\[
(8)\quad
\begin{cases}
\mathcal{M}_{X,x,K} = \pi_1(\mathfrak{X}_{g,\nu\,K}, x) \\
\mathcal{N}_{X,x,K} = \pi_1(M_{g,\nu\,K}, \xi) \\
\Gamma_{\overline{K}|K} = \mathrm{Gal}(\overline{K}|K) = \pi_1(\operatorname{Spec} K, \overline{k})
\end{cases}
\]

\[
(10)\qquad
\overbrace{\pi_{X,x} \subset \underbrace{\mathfrak{G}_{X,x} \subset \mathcal{M}_{X,x,K}}_{\Gamma_{\overline{K}|K}}}^{\mathcal{N}_{X,K}}
\]

\[
(11)\qquad \Gamma_{\overline{K}|K} \longrightarrow \Gamma_{\widetilde{\mathbb{Q}}/\mathbb{Q}}
\]

\[
(12)\quad
\begin{cases}
\mathcal{M}_{X,x,K} \simeq \mathcal{M}_{X,x} \times_{\Gamma_{\widetilde{\mathbb{Q}}/\mathbb{Q}}} \Gamma_{\overline{K}/K} \\
\mathcal{N}_{X,x,K} \simeq \mathcal{N}_{X,x} \times_{\Gamma_{\widetilde{\mathbb{Q}}/\mathbb{Q}}} \Gamma_{\overline{K}/K}
\end{cases}
\]

\[
(15)\qquad \Gamma_{\overline{K}|K} \longrightarrow \operatorname{Aut}_{\mathrm{lac}} \pi_{X,x} \qquad \text{resp.}
\]

\[
(15')\qquad \Gamma_{\overline{K}/K} \longrightarrow \operatorname{Autext}_{\mathrm{lac}}(\pi_{X,x})
\]

\[
\operatorname{Hom}(\overline{K},\mathbb{C}) \xrightarrow{\ \text{surj}\ } \operatorname{Discspec}(\overline{K})
\]

\[
\pi_{1,\mathrm{ab}} \otimes_{\mathbb{Z}} \mathbb{C}
\]

\[
\boxed{\rho = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}}, \quad
\rho^{-1} = \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}
\qquad \rho^3 = \rho^{-3} = \omega \quad \text{donc } \rho^2 = \omega\rho^{-1}, \quad \rho^6 = 1
\]

\[
\omega = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \quad (\text{donc } \omega g = -g) \qquad \omega^2 = 1
\]

\[
\begin{cases}
\boxed{\sigma_\infty = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} \qquad
\sigma_\infty^{-1} = \omega\sigma_\infty = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \\[4pt]
\sigma_0 = \begin{pmatrix} -1 & 1 \\ -2 & 1 \end{pmatrix} \qquad \rho(\sigma_i) = \sigma_{i+1} \\[4pt]
\sigma_1 = \begin{pmatrix} -1 & 2 \\ -1 & 1 \end{pmatrix}
\end{cases}
\]

\[
\sigma^2 = \sigma^{-2} = \omega, \quad \sigma^{-1} = \omega\sigma \qquad\qquad \sigma_i^2 = \omega, \quad \sigma_i^{-1} = \omega\sigma_i
\]

\[
\begin{cases}
\boxed{\varepsilon_0 = \sigma_\infty^{-1}\rho = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}} \quad (\text{unipotent \& régulier}) \\[4pt]
\boxed{\varepsilon_1 = \rho\sigma_\infty^{-1} = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}} \\[4pt]
\varepsilon_\infty = \rho^2\sigma_\infty^{-1}\rho^{-1} = \begin{pmatrix} 2 & -1 \\ 1 & 0 \end{pmatrix}
\end{cases}
\qquad
\begin{cases}
\varrho_0 = \varepsilon_0^2 = \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix} \\[4pt]
\varrho_1 = \varepsilon_1^2 = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \\[4pt]
\varrho_\infty = \varepsilon_\infty^2 = \begin{pmatrix} 3 & -2 \\ 2 & -1 \end{pmatrix}
\end{cases}
\]

\[
\varepsilon_i = \sigma_i^{-1}\rho, \qquad \rho(\varepsilon_i) = \varepsilon_{i+1}, \qquad \rho(\varrho_i) = \varrho_{i+1}
\]

\[
\varrho_\infty\varrho_1\varrho_0 = \omega, \qquad
\varrho'_\infty\varrho'_1\varrho'_0 = 1 \quad \text{où } \varrho'_i = \omega\varrho_i
\]

\[
\begin{cases}
\sigma_i\sigma_{i+1} = \omega\rho\varrho_i = \omega\varrho_{i+1}\rho \\
\rho(\sigma_i) = \sigma_{i+1} = \omega\varrho_{i-1}\varepsilon_{i+1} \\
\sigma_i(\rho) = \rho^{-1}\varrho'_{i-1}
\end{cases}
\]

\[
\begin{cases}
\boxed{\tau_\infty = \begin{pmatrix} -1 & 0 \\ 0 & +1 \end{pmatrix}} \\[4pt]
\tau_0 = \begin{pmatrix} 1 & 0 \\ 2 & -1 \end{pmatrix} \\[4pt]
\tau_1 = \begin{pmatrix} 1 & -2 \\ 0 & -1 \end{pmatrix}
\end{cases}
\qquad
\tau_i^2 = 1 \qquad \rho(\tau_i) = \tau_{i+1}
\]

\[
\tau_i\tau_{i-1} = \underbrace{\omega\varrho_{i+1}}_{\varrho'_{i+1}} \quad \text{i.e.} \quad
\begin{cases}
\tau_1\tau_0 = \varrho'_\infty \\
\tau_\infty\tau_1 = \varrho'_0 \\
\tau_0\tau_\infty = \varrho'_1
\end{cases}
\]

\[
\begin{cases}
\boxed{\tau'_\infty = \sigma_\infty\tau_\infty = \omega\tau_\infty\sigma_\infty = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}} \\[4pt]
\tau'_0 = \sigma_0\tau_0 = \omega\tau_0\sigma_0 = \begin{pmatrix} 1 & -1 \\ 0 & -1 \end{pmatrix} \\[4pt]
\tau'_1 = \sigma_1\tau_1 = \omega\tau_1\sigma_1 = \begin{pmatrix} -1 & 0 \\ -1 & 1 \end{pmatrix}
\end{cases}
\]

\[
\tau_i'^{\,2} = 1 \qquad \rho(\tau'_i) = \tau'_{i+1}
\]

\[
\tau'_0\tau'_1 = \tau'_1\tau'_\infty = \tau'_\infty\tau'_0 = \omega\rho, \qquad
\tau'_1\tau'_0 = \tau'_\infty\tau'_1 = \tau'_0\tau'_\infty = \omega\rho^{-1}
\]

\[
\tau'_i = \sigma_i\tau_i = \omega\tau_i\sigma_i = \tau_i\sigma_i^{-1}, \qquad
\tau_i = \tau'_i\sigma_i = \omega\sigma_i\tau'_i = \sigma_i^{-1}\tau'_i,
\]

\[
\sigma_i = \tau'_i\tau_i = \omega\tau_i\tau'_i
\]

\[
\left.
\begin{aligned}
\tau_i(\varepsilon_j) &= \varepsilon_j^{-1} \\
\tau_i(\varrho_j) &= \varrho_j^{-1} \\
\tau_i(\varrho'_j) &= \varrho_j'^{\,-1}
\end{aligned}
\right| \ \text{si } i \neq j
\]

\[
\begin{cases}
\tau_i(\sigma_i) = \sigma_i^{-1} = \omega\sigma_i \qquad \text{[au-dessus :] } \tau'_i(\sigma_i) = \\
\tau'_i(\rho) = \rho^{-1} \\
\tau_i(\rho) = \varrho_{i-1}'^{\,-1}\rho = \sigma_i(\rho^{-1})
\end{cases}
\]

\[
\begin{aligned}
&\tau_\infty(\varepsilon_0) = \varepsilon_0^{-1} &\qquad& \tau_\infty(\varrho_0) = \varrho_0^{-1} \\
&\tau_\infty(\varepsilon_1) = \varepsilon_1^{-1} && \tau_\infty(\varrho_1) = \varrho_1^{-1} \\
&\tau_\infty(\varepsilon_\infty) = \varrho_0(\varepsilon_\infty^{-1}) && \tau_\infty(\varrho_\infty) = \varrho_0\varrho_1\omega \\
&\phantom{\tau_\infty(\varepsilon_\infty)} = \varrho_1^{-1}(\varepsilon_\infty^{-1}) && \phantom{\tau_\infty(\varrho_\infty)} = \varrho_1^{-1}(\varrho_\infty^{-1}) \\
& && \phantom{\tau_\infty(\varrho_\infty)} = \varrho_0^{-1}(\varrho_\infty^{-1})
\end{aligned}
\]

\[
\begin{aligned}
&\sigma_i(\varrho_{i+1}) = \varrho_{i+2} &\qquad& \sigma_i(\varepsilon_{i+1}) = \varepsilon_{i+2} \\
&\sigma_i(\varrho_{i+2}) = \varrho_{i+1} && \sigma_i(\varepsilon_{i+2}) = \varepsilon_{i+1}) \\
&\sigma_i(\varrho_i) = \omega\underbrace{(\varrho_{i+1}\varrho_{i+2})^{-1}}_{\omega\varrho_{i+2}^{-1}\varrho_{i+1}^{-1}} = \varrho_{i+1}(\varrho_i) && \\
& \phantom{\sigma_i(\varrho_i)} = \varrho_{i+2}^{-1}(\varrho_i) &&
\end{aligned}
\]

\[
\begin{cases}
\lambda_0 = \varepsilon_0, \ \lambda_1 = \sigma_\infty, \ \lambda_\infty = \rho^{-1} \\
\lambda_\infty\lambda_1\lambda_0 = 1 \qquad \lambda_1^4 = \lambda_\infty^6 = 1
\end{cases}
\qquad
\overbrace{\Gamma_0 = \tau'_\infty}^{= \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)}, \
\overbrace{\Gamma_1 = -\tau'_0}^{= \left(\begin{smallmatrix} -1 & 1 \\ 0 & 1 \end{smallmatrix}\right)}, \
\overbrace{\Gamma_\infty = \tau_\infty}^{= \left(\begin{smallmatrix} -1 & 0 \\ 0 & 1 \end{smallmatrix}\right)}
\]

\[
\Gamma_\infty\Gamma_1 = \lambda_0, \qquad \Gamma_0\Gamma_\infty = \lambda_1, \qquad \Gamma_1\Gamma_0 = \lambda_\infty
\]

\[
\mathrm{Sl}(2,\mathbb{Z}) = \{\rho, \sigma \mid \rho^3 = \rho^{-3} = \sigma^2 \ (= \sigma^{-2})\}
= \mathbb{Z}/6\mathbb{Z} *_{\mathbb{Z}/2\mathbb{Z}} \mathbb{Z}/4\mathbb{Z}
\]

\[
\begin{aligned}
\mathrm{Gl}(2,\mathbb{Z}) &= \{\rho, \sigma, \tau' \mid \rho^3 = \rho^{-3} = \sigma^2 \ (= \sigma^{-2}),\ \tau'(\rho) = \rho^{-1},\ \tau'(\sigma) = \sigma^{-1}\} \\
&= \{\Gamma_0, \Gamma_1, \Gamma_\infty \mid \Gamma_0^2 = \Gamma_1^2 = \Gamma_\infty^2 = 1,\
(\Gamma_0\Gamma_\infty)^2 = (\Gamma_1\Gamma_0)^3 = (\Gamma_1\Gamma_0)^{-3}\} \\
&= D_6 *_{D_2} D_4
\end{aligned}
\]

\[
\begin{cases}
\text{NB } \tau' = \Gamma_0, \quad \Gamma_0(\rho) = \rho^{-1}, \ \Gamma_0(\sigma) = \sigma^{-1} \\
\Gamma_1 = \rho^{-1}\Gamma_0 = \Gamma_0\rho \\
\Gamma_\infty = \sigma\Gamma_0 = \Gamma_0\sigma^{-1}
\end{cases}
\]

\[
1 \to \mathbb{Z} \xrightarrow{\ k\ } \mathrm{Gl}(2,\mathbb{Z})^{\sim} \to \mathrm{Gl}(2,\mathbb{Z}) \to 1
\]

\[
k(1) \overset{\mathrm{def}}{=} \ell'_0
\]

\[
\left.
\begin{aligned}
&\tilde{\rho} \text{ au dessus de } \rho, & \tilde{\rho}^{6} &= \ell_0'^{\,-1} \\
&\tilde{\sigma} \text{ — — } \sigma, & \tilde{\sigma}^{4} &= \ell'_0 \\
&\tilde{\omega}_0 \text{ — — } \omega_0 = -1, & \tilde{\omega}^{2} &= \ell'_0
\end{aligned}
\right|
\ \text{conditions sur } \tilde{\rho}, \tilde{\sigma}, \tilde{\omega}_0 \text{ donc :}
\ \left|\ \tilde{\rho}^{-3} = \tilde{\sigma}^{2} = \tilde{\omega}_0 \right.
\]

\[
\begin{cases}
\tilde{\tau}'(\tilde{\rho}) = \tilde{\rho}^{-1}, \quad \tilde{\tau}'(\tilde{\sigma}) = \tilde{\sigma}^{-1}, \quad \tilde{\tau}'^{\,2} = 1 \\
\tilde{\tau}'(\tilde{\omega}_0) = \tilde{\omega}_0^{-1}, \quad \tilde{\tau}'(\ell'_0) = \ell_0'^{\,-1}
\end{cases}
\]

\[
\tilde{\varepsilon}_0 = \tilde{\sigma}\tilde{\rho}, \qquad
\tilde{\varepsilon}_1 = \tilde{\rho}(\tilde{\varepsilon}_0) = \tilde{\sigma}^{-1}(\varepsilon_0) = \tilde{\sigma}(\varepsilon_0) = \tilde{\rho}\tilde{\sigma}
\]

\[
\tilde{\varepsilon}_1\tilde{\varepsilon}_0 = \tilde{\rho}\tilde{\sigma}^2\tilde{\rho} = \tilde{\rho}^{-1} \qquad (\text{car } \tilde{\sigma}^2 = \tilde{\rho}^{-3}), \ \text{donc}
\]

\[
\tilde{\rho} = (\tilde{\varepsilon}_1\tilde{\varepsilon}_0)^{-1} = \tilde{\varepsilon}_0^{-1}\tilde{\varepsilon}_1^{-1}, \qquad
\tilde{\sigma} = \tilde{\rho}^{-1}\tilde{\varepsilon}_1 = \tilde{\varepsilon}_1\tilde{\varepsilon}_0\tilde{\varepsilon}_1 = \tilde{\varepsilon}_0\tilde{\rho}^{-1} = \tilde{\varepsilon}_0\tilde{\varepsilon}_1\tilde{\varepsilon}_0
\]

\[
\mathrm{Sl}(2,\mathbb{Z})^{\sim} \ \left|\
\begin{aligned}
&\tilde{\rho}, \tilde{\sigma} : && \tilde{\rho}^3\tilde{\sigma}^2 = 1 \\
&\tilde{\varepsilon}_0, \tilde{\varepsilon}_1 : && \boxed{\tilde{\varepsilon}_0\tilde{\varepsilon}_1\tilde{\varepsilon}_0 = \tilde{\varepsilon}_1\tilde{\varepsilon}_0\tilde{\varepsilon}_1}
\end{aligned}
\right.
\]

\[
\mathrm{Gl}(2,\mathbb{Z})^{\sim} \ \left|\
\begin{aligned}
&\tilde{\rho}, \tilde{\sigma}, \tilde{\tau}' = \tilde{\Gamma}_0 && \tilde{\Gamma}_0^{2} = \tilde{\rho}^3\tilde{\sigma}^2 = 1, \ \tilde{\Gamma}_0(\tilde{\rho}) = \tilde{\rho}^{-1}, \ \tilde{\Gamma}_0(\tilde{\sigma}) = \tilde{\sigma}^{-1} \\
&\tilde{\varepsilon}_0, \tilde{\varepsilon}_1, \tilde{\Gamma}_0 && \tilde{\Gamma}_0^{2} = 1, \ \tilde{\varepsilon}_0\tilde{\varepsilon}_1\tilde{\varepsilon}_0 = \tilde{\varepsilon}_1\tilde{\varepsilon}_0\tilde{\varepsilon}_1, \\
& && \tilde{\Gamma}_0(\tilde{\varepsilon}_0) = \tilde{\varepsilon}_1^{-1}, \ \tilde{\Gamma}_0(\tilde{\varepsilon}_1) = \tilde{\varepsilon}_0^{-1} \\
&\tilde{\Gamma}_0, \tilde{\Gamma}_1, \tilde{\Gamma}_\infty && \tilde{\Gamma}_0^2 = \tilde{\Gamma}_1^2 = \tilde{\Gamma}_\infty^2 = (\Gamma_1\Gamma_0)^3(\Gamma_0\Gamma_\infty)^{2} = 1
\end{aligned}
\right.
\]

\[
\begin{aligned}
&\tilde{\varepsilon}_i \ (i \in \{0,1,\infty\}) \quad \rho(\tilde{\varepsilon}_i) = \tilde{\varepsilon}_{i+1} &\quad& \tilde{\ell}_i = \tilde{\varepsilon}_i^{\,2}, \ \rho(\tilde{\ell}_i) = \tilde{\ell}_{i+1} \\
&\tilde{\sigma}_i \ (i \in \{0,1,\infty\}) \quad \rho(\tilde{\sigma}_i) = \tilde{\sigma}_{i+1} = \tilde{\ell}_{i-1}\tilde{\varepsilon}_{i+1} && \tilde{\sigma}_i^{\,2} = \tilde{\rho}^{-3} = \tilde{\omega}_0, \quad \tilde{\sigma}_i^{\,4} = \tilde{\rho}^{-6} = \ell'_0
\end{aligned}
\]

\[
\tilde{\varepsilon}_i = \tilde{\sigma}_{i-1}\tilde{\rho} = \tilde{\rho}\tilde{\sigma}_{i+1}
\]

\[
\tilde{\sigma}_i\tilde{\sigma}_{i+1} = \tilde{\omega}_0\tilde{\rho}\tilde{\ell}_i = \tilde{\omega}_0\tilde{\ell}_{i+1}\tilde{\rho}
\qquad
\tilde{\sigma}_i(\tilde{\rho}) = \tilde{\rho}^{-1}\tilde{\lambda}_{i-1} = \tilde{\omega}_0^{-1}\tilde{\rho}^{-1}\tilde{\ell}_{i-1}
\]

\[
\tilde{\ell}_\infty\tilde{\ell}_1\tilde{\ell}_0 = \tilde{\omega}_0 \quad \text{i.e.} \quad
\tilde{\lambda}_\infty\tilde{\lambda}_1\tilde{\lambda}_0 = \tilde{\omega}_0^{-2} = \ell_0'^{\,-1}
\quad (\text{où } \tilde{\lambda}_i = \tilde{\omega}_0^{-1}\tilde{\ell}_i)
\]

\[
\text{i.e.} \quad \tilde{\lambda}_0^{-1}\tilde{\lambda}_1^{-1}\tilde{\lambda}_\infty^{-1} = \ell'_0
\]

\[
\tilde{\rho}^{3}(\tilde{\tau}'_\infty) = \text{\struck{\ill{}}}\,, \qquad \tilde{\omega}_0^{-1}(\tilde{\tau}'_\infty) = \ell_0'^{\,-1} \cdot \tilde{\tau}'_\infty
\]

\[
\begin{cases} \tilde{\Gamma}_0(\tilde{u}) = \tilde{v} \\ \tilde{\Gamma}_0(\tilde{v}) = \tilde{u} \end{cases}
\qquad
\begin{cases} \tilde{\Gamma}_\infty(\tilde{u}) = \tilde{u}^{-1} \\ \tilde{\Gamma}_\infty(\tilde{v}) = \tilde{u}(\tilde{v}) \end{cases}
\qquad
\begin{cases} \tilde{\Gamma}_1(\tilde{u}) = \tilde{u}^{-1} \\ \tilde{\Gamma}_1(\tilde{v}) = \tilde{u}\tilde{v} \end{cases}
\]

\[
\begin{cases} \tilde{\rho}(\tilde{u}) = \tilde{v}^{-1} \\ \tilde{\rho}(\tilde{v}) = \tilde{v}\tilde{u} \end{cases}
\qquad
\begin{cases} \tilde{\varepsilon}_0(\tilde{u}) = \tilde{u} \\ \tilde{\varepsilon}_0(\tilde{v}) = \tilde{v}\tilde{u}^{-1} \end{cases}
\qquad
\begin{cases} \tilde{\sigma}(\tilde{u}) = \tilde{u}(\tilde{v}) \\ \tilde{\sigma}(\tilde{v}) = \tilde{u}^{-1} \end{cases}
\qquad
\begin{cases} \tilde{\varepsilon}_1(\tilde{u}) = \tilde{u}\tilde{v} \\ \tilde{\varepsilon}_1(\tilde{v}) = \tilde{v} \end{cases}
\]

\[
\boxed{\mathfrak{S}_{0,3} \xrightarrow{\ \sim\ } \mathrm{Gl}(2,\hat{\mathbb{Z}})'} \qquad
\boxed{\mathfrak{S}^{+}_{0,3} \simeq \mathrm{Sl}(2,\hat{\mathbb{Z}})'} \qquad \text{formulaire (1981)}
\]

\[
\begin{aligned}
&{*}\ \rho = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix} \quad \rho^{-1} = \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}
&\quad& \rho^3 = -1 \ \text{i.e.}\ \rho^2 = -\rho^{-1} \qquad \rho^6 = 1 \\
&{*}\ \sigma_\infty = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} && \sigma_i^2 = -1 \ \text{i.e.}\ \sigma_i^{-1} = -\sigma_i \qquad \sigma_i^4 = 1 \\
&\phantom{*}\ \sigma_0 = \begin{pmatrix} 1 & -1 \\ 2 & -1 \end{pmatrix} \\
&\phantom{*}\ \sigma_1 = \begin{pmatrix} 1 & -2 \\ 1 & -1 \end{pmatrix} && \rho\sigma_i\rho^{-1} = \sigma_{i+1} \qquad \sigma_i\sigma_{i+1} = \rho\ell_i = -\ell_{i+1}\rho
\end{aligned}
\]

\[
\begin{cases}
\rho(\sigma_i) = \sigma_{i+1} = \ell_{i-1}\varepsilon_{i+1} \\
\sigma_i(\rho) = \rho^{-1}\lambda_{i-1}
\end{cases}
\]

\[
\left|
\begin{aligned}
&{*}\ \varepsilon_0 = \sigma_\infty\rho = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} = 1 + N_0 &\quad& \ell_0 = \varepsilon_0^2 = \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix} \\
&{*}\ \varepsilon_1 = \rho\sigma_\infty = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = 1 + N_1 && \ell_1 = \varepsilon_1^2 = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \\
&\phantom{*}\ \varepsilon_\infty = \rho^2\sigma_\infty\rho^{-1} = \begin{pmatrix} 2 & -1 \\ 1 & 0 \end{pmatrix} = 1 + N_\infty && \ell_\infty = \varepsilon_\infty^2 = \begin{pmatrix} 3 & -2 \\ 2 & -1 \end{pmatrix}
\end{aligned}
\right.
\]

\[
\rho\varepsilon_i\rho^{-1} = \varepsilon_{i+1}, \qquad \rho\ell_i\rho^{-1} = \ell_{i+1}
\]

\[
\ell_\infty\ell_1\ell_0 = -1 \quad \text{i.e.} \quad \lambda_\infty\lambda_1\lambda_0 = 1 \quad (\lambda_i = -\ell_i), \qquad
\varepsilon_i = \sigma_{i-1}\rho = \rho\sigma_{i+1}
\]

\[
\begin{aligned}
&{*}\ \tau_\infty = \begin{pmatrix} -1 & 0 \\ 0 & +1 \end{pmatrix} &\quad& \tau_i^2 = 1 \\
&\phantom{*}\ \tau_0 = \begin{pmatrix} -1 & 0 \\ -2 & -1 \end{pmatrix} && \tau_i\tau_{i+1} = -\ell_{i+2} \ \text{i.e.} \ 
\begin{cases} \tau_1\tau_0 = -\ell_\infty \\ \tau_\infty\tau_1 = -\ell_0 \\ \tau_0\tau_\infty = -\ell_1 \end{cases} \\
&\phantom{*}\ \tau_1 = \begin{pmatrix} -1 & -2 \\ 0 & -1 \end{pmatrix} && \rho\tau_i\rho^{-1} = \tau_{i+1}
\end{aligned}
\]

\[
\begin{aligned}
&{*}\ \tau'_\infty = \text{\struck{\ill{}}}\,\sigma_\infty\tau_\infty = +\tau_\infty\sigma_\infty = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\
&{*}\ \tau'_0 = -\sigma_0\tau_0 = +\tau_0\sigma_0 = \begin{pmatrix} 1 & -1 \\ 0 & -1 \end{pmatrix} \\
&\phantom{*}\ \tau'_1 = -\sigma_1\tau_1 = +\tau_1\sigma_1 = \begin{pmatrix} -1 & 0 \\ -1 & 1 \end{pmatrix}
\end{aligned}
\qquad
\begin{cases}
\tau_i'^{\,2} = 1 \\
\rho\tau'_i\rho^{-1} = \tau'_{i+1} \\
\tau'_0\tau'_1 = \tau'_1\tau'_\infty = \tau'_\infty\tau'_0 = -\rho \\
\tau'_i(\rho) = \rho^{-1} \quad \tau'_i(\sigma_i) = \tau_i(\sigma_i) = \sigma_i^{-1} = -\sigma_i
\end{cases}
\]

\[
\left.
\begin{aligned}
\tau_i(\sigma_i) &= \sigma_i^{-1} = -\sigma_i \\
\tau_i(\rho) &= -\ell_{i-1}^{-1}\rho = \sigma_i(\rho^{-1})
\end{aligned}
\right|
\quad
\begin{cases}
\tau'_i = -\sigma_i\tau_i = +\tau_i\sigma_i \\
\tau_i = \tau'_i\sigma_i = +\sigma_i\tau'_i \\
\sigma_i = -\tau'_i\tau_i = +\tau_i\tau'_i
\end{cases}
\]

\[
\boxed{\begin{cases} \sigma^4 = \rho^6 = 1 \\ \sigma^2 = \rho^3 \end{cases}}
\]

\[
\boxed{\begin{cases} \Gamma_0^2 = \Gamma_1^2 = \Gamma_\infty^2 = 1 \\ (\Gamma_0\Gamma_\infty)^4 = (\Gamma_1\Gamma_0)^6 = 1 \\ (\Gamma_0\Gamma_\infty)^2 = (\Gamma_1\Gamma_0)^3 \end{cases}}
\qquad \simeq \ D_6 \amalg_{D_2} D_4
\]

\[
\begin{cases}
\Gamma_0(\sigma_\infty) = \sigma_\infty^{-1} \\
\Gamma_0(\rho) = \rho^{-1} \\
\Gamma_1 = \rho^{-1}\Gamma_0 = \Gamma_0\rho \\
\Gamma_\infty = \sigma\Gamma_0 = \Gamma_0\sigma^{-1}
\end{cases}
\]

\[
\begin{cases}
\ell'_0 = \varepsilon_0, \ \ell'_1 = (-\sigma_\infty), \ \ell'_\infty = \rho^{-1} \\
\ell'_\infty\ell'_1\ell'_0 = 1
\end{cases}
\qquad
\Gamma_0 = \tau'_\infty, \ \Gamma_1 = -\tau'_0, \ \Gamma_\infty = \tau_\infty
\]

\[
\begin{cases}
\tau_\infty\tau'_0 = \varepsilon_0 \\
\tau'_\infty\tau_\infty = -\sigma_\infty \\
(-\tau'_0)\tau'_\infty = \rho^{-1}
\end{cases}
\ \text{i.e.} \
\begin{cases}
\Gamma_\infty\Gamma_1 = \ell'_0 \\
\Gamma_0\Gamma_\infty = \ell'_1 \\
\Gamma_1\Gamma_0 = \ell'_\infty
\end{cases}
\]

\[
\begin{cases}
\tau_i(\varepsilon_j) = \varepsilon_j^{-1} \ \text{si } i \neq j \\
\tau'_i(\varepsilon_{i+1}) = \varepsilon_{i-1}^{-1} \\
\sigma_i(\varepsilon_{i+1}) = \varepsilon_{i-1}
\end{cases}
\qquad
\begin{cases}
\sigma_i(\ell_{i+1}) = \ell_{i-1} \\
\sigma_i(\ell_{i-1}) = \ell_{i+1}
\end{cases}
\]

\[
\sigma_i(\ell_i) = -(\ell_{i+1}\ell_{i-1})^{-1} \ (\text{attention aux signes !})
\]

\[
\begin{cases}
\tau'_i(\ell_{i-1}) = \ell_{i+1}^{-1} \\
\tau'_i(\ell_{i+1}) = \ell_{i-1}^{-1}
\end{cases}
\ \tau'_i(\ell_i) = \ell_i^{-1}
\qquad
\begin{cases}
\tau_i(\ell_{i-1}) = \ell_{i-1}^{-1} \\
\tau_i(\ell_{i+1}) = \ell_{i+1}^{-1}
\end{cases}
\ \tau_i(\ell_i) = -(\ell_{i-1}^{-1}\ell_{i+1}^{-1})^{-1} = -\ell_{i+1}\ell_{i-1}
\]

\[
[\ell_0, \ell_1] = [\lambda_0, \lambda_1] = (\sigma_\infty(\rho)\rho^{-1})^3 \qquad \sigma_\infty(\rho)\rho^{-1} = \ell_0\rho = \rho\ell_\infty
\]

\[
[\sigma_\infty, \ell_\infty] = [\sigma_\infty, \lambda_\infty] = (\sigma_\infty(\rho^{-1})\rho)^3 = \mathrm{int}(\sigma)(\rho^{-1}) \cdot \underbrace{[\ell_0, \ell_1]}_{[\lambda_0, \lambda_1]}
\]

\[
[\lambda_0, \lambda_1] = 1 \iff [\sigma_\infty, \ell_\infty] = 1 \iff (\underbrace{\sigma_\infty(\rho)\rho^{-1}}_{\ell_0\rho = \rho\ell_\infty})^3 = 1
\]

\[
\boxed{\varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1} \quad (*)
\]

\[
\begin{cases}
\sigma = \varepsilon_0\varepsilon_1\varepsilon_0 \ (= \varepsilon_1\varepsilon_0\varepsilon_1) \\
\rho'_0 = \sigma\varepsilon_0 = \dots = (\varepsilon_1\varepsilon_0)^2 \\
\rho'_1 = \sigma\varepsilon_1 = \varepsilon_0\sigma = (\varepsilon_0\varepsilon_1)^2 \\
\rho_0 = \sigma^{-1}\varepsilon_0 = \varepsilon_1\sigma^{-1} \ (= \varepsilon_0^{-1}\varepsilon_1^{-1} = (\varepsilon_1\varepsilon_0)^{-1}) \\
\rho_1 = \varepsilon_0\sigma^{-1} = \sigma^{-1}\varepsilon_1 \ (= \varepsilon_1^{-1}\varepsilon_0^{-1} = (\varepsilon_0\varepsilon_1)^{-1})
\end{cases}
\]

\[
\text{(a)} \quad
\begin{cases}
\boxed{\begin{aligned} &\sigma(\varepsilon_0) = \varepsilon_1, \quad \sigma(\varepsilon_1) = \varepsilon_0 \\ &\sigma(\rho_0) = \rho_1, \quad \sigma(\rho_1) = \rho_0 \end{aligned}} \\
\sigma(\rho'_0) = \rho'_1, \quad \sigma(\rho'_1) = \rho'_1
\end{cases}
\]

\[
\text{(a')} \quad
\begin{cases}
\rho_0(\varepsilon_0) = \rho'_0(\varepsilon_0) = \varepsilon_1 \\
\rho_1(\varepsilon_1) = \rho'_1(\varepsilon_1) = \varepsilon_0 \\[4pt]
\varepsilon_0(\rho_0) = \rho_1, \quad \varepsilon_0(\rho'_0) = \rho'_1 \\
\varepsilon_1(\rho_1) = \rho_0, \quad \varepsilon_1(\rho'_1) = \rho'_0
\end{cases}
\]

\[
\text{(b)} \quad
\begin{aligned}
&\rho_0 = \sigma^{-1}\varepsilon_0 = \varepsilon_1\sigma^{-1}, \quad \rho_1 = \sigma^{-1}\varepsilon_1 = \varepsilon_0\sigma^{-1} \\
&\rho'_0 = \sigma\varepsilon_0 = \varepsilon_1\sigma, \quad \rho'_1 = \sigma\varepsilon_1 = \varepsilon_0\sigma
\end{aligned}
\ \left|\ \text{$\rho_i, \rho'_i$ en termes de $\sigma$, $\varepsilon$.}\right.
\]

\[
\text{(c)} \quad \rho_0^{-3} = \rho_1^{-3} = \sigma^{2} \ \Big(\overset{\mathrm{def}}{=} \omega \in Z(H)\Big)
\]

\[
\text{(d)} \quad \rho'_0 = \rho_0^{-2} = \omega\rho_0, \quad \rho'_1 = \rho_1^{-2} = \omega\rho_1
\qquad [\text{donc } \operatorname{int}(\rho_i) = \operatorname{int}(\rho'_i) = \operatorname{int}(\rho_i)^{-2}]
\]

\[
\text{d'où} \quad \rho_0 = \omega^{-1}\rho'_0 \ (= \rho'_0\omega^{-1}), \quad \rho_1 = \omega^{-1}\rho'_1 = \rho'_1\omega^{-1}
\]

\[
\sigma = \varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1, \quad
\rho_0 = (\varepsilon_1\varepsilon_0)^{-1}, \ \rho_1 = (\varepsilon_0\varepsilon_1)^{-1}, \quad
\rho'_0 = (\varepsilon_1\varepsilon_0)^2, \ \rho'_1 = (\varepsilon_0\varepsilon_1)^2
\]

\[
\boxed{\varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1 \quad \text{ou} \quad \varepsilon_0\varepsilon_1\varepsilon_0\varepsilon_1^{-1}\varepsilon_0^{-1}\varepsilon_1^{-1} = 1}
\]

\[
\varepsilon_1 = \sigma(\varepsilon_0) = \sigma\varepsilon_0\sigma^{-1}, \quad
\boxed{\begin{aligned} \rho_0 &= \sigma^{-1}\varepsilon_0 \\ \rho_1 &= \varepsilon_0\sigma^{-1} \end{aligned}}, \
\begin{aligned} \rho'_0 &= \sigma\varepsilon_0 \\ \rho'_1 &= \varepsilon_0\sigma \end{aligned}
\qquad
\boxed{\begin{aligned} &[\sigma^2, \varepsilon_0] = 1 \\ &(\sigma^{-1}\varepsilon_0)^3 = \sigma^{-2} \end{aligned}}
\]

\[
\varepsilon_0 = \sigma\rho_0, \ \varepsilon_1 = \rho_0\sigma, \quad
\rho'_0 = \rho_0^{-2}, \quad
\rho_1 = \sigma(\rho_0) = \sigma\rho_0\sigma^{-1},
\]

\[
\rho'_1 = \sigma(\rho_0^{-2}) = \sigma\rho_0^{-2}\sigma^{-1}
\qquad
\boxed{\rho_0^{-3} = \underbrace{\sigma^{2}}_{\omega}}
\]

\[
\varepsilon_0 = \sigma^{-1}\rho'_0, \ \varepsilon_1 = \rho'_0\sigma^{-1}, \quad
\rho_0 = \sigma^{-2}\rho'_0, \quad
\rho_1 = \sigma^{-1}\rho'_0\sigma^{-1} =, \quad
\rho'_1 = \sigma\rho'_0\sigma^{-1}
\qquad
\boxed{\begin{aligned} &[\sigma^2, \rho'_0] = 1 \ \text{et} \\ &\rho_0'^{\,3} = \sigma^4 \end{aligned}}
\]

\[
\varepsilon_1 = \rho_0(\varepsilon_0) = \rho_0\varepsilon_0\rho_0^{-1}, \quad
\sigma = \varepsilon_0\rho_0^{-1}, \quad
\rho'_0 = \rho_0^{-2},
\]

\[
\rho_1 = \varepsilon_0(\rho_0) = \varepsilon_0\rho_0\varepsilon_0^{-1}, \quad
\rho'_1 = \varepsilon_0(\rho_0^{-2}) = \varepsilon_0\rho_0^{-2}\varepsilon_0^{-1}
\]

\[
\boxed{\rho_0\varepsilon_0\rho_0^{-1}\varepsilon_0\rho_0 = 1}
\]

\[
\varepsilon_1 = \rho'_0(\varepsilon_0) = \rho'_0\varepsilon_0\rho_0'^{\,-1}, \quad
\sigma = \rho'_0\varepsilon_0^{-1}, \quad
\rho_0 = \varepsilon_0\rho_0'^{\,-1}\varepsilon_0, \quad
\rho_1 = \varepsilon_0^2\rho_0'^{\,-1}, \quad
\rho'_1 = \varepsilon_0\rho'_0\varepsilon_0^{-1}
\]

\[
\boxed{\begin{aligned}
&\rho'_0 = (\rho'_0(\varepsilon_0)\varepsilon_0)^2 \\
&\quad \text{i.e.} \ \rho'_0 = (\rho'_0\varepsilon_0\rho_0'^{\,-1}\varepsilon_0)^2 \\
&\text{et} \ [\rho_0'^{\,3}, \varepsilon_0] = 1
\end{aligned}}
\]

\[
\underbrace{\rho'_0\varepsilon_0\rho_0'^{\,-1}\varepsilon_0\rho'_0}\varepsilon_0\underbrace{\rho_0'^{\,-1}\varepsilon_0^{-1}\rho'_0\varepsilon_0^{-1}\rho_0'^{\,-1}}\varepsilon_0^{-1} = 1
\]

\[
[\underbrace{\rho'_0\varepsilon_0\rho_0'^{\,-1}\varepsilon_0\rho'_0}, \varepsilon_0] = 1
\]

\[
\varepsilon_1 = \rho_1^{-1}(\varepsilon_0) = \rho_1^{-1}\varepsilon_0\rho_1, \quad
\sigma = \rho_1^{-1}\varepsilon_0, \quad
\rho_0 = \varepsilon_0^{-1}(\rho_1) = \varepsilon_0^{-1}\rho_1\varepsilon_0,
\]

\[
\rho'_0 = \text{\struck{\ill{}}} = \varepsilon_0^{-1}(\rho_1^{-2}) = \varepsilon_0^{-1}\rho_1^{-2}\varepsilon_0, \quad
\rho'_1 = \rho_1^{-2}
\qquad
\boxed{\rho_1\varepsilon_0\rho_1^{-1}\varepsilon_0\rho_1 = 1}
\]

\[
\varepsilon_1 = \rho_1'^{\,-1}(\varepsilon_0) = \rho_1'^{\,-1}\varepsilon_0\rho'_1, \quad
\sigma = \varepsilon_0^{-1}\rho'_1, \quad
\rho_1 = \varepsilon_0\rho_1'^{\,-1}\varepsilon_0,
\]

\[
\rho'_0 = \varepsilon_0^{-1}(\rho'_1) = \varepsilon_0^{-1}\rho'_1\varepsilon_0, \quad
\rho_0 = \rho_1'^{\,-1}\varepsilon_0^2
\]

\[
\boxed{\begin{aligned}
&\rho'_1 = (\rho'_1(\varepsilon_0)\varepsilon_0)^2 \\
&\quad \text{i.e.} \ \rho'_1 = (\rho'_1\varepsilon_0\rho_1'^{\,-1}\varepsilon_0)^2 \\
&\text{et} \ [\rho_1'^{\,3}, \varepsilon_0] = 1
\end{aligned}}
\]

\[
\begin{aligned}
&1^\circ)\ g \mapsto \check{g} \ (\text{antiaut.}) &\ \Big|\ & \varepsilon_0 \mapsto \varepsilon_0^{-1}, \ \varepsilon_1 \mapsto \varepsilon_1^{-1}, \quad \sigma \mapsto \sigma^{-1}, \\
& & & \rho_0 \mapsto \rho_1^{-1}, \ \rho_1 \mapsto \rho_0^{-1}, \quad \rho'_0 \mapsto \rho_1'^{\,-1}, \ \rho'_1 \mapsto \rho_0'^{\,-1} \\
&2^\circ)\ g \mapsto g^{*} = \operatorname{int}(\sigma) &\ \Big|\ & \varepsilon_0 \mapsto \varepsilon_1, \ \varepsilon_1 \mapsto \varepsilon_0, \quad \sigma \mapsto \sigma, \\
& & & \rho_0 \mapsto \rho_1, \ \rho_1 \mapsto \rho_0, \quad \rho'_0 \mapsto \rho'_1, \ \rho'_1 \mapsto \rho'_0
\end{aligned}
\]

\[
\varepsilon_0, \quad \varepsilon_1 = \rho(\varepsilon_0), \quad \varepsilon_\infty = \rho(\varepsilon_1) = \rho^2(\varepsilon_0)
\]

\[
\begin{aligned}
\sigma_1 &= \varepsilon_\infty\varepsilon_0\varepsilon_\infty = \varepsilon_0\varepsilon_\infty\varepsilon_0, \qquad
\sigma_0 = \varepsilon_1\varepsilon_\infty\varepsilon_1 = \varepsilon_\infty\varepsilon_1\varepsilon_\infty, \\
\sigma_\infty &= \varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1 \ (= \sigma)
\end{aligned}
\]

\[
\rho = (\varepsilon_1\varepsilon_0)^{-1} = (\varepsilon_\infty\varepsilon_1)^{-1} = (\varepsilon_0\varepsilon_\infty)^{-1}
\]

\[
\rho_1 = (\varepsilon_0\varepsilon_1)^{-1} = \sigma_\infty(\rho), \quad
(\varepsilon_1\varepsilon_\infty)^{-1} = \sigma_0(\rho) = \rho(\rho_1), \quad
(\varepsilon_\infty\varepsilon_0)^{-1} = \sigma_1(\rho) = \rho^2(\rho_1)
\]

\[
\varepsilon_0, \ \varepsilon_1, \ \sigma, \ \rho \ (= \rho_0)
\qquad
\left|\
\begin{aligned}
&(\varepsilon_0, \varepsilon_1) \\
&(\varepsilon_0, \sigma) \leftrightarrow (\varepsilon_1, \sigma) \\
&(\varepsilon_0, \rho) \leftrightarrow (\varepsilon_1, \rho) \\
&(\sigma, \rho)
\end{aligned}
\right.
\]

\[
\begin{aligned}
&(1)\ (\varepsilon_0, \varepsilon_1), &&\boxed{\varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1}
&&\begin{cases} \sigma = \varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1 \\ \rho = (\varepsilon_1\varepsilon_0)^{-1} \end{cases} \\
&(2)\ (\varepsilon_0, \sigma), &&\boxed{[\sigma^2, \varepsilon_0] = 1, \ (\sigma^{-1}\varepsilon_0)^3 = \sigma^{-2}}
&&\begin{cases} \varepsilon_1 = \sigma(\varepsilon_0) = \sigma\varepsilon_0\sigma^{-1} \\ \rho = \sigma^{-1}\varepsilon_0 \end{cases} \\
&(3)\ (\varepsilon_0, \rho) &&\boxed{\rho\varepsilon_0\rho^{-1}\varepsilon_0\rho = 1}
&&\begin{cases} \varepsilon_1 = \rho(\varepsilon_0) = \rho\varepsilon_0\rho^{-1} \\ \sigma = \varepsilon_0\rho^{-1} \end{cases} \\
&(4)\ (\sigma, \rho) &&\boxed{\rho^{-3} = \sigma^2}
&&\begin{cases} \varepsilon_0 = \sigma\rho \\ \varepsilon_1 = \rho\sigma \end{cases}
\end{aligned}
\]

\[
\left[\varepsilon_0, \varepsilon_1, \sigma, \rho\right]
\quad
\begin{array}{|l|l|}
\hline
\begin{aligned}
&\varepsilon_0 = \sigma\rho, \quad \sigma = \varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1 \\
&\varepsilon_1 = \rho\sigma, \quad \rho = (\varepsilon_1\varepsilon_0)^{-1} = \varepsilon_0^{-1}\varepsilon_1^{-1} \\
&\sigma(\varepsilon_0) = \rho(\varepsilon_0) = \varepsilon_1 \\
&\sigma(\varepsilon_1) = \rho^{-1}(\varepsilon_1) = \varepsilon_0
\end{aligned}
&
\begin{aligned}
&\varepsilon_0\varepsilon_1\varepsilon_0 = \varepsilon_1\varepsilon_0\varepsilon_1 \\
&\rho^{-3} = \sigma^2 \ (\overset{\mathrm{def}}{=} \omega) \ \text{central} \\
&\rho\varepsilon_0\rho^{-1}\varepsilon_0\rho = 1
\end{aligned}
\\ \hline
\end{array}
\]

\[
  P\mathcal{D} \simeq \{z_{1}, z_{2} \in \mathbb{C} \mid z_{2}/z_{1} \in \mathcal{D}\} \simeq \mathrm{Hom}^{+}(\mathbb{Z}^{2}, \mathbb{C})
\]

\[
  \mathcal{D} \simeq \widetilde{M}_{1,1} \qquad \mathcal{D} = \{z \in \mathbb{C} \mid \Im z > 0\}
\]

\[
  \mathbb{Z}^{2} \to \qquad
  \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}
\]

\[
  \begin{pmatrix} a & b \\ c & d \end{pmatrix}(z) = \text{\struck{$\frac{dz + c}{bz + a}$}}\ \frac{az + b}{cz + d}
\]

\[
  \begin{pmatrix} a & b \\ c & d \end{pmatrix}(z) = \frac{az + b}{cz + d}
\]

\begin{align*}
  M_{1,1}^{\mathrm{an}} &\simeq (\mathcal{D}, \mathrm{Sl}(2, \mathbb{Z})) \\
  PM_{11}^{\mathrm{an}} &\simeq (P\mathcal{D}, \mathrm{Sl}(2, \mathbb{Z}))
\end{align*}

\begin{align*}
  M_{11}^{\mathrm{an}} &\simeq (\mathcal{D}, \mathrm{Gl}(2, \mathbb{Z})) \\
  PM_{11}^{\mathrm{an}} &\simeq (P\mathcal{D}, \mathrm{Gl}(2, \mathbb{Z}))
\end{align*}

\begin{align*}
  T_{11} &\simeq \mathrm{Sl}(2, \mathbb{Z}) \\
  ST_{11} &\simeq \widetilde{\mathrm{Sl}}(2, \mathbb{Z})
\end{align*}

\[
  (z_{1}, z_{2}) \mapsto (a\tau^{\alpha}z_{1} + b\tau^{\alpha}z_{2},\ c\tau^{\alpha}z_{1} + d\tau^{\alpha}z_{2})
\]

\[
  u_{g} = \tau^{\alpha(g)} \cdot g \qquad
  \begin{cases}
    \alpha(g) \in \mathbb{Z}/2\mathbb{Z} \\
    \alpha(g) = 0 & \text{si } \det g = 1 \\
    \phantom{\alpha(g)} = 1 & \text{si } \det g = -1
  \end{cases}
\]

\begin{align*}
  \mathcal{D} \times \mathbb{C}^{*} &\to P\mathcal{D} \\
  (\tau, \lambda) &\mapsto \lambda(\tau, 1) = (\lambda\tau, \lambda) \overset{\text{déf}}{=} [\tau, \lambda]
\end{align*}

\[
  \begin{cases}
    \lambda'[\tau, \lambda] = [\tau, \lambda'\lambda] \\
    \begin{pmatrix} a & b \\ c & d \end{pmatrix}([\tau, \lambda]) = \left[\dfrac{a\tau + b}{c\tau + d},\ \lambda(c\tau + d)\right]
  \end{cases}
\]

\begin{align*}
  \widetilde{PM}_{11}^{\mathrm{an}} \simeq \mathcal{D} \times \mathbb{C} &\to P\mathcal{D} \\
  (\tau, z) &\mapsto (\exp 2\pi i z)(\tau, 1) = [\tau, \exp 2i\pi z]
\end{align*}

\[
  \overset{\omega^{2}}{\mathbb{Z}} \subset \widetilde{\mathrm{Sl}}(2, \mathbb{Z})
\]

\[
  (\tau, z) \mapsto (\tau, z + 1)
\]

\[
  cz^{2} + (d - a)z - b = 0 \qquad \Delta = (d - a)^{2} + 4bc = (a + d)^{2} - 4 \quad \text{car } bc = ad - 1
\]

\[
  z \mapsto \frac{az + b}{cz - a} \qquad -a^{2} - bc = 1, \quad bc = -1 - a^{2}
\]

\[
  cz^{2} - 2az - b = 0 \qquad z = \frac{a \pm 2i}{2c} \qquad \text{On trouve (4)}
\]

\[
  \rho = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}, \qquad z = \frac{1}{2} + i\frac{\sqrt{3}}{2} = \exp\frac{2i\pi}{6} = -j
\]

\[
  \rho = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix} \qquad \rho^{3} = \omega = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
\]

\[
  \sigma = \begin{pmatrix} 0 & +1 \\ -1 & 0 \end{pmatrix} \qquad \sigma^{2} = \omega, \quad \sigma^{-1} = \omega\sigma
\]

\[
  (X, G) = \Pi_{1}(X, G)
\]

\[
  \mathop{\mathrm{Hom}}_{(X,G)}(x, y) = \{(\ell, g) \mid \ell : gx \to y,\ g \in G\}
\]

\[
  (\ell', g')(\ell, g) = (\ell' \circ g'(\ell),\ g'g)
\]

\[
  (1)\qquad \text{Cartes isot.} \xrightarrow{\ \approx\ } \mathfrak{G}_{2}\text{-ensembles finis spéciaux}
\]

\[
  (2)\qquad \mathfrak{G}_{2} \simeq \{\tau_{0}, \tau_{1}, \tau_{2} \mid \tau_{0}^{2} = \tau_{1}^{2} = \tau_{2}^{2} = (\tau_{0}\tau_{2})^{2} = 1\}
\]

\[
  \tau_{0}, \tau_{1}, \tau_{2} \quad \text{et} \quad \sigma = \tau_{0}\tau_{2}
\]

\[
  (3)\qquad \mathfrak{G}_{2}^{0} = \mathrm{Ker}\,(\mathfrak{G}_{2} \to \mathbb{Z}/2\mathbb{Z}), \quad \tau_{i} \mapsto 1 \bmod 2
\]

\[
  (4)\qquad \mathfrak{G}_{2}^{0} = \{\rho_{0}, \rho_{1}, \rho_{\infty} \mid \rho_{\infty}\rho_{1}\rho_{0} = \rho_{1}^{2} = 1\}
\]

\[
  (5)\qquad \rho_{\infty} = \tau_{1}\tau_{0}, \quad \rho_{1} = \tau_{0}\tau_{\infty}, \quad \rho_{0} = \tau_{\infty}\tau_{1}
\]

\[
  (5)\qquad \mathcal{R}(X) = \mathcal{R}^{+}(X) \amalg \mathcal{R}^{-}(X)
\]

\[
  (6)\qquad \sigma_{i}(\mathcal{R}^{+}(X)) \subset \mathcal{R}^{-}(X) \quad \text{pour } i \in \{0, 1, \infty\}
\]

\begin{align*}
  (7)&\qquad \sigma_{i}(\mathcal{R}^{+}(X)) = \mathcal{R}^{+}(X), \quad \sigma_{i}(\mathcal{R}^{-}(X)) = \mathcal{R}^{-}(X) \quad i \in \{0, 1, \infty\} \\
  (7\text{ bis})&\qquad \rho_{i}(\mathcal{R}^{+}(X)) = \mathcal{R}^{+}(X), \quad \rho_{i}(\mathcal{R}^{-}(X)) = \mathcal{R}^{-}(X) \quad i \in \{0, 1, \infty\}
\end{align*}

\[
  (8)\qquad \text{Cartes isot.\ or.} \xrightarrow{\ \approx\ } \mathfrak{G}_{2}^{0}\text{-ens.\ spéciaux finis}
\]

\begin{align*}
  & r \qquad \tau_{0}r, \tau_{1}r, \tau_{\infty}r \qquad \tau_{1}\tau_{0}r = \rho_{\infty}r, \ \tau_{0}\tau_{1}r = \rho_{\infty}^{-1}r \\
  & \phantom{r \qquad \tau_{0}r, \tau_{1}r, \tau_{\infty}r \qquad} \tau_{0}\tau_{\infty}r = \rho_{1}r = \tau_{\infty}\tau_{0}r = \rho_{1}^{-1}r \\
  & \phantom{r \qquad \tau_{0}r, \tau_{1}r, \tau_{\infty}r \qquad} \tau_{\infty}\tau_{1}r = \rho_{0}r, = \tau_{1}\tau_{\infty}r = \rho_{0}^{-1}r
\end{align*}

\[
  \tau_{0}\tau_{\infty} = \tau_{\infty}\tau_{0} \quad \text{i.e.} \quad \rho_{1} = \rho_{1}^{-1} \quad \text{i.e.} \quad \rho_{1}^{2} = 1 .
\]

\[
  (9)\qquad \mathcal{R}^{+}(\check C) = \mathcal{R}^{-}(C), \quad \mathcal{R}^{-}(\check C) = \mathcal{R}^{+}(C)
\]

\[
  (10)\qquad
  \begin{cases}
    r' = \tau_{\infty}r = \tau_{0}r \\
    \text{i.e.}\quad \tau_{0}\tau_{\infty}r = r \quad \text{i.e.}\quad \rho_{1}r = r
  \end{cases}
\]

\[
  (11)\qquad \tau_{1}r = \tau_{\infty}r \quad \text{i.e.} \quad \rho_{0}r = r
\]

\[
  \tau_{1}r = \tau_{\infty}r = \tau_{0}r
\]

\[
  \rho_{1}^{2} = 1 ,
\]

\[
  (12)\qquad \tilde\pi_{03} = \{\tau_{0}, \tau_{1}, \tau_{\infty} \mid \tau_{0}^{2} = \tau_{1}^{2} = \tau_{\infty}^{2} = 1\}
\]

\[
  (13)\qquad \pi_{03} = \{\rho_{0}, \rho_{1}, \rho_{\infty} \mid \rho_{\infty}\rho_{1}\rho_{0} = 1\}
\]

\[
  (14)\qquad \rho_{\infty} = \tau_{1}\tau_{0}, \quad \rho_{1} = \tau_{0}\tau_{\infty}, \quad \rho_{0} = \tau_{\infty}\tau_{1} .
\]

\[
  (15)\qquad \mathfrak{G}_{2} \simeq \tilde\pi_{03}/\langle\rho_{1}^{2}\rangle, \qquad \mathfrak{G}_{2}^{0} \simeq \pi_{03}/\langle\rho_{1}^{2}\rangle
\]

\[
  (16)\qquad S \longrightarrow \{0, 1, \infty\}
\]

\[
  (17)\qquad \text{cartes isot.\ triangulaires pondérées} \xrightarrow{\ \approx\ } \tilde\pi_{03}\text{-ensembles finis spéciaux}
\]

\[
  \mathcal{R}^{-}(X) = \mathcal{R}(X) \setminus \mathcal{R}^{+}(X) .
\]

\[
  (18)\qquad \text{Cartes (isotop.) triangulaires pondérées orientées} \xrightarrow{\ \approx\ } (\pi_{03}\text{-Ens}) .
\]

\[
  g \in \mathfrak{S}_{3} \overset{\text{déf}}{=} \mathfrak{S}_{\{0, 1, \infty\}}
\]

\[
  (18)\qquad \mathcal{R}^{+}(X^{g}) = \mathcal{R}^{\mathrm{sg}(g)}(X) \qquad g \in \mathfrak{S}_{3}
\]

\begin{align*}
  \mathfrak{G}_{2} &\simeq \tilde\pi_{03}/\langle\rho_{1}^{2}\rangle \\
  \mathfrak{G}_{2}^{+} &\simeq \pi_{03}/\langle\rho_{1}^{2}\rangle ,
\end{align*}

\[
  (19)\qquad \pi_{03} \simeq \pi_{1}(U_{0,3}, P_{+}^{*})
\]

\[
  (20)\qquad U_{0,3} = \mathbb{P}^{1}(\mathbb{C}) \setminus \{0, 1, \infty\}
\]

\[
  P_{+} = -\zeta, \qquad \zeta = \exp 2i\pi/3 .
\]

\[
  \rho_1^2 = 1 \;) ,
\]

\[
  \text{(21)} \qquad S_0 = \{0\} \subset K_\infty = [0, 1]
\]

\[
  \text{(22)} \qquad
  \left\{
  \begin{array}{l}
    S_1 = \{1\}, \quad S_\infty = \{\infty\}, \quad
    K = \mathbb{P}^1(\mathbb{R}) = K_0 \cup K_1 \cup K_\infty \\
    \text{où} \quad K_0 = [1, \infty], \quad K_1 = [\infty, 0], \quad
    \underbrace{K_\infty = [0, 1]}_{\text{comme avant, cf.\ (21)}}
  \end{array}
  \right.
\]

\[
  g : \mathbb{D} \xrightarrow{\ \sim\ } f \qquad
  \begin{array}{l}
    0 \longmapsto \infty \quad (\text{centre de } f) \\
    1 \longmapsto 0' \quad (\text{unique sommet de } f)
  \end{array}
\]

\[
  \left\{
  \begin{array}{l}
    g(-1) = 1' \\
    g(i) = (\tfrac12)'_+ \\
    g(-i) = (\tfrac12)'_-
  \end{array}
  \right.
\]

\[
  g : \quad
  \begin{array}{l}
    1 \longmapsto 0 \\
    -1 \longmapsto 1 \\
    0, \infty \longmapsto \infty
  \end{array}
\]

\[
  g' : \quad
  \begin{array}{l}
    0 \longmapsto 0 \\
    \infty \longmapsto \infty \\
    1, -1 \longmapsto 1
  \end{array}
\]

\[
  g(z) = \varphi\, g'(\psi(z))
\]

\[
  \psi : \left\{
  \begin{array}{l} 1 \\ -1 \\ 0, \infty \end{array}
  \right.
  \longrightarrow
  \left\{
  \begin{array}{l} 0 \\ \infty \\ 1, -1 \end{array}
  \right.
  \qquad
  \varphi : \left\{
  \begin{array}{ll} 0 & 0 \\ \infty & 1 \\ 1 & \infty \end{array}
  \right.
\]

\[
  \psi(z) = \frac{z-1}{z+1} \qquad
  \varphi(z) = \sigma_0(z) = \frac{z}{z-1}
\]

\[
  g(z) = \frac{\left(\frac{z-1}{z+1}\right)^2}
              {\left(\frac{z-1}{z+1}\right)^2 + 1}
       = \frac{(z-1)^2}{(z-1)^2 - (z+1)^2}
       = \frac{(z-1)^2}{-4z}
\]

\[
  \text{(23)} \qquad g(z) = -\frac{1}{4z}\,(z-1)^2
\]

\[
  \text{(24)} \qquad \Pi_{0,3}^\tau \simeq
  \Pi_1(U_{0,3}, \{1, \tau\}; P_+)
\]

\[
  \tau_i = (\underbrace{\lambda_i}_{\tau P_+ = P_- \to P_+},
  \underbrace{\tau}_{\{1, \tau\}})
\]

\[
  \text{(25)} \qquad U'^\tau = \emptyset \quad \text{i.e.} \quad
  X'^\tau = \emptyset
\]

\[
  \text{\struck{$U$}} \ U_{0,3\,\mathbb{R}} = \Sigma^*_{\mathbb{R}}
  = \Sigma_{\mathbb{R}} \smallsetminus \{0, 1, \infty\} .
\]

\[
  \text{(26)} \quad
  \begin{array}{c}
    \text{cartes triangulées pondérées \textit{finies}} \\
    \text{de type } \{0, 1, \infty\}
  \end{array}
  \hookrightarrow
  \begin{array}{l}
    \text{revêtements étales \textit{finis}} \\
    \text{du schéma } \mathbb{P}^1_{\mathbb{R}} \smallsetminus \{0,1,\infty\} \\
    (\simeq \text{rev.\ ramifiés de } \mathbb{P}^1_{\mathbb{R}}, \\
    \ \text{étales au-dessus de} \\
    \ U^*_{03\,\mathbb{R}} = \mathbb{P}^1_{\mathbb{R}} \smallsetminus \{0,1,\infty\})
  \end{array}
\]

\[
  \text{(27)} \qquad X'(\mathbb{R}) = \emptyset
\]

\[
  X'(\mathbb{R}) = \underbrace{X'(\mathbb{C})^\tau}_{\text{\struck{\ill{}} $X'$ dans (25)}} .
\]

\[
  \text{(28)} \qquad X = X'(\mathbb{C}) / \tau
\]

\[
  \text{(29)} \qquad \rho_i^{\nu_i} = 1
  \qquad \text{\struck{type}} \ \text{où} \quad
  \nu_i \in \overline{\mathbb{N}^*},
\]

\[
  \text{(30)} \qquad \rho_0^{\nu_0} = 1 \qquad
  (\text{resp.}\ \rho_\infty^{\nu_\infty} = 1)
\]

\[
  \text{(31)} \qquad \rho_1^{\nu_1} = 1 \qquad )
\]

\[
  \rho_1 = 1
\]

\[
  \text{(34)} \qquad \nu_i \in \overline{\mathbb{N}^*} = \mathbb{N}^* \cup \{+\infty\} .
\]

\[
  \text{(32)} \qquad \Pi_{0,3} / \langle \rho_0^{\nu_0}, \rho_1^{\nu_1},
  \rho_\infty^{\nu_\infty} \rangle \qquad (\text{cas orienté})
\]

\[
  \text{(32 bis)} \qquad \Pi_{0,3}^\tau / \langle \rho_0^{\nu_0}, \rho_1^{\nu_1},
  \rho_\infty^{\nu_\infty} \rangle \qquad (\text{cas non orienté})
\]

\[
  \text{(33)} \qquad \Pi_{0,3} / \langle (\rho_i^{\nu_i}) \rangle \simeq
  \Pi_1(\mathbb{P}^1(\mathbb{C}), \nu_*; P_+)
\]

\[
  \nu_* = (\nu_0 R_0 + \nu_1 R_1 + \nu_\infty R_\infty) \qquad
  \text{singularités sur } \mathbb{P}^1_{\mathbb{C}}
\]

\[
  \text{(34)} \qquad \mathcal{X}(\mathbb{P}^1_{\mathbb{C}}, \nu_*) =
  \text{\struck{catégorie}}\ \text{topos des \struck{espaces} \add{surfaces}
  topologiques}
\]

\[
  \text{(35)} \qquad \Pi_{03} / \langle (\rho_i^{\nu_i}) \rangle \simeq
  \Pi_1(\mathbb{P}^1(\mathbb{C}), \nu_*; P_+) \overset{\text{déf}}{=}
  \Pi_1(\mathcal{X}(\mathbb{P}^1(\mathbb{C}), \nu_*), P_+) ,
\]

\[
  \begin{aligned}
  \text{(36)} \qquad \Pi_{0,3}^\tau / \langle (\rho_i^{\nu_i}) \rangle
  &\simeq \Pi_1(\mathbb{P}^1(\mathbb{C}), \{1, \tau\}, \nu_*; P_+) \\
  &\overset{\text{déf}}{=}
  \Pi_1(\mathcal{X}(\mathbb{P}^1(\mathbb{C}), \nu_*, \{1, \tau\}), P_+)
  \end{aligned}
\]

\[
  \text{(37)} \qquad \mathcal{X}(\mathbb{P}^1(\mathbb{C}), \nu_*, \{1, \tau\})
  = (\mathcal{X}(\mathbb{P}^1(\mathbb{C}), \nu_*), \{1, \tau\})
\]

\[
  \begin{aligned}
  \text{(38)} \qquad \widehat{\Pi}_{0,3} / \langle (\rho_i^{\nu_i}) \rangle
  &\simeq \Pi_1(\underbrace{\mathbb{P}^1_{\mathbb{C}}}_{\text{schéma}}, \nu_*; P_+) \\
  &\simeq \Pi_1(\mathbb{P}^1_{\overline{\mathbb{Q}}}, \nu_*; P_+)
  \end{aligned}
\]

\[
  \begin{aligned}
  \text{(39)} \qquad \widehat{\Pi}_{0,3}^\tau / \langle (\rho_i^{\nu_i}) \rangle
  &\simeq \Pi_1(\underbrace{\mathbb{P}^1_{\mathbb{R}}}_{\text{schéma}}, \nu_*; P_+) \\
  &\simeq \Pi_1(\mathbb{P}^1_{\overline{\mathbb{Q}} \cap \mathbb{R}}, \nu_*; P_+)
  \end{aligned}
\]

\[
  \Pi_{0,3}^\tau / \langle \rho_1^2 \rangle \quad \text{et} \quad
  \Pi_{0,3} / \langle \rho_1^2 \rangle ,
\]

\[
  \Pi_{0,3}^\tau / \langle \rho_1^2, \rho_\infty^3 \rangle \quad \text{et} \quad
  \Pi_{0,3} / \langle \rho_1^2, \rho_\infty^3 \rangle ,
\]

\[
  \underbrace{\Pi_{0,3}^\tau / \langle \rho_1^2, \rho_\infty^3 \rangle}
    _{\text{groupes cartographiques}}
  \xleftarrow{\ \sim\ }
  \underbrace{\Pi_1(U_{0,3}, \mathfrak{S}_3 \times \{1, \tau\}; P_+)}
    _{\Pi_{0,3}^{\mathbb{D}_3, \tau}}
\]

\[
  \underset{\text{triangulés, non or.\ resp.\ orientés}}
    {\Pi_{0,3} / \langle \rho_1^2, \rho_\infty^3 \rangle}
  \xleftrightarrow{\ \sim\ }
  \overbrace{\Pi_1(U_{0,3}, \mathfrak{S}_3; P_+)}^{\Pi_{0,3}^{\mathbb{D}_3}}
\]

\[
  \text{(41)} \qquad
  \boxed{\rho \longmapsto \rho_\infty^{-1}, \quad \sigma_\infty \longmapsto \rho_1}
  , \quad
  \underset{\sigma_\infty \rho}{\varepsilon_0} \longmapsto \rho_0
\]

\[
  \text{(42)} \qquad
  \boxed{\tau_\infty \longmapsto \tau_\infty}, \quad
  \underset{\rho(\tau_\infty)}{\tau_0} \longmapsto
  \overbrace{\underbrace{\rho_\infty^{-1} \tau_\infty \rho_\infty}
    _{(\rho_\infty^{-1}\rho_1\rho_\infty^{-1}\rho_1)\tau_\infty}}
    ^{\tau_0\tau_1\tau_\infty\tau_1\tau_0} ,
\]

\[
  \underset{\rho^2(\tau_\infty)}{\tau_1} \longmapsto
  \overbrace{\rho_\infty^{-2} \tau_\infty \rho_\infty^{2}}
    ^{\tau_1\tau_0\tau_\infty\tau_0\tau_1}
  = \rho_\infty \tau_\infty \rho_\infty^{-1}
  \quad (\text{car } \rho_\infty^3 = 1 \text{ dans }
  \Pi_{03}^\tau / \langle \cdots \rangle)
\]

\[
  \rho, \sigma_i, \varepsilon_i \ (i \in \{0, 1, \infty\}) \quad \text{dans} \quad
  \Pi_1(U_{0,3}, \mathfrak{S}_3; P_+) = \Pi_{03}^{\mathbb{D}_3}
\]

\[
  \tau_i \ (i \in \{0, 1, \infty\}) \in
  \underbrace{\Pi_1(U_{0,3}, \{1, \tau\}, P_+)}_{\Pi_{0,3}^\tau}
  \subset
  \underbrace{\Pi_1(U_{0,3}, \mathfrak{S}_3 \times \{1, \tau\}; P_+)}
    _{\Pi_{0,3}^{\mathbb{D}_3, \tau}}
\]

\[
  \begin{aligned}
  \text{(43)} \qquad \mathrm{Gl}(2, \mathbb{Z})/\pm 1
  &\xrightarrow{\ \sim\ } \Pi_{0,3}^{\mathbb{D}_3, \tau}
  \quad (\xleftarrow{\ \sim\ } \Pi_{0,3}^\tau / \langle \rho_1^2, \rho_\infty^3 \rangle) \\
  \text{(44)} \qquad \mathrm{Sl}(2, \mathbb{Z})/\pm 1
  &\xrightarrow{\ \sim\ } \Pi_{0,3}^{\mathbb{D}_3}
  \quad (\xrightarrow{\ \sim\ } \Pi_{0,3} / \langle \rho_1^2, \rho_\infty^3 \rangle)
  \end{aligned}
\]

\[
  \text{(*)} \qquad X'^{+} \subset X'
\]

\[
  \text{(43)} \qquad \mathrm{cl}(X) \in H^1(X, \mathbb{Z}/3\mathbb{Z})
\]

\[
  \text{(44)} \qquad P = (S, A, R \subset S \times A)
\]

\[
  \text{(45)} \qquad \mathbb{D}_P = \mathbb{D}
\]

\[
  \text{(46)} \qquad 1 \longrightarrow \mathbb{D}^{+} \longrightarrow \mathbb{D}
  \longrightarrow \{\pm 1\} \longrightarrow 1 .
\]

\[
  \text{(47)} \qquad \underset{\{\omega, \omega'\}}{\underline{\omega}}
  \subset \mathbb{D}^{+}
\]

\[
  \text{(48)} \qquad S = S_P \hookrightarrow \Sigma = \Sigma_P
\]

\[
  \text{(49)} \qquad \pi_0(\Sigma \smallsetminus \Sigma_{\mathbb{R}}) \simeq
  \underline{\omega} .
\]

\[
  \Sigma^{\mathbb{D}} = \Sigma^{\mathbb{D}^+} = \Sigma^{\underline{\omega}}
  = \Sigma^{\{\omega\}} \subset \Sigma \smallsetminus \Sigma_{\mathbb{R}}
  \qquad (\omega \in \underline{\omega})
\]

\[
  \text{(50)} \qquad \Sigma^{\mathbb{D}} \simeq \underline{\omega} ,
\]

\[
  \Pi_1(\underbrace{\Sigma_P \smallsetminus S_P}_{\Sigma_P^{*} \text{ ou } \Sigma^{*}}) ,
\]

\[
  \text{(51)} \qquad
  \underset{\text{ou } \Pi_P \text{ ou } \Pi}{\Pi_{P,\omega}}
  = \Pi_1(\Sigma_P^{*}, P_\omega) .
\]

\[
  \text{(52)} \qquad \mathbb{D}_P \times \{1, \tau\} \qquad \text{d'ordre } 4n
\]

\[
  \text{(53)} \qquad
  \left\{
  \begin{aligned}
    \Pi_{P,\omega}^{\tau} &= \Pi_1(\Sigma^{*}, \{1, \tau\}; P_\omega) \\
    \Pi_{P,\omega}^{\mathbb{D}} &= \Pi_1(\Sigma^{*}, \mathbb{D}; P_\omega) \\
    \Pi_{P,\omega}^{\mathbb{D},\tau} &= \Pi_1(\Sigma^{*}, \mathbb{D} \times \{1, \tau\}; P_\omega)
  \end{aligned}
  \right.
\]

\[
  \text{(55)} \qquad \lambda_a : P_{\omega'} \longrightarrow P_\omega
  \quad \text{arc de demi-grand cercle passant par } Q_a
\]

\[
  \left\{
  \begin{array}{l}
    a(s, \omega) \text{ ou } a(s) \in A \text{ tel que } (s, a) \text{ soit un repère} \\
    \text{d'orientation } \omega, \text{ i.e.\ } s = \mathrm{or}_\omega(a)
  \end{array}
  \right.
\]

\[
  \text{(56)} \qquad \rho_s = \lambda_a \lambda_{\omega' a}^{-1} ,
  \qquad \text{où } a = a(s, \omega) ,\ \omega' = \omega^{-1}
\]

\[
  \text{donc} \quad \omega'(a) = a(\omega'(s), \omega) ,
\]

\[
  \underset{\Pi_P = \Pi_1(\Sigma_P^{*}, P_\omega)}{\cap}
\]

\[
  \text{(57)} \qquad \rho_{\omega^{n-1}s}\, \rho_{\omega^{n-2}s} \cdots
  \rho_{\omega s}\, \rho_s = 1 .
\]

\[
  \text{(58)} \qquad \rho_i = \rho_{\omega^i s_0}
  \qquad (i \in \mathbb{Z}/n\mathbb{Z})
\]

\[
  \text{(59)}\qquad \rho_{n-1}\rho_{n-2}\cdots\rho_1\rho_0 = 1 ,
\]

\[
  \text{(60)}\qquad \Pi_P \simeq \Pi_{P_0} \simeq
  \{\rho_0,\rho_1,\ldots,\rho_{n-1} \mid \rho_{n-1}\rho_{n-2}\cdots\rho_1\rho_0 = 1\} ,
\]

\[
  \text{(61)}\qquad \tau_a = (\lambda_a, \tau) \in \Pi_P^\tau \qquad (a \in A)
\]

\[
  \text{(62)}\qquad \Pi^\tau \simeq \{(\tau_a)_{a\in A} \mid \tau_a^2 = 1\} ,
\]

\[
  \text{(63)}\qquad \rho_s = \tau_a \tau_{a'} ,
\]

\[
  \text{(64)}\qquad \tau_i = \tau_{\omega^i a_0} \qquad (i \in \mathbb{Z}/n\mathbb{Z})
\]

\[
  \text{(65)}\qquad \rho_i = \tau_i \tau_{i-1}
\]

\[
  \text{(62 bis)}\qquad \Pi^\tau \simeq \{\tau_0,\ldots,\tau_{n-1} \mid \tau_0^2 = \tau_1^2 = \cdots = \tau_{n-1}^2 = 1\} .
\]

\[
  \text{(66)}\qquad \tilde g^{0} = (g, \tau^{\alpha(g)}) \qquad
  \alpha(g) \in \mathbb{Z}/2\mathbb{Z},\quad
  \alpha(g) = \begin{cases} 0 & \text{si } \operatorname{sg}(g) = 1 \\ 1 & \text{si } \operatorname{sg}(g) = -1 \end{cases}
\]

\[
  \text{(67)}\qquad D \xrightarrow{\ \sim\ } (D \times \{1,\tau\})_{P_\omega} \longrightarrow \Pi_P^{D,\tau}
\]

\[
  \text{(68)}\qquad g \longmapsto \tilde g = (1_{P_\omega}, \tilde g^{0})
\]

\[
  \text{(69)}\qquad \rho \overset{\text{déf}}{=} \tilde\omega_\pi \in \Pi_P^D \subset \Pi_P^{D,\tau} \qquad (\rho^n = 1)
\]

\[
  \text{(70)}\qquad \tilde\sigma_a = (\sigma_a^0)^{\sim} \in \Pi_P^{D,\tau} \quad \text{pour } a \in A \qquad (\tilde\sigma_a^2 = 1),
\]

\[
  \text{(71)}\qquad \sigma_a^0 \in D_P
\]

\[
  \text{(72)}\qquad \sigma_a = \tilde\sigma_a \tau_a = \tau_a \tilde\sigma_a = (\lambda_a, \sigma_a^0) \in \Pi_P^D \subset \Pi_P^{D,\tau}
\]

\[
  \text{(73)}\qquad
  \begin{cases}
    \rho(\tau_a) = \tau_{\rho(a)} \\
    \rho(\tilde\sigma_a) = \tilde\sigma_{\rho(a)} \\
    \rho(\sigma_a) = \sigma_{\rho(a)}
  \end{cases}
\]

\[
  \text{(74)}\qquad \varepsilon_s = \sigma_a \rho = (\lambda_a, \sigma_s^0) \in \Pi_P^D
  \qquad s \in S, \text{ où } a = a(s,\omega)
\]

\[
  \text{(75)}\qquad \text{où } \sigma_s^0 \in D_P \text{ est l'unique \ill{} } \neq 1 \text{ de } P \text{ qui fixe le sommet } s \in S
\]

\[
  \text{(76)}\qquad \rho_s = \varepsilon_s^2 = \sigma_a \rho \sigma_a \rho
\]

\[
  \text{(77)}\qquad \Pi_P^D \simeq (\rho, \sigma_{a_0} \mid \rho^n = \sigma_{a_0}^2 = 1) \simeq \mathbb{Z}/n\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}
\]

\[
  \text{(78)}\qquad \rho^n = 1, \quad \sigma_{a_0}^2 = 1 .
\]

\[
  \text{(79)}\qquad
  \begin{cases}
    \rho(\varepsilon_s) = \varepsilon_{\rho(s)} \\
    \rho(\rho_s) = \rho_{\rho(s)}
  \end{cases}
  \qquad (s \in S)
\]

\[
  \text{\struck{(80)}}\qquad
  \text{\struck{$\tilde\sigma_b(\varepsilon_s) = \varepsilon_{\sigma_b^0(s)}^{-1}$}}, \qquad
  \text{\struck{$\tilde\sigma_b(\rho_s) = \rho_{\sigma_b^0(s)}^{-1}$}}, \qquad
  \text{\struck{$\forall s \in S,\ b \in A$}}
\]

\[
  \text{(80)}\qquad
  \begin{aligned}
    &\tau_{a_0}(\rho_0) = \rho_0^{-1}, \quad \tau_{a_0}(\rho_1) = \rho_1^{-1}, \quad \tau_{a_0}(\rho_2) = \rho_1^{-1}(\rho_2^{-1}) \quad \cdots \\
    &\tau_{a_0}(\rho_i) = (\rho_{i-1}\rho_{i-2}\cdots\rho_1)^{-1}(\rho_i^{-1}) \\
    &\phantom{\tau_{a_0}(\rho_i)} = (\rho_{n-1}\cdots\rho_{i+1})(\rho_i^{-1}) \qquad 0 \leq i \leq n-1
  \end{aligned}
\]

\[
  \text{(81)}\qquad
  \begin{cases}
    \tilde g(\varepsilon_s) = (\varepsilon_{g^0(s)})^{\operatorname{sg}(g)} \\
    \tilde g(\rho_s) = (\rho_{g^0(s)})^{\operatorname{sg}(g)}
  \end{cases}
  \qquad (g \in D_P,\ s \in S),
\]

\[
  \text{(82)}\qquad
  \begin{cases}
    \tilde\sigma_a(\varepsilon_s) = \varepsilon_{\sigma_a^0(s)}^{-1} \\
    \tilde\sigma_a(\rho_s) = \rho_{\sigma_a^0(s)}^{-1}
  \end{cases}
  \qquad s \in S,\ a \in A
\]

\[
  \begin{aligned}
    &\sigma_{a_0}(\rho_0) = \rho_1, \quad \sigma_{a_0}(\rho_1) = \rho_0, \quad
    \sigma_{a_0}(\rho_i) = \tau_{a_0}(\rho_{n+1-i}^{-1}) \\
    &\phantom{\sigma_{a_0}(\rho_i)} = (\rho_{i-1}\cdots\rho_{n+3-i})(\rho_{n+1-i}) \qquad \text{pour } 2 \leq i \leq n-1 .
  \end{aligned}
\]

\[
  \text{(84)}\qquad \Pi_{P_\omega}^{D,\tau} \simeq \Pi_P^D \cdot \{1, \tau_{a_0}\}
\]

\[
  \text{(85)}\qquad \tau_{a_0}(\rho) = \rho^{-1}, \quad \tau_{a_0}(\sigma_{a_0}) = \sigma_{a_0} ,
\]

\[
  \tau_{a_0}\underbrace{\rho\,\tau_{a_0}\rho^{-1}}_{\tau_{a_1}} = \tau_{a_0}\tau_{a_1} = \rho_1^{-1}
  = (\rho\sigma_{a_0}\rho\sigma_{a_0})^{-1} = \sigma_{a_0}\rho^{-1}\sigma_{a_0}\rho^{-1}
\]

\[
  \tilde\sigma_{a_0}(\rho) = \rho^{-1}
\]

\[
  \text{(86)}\qquad
  \begin{cases}
    \Pi_P^{D,\tau} \simeq \Pi_P^D \cdot \{1, \tilde\sigma_{a_0}\} \\
    \tilde\sigma_{a_0}(\rho) = \rho^{-1}, \quad \tilde\sigma_{a_0}(\sigma_{a_0}) = \sigma_{a_0}
  \end{cases}
\]

\[
  \Bigl[ = \{\rho, \sigma_{a_0}, \tilde\sigma_{a_0} \mid \rho^n = \sigma_{a_0}^2 = \tilde\sigma_{a_0}^2 = (\sigma_{a_0}\tilde\sigma_{a_0})^2 = \tilde\sigma_{a_0}(\rho)\rho = 1\} \Bigr]
\]

\[
  \text{(87)}\qquad \Pi_P^{D,\tau} \simeq D(\rho) \underset{D_1}{*} D(\sigma_a)
\]

\[
  \text{(88)}\qquad
  \begin{cases}
    D(\rho) = D^+ \cdot \{1, \tilde\sigma_a\} \\
    D(\sigma_a) = \{1, \sigma_a\} \times \{1, \tilde\sigma_a\} \\
    D_1 = \{1, \tilde\sigma_a\}
  \end{cases}
\]

\[
  \text{(89)}\qquad \Pi_P^D \xrightarrow{\ \sim\ } \Pi_{0,3}/\langle \rho_1^2, \rho_\infty^n \rangle
  = \pi_1\bigl(\mathbb{P}^1(\mathbb{C}), \underbrace{\infty\cdot(0) + 2(1) + n(\infty)}_{\text{signature}} ; P_+\bigr)
\]

\[
  \text{(90)}\qquad
  \begin{cases}
    R_s \longmapsto 0 & (s \in S) \\
    Q_a \longmapsto 1 & (a \in A) \\
    P_\omega \longmapsto \infty & (\omega \in \underline{\omega})
  \end{cases}
  \qquad (\Sigma_P \to \mathbb{P}^1_{\mathbb{C}}) .
\]

\[
  \text{(91)}\qquad
  \left\{
  \begin{array}{l}
    P = P_0 = \text{polygone comb. formé des } S_0 = \mu_n(\mathbb{C}) \\
    \text{et des arcs de cercle sur } U = \{z \in \mathbb{C} \mid |z| = 1\} \\
    \text{qui les joignent (car } P \text{ \emph{épinglé})}
  \end{array}
  \right.
\]

\[
  \Sigma_P = \mathbb{P}^1_{\mathbb{C}} = \hat{\mathbb{C}}, \qquad
  D_P^+ = \mu_n(\mathbb{C}) \text{ opérant par homothéties}, \qquad
  \tau(z) = 1/\bar z
\]

\[
  z \longmapsto z^n \qquad \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})
\]

\[
  z \longmapsto -\frac{1}{4z}(z-1)^2 \qquad \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})
\]

\[
  \text{(92)}\qquad z \longmapsto -\frac{1}{4z^n}(z^n - 1)^2 \qquad
  \underbrace{X_n}_{\substack{\text{carte cyclotomique}\\ \text{type d'ordre } n}} \xrightarrow{\ g_n\ }
  \underbrace{X_0}_{\substack{\text{carte}\\ \text{universelle}}} . \;]
\]

\[
  \text{(93)}\qquad
  \begin{aligned}
    \pi_1(X_n^*, P_{n+}) &\xrightarrow{\ \sim\ } \pi_1\bigl(\underbrace{X_0^*}_{X_0 \setminus \{0\}}, 2\{1\} + n\{\infty\} ; \underbrace{P_{0+}}_{\exp 2i\pi/6}\bigr) \\
    &\simeq \pi_1\bigl(\underbrace{X_0}_{\mathbb{P}^1(\mathbb{C})}, \infty\{0\} + 2\{1\} + n\{\infty\} ; P_{0+}\bigr)
  \end{aligned}
\]

\[
  \text{(94)}\qquad r_0 = (R_0, Q_{a_0}, P_\omega) \qquad \bigl(\xrightarrow{\ g_n\ } X_0^+\bigr)
\]

\[
  g_n : \quad R_0 \longmapsto 0, \quad Q_{a_0} \longmapsto 1, \quad P_\omega \longmapsto \infty
\]

\[
  g_n : \quad P \longmapsto P_+ .
\]

\[
  \text{(95)}\qquad
  \begin{aligned}
    \pi_1(X_n^*, \text{\add{$\mathbb{D}_n;$}}\, P) &\xrightarrow{\ \sim\ } \pi_1\bigl(X_0^*, 2\{1\} + n\{\infty\} ; P_+\bigr) \qquad X_0^* \overset{\text{déf}}{=} X_0 \setminus \{0\} \\
    &\simeq \Pi_{0,3}/\langle \rho_1^2, \rho_\infty^n \rangle ,
  \end{aligned}
\]

\[
  \text{(96)}\qquad \pi_1(X_n^*, \text{\add{$\mathbb{D}_n;$}}\, P_+) \xrightarrow{\ \sim\ } \pi_1(X_n^*, \mathbb{D}_n ; P)
\]

\[
  (l, g) \longmapsto g_n(l)
\]

\[
  \text{(97)}\qquad
  \begin{aligned}
    \pi_1(X_n^*, \mathbb{D}_n ; P_+) &\xrightarrow{\ g_n\ } \pi_1\bigl(X_0, \infty\cdot\{0\} + 2\cdot\{1\} + n\{\infty\} ; P_+\bigr) \\
    &\simeq \Pi_{0,3}/\langle \rho_1^2, \rho_\infty^n \rangle
  \end{aligned}
\]

\[
  \rho(P) \xrightarrow{\ \alpha_\rho\ } P \qquad \text{dans } X_n^{**}
\]

\[
  (*)\qquad g_n(\rho) = \rho_\infty^{-1} .
\]

\[
  \alpha_\sigma : \sigma_{a_0}^0(P) \longrightarrow P
\]

\[
  (**)\qquad g_n(\sigma_{a_0}) = \rho_1 \quad (= \rho_1^{-1})
\]

\[
  g_n(\underbrace{\varepsilon_0}_{\sigma_{a_0}\rho}) = g_n(\sigma_{a_0})\, g_n(\rho) = \rho_1^{-1}\rho_\infty^{-1} = (\rho_\infty\rho_1)^{-1} = \rho_0
\]

\[
  \text{(98)}\qquad
  \begin{cases}
    g_n : \pi_1(X_n^*, \mathbb{D}_n ; P_{n+}) \xrightarrow{\ \sim\ } \Pi_{0,3}/\langle \rho_1^2, \rho_\infty^n \rangle \\
    \qquad\qquad = \pi_1\bigl(\underbrace{X_0}_{\mathbb{P}^1_{\mathbb{C}}}, \underbrace{\infty\{0\} + 2\{1\} + n\{\infty\}}_{\text{signature } \nu_n} ; P_+\bigr) \\
    \varepsilon_0 \longmapsto \rho_0 \\
    \sigma_{a_0} \longmapsto \rho_1 = \rho_1^{-1} \\
    \rho \longmapsto \rho_\infty^{-1}
  \end{cases}
\]

\[
  \text{(99)}\qquad
  \begin{aligned}
    g_n : \pi_1(X_n^*, \mathbb{D}_n \times \{1,\tau\} ; P_{n+}) &\xrightarrow{\ \sim\ } \pi_1(X_0, \nu_n, \{1,\tau\} ; P_+) \\
    \simeq \Pi_n^{D,\tau} \simeq \Pi_n^D \cdot \{1, \tilde\sigma_{a_0}\}
    &\simeq \bigl(\Pi_{0,3}/\langle \rho_1^2, \rho_\infty^n \rangle\bigr) \cdot \{1, \ill{}\}
  \end{aligned}
\]

\[
  P' = \tilde\sigma_{a_0}^0(P) \longrightarrow P
\]

\[
  g_n(\tilde\alpha_\sigma) : \quad g_n(P') = g_n(\tilde\sigma_{a_0}^0(P)) = \tau\, g_n(P) = \tau(P_+) = P_- \longrightarrow g_n(P) = P_+
\]

\[
  (**)\qquad g_n(\tilde\sigma_{a_0}) = \tau_0 \in \Pi_{0,3}^\tau/\langle \rho_1^2, \rho_\infty^n \rangle
\]

\[
  \tau_0(\rho_\infty^{-1}) = \rho_\infty, \qquad \tau_0(\rho_1) = \rho_1^{-1}
\]

\[
  g_n(\tau_{a_0}) = g_n(\tilde\sigma_{a_0} \cdot \sigma_{a_0}) = g_n(\tilde\sigma_{a_0})\, g_n(\sigma_{a_0}) = \tau_0 \rho_1 = \tau_\infty
\]

\[
  \text{(100)}\qquad g_n(\tilde\sigma_{a_0}) = \tau_0, \qquad g_n(\tau_{a_0}) = \tau_\infty .
\]

\[
  \text{(101)}\qquad
  \begin{array}{ccc}
    X & \longmapsto & R(X) \\[2pt]
    \begin{array}{c}\text{cartes $P$-pondérées}\\ \text{finies}\\ \text{(cat. isotopique)}\end{array}
    & \xrightarrow{\ \approx\ } &
    \begin{array}{c}\Pi_P^\tau\text{-ensembles finis}\\ \text{spéciaux}\\ \text{(i.e. où les $\tau_i$}\\ \text{opèrent ss pts fixes)}\end{array}
  \end{array}
\]

\[
  \text{(102)}\qquad
  \begin{array}{ccc}
    \begin{array}{c}\text{cartes $P$-pondérées}\\ \text{finies \emph{orientées}}\end{array}
    & \xrightarrow{\ \approx\ } & \Pi_P\text{-ens. finis} \\[2pt]
    X & \longmapsto & R^+(X)
  \end{array}
\]

\[
  \text{(103)}\qquad
  \begin{array}{ccccccccc}
    1 & \to & \Pi_P & \to & \Pi_P^\tau & \longrightarrow & \{\pm 1\} & \longrightarrow & 1 \\
      &     &       &     & \tau_a & \longmapsto & -1 & &
  \end{array}
\]

\[
  \text{(104)}\qquad
  \Pi_P\text{-ens. finis} \xrightarrow{\ \approx\ }
  \begin{array}{l}\text{revêtements ramifiés}\\ \text{finis de } \Sigma_P \text{, étales}\\ \text{au-dessus de } \Sigma_P^* ,\end{array}
\]

\[
  \text{(1)}\qquad \Sigma = \mathbb{P}(V_{\tilde P_{2n}})(\mathbb{C})
\]

\[
  \text{(2)}\qquad P_n = \tilde P_{2n}/\{\pm 1\}
\]

\[
  S = \tilde S/\pm 1, \qquad A = \tilde A/\pm 1, \qquad R = \tilde R/\pm 1 ,
\]

\[
  \text{(3)}\qquad S \hookrightarrow \Sigma
\]

\[
  \begin{array}{rcl}
    \tilde S & \longrightarrow & \Sigma \\
    R_{\tilde s} & \longmapsto & \mathbb{C}\cdot R_{\tilde s} \in \mathbb{P}(V_{\mathbb{C}})(\ldots) = \mathbb{P}(V)(\mathbb{C}) .
  \end{array}
\]

\[
  \text{(4)}\qquad \Sigma_{\mathbb{R}} = \Sigma_{\tilde P_{2n}}(\mathbb{R})
\]

\[
  \text{(5)}\qquad \underset{\textstyle \Sigma_{P_n}}{\Sigma_{\tilde P_{2n}}} = \mathbb{P}(V_{\tilde P_{2n}})
  \qquad \text{(ne dépend en fait que de } P_n = \tilde P_{2n}/\pm 1\text{)}
\]

\[
  \text{(6)}\qquad D_{\tilde P_{2n}} \overset{\text{déf}}{=} \operatorname{Aut}_{\text{comb}}(\tilde P_{2n}) \qquad (\simeq \mathbb{D}_{2n})
\]

\[
  \text{(7)}\qquad D_{P_n} \overset{\text{déf}}{=} \operatorname{Aut}_{\text{comb}}(P_n) \simeq D_{\tilde P_{2n}}/\{\pm 1\}
\]

\[
  \text{(8)}\qquad 1 \longrightarrow \{\pm 1\} \longrightarrow D_{\tilde P_{2n}} \longrightarrow D_{P_n} \longrightarrow 1
\]

\[
  \text{(9)}\qquad
  \begin{array}{ccc}
    \begin{array}{c}\text{Polygones combinatoires $\tilde P_{2n}$}\\ \text{d'ordre $2n$, avec un}\\ \text{sous-polygone d'ordre}\\ \text{$n$ marqué, soit $P'_n$}\end{array}
    & \xrightarrow{\ \approx\ } &
    \begin{array}{c}\text{polygones}\\ \text{combinatoires $P_n$}\\ \text{d'ordre $n$}\end{array} \\[2pt]
    (\tilde P_{2n}, P'_n) & \longmapsto & P'_n
  \end{array}
  \qquad (n \text{ \emph{impair}})
\]

\[
  \text{(10)}\qquad V_{\tilde P_{2n}} = \underbrace{V_{P'_n} \simeq V_{P_n}}_{\text{car } P'_n \simeq P_n}
\]

\[
  \text{(11)}\qquad \underset{\textstyle \Sigma_{P_n}}{\Sigma_{\tilde P_{2n}}} = \mathbb{P}(V_{\tilde P_{2n}}) \xrightarrow{\ \sim\ } \mathbb{P}(V_{P_n}) , \qquad (n \text{ \emph{impair}})
\]

\[
  \begin{cases}
    S = S_{P_n} \longrightarrow \mathbb{P}(V_{P_n})(\mathbb{R}) \\
    s \longmapsto \mathbb{R}s
  \end{cases}
\]

\[
  \begin{array}{ccc}
    D_{\tilde P_{2n}}^+ & \longrightarrow & D_{P_n}^+ \\
    \wr & & \wr \\
    \mathbb{Z}/2n & & \mathbb{Z}/n
  \end{array}
\]

\[
  1 \to \{\pm 1\} \to \operatorname{Sl}(V_{\tilde P_{2n}}) \to \underset{\textstyle \operatorname{Aut}(\Sigma_{P_n})}{\operatorname{GP}(V_{\tilde P_{2n}})} \longrightarrow 1
\]

\[
  \text{(12)}\qquad P\Sigma_{P_n} = \check{\mathbb{V}}(\mathcal{O}_{\Sigma_{P_n}}(1))^*
\]

\[
  \text{(13)}\qquad P\Sigma_{P_n}^* = P\Sigma_{P_n} \mid \Sigma_{P_n}^* ,
\]

\[
  K\otimes\overline{\mathbb{Q}} \xrightarrow{\ \sim\ } \overline{\mathbb{Q}}^{\,\mathrm{Hom}(K,\overline{\mathbb{Q}})}
\]

\[
  \det\mathrm{Lie} : K^* \longrightarrow \mathbb{G}_m \qquad (\text{défini sur } \overline{\mathbb{Q}}).
\]

\[
  \delta : K'^* \longrightarrow K^* : \quad k' \longmapsto \det{}_K(\text{multiplication par } k'),
\]

\[
  \delta : \text{classes d'idèles de } K'^* \longrightarrow \text{classes d'idèles de } K
\]

\[
  \sigma(E) \sim E\otimes_{\mathcal{O}}(\sigma)
\]

\[
  \mathcal{M}^{!\circ}_0 \simeq \mathcal{M}(0)^{\circ}\cdot\Pi_0
\]

\[
  D_\rho,\ D_\sigma,
\]

\[
  \mathcal{M} \longrightarrow \mathfrak{S}/\Pi_0 \times \mu \qquad (\varphi,\varepsilon_3)
\]

\[
  \mathcal{M} \longrightarrow \mathfrak{S}_3 \times\mu\times\mu \longrightarrow \mathfrak{S}_3\times\mu\times\mu,
  \qquad \mathfrak{S}_3\times\mu .
\]

\[
  \underline{\mathcal{M}}/\underline{\mathcal{M}}^{!\circ} \simeq \mathfrak{S}_3\times\mu\times\mu,
  \qquad
  \underline{\mathcal{M}}/\underline{\mathcal{M}}^{!\circ}_0 \longrightarrow \cdots
\]

\[
  1\to\mu\to\cdots\qquad
  1\to \mathrm{Sl}(2,\mathbb{Z}/4\mathbb{Z}) \to \mathrm{Gl}(2,\mathbb{Z}/4\mathbb{Z}) \to (\mathbb{Z}/4\mathbb{Z})^* \to 1
\]

\[
  \mathcal{M}''^!_\bullet \cap N^*_\rho = Z^{''!}_\rho
\]

\[
  \mathfrak{S}_{1,1} \simeq \mathcal{T}_{1,2}^{\,!}
\]

\[
  \begin{pmatrix}-1 & 1\\ 0 & 1\end{pmatrix}
\]

\[
  r_0\,u\,r_0 \qquad (u r_1)^{-1} = r_1 u^{-1} \qquad (u r_\infty)^{-1} = r_\infty u^{-1}
\]

\[
  (u r_\infty)^{-1} \qquad (v r_\infty)^{-1} = r_\infty v^{-1} \qquad r_1 v^{-1}
\]

\[
  \tilde v^{-1}\,\tilde u\,\tilde v\,\tilde u^{-1} \qquad \tilde u\,\tilde v \qquad \tilde\rho
\]

\[
  (u,l)(u',l') = \bigl(uu',\, u(l')\,l\bigr)
\]

\[
  l : a\to ua,\qquad l' : a\to u'a, \qquad
  u'(l) : u'a \to u'ua,\qquad u(l') : ua \to uu'a
\]

\[
  a \xrightarrow{\ l\ } ua \longrightarrow uu'a
\]

\[
  (\gamma,l)(\gamma',l')(\gamma'',l'')
\]

\[
  \tilde v = [\tilde v,\tilde u^{-1}] \ldots
\]

\[
  \tilde u,\ \tilde v \qquad [\tilde u,\tilde v] = l^{!\,-1}_0
\]

\[
  \tilde\rho,\ \tilde\sigma,\ \tilde\omega_0
\]

\[
  \tilde\rho^{\,3} = \tilde\omega_0,\qquad \tilde\sigma^{2} = \tilde\omega_0^{-1}
\]

\[
  \tilde\omega_0^{\,2} = l^!_0
\]

\[
  \left.
  \begin{aligned}
    \tilde r_0(\tilde\rho) &= \tilde\rho^{-1}\\
    \tilde r_0(\tilde\sigma) &= \tilde\sigma^{-1}
  \end{aligned}
  \right\}
  \Longrightarrow\ r_0(\tilde\omega_0) = \tilde\omega_0^{-1}
\]

\[
  \text{déf}
  \begin{cases}
    \tilde r_1 = \tilde r_0\,\tilde\rho & \text{i.e. } \tilde\rho = \tilde r_0\,\tilde r_1\\
    \tilde r_\infty = \tilde\sigma\,\tilde r_0 & \text{i.e. } \tilde\sigma = \tilde r_\infty\,\tilde r_0
  \end{cases}
\]

\[
  \boxed{\ \tilde r_0^{\,2} = \tilde r_1^{\,2} = \tilde r_\infty^{\,2} = 1,\quad
  (\tilde r_0\tilde r_1)^3(\tilde r_\infty\tilde r_0)^2 = 1\ }
\]

\[
  \mathfrak{S}_{1,1} \simeq \mathcal{T}_{1,2}^{\,!}
  \ \supset\ S\mathcal{T}_{1,1} \simeq \mathrm{Gl}(2,\mathbb{Z})^{\sim}
  = \mathrm{Norm}_{\mathfrak{S}_{1,1}}(L^!_0)
\]

\[
  \begin{aligned}
    \tilde\rho(\tilde u) &= (\tilde r_0\tilde r_1)(\tilde u) = \tilde v^{-1}\\
    \tilde\rho(\tilde v) &= (\tilde r_0\tilde r_1)(\tilde v) = \tilde v\,\tilde u\\
    \tilde\sigma(\tilde u) &= (\tilde r_\infty\tilde r_0)(\tilde u) = \tilde u(\tilde v) = \tilde u\,\tilde v\,\tilde u^{-1}\\
    \tilde\sigma(\tilde v) &= (\tilde r_\infty\tilde r_0)(\tilde v) = \tilde u^{-1}
  \end{aligned}
\]

\[
  \left.
  \begin{aligned}
    \tilde r_0(\tilde u) &= \tilde r_0\,\tilde u\,\tilde r_0 = \tilde v\\
    \tilde r_0(\tilde v) &= \tilde r_0\,\tilde v\,\tilde r_0 = \tilde u
  \end{aligned}
  \right\}
  \Longrightarrow\ \tilde r_0(l^!_0) = l^{!\,-1}_0
\]

\[
  \begin{aligned}
    \tilde r_\infty(\tilde u) &= \tilde r_\infty\,\tilde u\,\tilde r_\infty = \tilde u^{-1}\\
    \tilde r_\infty(\tilde v) &= \tilde r_\infty\,\tilde v\,\tilde r_\infty
    = \tilde v^{\,?}\,[\tilde v^{-1},\tilde u]\\
    &= \tilde u\,\tilde v\,\tilde u^{-1} = \tilde u(\tilde v)
  \end{aligned}
\]

\[
  \left.
  \begin{aligned}
    \tilde r_1(\tilde u) &= \tilde r_1\,\tilde u\,\tilde r_1 = \tilde u^{-1}\\
    \tilde r_1(\tilde v) &= \tilde r_1\,\tilde v\,\tilde r_1 = \tilde u\,\tilde v
  \end{aligned}
  \right\}
  \Longrightarrow\ \tilde r_1(l^!_0) = l^{!\,-1}_0
\]

\[
  \Bigl\{\tilde r_0,\tilde r_1,\tilde r_\infty,\tilde u,\tilde v \ \Bigm|\
  \begin{array}{l}
    \tilde r_0^{\,2} = \tilde r_1^{\,2} = \tilde r_\infty^{\,2}
      = 1\\
    (\tilde r_0\tilde r_1)^{6}\ \bigl(= (\tilde r_\infty\tilde r_0)^4\bigr) = [\tilde u,\tilde v]\\
    \tilde r_0(\tilde u) = \tilde v,\ \tilde r_0(\tilde v) = \tilde u\\
    \tilde r_\infty(\tilde u) = \tilde u^{-1},\ \tilde r_\infty(\tilde v) = \tilde u(\tilde v)\ \bigl(= \tilde u\tilde v\tilde u^{-1}\bigr)\\
    r_1(\tilde u) = \tilde u^{-1};\ r_1(\tilde v) = \tilde u\,\tilde v
  \end{array}
  \Bigr\}
\]

\[
  = \Bigl\{\tilde r_0,\tilde r_1,\tilde r_\infty,\tilde\omega_0,\tilde u,\tilde v \ \Bigm|\
  \begin{array}{l}
    r_0^2 = r_1^2 = r_\infty^2 = 1,\ (\tilde r_0\tilde r_1)^3 = (\tilde r_\infty\tilde r_0)^2 = \tilde\omega_0\\
    \tilde\omega_0^{\,2} = [\tilde u,\tilde v]\\
    \text{relations de conjugaison (C)}
  \end{array}
  \Bigr\}
\]

\[
  \begin{array}{ccccc}
    1 & & & & 1\\
    \uparrow & & & & \uparrow\\
    \mathfrak{S}_{0,3} & \longrightarrow & \mathcal{T}_{0,3} & = & \mathfrak{S}_{0,3}/\mathfrak{S}^{+!}_{0,3} \simeq \mathfrak{S}_3\times\mu_\tau\\
    \uparrow & & \uparrow & & \\
    \widetilde{\mathfrak{S}}^{+} & \longrightarrow & \widetilde{\mathcal{T}}_{0,3} & = & \cdots/(\mathfrak{S}^{!+}_{0,3})_0\\
    \uparrow & & \uparrow & & \\
    \mu & = & \mu & & \\
    \uparrow & & \uparrow & & \\
    1 & & 1 & &
  \end{array}
\]

\[
  \mathfrak{S} = \Bigl\{\rho,\sigma,\tau' \ \Bigm|\
  \begin{array}{l}
    \rho^6 = \sigma^4 = \tau'^2 = 1,\ \rho^3 = \sigma^2;\\
    \tau'(\rho) = \rho^{-1},\ \tau'(\sigma) = \sigma^{-1}
  \end{array}\Bigr\}
\]

\[
  = \Bigl\{\rho,\sigma,\tau \ \Bigm|\
  \begin{array}{l}
    \rho^6 = \sigma^4 = \tau^2 = 1,\ \rho^3 = \sigma^2\\
    \tau(\rho) = \cdots,\ \tau(\sigma) = \sigma^{-1}
  \end{array}\Bigr\}
\]

\[
  \text{ordre } 24\quad \mathfrak{S}/\Pi_0 = \Bigl\{\rho,\sigma,\tau \ \Bigm|\
  \begin{array}{l}
    \rho^6 = \sigma^4 = \tau^2 = 1,\ \rho^3 = \sigma^2\\
    \tau(\sigma) = \sigma^{-1},\ \tau(\rho) = \rho\\
    \sigma(\rho) = \rho^{-1}
  \end{array}\Bigr\}
\]

\[
  \text{ordre } 12\quad \mathfrak{S}^+/\Pi_0 = \Bigl\{\rho,\sigma \ \Bigm|\
  \begin{array}{l}
    \rho^6 = \sigma^4 = 1,\ \rho^3 = \sigma^2\\
    \sigma(\rho) = \rho^{-1}
  \end{array}\Bigr\}
\]

\[
  \underline\mu \simeq \mu_\tau \simeq \{\pm1\}
\]

\[
  \mu_\tau = \{1,\tau\},\quad \mu_{\tau'} = \{1,\tau'\}
\]

\[
  \mu = \{\pm1\}
\]

\[
  \mathfrak{S} =
\]

\[
  \mathfrak{S}^+ \qquad \rho\longmapsto\rho,\quad \sigma\longmapsto -\sigma
\]

\[
  \Pi \simeq \Pi_0\times\{\pm1\},\qquad
  \Pi' = \{1,\tau\}\cdot\Pi = D_\tau\cdot\Pi_0 = \Pi'_0\times\mu
\]

\[
  \Pi'_0 = \{1,\tau\}\cdot\Pi_0
\]

\[
  \Pi_0 \simeq \{\lambda_0,\lambda_1,\lambda_\infty \mid \lambda_\infty\lambda_1\lambda_0 = 1\}
  \hookrightarrow \mathfrak{S},\qquad \lambda_i\longmapsto -l_i
\]

\[
(1)\qquad X = \mathbb{U}\times\mathbb{I}, \quad\text{où}\quad
\mathbb{U}=\{z\in\mathbb{C} \mid |z|=1\},\quad \mathbb{I}=[-1,+1],
\]

\[
(2)\qquad \partial X = \mathbb{U}\times\partial\mathbb{I}
= \mathbb{U}\times\{-1,+1\} = \mathbb{U}_{-1}\sqcup\mathbb{U}_{1}.
\]

\[
(3)\qquad
\begin{cases}
\widetilde{\mathbb{U}}_\varepsilon \overset{\text{déf}}{=} \mathbb{R}_\varepsilon \longrightarrow \mathbb{U}_\varepsilon\\
(t,\varepsilon)\longmapsto (\exp(2i\pi t),\varepsilon).
\end{cases}
\]

\[
(4)\qquad D_{\mathbb{U}} \overset{\text{déf}}{=} \{1,\sigma\}.\mathbb{U}\qquad(\tfrac12\text{ direct})
\]

\[
(5)\qquad \widetilde{D}_{\mathbb{U}} \overset{\text{déf}}{=} D_{\mathbb{R}} = \{1,\sigma\}.\mathbb{R},
\]

\[
(6)\qquad \sigma^\nu.t \longmapsto \sigma^\nu \exp(2i\pi t),
\]

\[
(7)\qquad 1\longrightarrow\mathbb{Z}\longrightarrow D_{\mathbb{R}}\longrightarrow D_{\mathbb{U}}\longrightarrow 1.
\]

\[
(8)\qquad (\sigma^\nu.u).t_\varepsilon = (-1)^\nu(t+u)_\varepsilon
\]

\[
(9)\qquad (\sigma^\nu.\theta).z_\varepsilon = (z\theta)^{(-1)^\nu} =
\begin{cases}
z\theta & \text{si } \nu\equiv0\ (2)\\
z^{-1}\theta^{-1} & \text{si } \nu\equiv1\ (2).
\end{cases}
\]

\[
\widetilde{\partial X} \overset{\text{déf}}{=} \widetilde{\mathbb{U}}_{-1}\sqcup\widetilde{\mathbb{U}}_{1} = \mathbb{R}_{-1}\sqcup\mathbb{R}_{1}
\]

\[
(10)\qquad \widetilde{X} \overset{\text{déf}}{=} \mathbb{R}\times\mathbb{I}
\]

\[
(11)\qquad (t,s)\longmapsto(\exp 2i\pi t, s)\ ;
\]

\[
(12)\qquad
\begin{gathered}
D_{\mathbb{R}\times\mathbb{R}} \overset{\text{déf}}{=} \{1,\sigma\}.(\mathbb{R}\times\mathbb{R}) \hookrightarrow D_{\mathbb{R}}\times D_{\mathbb{R}}\qquad(\tfrac12\text{ direct})\\
\sigma^\nu.(u_{-1},u_1)\longmapsto(\sigma^\nu.u_{-1},\sigma^\nu.u_{+1})
\end{gathered}
\]

\[
(13)\qquad s\longmapsto u_s
\]

\[
(14)\qquad u_s = \frac{u_1-u_{-1}}{2}\,s + \frac{u_1+u_{-1}}{2},
\]

\[
(15)\qquad
\begin{cases}
(u_{-1},u_1).(t,s) \overset{\text{déf}}{=} (t+u_s,s)\\
\sigma.(t,s) \overset{\text{déf}}{=} (-t,s)
\end{cases}
\]

\[
\widetilde{X}=\mathbb{R}\times\mathbb{I}\longrightarrow\mathbb{I},\qquad (t,s)\longmapsto s
\]

\[
X=\mathbb{U}\times\mathbb{I}\longrightarrow\mathbb{I},\qquad (z,s)\longmapsto s
\]

\[
\mathbb{Z}\times\mathbb{Z}\subset D^{\circ}_{\mathbb{R}\times\mathbb{R}}.
\]

\[
(16)\qquad \sigma^\nu u_{\cdot}=\sigma^\nu.(u_{-1},u_1),\qquad \sigma^{\nu'}.u'_{\cdot}=\sigma^{\nu'}.(u'_{-1},u'_1)
\]

\[
(17)\qquad \nu=\nu' \quad\text{et}\quad u_s-u'_s\in\mathbb{Z}\quad \forall s\in\mathbb{I}
\]

\[
(18)\qquad
\begin{cases}
\dfrac{u_1-u_{-1}}{2}=\dfrac{u'_1-u'_{-1}}{2} \quad\text{i.e.}\quad u_1-u_{-1}=u'_1-u'_{-1}\\[2ex]
\dfrac{u_1+u_{+1}}{2}\equiv\dfrac{u'_1+u'_{-1}}{2}\ (\mathbb{Z}) \quad \text{\struck{\ill{}}}
\end{cases}
\]

\[
(19)\qquad f_{a,b}\overset{\text{déf}}{=}(u_{-1},u_1)\quad\text{où}\quad
\begin{cases} u_{-1}=-a+b\\ u_1=a+b.\end{cases}
\]

\[
(20)\qquad f_{a,b}=f_{a',b'} \iff a=a',\ b\equiv b'\ (\mathbb{Z})
\]

\[
(21)\qquad D_{\mathbb{R}\times\mathbb{U}} \overset{\text{déf}}{=} \{1,\sigma\}.(\mathbb{R}\times\mathbb{U})\qquad(\tfrac12\text{ direct})
\]

\[
\sigma(u,\theta)=(-u,\theta^{-1}).
\]

\[
(22)\qquad
\begin{cases}
\sigma(z,s)=(z^{-1},s)\\
g_{a,\theta}(z,s)\overset{\text{déf}}{=}(z\exp(2i\pi as)\,\theta,\ s)\qquad (a\in\mathbb{R},\ \theta\in\mathbb{U})
\end{cases}
\]

\[
(23)\qquad
\begin{cases}
\sigma(t,s)\overset{\text{déf}}{=}(-t,s)\\
f_{a,b}(t,s)\overset{\text{déf}}{=}(t+(as+b),\ s)\qquad (a\in\mathbb{R},\ b\in\mathbb{R}).
\end{cases}
\]

\[
(24)\qquad \{\pm1\}\times D_{\mathbb{R}\times\mathbb{U}}
\]

\[
(25)\qquad (\varepsilon\times\sigma^\nu.g_{a,\theta})(z,s)\overset{\text{déf}}{=}\bigl((z\exp(2i\pi as)\,\theta)^{(-1)^\nu},\ \varepsilon s\bigr) ;
\]

\[
(26)\qquad \{\pm1\}\times D_{\mathbb{R}\times\mathbb{R}}
\]

\[
(27)\qquad (\varepsilon\times\sigma^\nu f_{a,b})(t,s)\overset{\text{déf}}{=}\bigl((-1)^\nu(t+(as+b)),\ \varepsilon s\bigr).
\]

\[
\theta\exp(-2i\pi a),\ \theta\exp 2i\pi a\in\mu_\infty(\mathbb{C})
\]

\[
(28)\qquad \theta\in\mu_\infty(\mathbb{C}),\quad a\in\mathbb{Q}.
\]

\[
(29)\qquad G=(\mu\times\mu).(\mathbb{Q}\times\mu_\infty)\qquad \tfrac12\text{ direct}
\]

\[
\mu=\{\pm1\},\qquad \mu_\infty=\mu_\infty(\mathbb{C})\simeq\mathbb{Q}/\mathbb{Z}\qquad
(\exp 2i\pi q \longleftarrow q \bmod \mathbb{Z}),
\]

\[
(30)\qquad \widetilde{G}=(\mu\times\mu).(\mathbb{Q}\times\mathbb{Q})\qquad\tfrac12\text{ direct}
\]

\[
(31)\qquad
\begin{gathered}
1\longrightarrow\mathbb{Z}\longrightarrow\widetilde{G}\longrightarrow G\longrightarrow 1\\
n\longmapsto f_{0,n},\qquad (\varepsilon,\varepsilon').f_{a,b}\longmapsto(\varepsilon,\varepsilon').g_{a,\exp 2i\pi b}
\end{gathered}
\]

\[
(32)\qquad \bigl((\varepsilon,\varepsilon').f_{a,b}\bigr)(t,s)=\bigl(\varepsilon'(t+(as+b)),\ \varepsilon\varepsilon's\bigr)=\varepsilon'\bigl(t+(as+b),\ \varepsilon s\bigr)
\]

\[
(\varepsilon,\varepsilon'\in\mu,\quad a,b\in\mathbb{Q},\quad t\in\mathbb{R},\quad s\in\mathbb{I}),
\]

\[
(33)\qquad \bigl((\varepsilon,\varepsilon').g_{a,\theta}\bigr)(z,s)=\bigl((z\theta\exp 2i\pi as)^{\varepsilon'},\ \varepsilon\varepsilon's\bigr)
\qquad(\varepsilon,\varepsilon'\in\mu,\ a\in\mathbb{Q},\ \theta\in\mu_\infty).
\]

\[
\theta\exp(-2i\pi a),\quad \theta\exp(2i\pi a).
\]

\[
\text{\struck{$(34)\quad \theta\exp(-2i\pi a)=-1,\quad \theta\exp 2i\pi a=-1$}}
\]

\[
\text{\struck{$(35)\quad \theta=1,\ a\in\tfrac12+\mathbb{Z}\quad;\quad \theta=-1,\ a\in\mathbb{Z}$}}
\]

\[
\text{\struck{$(36)\quad a\equiv\tfrac12,\ b\equiv1\ (\mathbb{Z})\quad\text{ou}\quad a\equiv1,\ b\equiv\tfrac12\ (\mathbb{Z}).$}}
\]

\[
(34)\qquad g_a(z,s)=(z\exp 2i\pi as,\ s).
\]

\[
(35)\qquad \exp 2i\pi a=1\quad\text{i.e.}\quad a\in\mathbb{Z}.
\]

\[
\omega(X)\simeq\varepsilon_0(X)\wedge\varepsilon_1(X).
\]

\[
\tilde u^{S'}=\tau_n^{-1}\tilde u^{S}=\tau_{-n}\tilde u^{S}
\]

\[
H_1(X)\wedge_{\{\pm1\}}\varepsilon_0(X)=\bigl(\mathbb{Z}\wedge_{\pm1}\varepsilon_1(X)\bigr)\wedge_{\pm1}\varepsilon_0(X)
\simeq\mathbb{Z}\wedge_{\pm1}\bigl(\varepsilon_1(X)\wedge\varepsilon_0(X)\bigr)
\simeq\mathbb{Z}\wedge_{\pm1}\omega(X).
\]

\[
\text{\struck{$\tilde\rho^{3}(\tilde\tau'_\infty)=\tilde\omega_0^{-1}\tilde\tau'_\infty\tilde\omega_0=\tilde\omega_0^{-1}\tilde\tau'_\infty\tilde\omega_0\tilde\tau'^{-1}_\infty\tilde\tau'_0$}}
\]

\[
\text{\struck{$=\tilde\omega_0^{-2}\tilde\tau'_0=\ldots$}}
\]

\[
\tau'',\ \tau,\quad \xi_0,\ \xi_1,\ \xi_2
\]

\[
\begin{array}{lll}
\tau'',\ \xi_0,\ \xi_1 & \text{engendrent} & \mathrm{Gl}(2,\mathbb{Z})\\
\tau'',\ \xi_2,\ \xi_1^{-1} & \text{———} & \mathrm{Gl}(2,\mathbb{Z})\\
\tau'',\ \xi_1 & \text{———} & D_{\xi_1}
\end{array}
\]

\[
\mathcal{T}_{1,2}\ \overset{?}{\simeq}\ \mathrm{Gl}(2,\mathbb{Z})\underset{D_\infty}{*}\mathrm{Gl}(2,\mathbb{Z})
\]

\[
2g+2\,\frac{g(g-1)}{2}=2g\Bigl(1+\frac{g-1}{2}\Bigr)=2\,\frac{g(g+1)}{2}
\]

\[
\tau',\ \tau'',\ \tau^{*}\qquad \text{\struck{$\xi$}}\ \ \xi^{*}\ \ \bar\xi'
\]

\[
\tau'^2=\tau''^2=\tau^2=\text{\struck{\ill{}}}=1,\qquad (\tau''\tau)^2=(\tau\tau')^2=(\tau'\tau'')^2=1
\]

\[
\begin{aligned}
&\tau(\xi_0)=\xi_2\qquad(\tau(\xi_2)=\tau(\xi_1))\\
&\tau(\xi_1)=\xi_1^{-1}\\
&\tau'(\xi_0)=\tau''(\xi_0)=\xi_0^{-1}\\
&(\Rightarrow\ \tau'(\xi_2)=\tau''(\xi_2)=\xi_2^{-1})\\
&\tau'(\xi_1)=\tau''(\xi_1)=\xi_1^{-1}\\
&[\xi_0,\xi_2]=1
\end{aligned}
\]

\[
\rho_0\overset{\text{déf}}{=}(\xi_1\xi_0)^{-1}=\xi_0^{-1}\xi_1^{-1},\qquad
\tau(\rho_0)=(\xi_1^{-1}\xi_2)^{-1}\overset{\text{déf}}{=}\rho_2=\xi_2^{-1}\xi_1
\]

\[
\xi_0\underbrace{\xi_1\xi_0}_{\rho_0^{-1}}=\underbrace{\xi_1\xi_0}_{\rho_0^{-1}}\xi_1
\ \Longrightarrow\ (\underbrace{\xi_1\xi_0}_{\rho_0^{-1}})^3=(\underbrace{\xi_0\xi_1\xi_0}_{\sigma_0})^2
\]

\[
(\xi_1\xi_0)^6=1,\qquad (\xi_1\xi_0)^6=(\xi_0\xi_1\xi_0)^4
\]

\[
\xi_2\xi_1^{-1}\xi_2=\xi_1^{-1}\xi_2\xi_1^{-1}
\]

\[
\begin{array}{lll}
\mathrm{Sl}(2,\mathbb{Z}) & \rho,\sigma & \bigm| \ \rho^3=\sigma^2,\ \sigma^4\,(=\rho^6)=1\\[1ex]
& \xi_0\,\xi_1 & \bigm| \ \xi_0\xi_1\xi_0=\xi_1\xi_0\xi_1,\ (\xi_1\xi_0)^6=1\\[2ex]
\mathrm{Gl}(2,\mathbb{Z}) & \rho,\sigma,r_0 & \bigm| \ \rho^3=\sigma^2,\ \sigma^4\,(=\rho^6)=1,\ r_0^2=1,\\
& & \phantom{\bigm|}\ r_0(\rho)=\rho^{-1},\ r_0(\sigma)=\sigma^{-1}\\[1ex]
& r_0,r_\infty,r_1 & \bigm| \ r_0^2=r_1^2=r_\infty^2=1,\ (r_0r_\infty)^2=(r_1r_0)^3,\ (r_0r_\infty)^4=1\\[1ex]
& r_0,\xi_0,\xi_1 & \bigm| \ r_0^2=1,\ r_0(\xi_0)=\xi_1^{-1},\ r_0(\xi_1)=\xi_0^{-1},\\
& & \phantom{\bigm|}\ \xi_0\xi_1\xi_0=\xi_1\xi_0\xi_1,\ (\xi_1\xi_0)^6=1
\end{array}
\]

\[
\begin{aligned}
\tilde\rho&=(\tilde\xi_1\tilde\xi_0)^{-1}=\tilde\xi_0^{-1}\tilde\xi_1^{-1}\quad\text{i.e.}\quad\tilde\xi_1\tilde\xi_0=\tilde\rho^{-1}\\
\tilde\sigma&=\tilde\xi_0\tilde\rho^{-1}\qquad\text{d'où}\quad\tilde\xi_0=\tilde\sigma\tilde\rho\\
&=\tilde\xi_0\tilde\xi_1\tilde\xi_0
\end{aligned}
\]

\[
\begin{aligned}
&\tilde\sigma(\tilde\xi_0)=\sigma(\tilde\xi_0)=\tilde\xi_1\\
&\text{d'où}\quad \tilde\rho(\tilde\xi_0)=\tilde\sigma^{-1}(\tilde\xi_0)=\sigma^{-1}(\tilde\xi_0)=\sigma(\tilde\xi_0)=\tilde\xi_1
\end{aligned}
\]

\[
\tilde\rho^{-3}=(\tilde\xi_1\tilde\xi_0)^3,\qquad \tilde\sigma^2=(\tilde\xi_0\tilde\xi_1\tilde\xi_0)^2
\]

\[
\tilde\rho^{-3}=\tilde\sigma^2\quad\text{i.e.}\quad
\tilde\xi_1\tilde\xi_0\tilde\xi_1\tilde\xi_0\tilde\xi_1\tilde\xi_0=\tilde\xi_0\tilde\xi_1\tilde\xi_0^{2}\tilde\xi_1\tilde\xi_0\ \ ?
\]

\[
\tilde\rho^{-6}=\tilde\sigma^4
\]

\[
\text{\struck{$\xi_0\ldots\tilde\xi_0$}}
\]

\[
\tilde\rho',\qquad \tilde\rho''=\sigma(\tilde\rho')
\]

\[
\begin{cases}
\tau'e'=e' & \tau''e'=-e'\\
\tau'e''=-e'' & \tau''e''=e''\\
\sigma e'=e'' & \sigma e''=-e'
\end{cases}
\]

\[
\tau'\tau''=\tau''\tau'=\sigma^2=\omega_0,\qquad \sigma^4=\omega_0^2=1
\qquad
\begin{cases}\omega_0e'=-e'\\ \omega_0e''=-e''\end{cases}
\]

\[
\sigma\tau'=\tau'_\infty=r_0\ :\quad e'\mapsto e'',\ \ e''\mapsto e'
\]

\[
\sigma\tau'=\tau'\sigma^{-1}=\sigma^{-1}\tau''=\tau''\sigma\qquad(\tau'=\sigma^2\tau'')
\]

\[
\begin{aligned}
\tau''&\longmapsto\tau_\infty=r_\infty\\
\sigma&\longmapsto\sigma=\sigma_\infty\\
\xi'&\longmapsto\xi_0\\
\xi''=\sigma(\xi')&\longmapsto\xi_1
\end{aligned}
\qquad\qquad
\begin{aligned}
\xi'(e')&=e'\\
\xi'e''&=e''-e'
\end{aligned}
\]

\[
\xi_0=\sigma\rho,\quad \xi_1=\rho\sigma,\qquad \xi_1\xi_0=\rho\sigma^2\rho=-\rho^2=\rho^{-1}
\]

\[
\text{\struck{$\xi_0\ \xi_1\ \xi_0=\rho\sigma$}}\qquad
\begin{cases}
\sigma(\xi')=\xi''\\
\tau'(\xi')=\tau''(\xi')=\xi'^{-1}\\
\tau'(\xi'')=\tau''(\xi'')=\xi''^{-1}
\end{cases}
\]

\[
\begin{cases}
\rho=(\xi_1\xi_0)^{-1}=\xi_0^{-1}\xi_1^{-1}\\
\sigma_\infty=\xi_0\rho^{-1}=\xi_0\xi_1\xi_0=\rho^{-1}\xi_1=\xi_1\xi_0\xi_1\\
r_0(\xi_0)=\xi_1^{-1}\\
r_0(\xi_1)=\xi_0^{-1}
\end{cases}
\qquad
\begin{bmatrix}
r_0(\rho)=\xi_1\xi_0=\rho^{-1}\\
r_0(\sigma)=\xi_1^{-1}\xi_0^{-1}\xi_1^{-1}=\xi_0^{-1}\xi_1^{-1}\xi_0^{-1}
\end{bmatrix}
\]

\[
\text{i.e.}\quad \xi_1\xi_0\xi_1=\xi_0\xi_1\xi_0
\]

\[
\text{\struck{$\rho\sigma\sigma\rho\rho\sigma=\sigma\rho\rho\sigma\sigma\rho$}}
\]

\[
\rho\sigma^2\rho^2\sigma=\sigma\rho^2\sigma^2\rho,\qquad -\rho^3\sigma=-\sigma\rho^3
\]

\[
\begin{array}{ll}
X'_{\xi} & \varepsilon_0(X') \xrightarrow{\ \sim\ } \varepsilon_1(X'') \\
X''_{\xi} & \varepsilon_1(X') \xleftarrow{\ \sim\ } \varepsilon_0(X'') \\
\hline
 & \omega(X') \simeq \omega(X'')
\end{array}
\qquad \Longrightarrow
\]

\[
\begin{aligned}
\tau' e' &= e' & \qquad \tau'' e' &= -e' \\
\tau' e'' &= -e'' & \qquad \tau'' e'' &= e''
\end{aligned}
\]

\[
\left\{
\begin{aligned}
&\tau'(\rho') = \tau''(\rho') = \rho'^{-1} \\
&\tau'(\rho'') = \tau''(\rho'') = \rho''^{-1} \\
&\sigma(\rho') = \rho'', \quad \sigma\rho'' = \rho'
\end{aligned}
\right.
\qquad
\begin{aligned}
\rho'(e') &= e' \\
\rho'(e'') &= e' + e''
\end{aligned}
\]

\[
\begin{aligned}
\rho' &\longmapsto \varepsilon_0 \\
\rho'' &\longmapsto \varepsilon_1 \\
\tau'' &\longmapsto \tau_\infty = \tau = r_\infty \\
\tau' &\longmapsto \omega_0 \tau_\infty = -\tau \\
\sigma &\longmapsto \sigma_\infty = \sigma
\end{aligned}
\qquad
\left\{
\begin{aligned}
&\tau'^2 = \tau''^2 = \sigma^4 = 1, \quad \sigma^2 = \tau'\tau'' \\
&\tau'(\rho') = \tau''(\rho') = \rho'^{-1}, \quad \tau'(\rho'') = \tau''(\rho'') = \rho''^{-1} \\
&\sigma(\rho') = \rho'' \quad (\Rightarrow \sigma(\rho'') = \rho')
\end{aligned}
\right.
\]

\[
\begin{aligned}
\sigma^{-1}\rho' = \sigma^3\rho' &\longmapsto \sigma^3\varepsilon_0 = \sigma^3(\sigma\rho) = \rho \\
\tau''\sigma &\longmapsto \tau'_\infty = r_0 \\
\tau''\rho' = \underbrace{(\tau''\sigma)}_{\mapsto r_0}\underbrace{(\sigma^{-1}\rho')}_{\mapsto \rho} &\longmapsto r_1
\end{aligned}
\]

\[
\begin{array}{l|l}
\tau'' \longmapsto r_\infty & \rho' \longmapsto \varepsilon_0 \\
\tau''\sigma \longmapsto r_0 & \sigma \longmapsto \sigma \\
\tau''\rho' \longmapsto r_1 & \sigma^{-1}\rho' \longmapsto \rho
\end{array}
\]

\[
\begin{aligned}
\varepsilon_0(X') \times \varepsilon_1(X') &\simeq \text{ens. des \struck{côtés} \add{\uncertain{sommets}} du carré} \\
&\simeq \varepsilon_1(X'') \times \varepsilon_0(X'') \\
&\simeq \varepsilon_0(X') \times \varepsilon_0(X'') \\
&\simeq \varepsilon_1(X') \times \varepsilon_1(X'')
\end{aligned}
\]

\[
\left\{
\begin{aligned}
&\sigma_1^2 = \sigma_2^2 = (\sigma_1\sigma_2)^2 = 1 \\
&\rho_1, \rho_2
\end{aligned}
\right.
\qquad
\sigma_i\rho_j = \rho_j^{-1}
\]

\[
\tilde\tau\tilde\rho = \tilde\rho^{-1}, \qquad \tilde\tau\tilde\sigma = \tilde\sigma^{-1}, \qquad \tilde\rho^3 -
\]

\[
x \longmapsto -x, \qquad y \longmapsto y, \qquad (x, y, z) \longmapsto ix,
\]

\[
(z, t) \longrightarrow (iz, -t)
\]

\[
\sigma^2 = \sigma_1\sigma_2 = \sigma_2\sigma_1
\]

\[
\sigma_1^2 = \sigma_2^2 \ (= \sigma^4) = 1
\]

\[
\sigma_1(\sigma) = \sigma^{-1}, \qquad \sigma_2(\sigma) = \sigma^{-1} \quad ?
\]

\[
\text{\struck{$(\sigma_1, \sigma_2)$}} \qquad \text{\struck{$(\sigma_1\sigma_2)^2$}}
\]

\[
\sigma_2 = \sigma_1\sigma^2 = \sigma^2\sigma_1
\]

\[
\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}
\]

\[
\begin{aligned}
\sigma(z, t) &= (iz, -t) \\
\sigma_1(z, t) &= (\bar z, t) \\
\sigma_2(z, t) &= (-\bar z, t)
\end{aligned}
\]

\[
(z, t) \longmapsto (\bar z, t) \longmapsto (i\bar z, -t) \longmapsto (-iz, -t)
\]

\[
\sigma_1, \sigma_2, \sigma, \rho_1, \rho_2
\]

\[
\left.
\begin{aligned}
&\sigma_1^2 = \sigma_2^2 = (\sigma_1\sigma_2)^2 = 1 \\
&\sigma_1\sigma_2 = \sigma^2, \quad \sigma_1(\sigma) = \sigma_2(\sigma) = \sigma^{-1}
\end{aligned}
\right\}
\quad D_\sigma \simeq D_4
\]

\[
\left.
\begin{aligned}
&\sigma_i(\rho_j) = \rho_j^{-1} \\
&\sigma(\rho_1) = \rho_2
\end{aligned}
\right|
\Longrightarrow \sigma(\rho_2) = \rho_1
\]

\[
\sigma, \sigma_1, \rho_1 \;\Big|\; \sigma^4 = \sigma_1^2 = 1, \qquad
\begin{aligned}
\sigma_1(\sigma) &= \sigma^{-1} \\
\sigma_1(\rho_1) &= \rho_1^{-1} \\
\sigma^2(\rho_1) &= \rho_1
\end{aligned}
\]

\[
\left\{
\begin{aligned}
\sigma_1 e_1 &= e_1 \\
e_1 e_2 &= -e_2
\end{aligned}
\right.
\qquad
\left\{
\begin{aligned}
\sigma_2 e_1 &= -e_1 \\
\sigma_2 e_2 &= e_2
\end{aligned}
\right.
\qquad
\left\{
\begin{aligned}
\sigma e_1 &= e_2 \\
\sigma e_2 &= -e_1
\end{aligned}
\right.
\]

\[
\rho_1 e_1 = e_1, \qquad \rho_1 e_2 = e_2 - e_1
\]

\[
\tau' = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad
\tau'' = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = \tau_\infty, \qquad
\sigma = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}
\]

\[
\rho_0 = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}, \qquad
\sigma(\varepsilon_0) = \rho(\varepsilon_0) = \varepsilon_1
\]

\[
(z, s) \longrightarrow (\varepsilon z \exp 2i\pi a s, \ \varepsilon' s), \qquad \varepsilon, \varepsilon' \in \mu, \quad a \in \mathbb{Q}
\]

\[
\bigl(z \exp 2i\pi a s\bigr)^{(-1)^\nu}, \quad \varepsilon s
\]

\[
\begin{aligned}
\sigma_1(z, s) &= (z^{-1}, s) \\
\sigma_2(z, s) &= (z, -s) \\
\theta(z, s) &= (z \exp 2i\pi s, s)
\end{aligned}
\qquad
\mathbf{a}(z, s) = (-z, -s)
\]

\[
\begin{gathered}
\mathbf{a}^2 = \sigma_1^2 = \sigma_2^2 = 1, \qquad \sigma_1\sigma_2 = \sigma_2\sigma_1, \quad \mathbf{a}\sigma_1 = \sigma_1\mathbf{a} \\
\sigma_1(\theta) = \theta^{-1}, \qquad \sigma_2(\theta) = \theta^{-1} \\
\mathbf{a}(\theta) = \theta^{-1}
\end{gathered}
\]

\[
\mathbf{a}, \ \alpha, \ \beta, \ \theta
\]

\[
\begin{gathered}
(z^{-1} \exp 2i\pi s, \ s) \\
(z \exp -2i\pi s, \ s) \\
z \exp -2i\pi s, \ {+s} \\
-z \exp -2i\pi s, \ -s \\
z \exp -2i\pi s, \ s)
\end{gathered}
\]

\[
a \xrightarrow{\ l\ } \gamma^{-1}(a), \qquad a \xrightarrow{\ l'\ } \gamma'^{-1}(a)
\]

\[
(\gamma, l)(\gamma', l') = \bigl(\gamma\gamma', \ \gamma'^{-1}(l) \circ l'\bigr)
\]

\[
(\gamma . l)(\gamma' . l') = (\gamma\gamma')\,\underbrace{\gamma'^{-1}\, l\, \gamma'}\, l'
\]

\[
a \longrightarrow (\gamma\gamma')^{-1}a = \gamma'^{-1}(\gamma^{-1}a), \qquad
a \xrightarrow{\ l'\ } \gamma'^{-1}a \xrightarrow{\ \gamma'^{-1}(l)\ } (\gamma\gamma')^{-1}a
\]

\[
\tilde u\,\tilde v\,\tilde u^{-1}\,\tilde v^{-1}
\]

\[
(\gamma, l)(\gamma', l') = \bigl(\gamma\gamma', \ l \circ \gamma(l')\bigr)
\]

\[
1 \longrightarrow \pi \longrightarrow E \longrightarrow G \longrightarrow 1
\]

\[
(l, \gamma)(l', \gamma') = (l . \gamma(l'))
\]

\[
(l, \gamma)(l', \gamma') = \bigl(l . \gamma(l'), \ \gamma\gamma'\bigr)
\]

\[
\text{\struck{$u \longmapsto u$, $a = \varepsilon_3(\tau')\alpha^{-1}u$, $b = \beta^{-1}u$}}
\qquad
\begin{aligned}
\alpha &= u\,\mathbf{a}^{-1}\varepsilon_3(\tau') \\
\beta &= u\,\mathbf{b}^{-1}
\end{aligned}
\]

\[
\begin{aligned}
\tilde\eta &= \tilde\beta\,\tilde\sigma\,(\tilde\beta)^{-1} &&= \eta_0\,\mathfrak{z}(6\nu) \\
\tilde\sigma(\tilde\eta) &= \tilde\sigma(\tilde\beta)\,\tilde\beta^{-1} = \tilde\eta^{-1} &&= \tilde\sigma(\eta_0)\,\mathfrak{z}(6\nu) = \eta_0^{-1}\mathfrak{z}(6\nu)
\end{aligned}
\]

\[
\begin{aligned}
\tilde\eta = \eta_0 = [\beta_0, \sigma_0] &= \beta_0(\sigma_0)\sigma_0^{-1} = \mathbf{u}(\sigma_0)\sigma_0^{-1}, \qquad \tilde\sigma = \sigma_0\,\mathfrak{z}(3)
\end{aligned}
\]

\[
\text{\struck{$\tilde\eta = u(\tilde\sigma)\tilde\sigma^{-1} = u($}}
\]

\[
\begin{aligned}
\tilde\eta = \mathbf{u}(\tilde\sigma)\,\tilde\sigma^{-1}
&= \mathbf{u}(\sigma_0)\,\mathfrak{z}(3\mu)\,\sigma_0^{-1}\,\mathfrak{z}(-3) \\
&= \mathbf{u}(\sigma_0)\,\sigma_0^{-1}\,\mathfrak{z}\bigl(\underbrace{3(\mu - 1)}_{6\nu}\bigr)
\end{aligned}
\]

\[
\beta(\tilde\sigma) = u(\tilde\sigma), \qquad \beta(\tilde\sigma) = \beta(\sigma_0)\,\mathfrak{z}(3)
\]

\[
\tilde\eta = \eta_0, \qquad \tilde\xi = \xi_0
\]

\[
\begin{aligned}
\tilde u(\tilde\varepsilon_0) = \tilde u(\tilde\sigma\tilde\rho)
&= \mathfrak{z}(\nu')\,\tilde\eta\,\tilde\sigma\,\mathfrak{z}(-\nu'')\,\tilde\xi\,\tilde\rho \\
&= \mathfrak{z}(\nu' - \nu'')\,\underbrace{\eta_0\,\sigma(\xi_0)}_{l_0^{\nu}}\,
\underbrace{\tilde\sigma\tilde\rho}_{\sigma_0\rho_0\mathfrak{z}(1) = \varepsilon_0\mathfrak{z}(1)} \\
&= \mathfrak{z}(\nu' - \nu'')\,\underbrace{l_0^{\nu}}_{\varepsilon_0^{2\nu}}\,\varepsilon_0\,\mathfrak{z}(1) \\
&= \mathfrak{z}(1 + \nu' - \nu'')\,\varepsilon_0^{2\nu+1} \\
&= \mathfrak{z}(1 + \nu' - \nu'')\,\underbrace{\varepsilon_0^{\mu}}_{\tilde\varepsilon_0^{\mu}\mathfrak{z}(-\mu)}
\end{aligned}
\]

\[
\tilde u(\tilde\varepsilon_0) = \mathfrak{z}(\nu' - \nu'' - 2\nu)\,\tilde\varepsilon_0^{\mu}
\]

\[
\begin{aligned}
&\underbrace{\nu' - \nu''}_{-\nu_2\nu_3 + \nu_0} = 2\nu \\
&-\nu_2\nu_3 + \nu_0 = 2(2\nu_3 - 3\nu_2 - 6\nu_2\nu_3 + 6\nu_0) \\
&11\nu_0 = 0, \qquad \nu_0 = 0
\end{aligned}
\]

\[
\begin{aligned}
\tilde u_1(\tilde\varepsilon_0) &= \mathfrak{z}(\nu_1' - \nu_1'' - 2\nu_1)\,\tilde\varepsilon_0^{\mu_1} \\
\tilde u_1\tilde u(\tilde\varepsilon_0) &= \mathfrak{z}\bigl(\mu_1(\nu' - \nu'' - 2\nu)\bigr)\,\tilde u_1(\tilde\varepsilon_0^{\mu})
\end{aligned}
\]

\[
(1) \qquad G = Gl(2, \widehat{\mathbb{Z}}) . \widehat{\mathbb{Z}} \qquad \text{(produit semi-direct)}
\]

\[
(2) \qquad u . \lambda = \det(u)\,\lambda
\]

\[
(3) \qquad
\left\{
\begin{aligned}
&Sl(2, \mathbb{Z})^{\sim} \longrightarrow G \\
&\tilde\rho \longmapsto \rho_0 . \mathfrak{z}(-2) \\
&\tilde\sigma \longmapsto \sigma_0 . \mathfrak{z}(3) \\
&\Bigl[\ \tilde\omega_0 = \tilde\rho^{-3} = \tilde\sigma^2 \longmapsto \omega_0 . \mathfrak{z}(6), \quad
l'_0 = \tilde\omega_0^{2} \longmapsto \mathfrak{z}(12)\ \Bigr] \\
&\tilde r_0 = \tilde\tau' \longmapsto r_0
\end{aligned}
\right.
\]

\[
(4) \qquad SG = Sl(2, \widehat{\mathbb{Z}}) . \widehat{\mathbb{Z}}
\]

\[
(5) \qquad \alpha(x) \text{ ou } \alpha x \alpha^{-1} \overset{\text{déf}}{=} \tilde\alpha(x) = \tilde\alpha x \tilde\alpha^{-1} = \alpha_0(x) = \alpha_0 x \alpha_0^{-1}.
\]

\[
(6) \qquad
\left\{
\begin{aligned}
&\beta \in Sl(2, \widehat{\mathbb{Z}}), \qquad \beta = \mathbf{u}\,b^{-1}, \quad \mathbf{u} = \begin{pmatrix} \mu & \lambda \\ 0 & 1 \end{pmatrix}, \quad b \in \mathrm{Cent}(L_\sigma) \\
&\text{i.e.}\quad \beta = \mathbf{u}\,(t\sigma + u)^{-1}
\end{aligned}
\right.
\]

\[
\left|
\begin{aligned}
&\nu_2 \in \{0, 1\}, \quad t^2 + u^2 = \mu \in \widehat{\mathbb{Z}}^{\ast} \\
&t, u \in \widehat{\mathbb{Z}}
\end{aligned}
\right.
\]

\[
(7) \qquad \tilde\eta \overset{\text{déf}}{=} [\tilde\beta, \tilde\sigma] = \tilde\beta\,\tilde\sigma\,(\tilde\beta)^{-1}
= \tilde\beta(\tilde\sigma)\,\tilde\sigma^{-1} = \beta(\tilde\sigma)\,\tilde\sigma^{-1}
\]

\[
\sigma(\tilde\eta) = \tilde\sigma(\tilde\eta) = \tilde\sigma(\tilde\beta)\,\underbrace{\tilde\sigma^2(\tilde\beta)^{-1}} =
\]

\[
\left\{
\begin{aligned}
&\tilde u(\tilde\rho) = \lambda'\,\tilde\xi\,\tilde\rho \\
&\tilde u(\tilde\sigma) = \lambda''\,\tilde\eta\,\tilde\sigma \\
&\tilde u(\tilde\omega_0) = \tilde\omega_0^{\mu} \\
&\tilde u(\tilde\varepsilon_0) \overset{?}{=} \tilde\varepsilon_0^{\mu}
\end{aligned}
\right.
\quad\Longleftrightarrow\quad
\underbrace{\nu' - \nu''}_{-\nu_2\nu_3 + \nu_0} = 2\nu
\quad (\text{i.e. } \nu_0 = 0 \text{ si } \nu_2 = \nu_3 = 0)
\]

\[
\left\{
\begin{aligned}
&\tilde\rho^{-3} = \tilde\sigma^2 = \tilde\omega_0 \\
&\tilde\varepsilon_0 = \tilde\sigma\tilde\rho
\end{aligned}
\right.
\qquad \text{N.B. on pose } l'_0 = \tilde\omega_0^2
\]

\[
\text{\struck{$\underbrace{\tilde\eta\,\tilde\sigma(\tilde\eta)}_{\in \widehat{Z}_{1,1}}\,\lambda''^2 = l_0\,\tilde\omega_0^{2\nu}$,
\quad $\underbrace{\tilde\xi\,\rho(\tilde\xi)\,\rho^2(\tilde\xi)}_{\in Z_{1,1}}\,\lambda'^3 = \tilde\omega_0^{2\nu}$}}
\]

\[
\begin{aligned}
\tilde u(\tilde\rho) &= \lambda'\,\mathrm{int}(\tilde\alpha)(\tilde\rho) \\
\tilde u(\tilde\rho^3) &= \lambda'^3\,\mathrm{int}(\tilde\alpha)(\underbrace{\tilde\rho^3}_{\tilde\omega_0^{-1}})
= \lambda'^3\,\tilde\omega_0^{-\varepsilon_3} \overset{?}{=} \tilde u(\tilde\omega_0^{-1}) = \tilde\omega_0^{-\mu} \\
\lambda'^3 &= \tilde\omega_0^{-(\mu - \varepsilon_3)}
\end{aligned}
\]

\[
\begin{aligned}
\lambda' &= \tilde\omega_0^{-\frac{\mu - \varepsilon_3}{3}} = \tilde\omega_0^{-2\nu''} = l'^{-\nu''}_0 \\
\lambda'' &= \tilde\omega_0^{\frac{\mu - \varepsilon_2}{2}} = \omega_0^{2\nu'} = l'^{\nu'}_0
\end{aligned}
\]

\[
\lambda'\lambda'' = \omega_0^{\frac{\mu - 3\varepsilon_2 + 2\varepsilon_3}{6}}
\]

\[
\text{\struck{$\tilde u(\tilde\sigma) = \lambda''\,\mathrm{int}(\tilde\beta)(\tilde\sigma)$}}
\]

\[
\text{\struck{$\tilde u(\tilde\sigma^2) = \lambda''^2\,\mathrm{int}(\tilde\beta)(\tilde\sigma^2) = \lambda''^2\,\tilde\omega_0^{\varepsilon_2}$}}
\;\text{\struck{$\overset{?}{=} \tilde u(\tilde\omega_0) = \tilde\omega_0^{\mu}$}}
\]

\[
\begin{aligned}
\tilde u(\tilde\varepsilon_0) &= \tilde u(\tilde\sigma)\,\tilde u(\tilde\rho)
= \lambda''\,\tilde\eta\,\tilde\sigma\,\lambda'\,\tilde\xi\,\tilde\rho
= \lambda''\,\tilde\eta\,\tilde\sigma(\lambda'\tilde\xi)\,\tilde\sigma\tilde\rho \\
&= \lambda''\,\tilde\eta . \sigma(\lambda'\tilde\xi)\,\tilde\varepsilon_0 \overset{?}{=} \tilde\varepsilon_0^{\mu}
\end{aligned}
\]

\[
\lambda''\,\tilde\eta\,\sigma(\lambda'\tilde\xi) \overset{?}{=} \tilde l_0^{\nu}
\]

\[
(\lambda'\lambda'')\,\underbrace{\sigma(\tilde\eta)\,\tilde\xi}_{\tilde\eta^{-1}\tilde\xi} \overset{?}{=} \tilde l_1^{\nu}
\qquad
\begin{aligned}
\tilde\eta &= \tilde\beta\,\sigma(\tilde\beta^{-1}) \\
\sigma(\tilde\eta) &= \sigma(\tilde\beta)\,\tilde\beta^{-1} = \tilde\eta^{-1}
\end{aligned}
\]

\[
(\lambda'\lambda'') \overset{?}{=} \tilde\xi^{-1}\,\tilde\eta\,\tilde l_1^{\nu} \quad ?
\]

\[
\begin{array}{ccccccccc}
\mathfrak{S}_\nu & & \mathfrak{S}_{\nu-1} & & \mathfrak{S}_5 & & \mathfrak{S}_4 & & \\
\Gamma^{!+}_{0,\nu} & \longrightarrow & \Gamma^{!+}_{0,\nu-1} & \cdots & \Gamma^{!+}_{0,5} & \longrightarrow & \Gamma^{!+}_{0,4} & \longrightarrow & \Gamma^{!+}_{0,3} \\
\cup & & \cup & & \cup & & \wr & & \| \\
 & & & & & & Sl(2,\mathbb{Z})/\pm 1 & & \{1\} \\
 & & & & & & \cup & & \\
\Pi_{0,\nu-1} & & \Pi_{0,\nu-2} & \cdots & \Pi_{0,4} & & \Pi_{0,3} & & \\
\cup & & \cup & & \cup & & \cup & & \\
L_0^{(0,\nu-1)} & & L_0^{(0,\nu-1)} & & L_0^{(0,4)} & & L_0^{(0,3)} & &
\end{array}
\]

\[
k[x_1, x_2, \ldots, x_\nu] \subset k\{x_1\}\{x_2\} \cdots \{x_\nu\}
\]

\[
\begin{array}{ccccccc}
\Gamma^{!+}_{1,\nu} & \longrightarrow & \cdots & \Gamma^{!+}_{1,3} & \longrightarrow & \Gamma^{!+}_{1,2} \longrightarrow & \Gamma^{!+}_{1,1} \\
 & & & \cup & & \cup & \wr \\
\Pi_{1,\nu-1} & & & \Pi_{1,2} & & \Pi_{1,1} & Sl(2,\mathbb{Z}) \\
\cup & & & \cup & & \cup & \cup \\
L_0^{(1,\nu-1)} & & & L_0^{(1,2)} & & L_0^{(1,1)} & L_0^{\ill{}}
\end{array}
\]

\[
\begin{array}{cccccc}
\Gamma^{!+}_{g,\nu} \longrightarrow \cdots & & \Gamma^{!+}_{g,2} & \longrightarrow \Gamma^{!+}_{g,1} & \longrightarrow \Gamma^{!+}_{g,0} & \\
\cup & & \cup & \cup & \cup & \\
\Pi_{g,\nu-1} & \cdots & \Pi_{g,1} & \Pi_g & \Delta_g \simeq \mathbb{Z}^g & \\
\cup & & \cup & \cup & & \\
L_0^{(g,\nu-1)} & & L_0^{(g,1)} & \Lambda_g\ (\simeq \mathbb{Z}) & & \\
 & & & ? & &
\end{array}
\]

\[
  g-1 = \sum (g_i - 1) + \mu
\]

\[
  g = \sum g_i + \mu + 1 - s
\]

\[
  \operatorname{card} S = \nu, \qquad \sum_{\delta \in \Delta} (g_\delta - 1) + \underbrace{\mu}_{\operatorname{card} \widetilde{D}} = g - 1,
\]

\[
  \operatorname{card}\bigl(D \amalg_{\widetilde{D}} \Delta\bigr) = 1
\]

\begin{align*}
  \widetilde{D} &= \text{image inv.\ de } D \text{ dans } \widetilde{X} \\
  \widetilde{S} &= \text{image inv.\ de } S \text{ dans } \widetilde{X} \qquad (\text{NB } \widetilde{S} \xrightarrow{\sim} S) \\
  \Delta &= \pi_0(\widetilde{X}) \simeq \text{ens des comp.\ irréd.\ de } X \\
  D &= (\text{pour mémoire}) \text{ ens des pts doubles}
\end{align*}

\[
  \mu(\Sigma) = \sum_{\delta \in \Delta} \mu(\delta, \Sigma)
\]

\[
  \mu(\delta, \Sigma) = \mu(g_\delta, \nu_\delta) \overset{\text{déf}}{=} 3(g_\delta - 1) + \nu_\delta
\]

\begin{align*}
  \mu(\Sigma) &= \sum_{\delta} \mu(\delta, \Sigma) = 3 \sum (g_\delta - 1) + \underbrace{\sum \nu_\delta}_{\operatorname{card}\widetilde{D} + \operatorname{card}\widetilde{S}} \\
  &= 3 \sum (g_\delta - 1) + 2\mu + \nu \\
  &= 3 \underbrace{\Bigl(\sum_{\delta} (g_\delta - 1) + \mu\Bigr)}_{g-1} + \nu - \mu \\
  &= \underbrace{\bigl(3(g-1) + \nu\bigr)}_{\mu(g, \nu)} - \mu = \mu(g, \nu) - \mu
\end{align*}