\[
(\mathrm{A}^f_n) \quad
\left.
\begin{array}{l}
  H^i(U, F) \text{ de dim finie si } i \leq n \\
  \Updownarrow \\
  F \text{ de prof} > n+1 \text{ sur } U \text{ au voisinage de } Z \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ pt isolé de } \operatorname{Supp} E^{r-i-1} \cup \lbrace z \rbrace \text{ si } i \leq n \\
  \Updownarrow \\
  \alpha_i \text{ injectif pour } \underline{m \text{ grand}} \text{ si } i \leq n+1 \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  R^i g_*(F) \text{ coh.\ pour } i \leq n \iff \underline{H}^i_z(F) \text{ de dim finie pour } z \in Z,\ i \leq n+1 \\
  \Updownarrow \\
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S \text{ pour } i \leq n
\end{array}
\right.
\]

\[
\begin{array}{rll}
6 & \textit{Cas tétraédraux} &
(3,3,2)_{(4)}\ \underline{(3,2,3)}_{(2)}\ \underline{(2,3,3)}_{(2)}\
(3,3,3)_{(4)} \\[4pt]
13 & \textit{Cas octaédraux} &
(4,3,2)_{(4)}\ \underline{(3,2,4)}_{(2)}\ \underline{(2,4,3)}_{(2)}\
\underline{(2,3,4)}_{(2)}\ (3,4,2)_{(4)}\ \underline{(4,2,3)}_{(2)} \\
 & & (4,4,3)_{(4)}\ (4,3,4)_{(4)}\ (3,4,4)_{(4)} \\[4pt]
10 & \textit{Cas icosaédraux} &
(5,3,2)_{(8)}\ \underline{(3,2,5)}_{(2)}\ \underline{(2,5,3)}_{(4)}\
\underline{(2,3,5)}_{(2)}\ (3,5,2)_{(8)}\ \underline{(5,2,3)}_{(4)} \\
+4 & & \underline{(3,2,5')}_{(2)}\ \underline{(2,3,5')}_{(2)} \\
+4 & & (5,5,3)_{(16)}\ (5,3,5)_{(8)}\ (3,5,5)_{(8)} \\
2 & & (5,3,5')_{(8)}\ (3,5,5')_{(8)} \\
4 & & \underline{(2,5,5')}_{(4)}\ \underline{(5,2,5')}_{(4)} \\
1 & & (5,5,5)_{(16)} \\
\hline
25 & &
\end{array}
\]

\begin{align*}
  S &\simeq \operatorname{Pent}_\omega(E) && \text{struct.\ pent.\ comp.\ avec } \omega \\
  A &\simeq \operatorname{Car}(E) && \text{partitions du type } (2, 2, 1) \\
    & && \simeq \text{structures carrées sur une partie à 4 él.\ de } E \\
  F &\simeq \operatorname{Tr}_3(E) && \text{triangles} \subset E \\
  \vec{A} &\simeq \operatorname{Pent}^{s}_\omega(E) && \omega\text{-structures pentagonales à somm.\ marqué} \\
  A^{\uparrow} &\simeq \operatorname{Tr}_3(E) && \text{triangles pointés} \\
  A^{\wedge} &\simeq \operatorname{Pent}^{a}_\omega(E) \simeq \operatorname{Pent}^{s}_\omega(E)
    && \text{structures pol.\ à arête marquée} \\
  A^{\uparrow\to} &\simeq \operatorname{Rep}_\omega(E) && \text{ordres totaux sur } E \text{ comp.\ avec } \omega \\
  \tilde{S} &\simeq \overrightarrow{\operatorname{Pent}}_\omega(E) && \text{permutations circulaires comp.\ avec } \omega \\
  \tilde{A} &\simeq \overrightarrow{\operatorname{Car}}(E) && \text{carrés orientés} \subset E \\
  \tilde{F} &\simeq \overrightarrow{\operatorname{Tr}}_3(E) && \text{triangles orientés} \subset E
\end{align*}

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

\[
\begin{array}{rll}
180 & \left\lbrace \begin{array}{l}
  60 = 3\times 20 = 2\times 30 \\
  120 = 4\times 30 = 2\times 60
  \end{array}\right. &
  \begin{array}{l} f_{B,s} \\ f_{\vec{Q},\alpha} \end{array} \\[1ex]
420 & \left\lbrace \begin{array}{l}
  60 = 1\times 60 = 1\times 60 \\
  120 = 1\times 120 = 1\times 120 \\
  60 = 1\times 60 = 5\times 12 \\
  60 = 1\times 60 = 3\times 20 \\
  120 = 1\times 120 = 6\times 20
  \end{array}\right. &
  \begin{array}{l} \tau_{\pi,s} \\ \tau^{0}_{r} \\ \gamma_{\pi,s} \\
  \gamma'_{\pi,s} \\ \gamma^{0}_{r} \end{array} \\[1ex]
1020 & \left\lbrace \begin{array}{l}
  120 = 2\times 60 = 1\times 120 \\
  120 = 2\times 60 = 2\times 60 \\
  120 = 1\times 120 = 2\times 60 \\
  120 = 1\times 120 = 2\times 60 \\
  3\times 120 = 3\times 40 = 3\times 40 \\
  120 = 2\times 60 = 2\times 60 \\
  60 = 1\times 60 = 2\times 30
  \end{array}\right. &
  \begin{array}{l} \tau_{\pi,\rho} \\ \gamma_{\pi,\rho} \\ \tau'_{r} \\
  \gamma^{1}_{r} \\ \gamma^{1\,\prime}_{r} \\ \gamma^{2}_{r} \\
  \gamma^{2\,\prime}_{r} \end{array}
\end{array}
\]

\[
\begin{array}{l||c||c|c|c}
n & 2 & 3 & 4 & 5\\ \hline
[\mathrm{SL}(2,\mathbb{Z}/n\mathbb{Z})] & 6 & 24 & 48 & 120\\
{}[\mathrm{GL}(2,\mathbb{Z}/n\mathbb{Z})] & 6 & 48 & 96 & 480\\ \hline
{}[\mathrm{SL}(2,\mathbb{Z}/n\mathbb{Z})'] & 6 & 12 & 24 & 60\\
\mathrm{SL}(2,\mathbb{Z}/n\mathbb{Z})' & \mathfrak{S}_3=D_3 & \mathfrak{S}_4^{+} & \mathfrak{S}_4 & \mathfrak{S}_5^{+}\\ \hline
{}[\mathrm{GP}(1,\mathbb{Z}/n\mathbb{Z})] & 6 & 24 & 48 & 120\\
\mathrm{GP}(1,\mathbb{Z}/n\mathbb{Z}) & \mathfrak{S}_3=D_3 & \mathfrak{S}_4 & \mathfrak{S}_4\times\{\pm1\}\,? & \mathfrak{S}_5\,?\\ \hline
{}[M^{*}] & 3 & 8 & 12 & 24\\
{}[\underbrace{M^{*}/\{\pm1\}}_{S_M}] & 3 & 4 & 6 & 12\\ \hline
\text{type } M^{*}/\{\pm1\} & \text{trièdre} & \text{tétraèdre} & \text{octaèdre} & \text{icosaèdre}\\ \hline
{}[P(M)] & 3 & 4 & 6 & 6\\ \hline
{}[F_M^{*}] & 1 & 4 & 8 & 40\\
{}[F_M^{\circ}] & 1 & 4 & 8 & 20
\end{array}
\]

\[
(58) \qquad
\begin{array}{l}
\text{conditions} \\
\text{équivalentes} \\
\text{sur } \mathcal{M} = (\Gamma, Z, \pi)
\end{array}
\quad
\left\{
\begin{array}{l}
\text{a) L'hom } \delta \text{ est surjectif} \\
\text{b) \struck{\ill{}} $G$ opère transitivement sur } \pi \\
\text{c) } \forall \lambda \in Z, \text{ \struck{\ill{}} la structure de graphe définie par} \\
\qquad A_{\lambda} \subset \mathfrak{P}_{\{1,2\}}(S), \\
\qquad A_{\lambda} = \{\{s, \lambda t\} \mid s, t \in S,\ \text{t.q. } \{s,t\} \in A_0\} \\
\qquad \text{est isomorphe à } \gamma(S, A_0), \\
\qquad \text{par un } f \in \mathfrak{S}_S \text{ qui centralise } Z_S \\
\text{d) } \forall \lambda \in Z,\ \exists f_{\lambda} \in \mathfrak{S}_S, \text{ centralisant } Z_S, \text{ telle que} \\
\qquad \forall s, t \in S, \ \ldots \\
\qquad \{s,t\} \in A_0 \iff \{f_{\lambda}^{-1} s, f_{\lambda}^{-1} \lambda t\} \in A_0
\end{array}
\right.
\]

\[
(\mathrm{B}_n) \quad
\left.
\begin{array}{l}
  \alpha_i \text{ bijectif pour } m \text{ grand},\ i \leq n, \\
  \quad \text{et } \alpha_i \text{ injectif pour } m \text{ grand},\ i = n+1 \\
  \Updownarrow \\
  \operatorname{Supp} E^{r-i} \subset Z \text{ pour } i \leq n,\ \operatorname{Supp} E^{r-n-1} \ldots Z \\
  \Downarrow \\
  \operatorname{prof}(F_x) > n \text{ si } x \text{ fermé dans } U, \text{ et} \\
  \operatorname{prof}(F_x) > n+1 \text{ si de plus } x \in V \cap U,\ V \text{ voisinage ouvert convenable de } Z \\
  \Updownarrow \\
  H^i(U, F(-m)) = 0 \text{ pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S(-h) \text{ pour } i \leq n
\end{array}
\right.
\]

\[
\begin{array}{lll}
1 & \begin{cases} r_0 = 2\eta - \sum \xi_i = \lambda - \eta \\ r_0 = \xi'_1 - \xi_1 = \xi'_2 - \xi_2 = \cdots = \xi'_6 - \xi_6 \end{cases}
& \text{déduite des } \begin{cases} (\xi_1, \dots, \xi_6) \\ (\xi'_1, \dots, \xi'_6) \end{cases} \\[3ex]
15 & \begin{cases} r_{ij} = \xi_i - \xi_j \\ 1 \leq i < j \leq 6 \end{cases}
& \text{déduites des } \begin{cases} (\xi_j, \xi'_j, (\xi_{ik})_{k \neq i,j}) \\ (\xi_i, \xi'_i, (\xi_{jk})_{k \neq i,j}) \end{cases} \\[3ex]
20 & \begin{cases} r_{ijk} = \eta - \xi_i - \xi_j - \xi_k \\ 1 \leq i < j < k \leq 6 \end{cases}
& \text{déduites des } \begin{cases} (\xi_i, \xi_j, \xi_k, \xi_{lm}, \xi_{mn}, \xi_{nl}) \\ (\xi_{jk}, \xi_{ki}, \xi_{ij}, \xi'_l, \xi'_m, \xi'_n) \end{cases}
\end{array}
\]

\[
(19)\quad\left\lbrace
\begin{array}{l}
\hat X=X^{*}\amalg_{\dot X=\coprod\dot X_\beta}\underbrace{\mathcal C\bigl(p=\textstyle\coprod p_\beta\bigr)}_{\coprod_\beta\mathcal C(p_\beta)},\\[10pt]
T_\beta=\mathcal C(p_\beta)\\[4pt]
X_\beta\ \text{pour}\ \beta\in I_0\ \text{clair}\\[4pt]
X_i\ \text{pour}\ i\in I^{*}\ \text{est donné par}\\[4pt]
\qquad\bar X_i=X^{*}_i\cup\bigcup_{\beta\in I_0,\ \beta\le i}\bigl(\mathbf{I}.\dot X_{i\beta}\cup X_\beta\bigr),\qquad \dot X_{i\beta}\overset{\text{déf}}{=}X^{*}_i\cap\dot X_\beta\\[10pt]
B_\alpha=B^{*}_\alpha\cup\bigcup_{\beta\in I_0}\mathbf{I}.\dot B_\alpha,\qquad \dot B_\alpha\overset{\text{déf}}{=}B^{*}_\alpha\cap\dot X_\beta\\[10pt]
\mathcal E_\alpha\ \text{défini par}\ \mathcal E^{*}_\alpha\ \text{et les}\ (B_{\beta,\alpha},\mathcal E_{\beta\alpha})
\end{array}\right.
\]

\[ (277)\quad \left\{\begin{array}{ll} D_0^*=\mathbb{C}^*\times\mathbb{C}^* & (D_0=X=\mathbb{C}\times\mathbb{C})\\[2pt] \left.\begin{array}{l}\Delta_1^*=\mathbb{C}^*\times\mathbb{P}\\ \Delta_2^*=\mathbb{P}\times\mathbb{C}^*\end{array}\right\} & D_1^*=\Delta_1^*\amalg\Delta_2^*\quad (D_1=\Delta_1\amalg\Delta_2)\\ & \quad\text{où } \Delta_1=\mathbb{C}\times\mathbb{P},\ \Delta_2=\mathbb{P}\times\mathbb{C}\\[2pt] \mathbb{P}=\mathbb{P}\times\mathbb{P} & D_2^*=D_2=\mathbb{P}\\[6pt] \mathcal{V}_{\Delta_1^*,X}=\mathbb{C}^*\times\mathbb{D} & \mathcal{V}^*_{\Delta_1^*,X}=\mathbb{C}^*\times\mathbb{D}^*\\ \mathcal{V}_{\Delta_2^*,X}=\mathbb{D}\times\mathbb{C}^* & \mathcal{V}^*_{\Delta_2^*,X}=\mathbb{D}^*\times\mathbb{C}^*\\[2pt] \mathcal{V}_{\mathbb{P},\Delta_1}=\mathbb{D}\times\mathbb{P} & \mathcal{V}^*_{\mathbb{P},\Delta_1}=\mathbb{D}^*\times\mathbb{P}\\ \mathcal{V}_{\mathbb{P},\Delta_2}=\mathbb{P}\times\mathbb{D} & \mathcal{V}^*_{\mathbb{P},\Delta_2}=\mathbb{P}\times\mathbb{D}^*\\[6pt] \mathcal{V}_{\mathbb{P},X}=\mathbb{D}\times\mathbb{D}, & \\ \mathcal{V}^{\Delta_1}_{\mathbb{P},X}=\mathbb{D}\times\mathbb{D}^*, & \mathcal{V}^{\Delta_2}_{\mathbb{P},X}=\mathbb{D}^*\times\mathbb{D}\\ \mathcal{V}^*_{\mathbb{P},X}=\mathbb{D}^*\times\mathbb{D}^* & \end{array}\right. \]

\[
\begin{aligned}
\alpha_n(\omega_1) &= \omega_1^n\\
\beta_n(\underset{\substack{\|\\ f\,dt^n}}{\omega_n}) &= \beta_1(\omega_n^{1/n}) = \beta_1(f^{1/n}\,dt) = D_{dt} + \frac{1}{n}\frac{df}{f} = D_{dt} + \frac{1}{n}\frac{f'}{f}\,dt\\
\gamma(D_{dt} + \underset{\substack{\|\\ f\,dt}}{\omega_1}) &= \underset{\substack{\|\\ c_{\mathrm{pr}}(t)}}{\gamma(D_{dt})} + 2D_{dt}(\omega_1) + \omega_1^2\\
&= c_{\mathrm{pr}}(t) + 2\frac{df}{dt}\,dt + f^2\,dt^2 = c_{\mathrm{pr}}(t) + (2f' + f^2)\,dt^2\\
\gamma\beta_n(\omega_n) &= c_{\mathrm{pr}}(t) + \Bigl(2\Bigl(\frac{1}{n}\frac{f'}{f}\Bigr)' + \Bigl(\frac{1}{n}\frac{f'}{f}\Bigr)^2\Bigr)dt^2\\
&= c_{\mathrm{pr}}(t) + \frac{1}{n^2}\Bigl(-2n\frac{f''}{f}\ \ldots\ \Bigl(\frac{f'}{f}\Bigr)^2\Bigr)dt^2\\
&= c_{\mathrm{pr}}(t) + \Bigl(-2\frac{f''}{f} + 3\Bigl(\frac{f'}{f}\Bigr)^2\Bigr)dt^2\\
\gamma\beta_1\,d(f) &= c_{\mathrm{pr}}(t) + \Bigl(-2\frac{f'''}{f'} + 3\Bigl(\frac{f''}{f'}\Bigr)^2\Bigr)dt^2
\end{aligned}
\]

\[
\begin{aligned}
\text{Écad}(f,g) &= \frac{dg}{df} = \frac{g'}{f'} \in \Gamma\,\underline{O}^*_X\\
\text{Écaf}_n(f,g) &= \Bigl(\frac{dg}{df}\Bigr)^n = \Bigl(\frac{g'}{f'}\Bigr)^n \in \Gamma(\underline{O}^*_X)\\
\text{Écaf}(f,g) &= \frac{d(dg/df)}{dg/df} = \frac{d(g'/f')}{g'/f'} = \frac{dg'}{g'} - \frac{df'}{f'}\\
&= \Bigl(\frac{g''}{g'} - \frac{f''}{f'}\Bigr)dt \in \Gamma(\underline{\omega}^1_X)\\
\text{Écmu}(f,g) &= \frac{dg/g}{df/f} = \frac{g'/g}{f'/f} \in \Gamma(\underline{O}^*_X)\\
\text{Écmu}_2(f,g) &= \Bigl(\frac{dg/g}{df/f}\Bigr)^2 = \Bigl(\frac{g'/g}{f'/f}\Bigr)^2 \in \Gamma(\underline{O}^*_X)\\
\text{Écpr}(f,g) &= -2\,\frac{d^3g/df^3}{dg/df} + 3\Bigl(\frac{d^2g/df^2}{dg/df}\Bigr)^2\\
&= \Bigl[\Bigl(-2\frac{g'''}{g'} + 3\Bigl(\frac{g''}{g'}\Bigr)^2\Bigr) - \Bigl(-2\frac{f'''}{f'} + 3\Bigl(\frac{f''}{f'}\Bigr)^2\Bigr)\Bigr]dt^2\\
&\in \Gamma(\underline{\omega}^2_X)
\end{aligned}
\]

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

\[
(62)\qquad
\begin{array}{l}
\text{Homothéties de } \Gamma \text{ i.e.\ couples } \lambda_S, \lambda_{S^{\vee}} \text{ satisfaisant (59)} \\
\quad \simeq\ \underbrace{\text{automorphismes de la structure } {}^{s}\Gamma = (S, \check{S}, \mathcal{R})}_{\text{soit } Z} \\[4pt]
\quad \underset{\text{soit } Z_S}{\simeq} \{\lambda \in \mathfrak{S}_S \mid \underbrace{\lambda(\Sigma)}_{\text{transport de str.}} = \Sigma\} \\
\qquad \text{où } \Sigma \subset \mathfrak{P}(S) \text{ est l'image de } S^{\vee} \xrightarrow{{}^{s}\mathcal{R}} \mathfrak{P}(S) \\[4pt]
\quad \underset{\text{soit } Z_{S^{\vee}}}{\simeq} \{\lambda^{\vee} \in \mathfrak{S}_{S^{\vee}} \mid \underbrace{\lambda^{\vee}(\Sigma^{\vee})}_{\text{transp. de str.}} = \Sigma^{\vee}\} \\
\qquad \text{où } \Sigma^{\vee} \subset \mathfrak{P}(S^{\vee}) \text{ est l'image de } S \xrightarrow{\mathcal{R}} \mathfrak{P}(S^{\vee})
\end{array}
\]

\[
(11)\quad
\begin{cases}
\sigma_i e_j = e_j & \text{si } 1\le j< i\\
\sigma_i e_i = e_{i+1} \quad \struck{(\text{si } i\ge 1)} & \\
\sigma_i e_{i+1} = \struck{\ldots}
  \begin{cases} u_0 - s_1 = (u_0-s_0) - e_1 = (u_{01}-1)\,e_1 & \text{si } i=0\\
  u_i - s_0 = \sum_{1\le k\le i+1} u_{ik}\, e_k & \text{si } i\ge 1\end{cases}\\
\struck{\sigma_i e_{i+2} = \ldots}\\
\Bigl[\sigma_0 e_{i+2} = \Bigr]
  \begin{cases} v_0 - s_1 = (v_0-s_0) - e_1 = (v_{01}-1)\,e_1 + v_{0,2}\, e_2 & \text{si } i=0\\
  v_i - s_0 = \sum_{1\le k\le i+2} v_{i,k}\, e_k & \text{si } i\ge 1\end{cases}\\
\sigma_i e_j = \begin{cases} (v_{01}-1)\,e_1 + v_{0j}\, e_j & \text{si } i=0\\
  \sum_{1\le k\le i+1} v_{ik}\, e_k + v_{i,i+2}\, e_j & \text{si } i\ge 1\end{cases}\\
\qquad \text{si } i+2\le j\le n+1
\end{cases}
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c}
 & \mu_p & W_{11} & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \mathbb{G}_m & \mathbb{G}_a & Ab & \mathbb{Z} \\ \hline
{}_p\mathbb{G}_m = \mu_p & \mathbb{Z}/p & 0 & 0\ \ast & \ast\ 0 & ? & 0 & [a] & 0\ \ast \\
{}_F\mathbb{G}_a = W_{11} & 0 & \mathbb{G}_a^2\ (?) & 0\ \ast & \ast\ 0 & 0 & (?) & [b] & 0\ \ast \\
\mathbb{Z}/\ell\mathbb{Z} & 0\ \times & 0\ \times & \mathbb{Z}/\ell\mathbb{Z} & 0 & \emptyset\ 0 & \emptyset\ 0 & \emptyset\ 0 & \emptyset\ \mathbb{Z}/\ell \\
\mathbb{Z}/p\mathbb{Z} & 0\ \ast & 0\ \times & 0 & \mathbb{Z}/p\mathbb{Z} & \emptyset\ 0 & \mathbb{G}_a & \emptyset\ 0 & \emptyset\ \mathbb{Z}/p\mathbb{Z} \\
\mathbb{G}_m & \mathbb{Z}/p\mathbb{Z} & 0 & [c] & 0 & 0 & 0 & [d] & \\
\mathbb{G}_a & 0 & (?) & 0\ 0 & (?) & 0 & (?) & [e] & \\
Ab & [f] & \longleftarrow & \longrightarrow & & (Ab)' & H^1(Ab, \underline{O}) & & \\
\mathbb{Z} & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast
\end{array}
\]

\[
\begin{array}{llll}
\operatorname{Conf}_{D_0} & 1 & \text{---} \operatorname{Aut} D_0 & e \\
\operatorname{Conf}_{D_1} & 27 & \text{---} \operatorname{Aut} D_1 & e \\
\operatorname{Conf}_{D_2} & 27 \cdot 10 : 2 = 135 & \text{---}
\operatorname{Aut} D_2 & \mathfrak{S}_2 \\
\operatorname{Conf}_{D_3} & 27 \cdot 10 \cdot 1 : 6 = 45 & \text{---}
\operatorname{Aut} D_3 & \mathfrak{S}_3 \\
\operatorname{Conf}_{D_4} & 27 \cdot 10 \cdot 1 \cdot 8 : 2 = 2^3 \cdot 3^3
\cdot 5 = 1080 & \text{---} \operatorname{Aut} D_4 & \mathfrak{S}_2 \\
\operatorname{Conf}_{D_5} & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 : 2 = 2^5
\cdot 3^3 \cdot 5 = 4320 & \text{---} \operatorname{Aut} D_5 &
\mathfrak{S}_2 \\
\operatorname{Conf}_{D_6} & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1 : 12
= 2^4 \cdot 3^2 \cdot 5 = 720 & \text{---} \operatorname{Aut} D_6 &
\mathfrak{S}_2 \times \mathfrak{S}_3 \\
\operatorname{Conf}_{D_7} & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1
\cdot 1 : 72 = 2^3 \cdot 3 \cdot 5 = 120 & \text{---} \operatorname{Aut} D_7 &
\mathfrak{S}_2 . (\mathfrak{S}_3 \times \mathfrak{S}_3)
\end{array}
\]

\[
\begin{array}{rlll}
4 & \mathfrak{A}_4 & 3 & (3, 3', 2)\,(3', 2, 3)\,(2, 3, 3')\
  \bigl[(2, 3', 3)\,(3', 3, 2)\,(3, 2, 3')\bigr] \\
  & & 1 & (3, 3, 3)\ \bigl[(3', 3', 3')\bigr] \\[4pt]
9 & \mathfrak{S}_4 & 6 & (4, 3, 2)\,(3, 2, 4)\,(2, 4, 3)\,(2, 3, 4)\,(3, 4, 2)\,(4, 2, 3)
  \ \text{standard} \\
  & & 3 & (4, 4, 3)\,(4, 3, 4)\,(3, 4, 4) \\[4pt]
32 & \mathfrak{A}_5 & 6 & (5, 3, 2)\,(3, 2, 5)\,(2, 5, 3)\,(2, 3, 5)\,(3, 5, 2)\,(5, 2, 3)
  \ \text{standard } \alpha \\
  & & 6 & (5', 3, 2)\,(3, 2, 5')\,(2, 5', 3)\,(2, 3, 5')\,(3, 5', 2)\,(5', 2, 3)
  \ \text{id. } \alpha' \\
  & & 3 & (5, 5, 3)\,(5, 3, 5)\,(3, 5, 5) \\
  & & 3 & (5', 5', 3)\,(5', 3, 5')\,(3, 5', 5') \\
  & & 6 & (5, 5', 3)\,(5', 3, 5)\,(3, 5, 5')\,(3, 5', 5)\,(5', 5, 3)\,(5, 3, 5') \\
  & & 6 & (5, 5', 2)\,(5', 2, 5)\,(2, 5, 5')\,(2, 5', 5)\,(5', 5, 2)\,(5, 2, 5') \\
  & & 1 & (5, 5, 5) \\
  & & 1 & (5', 5', 5') \\[4pt]
\hline
45 & & &
\end{array}
\]

\[
(10)\ \left\lbrace
\begin{array}{lll}
{}^{*}\ \mathrm{cn}(-z) = \mathrm{cn}\,z & \mathrm{dn}(-z) = \mathrm{dn}\,z &
\text{$\mathrm{cn}\,z$ et $\mathrm{dn}\,z$ sont paires}\\[4pt]
{}^{*}\ \mathrm{cn}(z+2\omega_1) = -\mathrm{cn}\,z & \mathrm{dn}(z+2\omega_3) = -\mathrm{dn}\,z & \\
\phantom{{}^{*}}\ \mathrm{cn}(z+2\omega_3) = -\mathrm{cn}\,z & \mathrm{dn}(z+2\omega_1) = -\mathrm{dn}\,z & \\[4pt]
{}^{*}\ \mathrm{cn}(z+2\omega_2) = \mathrm{cn}\,z & \mathrm{dn}(z+2\omega_2) = \mathrm{dn}\,z &
\text{$\mathrm{cn}$ admet pour périodes}\\
\phantom{{}^{*}}\ \mathrm{cn}(z+4\omega_1) = \mathrm{cn}\,z & \mathrm{dn}(z+4\omega_3) = \mathrm{dn}\,z &
\quad 2\omega_2,\ 4\omega_1,\ 4\omega_3\\
\phantom{{}^{*}}\ \mathrm{cn}(z+4\omega_3) = \mathrm{cn}\,z & \mathrm{dn}(z+4\omega_1) = \mathrm{dn}\,z &
\text{$\mathrm{dn}$ admet pour périodes}\\
& & \quad 2\omega_3,\ 4\omega_1,\ 4\omega_2
\end{array}
\right.
\]

\[
\left\{
\begin{aligned}
B_0 = T . U_0 &= \{ e_0(\beta)\, u(\lambda) \mid \beta \in \Lambda,\ \lambda \in \Lambda^* \}
= \Bigl\{ \begin{pmatrix} \lambda & \lambda^{-1}\beta \\ 0 & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^*,\ \beta \in \Lambda \Bigr\} \\
&\qquad \Bigl( = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| c = 0 \Bigr\} \Bigr) \\
B_1 = T . U_1 &= \{ u(\lambda)\, e_1(\gamma) \mid \gamma \in \Lambda,\ \lambda \in \Lambda^* \}
= \Bigl\{ \begin{pmatrix} \lambda & 0 \\ \lambda^{-1}\gamma & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^*,\ \gamma \in \Lambda \Bigr\} \\
&= \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| b = 0 \Bigr\}
\end{aligned}
\right. \tag{17}
\]

\[
\begin{array}{lll}
B_1(C) & 27:1=27=3^3 & \operatorname{aut}L_1=e\\
B_2(C) & 27\cdot16:2!=27\cdot8=2^3 3^3=216 & \operatorname{aut}L_2=\mathfrak{S}_2\\
B_3(C) & 27\cdot16\cdot10:3!=2^4\cdot3^2\cdot5=720=6! & \operatorname{aut}L_3=\mathfrak{S}_3\\
B_4(C) & 27\cdot16\cdot10\cdot6:4!=2^3\cdot3^3\cdot5=1080 & \operatorname{aut}L_4=\mathfrak{S}_4\\
\quad\simeq B''_4(C) & & \\
B'_5(C) & 27\cdot16\cdot10\cdot6\cdot2:5!=2^4\cdot3^3=432 & \operatorname{aut}L_5=\mathfrak{S}_5\\
B(C)=B_6(C) & 27\cdot16\cdot10\cdot6\cdot2\cdot1:6!=2^3 3^2=72 & \operatorname{aut}L_5\simeq\mathfrak{S}_6\\
\operatorname{Bib}(C) & 36 & \operatorname{aut}(L'_5)\simeq\mathfrak{S}_2\times\mathfrak{S}_6
\end{array}
\]

\[
\begin{array}{lll|l}
D_0 = \emptyset & & & 1 \\
\cap \ D_1 = \bullet\, u & \text{pt} & & 27 = 3^3 . \\
D_2 = u \,\text{---}\, v & \text{arête} & & 27 \cdot 10 = 2 \cdot 3^3 \cdot 5 = 270 \\
\cap \ D_3 = \text{triangle } u, v, w & \text{triangle} & & 27 \cdot 10 \cdot 1 = 2 \cdot 3^3 \cdot 5 = 270 \\
D_4 & \text{\struck{triangle} \add{balise}} & & 27 \cdot 10 \cdot 1 \cdot 8 = 2^4 \cdot 3^3 \cdot 5 = 2160 \\
\cap \ D_5 & \text{préprisme (triangulaire)} & & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 = 2^6 \cdot 3^3 \cdot 5 = 8640 \\
\cap \ D_6 & \text{prisme (triangulaire)} & & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1 = 2^6 \cdot 3^3 \cdot 5 = 8640 \\
\cap \ D_7 & \text{bitriangle} & & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1 \cdot 1 = 2^6 \cdot 3^3 \cdot 5 = 8640
\end{array}
\]

\[
\text{(40)}\qquad
\left\{
\begin{array}{l}
S' = \coprod_{\alpha\in S} S(\alpha) \\[2pt]
\tilde{A}' = \tilde{A} \sqcup \coprod_{\alpha\in S} \tilde{A}(\alpha) \\[2pt]
\sigma_{\tilde{A}'} \text{ induit par } \sigma_{\tilde{A}} \text{ et par les } \sigma_{\tilde{A}(\alpha)} \\[2pt]
I' = I \quad \text{donc} \quad \hat{I}' \overset{\text{déf}}{=} \tilde{A}' \sqcup I' \simeq \underbrace{(\tilde{A} \sqcup I)}_{\hat{I} = \coprod_{\alpha\in S} \hat{I}_\alpha = \coprod_{\alpha\in S} I(\alpha)} \sqcup \coprod_{\alpha\in S} \tilde{A}(\alpha) \\[2pt]
\phantom{I' = I \quad \text{donc} \quad \hat{I}'} \simeq \coprod_{\alpha\in S} \underbrace{\hat{I}_\alpha \sqcup \tilde{A}(\alpha)}_{\hat{I}(\alpha)} \\[2pt]
\hat{I}' \to S' \text{ induit par les } \hat{I}(\alpha) \to S(\alpha) \\[2pt]
S' \xrightarrow{g'} \mathbb{N} \text{ induit par les } S(\alpha) \xrightarrow{g_\alpha} \mathbb{N}
\end{array}
\right.
\]

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

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

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

\[
  \left\lbrace
  \begin{array}{l}
    L_{E_\eta}\Bigl(\dfrac{1}{pt}\Bigr) = \prod_x \lambda_x(E)\,
      (-pt)^{\chi^{\natural}(E_\eta)}\, \delta^{\natural}(E_\eta)^{-1}\, L_{\check E_\eta}(t) \\[10pt]
    \chi^{\natural}(E_\eta) = \chi(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
    \delta^{\natural}(E_\eta) = \delta(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
    \lambda_x(E) = L_{\mu_x(E)}(t) = L_{\alpha_x(E_\eta)^{\vee}}(t) / L_{\alpha_x(\check E_\eta)}(p^{-1} t)
      = (-t)^{\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
      \dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(\check E_\eta)}(t)}
  \end{array}
  \right.
\]

\[
(62) \qquad
\left\{
\begin{array}{l}
\text{a) graphe } \gamma_0 = (S, A_0), \text{ muni de \struck{\ill{}}} \\
\text{b) groupe \ill{} d'automorphismes de } (S, A_0), \text{ transitif sur } \overrightarrow{A_0} \\
\qquad \text{[et sur } S, \text{ i.e. } \forall s \in S,\ \mathrm{Et}_{A_0}(s) \neq \emptyset] \\
\text{c) et d'une « origine » } (s_0, t_0), \text{ \uncertain{avec} donc } \{s_0, t_0\} \in A_0 \\
\qquad \text{et enfin} \\
\text{d) d'un sous-groupe } Z \subset \mathfrak{S}_S \text{ commutatif,} \\
\qquad \text{formé d'homothéties pour le graphe } \gamma_0 \ \text{\ill{}} \\
\qquad \text{commutant \struck{\ill{}} : l'action de \ill{}}
\end{array}
\right.
\]

\[
\begin{aligned}
&\xi_{n-1} - 2\xi_n = 0
&& \boxed{\xi_n = 1,\ \xi_{n-1} = 2} \\
&\xi_{n-2} - 2\xi_{n-1} + (\alpha_{n-1} + 2)\xi_n = 0
&& \boxed{\xi_{n-2} = -(\alpha_{n-1} - 2)} \\
&\xi_{n-3} - 2\xi_{n-2} + (\alpha_{n-2} + 2)\xi_{n-1} = 0
&& \boxed{\xi_{n-3} = -2(\alpha_{n-1} + \alpha_{n-2})} \\
&\qquad 2(\alpha_{n-1} - 2) + (\alpha_{n-2} + 2)2 = 2(\alpha_{n-1} + \alpha_{n-2}) \\
&\xi_{n-4} - 2\xi_{n-3} + (\alpha_{n-3} + 2)\xi_{n-2} = 0
\\
&\qquad \boxed{\xi_{n-4} = -4 - 2(\alpha_{n-1} + 2\alpha_{n-2} + \alpha_{n-3})
+ \alpha_{n-1}\alpha_{n-3}} \\
&\qquad 4(\alpha_{n-1} + \alpha_{n-2}) - (\alpha_{n-3} + 2)(\alpha_{n-1} - 2) \\
&\qquad 2\alpha_{n-1} + 4\alpha_{n-2} + 4 - \alpha_{n-1}\alpha_{n-3} + 2\alpha_{n-3} \\
&\xi_{n-5} = \cdots
\end{aligned}
\]

\[
\begin{aligned}
\sigma_i(\sigma_i s_j) &= \sigma_i(s_0+\lambda_{i1}e_1+\dots+\lambda_{i,i+1}e_{i+1}+\lambda_{i,i+2}e_j) \\
&= \underbrace{\sigma_i(s_0+\dots+\lambda_{i,i-1}e_{i-1})}_{s_0+\lambda_{i1}e_1+\dots+\lambda_{i,i-1}e_{i-1}} + (\lambda_{i,i}e_{i+1}+\lambda_{i,i+1}e_i) \\
&\quad + \lambda_{i,i+2}\underbrace{\sigma_i e_j}_{(\lambda_{i1}e_1+\dots+\lambda_{i,i+1}e_{i+1})+\lambda_{i,i+2}e_j} \qquad (i\geq 1) \\
&= s_0+(\lambda_{i1}+\lambda_{i,i+2}\lambda_{i1})e_1+(\lambda_{i2}+\lambda_{i,i+2}\lambda_{i2})e_2+\dots \\
&\quad +(\lambda_{i,i-1}+\lambda_{i,i+2}\lambda_{i,i-1})e_{i-1} \\
&\quad +(\lambda_{i,i+1}+\lambda_{i,i+2}\lambda_{i,i})e_i \\
&\quad +(\lambda_{i,i}+\lambda_{i,i+2}\lambda_{i,i+1})e_{i+1} \\
&\quad +(\lambda_{i,i+2})^2e_j
\end{aligned}
\]

\[
\begin{aligned}
(y_2)\,\lambda_2 &= \lambda_1 (y_0 + y_1) - \lambda_0 y_{n-1}
 = \lambda_0 (-y_{n-1}) + \lambda_1 (y_0 + y_1) \qquad
 (= \lambda_4 y_3 - \lambda_5 y_5) \\
(y_2 y_3)\,\lambda_3 &= \underbrace{(y_2 \lambda_2)}\, y_1
 - y_2 (\lambda_0 y_{n-1} - \lambda_1 y_1) \\
 &= \lambda_0 \bigl( -y_{n-1} (y_1 + y_2) \bigr)
 + \lambda_1 y_1 [y_0 + y_1 + y_2] \\
(y_2 y_3 y_4)\,\lambda_4 &= (y_2 y_3 \lambda_3)\, y_3
 - y_2 y_3 (\lambda_0 y_{n-1} - \lambda_1 y_1) \\
 &= \lambda_0 \bigl( -y_2 y_{n-1} (y_1 + y_2 + y_3) \bigr)
 + \lambda_1 y_1 y_2 (y_0 + y_1 + y_2 + y_3) \\
(y_2 y_3 y_4 y_5)\,\lambda_5 &= \underbrace{(y_2 y_3 y_4 \lambda_4)}\, y_3
 - y_2 y_3 y_4 (\lambda_0 y_{n-1} - \lambda_1 y_1)
\end{aligned}
\]

\[
\boxed{11}\quad
\begin{cases}
\text{(a) } \forall a \in A, \struck{\text{il y a } \ill{} \text{ exactement}} \text{ l'ens.\ des } s \in S \text{ incidents à } a \\
\quad \text{est de cardinal 2 exactement —} \\
\quad \text{et ceci reste vrai après tt chgt de base } S' \to S,\ S' \text{ non vide} \\
\text{(b) } \forall s \in S,\ f \in F, \text{ l'ens.\ des } a \text{ incidents à } s \text{ et } f \text{ est \add{loc.} de cardinal égal à } 2, \\
\quad \text{et ceci reste vrai après tt chgt de base } S' \to S,\ S' \neq \varnothing \\
\text{(c) } \forall a \in A, \text{ l'ens.\ des } f \in F \text{ incidents \struck{\ill{}} est de cardinal } = 2, \\
\quad \text{et ceci reste vrai après tt chgt de base } S' \to S,\ S' \neq \varnothing
\end{cases}
\]

\[
\begin{array}{ll}
  \text{\emph{prismes épinglés}}\ \ 27\cdot10\cdot8\cdot4\cdot1=[W]:6 &\\
  \text{prismes} & [W]:72\\[4pt]
  \text{A}_5\ \text{épinglés}\ \ 27\cdot10\cdot8\cdot4\cdot1=[W]:6 &\\
  \text{A}_5 & [W]:12\\[4pt]
  \text{A}_4\ \text{épinglé}\ \ 27\cdot10\cdot8^{\,4}=[W]:\uncertain{6} &\\
  \text{A}_4 & [W]:12\\[4pt]
  \text{A}_3\ \text{épinglés}\ \ 27\cdot10\cdot8=[W]:24 &\\
  \quad(\text{autres que les triangles épinglés}) &\\
  \text{A}_3\ \text{ép.} & [W]:48\\[4pt]
  \text{carrousels}\ \text{\add{simples}}\ \text{épinglés}\ \
  27\cdot10\cdot8\cdot4\cdot3=[W]:2 &\\
  \text{carrousels}\ \text{\add{simples}} & [W]:12
\end{array}
\]

\[
\begin{array}{ll}
F_1(U) = U-2 & \Phi_1(T) = T-1\\
F_2(U) = U+2 & \Phi_2(T) = T+1\\
F_3(U) = U+1 & \Phi_3(T) = T^2+T+1\\
F_4(U) = U & \Phi_4(T) = T^2+1\\
F_5(U) = U^2+U-1 & \Phi_5(T) = T^4+T^3+T^2+T+1\\
F_6(U) = U-1 & \Phi_6(T) = T^2-T+1\\
F_7(U) = U^3+U^2-2U-1 & \Phi_7(T) = T^6+T^5+T^4+T^3+T^2+T+1\\
F_8(U) = U^2-2 & \Phi_8(T) = T^4+1\\
F_9(U) = U^3-3U+1 & \Phi_9(T) = T^6+T^3+1\\
F_{10}(U) = U^2-U-1 & \Phi_{10}(T) = T^4-T^3+T^2-T+1\\
\cdots & \cdots
\end{array}
\]

\[
(20)\quad
\begin{array}{ll}
(a) & \text{Les } X_\beta \text{ disjoints}\\
(b) & \text{Chaque } \dot X_\beta \text{ transverse : } (B^{*}_\alpha,\mathcal E^{*}_\alpha)\ \Longrightarrow\ \text{le syst. induit dans } (\dot X_\beta,\ldots)\ (\dot B^{\alpha},\dot{\mathcal E}^{\alpha})\\
(c) & (\dot X_\beta,\underbrace{\dot B_{\beta\alpha},\dot{\mathcal E}_{\beta\alpha}}_{\text{induit sur }\dot X_\beta\text{ par }(B_\alpha,\mathcal E_\alpha)})\longrightarrow(X_\beta,B_{\beta,\alpha},\mathcal E_{\beta,\alpha})\ \text{est transverse}\\
(d) & \text{Les } X^{*}_i \text{ transverses aux } \bigl((\dot X_\beta,\mathcal E_\beta)_{\beta\in I_0},(B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}\bigr)
\end{array}
\]

\begin{gather*}
  f_1 \dots f_n \text{ régulier} \Longleftrightarrow f_1 \dots f_n \text{ quasi-régulier} \Longleftrightarrow \\
  A/I[T_1 \dots T_n] \otimes_{A/I} M/I \to \operatorname{gr}_I(A) \otimes_{\operatorname{gr}^0_I(A)} \operatorname{gr}^0_I(M) \to \operatorname{gr}_I(M) \text{ bijectif} \\
  \Longleftrightarrow \operatorname{gr}_I(A) \otimes_{A/I} M/I \to \operatorname{gr}_I(M) \text{ bijectif et} \\
  A/I[T_1 \dots T_n] \otimes A/\operatorname{Ann}(M/I) \to \operatorname{gr}_I(A) \otimes A/\operatorname{Ann}(M/I) \text{ bijectif.} \\
  \Longleftrightarrow \operatorname{gr}_I(A) \otimes M/I \xrightarrow{\sim} \operatorname{gr}_I(M),
  \text{ et modulo } \operatorname{Ann}(M/I),\ I/I^2 \text{ libre de rg } n \text{ et}
  \operatorname{Sym}_{A/I}(I/I^2) \simeq \operatorname{gr}_I(A).
\end{gather*}

\[
  (1)\quad
  \left\lbrace
  \begin{array}{l}
    \boxed{L_{DE}(t) = (-t)^{-\chi(E)}\, \delta(E)\, L_E(t^{-1})} \\[6pt]
    \chi(E) = \operatorname{rang} R f_{*} E = \sum (-1)^i \operatorname{rg} H^i(\bar X, \bar E) \\[6pt]
    \delta(E) = \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^i}
      = \varepsilon(E)\, p^{\sum_i (-1)^i \frac{i + \rho}{2} b_i}
  \end{array}
  \right.
\]

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

\[
\begin{cases}
\psi(a_i) = -(\beta_0 \cdots \beta_{i-1})\, a'_i & 0 \leqslant i \leqslant n \\
\psi(a_{-1}) = -a^{*}_0 = -\bigl[a'_1 + 2a'_2 + \rho_{11} a'_3 + \cdots
+ \overbrace{\rho_{1,n-2}}^{\gamma} a'_n + \overbrace{\rho_{1,n-1}}^{\sigma} a'_{n+1}\bigr] \\
f(a_i) = \beta_0 \cdots \beta_{i-1} & 0 \leqslant i \leqslant n \\
\qquad f(a_0) = 1,\ f(a_1) = \beta_0, \ldots, f(a_n) = \beta_0 \cdots \beta_{n-1} = \beta \\
f(a_{-1}) = 0 \qquad \varphi(a_i, a_{i+1}) = -\beta_0 \cdots \beta_i & 0 \leqslant i \leqslant n-1 \\
\psi'(\pi) = -\sigma\pi' \qquad f(\pi) = -\tilde{\sigma}\tilde{\rho} & (0
\end{cases}
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c|c}
 & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mathbb{G}_m & \mathbb{G}_a & \mathbb{Z} & \mathbb{V} & A \\ \hline
\mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/\ell\mathbb{Z} & 0 & 0 & 0 & \mu_\ell & 0 & 0 & 0 & {}_\ell A \\
\mathbb{Z}/p\mathbb{Z} & 0 & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mu_p & \mathbb{G}_a & 0 & \mathbb{V} & {}_p A \\
\alpha_p & 0 & 0 & \mathbb{G}_a & \alpha_p & \alpha_p & \mathbb{G}_a & 0 & (\mathrm{gros}) & ? \\
\mu_p & 0 & 0 & 0 & \mathbb{Z}/p\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & 0 & 0 & 0 & ? \\
\mathbb{G}_m & 0 & 0 & 0 & 0 & \mathbb{Z} & 0 & 0 & 0 & 0 \\
\mathbb{G}_a & 0 & 0 & (\mathrm{gros}) & \mathbb{V} & \mathbb{V} & ? & 0 & (\mathbb{G}_a ?) & \mathrm{gros} \\
\mathbb{Z} & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mathbb{G}_m & \mathbb{G}_a & \mathbb{Z} & \mathbb{V} & A \\
\mathbb{V} & 0 & 0 & \mathbb{G}_a & \mathbb{G}_a & \mathbb{G}_a & \mathbb{G}_a ? & 0 & (\mathrm{Imm}) & \underline{V}(\omega_A) \\
A & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ? & \mathbb{Z}^n
\end{array}
\]

\begin{align*}
\Phi\bigl(F \overset{L}{*} G\bigr)
  &\underset{\text{déf}}{\simeq}
    \mathbb{R}pr_{2*}\bigl(\mathbb{L}pr_1^{*}(F \overset{L}{*} G) \otimes L\bigr)
  \underset{(*)}{=}
    \mathbb{R}pr_{2*}\bigl(\mathbb{L}pr_1^{*}(\pi_{*}(F \overset{L}{\otimes}_S G)) \otimes L\bigr) \\
  &\underset{\text{chgt. de base}}{\simeq}
    \mathbb{R}pr_{2*}\bigl(\mathbb{R}(\pi \times \mathrm{id}_B)_{*}(\mathbb{L}pr_{12}^{*}(F \overset{L}{\otimes}_S G)) \otimes L\bigr) \\
  &\underset{\text{formule de proj.}}{\simeq}
    \mathbb{R}pr_{2*}\bigl(\mathbb{R}(\pi \times \mathrm{id}_B)_{*}(\mathbb{L}pr_{12}^{*}(F \overset{L}{\otimes}_S G) \\
  &\hphantom{\underset{\text{formule de proj.}}{\simeq}\mathbb{R}pr_{2*}\bigl(}
    \otimes (\pi \times \mathrm{id}_B)^{*}(L))\bigr) \\
  &\underset{\text{transit.}}{\simeq}
    \mathbb{R}pr_{3*}\bigl(\mathbb{L}pr_{12}^{*}(F \overset{L}{\otimes}_S G) \otimes (\pi \times \mathrm{id}_B)^{*}(L)\bigr)
\end{align*}

\[
(144)\qquad
\left\{
\begin{array}{l}
T_0 \text{ opère fidèlement sur } H, \text{ et via cette opération s'identifie}\\
\text{au sous-groupe } \widetilde{Z} \text{ de } \mathrm{Aut}(H) \text{ formé des } \bar g\in\mathrm{Aut}\,H\\
\text{satisfaisant les conditions}\\
\qquad \begin{cases}
\bar g \text{ normalise } U_0 \text{ et centralise } T_H\\
\bar g(\sigma)\in\sigma T_H .
\end{cases}\\
\text{De plus, on aura alors}\\
\qquad \bar g(\sigma)=\sigma\,u(\delta(g)) \qquad \delta(g)=c(\tilde\delta(g)) .
\end{array}
\right.
\]

\[
\begin{cases}
a^{*}_{-1} = \pi' = a'_0 + 2a'_1 + \overbrace{\rho_{0,0}}^{4 - \beta_0 = 2 - \alpha_0} a'_2
+ \rho_{0,1} a'_3 + \cdots + \overbrace{\rho_{0,n-2}}^{\sigma'} a'_n
+ \overbrace{\rho_{0,n-1}}^{\rho} a'_{n+1} \\
a^{*}_0 = -\psi(a_{-1}) = -\psi(s_0) = a'_1 + 2a'_2 + \rho_{1,1} a'_3 + \cdots
+ \overbrace{\rho_{1,n-2}}^{\gamma} a'_n + \overbrace{\rho_{1,n-1}}^{\sigma} a'_{n+1} \\
\bigl[\, a^{*}_1 = a'_2 + 2a'_3 + \rho_{2,2} a'_3 + \cdots + \rho_{2,n-2} a'_n
+ \rho_{2,n-1} a'_{n+1} \\
\cdots \\
a^{*}_{n-2} = a'_{n-1} + 2a'_n + \underbrace{\rho_{n-1,n-1}}_{4 - \beta_{n-1} = 2 - \alpha_{n-1}} a'_{n+1} \\
a^{*}_{n-1} = a'_n + 2a'_{n+1} \\
a^{*}_n = a'_{n+1}
\end{cases}
\]

\[
  \bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr) \quad
  \bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
  \bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
  \bigl(\begin{smallmatrix}5\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr)
\]

\[
\begin{gathered}
\ldots\ \tfrac{6}{9} > \tfrac{11}{17}, \qquad \tfrac{2}{3} > \tfrac{11}{17}, \qquad
\tfrac{5}{12} > \tfrac{7}{17}, \qquad \tfrac{1}{4} < \tfrac{?}{11} < \tfrac{2}{7}, \qquad
\tfrac{2}{7} < \tfrac{5}{17}, \\
\tfrac{4}{17} < \tfrac{1}{4}, \qquad \tfrac{1}{9} < \tfrac{2}{17}, \qquad
\tfrac{1}{6} < \tfrac{3}{17}, \qquad \tfrac{7}{16} < \tfrac{12}{17}, \qquad
\tfrac{5}{7} \mathrel{?} \tfrac{13}{18}, \\
\tfrac{4}{13} > \tfrac{7}{23}, \qquad {<}\ \tfrac{51}{13}\,(?), \qquad
\tfrac{3}{10} > \tfrac{7}{24}, \qquad \tfrac{2}{7} < \tfrac{7}{24}, \qquad
\tfrac{1}{3} > \tfrac{7}{23}, \qquad \tfrac{2}{7} < \tfrac{1}{3}, \\
\tfrac{7}{9} < \tfrac{7}{8}\,(?), \qquad \tfrac{4}{5} < \tfrac{14}{17}\,(?), \qquad
\tfrac{11}{17} > \tfrac{13}{23}, \qquad \tfrac{?}{13} > \tfrac{15}{23}, \qquad
\tfrac{10}{14} < \tfrac{13}{18}, \\
\tfrac{13}{17} < \tfrac{10}{13}, \qquad \tfrac{3}{5} < \tfrac{14}{23}, \qquad
\tfrac{3}{4} < \tfrac{13}{17}
\end{gathered}
\]

\[
\begin{aligned}
&\operatorname{Hom}(\Psi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } \mathrm{id})
\end{cases} \\
&\operatorname{Hom}(\Phi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } u_i)
\end{cases} \\
&\operatorname{Hom}(\Psi^{i}, \Phi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j})) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \ (i \neq 0) \\
k & \text{si } j = i+1 \ (\text{base } \partial_i) \\
k & \text{si } j = i = 0 \ (\text{base } \mathrm{id})
\end{cases}
\end{aligned}
\]

\[
(29) \qquad
\left\{
\begin{array}{l}
\text{Soit } \varphi_0 \in \sqcap_Z \text{ une $Z$-jauge stricte (i.e. bij.) de } \Gamma, \ \text{\struck{alors}} \text{ soit} \\
\text{\struck{$\mathcal{G}_{\varphi_0}$}} \ \mathcal{G}_{\varphi_0} = \{g \in \mathcal{G} \mid g_{\check{S}} \varphi = \varphi\, g_S^{-1} \text{ i.e. } \varphi_0 \text{ est une $g$-jauge}\} \\
\text{(NB } \mathcal{G}_{\varphi_0} \text{ n'est pas nécessairement un sous-groupe de } \mathcal{G}) \\
\text{alors} \\
\qquad g \longmapsto T_g \varphi \overset{\mathrm{def}}{=} g_{\check{S}} \varphi_0 = \varphi_0 g_S^{-1} \\
\text{est une bijection } \mathcal{G}_{\varphi_0} \xrightarrow{\ \sim\ } \sqcap_Z
\end{array}
\right.
\]

\begin{align*}
  &\pi_{1}^{\ell}(\underline{\mathrm{Alb}}^{0}_{X}) \xleftarrow[\ \sim\ ]{(t)} \pi_{1}^{\ell}(X)^{\mathrm{ab}}(\ell)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours surjectif ?}] \\
  &\pi_{1}(\underline{\mathrm{Pic}}^{00}_{X}) \simeq H^{1}(\underline{\mathrm{Alb}}^{0}_{X})(1)
  \xrightarrow[\ \sim\ ]{t} H^{1}(X)(1)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours injectif}] \\
  &H^{1}(\underline{\mathrm{Alb}}^{0}_{X}) \xrightarrow[\ \sim\ ]{t} H^{1}(X)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours injectif}] \\
  &H^{1}(\underline{\mathrm{Pic}}^{00}_{X}) \simeq \pi_{1}(\underline{\mathrm{Alb}}_{X})(-1)
  \xleftarrow[\ \sim\ ]{(t)} H^{2n-1}(X)(n-1)
  && \text{toujours}
\end{align*}

\[
\begin{array}{c}
\text{(A)}\\
\text{catégorie-groupoïde}\\
\text{des graphes stricts}\\
(S, A_0) \text{ satisfaisant (13)}\\
\text{(sommets déterminés}\\
\text{par leurs étoiles)}
\end{array}
\ \xrightarrow{\ \approx\ }\
\begin{array}{c}
\text{(B)}\\
\text{catégorie-groupoïde des}\\
\text{systèmes } (S, S^{*}, R_{*})\\
\text{« sommets-cosommets-incidence »}\\
\text{satisfaisant les conditions d'inj. symétriques,}\\
\text{munis d'une application bijection}\\
S \xrightarrow{\varphi_0} S^{*} \text{ « symétrique »}\\
\text{et « antiréflexive » } (s, \varphi_0(s)) \notin R_0
\end{array}
\]

\[
(82)\qquad
\begin{cases}
\text{a) } S, S^{\vee} \text{ deux ensembles, } \mathcal{R} \subset S \times S^{\vee} \text{ une relation} \\
\text{b) } Z \text{ un sous-groupe \emph{commutatif} de } \mathrm{Aut}(\Gamma) \\
\text{c) } \Pi \text{ un ensemble de bijections } S \to S^{\vee}, \text{ satisfaisant} \\
\quad \text{(1) condition de symétrie (cf.\ (70)),} \\
\quad \text{(2) } \lambda_{S^{\vee}} \varphi = \varphi \lambda_S^{-1} \text{ i.e.\ } \lambda_{S^{\vee}} \varphi \lambda_S = \varphi \quad \forall \lambda \in Z, \\
\quad \text{(3) } \Pi \text{ est un sous-}Z\text{-torseur de } \Pi_Z
\end{cases}
\]

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

\[
  (\alpha)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, \begin{smallmatrix}3_2\\3\end{smallmatrix}\bigr) \qquad
  (\beta)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}[\ldots]\\2_3\end{smallmatrix}, \begin{smallmatrix}3\\2\\1\end{smallmatrix}\bigr) \qquad
  (\gamma)\ \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\2_1\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)
\]

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

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

\[
(74)\qquad
\left\{
\begin{array}{lll}
\widehat{M}^{!}_{0,5} & \Gamma \text{ opère via} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0}} \ (\simeq \mathfrak{S}_5) \\[2pt]
\widehat{M}^{!}_{1,2} & \Gamma \text{ ------} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0}} \simeq \mathbb{Z}/2\mathbb{Z} \\[2pt]
\widehat{M}^{!}_{0,4} \times \widehat{M}^{!}_{0,4} & \Gamma \text{ ------} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0} \amalg \hat{I}_{\alpha_1}} \simeq \mathfrak{S}_8 \\[2pt]
\widehat{M}^{!}_{0,4} \times \widehat{M}^{!}_{1,1} & \Gamma \text{ ------} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0}} \simeq \mathfrak{S}_4 \\[2pt]
\widehat{M}^{!}_{1,1} \times \widehat{M}^{!}_{1,1} & \Gamma \text{ opère trivialement.} &
\end{array}
\right.
\]

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

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

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

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

\[
\begin{aligned}
\overline{\sigma}_0(\underbrace{x_1, \dots, x_{n+1}, \overset{x_0}{\overset{\|}{z}}}_{\overline{x}}) &= (x_0 - x_1 + \alpha_0(x_2 + \cdots + x_{n+1}),\ x_2, \dots, x_{n+1},\ x_0)\\
&= \overline{x} + \struck{\ill{}}\ [e'_0 - 2e'_1 + \alpha_0(e'_2 + \cdots + e'_{n+1})] \otimes e_1\\
\overrightarrow{\sigma_i}(\overline{x}) &= \overline{x} + [e'_i - e'_{i+1} + (1+\alpha_i)(e'_{i+2} + \cdots + e'_{n+1})] \otimes (-e_i + e_{i+1}) \quad (1\le i\le n)\\
\overrightarrow{\sigma_n}(\overline{x}) &= \overline{x} + (e'_n - e'_{n+1}) \otimes (-e_n + e_{n+1})
\end{aligned}
\]

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

\[
(51)\quad\left\{
\begin{array}{l}
\tilde{U}=G_{0s_0}=\{g\in G_0\mid g.s_0=s_0\}\\
\tilde{B}''=\{g\in G_0\mid gs_0\in Z.s\}\\
Z'=G/G_0,\quad Z\to Z' \text{ le composé } Z\xrightarrow[\text{incl.}]{}G\xrightarrow{\text{can}}G/G_0=Z'\\
B'=G_{s_0}=\{g\in G\mid gs_0=s_0\} \text{ de sorte que}\\
\phantom{B'=}\ \tilde{U}=B'\cap G_0,\quad B'\to Z' \text{ composé } B'\xrightarrow[\text{incl}]{}G\xrightarrow{\text{can}}G/G_0=Z'\\
\text{op. de } B' \text{ sur } G_0 \text{ induite par celle de } G \text{ sur le sous-groupe distingué } G_0
\end{array}\right.
\]

\[
\begin{aligned}
(3.2) \qquad 0 &\to \underbrace{[f^{\mathrm{fl}}_{*}(\underline{\mathcal{O}}^{*}_{X/S}) \to f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S}) \to f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1\otimes p}_{X/S})]}_{\underline{\mathrm{Cart}}_{X/S} \text{ « faisceau de Cartier »}} \to P^{cip}_{X/S} \\
&\to \underline{\mathrm{Pic}}^{ci}_{X/S} \to f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})/\mathrm{Im}\, f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S}),
\end{aligned}
\]

\[
\begin{aligned}
\xi^\circ &\in \mathbb{R}^0\mathrm{Hom}\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) = \mathbb{R}^0\Gamma_{X\times_S Y}\,\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr)\\
\eta^\circ &\in \mathbb{R}^0\mathrm{Hom}\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr) = \mathbb{R}^0\Gamma_{X\times_S Y}\,\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr)
\end{aligned}
\]

\[
\begin{cases}
f(u) = \struck{f(s_0) + f} 4 f(\underbrace{a_{-1}}_{0}) + \underset{1}{f(a_0)} + 2\varphi(\underbrace{a_{-1}, a_0}_{-1}) = -1 \\
f(v) = 4 \underbrace{f(t_{n+1})}_{2\beta\sigma\gamma} + \sigma^2 \underbrace{f(a_n)}_{\beta} - 2\sigma \underbrace{\varphi(a_n, t_{n+1})}_{\beta\sigma} = \beta\sigma(2\gamma - \sigma) \\[1ex]
\left.
\begin{array}{l}
f(s_0) = 0 \\
f(t_{n+1}) = \beta\tilde{\gamma}\tilde{\sigma}
\end{array}
\right|\ \text{pour mémoire}
\end{cases}
\]

\[
(68)\qquad
\begin{cases}
\text{l'ens.\ } \Pi_Z \text{ des jauges de } \Gamma = (S, S^{\vee}, \mathcal{R}) \text{ compatibles avec le groupe} \\
\text{d'automorphismes } Z \subset \mathrm{Aut}(\Gamma) \text{, i.e.\ satisfaisant } \check{\lambda}\varphi\lambda = \varphi\ \forall (\lambda, \check{\lambda}) \in Z_{\Gamma}, \\
\text{est un \struck{torseur} sous le \emph{centralisateur} de } Z \text{ dans } \mathrm{Aut}(\Gamma), \text{ opérant par} \\
\quad T_g \varphi_0 = g_{S^{\vee}} \varphi_0 = \varphi_0 g_S^{-1}
\end{cases}
\]

\[
(\mathrm{A}'_i) \quad
\left.
\begin{array}{l}
  H^i(U, F) \text{ de dim finie sur } k \\
  \Updownarrow \\
  H^{i+1}_z(F_z) \text{ de dim finie [i.e.\ de long.\ finie] pour tt } z \in Z \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ pt isolé de } \operatorname{Supp} E^{r-i-1} \cup \lbrace z \rbrace \\
  \Updownarrow \\
  \alpha_{i+1} \text{ injectif pour } m \text{ grand} \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand}
\end{array}
\right.
\]

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

\[
(7)\ \left\lbrace
\begin{array}{ll}
\mathrm{cn}_1(-z) = \mathrm{cn}_1 z & \mathrm{dn}_1(-z) = \mathrm{dn}_1 z\\[4pt]
\mathrm{cn}_1(z+2\omega_1) = -\mathrm{cn}_1 z & \mathrm{dn}_1(z+2\omega_1) = -\mathrm{dn}_1 z\\
\mathrm{cn}_1(z+2\omega_2) = -\mathrm{cn}_1 z & \mathrm{dn}_1(z+2\omega_3) = -\mathrm{dn}_1 z\\[4pt]
\mathrm{cn}_1(z+2\omega_3) = \mathrm{cn}_1 z & \mathrm{dn}_1(z+2\omega_2) = \mathrm{dn}_1 z\\
\mathrm{cn}_1(z+4\omega_1) = \mathrm{cn}_1 z & \mathrm{dn}_1(z+4\omega_3) = \mathrm{dn}_1 z\\
\mathrm{cn}_1(z+4\omega_2) = \mathrm{cn}_1 z & \mathrm{dn}_1(z+4\omega_1) = \mathrm{dn}_1 z
\end{array}
\right.
\]

\[ (268)\quad \left\{\begin{array}{l} I=\underbrace{I'}_{\substack{\text{correspond aux facteurs}\\ \text{du type « global »}}}\amalg\underbrace{I''}_{\substack{\text{correspond aux facteurs}\\ \text{du type local}}}\\[18pt] I'=\underbrace{I'_0}_{\substack{\text{correspond aux facteurs } \mathbb{P}\\ \text{(de dim 0)}}}\amalg\underbrace{I'_1}_{\substack{\text{correspond aux facteurs } \mathbb{C}^*\\ \text{(de dim 1)}}}\\[18pt] I''=\underbrace{I''_0}_{\text{facteurs } \mathbb{D}}\amalg\underbrace{I''^*}_{\text{facteurs } \mathbb{D}^* \text{ (épointés)}} \end{array}\right. \]

\[
  \left\{
  \begin{array}{ll}
    (\mathrm{vii}) \not\Rightarrow (\mathrm{vi}) & = \text{si } \psi
      \text{ birationnel} : \mathrm{Sl}(2, k), \text{ car. } 2 \\[3pt]
    (\mathrm{x}) \not\Rightarrow (\mathrm{v}) & [\mathrm{x} \not\Rightarrow
      (\mathrm{i}),\ (\mathrm{x}) \not\Rightarrow (\mathrm{xiii}),\
      (\mathrm{vi}) \Rightarrow (\mathrm{v})] \\
      & \text{Cas d'un système de } G \text{ semi-simple} \ldots \\[3pt]
    (\mathrm{i}) \not\Rightarrow (\mathrm{x}) & [(\mathrm{v}) \not\Rightarrow
      (\mathrm{x}),\ (\mathrm{vi}) \not\Rightarrow (\mathrm{x}),\
      (\mathrm{i}) \not\Rightarrow (\mathrm{xiii})] \\
      & GP(2, k), \text{ car. } 2
  \end{array}
  \right.
\]

\[
(82)\qquad
\begin{cases}
\tau_0^r \begin{cases} x \mapsto y \\ y \mapsto x \\ z \mapsto z \end{cases} \quad \text{matrice } \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = r_0 \\[14pt]
\tau_1^r \begin{cases} x \mapsto x \\ y \mapsto z \\ z \mapsto y \end{cases} \quad \text{matrice } \begin{pmatrix} 1 & -1 \\ 0 & -1 \end{pmatrix} = -r_1 \\[14pt]
\tau_\infty^r \begin{cases} x \mapsto x \\ y \mapsto -y \\ z \mapsto y - x \end{cases} \quad \text{matrice } \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = -r_\infty
\end{cases}
\]

\[
\begin{array}{lll}
H_0 : & \overset{\substack{z\ \add{\text{variable d'homogénéité}}\\ \|}}{\struck{x_0}} - 2x_1 + \alpha_0(x_2 + \cdots + x_{n+1}) = 0 &\\
\delta_0 : & \text{pt à l'infini dans la direction de } e_1 = s_1 - s_0 &\\[4pt]
H_i : & x_i - x_{i+1} + (1+\alpha_i)(x_{i+2} + \cdots + x_{n+1}) = 0 & (1\le i\le n)\\
\delta_i : & \text{pt à l'infini dans la direction de } e_{i+1} - e_i = s_{i+1} - s_i &
\end{array}
\]

\[ (\mathrm{M}11) \qquad \begin{cases}
\text{a) } 0 \neq 1 \text{ sont } \in R_0 \text{ i.e.\ } \lbrace 0 \rbrace,
\lbrace 1 \rbrace \in \mathcal{M}_1 \\
\text{b) } \forall A \in \mathcal{M}_2, \text{ les fonctions } (x,y) \mapsto
x - y, \ (x,y) \mapsto xy \\ \qquad \text{sur } A \text{ sont modérées, i.e.\ les} \\
\qquad \lbrace (x, y, x-y) \mid (x,y) \in A \rbrace \text{ et } \lbrace (x,
y, xy) \mid (x,y) \in A \rbrace \\ \qquad \text{sont modérées dans } R^3 \\
\text{c) } \forall A \subset R^*,\ A \in \mathcal{M}_1,\ \exists B \in
\mathcal{M}_1 \text{ avec } A^{-1} \subset B
\end{cases} \]

\[
  (A) \;
  \begin{cases}
    (0)\ x^{(i)} \ (i \geq 0) \quad x^{(0)} = 1,\ x^{(1)} = x,\ x^{(i)} \in J^0 \ (i \geq 1) \\
    (1)\ (x + y)^{(i)} = \sum_{j+k=i} x^{(j)} y^{(k)} & x, y \in J^0,\ i \geq 0 \\
    (2)\ (\lambda x)^{(i)} = \lambda^i x^{(i)} & \lambda \in A^0,\ x \in J^0 \\
    (3)\ x^{(i)} x^{(j)} = c_{ij}\, x^{(i+j)} & x \in J^0,\ i, j \geq 0, \quad c_{ij} = \frac{(i+j)!}{i!\,j!} \\
    (4)\ (x^{(i)})^{(j)} = d_{ij}\, x^{(ij)} & x \in J^0,\ i, j \geq 1, \quad d_{ij} = \frac{(ij)!}{(i!)^j\, j!}
  \end{cases}
\]

\[
\text{(19)}\quad
\begin{cases}
\text{a) les applications}\\
\qquad s^{*} \longmapsto {}^{s}R_{*}(s^{*}) : S^{*} \longrightarrow \mathfrak{P}(S)\\
\qquad s \longmapsto R_{*}(s) : S \longrightarrow \mathfrak{P}(S^{*})\\
\qquad \text{sont injectives}\\
\text{b) L'application } \varphi_0 : S \longrightarrow S^{*} \text{ est bijection, et}\\
\qquad \text{satisfait la condition de symétrie :}\\
\qquad \text{la relation } (s, \varphi_0(t)) \in R_{*} \text{ en } (s,t) \text{ est symétrique}\\
\qquad \text{i.e. } \forall s,t \in S , \ \text{équivalence}\\
\qquad (s, \varphi_0(t)) \in R_{*} \iff (t, \varphi_0(s)) \in R_{*} .\\
\text{c) } \forall s \in S , \ (s, \varphi_0(s)) \notin R
\end{cases}
\]

\[
\begin{aligned}
  0 &\to \underset{\substack{\parallel \\ \mathbb{Z}^{*} = \{+1, -1\}}}{H^{0}(\overline{X}, \mathbb{G}_{m})}
  \to H^{0}(\overline{U}, \mathbb{G}_{m}) \to \mathbb{Z}^{S}
  \to \underset{\substack{\parallel \\ 0}}{H^{1}(\overline{X}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0\,!}}{H^{1}(\overline{U}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0}}{H^{2}_{S}(\overline{X}, \mathbb{G}_{m})} \\
  0 &\to \underset{\substack{\parallel \\ 0}}{H^{2}(\overline{X}, \mathbb{G}_{m})}
  \to H^{2}(\overline{U}, \mathbb{G}_{m})
  \to \underset{\substack{\parallel \\ (\mathbb{Q}/\mathbb{Z})^{S}}}{H^{1}(S, \mathbb{Q}/\mathbb{Z})}
  \to \underset{\substack{\parallel \\ \mathbb{Q}/\mathbb{Z}}}{H^{3}(\overline{X}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0\,?}}{H^{3}(\overline{U}, \mathbb{G}_{m})} \to 0
\end{aligned}
\]

\[
(72)\qquad
\left\{
\begin{array}{lll}
\overset{3}{\underset{\alpha_0}{\bullet}}\!\!-\!\!\!-\!\!\overset{2}{\underset{\alpha}{\bullet}} & \text{i.e.\ deux droites } (a, b, c;\ d, e) & \text{type } (0,5) \\[6pt]
\text{boucle sur } \alpha_0, \text{ de poids } 2 & \text{i.e.\ courbe nodale } (a, b) & \text{type } (1,2) \\[6pt]
\overset{}{\underset{(\alpha_0),\ g_{\alpha_0} = 1}{\bullet}}\!\!-\!\!\!-\!\!\overset{2}{\underset{g_\alpha = 0}{\bullet}} & \text{i.e.\ genre } 1 + \text{droite } (a, b) & \text{type } (1,2)
\end{array}
\right.
\]

\[
  (4)\qquad
  \begin{cases}
    \pi_{X,x}\simeq P_{X,x}\wedge^{\mathcal{M}_{g,\nu}}\hat\pi_{g,\nu}\\
    \mathfrak{S}_{X,x}\simeq P_{X,x}\wedge^{\mathcal{M}_{g,\nu}}\hat{\mathfrak{S}}_{g,\nu}\\
    \mathcal{T}_{X,x}\simeq P_{X,x}\wedge^{\mathcal{M}_{g,\nu}}\hat{\mathcal{T}}_{g,\nu}
      \simeq Q_X\wedge^{\mathcal{N}_{g,\nu}}\hat{\mathcal{T}}_{g,\nu}\\
    \mathcal{M}_{X,x}\simeq P_{X,x}\wedge^{\mathcal{M}_{g,\nu}}\mathcal{M}_{g,\nu}\\
    \mathcal{N}_X\simeq P_{X,x}\wedge^{\mathcal{M}_{g,\nu}}\mathcal{N}_{g,\nu}
      \simeq Q_X\wedge^{\mathcal{N}_{g,\nu}}\mathcal{N}_{g,\nu}\\
    \tilde{\mathbb{Q}}\simeq P_{X,x}\wedge^{\mathcal{M}_{g,\nu}}\bar{\mathbb{Q}}
      \simeq Q_X\wedge^{\mathcal{N}_{g,\nu}}\bar{\mathbb{Q}}
      \simeq R\wedge^{\Gamma_{\bar{\mathbb{Q}}/\mathbb{Q}}}\bar{\mathbb{Q}}
  \end{cases}
\]

\[
\begin{cases}
a'^{*}_{n+1} = \pi = a_n + 2a_{n-1} + \underbrace{\rho'_{n-1,n-1}}_{4 - \beta_{n-1} = 2 - \alpha_{n-1}} a_{n-2}
+ \cdots + \overbrace{\rho'_{n-1,1}}^{\sigma} a_0 + \overbrace{\rho'_{n-1,0}}^{\rho} a_{-1} \\
a'^{*}_n = -\psi'(a'_{n+1}) = a_{n-1} + 2a_{n-2}
+ \underbrace{\rho'_{n-2,n-2}}_{4 - \beta_{n-2} = 2 - \alpha_{n-2}} a_{n-3} + \cdots
+ \overbrace{\rho'_{n-2,1}}^{\gamma} a_0 + \overbrace{\rho'_{n-2,0}}^{\sigma'} a_{-1} \\
\cdots \\
a'^{*}_2 = a_1 + 2a_2 + \underbrace{\rho'_{0,0}}_{4 - \beta_0 = 2 - \alpha_0} a_{-1} \\
a'^{*}_1 = a_0 + 2a_{-1} \\
a'^{*}_0 = a_{-1}
\end{cases}
\]

\[
\begin{aligned}
& f(a_i) = (\alpha_0 + 2) \cdots (\alpha_{i-1} + 2) && 0 \leqslant i \leqslant n \\
& \varphi(a_i, a_j) = 0 && 0 \leqslant i < j \leqslant n,\ i, j \text{ non consécutifs} \\
& \varphi(a_i, a_{i+1}) = -(\alpha_0 + 2) \cdots (\alpha_i + 2) \\
& \varphi(s_0, a_i) = 0 && 1 \leqslant i \leqslant n \\
& \varphi(s_0, a_0) = -1 \\
& \varphi(s_j, a_i) = 0 && 0 \leqslant j < i \leqslant n \\
& \varphi(s_j, a_j) = -(\alpha_0 + 2) \cdots (\alpha_{j-1} + 2) && 0 \leqslant j \leqslant n \\
& \varphi(s_j, a_{j-1}) = +(\alpha_0 + 2) \cdots (\alpha_{j-2} + 2) && 1 \leqslant j \leqslant n+1 \\
& \varphi(s_j, a_i) = 0 && 0 \leqslant i < j-1 \leqslant n
\end{aligned}
\]

\[
\begin{array}{ll}
s_0 = (0, 0) & u_0 = (1, 0) \\
s_1 = (1, 0) & u_1 = (x - 1, y) \\
{[s_4 = (0, f_0)]} & u_2 = (x' - x, y' - y) \\
s_2 = (x, y) & u_3 = (-x', f_0 - y') \\
s_3 = (x', y') & u_4 = (0, -f_0)
\end{array}
\qquad
\begin{aligned}
f_1 &= y \\
f_2 &= (x - 1)(y' - y) - y(x' - x) \\
 &= (xy' - yx') - (y' - y) \\
f_3 &= (x' - x)(f_0 - y') + x'(y' - y) \\
 &= (xy' - yx') + f_0 (x' - x) \\
f_4 &= f_0 x' \\
f_0 &= f_0
\end{aligned}
\]

\[
(41'') \qquad
\begin{array}{l}
\text{(41) signifie que pour deux rayons } D, D' \in S^{\flat} \text{ incidents} \\
\text{(i.e. tels qu'il existe } (s,s') \in D \times D' \text{ avec } (s,s') \in A_0\text{),} \\
\forall s \in D, \ \exists!\, s' \in D' \text{ adjacent à } s \text{ --- cette bijection } D \simeq D' \\
\text{est un \emph{anti}-isomorphisme pour les structures de torseur,} \\
\text{donc établit un iso.} \\
\qquad D \wedge^{Z} D' \simeq \mathbf{1}_Z
\end{array}
\]

\[
\begin{aligned}
\pi^3(x_1, \dots, x_n) = \ &1 + \alpha_0(1+\alpha_0) + \alpha_0(\alpha_0^2 - 1)x_1 - \alpha_0^2 x_2 + \alpha_0(a\alpha_0 + a - b)\Sigma_1\\
&-1\ \struck{\ill{}}\quad - \alpha_0 x_1 \quad + x_2 \quad - (a+b)\Sigma_1 + a\Sigma_1\\
&\uncertain{(1+\alpha_0) + (\alpha_0^2-1)x_1 - \alpha_0 x_2 + [a(\alpha_0 + 1)]\Sigma_1,\ x_3 \ldots}\\
= \ &\struck{(1+}\ \alpha_0(1+\alpha_0) + \alpha_0(\alpha_0^2 - 2)x_1 + (1 - \alpha_0^2)x_2 + \struck{(a\alpha_0^2 + a\alpha_0 \ldots}\\
&(\alpha_0+1)(a\alpha_0 - b)\ \struck{\ill{}}\ \Sigma_1,\ \ldots
\end{aligned}
\]

\[
\begin{array}{l|c|c|c|c|c}
E_{2}^{i,0} & 2(i-n+1) & 2(i-n+1)+1 & \cdots & & i \\
 & 0 & 1 & 2 \;\cdots & & 2(n-1)-i \\
\hline
E_{2}^{i-1,1} & 2(i-n)+2 & 2(i-n)+3 & \cdots & i & i+1 \\
 & 0 & 1 & \cdots & 2n-i-2 & 2n-i-1 \\
\hline
E_{2}^{i-2,2} & 2(i-n)+4 & 2(i-n)+5 & \cdots & i \;\; i+1 & i+2 \\
 & 0 & 1 & \cdots & 2n-i-2 \;\; 2n-i-3 & 2n-i-2 \\
\hline
E_{2}^{i-3,3} & 2(i-n)+6 & 2(i-n)+7 & \cdots & i \;\; i+1 \;\; i+2 & i+3 \\
 & 0 & 1 & \cdots & 2n-i-6 \;\; \ldots & 2n-i-3 \\
\hline
E_{2}^{0,i} & 2i & & & & \\
 & 0 & & & &
\end{array}
\]

\begin{align*}
\mathbb{R}\mathcal{H}om\bigl(\mathbb{R}i_!(F'),G^{\cdot}\bigr)&\simeq\mathbb{R}i_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(F',\mathbb{R}i^!(G^{\cdot})\bigr)\Bigr)
&&\text{(dualité pour $i$)}\\
\mathbb{L}f^{*}(\ \cdots\ )&\simeq\mathbb{L}f^{*}\,\mathbb{R}i_{*}(\ \cdots\ )\\
&\simeq\mathbb{R}j_{*}\,\mathbb{L}f'^{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(F',\mathbb{R}i^!(G^{\cdot})\bigr)\Bigr)
&&\text{(chang.\ de base lisse)}\\
&\simeq\mathbb{R}j_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(f'^{*}(F'),f'^{*}(\mathbb{R}i^!(G^{\cdot}))\bigr)\Bigr)\\
&\simeq\mathbb{R}j_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(F'_{X'},\mathbb{R}j^!G^{\cdot}_{X}\bigr)\Bigr)\\
&\simeq\mathbb{R}\mathcal{H}om\bigl(j_!(F'_{X'}),G^{\cdot}_{X}\bigr)
&&\text{(dualité pour $j$)}\\
&\simeq\mathbb{R}\mathcal{H}om\bigl(f^{*}(F_{X}),f^{*}(G)\bigr)
\end{align*}

\[
\left.
\begin{array}{l}
  \alpha_i \text{ bijectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \operatorname{Supp} E^{r-i} \subset Z \text{ pour } i \leq n \\
  \Updownarrow \\
  \operatorname{prof}(F_x) > n \text{ pour tt } x \text{ fermé dans } U \\
  \Updownarrow \\
  H^i(U, F(-m)) \leftarrow H^i(X, F(-m)) \text{ pour } m \text{ grand},\ i < n \text{ et} \\
  H^n(U, F(-m)) \to H^{n+1}_Z(X, F(-m)) \text{ injectif}
\end{array}
\right.
\]

\begin{align*}
  &\simeq s^{k'}\{\{X_0,\dots,X_{n-1}\}\}[dX_0,\dots,dX_{n-1}]\\
  &\simeq \prod\bigl(s^{k'}\bigr)^{\mathbb{N}^n\times\mathfrak{P}([0,n-1])}
  \qquad\text{comme foncteur en } k'\\
  &\simeq \operatorname{Hom}_k(s,k')^{\mathbb{N}^n\times\mathfrak{P}([0,\dots])}
  \qquad\text{comme foncteur en } k'\\
  &\simeq \operatorname{Hom}_k\bigl(s^{(\mathbb{N}^n\times\mathfrak{P}([0,\dots]))},k'\bigr)\\
  &= V\bigl(s^{(\mathbb{N}^n\times\mathfrak{P}([0,\dots]))}\bigr)(k')\\
  &= V\bigl(s\otimes_k k[Y_0,\dots,Y_{n-1}]\otimes_k
     {\textstyle\Lambda^{*}}(y_0,\dots,y_{n-1})\bigr)(k')
\end{align*}

\[
  \bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, 6\bigr) \quad
  \bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, [\ldots]\bigr) \quad
  \bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
  \bigl(\begin{smallmatrix}3\\2\\1\end{smallmatrix}, 2_3, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)
\]

\[
(73)\qquad
\left\{
\begin{array}{ll}
\overset{3}{\underset{\alpha_0}{\bullet}}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \Gamma = \mathfrak{S}_3 \times \mathfrak{S}_2, \ \Gamma^{!} = 1, \ \operatorname{Im}(\Gamma \to \mathfrak{S}_4) \simeq \mathfrak{S}_3 \\[6pt]
\text{boucle sur } (\alpha_0), \text{ de poids } 2 & \Gamma \simeq \mathfrak{S}_2 \times \mathfrak{S}_2, \ \Gamma^{!} \simeq \mathfrak{S}_2, \ \operatorname{Im}(\Gamma \to \mathfrak{S}_4) \simeq \Gamma \\ & \qquad = \mathfrak{S}_2 \times \mathfrak{S}_2 \\[6pt]
\underset{g_{\alpha_0} = 1}{\bullet}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \Gamma \simeq \mathfrak{S}_2, \ \Gamma^{!} \simeq 1
\end{array}
\right.
\]

\[
(3)\quad \left\lbrace
\begin{aligned}
\mathrm{sn}\,u &= u - \frac{2\alpha k}{3!}u^3 + \frac{4k^2(\alpha^2+3)}{5!}u^5
- \frac{8k^3(\alpha^3+33\alpha)}{7!}u^7 + \cdots\\
\mathrm{cn}\,u &= 1 - \frac{u^2}{2!} + \frac{1+4k^2}{4!}u^4
- \frac{1+44k^2+16k^4}{6!}u^6 + \cdots\\
\mathrm{dn}\,u &= 1 - \frac{k^2u^2}{2!} + \frac{k^2(4+k^2)}{4!}u^4
- k^2\,\frac{16+44k^2+k^4}{6!}u^6 + \cdots
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
B' &= G_{s_0} = \{ g \in G \mid g s_0 = s_0 \} \\
\tilde U &= B' \cap G_0 = \{ g \in G_0 \mid g s_0 = s_0 \} = \{ b' \in B' \mid \delta(b') = 1 \} \\
U &= B' \cap G_0^+ = \{ g \in G_0^+ \mid g s_0 = s_0 \} = \{ b' \in B' \mid \tilde\delta(b') = 1 \} \\
\tilde B'' &= \{ g \in G_0 \mid g s_0 \in Z.s_0 \} \\
B'' &= \tilde B'' \cap G_0^+ \qquad [\mathrm{NB}\ \ \tilde B'' = \tilde U . B'']^{!} \\
B &= B' . B'' \ (= B' . \tilde B'')^{!}
\end{aligned}
\right. \tag{13}
\]

\[
\left\{
\begin{aligned}
&\text{a)}\ \sigma_0^{2} = \tau_0^{2} = 1, \quad \tau_0\sigma_0 =
\sigma_0\tau_0 \quad (\text{i.e.\ } (\underbrace{\sigma_0\tau_0}_{\tilde\sigma_0})^{2} = 1)\\
&\text{b)}\ \rho^{3} = 1\\
&\underbrace{\sigma_0\rho\sigma_0^{-1}\rho^{-1}}_{\uncertain{\sigma_1}}
= \underbrace{\rho^{2}\tau_0\rho^{-2}}_{\tau_\infty}\,
\underbrace{\rho\tau_0\rho^{-1}}_{\tau_1}\,\rho
\quad \text{i.e.} \quad \sigma_0\rho\sigma_0^{-1}\rho^{-1} =
\rho^{-1}\tau_0\rho\tau_0\\
&\rho\sigma_0\rho^{-1}\sigma_0 = \tau_0\rho^{-1}\tau_0\rho
\quad \text{\uncertain{soit}} \quad \struck{\operatorname{int}(\rho)\sigma_0}\
[\rho, \sigma_0] = [\tau_0, \rho^{-1}]\\
&\text{\uncertain{soit}} \quad [\rho, \sigma_0][\rho^{-1}, \tau_0] = 1
\end{aligned}
\right.
\]

\[
\begin{array}{l|c|c|c}
n & 4 & 5 & 6 \\ \hline
(n-1)(n-2) & 6 & 12 & 20 \\ \hline
\text{meilleur exposant pour trivialiser } L & 3 & 12\,? & 20\,? \\ \hline
\text{meilleur exposant pour trivialiser } L_i & 6 & 12\,? & 60\,? \\ \hline
\text{exposant pour trivialiser } \text{\struck{$H^1(M_{0,I}, \mathfrak{S}_I;\, \mathbb{G}_m)$}} & & & \\ \hline
\text{dividende (?)} \ H^1(M_{0,I}, \mathfrak{S}_{I \setminus \{i\}};\, \mathbb{G}_m) & & &
\end{array}
\]

\[
  \begin{aligned}
    &\text{si } i \neq 2j, 2j+1, && H^{i}(X, \mathbb{Q}_{\ell}(j)) = 0 \\
    &\text{si } i = 2j, && H^{2j}(X, \mathbb{Q}_{\ell}(j)) = H^{2j}(\overline{X}, \mathbb{Q}_{\ell}(j))^{\pi} \underset{\text{conj.\ de Tate}}{\simeq} a^{j}(X/k) \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell} \\
    &\text{si } i = 2j+1, && H^{2j+1}(X, \mathbb{Q}_{\ell}(j)) = H^{2j}(\overline{X}, \mathbb{Q}_{\ell}(j))_{\pi} \underset{\text{Tate} + \text{action semi-simple}}{\simeq} a^{j}(X/k) \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell}
  \end{aligned}
\]

\[
  \left.
  \begin{array}{l}
    \text{Géom.\ de drapeaux } D \\
    + \text{« fonction dimension »} \\
    \delta : \Phi \longrightarrow I \\
    \quad \text{\scriptsize él.\ min.}
  \end{array}
  \right|
  \Longleftrightarrow
  \left|
  \begin{array}{l}
    \text{foncteur } D' : \mathfrak{P}_f^*(I)^\circ \to (\mathrm{Ens}) \\
    \text{\emph{NB} Posant } I_0 = \lbrace i \in I \mid D'(\lbrace i
    \rbrace) \neq \emptyset \rbrace \\
    \text{on a sur } I_0 \text{ une structure } \dots
  \end{array}
  \right.
\]

\[
\begin{array}{lcl}
\widetilde{(\mathrm{Aff}_k)} \simeq \underline{\operatorname{Hom}}^{\sim}(\mathrm{Alg}_k, \mathrm{Ens}) & & \\
\widetilde{(\mathrm{Sch})_k} \xrightarrow{\ \mathrm{restr.}\ } \widetilde{\mathrm{Aff}_k} \simeq \underline{\operatorname{Hom}}^{\sim}(\mathrm{Alg}_k, \mathrm{Ens}) & & \text{(faisceaux pour la top. « de Zariski »)} \\
\qquad \downarrow \text{inc.} \qquad\qquad \downarrow \text{inc.} & & \\
\widehat{(\mathrm{Sch})_k} \xrightarrow{\ \mathrm{restr.}\ } \widehat{\mathrm{Aff}_k} \simeq \underline{\operatorname{Hom}}(\mathrm{Alg}_k, \mathrm{Ens}) & & \text{(pas plein\uncertain{fid.})} \\
\widehat{(\mathrm{Aff}_k)} \simeq \underline{\operatorname{Hom}}(\mathrm{Alg}_k, \mathrm{Ens}) & &
\end{array}
\]

\[
(6)\ \left\lbrace
\begin{aligned}
\mathrm{cn}_1 z &= \frac{\sigma\omega_1}{\sigma\omega_2}\,e^{(\eta_2-\eta_1)z}\,
\frac{\sigma(z-\omega_2)}{\sigma(z-\omega_1)}
= \frac{\sigma\omega_1}{\sigma\omega_2}\,e^{-(\eta_2-\eta_1)z}\,
\frac{\sigma(z+\omega_2)}{\sigma(z+\omega_1)}\\[4pt]
\mathrm{dn}_1 z &= \frac{\sigma\omega_1}{\sigma\omega_3}\,e^{(\eta_3-\eta_1)z}\,
\frac{\sigma(z-\omega_3)}{\sigma(z-\omega_1)}
= \frac{\sigma\omega_1}{\sigma\omega_3}\,e^{-(\eta_3-\eta_1)z}\,
\frac{\sigma(z+\omega_3)}{\sigma(z+\omega_1)}
\end{aligned}
\right.
\]

\[
(39) \qquad
\begin{array}{c}
R(s^{\flat}, s^{\vee\flat}) \\
\uparrow \ (\lambda,\mu) \mapsto (\lambda s_0, \mu s_0^{\vee}) \\
E(s_0, s_0^{\vee}) \overset{\mathrm{def}}{=} \Bigl\{(\lambda,\mu) \in Z \times Z \ \Bigm|\ 
\begin{array}{l}
(\lambda s_0, \mu \check{s}_0) \in R \text{ i.e.} \\
((\mu^{-1}\lambda) s_0, \check{s}_0) \in R \text{ i.e.} \\
\mu^{-1}\lambda s_0 \in \underbrace{E(\check{s}_0)}_{\text{étoile dans $S$ définie par } \check{s}_0}
\end{array}
\Bigr\} .
\end{array}
\]

\begin{align*}
  &=\kappa_{n-1}\Bigl(\sum_{\lambda+\sum_0^{n-2}l_\alpha=i_{n-1}+1}
    (-1)^{\sum l_\alpha}c_{i_0,l_0}c_{i_1,l_1}\cdots c_{i_{n-2},l_{n-2}}\\
  &\qquad\qquad t^{(\lambda)}X_0^{(i_0+l_0)}\cdots X_{n-2}^{(i_{n-2}+l_{n-2})}\,
    dX_0\wedge\dots\wedge dX_{n-2}\Bigr)\\
  &=t^{(n+\sum i_\alpha)}\Bigl[(-1)^{i_{n-1}+1}
    \sum_{\lambda+\sum l_\alpha=i_{n-1}+1}(-1)^{\sum l_\alpha}
    c_{i_0,l_0}\cdots c_{i_{n-2},l_{n-2}}\\
  &\qquad\qquad c_{\lambda,\,n-1+\sum_0^{n-2}(l_\alpha+i_\alpha)}\,
    \gamma_{i_0+l_0,\dots,i_{n-2}+l_{n-2}}\Bigr]
\end{align*}

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

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

\[
\begin{cases}
\delta'_{2m}(B_0, \ldots, B_{2m-1}) \equiv
\;
\underbrace{(B_0 B_1)^{2m} (B_2 B_3)^{2(m-1)} \cdots (B_{2m-2} B_{2m-1})}_{[(B_0 B_1)^m (B_2 B_3)^{m-1} \cdots (B_{2m-2} B_{2m-1})]^2} \\[2ex]
\delta'_{2m+1}(B_0, \ldots, B_{2m-1}) \equiv B_0^{2m} B_1^{2m-1} \cdots B_{2m-1}
\bigl[B_0 B_2 \cdots B_{2m-2} + B_1 B_3 \cdots B_{2m-1} \\
\qquad\qquad + \sum_{1 \leqslant i \leqslant m-1} (B_0 \cdots B_{2i-2})(B_{2i+1} \cdots B_{2m-1})\bigr]
\end{cases}
\]

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

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

\[
  (10) \qquad
  \begin{cases}
    X_{\Delta_r} = \bigl\{ (x_0, \ldots, x_r) \in X_{\Delta_0}^{\,r+1}
      \ \big|\ x_0 \leq \cdots x_r \bigr\} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \bigl\{ (x, i_0, \ldots, i_r)
      \in X \times I^{\,r+1} \ \big| \\
    \qquad\qquad x \in X_{i_0},\ i_0 \leq \cdots \leq i_r \bigr\} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \coprod_{i_{*} \in I(\Delta_r)} X_{i_{*}} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \coprod_{i_0 \in I} X_{i_0} \times I_i(\Delta_{r-1})
  \end{cases}
\]

\[
\begin{aligned}
\delta_n(B_0, \ldots, B_{n-1}) &= 2B_0\, \delta_{n-1}(B_0B_1, B_2, \ldots, B_{n-1})
- (B_0B_1)^2\, \delta_{n-2}(B_0B_1B_2, B_3, \ldots, B_{n-1}) \\
&= 2B_0 (B_0B_1)^{n-1} B_2^{n-2} B_3^{n-3} \cdots B_{n-1}\,
L_{n-1}(B_2, B_3, \ldots, B_{n-1}) \\
&\qquad - B_0^2 B_1^2 (B_0B_1B_2)^{n-2} B_3^{n-3} \cdots B_{n-1}\,
L_{n-2}(B_3, \ldots, B_{n-1}) \\
&= B_0^n B_1^{n-1} \cdots B_{n-1}\,
\bigl[2L_{n-1}(B_2, \ldots, B_{n-1}) - B_1 L_{n-2}(B_3, \ldots, B_{n-1})\bigr]
\end{aligned}
\]

\[
  \begin{cases}
    c^{i}(1) = 0 & i \geqslant 1, \\[3pt]
    c^{n}(x+y) = \displaystyle\sum_{i+j=n} c^{i}(x)\,c^{j}(y)
      & x, y \in K,\ n \geqslant 1, \\[8pt]
    c^{n}(x \ast y) = Q_{n}\bigl(c^{1}(x), \ldots, c^{n}(x);\;
      c^{1}(y), \ldots, c^{n}(y)\bigr)
      & x, y \in I(K),\ n \geqslant 1, \\[5pt]
    c^{n}(E_{s}) = 0 & n > n_{s},
  \end{cases}
\]

\[
\begin{aligned}
\underbrace{f_V^{0}}\Bigl(\sum_{0}^{n} x_i a_i\Bigr)
&= \sum_{i=0}^{n} (\beta_0 \cdots \beta_{i-1}) x_i^{2}
+ \sum_{0 \leq i \leq n-1} (\beta_0 \cdots \beta_i) x_i x_{i+1} \\
&= x_0^{2} + \beta_0 x_1^{2} + (\beta_0 \beta_1) x_2^{2} + \cdots
+ (\beta_0 \beta_1 \cdots \beta_{n-1}) x_n^{2} \\
&\quad + \beta_0 x_0 x_1 + \beta_0 \beta_1 x_1 x_2 + \cdots
+ (\beta_0 \cdots \beta_{n-1}) x_{n-1} x_n \\
&= x_0^{2} + \beta_0 x_1 (x_0 + x_1) + (\beta_0 \beta_1) x_2 (x_1 + x_2) + \cdots \\
&\quad + (\beta_0 \cdots \beta_{n-1}) x_n (x_{n-1} + x_n).
\end{aligned}
\]

\[
(67)\qquad
\begin{cases}
\text{les } \varphi_\alpha : S \to \check{S} \text{ telles qu'on ait} \\
\quad \check{\lambda} \varphi_\alpha \lambda = \varphi_\alpha \text{ pour tout } \lambda \in Z_{\Gamma} \subset \mathrm{Aut}_S(\Gamma), \\
\text{forment un } \text{\struck{torseur sous le groupe } $Z_{\Gamma}$}\ \text{ensemble } \Pi_Z, \text{ sur} \\
\text{lequel le groupe } Z_{\Gamma} \text{ y opère \emph{librement} par l'opération} \text{\struck{\ill{}}} \\
\quad T_{\lambda, \check{\lambda}}\, \varphi = \check{\lambda} \varphi = \varphi \lambda^{-1} .
\end{cases}
\]

\[
(18) \qquad
\begin{array}{l}
\text{Si } \Gamma \text{ est} \\
\text{séparé en } S^{\vee} \\
\text{i.e. } S^{\vee} \hookrightarrow \mathfrak{P}(S) \\
\text{alors}
\end{array}
\quad
\begin{array}{l}
\lambda \mapsto \lambda_S \\
\mathcal{A} \to \mathfrak{S}_S
\end{array}
\quad
\begin{array}{l}
\text{induit un \struck{bijection} \add{isom.} du groupe} \\
\mathcal{A} \text{ avec le groupe des} \\
\text{permutations } f \text{ de } S \\
\text{telles que } f(\Sigma) = \Sigma \text{ (tr. de str.) i.e.} \\
\text{qui transforment étoile en} \\
\text{étoile,}
\end{array}
\]

\[
\begin{array}{lll}
(i, \nu) \text{ et } (j, \mu) \ (\text{distincts}) & \text{sont liés ssi} &
i = j \text{ \emph{ou} } \nu = \mu \\
(i, \nu) \text{ et } (j, \pi, \rho) & \text{liés dans les cas} &
\left\lbrace
\begin{array}{ll}
i = j : & \nu = 0 \\
i \neq j & \left\lbrace \begin{array}{l} \nu = 1, \ j = \rho i \\ \nu =
2, \ i = \rho j \end{array} \right.
\end{array}
\right. \\
(j, \pi, \rho) \text{ et } (j', \pi', \rho') \ \text{distincts} & \text{liés
dans les cas} &
\left\lbrace
\begin{array}{ll}
j = j' & \pi = \pi', \ (\rho \neq \rho') \\
j \neq j' & \left\lbrace \begin{array}{ll} i \neq j, & \rho = \rho' \\ i =
j & \rho \neq \rho' \end{array} \right.
\end{array}
\right.
\end{array}
\]

\[
\begin{array}{c|c|c|l}
n & \text{ordre } \Gamma^0_{n,3} & \text{ordre } \mathrm{Sl}(2,\mathbb{Z}_n)/\pm 1 & \\ \hline
2 & (\mathfrak{S}_3)\ 6 & 6 & \text{cas 3-diédral} \\
3 & (\mathfrak{A}_4)\ 12 & 12 & \text{tétraèdre} \\
4 & (\mathfrak{S}_4)\ 24 & 24 & \text{cube orienté} \\
5 & (\mathfrak{A}_5)\ 60 & 60 & \text{icos. orienté} \\
6 & (\mathbb{Z}_6\cdot\mathbb{Z}^2)\ \infty & 72 & \text{pavage par triangles orienté du plan}
\end{array}
\]

\[
(137)\qquad
\begin{cases}
T_0=\left\{g\in\mathfrak{S}_S \;\middle|\;
\begin{array}{l}
g \text{ normalise } H\subset\mathfrak{S}_S\\
g \text{ centralise } Z\subset\mathfrak{S}_S\\
gs_0=s_0\\
gt_0\in Z.t_0
\end{array}\right\}\\[4ex]
T_1=\left\{g\in\mathfrak{S}_S \;\middle|\;
\begin{array}{l}
g \text{ normalise } H\\
g \text{ centralise } Z\\
gt_0=t_0\\
gs_0\in Z.s_0
\end{array}\right\}
\end{cases}
\]

\[
  \left\lbrace
  \begin{array}{l}
    \mathbb{F}_2 = \mathfrak{Z}(\mathbb{D}_4) = \mathfrak{Z}(\mathbb{D}_4, \mathfrak{S}_4) \\
    \mathbb{D}_4 = \mathcal{N}(\mathbb{F}_2, \mathfrak{S}_4) \\[4pt]
    \mathbb{F}_2 = 2\cdot\mathbb{Z}_4 = {}_2(\mathbb{Z}_4) \\
    \mathbb{D}_4 = \mathcal{N}(\mathbb{Z}_4, \mathfrak{S}_4) \\[4pt]
    \mathbb{Z}_4 = \text{groupe engendré par \ill{} \ill{} des 2 él.\ d'ordre 4 de } \mathbb{D}_4 \\[4pt]
    \mathfrak{S}_4 = \mathcal{N}(\mathbb{F}_2^{2}, \mathfrak{S}_5) \\
    \mathbb{F}_2^{2} = \text{seul sous-groupe invariant de } \mathfrak{S}_4 \text{ d'indice } 6
  \end{array}
  \right.
\]

\[
  \left\lbrace
  \begin{array}{l}
    A(t) = L^{\infty}_E\Bigl(\dfrac{1}{p^{n+\rho}\, t}\Bigr)^{-1} \\[8pt]
    \chi(E) = \sum (-1)^i \operatorname{rang} H^i_!(\bar X, \bar F) \\[6pt]
    \delta(E) = \prod_i \bigl(\det f_{H^i_!(\bar X, \bar F)}\bigr)^{(-1)^i}
  \end{array}
  \right.
\]

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

\[
\begin{array}{ccc}
\mathrm{Hom}_{A^{\wedge}}\bigl(a, \Gamma''_M(X)\bigr) & \simeq &
\mathrm{Hom}_{A^{\wedge}}\bigl(a, p_{B*}\, \underline{\mathrm{Hom}}(p_A^{*}(M), X)\bigr)\\
\| & & \wr\\
\Gamma''_M(X)(a) & &
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a), \underline{\mathrm{Hom}}(p_A^{*}(M), X)\bigr)\\
\| & & \wr\\
\mathrm{Hom}_{B^{\wedge}}\bigl(M, X''(a)\bigr) & &
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a) \times p_A^{*}(M), X\bigr)\\
\wr & &\\
p_{A*}\bigl(\underline{\mathrm{Hom}}(p_B^{*}(a), X)\bigr) & &\\
\| & &\\
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_A^{*}(M), \underline{\mathrm{Hom}}(p_B^{*}(a), X)\bigr) & &\\
\| & &\\
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_A^{*}(M) \times p_B^{*}(a), X\bigr) & &
\end{array}
\]

\[
\begin{aligned}
  H^{1}(\pi, \mathbb{Z}_{\ell}(n)) &\underset{\text{non canoniquement}}{\simeq}
  \mathbb{Z}_{\ell}/(1-q^{n})\mathbb{Z}_{\ell} \\
  &= \begin{cases}
    \mathbb{Z}_{\ell} & \text{si } n = 0 \\
    0 & \text{si } \ell \nmid (1-q^{n}) \ \text{i.e. } n \not\equiv 0 \ (w_{\ell}(q)) \\
    \mathbb{Z}/\ell^{v_{\ell}(1-q^{n})} & \text{si } \ell \mid 1-q^{n},\ n \neq 0 \ \text{i.e. } n \equiv 0 \ (w_{\ell}(q))
  \end{cases}
\end{aligned}
\]

\begin{align*}
L_{x_1 \dots x_{n+1}} &= L_{(x_1 \dots x_{n-1}) x_n x_{n+1}} \\
&= L_{x_1 \dots x_n}\, L_{x_1 \dots x_{n-1} x_{n+1}}\, L_{x_n x_{n+1}}\,
L_{x_1 \dots x_{n-1}}^{-1} L_{x_n}^{-1} L_{x_{n+1}}^{-1} L_{\emptyset} \\
&= \prod_{1 \leq i < j \leq n} L_{x_i x_j} \prod_{1 \leq i \leq n} L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n}
\prod_{\substack{1 \leq i < j \leq n+1 \\ i, j \neq n}} L_{x_i x_j}
\prod_{\substack{1 \leq i \leq n+1 \\ i \neq n}} L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n} \\
&\qquad L_{x_n x_{n+1}} \prod_{1 \leq i < j \leq n-1} L_{x_i x_j}^{-1}
\prod_{1 \leq i \leq n-1} L_{x_i}^{+(n-3)} L_{\emptyset}^{-\alpha_{n-1}}\,
L_{x_n}^{-1} L_{x_{n+1}}^{-1} L_{\emptyset}
\end{align*}

\[
(6)\quad \left\lbrace
\begin{aligned}
\Theta(v|t) &= -\frac{1}{i}\sum_{-\infty}^{\infty}(-1)^n\, h(t)^{(n-\frac12)^2}
e^{(2n-1)i\pi v}\\
&= 2\sum_{1}^{\infty}(-1)^{n-1}\, h(t)^{(n-\frac12)^2}\sin(2n-1)\pi v\\
&= 2\bigl[h(t)^{\frac14}\sin\pi v - h(t)^{\frac94}\sin 3\pi v + \cdots\bigr]
\end{aligned}
\right.
\]

\[
\begin{array}{ll}
H_0 : & 2x_1 \struck{=} - \alpha_0(x_2 + \cdots + x_{n+1}) = z\\
D_0 : & x_2 = x_3 = \cdots = x_{n+1} = z = 0 \qquad (\text{pt à l'infini de la } \struck{\ill{}}\ \mathrm{dr}(s_0 s_1))\\[4pt]
H_i : & x_i - x_{i+1} + (1+\alpha_i)(\struck{x_{i+2} =}\ x_{i+2} + \cdots + x_{n+1}) = 0\\
D_i : & x_1 = \cdots = x_{i-1} = x_{i+2} = \cdots = x_{n+1} = z = 0
\end{array}
\]

\[
(57)\quad
\begin{cases}
\beta = \sigma_\infty(\eta)\eta^{-1} & (u(b_i^{\omega}) = b_i^{\omega} \varphi^{\omega}_{\omega i}(\beta)) \\
\alpha = \sigma_\infty(\lambda)\beta & (u(\tilde b_i^{\omega}) = \tilde b_i^{\omega} \varphi^{\omega}_{\omega i}(\alpha)) \\
\tilde\alpha = \eta^{-1}\varepsilon_0(\lambda\eta) = \zeta\rho^{-1}(\zeta)^{-1} & (u(\tilde a_i^{\omega}) = \psi^{\omega}_{\omega i}(\tilde\alpha)\, \tilde a_i^{\omega})
\end{cases}
\]

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

\[
  (23) \qquad
  \begin{cases}
    \text{a) } X_{\Delta_1} \text{ est un sous-espace top. fermé} \\
    \qquad \text{de } X_{\Delta_0} \times X_{\Delta_0}
      \text{ (avec la top. induite)} \\[2pt]
    \text{b) } \forall (i,j) \in I \times I,\
      \Gamma_{ij} \underset{\text{déf}}{=}
      X_{\Delta_1} \cap (X_i \times X_j) \\
    \qquad \text{est ou bien vide, ou bien le graphe} \\
    \qquad \text{d'une immersion fermée } X_i \hookrightarrow X_j .
  \end{cases}
\]

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

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

\[
\boxed{
\begin{aligned}
t_{n+1} = \Theta(s'_{n+1}) &= -\sigma a'^{*}_n + \gamma\pi = -\sigma a'^{*}_n + \gamma a'^{*}_{n+1} \\
&= \gamma a_n + (\uncertain{2\gamma} - \sigma)\, a_{n-1} + (\rho'_{n-1,n-1}\gamma - 2\sigma)\, a_{n-2} + \cdots \\
&\quad + (\rho'_{n-1,n-1-j}\gamma - \rho'_{n-2,n-1-j}\sigma)\, a_{n-j-2} + \cdots \\
&\quad + (\rho'_{n-1,2}\gamma - \rho'_{n-2,2}\sigma)\, a_1 + 0 \cdot a_0 - \beta a_{-1}
\end{aligned}}
\]

\begin{align*}
  (\widetilde{v \circ u})(x)
  &\overset{\text{déf.\ de }\sim}{=} \struck{\ill{}}\; \beta''_{*}\bigl[(v \circ u)\, \beta^{*}(x)\bigr] \\
  &\overset{\text{déf.\ de }v \circ u}{=} \beta''_{*}\bigl[\underbrace{p_{31*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\bigr) \beta^{*}(x)}_{=\ \text{formule de proj.}}\bigr] \\
  &= \beta''_{*}\, p_{31*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\, \underbrace{p_{31}^{*} \beta^{*}(x)}_{p_{1}^{*}(x)}\bigr) \\
  &\overset{\text{transitivité}}{=} p_{3*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\, p_{1}^{*}(x)\bigr) \\
  &\struck{\overset{\text{projection}}{=} p_{3*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\bigr) x}
\end{align*}

\[
  (33)\quad
  \left\{
  \begin{array}{l}
    \varepsilon_0 \overset{\mathrm{def}}{=} \sigma_\infty \rho, \quad
    \varepsilon_1 \overset{\mathrm{def}}{=} \rho(\varepsilon_0)
      \ (= \rho\, \varepsilon_0 \rho^{-1}), \\[4pt]
    \varepsilon_\infty \overset{\mathrm{def}}{=} \rho(\varepsilon_1)
      = \rho^2(\varepsilon_0) = \rho^{-1}(\varepsilon_0) \\[4pt]
    l_0 \overset{\mathrm{def}}{=} \varepsilon_0^2, \quad
    l_1 \overset{\mathrm{def}}{=} \varepsilon_1^2, \quad
    l_\infty \overset{\mathrm{def}}{=} \varepsilon_\infty^2, \quad
    l_\infty l_1 l_0 = 1 \\[4pt]
    \rho(l_i) = l_{\rho(i)}
      \ \ \{\rho(l_0) = l_1,\ \rho(l_1) = l_\infty,\ \rho(l_\infty) = l_0\}, \quad
    \rho(\varepsilon_i) = \varepsilon_{\rho(i)}
  \end{array}
  \right.
\]

\[
    \left\{
    \begin{array}{l}
      \dim \mathfrak{g}_\alpha = 1 \\
      \exists\,! \; x_\alpha : \mathbf{G}_a \longrightarrow Z_\chi \subset G,
        \quad x_\alpha(\mathbf{G}_a) = U_\alpha, \;
        \mathrm{Lie}(U_\alpha) = \mathfrak{g}_\alpha \\
      \phantom{\exists\,! \;} t\,x_\alpha(\lambda)\,t^{-1} =
        x_\alpha\bigl(\alpha(t)\lambda\bigr) \\
      \rho_\alpha : \mathbf{G}_m \longrightarrow T, \quad
        f\bigl(x_\alpha(\lambda)\bigr) = \rho_\alpha(1+\lambda), \quad
        \langle \rho_\alpha, \alpha \rangle = 1
    \end{array}
    \right.
  \]

\[
(48.7) \qquad
\left\{
\begin{array}{l}
\text{a) une \struck{\ill{}} \ill{} $S^{\flat}$, une relation symétrique } R^{\flat} \text{ (entre } p_1, p_2 : R^{\flat} \rightrightarrows S^{\flat}) \\
\qquad \text{(i.e. un \emph{graphe} } \gamma^{\flat}) \\
\text{b) un $Z$-torseur relatif } S/S^{\flat} \text{ sur } S^{\flat} \\
\text{c) un isomorphisme} \\
\qquad p_1^{*}(S/S^{\flat}) \wedge p_2^{*}(S/S^{\flat}) \simeq \underbrace{\mathbf{1}_{Z_{S^{\flat}}}}_{\text{torseur trivial}} .
\end{array}
\right.
\]

\[
  \Sigma_\nu
  \begin{cases}
    T = \hat{\mathbb{Z}} \\
    \Pi = \text{groupe profini } \Pi_\nu = \widehat{L_\nu}, \text{ où }
      L_\nu = \{\lambda_0, \ldots, \lambda_\nu \mid \lambda_0 \lambda_1 \cdots \lambda_\nu = 1\} \\
    \qquad \text{(donc $\Pi$ profini libre : $\nu$ générateurs)} \\
    I = [0, \nu]_{\mathbb{N}} \\
    \Phi_i = \text{classe de l'hom. } \hat{\mathbb{Z}} \xrightarrow{\varphi_i} \Pi
      \text{ tel que } \varphi_i(1) = \lambda_i
  \end{cases}
\]

\[
(18)\qquad \left\{\begin{array}{l}
(S,F_0) \text{ équivalente à } (S,F_1), \text{ \struck{si il existe} s'il existe}\\
\text{\struck{$\lambda$}}\ \lambda\in Z_{F_0} \text{ tel que } F_1=\lambda(F_0), \text{ i.e.\ \struck{s'il existe}}\\
\quad \text{a) } (S,F_1) \text{ isom.\ à } (S,F_0)\\
\quad \text{b) } \exists\lambda\in\mathfrak{S}_S, \text{ telle que } \lambda \text{ commute à}\\
\qquad G_0=\mathrm{Aut}(S,F_0), \text{ et que l'on ait}\\
\qquad \{s,t\}\in A_{F_0} \Longleftrightarrow \{s,\lambda t\}\in A_{F_1}.
\end{array}\right.
\]

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

\[ \struck{(18)} \quad \begin{cases}
\underline{a}\, s_0 = \bigl(1, -\frac{1}{\alpha}, -\frac{1}{\alpha}\bigr) = (1, -1-\alpha, -1-\alpha) = (1, \alpha', \alpha') \\
\underline{a}\, s_1 = \bigl(0, -\frac{1}{\alpha}, -\frac{1}{\alpha}\bigr) = (0, -1-\alpha, -1-\alpha) = (0, \alpha', \alpha') \\
\underline{a}\, s_2 = \bigl(1, -\frac{1}{\alpha} - 1, -\frac{1}{\alpha}\bigr) = (1, -2-\alpha, -1-\alpha) = (1, \alpha' - 1, \alpha') \\
\underline{a}\, s_3 = \bigl(1, -\frac{1}{\alpha}, -\frac{1}{\alpha} - 1\bigr) = (1, \uncertain{-1-\alpha}, -2-\alpha) = (1, \alpha', \alpha' - 1) \\
\underline{a}\, s'_3 = \bigl(2 + \alpha, -2(1+\alpha), -(2+\alpha)\bigr) = (1 - \alpha', -2\alpha', \alpha' - 1) \\
\underline{a}\, s''_3 = \bigl(1, -2(1+\alpha), -(2+\alpha)\bigr) = (1, 2\alpha', \alpha' - 1)
\end{cases} \]

\[
\text{(19)}\quad
\underset{\substack{\text{groupoïde des modules libres de}\\ \text{rang 2 sur } \Lambda_n\ \text{\struck{($n\in\mathbb{N}$, …)}}}}{\phantom{x}}
\xrightarrow{\ \approx\ }
\underset{\substack{\text{groupoïde des structures } (S,F,\Pi,\tau,a)\\ \text{où } F\subset\mathfrak{P}_3(S) \text{ une str. triangulée sur } S,\\ \text{décomposée en famille \add{disjointe} indexée par } \Pi\\ \text{(i.e. on a } \tau:F\to\Pi \text{ surj.),}\\ a \text{ est un automorphisme involutif}\\ \text{(antipodisme) de la structure précédente}}}{\phantom{x}}
\]

\[
\left\{
\begin{aligned}
\sigma_0(\beta)^2 &= \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = u(-1) \overset{\mathrm{déf}}{=} \omega \qquad (\text{élément \emph{central} de } H) \\
\sigma_0(\beta)\bigl(u(\lambda)\bigr) &= u(\lambda^{-1}) \qquad \text{donc } \sigma(\beta) \text{ normalise } T \\
\sigma_0(\lambda\beta) &= u(\lambda)\, \sigma_0(\beta) = \sigma_0(\beta)\, u(\lambda^{-1}) ,
\end{aligned}
\right. \tag{20}
\]

\[
\begin{array}{lll}
1 \leftrightarrow \lambda_{\widetilde{P}^\omega_a}\ \text{lacet en}\ \widetilde{P}^\omega_a & & (a,\omega)\in A\times\underline{\omega}\simeq R\\
1 \leftrightarrow \lambda_{\widetilde{Q}_a}\ \text{lacet en}\ \widetilde{Q}_a & & a \in A\\
1 \leftrightarrow \lambda_{\widetilde{R}_r}\ \text{lacet en}\ \widetilde{R}_r & & r\in R\\
\mu_{\widetilde{Q}_a} : \widetilde{Q}_a \to \widetilde{Q}_a^\vee & & a\in A\\
\mu_{\widetilde{R}_r} : \widetilde{R}_r \to \widetilde{R}_r^\vee & & r\in R\\
1 \leftrightarrow \tilde d^\omega_r : \widetilde{P}^\omega_{a_1} \to \widetilde{P}^\omega_{\bullet}\ \text{ou}\ \tilde d^\omega_s & & r = (s,\omega)\in S\times\underline{\omega}\simeq R
\end{array}
\]

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

\[
(93)\quad \text{b}'')\quad
\underbrace{\overset{2}{\underset{(a_0)}{\bullet}}\!-\!\overset{1}{\underset{(a_1)}{\bullet}}\!-\!\overset{1}{\bullet}\cdots\overset{1}{\underset{(a_{k-1})}{\bullet}}\!-\!\overset{1}{\underset{a_k}{\bullet}}}_{\substack{k \text{ segments},\ k+1 \text{ sommets} \\ \text{ayant un poids marqué total} \\ 2 + k = \operatorname{card} I'}}
\!-\!
\underbrace{\overset{1}{\underset{(b_l)}{\bullet}}\!-\!\overset{1}{\underset{(b_{l-1})}{\bullet}}\cdots\overset{1}{\underset{(b_2)}{\bullet}}\!-\!\overset{1}{\underset{(b_1)}{\bullet}}\!-\!\overset{2}{\underset{(b_0)}{\bullet}}}_{\substack{l \text{ segments},\ l+1 \text{ sommets} \\ \text{poids marqué total} \\ 2 + l = \operatorname{card} I''}}
\]

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

\[
\begin{array}{lll}
  u(e_{0}) = (z \mapsto -z) = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}
    & & \text{pts fixes } 0, \infty \\[1ex]
  u(e_{1}) = (z \mapsto 1/z) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
    & & \text{pts fixes } \pm 1 \\[1ex]
  u(e_{0}+e_{1}) = (z \mapsto -1/z) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}
    & & \text{pts fixes } \pm i
\end{array}
\]

\[
\text{(158)}\quad
\left\{
\begin{array}{l}
\text{\struck{$\widetilde{A}_M = \mathrm{Rep}(M)/\text{symétrie }(x,y)\mapsto(y,x)$}}\\
\text{\struck{$\phantom{\widetilde{A}_M} = \{\{x,y\}\in\mathfrak{P}_2(M)\mid \{x,y\}\text{ base de }M\}$}}\\[4pt]
A_M = \text{\struck{$\mathrm{Rep}(M)/\pm\mathrm{id}_M$}}\ \text{\struck{$\widetilde{A}_M/\pm\mathrm{id}_M$}}\\
\phantom{A_M ={}} \{\{s,t\}\in\mathfrak{P}_2(S_M)\mid \exists\,(x,y)\in\mathrm{Rep}(M),\\
\phantom{A_M ={}\{\{s,t\}\in\mathfrak{P}_2(S_M)\mid{}} \{s,t\}=\{\dot{x},\dot{y}\}\}
\end{array}
\right.
\]

\[
(*)\left\{
\begin{aligned}
&\rho_0' = \sigma \varepsilon_0, \quad \rho_0 = \sigma^{-1}\varepsilon_0 = (\varepsilon_1 \varepsilon_0)^{-1},
\quad \rho_0' = \rho_0^{-2}, \quad \rho_0'^3 = 1, \quad \rho_0^6 = 1 \\
&\rho_1' = \sigma \varepsilon_1, \quad \rho_1 = \sigma^{-1}\varepsilon_1 = (\varepsilon_0 \varepsilon_1)^{-1},
\quad \rho_1' = \rho_1^{-2}, \quad \rho_1'^3 = 1, \quad \rho_1^6 = 1 \\
&\varepsilon_1 = \sigma(\varepsilon_0) = \rho_0(\varepsilon_0) = \rho_0'(\varepsilon_0)
\quad [= \varepsilon_1(\varepsilon_0)] \\
&\varepsilon_0 = \sigma(\varepsilon_1) = \rho_1(\varepsilon_1) = \rho_1'(\varepsilon_1)
\quad [= \varepsilon_0(\varepsilon_1)] \\
&\sigma = \varepsilon_1 \varepsilon_0 \varepsilon_1 = \varepsilon_0 \varepsilon_1 \varepsilon_0
\end{aligned}
\right.
\]

\[
  (8) \qquad
  \begin{array}{l}
    \text{Catégorie des ens. } E \text{ de} \\
    \text{cardinal } 6, \text{ munis d'une} \\
    \text{partition } (P_i)_{i \in T} \text{ du type } (3,3), \\
    \text{indexée par } T, \text{ et d'une} \\
    \struck{\text{famille}}\ \text{paire de bijections} \\
    \qquad D_i^{*} \simeq \omega_{P_i}
  \end{array}
  \quad \xrightarrow{\ \approx\ } \quad
  \begin{array}{l}
    \text{Catégorie analogue} \\
    \text{définie en termes} \\
    \text{de } T' \text{ et de} \\
    \quad (D_i^{*})_{i \in T'}
  \end{array}
\]

\[ (\mathrm{M}6) \qquad \begin{cases}
\forall\, Z, X \in \mathcal{M}_n,\ Z \subset X,\ \exists\,
(f_\alpha)_{\alpha \in A} \text{ famille finie de fonctions modérées } X \to
R \\
\text{et des } (e_\alpha)_{\alpha \in A},\ e_\alpha \in R_0, \text{ telles
que } Z = \bigcap f_\alpha^{-1}(e_\alpha) . \\
\text{De plus OPS que les } f_\alpha \text{ séparent les points de } X
\smallsetminus Z
\end{cases} \]

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

\[
  \left\lbrace
  \begin{array}{l}
    \chi(E) = \sum (-1)^i \operatorname{rang} R^i f_{*}(E) \\[6pt]
    \delta(E) = p^{+(\rho+n)\chi(E)}\, \delta\bigl(R f_{*}(E)\bigr)^{-1}
      = p^{(n+\rho)\chi(E)} \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^{i+1}}
  \end{array}
  \right.
\]

\[
(18)\quad\left\lbrace
\begin{array}{l}
\overbrace{X^{*},\ (\dot X_\beta)_{\beta\in I_0},\ (B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}}^{\text{syst. de cylindres-bords transversaux}},\ (X^{*}_i)_{i\in I^{*}}\ ;\ (X_\beta)_{\beta\in I},\ (B_{\beta,\alpha},\mathcal E_{\beta,\alpha})_{\alpha\in\Lambda,\ \beta\in I_0},\\[6pt]
(\dot X_\beta\xrightarrow{\ p_\beta\ }X_\beta)_{\beta\in I_0}
\end{array}\right\rbrace
\]

\[
(69)\qquad \varphi
\begin{cases}
\rho_1 \longmapsto \lambda_1 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \sigma \\[4pt]
\rho_\infty \longmapsto \lambda_\infty = \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix} = \rho^{-1} \\[4pt]
\rho_0 \longmapsto \lambda_0 = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} = \varepsilon_0
\end{cases}
\quad \bmod \pm 1
\]

\[
\left\lbrace
\begin{array}{l}
\mathcal{A}^{0} = \prod_{x \in X} \hat{\mathcal{O}}_{x}
\quad \text{\uncertain{d'où} \uncertain{des}}\ \pi_{x} \in \mathcal{A}_0 \quad (x \in X) \\
\qquad \text{(\uncertain{comme} \uncertain{A-algèbre} \uncertain{topologique})} \\[4pt]
\Theta \in \mathcal{A}^{1}, \quad \text{avec}\ \underline{\Theta^{2} = 0}, \quad
\underline{\pi_{x'} \Theta \pi_{x} = 0\ \text{si}\ x'\ \text{\uncertain{ne} \uncertain{précède} \uncertain{pas}}\ x}
\end{array}
\right.
\]

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

\[
\begin{array}{ll}
\operatorname{Pl}(L_1,C)=C & 27=3^3\\
\uparrow\ \operatorname{Pl}(L_2,C) & 27\cdot 16=2^4 3^3=432\\
\uparrow\ \operatorname{Pl}(L_3,C) & 27\cdot16\cdot10=2^5\cdot3^3\cdot5=4320\\
\uparrow\ \operatorname{Pl}(L_4,C)\xleftarrow{\ \approx\ }\operatorname{Pl}''(L_5,C) & 27\cdot16\cdot10\cdot6=2^6\cdot3^4\cdot5=25\,920\\
\uparrow\ \operatorname{Pl}'(L_5,C) & 27\cdot16\cdot10\cdot6\cdot2=2^7\cdot3^4\cdot5=51\,840\\
\uparrow\ \operatorname{Pl}(L_6,C) & 
\end{array}
\]

\[
\begin{cases}
D_0 = H_1 \cap H_2 : y = z = \alpha' x \\ \qquad D_0 = \mathcal{O}_S(1, \alpha', \alpha') = \mathcal{O}_S(-\alpha, 1, 1) \\
D_1 = H_2 \cap H_0 : y = z = \tfrac12 - x \ (\text{car.} \neq 2) \\ \qquad \mathrm{hom}\,D_1 : (y = z = -x) = \mathcal{O}_S(1, -1, -1) \\
D_2 = H_0 \cap H_1 : y = (1-3\alpha)x + \alpha,\ z = \ldots(1-\alpha)x + (1-\alpha) \\ \qquad \mathrm{hom}(D_2) = \mathcal{O}_S\bigl(1, 1-3\alpha, \ldots(1-\alpha)\bigr)
\end{cases}
\]

\[
\begin{array}{lll}
s_0 = (0, 0) & & \\
s_1 = (1, 0) & & \\
s_2 = (1, 1) & & (y_2 = 1) \\
s_3 = (\beta_1, \beta_1) & \beta_1 \neq 1, 0 & \\
s_4 = (\beta_1, y_4) & y_4 \neq \beta_1 & \\
s_5 = (\beta_2 \beta_1, \beta_2 y_4) & \beta_2 \neq 1, 0 & \\
s_6 = (\beta_2 \beta_1, y_6) & y_6 \neq \beta_2 y_4 & \\
s_7 = (\beta_3 \beta_2 \beta_1, \beta_3 y_6) & \beta_3 \neq 1, 0 & \\
s_8 = (\beta_3 \beta_2 \beta_1, y_8) & y_8 \neq \beta_3 y_6 & \\
{[s_9 = s_0]} & [\beta_4 = 0] &
\end{array}
\qquad
\begin{array}{l}
\beta_1\ \beta_2\ \beta_3 \\
y_4\ y_6\ y_8
\end{array}
\]

\[
H \overset{\mathrm{déf}}{=} \underbrace{\mathrm{SL}(M)}_{= \mathrm{Aut}(M, e_M)} \simeq
\Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Bigm|
\begin{matrix} a, d \in \Lambda,\ b \in \mathrm{Hom}(L'_1, L'_0) \simeq L_1'^{-1} L'_0 \\ c \in \mathrm{Hom}(L'_0, L'_1) \simeq L_0'^{-1} L'_1,\ ad - bc = 1 \end{matrix} \Bigr\} \tag{8}
\]

\[
(66)\qquad
\left\{
\begin{array}{ll}
\text{boucle à un sommet de poids } 1 : & \Gamma = \Gamma^{!} \simeq \mathbb{Z}/2 \quad (\dim \operatorname{amb} 1) \\[2pt]
\text{deux sommets de poids } 1, \text{ deux arêtes} : & \Gamma \simeq D_2, \ \Gamma^{!} \simeq \mathbb{Z}/2 \quad (\dim \operatorname{amb} 2) \\[2pt]
\text{boucle avec queue, bout de poids } 2 : & \Gamma \simeq \mathbb{Z}/2 \times \mathbb{Z}/2, \ \Gamma^{!} \simeq \mathbb{Z}/2 \quad (\dim \operatorname{amb} 2)
\end{array}
\right.
\]

\[
  \begin{aligned}
    D(E_\eta^{\natural}) &= \bigl(R i_{*}(\check E_U)\bigr)(1)
      + \sum_{x \in Y} j_{x*}\bigl(\check E_\eta^{\natural(x)}\bigr) \\
    &= R i_{!}(\check E_U)(1) + \sum_{x \in Y} j_{x*}\Bigl[\check E_\eta^{\natural(x)}
      + j_x^{*} R i_{*}(\check E_U)(1)\Bigr] \\
    &= \check E_\eta^{\natural}(1) + \sum_{x \in Y} j_{x*}\bigl(\mu_x(E)\bigr)
  \end{aligned}
\]

\[
  \left\lbrace
  \begin{array}{l}
    q = p^{1+\rho} \\[4pt]
    \lambda_x(E_\eta) = L_{\mu_x(E_\eta)}(t)
      = (-t)^{-\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
      \dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(E_\eta)}(p^{-\rho} t)} \\[10pt]
    \chi_x(E_\eta) = \chi\bigl(\alpha_x(E_\eta)\bigr) \\[4pt]
    \delta_x(E_\eta) = \delta\bigl(\alpha_x(E_\eta)\bigr)
  \end{array}
  \right.
\]

\begin{align*}
&\sum_{\substack{\text{perm. circ.}\\ \text{de } X,Y,Z}}
\Big[ -\langle \delta\omega, [X,Y] \wedge Z \rangle
+ \theta_{[X,Y]} \langle \omega, Z \rangle
- \theta_Z \langle \omega, [X,Y] \rangle \Big] \\
&= \sum \Big[ -\langle i_Z \delta\omega, [X,Y] \rangle
+ (\theta_X\theta_Y - \theta_Y\theta_X) \langle \omega, Z \rangle \\
&\qquad + \theta_Z \langle \delta\omega, X \wedge Y \rangle
- \theta_Z\theta_X \langle \omega, Y \rangle
+ \theta_Z\theta_Y \langle \omega, X \rangle \Big] \\
&= \sum -\langle i_Z \delta\omega, [X,Y] \rangle
+ \theta_Z \langle \delta\omega, X \wedge Y \rangle \\
&= \sum \langle \delta i_Z \delta\omega, X \wedge Y \rangle
- \theta_X i_Y i_Z \delta\omega + \theta_Y i_X i_Z \delta\omega
+ \theta_Z \langle \delta\omega, X \wedge Y \rangle \\
&= \sum \big( i_X i_Y \delta i_Z + \theta_Z i_X i_Y
- \theta_X i_Y i_Z + \theta_Y i_X i_Z \big) . (\delta\omega)
\end{align*}

\[
\begin{cases}
a_{i+1} = \xi_{i+1}\,b_{i+1} = -\dfrac{1}{f(b_{i+1})}\,b_{i+1}\\[1ex]
\xi_{i+1} = -f(a_{i+1}) = -\dfrac{1}{f(b_{i+1})}\\
\phantom{\xi_{i+1}} = -\xi_{i+1}^2 f(b_{i+1})\\
1 = -\xi_{i+1} f(b_{i+1})
\end{cases}
\qquad
\begin{aligned}
a'_{i+1} &= -\frac{1}{f(a_{i+1})}\,\psi(a_{i+1})\\
&= \psi\Bigl[\underbrace{-\frac{1}{f(a_{i+1})}\,a_{i+1}}_{b_{i+1}}\Bigr]
\end{aligned}
\]

\[
(3.6) \qquad \mathrm{Ker}(V_{\underline{\mathrm{Pic}}_{X/S}}) \simeq
\left\{
\begin{array}{l}
\text{faisceau des sections de } f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1}_{X/S}) \text{ telles} \\
\text{que } d\omega = 0,\ C\omega = \tilde{\omega}, \text{ modulo les sous-} \\
\text{faisceau des } df/f,\ f \text{ dans } f^{\mathrm{fl}}_{*}(\underline{\mathcal{O}}^{*}_{X/S})
\end{array}
\right.
\]

\[
(70)\qquad
\begin{cases}
\varphi : S \xrightarrow{\ \sim\ } S^{\vee} \text{ satisfaisant} \\
\text{1) } \text{\struck{\ill{}}}\ (\mathrm{id} \times \varphi)^{-1}(\mathcal{R}) \text{ est une relation \emph{symétrique} sur } S \\
\qquad \text{i.e.\ } \forall s, t \in S,\ (s, \varphi(t)) \in \mathcal{R} \Longleftrightarrow (t, \varphi(s)) \in \mathcal{R} \\
\text{2) } \forall (\lambda, \check{\lambda}) \in Z, \text{ on a} \\
\qquad \check{\lambda} \varphi \lambda = \varphi \quad \text{i.e.} \quad \check{\lambda} \varphi = \varphi \lambda^{-1}
\end{cases}
\]

\[
\begin{aligned}
&\Delta_1(c_0) = 2c_0, \quad \Delta'_1(c_0) = c_0 \quad \overline{\Delta'_1}(c_0) = 1 \\
&\Delta_2(c_0, c_1) = 4c_0c_1 - c_1^2 = c_1(4c_0 - c_1) \quad
\overline{\Delta'_2}(c_0, c_1) = 4c_0 - c_1 \\
&\Delta_3(c_0, c_1, c_2) = 2c_2(4c_0c_1 - c_0c_2 - c_1^2), \quad
\Delta'_3 = c_2(4c_0c_1 - c_0c_2 - c_1^2), \\
&\qquad \overline{\Delta'_3}(c_0, c_1, c_2) = 4c_0c_1 - c_0c_2 - c_1^2 \\
&\cdots
\end{aligned}
\]

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

\[
\begin{cases}
c_0 = 1\\
c_1 = N(\mu_1)\,\xi_0\\
c_2 = N(\mu_1\mu_2)\,\xi_0\xi_1\\
c_3 = N(\mu_1\mu_2\mu_3)\,\xi_0\xi_1\xi_2\\
\cdots\\
c_n = N(\mu_1 \cdots \mu_n)\,\xi_0 \cdots \xi_{n-1}
\end{cases}
\qquad
\begin{aligned}
\varphi(a_{i-1}, a_i) &= \frac{c_i}{\mu_i} = N(\mu_0 \cdots \mu_{i-1})\,\bar{\mu}_i\,(\xi_0 \cdots \xi_{i-1})\\
\varphi(a_i, a_{i+1}) &= N(\mu_0 \cdots \mu_{i-1})\,\mu_i\,(\xi_0 \cdots \xi_{i-1})
\end{aligned}
\]

\[
\begin{aligned}
  (E, q) \longmapsto \bigl(&\Delta = \det E,\ \varphi = \delta'(q)^{-1},\
  E_0 = E \otimes \Delta^{-1}, \\
  &q_0 = \varepsilon q \otimes \Delta^{-1} : E_0 \to \Delta^{-2}
  \xrightarrow{\ {}^t\varphi\ } \underline{O}_S, \\
  &\omega_0 \in \Gamma(\det E_0) \text{ déduit de }
  \det E_0 \simeq (\det E) \otimes \Delta^{-n} \simeq \underline{O}_S\bigr)
\end{aligned}
\]

\[
\left\lbrace
\begin{aligned}
&\mathrm{sn}(u+K) = \frac{\mathrm{cn}\,u}{\mathrm{dn}\,u}
&& \mathrm{cn}(u+K) = -k'\frac{\mathrm{sn}\,u}{\mathrm{dn}\,u}
&& \mathrm{dn}(u+K) = \frac{k'}{\mathrm{dn}\,u}\\
&\mathrm{sn}(u+iK') = \frac{1}{k\,\mathrm{sn}\,u}
&& \mathrm{cn}(u+iK') = -\frac{i}{k}\frac{\mathrm{dn}\,u}{\mathrm{sn}\,u}
&& \mathrm{dn}(u+iK') = -i\,\frac{\mathrm{cn}\,u}{\mathrm{sn}\,u}\\
&\mathrm{sn}(u+K'') = -\frac{1}{k}\frac{\mathrm{dn}\,u}{\mathrm{cn}\,u}
&& \mathrm{cn}(u+K'') = -\frac{ik'}{k\,\mathrm{cn}\,u}
&& \mathrm{dn}(u+K'') = ik'\frac{\mathrm{sn}\,u}{\mathrm{cn}\,u}
\end{aligned}
\right.
\]

\[
  \begin{array}{c}
    (p_1,p_2) \\ \downarrow \\ (p_1',p_2') \\ \downarrow \\ (p_1'',p_2'') \\[2pt] \hline
    (p_1,p_2) \\ \downarrow \\ (p_1'',p_2'')
  \end{array}
  \quad
  \left\{
  \begin{aligned}
    u &= b_1 \otimes u_2 + u_1 \otimes b_2 + \chi\, u_1 \otimes u_2 \\
    u' &= b_1' \otimes u_2' + u_1' \otimes b_2' + \chi\, u_1' \otimes u_2' \\
    u'' &= b_1 \otimes u_2'' + u_1'' \otimes b_2 + \chi\, u_1'' \otimes u_2'' \\
    &\quad (\text{si } u_1'' = u_1 + u_1',\ u_2'' = u_2 + u_2')
  \end{aligned}
  \right.
\]

\[
\begin{array}{ll}
(1) & \tilde a_r\tilde b_r\tilde c_r\tilde d_r = \tilde b_{r_1}\tilde c_{r_2}\\[2pt]
 & \quad r\in R,\ r_1 = \sigma_0(r),\ r_2 = \sigma_1\sigma_0(r) = \omega^{-1}(r)\\[4pt]
(2') & \widetilde{A}_{\omega^m r''}\widetilde{A}_{\omega^{m-1} r''}\cdots\widetilde{A}_{r''} = \widetilde{A}_{\omega^m r_1}\cdots\widetilde{A}_{r_1}\\[2pt]
 & \quad r\in R,\ r_1 = \sigma_0 r,\ r'' = \sigma_0\sigma_1(r) = \omega(r)
\end{array}
\]

\begin{gather*}
e_1 e_2 = e_1^2 = 0 \\
(x e_1 + y e_2)(x' e_1 + y' e_2) = y y' \\
(e_1 + e_2)(e_1 + e_2) \qquad e_1 + e_2 - \ldots + e_n \qquad
\Sigma x_i^2 \\
\underset{\text{carré } 1}{e_1}\ \ \underbrace{e_2\ e_3 - \ldots\ e_n}_{\text{carré nul}} \\
e_1,\ e_1 - e_2,\ e_1 - e_3,\ \ldots,\ e_1 - e_n \\
e'_i e'_j = 0 \quad i, j \neq 1 \\
a_{11} x_1 x'_1 + \sum_{1 \leq i < j} a_{ij} (x_i x'_j + x_j x'_i)
\qquad (e_1 - e_i)(e_1 - e_j) = 1 \\
a_{11} x_1 x'_1
\end{gather*}

\[
  \left\{
  \begin{array}{ll}
    H^i(E) = 0 & \text{si}\ i \neq 0, 2\nu - 2, 4\nu - 1 \\[4pt]
    H^{2(\nu-1)}(E) \simeq
      \left\{ \begin{array}{ll} 0, & \ell \neq 2 \\ \mathbf{Z}/2\mathbf{Z} & \text{si}\ \ell = 2 \end{array} \right. \\[10pt]
    H^{4\nu-1}(E) \simeq \mu(-2\nu)
  \end{array}
  \right.
  \qquad \boxed{n + 1 = 2\nu} \ \ n = 2\nu - 1
\]

\[
(14)\quad \left\lbrace
\begin{aligned}
\Theta_1(v|t) &= \sum_{-\infty}^{\infty} e^{(n-\frac12)^2 i\pi t + (2n-1)i\pi v}\\
&= 2\sum_{1}^{\infty} e^{(n-\frac12)^2 i\pi t}\cos(2n-1)\pi v = \vartheta_2(v)\\
\Theta_2(v|t) &= \sum_{-\infty}^{+\infty}(-1)^n e^{n^2 i\pi t + 2ni\pi v}\\
&= 1 + 2\sum_{1}^{\infty} e^{n^2 i\pi t}\cos 2n\pi v = \vartheta_0(v)
\end{aligned}
\right.
\]

\[
(20)\qquad \begin{cases} \widetilde{S}_M = \{ x \in M \mid \exists y \in M \text{ tel que } (x,y) \in \mathrm{Rep}(M) \} \\ \text{\struck{$S_M = \widetilde{S}_M/\{\pm \mathrm{id}_M\}$}} \\ \widetilde{F}_M = \Bigl\{ \{x,y,z\} \in \mathfrak{P}_3(M) \Bigm| \begin{array}{l} x + y + z = 0 \\ (x,y) \in \mathrm{Rep}(M) \\ (\text{i.e. } (y,z) \in \mathrm{Rep}(M), \\ \text{i.e. } (z,x) \in \mathrm{Rep}(M)) \end{array} \Bigr\} \end{cases}
\]

\[
\text{(1)}\quad
\left\{
\begin{array}{l}
\widetilde{S}_M = M^{**} = \{x\in M\mid \exists\, y\in M,\ (x,y)\in\mathrm{Rep}(M)\}\\[4pt]
\widetilde{F}_M = \Bigl\{\{x,y,z\}\in\mathfrak{P}_3(\widetilde{S}_M) \Bigm| \begin{array}{l} x+y+z=0\\ x\wedge y\ (=y\wedge z=z\wedge x)\in(\det M)^{*}\end{array}\Bigr\}
\end{array}
\right.
\]

\[
(10) \qquad
\left\{
\begin{array}{lll}
P = (S, A, F) & & \text{des relations « d'incidence »} \\[2pt]
R_1 \subset S \times A & & R_1 = \{ (s, a) \in S \times A \mid s \in a \} \\[2pt]
R_2 \subset A \times F & & R_2 = \{ (a, f) \in A \times F \mid a \subset f \} \\[2pt]
R_3 = R_2 \circ R_1 \subset S \times F & & R_3 = \{ (s, f) \in S \times F \mid s \in f \} ,
\end{array}
\right.
\]

\begin{gather}
  (\Lambda^{p} u) \wedge (\Lambda^{q} v) = \frac{1}{(p+q)!} \sum_{\substack{i_1 < \cdots < i_p \\ j_1 < \cdots < j_q}} \rho_{i_1} \cdots \rho_{i_p}\, \sigma_{j_1} \cdots \sigma_{j_q}\; \times \\
  \overline{b_{i_1} \wedge \cdots \wedge b_{i_p} \wedge d_{j_1} \wedge \cdots \wedge d_{j_q}} \otimes \bigl(a_{i_1} \wedge \cdots \wedge a_{i_p} \wedge c_{j_1} \wedge \cdots \wedge c_{j_q}\bigr)
\end{gather}

\[
(8)\ \left\lbrace
\begin{aligned}
\mathrm{cn}^2 z &= 1 - \mathrm{sn}^2 z = \frac{\wp z - e_2}{\wp z - e_1}
= -\frac{1}{k^2}\bigl[\wp(z-\omega_1) - e_3\bigr]
&\qquad& \boxed{\mathrm{cn}\,0 = +1}\\
\mathrm{dn}^2 z &= 1 - k^2\mathrm{sn}^2 z = \frac{\wp z - e_3}{\wp z - e_1}
= -\bigl[\wp(z-\omega_1) - e_2\bigr]
&& \boxed{\mathrm{dn}\,0 = +1}
\end{aligned}
\right.
\]

\[
  (6, 1_6, 6) \;/\!/\; \bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
  \bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr) \quad
  \bigl(6, \begin{smallmatrix}2\\1_4\end{smallmatrix}, \begin{smallmatrix}3\\3\end{smallmatrix}\bigr) \;/\!/\; [\ldots]
\]

\[
  \left\{
  \begin{array}{l}
    \rho_{\mathrm{aff}} \geqslant \rho'_{\mathrm{aff}} \qquad
      d \geqslant d' \\[2pt]
    (5)\ \rho_{\mathrm{n}} \geqslant \rho'_{\mathrm{n}}, \quad
    (6)\ \rho_{\mathrm{u}} \geqslant \rho'_{\mathrm{u}}, \quad
    (7)\ \rho_{\mathrm{na}} \geqslant \rho'_{\mathrm{na}} \\[4pt]
    (1)\ \rho_{\mathrm{ab}} \leqslant \rho'_{\mathrm{ab}}, \quad
    (2)\ \rho_{\mathrm{ab}} + \rho_{\mathrm{r}} \leqslant \rho'_{\mathrm{ab}}
      + \rho'_{\mathrm{r}}, \quad
    (3)\ \rho_{\mathrm{s}} \leqslant \rho'_{\mathrm{s}}
  \end{array}
  \right.
\]

\[
  \boxed{
  \left\lbrace
  \begin{array}{l}
    D(E_\eta^{\natural}) = \check E_\eta^{\natural}(1)
      + \displaystyle\sum_{x \in X^{(0)}} j_{x*}\bigl(\mu_x(E)\bigr) \\[10pt]
    \mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(\check E_\eta)(1) \qquad
    \alpha_x(E_\eta) = E_\eta^{\natural(x)} - E_\eta(x)
  \end{array}
  \right.}
\]

\[
\begin{cases}
\sigma_0 s_0 = s_1 \\
\sigma_0 s_1 = s_0 \\
\sigma_0 s_2 = s_2^0 \in f_2\smallsetminus f_1 \\
\sigma_0 s_3 = s_3^0 \in f_3\smallsetminus f_2 \\
\dots \\
\sigma_0 s_{n+1} = s_{n+1}^0 \in f_{n+1}\smallsetminus f_n
\end{cases}
\qquad
\begin{aligned}
s_2^0 &= s_0+\lambda_2^{01}e_1+\lambda_2^{02}e_2 \\
s_3^0 &= s_0+\lambda_3^{01}e_1+\lambda_3^{02}e_2+\lambda_3^{0\,\uncertain{3}}e_3 \\
&\dots \\
s_{n+1}^0 &= s_0+\lambda_{n+1}^{01}e_1+\lambda_{n+1}^{02}e_2+\dots+\lambda_{n+1}^{0,n+1}e_{n+1}
\end{aligned}
\]

\[
(59)\qquad
\begin{cases}
SA_M \simeq \mathrm{Rep}(M)'/\text{symétrie} \quad \{\dot x, \dot y\} \mapsto \{\dot y, \dot x\}, \quad (x,y) \mapsto (-x,-y) \\
\text{de cardinal } \underbrace{\tfrac{1}{4} \prod_\alpha p_\alpha (p_\alpha - 1)(p_\alpha + 1)\, p_\alpha^{3(\nu_\alpha - 1)}}_{N'/2}
\end{cases}
\]

\[
(48.6) \qquad
\left\{
\begin{array}{l}
\text{i) une \struck{\ill{}} \add{bijection} } S^{\flat} \xrightarrow{\ \varphi^{\flat}\ } S^{\vee\flat} \text{ qui soit une $R^{\flat}$-jauge} \\
\qquad \text{(i.e. donnant lieu à une relation symétrique sur } S^{\flat} \times S^{\flat}, \\
\qquad \text{soit \struck{\ill{}}} \ R^{\flat}_{S^{\flat}}) \\
\text{ii) un \struck{\ill{}} \add{isomorphisme}} \\
\qquad (S/S^{\flat}) \wedge^{Z_{S^{\flat}}} (\varphi^{\flat *}(S^{\vee}/S^{\vee\flat})) \simeq \mathbf{1}_{Z_{S^{\flat}}}
\end{array}
\right.
\]

\[
  (33)\quad
  \left\{
  \begin{array}{l}
    \tau_\infty \overset{\mathrm{def}}{=} \sigma_\infty \widetilde{\sigma}_\infty
      = \widetilde{\sigma}_\infty \sigma_\infty, \quad \tau_\infty^2 = 1 \\[4pt]
    \tau_\infty(l_0) = l_0^{-1}, \quad \tau_\infty(l_1) = l_1^{-1}, \quad
    \tau_\infty(l_\infty) = l_0^{-1} l_\infty^{-1} l_0 \\[4pt]
    \widetilde{\sigma}_\infty(l_0) = l_1^{-1}, \quad
    \widetilde{\sigma}_\infty(l_1) = l_0^{-1}, \quad
    \widetilde{\sigma}_\infty(l_\infty) = l_\infty^{-1} \\[4pt]
    \sigma_\infty(l_0) = l_1, \quad \sigma_\infty(l_1) = l_0, \quad
    \sigma_\infty(l_\infty) = l_1^{-1} l_0^{-1}
  \end{array}
  \right.
\]

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

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

\begin{align*}
\chi_!(M^{!\,\mathrm{an}}_{0,4}) &= \chi_!(U^{\mathrm{an}}_{0,3}) = 2 - 2\underset{0}{g} - \underset{3}{\nu} = -1 \\
\chi_!(\widehat{M}^{!\,\mathrm{an}}_{0,4}) &= \chi_!(\mathbb{P}^{1\,\mathrm{an}}) = 2 \\
\chi_!(M^{\mathrm{an}}_{0,4}) &= \chi_!(\mathbb{B}^{\mathrm{an}}_{0,3}, \mathfrak{S}_4)
= \chi_!\bigl(U_{0,3} \setminus \{-j, -\bar{j}\} \setminus \{\tfrac12, 2, -1\}, \mathfrak{S}_4\bigr) \\
&\qquad + \chi_!(\{-j, -\bar{j}\}, \mathfrak{S}_4) + \chi_!(\{\tfrac12, 2, -1\}, \mathfrak{S}_4) \\
&= -1 + \tfrac13 + \tfrac12 = -\tfrac{1}{24}
\end{align*}

\[
(*) \quad
\left.
\begin{array}{l}
  \alpha_i \text{ injectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \exists \text{ voisinage ouvert } V \text{ de } Z, \text{ tel que } F \text{ soit de profondeur } > n\,? \text{ dans } U \cap V \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand},\ i < n
\end{array}
\right.
\]

\[
\begin{cases}
\text{Matrice de } f_0 \ (\text{quadratique}) = \begin{pmatrix} 2 & -\beta & -\beta \\ -\beta & 2 & -\gamma \\ -\beta & -\gamma & 2 \end{pmatrix} \\[4ex]
\delta'\ (\text{discrim.\ divisé}) = 4 - \beta^2(\gamma+2) - \gamma^2 = -(\alpha+2)(\beta+2)^2(\alpha-\beta) \\[1ex]
\gamma = 2(1+\alpha+\beta) + \alpha\beta
\end{cases}
\]

\[
  \begin{array}{l|c|c|c|c}
  \text{NB des sommets antiliés}
   & 0 & 4=2^2 & 8=2^3 & 16=2^4\\
  \text{à un sommet } 4c=2^{\nu+2} & & =2^e & =2^e & =2^{e-1}\\\hline
  \text{Ordre de } \mathrm{Aut}(S,A) : N_\omega & 6=2.3 & 72=2^3.3^2 & \cdot & 51840\\
   & & & & =2^7.3^4.5
  \end{array}
\]

\[
  (138)\qquad Z_0 = \left\{ \lambda\in\mathfrak{S}_S \;\middle|\;
  \begin{array}{l}
    \lambda \text{ centralise } G_0=\mathrm{Aut}(S,\Sigma_0),\\
    \exists f_{\lambda}\in\mathfrak{S}_S \text{ tel qu'on ait, }
      \forall (s,t)\in S\times S\\
    \{s,t\}\in A_0 \Longleftrightarrow
      \{\underbrace{f_{\lambda}(s)}_{s'}, \underbrace{\lambda^{-1}f_{\lambda}(t)}_{t'}\}\in A_0
  \end{array}
  \right\}
\]

\[
\tag{49}
\text{A)}\quad
\begin{cases}
G = \mathrm{Sl}(2,\Lambda) = \Bigl\{\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in M_2(\Lambda) \Bigm| ad-bc=1\Bigr\}\\[4pt]
H = \Bigl\{\begin{pmatrix} 1 & t\\ 0 & 1\end{pmatrix} \Bigm| t\in\Lambda\Bigr\}
\end{cases}
\]

\[
\begin{aligned}
\delta(\omega_2) &= \gamma\beta_2(\omega_2) + \omega_2\\
&= c_{\mathrm{pr}}(t) + \Bigl[\ldots\frac{f''}{f} + \frac{5}{4}\Bigl(\frac{f'}{f}\Bigr)^2\Bigr]dt^2\\
\delta\alpha'_2(\omega_1) &= \gamma\beta_1(\omega_1) + \omega_1^2\\
&= c_{\mathrm{pr}}(t) + \Bigl[f^2 - 2\frac{f''}{f} + 3\Bigl(\frac{f'}{f}\Bigr)^2\Bigr]dt^2
\end{aligned}
\]

\[
  \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2\\1_4\end{smallmatrix}, 6\bigr) \quad
  \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}5\\1\end{smallmatrix}\bigr) \quad
  \bigl(\begin{smallmatrix}4\\2\end{smallmatrix}, \begin{smallmatrix}2_2\\1_2\end{smallmatrix}, \begin{smallmatrix}4\\2\end{smallmatrix}\bigr)
\]

\[
  (2)\qquad \text{déf}\quad
  \begin{cases}
    \pi_{X,x}=\pi_1(X,x) & (\simeq\hat\pi_{g,\nu}\ \text{\uncertain{non can.}})\\
    \mathfrak{S}_{X,x}=\pi_1(\mathcal{X}_{g,\nu\,k},x) & (\simeq\hat{\mathfrak{S}}_{g,\nu}\ \text{---})\\
    \mathcal{T}_X=\pi_1(M_{g,\nu\,k},\xi) & (\simeq\hat{\mathcal{T}}_{g,\nu}\ \cdots)\\
    M_{X,x}=\pi_1(\mathcal{X}_{g,\nu},x) & (\simeq M_{g,\nu}\ \cdots)\\
    N_X=\pi_1(M_{g,\nu},\xi) & (\simeq N_{g,\nu}\ \cdots)
  \end{cases}
\]

\[
\begin{cases}
\sigma_0 =
\begin{pmatrix}
-1 & \alpha_0 & \alpha_0 & \cdots & \alpha_0 & 1 \\
0 & 1 & 0 & \cdots & 0 & 0 \\
0 & 0 & 1 & \cdots & 0 & 0 \\
\cdots & & & & & \\
0 & 0 & 0 & \cdots & 1 & 0 \\
0 & 0 & 0 & \cdots & 0 & 1
\end{pmatrix} \\[2ex]
\sigma_0(x_1,\dots,x_{n+1}) = \bigl(1-x_1+\alpha_0(x_2+\dots+x_{n+1}),\ x_2,\ \dots,\ x_{n+1}\bigr) \\[1ex]
H_0 : 2x_1\ \uncertain{-}\ \alpha_0(x_2+\dots+x_{n+1}) = 1
\end{cases}
\]

\[
(9)\quad \begin{cases}
\sigma_i s_j = s_j & \text{si } 0\le j< i\\
\sigma_i s_i = s_{i+1}\\
\sigma_i s_{i+1} = u_i \in E_{i+1} & E_{i+1} = \mathrm{Env}(E_{i-1}, s_{i+1}, u_i)\\
\sigma_i s_{i+2} = v_i \in E_{i+2} & E_{i+2} = \mathrm{Env}(E_{i+1}, v_i) \quad (0\le i\le n-1)\\
\sigma_i s_j = \sigma_{j-1}\cdots\sigma_{i+2}(v_i) & \text{si } \uncertain{i+2< j\le n+1}
\end{cases}
\]