j, k\}}
\]

\[
\struck{\ill{}}\ \mathrm{Sym}^{*}(V_J)^{\mathfrak{S}_3} \simeq
\mathcal{O}[\gamma_2, \gamma_3]/(2\gamma_3^2 - \gamma_2^3)
\]

\[
\begin{cases}
\gamma_2 = \sum_{i \in J} (e_j - e_k)^2 \\
\gamma_3 = \prod_{i \in J} (e_j + e_k - 2e_i)
\end{cases}
\qquad \text{et} \quad \Delta = \gamma_2^{\,3}
\]

\[
\struck{x \longmapsto x^2 \wedge \omega_0}
\]

\[
PU(TU_{03}, \mathbb{D}_3; \uncertain{\mu_2})
\]

\[
\begin{array}{ccccccccc}
1 & \to & \mathbb{Z} & \to & \widetilde{\Pi}^{D}_{0,3} & \to & \Pi^{D}_{0,3} & \to & 1 \\
 & & \downarrow{\scriptstyle 4} & & \downarrow & & \downarrow & & \\
1 & \to & \mathbb{Z} & \to & \widetilde{\Pi}^{D}_{0,3}(4) & \to & \Pi^{D}_{0,3} & \to & 1
\end{array}
\]

\[
\begin{array}{ccccccccc}
 & & & & & & \Pi^{D}_{0,3} & & \\
 & & & & & & \uparrow & & \\
1 & \to & \overset{\omega}{\mathbb{Z}} & \to & \widetilde{\Pi}^{D}_{0,3}(4) & \to & \Pi^{D}_{0,3} \times \mathbb{Z}/4\mathbb{Z} & \to & 1 \\
 & & & & & & \uparrow & & \\
1 & \to & \mathbb{Z} & \to & ? & \to & \struck{\ill{}}\ \mathrm{SL}(2, \mathbb{Z}) & \to & 1
\end{array}
\]

\[
\struck{e_i - e_j \quad \sigma(e)} \qquad e_i - e_j =
\qquad\qquad
\underline{\mathcal{O}}_X(U) \ni \xi_{ij}(x)
\]

\[
\xi_i \qquad\qquad
\lambda \qquad \lambda^6 = \prod_{i,j} \xi_{ij}(x)
\]

\[
(e_i - e_j)(\underline{Q_k}) \qquad \struck{e_i - e_j}
\]

\[
\mathbb{Z} \to \widetilde{\Pi}_{0,3} \to \Pi^{\struck{\ill{}}}_{03} \to 1
\]

\[
V(\omega) \overset{!!}{\simeq} V \qquad \varphi \in V \qquad
e_i - e_j(\struck{\ill{}}_K
\]

\[
e_i - e_j \bmod L_{Q_K}
\qquad
\begin{array}{ccc}
\frac12 & \frac12 & 1 \\[2pt]
-\frac12 & -\frac12 & -1
\end{array}
\qquad \Bigl(\frac12\Bigr)^4 \qquad \sqrt[3]{1/2}
\]

\[
\vec\rho_s^{\,-1}\,\vec\rho_f = \sigma .
\]

\[
\rho_s(\sigma_n a) = \sigma_n(\sigma_n \underbrace{\rho_s}_{\sigma_n \varepsilon = \rho_s^{-1}} \sigma_n\, a)
= \sigma_n(\rho_s^{-1} a) .
\]

\[
\struck{\rho_f(\sigma_n a) = \sigma_n(\sigma_n \rho_f \sigma_n\, a)}
\]

\[
\vec\rho\,'_s = \vec\rho_f^{\,-1}
\]

\[
\vec\rho\,'_f = \vec\rho\,'_s\,\sigma = \vec\rho_s^{\,-1}\sigma
= \sigma\,\vec\rho_f^{\,-1}\,\sigma = \mathrm{int}(\sigma)\,\vec\rho_f^{\,-1}
\]

\[
1 \hookrightarrow \mathbb{Z} \to \widetilde\Pi \to \Pi \to 1
\]

\[
\begin{array}{ccccccccc}
 & & & & \Pi_0 & & & & \\
 & & & & \downarrow & & & & \\
1 & \to & \mathbb{Z} & \to & \widetilde\Pi(4) & \to & \Pi \times \mathbb{Z}/4\mathbb{Z} & \to & 1
\end{array}
\]

\[
\begin{array}{ccccccccc}
1 & \to & \Pi_0 & \to & \Pi \times \mathbb{Z}/4\mathbb{Z} & \to & \mathbb{D}_3 \times \mathbb{Z}/4\mathbb{Z} & \to & 1 \\
 & & \parallel & & \uparrow & & \updownarrow & & \\
1 & \to & \Pi_0 & \to & \Pi' & \longrightarrow & \mathbb{D}'_3 & \to & 1
\end{array}
\]

\[
\Pi' \subset \Pi \qquad\qquad
\struck{\widetilde\Pi \times \mathbb{Z}/4} \quad
\widetilde\Pi(4) \to \Pi \times \mathbb{Z}/4\mathbb{Z}
\]

\[
A' \cup \sigma A' = A .
\]

\[
\varphi_p : \mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \quad \text{homom. can.}
\qquad
\alpha \in (\mathbb{Z}/p\mathbb{Z})^{*}
\qquad
\struck{p \in (\mathbb{Z}/q\mathbb{Z})^{*}}
\]

\[
\begin{array}{ccccccccc}
0 & \to & \mathbb{Z} & \to & \widetilde\Pi_0(q) & \to & \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z} & \to & 0 \\
 & & \parallel & & \updownarrow & & \uparrow{\scriptstyle\delta} & & \\
0 & \to & \mathbb{Z} & \to & \widetilde\Pi'_0 & \longrightarrow & \mathbb{Z}/\mu\mathbb{Z} & \to & 0
\end{array}
\]

\[
\xi' = \mathrm{cl}(\widetilde\Pi'_0) \in
\mathrm{Ext}^1(\mathbb{Z}/\mu\mathbb{Z}, \mathbb{Z}) \simeq \mathbb{Z}/\mu\mathbb{Z}
\]

\[
\xi' = \delta^{*}\bigl(\underbrace{\mathrm{cl}(\widetilde\Pi_0(q))}_{\xi \in
\mathrm{Ext}^1(\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z},\, \mathbb{Z})}\bigr)
\]

\[
\xi = \mathrm{pr}_1^{*}\bigl(\underbrace{\mathrm{cl}(\widetilde\Pi_0)}_{\alpha\eta_p}\bigr)
+ \mathrm{pr}_2^{*}\bigl(\underbrace{\mathrm{cl}(\mathbb{Z})}_{\eta_q}\bigr)
\]

\[
\delta_p : \mathbb{Z}/\mu\mathbb{Z} \xrightarrow{\ \mathrm{can}\ } \mathbb{Z}/p\mathbb{Z}
\qquad
\delta_q : \mathbb{Z}/\mu\mathbb{Z} \xrightarrow{\ \mathrm{can}\ } \mathbb{Z}/q\mathbb{Z}
\qquad
\xi' \struck{= q'\eta_\mu + p'\eta_\mu}
\]

\[
\delta_p^{*}(\eta_p) = q'\eta_\mu
\qquad
\delta_q^{*}(\eta_q) = p'\eta_\mu
\qquad
\delta_p^{*}(\alpha\eta_p) = q'\alpha\,\eta_\mu
\]

\[
S\mathcal{T}_{11} \simeq \widetilde{\mathrm{SL}}(2, \mathbb{Z})
\simeq \{\rho, \sigma \mid \underbrace{\rho^3 = \sigma^2}_{\omega}\}
\simeq \underset{\{\rho, \sigma \mid \rho^3 = \sigma^2 = 1\}}{\underset{\parallel}{\Pi^{D}_{03}}}
\times_{\mathbb{Z}/6\mathbb{Z}} \mathbb{Z}
\]

\[
\simeq S\Pi^{D}_{0,3}
\]

\[
\begin{array}{ccccccccc}
 & & \mathbb{Z} & \to & \widetilde\Pi(4) & \to & \Pi \times \mathbb{Z}/4\mathbb{Z} & \to & 1 \\
 & & & & \uparrow{\scriptstyle ?} & & \downarrow & & \\
 & & & & \widetilde\Pi & \longrightarrow & \Pi' & &
\end{array}
\]

\[
(p' + q'\alpha)\,\eta_\mu
\]

\[
p = 6, \quad q = 4 \qquad (\alpha = -1\ ?)
\]

\[
\struck{p'}\ \mu = 12 \qquad p' = 3, \quad q' = 2
\]

\[
\struck{p'+}\ (3 - 2)\,\eta_\mu \quad !
\]

\[
\begin{array}{rcl}
\Pi^{D}_{0,3} & \xrightarrow{\ -\alpha'\ } & \mathbb{Z}/6\mathbb{Z} \\
\rho & \longmapsto & -2 \bmod 6 \\
\sigma & \longmapsto & 3 \bmod 6 \\
\varepsilon_0 = \sigma\rho & \longmapsto & 1 \bmod 6 \\
\rho_0 = \varepsilon_0^{\,2} & \longmapsto & 2 \bmod 6
\end{array}
\]

\[
\widetilde\Pi^{D}_{03}
\]

\[
\struck{\ill{}}\quad U'_{03} \subset PU_{03}
\qquad
\mu_6\text{-}\uncertain{\text{revêt.}}\ \struck{\ill{}}\,/U_{03}
\]

\[
\begin{array}{ccccccccc}
1 & \to & \mathbb{Z} & \to & \widetilde\Pi^{D}_{03}(4) & \to & \Pi^{D}_{0,3} \times \mathbb{Z}/4\mathbb{Z} & \to & 1 \\
 & & \parallel & & \parallel & & \updownarrow & & \\
1 & \to & \mathbb{Z} & \to & \widetilde{\mathcal{T}}_{11} & \Longrightarrow & \mathcal{T}_{1,1} = \Pi^{D}_{0,3} \times_{\mathbb{Z}/12\mathbb{Z}} \mathbb{Z}/4\mathbb{Z} & &
\end{array}
\]

\[
\struck{\ill{}}\qquad
\widetilde\Pi^{D}_{0,3} \ni
\begin{cases}
\tilde\rho = (\rho, -2) \\
\tilde\sigma = (\sigma, -3)
\end{cases}
\qquad
\tilde\rho^{\,3} = \tilde\sigma^{2} = \varpi'
\]

\[
\underset{\substack{\downarrow \\ \rho}}{\tilde\rho},\ \underset{\substack{\downarrow \\ \sigma}}{\tilde\sigma}
\in \widetilde\Pi^{D}_{0,3}
= \{\tilde\rho, \tilde\sigma \mid \underbrace{\tilde\rho^{\,3} = \tilde\sigma^{2}}_{\varpi' = \omega}\ \struck{\ill{}}\}
\simeq \widetilde{\mathcal{T}}_{11}
\]

\[
S\Pi^{D}_{03}(4) = \{\rho, \sigma, \varpi \mid
\rho^3 = \sigma^2 = \dot\varpi^{4},\ \varpi \text{ central}\}
\]

\[
\overbrace{S\Pi^{D}_{0\,3}(4)}^{\pi_1(U_{03}, \mathfrak{S}_3 \times \mu_4)}
\longrightarrow \Pi^{D}_{0,3}
\]