Cote n° 138 · pages 4–126
· 432 displayed formulas · Cartes. Etude arithmétique (1976, 77 ?) : bon de commande (s.d.), notes manuscrites (s.d.).
Inventory dating : [vers 1976-1978]
Édition de démonstration
\[\mathcal{G} = \mathrm{Aut}_{k}(\mathfrak{X}), \qquad
\mathfrak{G} = \mathrm{Aut}_{\bar{k}}(\mathfrak{X})\]
LaTeX source
\[
\mathcal{G} = \mathrm{Aut}_{k}(\mathfrak{X}), \qquad
\mathfrak{G} = \mathrm{Aut}_{\bar{k}}(\mathfrak{X})
\]\[1 \to \mathfrak{G} \to \mathcal{G} \to \Gamma .\]
LaTeX source
\[
1 \to \mathfrak{G} \to \mathcal{G} \to \Gamma .
\]\[\mathcal{G} \simeq \Gamma \underset{1/2}{\cdot} \mathfrak{G}\]
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\[
\mathcal{G} \simeq \Gamma \underset{1/2}{\cdot} \mathfrak{G}
\]\[X \otimes_k \bar{k} \simeq \mathfrak{X}\]
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\[
X \otimes_k \bar{k} \simeq \mathfrak{X}
\]\[1 \to \mathfrak{G} \to \mathcal{G}_1 \to \Gamma_1\]
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\[
1 \to \mathfrak{G} \to \mathcal{G}_1 \to \Gamma_1
\]\[\mathcal{G}_1 = \Gamma_1 \underset{1/2}{\cdot} \mathfrak{G} .\]
LaTeX source
\[
\mathcal{G}_1 = \Gamma_1 \underset{1/2}{\cdot} \mathfrak{G} .
\]\[\mathcal{E} = \mathrm{Aut}_k(\widetilde{\mathfrak{X}} \to \mathfrak{X})\]
LaTeX source
\[
\mathcal{E} = \mathrm{Aut}_k(\widetilde{\mathfrak{X}} \to \mathfrak{X})
\]\[1 \to \pi_1 \to \mathcal{E} \to \mathcal{G} \to 1\]
LaTeX source
\[
1 \to \pi_1 \to \mathcal{E} \to \mathcal{G} \to 1
\]\[\pi_1 = \mathrm{Aut}(\widetilde{\mathfrak{X}}/\mathfrak{X})\]
LaTeX source
\[
\pi_1 = \mathrm{Aut}(\widetilde{\mathfrak{X}}/\mathfrak{X})
\]\[N \subset \pi_1, \qquad \pi_1/N = G = \mathrm{Aut}(\mathfrak{X}'/\mathfrak{X}),
\qquad \mathfrak{X}' = \widetilde{\mathfrak{X}}/N\]
LaTeX source
\[
N \subset \pi_1, \qquad \pi_1/N = G = \mathrm{Aut}(\mathfrak{X}'/\mathfrak{X}),
\qquad \mathfrak{X}' = \widetilde{\mathfrak{X}}/N
\]\[\mathcal{E}' = \mathrm{Norm}_{\mathcal{E}}(N) \supset \pi_1\]
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\[
\mathcal{E}' = \mathrm{Norm}_{\mathcal{E}}(N) \supset \pi_1
\]\[\mathcal{G}' = \mathcal{E}'/N\]
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\[
\mathcal{G}' = \mathcal{E}'/N
\]\[\mathcal{G}' \xrightarrow{\ \sim\ } \mathrm{Aut}_k(\mathfrak{X}' \to \mathfrak{X})\]
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\[
\mathcal{G}' \xrightarrow{\ \sim\ } \mathrm{Aut}_k(\mathfrak{X}' \to \mathfrak{X})
\]\[\mathfrak{G}' = \mathrm{Aut}_{\bar{k}}(\mathfrak{X}' \to \mathfrak{X})\]
LaTeX source
\[
\mathfrak{G}' = \mathrm{Aut}_{\bar{k}}(\mathfrak{X}' \to \mathfrak{X})
\]\[1 \to G \to \mathfrak{G}' \to \mathfrak{G} ,\]
LaTeX source
\[
1 \to G \to \mathfrak{G}' \to \mathfrak{G} ,
\]\[\begin{aligned}
\mathfrak{G}' &\simeq \text{image inverse de } \mathfrak{G} \subset \mathcal{G} \text{ par } \mathcal{G}' \to \mathcal{G} \\
&\simeq \mathrm{Ker}(\mathcal{G}' \to \Gamma)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\mathfrak{G}' &\simeq \text{image inverse de } \mathfrak{G} \subset \mathcal{G} \text{ par } \mathcal{G}' \to \mathcal{G} \\
&\simeq \mathrm{Ker}(\mathcal{G}' \to \Gamma)
\end{aligned}
\]\[\mathcal{G}' \simeq \Gamma \underset{1/2}{\cdot} \mathfrak{G}' .\]
LaTeX source
\[
\mathcal{G}' \simeq \Gamma \underset{1/2}{\cdot} \mathfrak{G}' .
\]\[\begin{aligned}
\mathcal{E}_1 &= \mathrm{Aut}_{k_1}(\widetilde{\mathfrak{X}} \to \mathfrak{X})
= \text{l'image inverse de } \Gamma_1 \subset \Gamma \\
&\qquad \text{par } \mathcal{E} \to \mathcal{G} \to \Gamma \text{ (composé)} \\
\mathcal{E}_1' &= \ldots\ldots = \mathcal{E}' \cap \mathcal{E}_1 \\
\mathcal{G}_1' &= \mathrm{Aut}_{k_1}(\mathfrak{X}' \to \mathfrak{X})
= \text{l'image inverse de } \Gamma_1 \subset \Gamma \\
&\qquad \text{par } \mathcal{G}' \to \mathcal{G} \to \Gamma
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\mathcal{E}_1 &= \mathrm{Aut}_{k_1}(\widetilde{\mathfrak{X}} \to \mathfrak{X})
= \text{l'image inverse de } \Gamma_1 \subset \Gamma \\
&\qquad \text{par } \mathcal{E} \to \mathcal{G} \to \Gamma \text{ (composé)} \\
\mathcal{E}_1' &= \ldots\ldots = \mathcal{E}' \cap \mathcal{E}_1 \\
\mathcal{G}_1' &= \mathrm{Aut}_{k_1}(\mathfrak{X}' \to \mathfrak{X})
= \text{l'image inverse de } \Gamma_1 \subset \Gamma \\
&\qquad \text{par } \mathcal{G}' \to \mathcal{G} \to \Gamma
\end{aligned}
\]\[\mathcal{G}_1' \simeq \Gamma_1 \underset{1/2}{\cdot} \mathfrak{G}'\]
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\[
\mathcal{G}_1' \simeq \Gamma_1 \underset{1/2}{\cdot} \mathfrak{G}'
\]\[\overbrace{G \quad \mathfrak{G} \quad \Gamma}^{\mathcal{G}'}\]
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\[
\overbrace{G \quad \mathfrak{G} \quad \Gamma}^{\mathcal{G}'}
\]\[\underbrace{G \quad \mathfrak{G}}_{\mathfrak{G}'} \qquad
\underbrace{\mathfrak{G} \quad \Gamma}_{\mathcal{G}}\]
LaTeX source
\[
\underbrace{G \quad \mathfrak{G}}_{\mathfrak{G}'} \qquad
\underbrace{\mathfrak{G} \quad \Gamma}_{\mathcal{G}}
\]\[1 \to \mathfrak{G}' \to \mathcal{G}_i' \to \gamma_i \to 1\]
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\[
1 \to \mathfrak{G}' \to \mathcal{G}_i' \to \gamma_i \to 1
\]\[(\mathfrak{X}' \to \mathfrak{X}) \simeq (X_i' \to X_i) \otimes_{k_i} \bar{k} ;\]
LaTeX source
\[
(\mathfrak{X}' \to \mathfrak{X}) \simeq (X_i' \to X_i) \otimes_{k_i} \bar{k} ;
\]\[1 \to G_i \to \mathfrak{G}_i' \to \mathfrak{G}_i \to 1\]
LaTeX source
\[
1 \to G_i \to \mathfrak{G}_i' \to \mathfrak{G}_i \to 1
\]\[Z = \mathrm{Centr}_{\mathcal{G}'}(\mathfrak{G}')\]
LaTeX source
\[
Z = \mathrm{Centr}_{\mathcal{G}'}(\mathfrak{G}')
\]\[\mathfrak{z} \overset{\mathrm{df}}{=} Z \cap \mathfrak{G}' = \text{Centre de } \mathfrak{G}'\]
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\[
\mathfrak{z} \overset{\mathrm{df}}{=} Z \cap \mathfrak{G}' = \text{Centre de } \mathfrak{G}'
\]\[1 \to \mathfrak{z} \longrightarrow Z \longrightarrow \Gamma\]
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\[
1 \to \mathfrak{z} \longrightarrow Z \longrightarrow \Gamma
\]\[1 \to \mathfrak{z} \to Z \to \Gamma_0 \to 1 , \qquad \mathfrak{z} = \mathrm{Centre}(\mathfrak{G}')\]
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\[
1 \to \mathfrak{z} \to Z \to \Gamma_0 \to 1 , \qquad \mathfrak{z} = \mathrm{Centre}(\mathfrak{G}')
\]\[H^1(\Gamma_i, \mathfrak{z}) = \mathrm{Hom}(\Gamma_i, \mathfrak{z}) \ \ldots\ldots\]
LaTeX source
\[
H^1(\Gamma_i, \mathfrak{z}) = \mathrm{Hom}(\Gamma_i, \mathfrak{z}) \ \ldots\ldots
\]\[\hat{X} = \mathbb{P}^1_{\mathbb{Q}}, \qquad
X = \mathbb{P}^1_{\mathbb{Q}} \setminus \{0,\infty\} = \mathbb{G}_{m\,\mathbb{Q}} .\]
LaTeX source
\[
\hat{X} = \mathbb{P}^1_{\mathbb{Q}}, \qquad
X = \mathbb{P}^1_{\mathbb{Q}} \setminus \{0,\infty\} = \mathbb{G}_{m\,\mathbb{Q}} .
\]\[\mathcal{G} = \Gamma \underset{1/2}{\cdot} \bar{\mathbb{Q}}^{*}\]
LaTeX source
\[
\mathcal{G} = \Gamma \underset{1/2}{\cdot} \bar{\mathbb{Q}}^{*}
\]\[X_n \xrightarrow{\ t \mapsto t^n\ } X ,\]
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\[
X_n \xrightarrow{\ t \mapsto t^n\ } X ,
\]\[\mathcal{G}_n = \mathrm{Aut}_k(\mathfrak{X}_n \to \mathfrak{X}) \xrightarrow{\ \sim\ } \mathrm{Aut}_k(\mathfrak{X}_n) ,
\qquad
\mathcal{G}_n' = \Gamma \underset{1/2}{\cdot} \bar{\mathbb{Q}}^{*}\]
LaTeX source
\[
\mathcal{G}_n = \mathrm{Aut}_k(\mathfrak{X}_n \to \mathfrak{X}) \xrightarrow{\ \sim\ } \mathrm{Aut}_k(\mathfrak{X}_n) ,
\qquad
\mathcal{G}_n' = \Gamma \underset{1/2}{\cdot} \bar{\mathbb{Q}}^{*}
\]\[\mathcal{G}_n \longrightarrow \mathcal{G}\]
LaTeX source
\[
\mathcal{G}_n \longrightarrow \mathcal{G}
\]\[\mathcal{G}_\infty = \mathrm{Aut}_k(\mathfrak{X}_\infty \to \mathfrak{X}) \simeq \Gamma \cdot T_\infty(\bar{\mathbb{Q}}^{*})\]
LaTeX source
\[
\mathcal{G}_\infty = \mathrm{Aut}_k(\mathfrak{X}_\infty \to \mathfrak{X}) \simeq \Gamma \cdot T_\infty(\bar{\mathbb{Q}}^{*})
\]\[\mathcal{G}_n = \bigl(\Gamma \times \mathfrak{S}_S\bigr) \cdot \pi_n ,
\qquad \pi_n = \mu_n^{S}/\mu_n\]
LaTeX source
\[
\mathcal{G}_n = \bigl(\Gamma \times \mathfrak{S}_S\bigr) \cdot \pi_n ,
\qquad \pi_n = \mu_n^{S}/\mu_n
\]\[\hat{\mathfrak{X}} \underset{\varphi_s}{\simeq} \mathbb{P}^1_{\bar{\mathbb{Q}}}\]
LaTeX source
\[
\hat{\mathfrak{X}} \underset{\varphi_s}{\simeq} \mathbb{P}^1_{\bar{\mathbb{Q}}}
\]\[f(z) = z^n\]
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\[ f(z) = z^n \]
\[\mathcal{E}_n = \Gamma \cdot \mu_n(\bar{\mathbb{Q}})\]
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\[
\mathcal{E}_n = \Gamma \cdot \mu_n(\bar{\mathbb{Q}})
\]\[\begin{cases}
S_0 = \{0\}, \quad S_1 = \mu_n(\bar{\mathbb{Q}}), \quad S_\infty = \{\infty\} \\
E^{+} = \{z \mid z^n = -j\} \\
E^{-} = \{z \mid z^n = -\bar{j}\}
\end{cases}\]
LaTeX source
\[
\begin{cases}
S_0 = \{0\}, \quad S_1 = \mu_n(\bar{\mathbb{Q}}), \quad S_\infty = \{\infty\} \\
E^{+} = \{z \mid z^n = -j\} \\
E^{-} = \{z \mid z^n = -\bar{j}\}
\end{cases}
\]\[E = \bigl\{z \mid z^{3n} = -1,\ z^n \neq -1\bigr\}\]
LaTeX source
\[
E = \bigl\{z \mid z^{3n} = -1,\ z^n \neq -1\bigr\}
\]\[\begin{aligned}
R' &= \{z \mid z^n = \tfrac{1}{2}\} \\
R'' &= \{z \mid z^n = 2\} \\
R &= \{z \mid z^n = -1\}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
R' &= \{z \mid z^n = \tfrac{1}{2}\} \\
R'' &= \{z \mid z^n = 2\} \\
R &= \{z \mid z^n = -1\}
\end{aligned}
\]\[E \cup R = \{z \mid z^{3n} = -1\}\]
LaTeX source
\[
E \cup R = \{z \mid z^{3n} = -1\}
\]\[\mathbb{Q}[E \cup R] = \mathbb{Q}[\mu_{6n}(\bar{\mathbb{Q}})] \supset \mathbb{Q}[S_0 \cup S_1 \cup S_\infty]\]
LaTeX source
\[
\mathbb{Q}[E \cup R] = \mathbb{Q}[\mu_{6n}(\bar{\mathbb{Q}})] \supset \mathbb{Q}[S_0 \cup S_1 \cup S_\infty]
\]\[\mathbb{Q}[R', R''] = \mathbb{Q}[R'] = \mathbb{Q}[\sqrt[n]{2}]\]
LaTeX source
\[
\mathbb{Q}[R', R''] = \mathbb{Q}[R'] = \mathbb{Q}[\sqrt[n]{2}]
\]\[1 \to \mu_{n}(\bar{\mathbb{Q}}) \longrightarrow \Gamma' \longrightarrow (\mathbb{Z}/n\mathbb{Z})^{*} \to 1\]
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\[
1 \to \mu_{n}(\bar{\mathbb{Q}}) \longrightarrow \Gamma' \longrightarrow (\mathbb{Z}/n\mathbb{Z})^{*} \to 1
\]\[\begin{aligned}
0, 1, \infty &\longmapsto 0 \quad \text{ordre } 2 \\
\tfrac{1}{2}, 2, -1 &\longmapsto 1 \quad \text{ordre } 2 \\
-j, -\bar{j} &\longmapsto \infty \quad \text{ordre } 3
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
0, 1, \infty &\longmapsto 0 \quad \text{ordre } 2 \\
\tfrac{1}{2}, 2, -1 &\longmapsto 1 \quad \text{ordre } 2 \\
-j, -\bar{j} &\longmapsto \infty \quad \text{ordre } 3
\end{aligned}
\]\[g(z) = \frac{27\, z^2 (z-1)^2}{4\,(z^2 - z + 1)^3}
\qquad
g_n(z) = \frac{27}{4}\,\frac{z^{2n}(z^n - 1)^2}{(z^{2n} - z^n + 1)^3}\]
LaTeX source
\[
g(z) = \frac{27\, z^2 (z-1)^2}{4\,(z^2 - z + 1)^3}
\qquad
g_n(z) = \frac{27}{4}\,\frac{z^{2n}(z^n - 1)^2}{(z^{2n} - z^n + 1)^3}
\]\[\lambda \, \frac{4}{27} = 1\]
LaTeX source
\[
\lambda \, \frac{4}{27} = 1
\]\[g(z) = 1 \iff 4(z^2 - z + 1)^3 - 27 z^2 (z-1)^2 = 0\]
LaTeX source
\[ g(z) = 1 \iff 4(z^2 - z + 1)^3 - 27 z^2 (z-1)^2 = 0 \]
\[\begin{aligned}
4(z^2 - z + 1)^3 - 27 z^2 (z-1)^2
&= 4\bigl[(z+1)(z - \tfrac{1}{2})(z-2)\bigr]^2 \\
&= 4\bigl[(z+1)(z^2 - \tfrac{5}{2} z + 1)\bigr]^2 \\
&= \bigl[(z+1)(2z^2 - 5z + 2)\bigr]^2
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
4(z^2 - z + 1)^3 - 27 z^2 (z-1)^2
&= 4\bigl[(z+1)(z - \tfrac{1}{2})(z-2)\bigr]^2 \\
&= 4\bigl[(z+1)(z^2 - \tfrac{5}{2} z + 1)\bigr]^2 \\
&= \bigl[(z+1)(2z^2 - 5z + 2)\bigr]^2
\end{aligned}
\]\[\begin{aligned}
\underline{\mathrm{Hom}}(D, \mathbb{G}_m) &= \mathcal{U} \\
\downarrow n \qquad & \qquad \downarrow n \\
\underline{\mathrm{Hom}}(D, \mathbb{G}_m) &= \mathcal{U}
\end{aligned}
\qquad \xi\]
LaTeX source
\[
\begin{aligned}
\underline{\mathrm{Hom}}(D, \mathbb{G}_m) &= \mathcal{U} \\
\downarrow n \qquad & \qquad \downarrow n \\
\underline{\mathrm{Hom}}(D, \mathbb{G}_m) &= \mathcal{U}
\end{aligned}
\qquad \xi
\]\[T(\mathbb{Q}) \ni \xi \qquad H^1(\mathbb{Q}, T_\infty(\mathcal{U}))\]
LaTeX source
\[
T(\mathbb{Q}) \ni \xi \qquad H^1(\mathbb{Q}, T_\infty(\mathcal{U}))
\]\[\mathbb{Q}^{*} \qquad Z^1(\Gamma, \mathbb{Z}/n\mathbb{Z})\]
LaTeX source
\[
\mathbb{Q}^{*} \qquad Z^1(\Gamma, \mathbb{Z}/n\mathbb{Z})
\]\[\frac{\gamma \cdot \sqrt[n]{\xi}}{\sqrt[n]{\xi}} = , \qquad
\gamma(\sqrt[n]{\xi}) = \sqrt[n]{\xi} \quad \forall \gamma \in \Gamma\]
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\[
\frac{\gamma \cdot \sqrt[n]{\xi}}{\sqrt[n]{\xi}} = , \qquad
\gamma(\sqrt[n]{\xi}) = \sqrt[n]{\xi} \quad \forall \gamma \in \Gamma
\]\[\Updownarrow\]
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\[ \Updownarrow \]
\[\sqrt[n]{\xi} \in \mathbb{Q} \qquad \xi \in \mathbb{Q}^{*n}\]
LaTeX source
\[
\sqrt[n]{\xi} \in \mathbb{Q} \qquad \xi \in \mathbb{Q}^{*n}
\]\[\varprojlim \mathbb{Q}^{*}/\mathbb{Q}^{*n} \longrightarrow Z^1(\Gamma, \hat{\mathbb{Z}})\]
LaTeX source
\[
\varprojlim \mathbb{Q}^{*}/\mathbb{Q}^{*n} \longrightarrow Z^1(\Gamma, \hat{\mathbb{Z}})
\]\[0 \to \hat{\mathbb{Z}} \to Z^1(\Gamma, \hat{\mathbb{Z}})\]
LaTeX source
\[
0 \to \hat{\mathbb{Z}} \to Z^1(\Gamma, \hat{\mathbb{Z}})
\]\[\begin{aligned}
\mathbb{Q}^{*}/\mathbb{Q}^{*n} &\xrightarrow{\ \sim\ } H^1(\Gamma, \mathbb{Z}/n\mathbb{Z}) \\
\mathbb{Q}^{*}/\mathbb{Q}^{*n} \oplus (\mathbb{Z}/n\mathbb{Z}) &\simeq Z^1(\Gamma, \mathbb{Z}/n\mathbb{Z})
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\mathbb{Q}^{*}/\mathbb{Q}^{*n} &\xrightarrow{\ \sim\ } H^1(\Gamma, \mathbb{Z}/n\mathbb{Z}) \\
\mathbb{Q}^{*}/\mathbb{Q}^{*n} \oplus (\mathbb{Z}/n\mathbb{Z}) &\simeq Z^1(\Gamma, \mathbb{Z}/n\mathbb{Z})
\end{aligned}
\]\[\boxed{\ \Gamma \longrightarrow \mathrm{Aff}(1, \hat{\mathbb{Z}})\ }\]
LaTeX source
\[
\boxed{\ \Gamma \longrightarrow \mathrm{Aff}(1, \hat{\mathbb{Z}})\ }
\]\[Z^1(\Gamma, \hat{\mathbb{Z}}) \simeq \varprojlim \mathbb{Q}^{*}/\mathbb{Q}^{*n} \oplus \hat{\mathbb{Z}}\]
LaTeX source
\[
Z^1(\Gamma, \hat{\mathbb{Z}}) \simeq \varprojlim \mathbb{Q}^{*}/\mathbb{Q}^{*n} \oplus \hat{\mathbb{Z}}
\]\[\varprojlim \mathbb{Q}^{*}/\mathbb{Q}^{*n}
= \{\pm 1\} + \varprojlim (\mathbb{Z}/n\mathbb{Z})^{P}
\qquad
\varprojlim (\mathbb{Z}/n\mathbb{Z})^{P} \to \hat{\mathbb{Z}}^{P}\]
LaTeX source
\[
\varprojlim \mathbb{Q}^{*}/\mathbb{Q}^{*n}
= \{\pm 1\} + \varprojlim (\mathbb{Z}/n\mathbb{Z})^{P}
\qquad
\varprojlim (\mathbb{Z}/n\mathbb{Z})^{P} \to \hat{\mathbb{Z}}^{P}
\]\[\exp\frac{2i\pi}{2n} = \exp\frac{i\pi}{n}\]
LaTeX source
\[
\exp\frac{2i\pi}{2n} = \exp\frac{i\pi}{n}
\]\[\mathbb{Q}^{*} \longrightarrow Z^{1}(\Gamma, \widehat{\mathbb{Z}}^{*}) \longrightarrow H^{1}(\Gamma, \widehat{\mathbb{Z}})\]
LaTeX source
\[
\mathbb{Q}^{*} \longrightarrow Z^{1}(\Gamma, \widehat{\mathbb{Z}}^{*}) \longrightarrow H^{1}(\Gamma, \widehat{\mathbb{Z}})
\]\[\mathbb{Q}^{*} \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}^{(\mathbb{P})}, \qquad
\varphi(\gamma\gamma') = \varphi(\gamma) + \gamma\varphi(\gamma'), \qquad
\varphi(\gamma) = a - \gamma a\]
LaTeX source
\[
\mathbb{Q}^{*} \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}^{(\mathbb{P})}, \qquad
\varphi(\gamma\gamma') = \varphi(\gamma) + \gamma\varphi(\gamma'), \qquad
\varphi(\gamma) = a - \gamma a
\]\[\mathbb{Q}^{*} \longrightarrow Z^{1}(\Gamma, \widehat{\mathbb{Z}}_{\ell}), \qquad
\widehat{\mathbb{Z}}^{\mathbb{P}} \longrightarrow Z^{1}, \qquad
\varprojlim_{n} Z^{1}(\Gamma, \mathbb{Z}/n\mathbb{Z})\]
LaTeX source
\[
\mathbb{Q}^{*} \longrightarrow Z^{1}(\Gamma, \widehat{\mathbb{Z}}_{\ell}), \qquad
\widehat{\mathbb{Z}}^{\mathbb{P}} \longrightarrow Z^{1}, \qquad
\varprojlim_{n} Z^{1}(\Gamma, \mathbb{Z}/n\mathbb{Z})
\]\[\mathbb{Q}^{*}/\mathbb{Q}^{*n} \simeq H^{1}(\Gamma, \mu_n) \simeq Z^{1}(\Gamma, \mathbb{Z}/n\mathbb{Z})\]
LaTeX source
\[
\mathbb{Q}^{*}/\mathbb{Q}^{*n} \simeq H^{1}(\Gamma, \mu_n) \simeq Z^{1}(\Gamma, \mathbb{Z}/n\mathbb{Z})
\]\[\mathbb{G}_{m,\mathbb{Q}} \xrightarrow{\ \lambda \mapsto \lambda^{n}\ } \mathbb{G}_{m,\mathbb{Q}}
\qquad\qquad \mathbb{Q} \longrightarrow \overline{\mathbb{Q}}\]
LaTeX source
\[
\mathbb{G}_{m,\mathbb{Q}} \xrightarrow{\ \lambda \mapsto \lambda^{n}\ } \mathbb{G}_{m,\mathbb{Q}}
\qquad\qquad \mathbb{Q} \longrightarrow \overline{\mathbb{Q}}
\]\[1 \longrightarrow \mu_n \longrightarrow \mathcal{E}'_n \rightleftarrows \Gamma' \longrightarrow 1\]
LaTeX source
\[
1 \longrightarrow \mu_n \longrightarrow \mathcal{E}'_n \rightleftarrows \Gamma' \longrightarrow 1
\]\[\mathcal{E}'_n = \bigl[(\mathbb{Z}/6n\mathbb{Z})^{*} \cdot \mu_\nu\bigr] \cdot \mu_n
\qquad \bigl[(\mathbb{Z}/6n\mathbb{Z})^{*} \cdot \mu_\nu \to \mathrm{Aff}(\mu_n)\bigr]\]
LaTeX source
\[
\mathcal{E}'_n = \bigl[(\mathbb{Z}/6n\mathbb{Z})^{*} \cdot \mu_\nu\bigr] \cdot \mu_n
\qquad \bigl[(\mathbb{Z}/6n\mathbb{Z})^{*} \cdot \mu_\nu \to \mathrm{Aff}(\mu_n)\bigr]
\]\[\begin{array}{cccccl}
E^{+} & R' & R & S_1 & S_0 & \\
\wr & \wr & \wr & \wr & \wr & \bigr\}\ z \mapsto \tfrac{1}{z} \\
E^{-} & R'' & R & S_1 & S_2 &
\end{array}\]
LaTeX source
\[
\begin{array}{cccccl}
E^{+} & R' & R & S_1 & S_0 & \\
\wr & \wr & \wr & \wr & \wr & \bigr\}\ z \mapsto \tfrac{1}{z} \\
E^{-} & R'' & R & S_1 & S_2 &
\end{array}
\]\[1 \longrightarrow \mu_\infty(\overline{\mathbb{Q}}) \longrightarrow \mathcal{E}_\infty \longrightarrow \Gamma \longrightarrow 1\]
LaTeX source
\[
1 \longrightarrow \mu_\infty(\overline{\mathbb{Q}}) \longrightarrow \mathcal{E}_\infty \longrightarrow \Gamma \longrightarrow 1
\]\[\text{(1)} \qquad x^{n} + y^{n} + z^{n} = 0 .\]
LaTeX source
\[
\text{(1)} \qquad x^{n} + y^{n} + z^{n} = 0 .
\]\[\text{(2)} \qquad \Pi_{n\,\mathbb{Q}} = \mu_n^{3}/\mu_n \ (\text{diag.}) \simeq \mu_n \otimes_{\mathbb{Z}} (\mathbb{Z}^{3}/\mathbb{Z})\]
LaTeX source
\[
\text{(2)} \qquad \Pi_{n\,\mathbb{Q}} = \mu_n^{3}/\mu_n \ (\text{diag.}) \simeq \mu_n \otimes_{\mathbb{Z}} (\mathbb{Z}^{3}/\mathbb{Z})
\]\[\text{(3)} \qquad X + Y + Z = 0\]
LaTeX source
\[
\text{(3)} \qquad X + Y + Z = 0
\]\[\text{(3')} \qquad \xi + \eta + 1 = 0\]
LaTeX source
\[
\text{(3')} \qquad \xi + \eta + 1 = 0
\]\[\eta = -1 - \xi ,\]
LaTeX source
\[ \eta = -1 - \xi , \]
\[\text{(3'')} \qquad t = -\xi \quad \text{d'où} \quad \xi = -t, \ \eta = t - 1, \ \zeta = 1\]
LaTeX source
\[
\text{(3'')} \qquad t = -\xi \quad \text{d'où} \quad \xi = -t, \ \eta = t - 1, \ \zeta = 1
\]\[\text{(5)} \qquad C_\infty = \varprojlim C_n, \qquad C_\infty^{*} = \varprojlim C_n^{*}\]
LaTeX source
\[
\text{(5)} \qquad C_\infty = \varprojlim C_n, \qquad C_\infty^{*} = \varprojlim C_n^{*}
\]\[\text{(6)} \qquad \Pi_\infty = \varprojlim \Pi_n \simeq \mu^{\leftarrow}_\infty \otimes_{\widehat{\mathbb{Z}}} (\widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}}) ,\]
LaTeX source
\[
\text{(6)} \qquad \Pi_\infty = \varprojlim \Pi_n \simeq \mu^{\leftarrow}_\infty \otimes_{\widehat{\mathbb{Z}}} (\widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}}) ,
\]\[\text{(7)} \qquad \Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\]
LaTeX source
\[
\text{(7)} \qquad \Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})
\]\[\text{(8)} \qquad \overline{\Pi}_n = \Pi_n(\overline{\mathbb{Q}}) = \mu_n^{3}/\mu_n \simeq \mu_n \otimes_{\mathbb{Z}} \mathbb{Z}^{3}/\mathbb{Z} \simeq \mu_n \otimes_{\widehat{\mathbb{Z}}} (\widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}})\]
LaTeX source
\[
\text{(8)} \qquad \overline{\Pi}_n = \Pi_n(\overline{\mathbb{Q}}) = \mu_n^{3}/\mu_n \simeq \mu_n \otimes_{\mathbb{Z}} \mathbb{Z}^{3}/\mathbb{Z} \simeq \mu_n \otimes_{\widehat{\mathbb{Z}}} (\widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}})
\]\[\text{(9)} \qquad \mathcal{E}_n = (\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \Pi_n\]
LaTeX source
\[
\text{(9)} \qquad \mathcal{E}_n = (\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \Pi_n
\]\[\text{(10)} \qquad \Gamma \xrightarrow{\ \chi_n\ } (\mathbb{Z}/n\mathbb{Z})^{*} .\]
LaTeX source
\[
\text{(10)} \qquad \Gamma \xrightarrow{\ \chi_n\ } (\mathbb{Z}/n\mathbb{Z})^{*} .
\]\[\text{(11)} \qquad
\begin{cases}
\Pi_\infty = \mathrm{T}^{3}/\mathrm{T} \simeq \mathrm{T} \otimes \mathbb{Z}^{3}/\mathbb{Z} \simeq \mathrm{T} \otimes \widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}}, \\
\text{où } \mathrm{T} = \varprojlim \mu_n = \mu^{\leftarrow}_\infty(\overline{\mathbb{Q}})
\end{cases}\]
LaTeX source
\[
\text{(11)} \qquad
\begin{cases}
\Pi_\infty = \mathrm{T}^{3}/\mathrm{T} \simeq \mathrm{T} \otimes \mathbb{Z}^{3}/\mathbb{Z} \simeq \mathrm{T} \otimes \widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}}, \\
\text{où } \mathrm{T} = \varprojlim \mu_n = \mu^{\leftarrow}_\infty(\overline{\mathbb{Q}})
\end{cases}
\]\[\text{(12)} \qquad \Gamma \xrightarrow{\ \chi_\infty = \chi\ } \widehat{\mathbb{Z}}^{*} .\]
LaTeX source
\[
\text{(12)} \qquad \Gamma \xrightarrow{\ \chi_\infty = \chi\ } \widehat{\mathbb{Z}}^{*} .
\]\[\text{(13)} \qquad
\begin{cases}
\mathfrak{S}_3 \times \Gamma = \mathrm{Aut}_{\mathbb{Q}}\, C_1 \\
(\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \mu_n \overset{\mathrm{df}}{=} \mathcal{E}_n \simeq \mathrm{Aut}_{\mathbb{Q}}(C_n) \\
(\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \mu_\infty \overset{\mathrm{df}}{=} \mathcal{E}_\infty \simeq \mathrm{Aut}_{\mathbb{Q}}(C_\infty)
\end{cases}\]
LaTeX source
\[
\text{(13)} \qquad
\begin{cases}
\mathfrak{S}_3 \times \Gamma = \mathrm{Aut}_{\mathbb{Q}}\, C_1 \\
(\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \mu_n \overset{\mathrm{df}}{=} \mathcal{E}_n \simeq \mathrm{Aut}_{\mathbb{Q}}(C_n) \\
(\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \mu_\infty \overset{\mathrm{df}}{=} \mathcal{E}_\infty \simeq \mathrm{Aut}_{\mathbb{Q}}(C_\infty)
\end{cases}
\]\[\pi_1(\overline{C}_1^{*}) \longrightarrow \Pi_\infty\]
LaTeX source
\[
\pi_1(\overline{C}_1^{*}) \longrightarrow \Pi_\infty
\]\[\text{(14)} \qquad \pi_1(\overline{C}_1^{*})^{\mathrm{ab}} \simeq \Pi_\infty \quad (\simeq H_1(\overline{C}_1^{*}))\]
LaTeX source
\[
\text{(14)} \qquad \pi_1(\overline{C}_1^{*})^{\mathrm{ab}} \simeq \Pi_\infty \quad (\simeq H_1(\overline{C}_1^{*}))
\]\[\begin{aligned}
E_0(n) &= \{(x, y, z) \in \overline{\mathbb{Q}}^{3} \mid x = 0,\ y^{n} + z^{n} = 0\}/\mu_n \\
&\simeq \{(y, z) \in \overline{\mathbb{Q}}^{2} \mid (y/z)^{n} = -1\}/\mu_n \\
&\simeq \{\lambda \in \overline{\mathbb{Q}} \mid \lambda^{n} = -1\}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
E_0(n) &= \{(x, y, z) \in \overline{\mathbb{Q}}^{3} \mid x = 0,\ y^{n} + z^{n} = 0\}/\mu_n \\
&\simeq \{(y, z) \in \overline{\mathbb{Q}}^{2} \mid (y/z)^{n} = -1\}/\mu_n \\
&\simeq \{\lambda \in \overline{\mathbb{Q}} \mid \lambda^{n} = -1\}
\end{aligned}
\]\[\text{(15)} \qquad E_0 \overset{\varphi_0}{\simeq} \text{image inverse de } -1 \in \mu_2 \text{ par } \mu^{\leftarrow}_{\infty} \to \mu_2 .\]
LaTeX source
\[
\text{(15)} \qquad E_0 \overset{\varphi_0}{\simeq} \text{image inverse de } -1 \in \mu_2 \text{ par } \mu^{\leftarrow}_{\infty} \to \mu_2 .
\]\[\text{(16)} \qquad \Pi_\infty \longrightarrow \Pi_2\]
LaTeX source
\[
\text{(16)} \qquad \Pi_\infty \longrightarrow \Pi_2
\]\[\text{(17)} \qquad \Pi_2^{*} = \Pi_2 \setminus \{1\} \simeq (\mathbb{F}_2^{3}/\mathbb{F}_2) \setminus \{0\} .\]
LaTeX source
\[
\text{(17)} \qquad \Pi_2^{*} = \Pi_2 \setminus \{1\} \simeq (\mathbb{F}_2^{3}/\mathbb{F}_2) \setminus \{0\} .
\]\[\text{(18)} \qquad 0 \longrightarrow \Pi_\infty \xrightarrow{\ 2\ } \Pi_\infty \xrightarrow{\ p\ } \Pi_2 \longrightarrow 0, \qquad \Pi_2 \supset I = \{0, 1, \infty\},\]
LaTeX source
\[
\text{(18)} \qquad 0 \longrightarrow \Pi_\infty \xrightarrow{\ 2\ } \Pi_\infty \xrightarrow{\ p\ } \Pi_2 \longrightarrow 0, \qquad \Pi_2 \supset I = \{0, 1, \infty\},
\]\[\text{(19)} \qquad Z_i \subset \Pi_\infty \ (\simeq \mathrm{T})\]
LaTeX source
\[
\text{(19)} \qquad Z_i \subset \Pi_\infty \ (\simeq \mathrm{T})
\]\[\text{(20)} \qquad E_i \simeq p^{-1}(\{i\})/Z_i .\]
LaTeX source
\[
\text{(20)} \qquad E_i \simeq p^{-1}(\{i\})/Z_i .
\]\[\text{(21)} \qquad \alpha = -j, \qquad \overline{\alpha} = -\overline{j}\]
LaTeX source
\[
\text{(21)} \qquad \alpha = -j, \qquad \overline{\alpha} = -\overline{j}
\]\[\text{(22)} \qquad E(\alpha) \sqcup E(\overline{\alpha})\]
LaTeX source
\[
\text{(22)} \qquad E(\alpha) \sqcup E(\overline{\alpha})
\]\[\text{(23)} \qquad
\begin{cases}
E(\alpha)_n = \{(\xi, \eta) \in \overline{\mathbb{Q}}^{2} \mid \xi^{n} = \overline{j},\ \eta^{n} = j\} \\
E(\overline{\alpha})_n = \{(\xi, \eta) \mid \xi^{n} = j,\ \eta^{n} = \overline{j}\}
\end{cases}\]
LaTeX source
\[
\text{(23)} \qquad
\begin{cases}
E(\alpha)_n = \{(\xi, \eta) \in \overline{\mathbb{Q}}^{2} \mid \xi^{n} = \overline{j},\ \eta^{n} = j\} \\
E(\overline{\alpha})_n = \{(\xi, \eta) \mid \xi^{n} = j,\ \eta^{n} = \overline{j}\}
\end{cases}
\]\[\text{(24)} \qquad \widetilde{C}_1 = \mathrm{Isom}(\underbrace{\{1, 2, \infty\}}_{I}, \mu_3) \simeq \{(\zeta_1, \zeta_2, \zeta_3) \in \mu_3^{3} \mid \zeta_1 \neq \zeta_2 \neq \zeta_3 \neq \zeta_1\}\]
LaTeX source
\[
\text{(24)} \qquad \widetilde{C}_1 = \mathrm{Isom}(\underbrace{\{1, 2, \infty\}}_{I}, \mu_3) \simeq \{(\zeta_1, \zeta_2, \zeta_3) \in \mu_3^{3} \mid \zeta_1 \neq \zeta_2 \neq \zeta_3 \neq \zeta_1\}
\]\[\widetilde{C}_1/\mu_3 \longrightarrow \overline{C}_1\]
LaTeX source
\[
\widetilde{C}_1/\mu_3 \longrightarrow \overline{C}_1
\]\[\text{(25)} \qquad \widetilde{C}_1/\mu_3 \xrightarrow{\ \sim\ } \overline{C}_1^{\,\mathfrak{S}_3^{+}} = \{\alpha, \overline{\alpha}\}\]
LaTeX source
\[
\text{(25)} \qquad \widetilde{C}_1/\mu_3 \xrightarrow{\ \sim\ } \overline{C}_1^{\,\mathfrak{S}_3^{+}} = \{\alpha, \overline{\alpha}\}
\]\[\text{(26)} \qquad
\begin{aligned}
\widetilde{C}_n &= \{(x, y, z) \in \overline{\mathbb{Q}}^{3} \mid (x^{n}, y^{n}, z^{n}) \in \widetilde{C}_1\} \\
&= \{(x, y, z) \in \overline{\mathbb{Q}}^{3} \mid x^{3n} = y^{3n} = z^{3n} = 1,\ x^{n} \neq y^{n} \neq z^{n} \neq x^{n}\}
\end{aligned}\]
LaTeX source
\[
\text{(26)} \qquad
\begin{aligned}
\widetilde{C}_n &= \{(x, y, z) \in \overline{\mathbb{Q}}^{3} \mid (x^{n}, y^{n}, z^{n}) \in \widetilde{C}_1\} \\
&= \{(x, y, z) \in \overline{\mathbb{Q}}^{3} \mid x^{3n} = y^{3n} = z^{3n} = 1,\ x^{n} \neq y^{n} \neq z^{n} \neq x^{n}\}
\end{aligned}
\]\[\widetilde{C}_n \longrightarrow \widetilde{C}_1\]
LaTeX source
\[
\widetilde{C}_n \longrightarrow \widetilde{C}_1
\]\[\mu_{3n} \xrightarrow{\ \zeta \mapsto \zeta^{n}\ } \mu_n\]
LaTeX source
\[
\mu_{3n} \xrightarrow{\ \zeta \mapsto \zeta^{n}\ } \mu_n
\]\[\widetilde{C}_n/\mu_{3n} \longrightarrow \widetilde{C}_1/\mu_n .\]
LaTeX source
\[
\widetilde{C}_n/\mu_{3n} \longrightarrow \widetilde{C}_1/\mu_n .
\]\[\text{(28)} \qquad \widetilde{C}_\infty = \{(x, y, z) \in \mathrm{T}^{3} \ (= \mu_\infty^{3}) \mid x_3 \neq y_3 \neq z_3 \neq x_3\}\]
LaTeX source
\[
\text{(28)} \qquad \widetilde{C}_\infty = \{(x, y, z) \in \mathrm{T}^{3} \ (= \mu_\infty^{3}) \mid x_3 \neq y_3 \neq z_3 \neq x_3\}
\]\[\widetilde{C}_\infty \subset \mathrm{T}^{3} \quad \text{stable par } \mathrm{T}\]
LaTeX source
\[
\widetilde{C}_\infty \subset \mathrm{T}^{3} \quad \text{stable par } \mathrm{T}
\]\[\text{(29)} \qquad \widetilde{C}_\infty/\mathrm{T} \hookrightarrow \mathrm{T}^{3}/\mathrm{T} \simeq \Pi_\infty ,\]
LaTeX source
\[
\text{(29)} \qquad \widetilde{C}_\infty/\mathrm{T} \hookrightarrow \mathrm{T}^{3}/\mathrm{T} \simeq \Pi_\infty ,
\]\[\text{(30)} \qquad \widetilde{C}_\infty/\mu_\infty \simeq \text{image inverse de } \overbrace{\widetilde{C}_1/\mu_3}^{\omega} \subset \mu_3^{3}/\mu_3 = \Pi_3 \ \text{par } \Pi_\infty \to \Pi_3 .\]
LaTeX source
\[
\text{(30)} \qquad \widetilde{C}_\infty/\mu_\infty \simeq \text{image inverse de } \overbrace{\widetilde{C}_1/\mu_3}^{\omega} \subset \mu_3^{3}/\mu_3 = \Pi_3 \ \text{par } \Pi_\infty \to \Pi_3 .
\]\[\text{(31)} \qquad 0 \longrightarrow \Pi_\infty \xrightarrow{\ 3\ } \Pi_\infty \xrightarrow{\ p_3\ } \Pi_3 \longrightarrow 0, \qquad \omega \subset \Pi_3,\]
LaTeX source
\[
\text{(31)} \qquad 0 \longrightarrow \Pi_\infty \xrightarrow{\ 3\ } \Pi_\infty \xrightarrow{\ p_3\ } \Pi_3 \longrightarrow 0, \qquad \omega \subset \Pi_3,
\]\[\text{(32)} \qquad \alpha'(0) = 2, \quad \alpha'(1) = -1, \quad \alpha'(\infty) = \tfrac12\]
LaTeX source
\[
\text{(32)} \qquad \alpha'(0) = 2, \quad \alpha'(1) = -1, \quad \alpha'(\infty) = \tfrac12
\]\[\text{(33)} \qquad \alpha'_{*} = \{\alpha'(0), \alpha'(1), \alpha'(2)\} .\]
LaTeX source
\[
\text{(33)} \qquad \alpha'_{*} = \{\alpha'(0), \alpha'(1), \alpha'(2)\} .
\]\[\text{(34)} \qquad
\begin{cases}
E_n(\alpha'(i)) \simeq \{(\xi_j) \in \overline{\mathbb{Q}}^{I \setminus \{i\}} \mid \xi_j^{n} = -\tfrac12 \ \ \forall j \in I \setminus \{i\}\} \simeq K_n(-\tfrac12)^{I \setminus \{i\}} \\
E(\alpha'(i)) \simeq K(-\tfrac12)^{I \setminus \{i\}}
\end{cases}\]
LaTeX source
\[
\text{(34)} \qquad
\begin{cases}
E_n(\alpha'(i)) \simeq \{(\xi_j) \in \overline{\mathbb{Q}}^{I \setminus \{i\}} \mid \xi_j^{n} = -\tfrac12 \ \ \forall j \in I \setminus \{i\}\} \simeq K_n(-\tfrac12)^{I \setminus \{i\}} \\
E(\alpha'(i)) \simeq K(-\tfrac12)^{I \setminus \{i\}}
\end{cases}
\]\[\text{(35)} \qquad
\begin{cases}
K_n(g) = \{\xi \in \overline{\mathbb{Q}} \mid \xi^{n} = g\} \\
K_\infty(g) = K(g) = \varprojlim_{n} K_n(g)
\end{cases}\]
LaTeX source
\[
\text{(35)} \qquad
\begin{cases}
K_n(g) = \{\xi \in \overline{\mathbb{Q}} \mid \xi^{n} = g\} \\
K_\infty(g) = K(g) = \varprojlim_{n} K_n(g)
\end{cases}
\]\[K(gg') \simeq K(g) \wedge_{\mathrm{T}} K(g')\]
LaTeX source
\[
K(gg') \simeq K(g) \wedge_{\mathrm{T}} K(g')
\]\[\text{(36)} \qquad K(-\tfrac12) \simeq K(-1) \wedge_{\mathrm{T}} K(\tfrac12) \simeq K(-1) \wedge_{\mathrm{T}} K(2)^{-1}\]
LaTeX source
\[
\text{(36)} \qquad K(-\tfrac12) \simeq K(-1) \wedge_{\mathrm{T}} K(\tfrac12) \simeq K(-1) \wedge_{\mathrm{T}} K(2)^{-1}
\]\[\text{(37)} \qquad K(-1) \hookrightarrow \mathrm{T}, \qquad (1 \to \mathrm{T} \xrightarrow{\ 2\ } \mathrm{T} \xrightarrow{\ p_2\ } \mu_2 \to 1)\]
LaTeX source
\[
\text{(37)} \qquad K(-1) \hookrightarrow \mathrm{T}, \qquad (1 \to \mathrm{T} \xrightarrow{\ 2\ } \mathrm{T} \xrightarrow{\ p_2\ } \mu_2 \to 1)
\]\[\text{(38)} \qquad A = \mathrm{Aff}(\mathrm{T}, K(\tfrac12))\]
LaTeX source
\[
\text{(38)} \qquad A = \mathrm{Aff}(\mathrm{T}, K(\tfrac12))
\]\[\text{(39)} \qquad 1 \longrightarrow \mathrm{T} \longrightarrow A \longrightarrow \widehat{\mathbb{Z}}^{*} \longrightarrow 1 \qquad (\widehat{\mathbb{Z}}^{*} \simeq \mathrm{Aut}(\mathrm{T})) .\]
LaTeX source
\[
\text{(39)} \qquad 1 \longrightarrow \mathrm{T} \longrightarrow A \longrightarrow \widehat{\mathbb{Z}}^{*} \longrightarrow 1 \qquad (\widehat{\mathbb{Z}}^{*} \simeq \mathrm{Aut}(\mathrm{T})) .
\]\[\widetilde{\mathcal{E}} = (\mathfrak{S}_3 \times A) \cdot_{1/2} \Pi_\infty\]
LaTeX source
\[
\widetilde{\mathcal{E}} = (\mathfrak{S}_3 \times A) \cdot_{1/2} \Pi_\infty
\]\[\widetilde{\mathcal{E}}/\mathrm{T} \simeq (\mathfrak{S}_3 \times \widehat{\mathbb{Z}}^{*}) \cdot_{1/2} \Pi_\infty .\]
LaTeX source
\[
\widetilde{\mathcal{E}}/\mathrm{T} \simeq (\mathfrak{S}_3 \times \widehat{\mathbb{Z}}^{*}) \cdot_{1/2} \Pi_\infty .
\]\[\text{(40)} \qquad \Gamma \longrightarrow A\]
LaTeX source
\[
\text{(40)} \qquad \Gamma \longrightarrow A
\]\[\text{(41)} \qquad \mathcal{E} = (\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \Pi_\infty \longrightarrow \widetilde{\mathcal{E}} = (\mathfrak{S}_3 \times A) \cdot_{1/2} \Pi_\infty\]
LaTeX source
\[
\text{(41)} \qquad \mathcal{E} = (\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} \Pi_\infty \longrightarrow \widetilde{\mathcal{E}} = (\mathfrak{S}_3 \times A) \cdot_{1/2} \Pi_\infty
\]\[\text{(42)} \qquad E(\alpha) \subset \Pi_\infty \xrightarrow{\ p_3\ } \Pi_3, \qquad E(\alpha) = p_3^{-1}(\alpha)\]
LaTeX source
\[
\text{(42)} \qquad E(\alpha) \subset \Pi_\infty \xrightarrow{\ p_3\ } \Pi_3, \qquad E(\alpha) = p_3^{-1}(\alpha)
\]\[\alpha = (\overline{j}, j, 1) \bmod \mu_3 = \{(\overline{j}, j, 1), (j, 1, \overline{j}), (1, \overline{j}, j)\} \in \Pi_3\]
LaTeX source
\[
\alpha = (\overline{j}, j, 1) \bmod \mu_3 = \{(\overline{j}, j, 1), (j, 1, \overline{j}), (1, \overline{j}, j)\} \in \Pi_3
\]\[\Bigl( \quad j = \exp\frac{2i\pi}{3}, \qquad \overline{j} = \exp -\frac{2i\pi}{3}\]
LaTeX source
\[
\Bigl( \quad j = \exp\frac{2i\pi}{3}, \qquad \overline{j} = \exp -\frac{2i\pi}{3}
\]\[\text{(43)} \qquad \zeta_n = \exp\frac{2i\pi}{3n}\]
LaTeX source
\[
\text{(43)} \qquad \zeta_n = \exp\frac{2i\pi}{3n}
\]\[\begin{cases}
\alpha_n = (\overline{\zeta}_n, \zeta_n, 1) \\
\alpha_\infty = (\overline{\zeta}_\infty, \zeta_\infty, 1)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\alpha_n = (\overline{\zeta}_n, \zeta_n, 1) \\
\alpha_\infty = (\overline{\zeta}_\infty, \zeta_\infty, 1)
\end{cases}
\]\[\text{(44)} \qquad \zeta_\infty = (\zeta_n) \in \mathrm{T}\]
LaTeX source
\[
\text{(44)} \qquad \zeta_\infty = (\zeta_n) \in \mathrm{T}
\]\[\text{(45)} \qquad \widehat{\mathbb{Z}} \overset{\varphi}{\simeq} \mathrm{T} .\]
LaTeX source
\[
\text{(45)} \qquad \widehat{\mathbb{Z}} \overset{\varphi}{\simeq} \mathrm{T} .
\]\[\text{(46)} \qquad
\begin{cases}
\Pi_\infty \simeq \widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}} \\
\mathcal{E} \simeq (\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} (\widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}})
\end{cases}\]
LaTeX source
\[
\text{(46)} \qquad
\begin{cases}
\Pi_\infty \simeq \widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}} \\
\mathcal{E} \simeq (\mathfrak{S}_3 \times \Gamma) \cdot_{1/2} (\widehat{\mathbb{Z}}^{3}/\widehat{\mathbb{Z}})
\end{cases}
\]\[\Pi_n \simeq (\mathbb{Z}/n)^{3}/(\mathbb{Z}/n)\]
LaTeX source
\[
\Pi_n \simeq (\mathbb{Z}/n)^{3}/(\mathbb{Z}/n)
\]\[\text{(47)} \qquad \alpha_\infty \simeq (-1, +1, 0) \bmod \widehat{\mathbb{Z}}\]
LaTeX source
\[
\text{(47)} \qquad \alpha_\infty \simeq (-1, +1, 0) \bmod \widehat{\mathbb{Z}}
\]\[\text{(47')} \qquad \alpha'_\infty = \overline{\alpha}_\infty = (1, -1, 0) \bmod \widehat{\mathbb{Z}} .\]
LaTeX source
\[
\text{(47')} \qquad \alpha'_\infty = \overline{\alpha}_\infty = (1, -1, 0) \bmod \widehat{\mathbb{Z}} .
\]\[\text{(48)} \qquad
\begin{cases}
\rho_0, \rho_1, \rho_\infty \in \Pi_\infty \\
\rho_0 = (1, 0, 0) \bmod \widehat{\mathbb{Z}} \\
\rho_1 = (0, 1, 0) \bmod \widehat{\mathbb{Z}} \\
\rho_\infty = (0, 0, 1) \bmod \widehat{\mathbb{Z}}
\end{cases}\]
LaTeX source
\[
\text{(48)} \qquad
\begin{cases}
\rho_0, \rho_1, \rho_\infty \in \Pi_\infty \\
\rho_0 = (1, 0, 0) \bmod \widehat{\mathbb{Z}} \\
\rho_1 = (0, 1, 0) \bmod \widehat{\mathbb{Z}} \\
\rho_\infty = (0, 0, 1) \bmod \widehat{\mathbb{Z}}
\end{cases}
\]\[\text{(49)} \qquad \rho_0 \rho_1 \rho_\infty = 1 .\]
LaTeX source
\[
\text{(49)} \qquad \rho_0 \rho_1 \rho_\infty = 1 .
\]\[\text{(50)} \qquad \theta_\infty = (\underbrace{\sqrt[n]{2}}_{\theta_n})_{n \in \mathbb{N}^{*}}\]
LaTeX source
\[
\text{(50)} \qquad \theta_\infty = (\underbrace{\sqrt[n]{2}}_{\theta_n})_{n \in \mathbb{N}^{*}}
\]\[\text{(51)} \qquad A \simeq \mathrm{Aff}(1, \widehat{\mathbb{Z}}) \simeq \widehat{\mathbb{Z}}^{*} \cdot_{1/2} \widehat{\mathbb{Z}}\]
LaTeX source
\[
\text{(51)} \qquad A \simeq \mathrm{Aff}(1, \widehat{\mathbb{Z}}) \simeq \widehat{\mathbb{Z}}^{*} \cdot_{1/2} \widehat{\mathbb{Z}}
\]\[\text{(52)} \qquad (a, b)(x) = ax + b \qquad (a \in \widehat{\mathbb{Z}}^{*},\ b \in \widehat{\mathbb{Z}},\ x \in K(2)) .\]
LaTeX source
\[
\text{(52)} \qquad (a, b)(x) = ax + b \qquad (a \in \widehat{\mathbb{Z}}^{*},\ b \in \widehat{\mathbb{Z}},\ x \in K(2)) .
\]\[\text{(53)} \qquad K(\tfrac12) \simeq \widehat{\mathbb{Z}}\]
LaTeX source
\[
\text{(53)} \qquad K(\tfrac12) \simeq \widehat{\mathbb{Z}}
\]\[\text{(54)} \qquad (a, b)(x) = ax - b \qquad (a \in \widehat{\mathbb{Z}}^{*},\ b \in \widehat{\mathbb{Z}},\ x \in K(\tfrac12)) .\]
LaTeX source
\[
\text{(54)} \qquad (a, b)(x) = ax - b \qquad (a \in \widehat{\mathbb{Z}}^{*},\ b \in \widehat{\mathbb{Z}},\ x \in K(\tfrac12)) .
\]\[E(\alpha) \longrightarrow E_0,\ E_1,\ E_\infty
\qquad \text{et} \qquad
E(\alpha) \xrightarrow{\ \sim\ } E'_0 = E_2,\ \ E'_1 = E_{-1},\ \ E'_\infty \simeq E_{1/2}\]
LaTeX source
\[
E(\alpha) \longrightarrow E_0,\ E_1,\ E_\infty
\qquad \text{et} \qquad
E(\alpha) \xrightarrow{\ \sim\ } E'_0 = E_2,\ \ E'_1 = E_{-1},\ \ E'_\infty \simeq E_{1/2}
\]\[E(\alpha) \longrightarrow E'_\infty \longrightarrow E_0\]
LaTeX source
\[ E(\alpha) \longrightarrow E'_\infty \longrightarrow E_0 \]
\[\begin{aligned}
\alpha'_n(\infty) &= \Bigl(\sqrt[n]{\tfrac12}\, \exp\frac{-\pi i}{n},\ \sqrt[n]{\tfrac12}\, \exp\frac{\pi i}{n},\ 1\Bigr) \\
&= (\theta_n^{-1} \overline{\zeta}_n,\ \theta_n^{-1} \zeta_n,\ 1) \\
\alpha'_\infty(\infty) &= (\theta_\infty \wedge \overline{\zeta}_\infty,\ \theta_\infty \wedge \zeta_\infty,\ 1)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\alpha'_n(\infty) &= \Bigl(\sqrt[n]{\tfrac12}\, \exp\frac{-\pi i}{n},\ \sqrt[n]{\tfrac12}\, \exp\frac{\pi i}{n},\ 1\Bigr) \\
&= (\theta_n^{-1} \overline{\zeta}_n,\ \theta_n^{-1} \zeta_n,\ 1) \\
\alpha'_\infty(\infty) &= (\theta_\infty \wedge \overline{\zeta}_\infty,\ \theta_\infty \wedge \zeta_\infty,\ 1)
\end{aligned}
\]\[\pi(I) = \hat{\mathbb{Z}}^{I} / \hat{\mathbb{Z}},
\qquad
\pi_n(I) = (\mathbb{Z}/n)^{I} / (\mathbb{Z}/n)\]
LaTeX source
\[
\pi(I) = \hat{\mathbb{Z}}^{I} / \hat{\mathbb{Z}},
\qquad
\pi_n(I) = (\mathbb{Z}/n)^{I} / (\mathbb{Z}/n)
\]\[\mathcal{E}''_{1}(I) \simeq \bigl(\mathfrak{S}_I \times \hat{\mathbb{Z}}^{\times}\bigr) \cdot_{1/2} \pi(I)\]
LaTeX source
\[
\mathcal{E}''_{1}(I) \simeq \bigl(\mathfrak{S}_I \times \hat{\mathbb{Z}}^{\times}\bigr) \cdot_{1/2} \pi(I)
\]\[E(I, i) \simeq \hat{\mathbb{Z}}^{(\pm 1)}_{\wedge}\bigl(I \setminus \{i\}\bigr)
\qquad (i \in I)\]
LaTeX source
\[
E(I, i) \simeq \hat{\mathbb{Z}}^{(\pm 1)}_{\wedge}\bigl(I \setminus \{i\}\bigr)
\qquad (i \in I)
\]\[E(I; I) \simeq \coprod_{i \in I} E(I, i)\]
LaTeX source
\[
E(I; I) \simeq \coprod_{i \in I} E(I, i)
\]\[E(I, \omega) = p_3^{-1}(\omega) \subset \pi_\infty(I)\]
LaTeX source
\[
E(I, \omega) = p_3^{-1}(\omega) \subset \pi_\infty(I)
\]\[0 \to \pi_\infty(I) \to \pi_\infty(I) \to \pi_3(I) \to 0,
\qquad \omega \in \pi_3(I)\]
LaTeX source
\[ 0 \to \pi_\infty(I) \to \pi_\infty(I) \to \pi_3(I) \to 0, \qquad \omega \in \pi_3(I) \]
\[g_n = \frac{(n-1)(n-2)}{2}.\]
LaTeX source
\[
g_n = \frac{(n-1)(n-2)}{2}.
\]\[C_2 : x^2 + y^2 + z^2 = 0\]
LaTeX source
\[ C_2 : x^2 + y^2 + z^2 = 0 \]
\[\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}\]
LaTeX source
\[
\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}
\]\[\left(\begin{matrix} 2 & 2 & n \\ 2 & 2 & n \end{matrix} \;\text{pour } n = 2\right),\]
LaTeX source
\[
\left(\begin{matrix} 2 & 2 & n \\ 2 & 2 & n \end{matrix} \;\text{pour } n = 2\right),
\]\[\begin{array}{ll}
X_n = \widehat{\mathrm{Spec}}\, k[x_n] & x_n^{\,n} = x_1 \\
\downarrow & \\
X_1 = \widehat{\mathrm{Spec}}\, k[x_1 = z] & \\
\downarrow & \\
X_0 = \widehat{\mathrm{Spec}}\, k[x_0] &
x_0 = \frac12 - \frac14\bigl(z + z^{-1}\bigr)
\end{array}\]
LaTeX source
\[
\begin{array}{ll}
X_n = \widehat{\mathrm{Spec}}\, k[x_n] & x_n^{\,n} = x_1 \\
\downarrow & \\
X_1 = \widehat{\mathrm{Spec}}\, k[x_1 = z] & \\
\downarrow & \\
X_0 = \widehat{\mathrm{Spec}}\, k[x_0] &
x_0 = \frac12 - \frac14\bigl(z + z^{-1}\bigr)
\end{array}
\]\[\mathbb{D}_\infty = (\mathbb{Z}/2) \cdot_{1/2} \mu_\infty\]
LaTeX source
\[
\mathbb{D}_\infty = (\mathbb{Z}/2) \cdot_{1/2} \mu_\infty
\]\[z' = z^3, \qquad
y' = y\,(y^2 - 3x^2), \qquad
x' = -x\,(x^2 - 3y^2)\]
LaTeX source
\[ z' = z^3, \qquad y' = y\,(y^2 - 3x^2), \qquad x' = -x\,(x^2 - 3y^2) \]
\[\begin{align*}
& z^6 + y^2\,(y^4 + 9x^4 - 6x^2y^2) + x^2\,(x^4 + 9y^4 - 6x^2y^2) \\
&\quad = x^6 + y^6 + z^6 + 3\,x^2y^2\,(x^2 + y^2) = \text{\struck{$(x^2 + y^2 + z^2)^3$}} \\
&\quad = x^6 + y^6 + z^6 - 3\,x^2y^2z^2
\end{align*}\]
LaTeX source
\begin{align*}
& z^6 + y^2\,(y^4 + 9x^4 - 6x^2y^2) + x^2\,(x^4 + 9y^4 - 6x^2y^2) \\
&\quad = x^6 + y^6 + z^6 + 3\,x^2y^2\,(x^2 + y^2) = \text{\struck{$(x^2 + y^2 + z^2)^3$}} \\
&\quad = x^6 + y^6 + z^6 - 3\,x^2y^2z^2
\end{align*}\[\Bigl(\frac{y}{z}\Bigr)^2 + \Bigl(\frac{x}{z}\Bigr)^2 = -1
\qquad a + ib \qquad \text{\struck{$a^2 + b^2 = -1$}}\]
LaTeX source
\[
\Bigl(\frac{y}{z}\Bigr)^2 + \Bigl(\frac{x}{z}\Bigr)^2 = -1
\qquad a + ib \qquad \text{\struck{$a^2 + b^2 = -1$}}
\]\[\begin{gather*}
\frac{p(x)}{q(x)} = \lambda, \qquad p(x) - \lambda\, q(x) = 0, \qquad p'(x) - \lambda\, q'(x) = 0 \\
\lambda = \frac{p'}{q'}, \qquad p - \frac{p'}{q'}\, q = 0, \qquad \text{\struck{$q'p -$}}\; p'q - q'p = 0
\end{gather*}\]
LaTeX source
\begin{gather*}
\frac{p(x)}{q(x)} = \lambda, \qquad p(x) - \lambda\, q(x) = 0, \qquad p'(x) - \lambda\, q'(x) = 0 \\
\lambda = \frac{p'}{q'}, \qquad p - \frac{p'}{q'}\, q = 0, \qquad \text{\struck{$q'p -$}}\; p'q - q'p = 0
\end{gather*}\[\cos 2u = \frac{1 - \mathrm{tg}^2 u}{1 + \mathrm{tg}^2 u},
\qquad
\sin 2u = \frac{2\,\mathrm{tg}\, u}{1 + \mathrm{tg}^2 u}\]
LaTeX source
\[
\cos 2u = \frac{1 - \mathrm{tg}^2 u}{1 + \mathrm{tg}^2 u},
\qquad
\sin 2u = \frac{2\,\mathrm{tg}\, u}{1 + \mathrm{tg}^2 u}
\]\[2 \sin u \cos u = 2 \cos^2 u \;\mathrm{tg}\, u =\]
LaTeX source
\[
2 \sin u \cos u = 2 \cos^2 u \;\mathrm{tg}\, u =
\]\[\left\{
\begin{aligned}
x &= \frac{1 - t^2}{1 + t^2} \\
y &= \frac{2t}{1 + t^2}
\end{aligned}
\right.
\qquad t = y/x \quad \text{quand } z = 1\]
LaTeX source
\[
\left\{
\begin{aligned}
x &= \frac{1 - t^2}{1 + t^2} \\
y &= \frac{2t}{1 + t^2}
\end{aligned}
\right.
\qquad t = y/x \quad \text{quand } z = 1
\]\[x = 1 - t^2, \qquad y = 2t, \qquad z = 1 + t^2\]
LaTeX source
\[ x = 1 - t^2, \qquad y = 2t, \qquad z = 1 + t^2 \]
\[z = 0 \text{ i.e. } t = \mp i \;\Longrightarrow\; x = 2,\ y = \mp 2i \;\sim\; x = 1,\ y = \mp i\]
LaTeX source
\[
z = 0 \text{ i.e. } t = \mp i \;\Longrightarrow\; x = 2,\ y = \mp 2i \;\sim\; x = 1,\ y = \mp i
\]\[\sim\; x = \pm i,\ y = 1\]
LaTeX source
\[ \sim\; x = \pm i,\ y = 1 \]
\[y = 0 \text{ i.e. } t = 0 \qquad (1, 0, 1)\]
LaTeX source
\[
y = 0 \text{ i.e. } t = 0 \qquad (1, 0, 1)
\]\[x^2 + y^2 + z^2 = 0 \qquad C_2 = X \longrightarrow \mathbb{P}^1 = \widehat{\mathrm{Spec}}\,(k[t])\]
LaTeX source
\[
x^2 + y^2 + z^2 = 0 \qquad C_2 = X \longrightarrow \mathbb{P}^1 = \widehat{\mathrm{Spec}}\,(k[t])
\]\[(\pm i, 1, 0) \longrightarrow t = \mp i, \qquad
(\pm i, 0, 1) \longmapsto t = 0, \qquad
t = \frac{y}{x}\]
LaTeX source
\[
(\pm i, 1, 0) \longrightarrow t = \mp i, \qquad
(\pm i, 0, 1) \longmapsto t = 0, \qquad
t = \frac{y}{x}
\]\[x' = 1 - t^2 = 1 - \frac{y^2}{x^2} = \frac{x^2 - y^2}{x^2}, \qquad
y' = 2t = \frac{2y}{x}, \qquad
z' = 1 + t^2 = \frac{x^2 + y^2}{x^2}\]
LaTeX source
\[
x' = 1 - t^2 = 1 - \frac{y^2}{x^2} = \frac{x^2 - y^2}{x^2}, \qquad
y' = 2t = \frac{2y}{x}, \qquad
z' = 1 + t^2 = \frac{x^2 + y^2}{x^2}
\]\[\sim\; (x', y', z') = (x^2 - y^2,\ 2xy,\ x^2 + y^2)\]
LaTeX source
\[ \sim\; (x', y', z') = (x^2 - y^2,\ 2xy,\ x^2 + y^2) \]
\[\left\{
\begin{aligned}
x' &= x^2 - y^2 \\
y' &= 2xy \\
z' &= x^2 + y^2
\end{aligned}
\right.
\qquad
\begin{aligned}
x'^2 + y'^2 - z'^2 &= x^4 + y^4 - 2x^2y^2 + 4x^2y^2 \\
&\quad - (x^2 + y^2)^2 = 0
\end{aligned}\]
LaTeX source
\[
\left\{
\begin{aligned}
x' &= x^2 - y^2 \\
y' &= 2xy \\
z' &= x^2 + y^2
\end{aligned}
\right.
\qquad
\begin{aligned}
x'^2 + y'^2 - z'^2 &= x^4 + y^4 - 2x^2y^2 + 4x^2y^2 \\
&\quad - (x^2 + y^2)^2 = 0
\end{aligned}
\]\[s = -\frac{t + i}{t - i} = -\frac{(t + i)^2}{t^2 + 1}
= -\frac{t^2 - 1}{t^2 + 1} - \frac{2it}{t^2 + 1}\]
LaTeX source
\[
s = -\frac{t + i}{t - i} = -\frac{(t + i)^2}{t^2 + 1}
= -\frac{t^2 - 1}{t^2 + 1} - \frac{2it}{t^2 + 1}
\]\[\Bigl(\frac{t + i}{t - 1}\Bigr)^{n} =\]
LaTeX source
\[
\Bigl(\frac{t + i}{t - 1}\Bigr)^{n} =
\]\[\pi_n = \mu_n^{3} / \mu_n\]
LaTeX source
\[
\pi_n = \mu_n^{3} / \mu_n
\]\[\widetilde{X}_2(p, q, r) = X_2(2p, q, 2r)\]
LaTeX source
\[
\widetilde{X}_2(p, q, r) = X_2(2p, q, 2r)
\]\[\mu_{2p} \times \mu_{2q} \times \mu_{2r} \times \pi_n\]
LaTeX source
\[
\mu_{2p} \times \mu_{2q} \times \mu_{2r} \times \pi_n
\]\[\mu_2 \times \mu_2 \times \mu_2 \times \pi_2 \twoheadrightarrow \mu_2^{3}/\mu_2\]
LaTeX source
\[
\mu_2 \times \mu_2 \times \mu_2 \times \pi_2 \twoheadrightarrow \mu_2^{3}/\mu_2
\]\[X = \frac{y^2 - x^2}{z^2}, \qquad Y = -\frac{2xy}{z^2}\]
LaTeX source
\[
X = \frac{y^2 - x^2}{z^2}, \qquad Y = -\frac{2xy}{z^2}
\]\[X + iY = \frac{1}{z^2}\bigl(y^2 - x^2 - 2ixy\bigr) = \frac{1}{z^2}\,(y - ix)^2\]
LaTeX source
\[
X + iY = \frac{1}{z^2}\bigl(y^2 - x^2 - 2ixy\bigr) = \frac{1}{z^2}\,(y - ix)^2
\]\[(X + iY)^{n} = \frac{1}{z^{2n}}\,(y - ix)^{2n}
= \frac{1}{z^{2n}}\bigl(P_n(x, y) + i\,Q_n(x, y)\bigr)\]
LaTeX source
\[
(X + iY)^{n} = \frac{1}{z^{2n}}\,(y - ix)^{2n}
= \frac{1}{z^{2n}}\bigl(P_n(x, y) + i\,Q_n(x, y)\bigr)
\]\[\overset{?}{=}\; \frac{y'^2 - x'^2}{z'^2} \mp 2i\,\frac{x'y'}{z'^2}\]
LaTeX source
\[
\overset{?}{=}\; \frac{y'^2 - x'^2}{z'^2} \mp 2i\,\frac{x'y'}{z'^2}
\]\[\begin{align*}
B_n(x, y)^2 - A_n(x, y)^2 &= P_n(x, y) \\
-2\, A_n(x, y)\, B_n(x, y) &= Q_n(x, y)
\end{align*}\]
LaTeX source
\begin{align*}
B_n(x, y)^2 - A_n(x, y)^2 &= P_n(x, y) \\
-2\, A_n(x, y)\, B_n(x, y) &= Q_n(x, y)
\end{align*}\[X + iY = \Bigl(\frac{y - ix}{z}\Bigr)^2 = \Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)^2\]
LaTeX source
\[
X + iY = \Bigl(\frac{y - ix}{z}\Bigr)^2 = \Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)^2
\]\[\Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)
\qquad
\Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)^{2n+1}
\qquad
(-i)^3 = -i\,(i^2) = i\]
LaTeX source
\[
\Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)
\qquad
\Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)^{2n+1}
\qquad
(-i)^3 = -i\,(i^2) = i
\]\[\Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)^3
= \frac{1}{z^3}\,\bigl[\,y^3 - 3iy^2x - 3yx^2 + ix^3\,\bigr]\]
LaTeX source
\[
\Bigl(\frac{y}{z} - i\,\frac{x}{z}\Bigr)^3
= \frac{1}{z^3}\,\bigl[\,y^3 - 3iy^2x - 3yx^2 + ix^3\,\bigr]
\]\[= y^3 - 3yx^2 + i\,(x^3 - 3y^2x)\]
LaTeX source
\[ = y^3 - 3yx^2 + i\,(x^3 - 3y^2x) \]
\[= y\,(y^2 - 3x^2) + i\,x\,(x^2 - 3y^2)\]
LaTeX source
\[ = y\,(y^2 - 3x^2) + i\,x\,(x^2 - 3y^2) \]
\[\mathrm{Hom}(C, C') = \mathrm{Ob}\, \underline{\mathrm{Hom}}(C, C').\]
LaTeX source
\[
\mathrm{Hom}(C, C') = \mathrm{Ob}\, \underline{\mathrm{Hom}}(C, C').
\]\[\mathrm{Hom}(C, C') \longrightarrow \mathrm{Hom}\bigl(\underbrace{\mathrm{Ob}\,C}_{C_0},\, \underbrace{\mathrm{Ob}\,C'}_{C'_0}\bigr)\]
LaTeX source
\[
\mathrm{Hom}(C, C') \longrightarrow \mathrm{Hom}\bigl(\underbrace{\mathrm{Ob}\,C}_{C_0},\, \underbrace{\mathrm{Ob}\,C'}_{C'_0}\bigr)
\]\[f_0 : C_0 \to C'_0 ,\]
LaTeX source
\[ f_0 : C_0 \to C'_0 , \]
\[\mathrm{Hom}_{f_0}(C, C') \xrightarrow{\ \sim\ }
\prod_{x \in C_0} \mathrm{Isom}\bigl(f_0(a_{i(x)}), f_0(x)\bigr)\]
LaTeX source
\[
\mathrm{Hom}_{f_0}(C, C') \xrightarrow{\ \sim\ }
\prod_{x \in C_0} \mathrm{Isom}\bigl(f_0(a_{i(x)}), f_0(x)\bigr)
\]\[\times \prod_{i} \mathrm{Hom}\bigl(\mathrm{Aut}(a_i), \mathrm{Aut}(f_0(a_i))\bigr)\]
LaTeX source
\[
\times \prod_{i} \mathrm{Hom}\bigl(\mathrm{Aut}(a_i), \mathrm{Aut}(f_0(a_i))\bigr)
\]\[f \longmapsto \Bigl(\bigl(\underbrace{f(u_x)}_{\ell_x}\bigr)_{x \in C_0},\
\bigl(\underbrace{f_{a_i, a_i} : \mathrm{Aut}(a_i) \to \mathrm{Aut}(f_0(a_i))}_{\varphi_i}\bigr)\Bigr).\]
LaTeX source
\[
f \longmapsto \Bigl(\bigl(\underbrace{f(u_x)}_{\ell_x}\bigr)_{x \in C_0},\
\bigl(\underbrace{f_{a_i, a_i} : \mathrm{Aut}(a_i) \to \mathrm{Aut}(f_0(a_i))}_{\varphi_i}\bigr)\Bigr).
\]\[\mathrm{Isom}\bigl(f_0(x), f_0(y)\bigr) \xrightarrow{\ \sim\ } \mathrm{Hom}\bigl(f_0(x), f_0(y)\bigr) ;\]
LaTeX source
\[
\mathrm{Isom}\bigl(f_0(x), f_0(y)\bigr) \xrightarrow{\ \sim\ } \mathrm{Hom}\bigl(f_0(x), f_0(y)\bigr) ;
\]\[1 \to \mathrm{Aut}^0(C) \to \mathrm{Aut}\,C \to \mathfrak{S}_{C} \to 1\]
LaTeX source
\[
1 \to \mathrm{Aut}^0(C) \to \mathrm{Aut}\,C \to \mathfrak{S}_{C} \to 1
\]\[1 \to \mathrm{Aut}^0(C, a) \to \mathrm{Aut}^0(C) \to \mathrm{Aut}(\pi_a) \to 1,
\qquad \pi_a = \mathrm{Aut}_C(a)\]
LaTeX source
\[
1 \to \mathrm{Aut}^0(C, a) \to \mathrm{Aut}^0(C) \to \mathrm{Aut}(\pi_a) \to 1,
\qquad \pi_a = \mathrm{Aut}_C(a)
\]\[\mathrm{Aut}^0(C, a) = \bigl\{ f \in \mathrm{Aut}(C) \bigm| \mathrm{Ob} f = \mathrm{id},\
\bigl(f_a : \mathrm{Aut}(a) \to \mathrm{Aut}(\underbrace{f(a)}_{=a})\bigr) = \mathrm{id} \bigr\}\]
LaTeX source
\[
\mathrm{Aut}^0(C, a) = \bigl\{ f \in \mathrm{Aut}(C) \bigm| \mathrm{Ob} f = \mathrm{id},\
\bigl(f_a : \mathrm{Aut}(a) \to \mathrm{Aut}(\underbrace{f(a)}_{=a})\bigr) = \mathrm{id} \bigr\}
\]\[T_x = \mathrm{Isom}(a, x), \quad \pi_a\text{-torseur à droite}\]
LaTeX source
\[
T_x = \mathrm{Isom}(a, x), \quad \pi_a\text{-torseur à droite}
\]\[\mathrm{Aut}(C) \longrightarrow \mathfrak{S}_{C_0} \quad \text{\textit{surjective}}\]
LaTeX source
\[
\mathrm{Aut}(C) \longrightarrow \mathfrak{S}_{C_0} \quad \text{\textit{surjective}}
\]\[\operatorname{Aut}^{\circ}(C, a) \simeq \prod_{x \in C_0 \smallsetminus \{a\}} \pi_x\]
LaTeX source
\[
\operatorname{Aut}^{\circ}(C, a) \simeq \prod_{x \in C_0 \smallsetminus \{a\}} \pi_x
\]\[\begin{cases}
\rho_0^{+} = (p_{01}^{+})^{-1}\, p_{01}^{-}\, (p_{20}^{-})^{-1}\, p_{20}^{+} \\
\rho_1^{+} = (p_{12}^{+})^{-1}\, p_{12}^{-}\, (p_{01}^{-})^{-1}\, p_{01}^{+} \\
\rho_2^{+} = (p_{20}^{+})^{-1}\, p_{20}^{-}\, (p_{12}^{-})^{-1}\, p_{12}^{+}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\rho_0^{+} = (p_{01}^{+})^{-1}\, p_{01}^{-}\, (p_{20}^{-})^{-1}\, p_{20}^{+} \\
\rho_1^{+} = (p_{12}^{+})^{-1}\, p_{12}^{-}\, (p_{01}^{-})^{-1}\, p_{01}^{+} \\
\rho_2^{+} = (p_{20}^{+})^{-1}\, p_{20}^{-}\, (p_{12}^{-})^{-1}\, p_{12}^{+}
\end{cases}
\]\[\rho_2^{+} \rho_1^{+} \rho_0^{+} = \mathrm{id}_{E^{+}}\]
LaTeX source
\[
\rho_2^{+} \rho_1^{+} \rho_0^{+} = \mathrm{id}_{E^{+}}
\]\[\begin{aligned}
S_0 &\simeq E^{+}/\rho_0^{+} \simeq E^{-}/\rho_0^{-} \\
S_1 &\simeq E^{+}/\rho_1^{+} \simeq E^{-}/\rho_1^{-} \\
S_2 &\simeq E^{+}/\rho_2^{+} \simeq E^{-}/\rho_2^{-}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
S_0 &\simeq E^{+}/\rho_0^{+} \simeq E^{-}/\rho_0^{-} \\
S_1 &\simeq E^{+}/\rho_1^{+} \simeq E^{-}/\rho_1^{-} \\
S_2 &\simeq E^{+}/\rho_2^{+} \simeq E^{-}/\rho_2^{-}
\end{aligned}
\]\[\begin{cases}
\rho_0^{-} = (p_{01}^{-})^{-1}\, p_{01}^{+}\, (p_{20}^{+})^{-1}\, p_{20}^{-} \\
\rho_1^{-} = (p_{12}^{-})^{-1}\, p_{12}^{+}\, (p_{01}^{+})^{-1}\, p_{01}^{-} \\
\rho_2^{-} = (p_{20}^{-})^{-1}\, p_{20}^{+}\, (p_{12}^{+})^{-1}\, p_{12}^{-}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\rho_0^{-} = (p_{01}^{-})^{-1}\, p_{01}^{+}\, (p_{20}^{+})^{-1}\, p_{20}^{-} \\
\rho_1^{-} = (p_{12}^{-})^{-1}\, p_{12}^{+}\, (p_{01}^{+})^{-1}\, p_{01}^{-} \\
\rho_2^{-} = (p_{20}^{-})^{-1}\, p_{20}^{+}\, (p_{12}^{+})^{-1}\, p_{12}^{-}
\end{cases}
\]\[2 - 2g' = N(2 - 2g) - \underbrace{(N\nu - \nu')}_{\sum_{i,j} (d_{ij} - 1)}\]
LaTeX source
\[
2 - 2g' = N(2 - 2g) - \underbrace{(N\nu - \nu')}_{\sum_{i,j} (d_{ij} - 1)}
\]\[\phantom{2 - 2g'} = N(2 - 2g) - \sum_i \underbrace{\nu_i (N/\nu_i - 1)}_{N - \nu_i \,=\, N(1 - \frac{\nu_i}{N})}\]
LaTeX source
\[
\phantom{2 - 2g'} = N(2 - 2g) - \sum_i \underbrace{\nu_i (N/\nu_i - 1)}_{N - \nu_i \,=\, N(1 - \frac{\nu_i}{N})}
\]\[\boxed{\,2 - 2g' = N \Bigl[ (2 - 2g) - \sum_i \Bigl(1 - \frac{1}{d_i}\Bigr) \Bigr]\,}\]
LaTeX source
\[
\boxed{\,2 - 2g' = N \Bigl[ (2 - 2g) - \sum_i \Bigl(1 - \frac{1}{d_i}\Bigr) \Bigr]\,}
\]\[\boxed{\,\frac{2}{N} = 2 - \sum_i \Bigl(1 - \frac{1}{d_i}\Bigr) > 0\,}\]
LaTeX source
\[
\boxed{\,\frac{2}{N} = 2 - \sum_i \Bigl(1 - \frac{1}{d_i}\Bigr) > 0\,}
\]\[\sum_i \Bigl(1 - \frac{1}{d_i}\Bigr) < 2 .\]
LaTeX source
\[
\sum_i \Bigl(1 - \frac{1}{d_i}\Bigr) < 2 .
\]\[\frac{2}{N} = 2 - \Bigl(1 - \frac{1}{d}\Bigr) = 1 + \frac{1}{d}, \qquad
2 = N + \frac{N}{d} = N + \nu\]
LaTeX source
\[
\frac{2}{N} = 2 - \Bigl(1 - \frac{1}{d}\Bigr) = 1 + \frac{1}{d}, \qquad
2 = N + \frac{N}{d} = N + \nu
\]\[N \geqslant 1,\ \nu \geqslant 1 \quad\text{donc}\quad N = 1,\ \nu = 1,\ I = \varnothing\]
LaTeX source
\[
N \geqslant 1,\ \nu \geqslant 1 \quad\text{donc}\quad N = 1,\ \nu = 1,\ I = \varnothing
\]\[\frac{2}{N} = \frac{1}{d_1} + \frac{1}{d_2}, \qquad
2 = \frac{N}{d_1} + \frac{N}{d_2} = \nu_1 + \nu_2, \qquad
\nu_1, \nu_2 \geqslant 1, \quad \nu_1 = \nu_2 = 1\]
LaTeX source
\[
\frac{2}{N} = \frac{1}{d_1} + \frac{1}{d_2}, \qquad
2 = \frac{N}{d_1} + \frac{N}{d_2} = \nu_1 + \nu_2, \qquad
\nu_1, \nu_2 \geqslant 1, \quad \nu_1 = \nu_2 = 1
\]\[\frac{1}{d_1} + \frac{1}{d_2} + \frac{1}{d_3} > 1 \qquad
\Bigl( \frac{2}{N} = \frac{1}{p} + \frac{1}{q} + \frac{1}{r} - 1 \Bigr)\]
LaTeX source
\[
\frac{1}{d_1} + \frac{1}{d_2} + \frac{1}{d_3} > 1 \qquad
\Bigl( \frac{2}{N} = \frac{1}{p} + \frac{1}{q} + \frac{1}{r} - 1 \Bigr)
\]\[2p + 2q - pq > 0\]
LaTeX source
\[ 2p + 2q - pq > 0 \]
\[\underbrace{2(\alpha + 2) + 2(\beta + 2) - (\alpha + 2)(\beta + 2)}_{4 - \alpha\beta} > 0\]
LaTeX source
\[
\underbrace{2(\alpha + 2) + 2(\beta + 2) - (\alpha + 2)(\beta + 2)}_{4 - \alpha\beta} > 0
\]\[\begin{cases}
(2, 2, p) & p \geqslant 2 \text{ quelc.} \qquad N = 2p \\
(2, 3, 3) & N = 12 \\
(2, 3, 4) & N = 24 \\
(2, 3, 5) & N = 60
\end{cases}\]
LaTeX source
\[
\begin{cases}
(2, 2, p) & p \geqslant 2 \text{ quelc.} \qquad N = 2p \\
(2, 3, 3) & N = 12 \\
(2, 3, 4) & N = 24 \\
(2, 3, 5) & N = 60
\end{cases}
\]\[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{s} > 2
\qquad p \geqslant q \geqslant r \geqslant s\]
LaTeX source
\[
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{s} > 2
\qquad p \geqslant q \geqslant r \geqslant s
\]\[\frac{1}{s} + \frac{1}{p} + \underbrace{\frac{1}{3} + \frac{1}{2}}_{5/6} > 2
\qquad \frac{1}{s} + \frac{1}{p} > \frac{7}{6}\]
LaTeX source
\[
\frac{1}{s} + \frac{1}{p} + \underbrace{\frac{1}{3} + \frac{1}{2}}_{5/6} > 2
\qquad \frac{1}{s} + \frac{1}{p} > \frac{7}{6}
\]\[\frac{1}{p} + \frac{1}{q} > 2 - \underbrace{\Bigl(\frac{1}{r} + \frac{1}{s}\Bigr)}_{\leqslant 1} \geqslant 1
\quad\text{donc}\quad \frac{1}{p} + \frac{1}{q} \geqslant 1\]
LaTeX source
\[
\frac{1}{p} + \frac{1}{q} > 2 - \underbrace{\Bigl(\frac{1}{r} + \frac{1}{s}\Bigr)}_{\leqslant 1} \geqslant 1
\quad\text{donc}\quad \frac{1}{p} + \frac{1}{q} \geqslant 1
\]\[\frac{1}{d_1} + \frac{1}{d_2} + \frac{1}{d_3} + \frac{1}{d_4}
> 2 + \sum_{i \geqslant 5} \Bigl(1 - \frac{1}{d_i}\Bigr)\]
LaTeX source
\[
\frac{1}{d_1} + \frac{1}{d_2} + \frac{1}{d_3} + \frac{1}{d_4}
> 2 + \sum_{i \geqslant 5} \Bigl(1 - \frac{1}{d_i}\Bigr)
\]\[G \simeq \mu_N(k) \simeq \mathbb{Z}/N\mathbb{Z} \ldots\]
LaTeX source
\[
G \simeq \mu_N(k) \simeq \mathbb{Z}/N\mathbb{Z} \ldots
\]\[G \subset \text{Borel},\]
LaTeX source
\[
G \subset \text{Borel},
\]\[T = \mathbb{Z} \wedge_{\mathbb{Z}/2} \omega ,\]
LaTeX source
\[
T = \mathbb{Z} \wedge_{\mathbb{Z}/2} \omega ,
\]\[\hat{X} = \coprod \hat{X}_i .\]
LaTeX source
\[
\hat{X} = \coprod \hat{X}_i .
\]\[(\hat{X}, o, S, J),\]
LaTeX source
\[
(\hat{X}, o, S, J),
\]\[S \longrightarrow \operatorname{Homext}(T, \Pi) \quad (4^{\circ})\]
LaTeX source
\[
S \longrightarrow \operatorname{Homext}(T, \Pi) \quad (4^{\circ})
\]\[T \hookrightarrow \Bigl( \bigwedge\nolimits^{2} \Pi_{\wedge\,\mathrm{ab}} \Bigr)^{\vee}\]
LaTeX source
\[
T \hookrightarrow \Bigl( \bigwedge\nolimits^{2} \Pi_{\wedge\,\mathrm{ab}} \Bigr)^{\vee}
\]\[E = (R, q') \times_A (R, q'')
= \bigl\{ \bigl((s, f), (\bar{s}, \bar{f})\bigr) \bigm|
q'(s, f) = q''(\bar{s}, \bar{f}) \bigr\}\]
LaTeX source
\[
E = (R, q') \times_A (R, q'')
= \bigl\{ \bigl((s, f), (\bar{s}, \bar{f})\bigr) \bigm|
q'(s, f) = q''(\bar{s}, \bar{f}) \bigr\}
\]\[\begin{aligned}
\pi_{01}^{+}(s, f, \bar{s}, \bar{f}) &= (s, f) \\
\pi_{12}^{+}(s, f, \bar{s}, \bar{f}) &= (\bar{s}, \bar{f}) \\
\pi_{02}^{+}(s, f, \bar{s}, \bar{f}) &= (s, \bar{f})
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\pi_{01}^{+}(s, f, \bar{s}, \bar{f}) &= (s, f) \\
\pi_{12}^{+}(s, f, \bar{s}, \bar{f}) &= (\bar{s}, \bar{f}) \\
\pi_{02}^{+}(s, f, \bar{s}, \bar{f}) &= (s, \bar{f})
\end{aligned}
\]\[\begin{aligned}
p(s, f) &= s &\qquad \sigma_0(s, f) &= (s', f') \\
r(s, f) &= f & \sigma_2(s, f) &= (s'', f'') \\
q'(s, f) &= a & & \\
q''(s, f) &= b & \sigma_0 \sigma_2(s, f) &= \sigma_2 \sigma_0(s, f) = (s'', f'')
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
p(s, f) &= s &\qquad \sigma_0(s, f) &= (s', f') \\
r(s, f) &= f & \sigma_2(s, f) &= (s'', f'') \\
q'(s, f) &= a & & \\
q''(s, f) &= b & \sigma_0 \sigma_2(s, f) &= \sigma_2 \sigma_0(s, f) = (s'', f'')
\end{aligned}
\]\[E^{+} = \bigl\{ (s, f, \bar{s}, \bar{f}) \in E \bigm| (s, a, \bar{f})
\ \text{direct} \bigr\}\]
LaTeX source
\[
E^{+} = \bigl\{ (s, f, \bar{s}, \bar{f}) \in E \bigm| (s, a, \bar{f})
\ \text{direct} \bigr\}
\]\[a = q'(s, f) = q''(\bar{s}, \bar{f})\]
LaTeX source
\[
a = q'(s, f) = q''(\bar{s}, \bar{f})
\]\[\begin{cases}
\sigma_0(s, f, \bar{s}, \bar{f}) = (s, f_1, \bar{s}, \bar{f}) \\
\sigma_1(s, f, \bar{s}, \bar{f}) = (s, f_1, \bar{s}', \bar{f}) \\
\sigma_2(s, f, \bar{s}, \bar{f}) = (s, f, \bar{s}_1, f_1)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\sigma_0(s, f, \bar{s}, \bar{f}) = (s, f_1, \bar{s}, \bar{f}) \\
\sigma_1(s, f, \bar{s}, \bar{f}) = (s, f_1, \bar{s}', \bar{f}) \\
\sigma_2(s, f, \bar{s}, \bar{f}) = (s, f, \bar{s}_1, f_1)
\end{cases}
\]\[\Sigma
\begin{cases}
T \text{ groupe profini} \simeq \hat{\mathbb{Z}} \\
\Pi \text{ groupe profini} \\
I \text{ ensemble fini} \\
\Phi_i \in \operatorname{Homext}(T, \Pi) \quad \text{pour } i \in I
\end{cases}\]
LaTeX source
\[
\Sigma
\begin{cases}
T \text{ groupe profini} \simeq \hat{\mathbb{Z}} \\
\Pi \text{ groupe profini} \\
I \text{ ensemble fini} \\
\Phi_i \in \operatorname{Homext}(T, \Pi) \quad \text{pour } i \in I
\end{cases}
\]\[\Sigma_\nu
\begin{cases}
T = \hat{\mathbb{Z}} \\
\Pi = \text{groupe profini } \Pi_\nu = \widehat{L_\nu}, \text{ où }
L_\nu = \{\lambda_0, \ldots, \lambda_\nu \mid \lambda_0 \lambda_1 \cdots \lambda_\nu = 1\} \\
\qquad \text{(donc $\Pi$ profini libre : $\nu$ générateurs)} \\
I = [0, \nu]_{\mathbb{N}} \\
\Phi_i = \text{classe de l'hom. } \hat{\mathbb{Z}} \xrightarrow{\varphi_i} \Pi
\text{ tel que } \varphi_i(1) = \lambda_i
\end{cases}\]
LaTeX source
\[
\Sigma_\nu
\begin{cases}
T = \hat{\mathbb{Z}} \\
\Pi = \text{groupe profini } \Pi_\nu = \widehat{L_\nu}, \text{ où }
L_\nu = \{\lambda_0, \ldots, \lambda_\nu \mid \lambda_0 \lambda_1 \cdots \lambda_\nu = 1\} \\
\qquad \text{(donc $\Pi$ profini libre : $\nu$ générateurs)} \\
I = [0, \nu]_{\mathbb{N}} \\
\Phi_i = \text{classe de l'hom. } \hat{\mathbb{Z}} \xrightarrow{\varphi_i} \Pi
\text{ tel que } \varphi_i(1) = \lambda_i
\end{cases}
\]\[\widetilde{A}_\nu = \operatorname{Aut}(\Sigma_\nu) .\]
LaTeX source
\[
\widetilde{A}_\nu = \operatorname{Aut}(\Sigma_\nu) .
\]\[\widetilde{A}_\nu \longrightarrow \mathfrak{S}_{\nu+1} \quad (= \operatorname{Aut}(I_\nu)),
\qquad
\widetilde{A}_\nu \xrightarrow{\ \chi\ } \operatorname{Aut} \hat{\mathbb{Z}} \simeq \hat{\mathbb{Z}}^{*}\]
LaTeX source
\[
\widetilde{A}_\nu \longrightarrow \mathfrak{S}_{\nu+1} \quad (= \operatorname{Aut}(I_\nu)),
\qquad
\widetilde{A}_\nu \xrightarrow{\ \chi\ } \operatorname{Aut} \hat{\mathbb{Z}} \simeq \hat{\mathbb{Z}}^{*}
\]\[\widetilde{A}_\nu \longrightarrow \mathfrak{S}_{\nu+1} \times \hat{\mathbb{Z}}^{*}\]
LaTeX source
\[
\widetilde{A}_\nu \longrightarrow \mathfrak{S}_{\nu+1} \times \hat{\mathbb{Z}}^{*}
\]\[\Pi \longrightarrow A_\nu^{\circ} \subset A_\nu\]
LaTeX source
\[
\Pi \longrightarrow A_\nu^{\circ} \subset A_\nu
\]\[\begin{cases}
1 \to \Pi \to A_\nu^{\circ} \to \Gamma_\nu \to 1 \\
1 \to A_\nu \to \widetilde{A}_\nu \to \mathfrak{S}_{\nu+1} \to 1 \\
1 \to A_\nu^{\circ} \to A_\nu \to \hat{\mathbb{Z}}^{*} \to 1
\end{cases}\]
LaTeX source
\[
\begin{cases}
1 \to \Pi \to A_\nu^{\circ} \to \Gamma_\nu \to 1 \\
1 \to A_\nu \to \widetilde{A}_\nu \to \mathfrak{S}_{\nu+1} \to 1 \\
1 \to A_\nu^{\circ} \to A_\nu \to \hat{\mathbb{Z}}^{*} \to 1
\end{cases}
\]\[\overbrace{\Pi \quad \Gamma_\nu^{\circ} \quad \hat{\mathbb{Z}}^{*} \quad \mathfrak{S}_{\nu+1}}^{\widetilde{A}_\nu}
\qquad \text{filtration}\]
LaTeX source
\[
\overbrace{\Pi \quad \Gamma_\nu^{\circ} \quad \hat{\mathbb{Z}}^{*} \quad \mathfrak{S}_{\nu+1}}^{\widetilde{A}_\nu}
\qquad \text{filtration}
\]\[\Sigma(X/I/\xi/k) = (T, I, \pi, (\Phi_{i}))\]
LaTeX source
\[
\Sigma(X/I/\xi/k) = (T, I, \pi, (\Phi_{i}))
\]\[\Gamma \longrightarrow {}^{P(X/I/k)}\widetilde{\Gamma}_{\nu}
\simeq \mathrm{Aut}(\Sigma^{\mathrm{ext}})\]
LaTeX source
\[
\Gamma \longrightarrow {}^{P(X/I/k)}\widetilde{\Gamma}_{\nu}
\simeq \mathrm{Aut}(\Sigma^{\mathrm{ext}})
\]\[1 \to \pi_{1}(X \setminus I, x) \to \pi_{1}(X_{0} \setminus I_{0}, x_{0})
\to \pi_{1}(k_{0}, \overline{k}) \to 1\]
LaTeX source
\[
1 \to \pi_{1}(X \setminus I, x) \to \pi_{1}(X_{0} \setminus I_{0}, x_{0})
\to \pi_{1}(k_{0}, \overline{k}) \to 1
\]\[\dots\qquad \Gamma \xrightarrow{\ \chi\ } \mathrm{Aut}(T)
\simeq \widehat{\mathbb{Z}}^{*}\]
LaTeX source
\[
\dots\qquad \Gamma \xrightarrow{\ \chi\ } \mathrm{Aut}(T)
\simeq \widehat{\mathbb{Z}}^{*}
\]\[I = \mu_{\nu+1}(\overline{\mathbb{Q}}) = \mu_{\nu+1}(\mathbb{C}^{*})
\subset \overline{\mathbb{Q}} = \mathbb{E}^{1}_{\overline{\mathbb{Q}}}(\overline{\mathbb{Q}})
\subset \mathbb{P}^{1}_{\overline{\mathbb{Q}}}(\overline{\mathbb{Q}})\]
LaTeX source
\[
I = \mu_{\nu+1}(\overline{\mathbb{Q}}) = \mu_{\nu+1}(\mathbb{C}^{*})
\subset \overline{\mathbb{Q}} = \mathbb{E}^{1}_{\overline{\mathbb{Q}}}(\overline{\mathbb{Q}})
\subset \mathbb{P}^{1}_{\overline{\mathbb{Q}}}(\overline{\mathbb{Q}})
\]\[\struck{I \simeq [0,1]},\qquad
\Sigma(X/I/k)^{\mathrm{ext}} \simeq L_{\nu}^{\mathrm{ext}}\]
LaTeX source
\[
\struck{I \simeq [0,1]},\qquad
\Sigma(X/I/k)^{\mathrm{ext}} \simeq L_{\nu}^{\mathrm{ext}}
\]\[P(X, I/k) = \text{torseur trivial}\]
LaTeX source
\[
P(X, I/k) = \text{torseur trivial}
\]\[\Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \widetilde{\mathcal{T}}_{\nu}\]
LaTeX source
\[
\Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \widetilde{\mathcal{T}}_{\nu}
\]\[\Gamma \to \mathfrak{S}_{\nu+1}\]
LaTeX source
\[
\Gamma \to \mathfrak{S}_{\nu+1}
\]\[\Gamma \to \mu_{\nu+1}(\mathbb{C}^{*}) \simeq \mathbb{Z}/\nu{+}1
\xrightarrow{\ \mathrm{can}\ } \mathfrak{S}_{\nu+1}\ \dots\]
LaTeX source
\[
\Gamma \to \mu_{\nu+1}(\mathbb{C}^{*}) \simeq \mathbb{Z}/\nu{+}1
\xrightarrow{\ \mathrm{can}\ } \mathfrak{S}_{\nu+1}\ \dots
\]\[(X_{0}/I_{0}/\mathbb{Q})\quad \text{donc de } \overset{\mathbb{Z}/2}{\dots}\ \mu_{n}(\overline{\mathbb{Q}})\ (= D_{n})\]
LaTeX source
\[
(X_{0}/I_{0}/\mathbb{Q})\quad \text{donc de } \overset{\mathbb{Z}/2}{\dots}\ \mu_{n}(\overline{\mathbb{Q}})\ (= D_{n})
\]\[(\Gamma \times \mathbb{Z}/2) \cdot \mu_{n}(\overline{\mathbb{Q}})
\longrightarrow \widetilde{\mathcal{T}}_{\nu}\]
LaTeX source
\[
(\Gamma \times \mathbb{Z}/2) \cdot \mu_{n}(\overline{\mathbb{Q}})
\longrightarrow \widetilde{\mathcal{T}}_{\nu}
\]\[\mu_{n}(\overline{\mathbb{Q}}) \longrightarrow \widetilde{\mathcal{T}}_{\nu}\]
LaTeX source
\[
\mu_{n}(\overline{\mathbb{Q}}) \longrightarrow \widetilde{\mathcal{T}}_{\nu}
\]\[\mu_{n}(\overline{\mathbb{Q}}) \xrightarrow{\ \rho_{\nu}\ } \widetilde{A}_{\nu}\]
LaTeX source
\[
\mu_{n}(\overline{\mathbb{Q}}) \xrightarrow{\ \rho_{\nu}\ } \widetilde{A}_{\nu}
\]\[\mathbb{Z}/2 \longrightarrow \widetilde{A}_{\nu}\]
LaTeX source
\[
\mathbb{Z}/2 \longrightarrow \widetilde{A}_{\nu}
\]\[(\lambda_{0}, \lambda_{1}, \dots, \lambda_{\nu}) \xrightarrow{\ t_{\nu}\ }
\bigl(\lambda_{0},\ g_{\nu}\lambda_{\nu}g_{\nu}^{-1},\
g_{\nu-1}\lambda_{\nu-1}g_{\nu-1}^{-1}, \dots,\ g_{2}\lambda_{2}g_{2}^{-1},\ \lambda_{1}\bigr)\]
LaTeX source
\[
(\lambda_{0}, \lambda_{1}, \dots, \lambda_{\nu}) \xrightarrow{\ t_{\nu}\ }
\bigl(\lambda_{0},\ g_{\nu}\lambda_{\nu}g_{\nu}^{-1},\
g_{\nu-1}\lambda_{\nu-1}g_{\nu-1}^{-1}, \dots,\ g_{2}\lambda_{2}g_{2}^{-1},\ \lambda_{1}\bigr)
\]\[\Gamma \xrightarrow{\ i_{\nu}\ } \widetilde{\mathcal{T}}_{\nu}
\overset{\varepsilon_{\nu}}{\dashrightarrow}
\mathfrak{S}_{n} \supset (\mathbb{Z}/n)^{*}\]
LaTeX source
\[
\Gamma \xrightarrow{\ i_{\nu}\ } \widetilde{\mathcal{T}}_{\nu}
\overset{\varepsilon_{\nu}}{\dashrightarrow}
\mathfrak{S}_{n} \supset (\mathbb{Z}/n)^{*}
\]\[\varepsilon_{\nu}\, i_{\nu} = \chi_{n} \bmod n\]
LaTeX source
\[
\varepsilon_{\nu}\, i_{\nu} = \chi_{n} \bmod n
\]\[\begin{cases}
i_{\nu}(\Gamma) \text{ commute à } t_{\nu}\\[2pt]
i_{\nu}(\gamma)\, \rho_{\nu}(i)\, i_{\nu}(\gamma)^{-1} = \rho_{\nu}(\chi_{n}(\gamma)\, i)
\end{cases}\]
LaTeX source
\[
\begin{cases}
i_{\nu}(\Gamma) \text{ commute à } t_{\nu}\\[2pt]
i_{\nu}(\gamma)\, \rho_{\nu}(i)\, i_{\nu}(\gamma)^{-1} = \rho_{\nu}(\chi_{n}(\gamma)\, i)
\end{cases}
\]\[1 \to \mathcal{T}_{\nu}^{!\,0} \to \mathcal{T}_{\nu}^{!} \to
\widehat{\mathbb{Z}}^{\times} \to 1\]
LaTeX source
\[
1 \to \mathcal{T}_{\nu}^{!\,0} \to \mathcal{T}_{\nu}^{!} \to
\widehat{\mathbb{Z}}^{\times} \to 1
\]\[\mathcal{T}_{\nu}^{!\,0} \subset \mathcal{T}_{\nu}^{0},\qquad
\mathcal{T}_{\nu}^{!\,0} =
\Bigl\{\gamma \in \mathcal{T}_{\nu}^{0} \Bigm| \gamma \text{ commute aux } \rho_{\nu}(i)
\text{ et à } t_{\nu}\Bigr\}\]
LaTeX source
\[
\mathcal{T}_{\nu}^{!\,0} \subset \mathcal{T}_{\nu}^{0},\qquad
\mathcal{T}_{\nu}^{!\,0} =
\Bigl\{\gamma \in \mathcal{T}_{\nu}^{0} \Bigm| \gamma \text{ commute aux } \rho_{\nu}(i)
\text{ et à } t_{\nu}\Bigr\}
\]\[\Gamma \times \mathfrak{S}_{3} \longrightarrow \widetilde{\mathcal{T}}_{2}\]
LaTeX source
\[
\Gamma \times \mathfrak{S}_{3} \longrightarrow \widetilde{\mathcal{T}}_{2}
\]\[\widetilde{\mathcal{T}}_{2} \simeq \mathfrak{S}_{3} \cdot \mathcal{T}_{2}\]
LaTeX source
\[
\widetilde{\mathcal{T}}_{2} \simeq \mathfrak{S}_{3} \cdot \mathcal{T}_{2}
\]\[\Gamma \longrightarrow \struck{\ill{}}\ \widetilde{\mathcal{T}}_{2}
\xrightarrow{\ \varepsilon\ } \mathfrak{S}_{3}\quad \text{trivial}\]
LaTeX source
\[
\Gamma \longrightarrow \struck{\ill{}}\ \widetilde{\mathcal{T}}_{2}
\xrightarrow{\ \varepsilon\ } \mathfrak{S}_{3}\quad \text{trivial}
\]\[\boxed{\ \Gamma \longrightarrow \mathcal{T}_{2}^{\mathfrak{S}_{3}}\ }\]
LaTeX source
\[
\boxed{\ \Gamma \longrightarrow \mathcal{T}_{2}^{\mathfrak{S}_{3}}\ }
\]\[\mathcal{T}_{2}^{!} \simeq \mathcal{T}_{\nu}^{!}\]
LaTeX source
\[
\mathcal{T}_{2}^{!} \simeq \mathcal{T}_{\nu}^{!}
\]\[\begin{cases}
\pi \ \text{groupe profini}\\
T\ \ \widehat{\mathbb{Z}}\text{-module libre de rang } 1
\ \struck{\text{de } \widehat{\mathbb{Z}}\ (\text{isomorphisme non précisé})}\\
\Phi_{i} \in \mathrm{Homext}(L, \pi)\ \struck{\ill{}}
\qquad 0 \le i \le \nu \qquad (\nu \ge 2)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\pi \ \text{groupe profini}\\
T\ \ \widehat{\mathbb{Z}}\text{-module libre de rang } 1
\ \struck{\text{de } \widehat{\mathbb{Z}}\ (\text{isomorphisme non précisé})}\\
\Phi_{i} \in \mathrm{Homext}(L, \pi)\ \struck{\ill{}}
\qquad 0 \le i \le \nu \qquad (\nu \ge 2)
\end{cases}
\]\[\begin{cases}
\exists\, e \in T^{*} \text{ base de } T\\
(\varphi_{i})_{0 \le i \le \nu} \in \prod_{0 \le i \le \nu} \Phi_{i}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\exists\, e \in T^{*} \text{ base de } T\\
(\varphi_{i})_{0 \le i \le \nu} \in \prod_{0 \le i \le \nu} \Phi_{i}
\end{cases}
\]\[\widehat{L}_{\nu} \struck{\ill{}} \longrightarrow \pi,\qquad
\lambda_{i} \longmapsto l_{i}\]
LaTeX source
\[
\widehat{L}_{\nu} \struck{\ill{}} \longrightarrow \pi,\qquad
\lambda_{i} \longmapsto l_{i}
\]\[\pi_{0} = \widehat{L}_{\nu} = \struck{\text{groupe profini}}\
(\lambda_{0}, \dots, \lambda_{\nu} \mid \lambda_{0}\lambda_{1}\cdots\lambda_{\nu} = 1)^{\wedge}\]
LaTeX source
\[
\pi_{0} = \widehat{L}_{\nu} = \struck{\text{groupe profini}}\
(\lambda_{0}, \dots, \lambda_{\nu} \mid \lambda_{0}\lambda_{1}\cdots\lambda_{\nu} = 1)^{\wedge}
\]\[T_{0} = \widehat{\mathbb{Z}},\quad e_{0} = 1\]
LaTeX source
\[
T_{0} = \widehat{\mathbb{Z}},\quad e_{0} = 1
\]\[\Phi_{0i} \text{ classe de } \widehat{\mathbb{Z}} \longrightarrow \pi_{\nu},
\qquad e_{0} \longmapsto \lambda_{i}\]
LaTeX source
\[
\Phi_{0i} \text{ classe de } \widehat{\mathbb{Z}} \longrightarrow \pi_{\nu},
\qquad e_{0} \longmapsto \lambda_{i}
\]\[A_{\nu} = \mathrm{Aut}(\widehat{L}_{\nu}, T_{0}, (\Phi_{0i}))
\xrightarrow{\ \chi_{\nu}\ } \widehat{\mathbb{Z}}^{*} = \mathrm{Aut}(T_{0})\]
LaTeX source
\[
A_{\nu} = \mathrm{Aut}(\widehat{L}_{\nu}, T_{0}, (\Phi_{0i}))
\xrightarrow{\ \chi_{\nu}\ } \widehat{\mathbb{Z}}^{*} = \mathrm{Aut}(T_{0})
\]\[\widehat{L}_{\nu} \xrightarrow{\ i\ } \mathrm{Ker}\,\chi_{\nu} \subset A_{\nu}
\quad\text{injectif},\qquad
g \longmapsto (\mathrm{int}(g), \mathrm{id}_{T_{0}})\]
LaTeX source
\[
\widehat{L}_{\nu} \xrightarrow{\ i\ } \mathrm{Ker}\,\chi_{\nu} \subset A_{\nu}
\quad\text{injectif},\qquad
g \longmapsto (\mathrm{int}(g), \mathrm{id}_{T_{0}})
\]\[A_{\nu}/\widehat{L}_{\nu} = \Gamma_{\nu}\]
LaTeX source
\[
A_{\nu}/\widehat{L}_{\nu} = \Gamma_{\nu}
\]\[L_{\nu} = \{\lambda_{0}, \dots, \lambda_{\nu} \mid \lambda_{0}\lambda_{1}\cdots\lambda_{\nu} = 1\}
\quad \text{groupe discret}\]
LaTeX source
\[
L_{\nu} = \{\lambda_{0}, \dots, \lambda_{\nu} \mid \lambda_{0}\lambda_{1}\cdots\lambda_{\nu} = 1\}
\quad \text{groupe discret}
\]\[\Psi_{i} \in \mathrm{Homext}(\mathbb{Z}, L_{\nu})\ \text{classe de}\
\psi_{i} : \mathbb{Z} \to L_{\nu},\quad \psi_{i}(1) = \lambda_{i}\]
LaTeX source
\[
\Psi_{i} \in \mathrm{Homext}(\mathbb{Z}, L_{\nu})\ \text{classe de}\
\psi_{i} : \mathbb{Z} \to L_{\nu},\quad \psi_{i}(1) = \lambda_{i}
\]\[B_{\nu} = \mathrm{Aut}(L_{\nu}, \mathbb{Z}, (\Psi_{i}))\]
LaTeX source
\[
B_{\nu} = \mathrm{Aut}(L_{\nu}, \mathbb{Z}, (\Psi_{i}))
\]\[B_{\nu} \xrightarrow{\ \chi_{\nu}^{0}\ } \mathbb{Z}^{*} = \{\pm 1\}
\quad\text{surjectif}\]
LaTeX source
\[
B_{\nu} \xrightarrow{\ \chi_{\nu}^{0}\ } \mathbb{Z}^{*} = \{\pm 1\}
\quad\text{surjectif}
\]\[L_{\nu} \hookrightarrow \mathrm{Ker}(\chi_{\nu}^{0}),\qquad
g \longmapsto (\mathrm{int}(g), \mathrm{id}_{\mathbb{Z}})\]
LaTeX source
\[
L_{\nu} \hookrightarrow \mathrm{Ker}(\chi_{\nu}^{0}),\qquad
g \longmapsto (\mathrm{int}(g), \mathrm{id}_{\mathbb{Z}})
\]\[\mathcal{T}_{\nu} = B_{\nu}/L_{\nu}\]
LaTeX source
\[
\mathcal{T}_{\nu} = B_{\nu}/L_{\nu}
\]\[1 \to \mathcal{T}_{\nu}^{0} \to \mathcal{T}_{\nu} \to \mathbb{Z}/2 \to 1\]
LaTeX source
\[
1 \to \mathcal{T}_{\nu}^{0} \to \mathcal{T}_{\nu} \to \mathbb{Z}/2 \to 1
\]\[\widehat{B}_{\nu}/\widehat{L}_{\nu} = \widehat{\mathcal{T}}_{\nu}
\subset A_{\nu}/\widehat{L}_{\nu} = \Gamma_{\nu}\]
LaTeX source
\[
\widehat{B}_{\nu}/\widehat{L}_{\nu} = \widehat{\mathcal{T}}_{\nu}
\subset A_{\nu}/\widehat{L}_{\nu} = \Gamma_{\nu}
\]\[\varepsilon = (\varepsilon_{0}, \varepsilon_{1}, \dots, \varepsilon_{\nu})
\in (\overline{\mathbb{N}^{*}})^{\nu+1}\]
LaTeX source
\[
\varepsilon = (\varepsilon_{0}, \varepsilon_{1}, \dots, \varepsilon_{\nu})
\in (\overline{\mathbb{N}^{*}})^{\nu+1}
\]\[L_{\nu}^{\varepsilon} = \bigl\{\lambda_{0}, \dots, \lambda_{\nu} \bigm|
\lambda_{0}\lambda_{1}\cdots\lambda_{\nu}
= \lambda_{0}^{\varepsilon_{0}} = \lambda_{1}^{\varepsilon_{1}} = \dots
= \lambda_{\nu}^{\varepsilon_{\nu}} = 1\bigr\}\]
LaTeX source
\[
L_{\nu}^{\varepsilon} = \bigl\{\lambda_{0}, \dots, \lambda_{\nu} \bigm|
\lambda_{0}\lambda_{1}\cdots\lambda_{\nu}
= \lambda_{0}^{\varepsilon_{0}} = \lambda_{1}^{\varepsilon_{1}} = \dots
= \lambda_{\nu}^{\varepsilon_{\nu}} = 1\bigr\}
\]\[\left\{
\begin{array}{l}
\Gamma \text{ groupe},\ \mathcal{C}\ \Gamma\text{-torseur à droite}\\
\pi,\ \Gamma\text{-groupe}\quad \struck{\ill{}}\\
T_{i}\ (i \in I)\ (\pi, \Gamma)\text{-tors.}\\
\pi_{i}\ \text{commutant}\\
I_{i} \hookrightarrow \pi_{i}\ \text{sous-}\Gamma\text{-groupe}
\end{array}
\right.
\qquad\Bigg|\qquad
\begin{array}{l}
\widetilde{\pi}\ \text{groupe}\\
\widetilde{T}_{i}\ (i \in I)\ \widetilde{\pi}\text{-torseur}\\
\widetilde{\pi}_{i}\ \text{son commutant}\\
\widetilde{I}_{i} \subset \widetilde{\pi}_{i}
\end{array}\]
LaTeX source
\[
\left\{
\begin{array}{l}
\Gamma \text{ groupe},\ \mathcal{C}\ \Gamma\text{-torseur à droite}\\
\pi,\ \Gamma\text{-groupe}\quad \struck{\ill{}}\\
T_{i}\ (i \in I)\ (\pi, \Gamma)\text{-tors.}\\
\pi_{i}\ \text{commutant}\\
I_{i} \hookrightarrow \pi_{i}\ \text{sous-}\Gamma\text{-groupe}
\end{array}
\right.
\qquad\Bigg|\qquad
\begin{array}{l}
\widetilde{\pi}\ \text{groupe}\\
\widetilde{T}_{i}\ (i \in I)\ \widetilde{\pi}\text{-torseur}\\
\widetilde{\pi}_{i}\ \text{son commutant}\\
\widetilde{I}_{i} \subset \widetilde{\pi}_{i}
\end{array}
\]\[\mathcal{C} \xrightarrow{\ \rho\ } \mathrm{Isom}(\pi, \widetilde{\pi})
\qquad \Gamma\text{-homomorphisme}\]
LaTeX source
\[
\mathcal{C} \xrightarrow{\ \rho\ } \mathrm{Isom}(\pi, \widetilde{\pi})
\qquad \Gamma\text{-homomorphisme}
\]\[\pi \xrightarrow{\ \rho(\varphi)\ } \widetilde{\pi}
\quad\text{donc}\quad \widetilde{\pi} \simeq \mathcal{C} \wedge_{\Gamma} \pi,
\qquad
T_{i} \xrightarrow{\ \rho_{i}(\varphi)\ } \widetilde{T}_{i}\]
LaTeX source
\[
\pi \xrightarrow{\ \rho(\varphi)\ } \widetilde{\pi}
\quad\text{donc}\quad \widetilde{\pi} \simeq \mathcal{C} \wedge_{\Gamma} \pi,
\qquad
T_{i} \xrightarrow{\ \rho_{i}(\varphi)\ } \widetilde{T}_{i}
\]\[\begin{cases}
\widetilde{\pi} = \mathcal{C} \wedge_{\Gamma} \pi\\
(\widetilde{\pi}_{i}, \widetilde{T}_{i}, \widetilde{\pi}) = \mathcal{C} \wedge_{\Gamma} (\pi_{i}, T_{i}, \pi)\\
\struck{\ill{}}\ \widetilde{I}_{i} \subset \widetilde{\pi}_{i}
\qquad \widetilde{I}_{i} = \mathcal{C} \wedge_{\Gamma} I_{i}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\widetilde{\pi} = \mathcal{C} \wedge_{\Gamma} \pi\\
(\widetilde{\pi}_{i}, \widetilde{T}_{i}, \widetilde{\pi}) = \mathcal{C} \wedge_{\Gamma} (\pi_{i}, T_{i}, \pi)\\
\struck{\ill{}}\ \widetilde{I}_{i} \subset \widetilde{\pi}_{i}
\qquad \widetilde{I}_{i} = \mathcal{C} \wedge_{\Gamma} I_{i}
\end{cases}
\]\[\widetilde{I}_{i} \simeq \widehat{\mathbb{Z}}
\quad\text{d'où}\quad
\chi_{i} : \Gamma \to \widehat{\mathbb{Z}}^{*} \simeq \mathrm{Aut}(I_{i})\]
LaTeX source
\[
\widetilde{I}_{i} \simeq \widehat{\mathbb{Z}}
\quad\text{d'où}\quad
\chi_{i} : \Gamma \to \widehat{\mathbb{Z}}^{*} \simeq \mathrm{Aut}(I_{i})
\]\[\Gamma\ (= \mathrm{Aut}(\overline{\mathbb{Q}}))\ \text{groupe},\quad
\mathcal{C}\ \Gamma\text{-tors.\ à droite}
\quad (= \mathrm{Isom}(\overline{\mathbb{Q}}, \ill{}))\]
LaTeX source
\[
\Gamma\ (= \mathrm{Aut}(\overline{\mathbb{Q}}))\ \text{groupe},\quad
\mathcal{C}\ \Gamma\text{-tors.\ à droite}
\quad (= \mathrm{Isom}(\overline{\mathbb{Q}}, \ill{}))
\]\[\pi\ \bigl(= \pi_{1}(\mathbb{P}^{1}_{\overline{\mathbb{Q}}} - I_{\overline{\mathbb{Q}}})\bigr)
\ \Gamma\text{-groupe}\quad
(I \subset \mathbb{P}^{1}(\mathbb{Q})\ \text{fini})\]
LaTeX source
\[
\pi\ \bigl(= \pi_{1}(\mathbb{P}^{1}_{\overline{\mathbb{Q}}} - I_{\overline{\mathbb{Q}}})\bigr)
\ \Gamma\text{-groupe}\quad
(I \subset \mathbb{P}^{1}(\mathbb{Q})\ \text{fini})
\]\[\forall i \in I\quad T_{i}\ \text{un}\ (\pi, \Gamma)\text{-torseur à droite}
\quad (= \mathrm{Isom}(F_{\overline{\xi}}, F_{\overline{\xi}_{i}}))\]
LaTeX source
\[
\forall i \in I\quad T_{i}\ \text{un}\ (\pi, \Gamma)\text{-torseur à droite}
\quad (= \mathrm{Isom}(F_{\overline{\xi}}, F_{\overline{\xi}_{i}}))
\]\[\forall\ r_{i} : \mathcal{C} \to T_{i}\ \text{satisfaisant}\quad
r_{i}(\varphi\gamma) = \gamma^{-1} \cdot r_{i}(\varphi)
\qquad \Big/\ r_{\cdot} : \mathcal{C} \to \textstyle\prod T_{i}\]
LaTeX source
\[
\forall\ r_{i} : \mathcal{C} \to T_{i}\ \text{satisfaisant}\quad
r_{i}(\varphi\gamma) = \gamma^{-1} \cdot r_{i}(\varphi)
\qquad \Big/\ r_{\cdot} : \mathcal{C} \to \textstyle\prod T_{i}
\]\[\pi_{i} = \mathrm{Aut}_{\pi}(T_{i})\ \ \text{commutant de } \pi \text{ dans } T_{i}
\ \ \text{ou}\ \ T_{i} \wedge_{\pi} (\pi\ \mathrm{int})\]
LaTeX source
\[
\pi_{i} = \mathrm{Aut}_{\pi}(T_{i})\ \ \text{commutant de } \pi \text{ dans } T_{i}
\ \ \text{ou}\ \ T_{i} \wedge_{\pi} (\pi\ \mathrm{int})
\]\[I_{i} = T_{\cdot}(\overline{\mathbb{Q}})
= \varprojlim_{n} \mu_{n}(\overline{\mathbb{Q}})
\hookrightarrow \pi_{i}\]
LaTeX source
\[
I_{i} = T_{\cdot}(\overline{\mathbb{Q}})
= \varprojlim_{n} \mu_{n}(\overline{\mathbb{Q}})
\hookrightarrow \pi_{i}
\]\[\mathcal{C}_{2}(\pi\text{-ens}) \xrightarrow{\ f\ } (\mathrm{ens})
\quad\text{\uncertain{ou} \ill{} de $\pi$}\]
LaTeX source
\[
\mathcal{C}_{2}(\pi\text{-ens}) \xrightarrow{\ f\ } (\mathrm{ens})
\quad\text{\uncertain{ou} \ill{} de $\pi$}
\]\[\begin{array}{l}
\downarrow\ \overline{F}_{i}(E) = T_{i} \wedge_{\pi} E\\
\pi_{i}\text{-}(\mathrm{Ens})\\
\downarrow\ E_{i} \longmapsto E_{i}/I_{i} = \widehat{E}_{i}\\
\mathrm{Ens}
\end{array}\]
LaTeX source
\[
\begin{array}{l}
\downarrow\ \overline{F}_{i}(E) = T_{i} \wedge_{\pi} E\\
\pi_{i}\text{-}(\mathrm{Ens})\\
\downarrow\ E_{i} \longmapsto E_{i}/I_{i} = \widehat{E}_{i}\\
\mathrm{Ens}
\end{array}
\]\[\mathcal{C} \longrightarrow \mathrm{Isom}(\pi, \widetilde{\pi}).\]
LaTeX source
\[
\mathcal{C} \longrightarrow \mathrm{Isom}(\pi, \widetilde{\pi}).
\]\[\alpha(\varphi\gamma)(\lambda) = \gamma_{\pi}^{-1}\,\alpha(\varphi)\bigl(\gamma_{I}(\lambda)\bigr),
\qquad \gamma_{I}(\lambda) = \lambda^{\gamma}\]
LaTeX source
\[
\alpha(\varphi\gamma)(\lambda) = \gamma_{\pi}^{-1}\,\alpha(\varphi)\bigl(\gamma_{I}(\lambda)\bigr),
\qquad \gamma_{I}(\lambda) = \lambda^{\gamma}
\]\[\gamma_{\pi}^{-1}\,\alpha_{i}(\varphi_{0})(\lambda_{i}) = l_{i}\]
LaTeX source
\[
\gamma_{\pi}^{-1}\,\alpha_{i}(\varphi_{0})(\lambda_{i}) = l_{i}
\]\[l_{1}, \dots, l_{n}\qquad
\bigl(\gamma_{\pi}^{-1}(l_{1}^{\gamma}), \dots, \gamma_{\pi}^{-1}(l_{n}^{\gamma})\bigr)\]
LaTeX source
\[
l_{1}, \dots, l_{n}\qquad
\bigl(\gamma_{\pi}^{-1}(l_{1}^{\gamma}), \dots, \gamma_{\pi}^{-1}(l_{n}^{\gamma})\bigr)
\]\[l_{1} l_{2} \cdots l_{n} = 1\]
LaTeX source
\[
l_{1} l_{2} \cdots l_{n} = 1
\]\[\gamma_{\pi}^{-1}(l_{1})\]
LaTeX source
\[
\gamma_{\pi}^{-1}(l_{1})
\]\[\mathcal{C} \xrightarrow{\ \rho_{i}\ } \mathrm{Isom}(\widehat{\mathbb{Z}}, I_{i})
\qquad \rho_{i}(\varphi)(1) = \lambda_{i}(\varphi)\]
LaTeX source
\[
\mathcal{C} \xrightarrow{\ \rho_{i}\ } \mathrm{Isom}(\widehat{\mathbb{Z}}, I_{i})
\qquad \rho_{i}(\varphi)(1) = \lambda_{i}(\varphi)
\]\[\pi \xrightarrow[\text{indépendant de } i]{\ \chi\ } \widehat{\mathbb{Z}}^{*}
\qquad \lambda_{i}(\varphi\gamma) = \chi(\gamma)\,\lambda_{i}(\varphi)\]
LaTeX source
\[
\pi \xrightarrow[\text{indépendant de } i]{\ \chi\ } \widehat{\mathbb{Z}}^{*}
\qquad \lambda_{i}(\varphi\gamma) = \chi(\gamma)\,\lambda_{i}(\varphi)
\]\[\rho_{i}(\varphi\gamma)(n) = \rho_{i}(\varphi)\bigl(\struck{n^{\chi(\gamma)}}\ \chi(\gamma) \cdot n\bigr)
= \chi(\gamma)\,\rho_{i}(\varphi)(n)\]
LaTeX source
\[
\rho_{i}(\varphi\gamma)(n) = \rho_{i}(\varphi)\bigl(\struck{n^{\chi(\gamma)}}\ \chi(\gamma) \cdot n\bigr)
= \chi(\gamma)\,\rho_{i}(\varphi)(n)
\]\[\lambda_{i}(\varphi) \in I_{i} \xrightarrow{\ \widetilde{r}_{i}(\varphi)\ } \pi\]
LaTeX source
\[
\lambda_{i}(\varphi) \in I_{i} \xrightarrow{\ \widetilde{r}_{i}(\varphi)\ } \pi
\]\[(\varphi \in \mathcal{C}) \longmapsto
\pi \overset{\alpha_{i}(\varphi)}{\rightleftarrows} \pi_{i}
\qquad \text{d'où}\quad I_{i} \hookrightarrow \pi_{i}\]
LaTeX source
\[
(\varphi \in \mathcal{C}) \longmapsto
\pi \overset{\alpha_{i}(\varphi)}{\rightleftarrows} \pi_{i}
\qquad \text{d'où}\quad I_{i} \hookrightarrow \pi_{i}
\]\[\alpha_{i}(\varphi\gamma) = \gamma_{\pi}^{-1}\,\alpha_{i}(\varphi)\,\gamma_{\pi_{i}}\]
LaTeX source
\[
\alpha_{i}(\varphi\gamma) = \gamma_{\pi}^{-1}\,\alpha_{i}(\varphi)\,\gamma_{\pi_{i}}
\]\[\mathcal{C} \xrightarrow{\ \beta\ } \mathrm{Isom}(\pi, \widetilde{\pi})
\qquad \beta(\varphi\gamma) = \beta(\varphi) \circ \gamma_{\pi}\]
LaTeX source
\[
\mathcal{C} \xrightarrow{\ \beta\ } \mathrm{Isom}(\pi, \widetilde{\pi})
\qquad \beta(\varphi\gamma) = \beta(\varphi) \circ \gamma_{\pi}
\]\[\begin{aligned}
\alpha'_{i}(\varphi\gamma) &= \beta(\varphi\gamma)\,\alpha_{i}(\varphi\gamma)
= \beta(\varphi)\,\gamma_{\pi}\,\gamma_{\pi}^{-1}\,\alpha_{i}(\varphi)\,\gamma_{\pi_{i}}\\
&= \beta(\varphi)\,\alpha_{i}(\varphi)\,\gamma_{\pi_{i}} = \alpha'_{i}(\varphi)\,\gamma_{\pi_{i}}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\alpha'_{i}(\varphi\gamma) &= \beta(\varphi\gamma)\,\alpha_{i}(\varphi\gamma)
= \beta(\varphi)\,\gamma_{\pi}\,\gamma_{\pi}^{-1}\,\alpha_{i}(\varphi)\,\gamma_{\pi_{i}}\\
&= \beta(\varphi)\,\alpha_{i}(\varphi)\,\gamma_{\pi_{i}} = \alpha'_{i}(\varphi)\,\gamma_{\pi_{i}}
\end{aligned}
\]\[\widetilde{I}_{i} = \alpha'_{i}(\varphi)(I_{i})\quad \text{indépendant de } i\]
LaTeX source
\[
\widetilde{I}_{i} = \alpha'_{i}(\varphi)(I_{i})\quad \text{indépendant de } i
\]\[\struck{\mathrm{Isom}(I_{i}, \widetilde{I}_{i}) =}\qquad
\widetilde{I}_{i} = \mathcal{C} \wedge_{\pi} I_{i}\]
LaTeX source
\[
\struck{\mathrm{Isom}(I_{i}, \widetilde{I}_{i}) =}\qquad
\widetilde{I}_{i} = \mathcal{C} \wedge_{\pi} I_{i}
\]\[\text{Si}\quad \struck{\pi' = \pi/\ill{} \dots\ \ill{}}\quad
\pi' = \pi/U_{i},\quad U_{i} = \text{sous-groupe de } \pi \text{ opérant trivialement sur } I_{i}\]
LaTeX source
\[
\text{Si}\quad \struck{\pi' = \pi/\ill{} \dots\ \ill{}}\quad
\pi' = \pi/U_{i},\quad U_{i} = \text{sous-groupe de } \pi \text{ opérant trivialement sur } I_{i}
\]\[\widetilde{I}_{i} = \mathcal{C} \wedge_{\pi} I_{i}\]
LaTeX source
\[
\widetilde{I}_{i} = \mathcal{C} \wedge_{\pi} I_{i}
\]\[I_{i} \simeq T_{\cdot}(\overline{\mathbb{Q}})\qquad
\pi' \simeq \mathrm{Aut}(I_{i}) \simeq \widehat{\mathbb{Z}}^{*}\]
LaTeX source
\[
I_{i} \simeq T_{\cdot}(\overline{\mathbb{Q}})\qquad
\pi' \simeq \mathrm{Aut}(I_{i}) \simeq \widehat{\mathbb{Z}}^{*}
\]\[\mathcal{C} \simeq \varprojlim \mu_{n}(\overline{\mathbb{Q}})^{*}\]
LaTeX source
\[
\mathcal{C} \simeq \varprojlim \mu_{n}(\overline{\mathbb{Q}})^{*}
\]\[\struck{\ill{}\ S, A,}\qquad
S \xrightarrow{\ p\ } \mathbb{N}^{*}\qquad
A \xrightarrow{\ q\ } \mathbb{N}^{*}\qquad
F \xrightarrow{\ r\ } \mathbb{N}^{*}\]
LaTeX source
\[
\struck{\ill{}\ S, A,}\qquad
S \xrightarrow{\ p\ } \mathbb{N}^{*}\qquad
A \xrightarrow{\ q\ } \mathbb{N}^{*}\qquad
F \xrightarrow{\ r\ } \mathbb{N}^{*}
\]\[\struck{\varphi_{S} : S' \to S}\]
LaTeX source
\[
\struck{\varphi_{S} : S' \to S}
\]\[\varphi_{s} : S' \to S,\qquad \varphi_{a} : A' \to A,\qquad \varphi_{f} : F' \to F\]
LaTeX source
\[
\varphi_{s} : S' \to S,\qquad \varphi_{a} : A' \to A,\qquad \varphi_{f} : F' \to F
\]\[\left\{
\begin{array}{l}
p(s') = \nu(\varphi, s)\, p(s)\quad \struck{p(\varphi_{s}(s'))\ |\ p(\ill{})}\\
q(a') = \nu(\varphi, a)\, q(a)\quad \struck{q(\varphi_{a}(a'))\ |\ q(\ill{})}\\
r(f') = \nu(\varphi, f)\, r(f)\quad \struck{r(\varphi_{f}(f'))\ |\ r(\ill{})}
\end{array}
\right.
\qquad
\begin{array}{l}
\forall s' \in S',\ s = \varphi_{s}(s')\\
\forall a' \in A',\ a = \varphi_{a}(a')\\
\forall f' \in F',\ f = \varphi_{f}(f')
\end{array}\]
LaTeX source
\[
\left\{
\begin{array}{l}
p(s') = \nu(\varphi, s)\, p(s)\quad \struck{p(\varphi_{s}(s'))\ |\ p(\ill{})}\\
q(a') = \nu(\varphi, a)\, q(a)\quad \struck{q(\varphi_{a}(a'))\ |\ q(\ill{})}\\
r(f') = \nu(\varphi, f)\, r(f)\quad \struck{r(\varphi_{f}(f'))\ |\ r(\ill{})}
\end{array}
\right.
\qquad
\begin{array}{l}
\forall s' \in S',\ s = \varphi_{s}(s')\\
\forall a' \in A',\ a = \varphi_{a}(a')\\
\forall f' \in F',\ f = \varphi_{f}(f')
\end{array}
\]\[\mathcal{C} \longmapsto \mathrm{Fac}(\mathcal{C})\]
LaTeX source
\[
\mathcal{C} \longmapsto \mathrm{Fac}(\mathcal{C})
\]\[\mathcal{C}_{0} = (S, A, F, p, q, r)\ \text{une précarte}\]
LaTeX source
\[
\mathcal{C}_{0} = (S, A, F, p, q, r)\ \text{une précarte}
\]\[\struck{\mathrm{Fac}(\mathcal{C}) \to \mathcal{C}}\qquad
\mathcal{C}_{0} \xrightarrow[\sim]{\ u\ } \mathrm{Fac}(\mathcal{C})\]
LaTeX source
\[
\struck{\mathrm{Fac}(\mathcal{C}) \to \mathcal{C}}\qquad
\mathcal{C}_{0} \xrightarrow[\sim]{\ u\ } \mathrm{Fac}(\mathcal{C})
\]\[\sum_{s} p(s) = \sum_{a} q(a) = \sum_{f} r(f)\ \ (= \delta),\]
LaTeX source
\[
\sum_{s} p(s) = \sum_{a} q(a) = \sum_{f} r(f)\ \ (= \delta),
\]\[\prod_{n \in \mathbb{N}^{*}} \bigl(a_{n}!\, b_{n}!\, c_{n}!\bigr)\]
LaTeX source
\[
\prod_{n \in \mathbb{N}^{*}} \bigl(a_{n}!\, b_{n}!\, c_{n}!\bigr)
\]\[\struck{f^{-1}(0_{T})_{\ill{}} \simeq}\]
LaTeX source
\[
\struck{f^{-1}(0_{T})_{\ill{}} \simeq}
\]\[f^{-1}(0_{T})_{\mathrm{réd}} \simeq \struck{\ill{}}\ S_{T\,\mathrm{réd}}\]
LaTeX source
\[
f^{-1}(0_{T})_{\mathrm{réd}} \simeq \struck{\ill{}}\ S_{T\,\mathrm{réd}}
\]\[f^{-1}(1_{T})_{\mathrm{réd}} \simeq A_{T\,\mathrm{réd}}\]
LaTeX source
\[
f^{-1}(1_{T})_{\mathrm{réd}} \simeq A_{T\,\mathrm{réd}}
\]\[f^{-1}(\infty_{T})_{\mathrm{réd}} \simeq F_{T\,\mathrm{réd}}\]
LaTeX source
\[
f^{-1}(\infty_{T})_{\mathrm{réd}} \simeq F_{T\,\mathrm{réd}}
\]\[\rho_s\,\rho_f\,\sigma = 1, \qquad \sigma^2 = 1,\]
LaTeX source
\[ \rho_s\,\rho_f\,\sigma = 1, \qquad \sigma^2 = 1, \]
\[\Bigl[\, f(z) = \lambda z(z-1) \qquad
\begin{cases} f(t) = \lambda t(t-1) = 1 \\ f'(t) = \lambda(2t-1) = 0 \end{cases}
\quad \text{i.e. } t = \tfrac12,\ \lambda = -4 \,\Bigr]\]
LaTeX source
\[
\Bigl[\, f(z) = \lambda z(z-1) \qquad
\begin{cases} f(t) = \lambda t(t-1) = 1 \\ f'(t) = \lambda(2t-1) = 0 \end{cases}
\quad \text{i.e. } t = \tfrac12,\ \lambda = -4 \,\Bigr]
\]\[f(z) = -4z(z-1)\]
LaTeX source
\[ f(z) = -4z(z-1) \]
\[\Bigl[\, f(z) = \lambda \frac{z^2}{z-1}, \qquad f(t) = \lambda\frac{t^2}{t-1} = 1\]
LaTeX source
\[
\Bigl[\, f(z) = \lambda \frac{z^2}{z-1}, \qquad f(t) = \lambda\frac{t^2}{t-1} = 1
\]\[\Bigl[\, f'(z) = \lambda\Bigl(\frac{2z(z-1) - z^2}{(z-1)^2}\Bigr) = \lambda\frac{z^2 - 2z}{(z-1)^2} = 0 \,\Bigr]\]
LaTeX source
\[
\Bigl[\, f'(z) = \lambda\Bigl(\frac{2z(z-1) - z^2}{(z-1)^2}\Bigr) = \lambda\frac{z^2 - 2z}{(z-1)^2} = 0 \,\Bigr]
\]\[f'(t) = 0, \quad t = 2, \quad \lambda = \frac14 \,\Bigr]\]
LaTeX source
\[ f'(t) = 0, \quad t = 2, \quad \lambda = \frac14 \,\Bigr] \]
\[f(z) = \frac14\,\frac{z^2}{z-1}\]
LaTeX source
\[
f(z) = \frac14\,\frac{z^2}{z-1}
\]\[f(z) = \frac{4}{27}\,\frac{z^3}{z-1}\]
LaTeX source
\[
f(z) = \frac{4}{27}\,\frac{z^3}{z-1}
\]\[2x(x^2+y^2) + (-3x^2+y^2) = 0\]
LaTeX source
\[ 2x(x^2+y^2) + (-3x^2+y^2) = 0 \]
\[f(z) = \frac{27}{4}\,z(z-1)^2\]
LaTeX source
\[
f(z) = \frac{27}{4}\,z(z-1)^2
\]\[\begin{matrix} 3 & 2_2 & \\ 2 & 1_2 & 6 \\ 1 & & \end{matrix}\]
LaTeX source
\[
\begin{matrix} 3 & 2_2 & \\ 2 & 1_2 & 6 \\ 1 & & \end{matrix}
\]\[\sigma_2\sigma_1\sigma_1\sigma_0 = \qquad
\rho_s\,\rho_f = \sigma, \qquad \rho_f = \rho_s^{-1}\sigma\]
LaTeX source
\[
\sigma_2\sigma_1\sigma_1\sigma_0 = \qquad
\rho_s\,\rho_f = \sigma, \qquad \rho_f = \rho_s^{-1}\sigma
\]\[\begin{array}{llll}
r_1 = (a, ab) & \rho_s r_1 = r_2 & \sigma r_1 = r_4 & \rho_f r_1 = r_5 \\
r_2 = (a, axa) & \rho_s r_2 = r_3 & \sigma r_4 = r_1 & \rho_f r_5 = r_4 \\
r_3 = (a, ac) & \rho_s r_3 = r_1 & \sigma r_2 = r_2 & \rho_f r_4 = r_3 \\
r_4 = (b, ba) & \rho_s r_4 = r_5 & \sigma r_3 = r_6 & \rho_f r_3 = r_6 \\
r_5 = (b, byb) & \rho_s r_5 = r_4 & \sigma r_6 = r_3 & \rho_f r_6 = r_2 \\
r_6 = (c, ca) & \rho_s r_6 = r_6 & \sigma r_5 = r_5 & \rho_f r_2 = r_1
\end{array}\]
LaTeX source
\[
\begin{array}{llll}
r_1 = (a, ab) & \rho_s r_1 = r_2 & \sigma r_1 = r_4 & \rho_f r_1 = r_5 \\
r_2 = (a, axa) & \rho_s r_2 = r_3 & \sigma r_4 = r_1 & \rho_f r_5 = r_4 \\
r_3 = (a, ac) & \rho_s r_3 = r_1 & \sigma r_2 = r_2 & \rho_f r_4 = r_3 \\
r_4 = (b, ba) & \rho_s r_4 = r_5 & \sigma r_3 = r_6 & \rho_f r_3 = r_6 \\
r_5 = (b, byb) & \rho_s r_5 = r_4 & \sigma r_6 = r_3 & \rho_f r_6 = r_2 \\
r_6 = (c, ca) & \rho_s r_6 = r_6 & \sigma r_5 = r_5 & \rho_f r_2 = r_1
\end{array}
\]\[p = 6 \quad q = 2 \quad r = 6 \qquad N =\]
LaTeX source
\[ p = 6 \quad q = 2 \quad r = 6 \qquad N = \]
\[\begin{array}{llll}
r_1 = (a, ab) & \rho_s r_1 = r_2 & \sigma r_1 = r_4 & \rho_f r_1 = r_5 \\
r_2 = (a, axa) & \rho_s r_2 = r_3 & \sigma r_4 = r_1 & \rho_f r_5 = r_6 \\
r_3 = (a, aya) & \rho_s r_3 = r_1 & \sigma r_2 = r_2 & \rho_f r_6 = r_4 \\
r_4 = (b, ba) & \rho_s r_4 = r_5 & \sigma r_3 = r_3 & \rho_f r_4 = r_3 \\
r_5 = (b, bc) & \rho_s r_5 = r_4 & \sigma r_5 = r_6 & \rho_f r_3 = r_2 \\
r_6 = (c, cb) & \rho_s r_6 = r_6 & \sigma r_6 = r_5 & \rho_f r_2 = r_1
\end{array}\]
LaTeX source
\[
\begin{array}{llll}
r_1 = (a, ab) & \rho_s r_1 = r_2 & \sigma r_1 = r_4 & \rho_f r_1 = r_5 \\
r_2 = (a, axa) & \rho_s r_2 = r_3 & \sigma r_4 = r_1 & \rho_f r_5 = r_6 \\
r_3 = (a, aya) & \rho_s r_3 = r_1 & \sigma r_2 = r_2 & \rho_f r_6 = r_4 \\
r_4 = (b, ba) & \rho_s r_4 = r_5 & \sigma r_3 = r_3 & \rho_f r_4 = r_3 \\
r_5 = (b, bc) & \rho_s r_5 = r_4 & \sigma r_5 = r_6 & \rho_f r_3 = r_2 \\
r_6 = (c, cb) & \rho_s r_6 = r_6 & \sigma r_6 = r_5 & \rho_f r_2 = r_1
\end{array}
\]\[a = 0 \quad b = 1 \qquad [\ldots] = \infty\]
LaTeX source
\[ a = 0 \quad b = 1 \qquad [\ldots] = \infty \]
\[f(z) = \lambda\, z^3 (z-1)^2 (z-c)
\qquad
\begin{cases}
0 \text{ zéro triple} \\
1 \text{ zéro double} \\
c \text{ zéro simple} \\
\infty \text{ pôle d'ordre } 6
\end{cases}\]
LaTeX source
\[
f(z) = \lambda\, z^3 (z-1)^2 (z-c)
\qquad
\begin{cases}
0 \text{ zéro triple} \\
1 \text{ zéro double} \\
c \text{ zéro simple} \\
\infty \text{ pôle d'ordre } 6
\end{cases}
\]\[f'(z) = \lambda z^2(z-1)\bigl[\,\underbrace{3(z-1)(z-c) + 2z(z-c)}_{(5z-3)(z-c)} + z(z-1)\,\bigr]\]
LaTeX source
\[
f'(z) = \lambda z^2(z-1)\bigl[\,\underbrace{3(z-1)(z-c) + 2z(z-c)}_{(5z-3)(z-c)} + z(z-1)\,\bigr]
\]\[= \lambda z^2 (z-1)\bigl[\,6z^2 - (5c+4)z + 3c\,\bigr]\]
LaTeX source
\[ = \lambda z^2 (z-1)\bigl[\,6z^2 - (5c+4)z + 3c\,\bigr] \]
\[6z^2 - (5c+4)z + 3c = 0\]
LaTeX source
\[ 6z^2 - (5c+4)z + 3c = 0 \]
\[\begin{cases}
u, v \text{ distincts entre eux et } \neq 0, 1, c \\
f(u) = f(v) = 1
\end{cases}\]
LaTeX source
\[
\begin{cases}
u, v \text{ distincts entre eux et } \neq 0, 1, c \\
f(u) = f(v) = 1
\end{cases}
\]\[f(z) - 1 = [\ldots]\ \lambda (z-u)^2 (z-v)^2 (z-x)(z-y)\]
LaTeX source
\[ f(z) - 1 = [\ldots]\ \lambda (z-u)^2 (z-v)^2 (z-x)(z-y) \]
\[\frac{1}{\lambda} f(u) = u(\alpha u + \beta)\bigl((\alpha-2)u + (\beta+1)\bigr)(u-c)\]
LaTeX source
\[
\frac{1}{\lambda} f(u) = u(\alpha u + \beta)\bigl((\alpha-2)u + (\beta+1)\bigr)(u-c)
\]\[u^2 = \alpha u + \beta, \qquad \alpha = \frac{5c}{6} + \frac23, \quad \beta = -\frac{c}{2}\]
LaTeX source
\[
u^2 = \alpha u + \beta, \qquad \alpha = \frac{5c}{6} + \frac23, \quad \beta = -\frac{c}{2}
\]\[u(\alpha u + \beta) = (\alpha^2+\beta)u + \alpha\beta\]
LaTeX source
\[ u(\alpha u + \beta) = (\alpha^2+\beta)u + \alpha\beta \]
\[\bigl((\alpha-2)u + (\beta+1)\bigr)(u-c) = (\alpha-2)u^2 + \bigl[(\beta+1) - c(\alpha-2)\bigr]u - (\beta+1)c\]
LaTeX source
\[ \bigl((\alpha-2)u + (\beta+1)\bigr)(u-c) = (\alpha-2)u^2 + \bigl[(\beta+1) - c(\alpha-2)\bigr]u - (\beta+1)c \]
\[= \bigl(\underbrace{(\alpha-2)\alpha + \beta+1 - c(\alpha-2)}_{(\alpha-2)(\alpha-c) + (\beta+1)}\bigr)u + \bigl((\alpha-2)\beta - (\beta+1)c\bigr)\]
LaTeX source
\[
= \bigl(\underbrace{(\alpha-2)\alpha + \beta+1 - c(\alpha-2)}_{(\alpha-2)(\alpha-c) + (\beta+1)}\bigr)u + \bigl((\alpha-2)\beta - (\beta+1)c\bigr)
\]\[\alpha(\alpha^2+\beta)\bigl[(\alpha-2)(\alpha-c)+\beta+1\bigr]\ \struck{\ill{}}
+ \beta(\alpha^2+\beta)\bigl[(\alpha-2)(\alpha-c)+\beta+1\bigr]\]
LaTeX source
\[
\alpha(\alpha^2+\beta)\bigl[(\alpha-2)(\alpha-c)+\beta+1\bigr]\ \struck{\ill{}}
+ \beta(\alpha^2+\beta)\bigl[(\alpha-2)(\alpha-c)+\beta+1\bigr]
\]\[+ \alpha\beta\bigl[(\alpha-2)(\alpha-c)+\beta+1\bigr]u
+ (\alpha^2+\beta)\bigl[(\alpha-2)\beta - (\beta+1)c\bigr]u\]
LaTeX source
\[ + \alpha\beta\bigl[(\alpha-2)(\alpha-c)+\beta+1\bigr]u + (\alpha^2+\beta)\bigl[(\alpha-2)\beta - (\beta+1)c\bigr]u \]
\[+ \alpha\beta\bigl[(\alpha-2)\beta - (\beta+1)c\bigr]\]
LaTeX source
\[ + \alpha\beta\bigl[(\alpha-2)\beta - (\beta+1)c\bigr] \]
\[\alpha - 2 = \frac{5c}{6} - \frac43, \qquad \alpha - c = -\frac{c}{6} + \frac23\]
LaTeX source
\[
\alpha - 2 = \frac{5c}{6} - \frac43, \qquad \alpha - c = -\frac{c}{6} + \frac23
\]\[(\alpha-2)(\alpha-c) = -\frac{5}{36}c^2 + \frac79 c - \frac89\]
LaTeX source
\[
(\alpha-2)(\alpha-c) = -\frac{5}{36}c^2 + \frac79 c - \frac89
\]\[\beta + 1 = -\frac{c}{2} + 1 = -\frac{9c}{18} + \frac99\]
LaTeX source
\[
\beta + 1 = -\frac{c}{2} + 1 = -\frac{9c}{18} + \frac99
\]\[(\alpha-2)(\alpha-c) + \beta+1 = -\frac{5}{36}c^2 + \frac{5}{18}c + \frac19\]
LaTeX source
\[
(\alpha-2)(\alpha-c) + \beta+1 = -\frac{5}{36}c^2 + \frac{5}{18}c + \frac19
\]\[= -\frac{1}{36}(5c^2 - 10c - 4)\]
LaTeX source
\[
= -\frac{1}{36}(5c^2 - 10c - 4)
\]\[= \alpha(\alpha^2 + 2\beta) = \frac{1}{6^3}(5c+4)\bigl((5c+4)^2 - 36c\bigr)\]
LaTeX source
\[
= \alpha(\alpha^2 + 2\beta) = \frac{1}{6^3}(5c+4)\bigl((5c+4)^2 - 36c\bigr)
\]\[= \frac{1}{6^3}(5c+4)(25c^2 + 4c + 16)\]
LaTeX source
\[
= \frac{1}{6^3}(5c+4)(25c^2 + 4c + 16)
\]\[(5c+4)(-29c+4) \qquad\qquad -\frac16\Bigl(\frac56\Bigr)^4 c^5\]
LaTeX source
\[ (5c+4)(-29c+4) \qquad\qquad -\frac16\Bigl(\frac56\Bigr)^4 c^5 \]
\[\lambda\bigl(P(c)u + Q(c)\bigr) = 1, \qquad \lambda\bigl(P(c)v + Q(c)\bigr) = 1\]
LaTeX source
\[ \lambda\bigl(P(c)u + Q(c)\bigr) = 1, \qquad \lambda\bigl(P(c)v + Q(c)\bigr) = 1 \]
\[P(c)(u-v) = 0 \qquad \boxed{P(c) = 0} \qquad \lambda = \frac{1}{Q(c)} \qquad Q(c) \neq 0\]
LaTeX source
\[
P(c)(u-v) = 0 \qquad \boxed{P(c) = 0} \qquad \lambda = \frac{1}{Q(c)} \qquad Q(c) \neq 0
\]\[f(z) = \frac{1}{Q(c)}\, z^3 (z-1)^2 (z-c)\]
LaTeX source
\[
f(z) = \frac{1}{Q(c)}\, z^3 (z-1)^2 (z-c)
\]\[z^2 - \frac{5c+4}{6}z + \frac{c}{2} = 0\]
LaTeX source
\[
z^2 - \frac{5c+4}{6}z + \frac{c}{2} = 0
\]\[\frac{f(z)-1}{(z-u)^2(z-v)^2} = \frac{f(z)-1}{\bigl(z^2 - \frac{5c+4}{6}z + \frac{c}{2}\bigr)^2}\]
LaTeX source
\[
\frac{f(z)-1}{(z-u)^2(z-v)^2} = \frac{f(z)-1}{\bigl(z^2 - \frac{5c+4}{6}z + \frac{c}{2}\bigr)^2}
\]\[2\Bigl(\sum p + \sum q + \sum r\Bigr) = 6N\]
LaTeX source
\[ 2\Bigl(\sum p + \sum q + \sum r\Bigr) = 6N \]
\[f(z) - 1 = \lambda z^3 (z-1)^2 (z-c) - 1 \ \text{divisible par}\ (z^2 - \alpha z - \beta)^2\]
LaTeX source
\[
f(z) - 1 = \lambda z^3 (z-1)^2 (z-c) - 1 \ \text{divisible par}\ (z^2 - \alpha z - \beta)^2
\]\[\lambda z^3 (z-1)^2 (z-c) - 1 = \lambda (z^2 - \alpha z - \beta)^2 (z^2 + \ell z + m)\]
LaTeX source
\[ \lambda z^3 (z-1)^2 (z-c) - 1 = \lambda (z^2 - \alpha z - \beta)^2 (z^2 + \ell z + m) \]
\[\frac{1}{\lambda} f = \frac{z^4 (z-1)^2}{z^3 (z-1)^3}\]
LaTeX source
\[
\frac{1}{\lambda} f = \frac{z^4 (z-1)^2}{z^3 (z-1)^3}
\]\[z^2 - \frac32 z + \frac12 = 0, \qquad (z-1)\Bigl(z - \frac12\Bigr)\]
LaTeX source
\[ z^2 - \frac32 z + \frac12 = 0, \qquad (z-1)\Bigl(z - \frac12\Bigr) \]
\[P(c) = 0 \Longrightarrow c \neq 0, 1\]
LaTeX source
\[ P(c) = 0 \Longrightarrow c \neq 0, 1 \]
\[P(c) = 0 \qquad \underbrace{25c^2 - 32c + 16}_{\delta(c)}\]
LaTeX source
\[
P(c) = 0 \qquad \underbrace{25c^2 - 32c + 16}_{\delta(c)}
\]\[32(32 - 50) = -32 \cdot 18 = -(8 \cdot 3)^2 \qquad c = \frac{16}{25} \pm \frac{24}{25}\,i\]
LaTeX source
\[
32(32 - 50) = -32 \cdot 18 = -(8 \cdot 3)^2 \qquad c = \frac{16}{25} \pm \frac{24}{25}\,i
\]\[\mathbb{Q}[c]_{\delta(c)\,c(c-1)} / P(c)Q(c)
\qquad
\mathbb{Q}[c]_{\delta(c)Q(c)}\]
LaTeX source
\[
\mathbb{Q}[c]_{\delta(c)\,c(c-1)} / P(c)Q(c)
\qquad
\mathbb{Q}[c]_{\delta(c)Q(c)}
\]\[P(c) = Q(c) = 0 \Longrightarrow f(u) = f(v) = 0\]
LaTeX source
\[ P(c) = Q(c) = 0 \Longrightarrow f(u) = f(v) = 0 \]
\[f'(z) = 0 \Longrightarrow f(z) = 0\]
LaTeX source
\[ f'(z) = 0 \Longrightarrow f(z) = 0 \]
\[f(z) = \lambda\frac{\varphi}{\psi}\]
LaTeX source
\[
f(z) = \lambda\frac{\varphi}{\psi}
\]\[\frac{f'(z)}{f(z)} = [\ldots]\ \Bigl(\frac{\varphi'}{\varphi} - \frac{\psi'}{\psi}\Bigr)\]
LaTeX source
\[
\frac{f'(z)}{f(z)} = [\ldots]\ \Bigl(\frac{\varphi'}{\varphi} - \frac{\psi'}{\psi}\Bigr)
\]\[\sum \frac{p_i}{z - \alpha_i} - \sum \frac{q_j}{z - \beta_j}
= \frac{1}{\prod (z-\alpha_i) \prod (z-\beta_j)} \bigl(\qquad\bigr)\]
LaTeX source
\[
\sum \frac{p_i}{z - \alpha_i} - \sum \frac{q_j}{z - \beta_j}
= \frac{1}{\prod (z-\alpha_i) \prod (z-\beta_j)} \bigl(\qquad\bigr)
\]\[n = \sum_s q_s = \sum_f p_f\]
LaTeX source
\[ n = \sum_s q_s = \sum_f p_f \]
\[\bigl(2, \begin{smallmatrix}1\\1\end{smallmatrix}, 2\bigr) \quad
\bigl(\begin{smallmatrix}1\\1\end{smallmatrix}, 2, 2\bigr) \quad
\bigl(2, 2, \begin{smallmatrix}1\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(2, \begin{smallmatrix}1\\1\end{smallmatrix}, 2\bigr) \quad
\bigl(\begin{smallmatrix}1\\1\end{smallmatrix}, 2, 2\bigr) \quad
\bigl(2, 2, \begin{smallmatrix}1\\1\end{smallmatrix}\bigr)
\]\[\bigl(3, \begin{smallmatrix}1\\1\\1\end{smallmatrix}, 3\bigr) \quad
\bigl(3, \begin{smallmatrix}2\\1\end{smallmatrix}, \begin{smallmatrix}2\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}2\\1\end{smallmatrix}, \begin{smallmatrix}2\\1\end{smallmatrix}, 3\bigr)\]
LaTeX source
\[
\bigl(3, \begin{smallmatrix}1\\1\\1\end{smallmatrix}, 3\bigr) \quad
\bigl(3, \begin{smallmatrix}2\\1\end{smallmatrix}, \begin{smallmatrix}2\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}2\\1\end{smallmatrix}, \begin{smallmatrix}2\\1\end{smallmatrix}, 3\bigr)
\]\[\bigl(\begin{smallmatrix}3\\1\end{smallmatrix}, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, 4\bigr) \quad
\bigl(4, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, \begin{smallmatrix}3\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}3\\1\end{smallmatrix}, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, 4\bigr) \quad
\bigl(4, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, \begin{smallmatrix}3\\1\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}2\\2\end{smallmatrix}, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, 4\bigr) \quad
\bigl(4, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, \begin{smallmatrix}2\\2\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}2\\2\end{smallmatrix}, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, 4\bigr) \quad
\bigl(4, \begin{smallmatrix}2\\1\\1\end{smallmatrix}, \begin{smallmatrix}2\\2\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}2\\1\\1\end{smallmatrix}, \begin{smallmatrix}2\\2\end{smallmatrix}, 4\bigr) \quad
\bigl(4, \begin{smallmatrix}2\\2\end{smallmatrix}, \begin{smallmatrix}2\\1\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}2\\1\\1\end{smallmatrix}, \begin{smallmatrix}2\\2\end{smallmatrix}, 4\bigr) \quad
\bigl(4, \begin{smallmatrix}2\\2\end{smallmatrix}, \begin{smallmatrix}2\\1\\1\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}4\\1\end{smallmatrix}, \begin{smallmatrix}2\\1_3\end{smallmatrix}, 5\bigr) \quad
\bigl(5, \begin{smallmatrix}2\\1_3\end{smallmatrix}, \begin{smallmatrix}4\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}4\\1\end{smallmatrix}, \begin{smallmatrix}2\\1_3\end{smallmatrix}, 5\bigr) \quad
\bigl(5, \begin{smallmatrix}2\\1_3\end{smallmatrix}, \begin{smallmatrix}4\\1\end{smallmatrix}\bigr)
\]\[\bigl(5, \begin{smallmatrix}2\\1_3\end{smallmatrix}, \begin{smallmatrix}3\\2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\end{smallmatrix}, \begin{smallmatrix}2\\1_3\end{smallmatrix}, 5\bigr)\]
LaTeX source
\[
\bigl(5, \begin{smallmatrix}2\\1_3\end{smallmatrix}, \begin{smallmatrix}3\\2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\end{smallmatrix}, \begin{smallmatrix}2\\1_3\end{smallmatrix}, 5\bigr)
\]\[\bigl(5, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, 5\bigr)\]
LaTeX source
\[
\bigl(5, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, 5\bigr)
\]\[\bigl(5, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}3\\1_2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\1_2\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, 5\bigr)\]
LaTeX source
\[
\bigl(5, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}3\\1_2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\1_2\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, 5\bigr)
\]\[\bigl(\begin{smallmatrix}4\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}3\\2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}4\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}4\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}3\\2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1\end{smallmatrix}, \begin{smallmatrix}4\\1\end{smallmatrix}\bigr)
\]\[(6, 1_6, 6) \;/\!/\; \bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr) \;/\!/\; [\ldots]\]
LaTeX source
\[
(6, 1_6, 6) \;/\!/\; \bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr) \;/\!/\; [\ldots]
\]\[\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\1_2\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 2_3\bigr) \quad
\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\1_2\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr) \quad
\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 2_3\bigr) \quad
\bigl(6, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr)
\]\[\bigl(6, 2_3, \begin{smallmatrix}3\\1_3\end{smallmatrix}\bigr) \quad
\bigl(6, 2_3, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(6, 2_3, \begin{smallmatrix}3\\1_3\end{smallmatrix}\bigr) \quad
\bigl(6, 2_3, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}4\\1_2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}4\\1_2\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}4\\1_2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}4\\1_2\end{smallmatrix}, 2_3, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}4\\1_2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}4\\1_2\end{smallmatrix}, 2_3, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr)
\]\[\bigl(3_2, 2_3, \begin{smallmatrix}4\\1_2\end{smallmatrix}\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr)\]
LaTeX source
\[
\bigl(3_2, 2_3, \begin{smallmatrix}4\\1_2\end{smallmatrix}\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr)
\]\[\bigl(3_2, 2_3, 2_3\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(3_2, 2_3, 2_3\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
\bigl(3_2, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr)
\]\[(\alpha)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, \begin{smallmatrix}3_2\\3\end{smallmatrix}\bigr) \qquad
(\beta)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}[\ldots]\\2_3\end{smallmatrix}, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr) \qquad
(\gamma)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)\]
LaTeX source
\[
(\alpha)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, \begin{smallmatrix}3_2\\3\end{smallmatrix}\bigr) \qquad
(\beta)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}[\ldots]\\2_3\end{smallmatrix}, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr) \qquad
(\gamma)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)
\]\[(\delta)\ \bigl(\begin{smallmatrix}4\\2_1\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, 6\bigr) \qquad
(\eta)\ \bigl(\begin{smallmatrix}4\\2_1\end{smallmatrix}, 2_3, \begin{smallmatrix}3_2\\ {[\ldots]}\end{smallmatrix}\bigr)\]
LaTeX source
\[
(\delta)\ \bigl(\begin{smallmatrix}4\\2_1\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, 6\bigr) \qquad
(\eta)\ \bigl(\begin{smallmatrix}4\\2_1\end{smallmatrix}, 2_3, \begin{smallmatrix}3_2\\ {[\ldots]}\end{smallmatrix}\bigr)
\]\[\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, [\ldots]\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)\]
LaTeX source
\[
\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 6\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, [\ldots]\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
\bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)
\]\[(2_3, 2_3, 3_2) \qquad \bigl(2_3, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 6\bigr)\]
LaTeX source
\[
(2_3, 2_3, 3_2) \qquad \bigl(2_3, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 6\bigr)
\]\[\mathcal{F}^{\natural}(C) = \mathcal{F}(X)/\mathrm{Aut}^{0}(C).\]
LaTeX source
\[
\mathcal{F}^{\natural}(C) = \mathcal{F}(X)/\mathrm{Aut}^{0}(C).
\]\[\mathcal{F}(X)/\mathrm{Aut}(f)^{0} \longrightarrow \mathcal{F}(X')/\mathrm{Aut}(f)^{0},\]
LaTeX source
\[
\mathcal{F}(X)/\mathrm{Aut}(f)^{0} \longrightarrow \mathcal{F}(X')/\mathrm{Aut}(f)^{0},
\]\[\mathcal{F}(X)/\mathrm{Aut}(C)^{0} \to \mathcal{F}(X')/\mathrm{Aut}(f)^{0}
\to \mathcal{F}(X')/\mathrm{Aut}(C')^{0},\]
LaTeX source
\[
\mathcal{F}(X)/\mathrm{Aut}(C)^{0} \to \mathcal{F}(X')/\mathrm{Aut}(f)^{0}
\to \mathcal{F}(X')/\mathrm{Aut}(C')^{0},
\]\[\mathcal{F}^{\natural}(C, G) = \mathcal{F}(C)^{G}/\mathrm{Aut}(C, G)^{0}.\]
LaTeX source
\[
\mathcal{F}^{\natural}(C, G) = \mathcal{F}(C)^{G}/\mathrm{Aut}(C, G)^{0}.
\]\[\mathcal{F}^{\natural} : \text{Cartes finies}/\text{isomorphismes} \longrightarrow \text{ens.\ finis}\]
LaTeX source
\[
\mathcal{F}^{\natural} : \text{Cartes finies}/\text{isomorphismes} \longrightarrow \text{ens.\ finis}
\]\[\Gamma(\mathcal{F}^{\natural}) \simeq \varinjlim_{H_\alpha} \mathcal{F}^{\natural}(C_\alpha, G_\alpha)\]
LaTeX source
\[
\Gamma(\mathcal{F}^{\natural}) \simeq \varinjlim_{H_\alpha} \mathcal{F}^{\natural}(C_\alpha, G_\alpha)
\]\[\Gamma\mathcal{F}^{\natural} \simeq \mathfrak{P}(S)/\mathrm{Aut}(S),\]
LaTeX source
\[
\Gamma\mathcal{F}^{\natural} \simeq \mathfrak{P}(S)/\mathrm{Aut}(S),
\]\[\Gamma(\mathcal{F}^{\natural}) \simeq \mathfrak{P}(S)^{\underline{\sigma}}/\mathrm{Aut}(S, \underline{\sigma}, 0, 1, \infty)^{0}\]
LaTeX source
\[
\Gamma(\mathcal{F}^{\natural}) \simeq \mathfrak{P}(S)^{\underline{\sigma}}/\mathrm{Aut}(S, \underline{\sigma}, 0, 1, \infty)^{0}
\]\[z \longmapsto \frac{z}{cz + (1-c)}\]
LaTeX source
\[
z \longmapsto \frac{z}{cz + (1-c)}
\]\[\begin{pmatrix} 1 & 0 \\ c & 1-c \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ c & 1-c \end{pmatrix}
= \begin{pmatrix} 1 & 0 \\ 2c - c^{2} & (1-c)^{2} \end{pmatrix}\]
LaTeX source
\[
\begin{pmatrix} 1 & 0 \\ c & 1-c \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ c & 1-c \end{pmatrix}
= \begin{pmatrix} 1 & 0 \\ 2c - c^{2} & (1-c)^{2} \end{pmatrix}
\]\[(1-c)^{2} = 1, \qquad c - 1 = \pm 1, \qquad c = 0 \text{ ou } 2 .\]
LaTeX source
\[
(1-c)^{2} = 1, \qquad c - 1 = \pm 1, \qquad c = 0 \text{ ou } 2 .
\]\[\frac{z}{2z-1} = \frac{1}{2 - \frac{1}{z}}, \qquad
z = \frac{z}{2z-1}, \qquad 2z^{2} - 2z = 0, \qquad z = 0 \text{ ou } 1 .\]
LaTeX source
\[
\frac{z}{2z-1} = \frac{1}{2 - \frac{1}{z}}, \qquad
z = \frac{z}{2z-1}, \qquad 2z^{2} - 2z = 0, \qquad z = 0 \text{ ou } 1 .
\]\[f(z) \in [0,1], \qquad f(z) - \tfrac12, \qquad \bigl(2f(z) - 1\bigr)^{2},\]
LaTeX source
\[
f(z) \in [0,1], \qquad f(z) - \tfrac12, \qquad \bigl(2f(z) - 1\bigr)^{2},
\]\[g(z) = \bigl(2f(z) - 1\bigr)^{2},\]
LaTeX source
\[
g(z) = \bigl(2f(z) - 1\bigr)^{2},
\]\[g(z) = 0 \iff f(z) = \tfrac12, \qquad g(z) = 1 \iff f(z) = 0, 1 .\]
LaTeX source
\[ g(z) = 0 \iff f(z) = \tfrac12, \qquad g(z) = 1 \iff f(z) = 0, 1 . \]
\[2z - 1 \in [-1, +1], \qquad (2z-1)^{2} \in [0, 1] .\]
LaTeX source
\[
2z - 1 \in [-1, +1], \qquad (2z-1)^{2} \in [0, 1] .
\]