Cote n° 64 · pages 8–146
· 20 commutative diagrams · Modules courbes elliptiques en caractéristiques ≠2 : notes manuscrites (s.d.).
Inventory dating : [à partir de 1980- 1982]
Édition de démonstration
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& \mathfrak{S}_S \\
\mathfrak{D} \arrow[ur] \arrow[dr] & \\
& \mathfrak{S}_A
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\mathbb{F}_2^C \simeq V_S \arrow[r, hook] \arrow[d, hook] & \mathfrak{S}_S \\
\mathfrak{D} \arrow[ur, no head] \arrow[dr, hook] & \\
\mathbb{F}_2^D \simeq V_A \arrow[u, hook] \arrow[r, hook] & \mathfrak{S}_A
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=large, row sep=huge]
V^{*} \arrow[rr, leftrightarrow, "\sim"] \arrow[dr, leftrightarrow, "2"'] & &
\text{ss-gr.\ cycliques } \mathfrak{D}^+ \text{ d'ordre 4 de } G \arrow[dl, leftrightarrow, "\sim"] \\
& \text{2-ss-gr.\ de Sylow de } G &
\end{tikzcd}LaTeX source
\begin{tikzcd}
0 \arrow[r] & M_0 \arrow[r] & M \arrow[r, "\mathrm{can}"] & M_0 \arrow[r] & 0 \\
& & M \arrow[ul, dashed, "\mathrm{can}"] \arrow[u, dashed, "2\cdot"'] & &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
1 \arrow[r] & \operatorname{End}_{\mathbb{F}_2}(M_0) \arrow[r] & \operatorname{GL}_\Lambda(M) \arrow[r] & \operatorname{GL}(M_0) \simeq \mathfrak{S}_J \arrow[r] & 1 \\
1 \arrow[r] & M_0 \times \mathbb{F}_2^{\underline{\omega}} \arrow[r] \arrow[u, "\wr"'] & \operatorname{Aut}(M_0, I_M, S_M) \arrow[r] \arrow[u, "\wr"'] & \operatorname{GL}(M_0) \arrow[r] \arrow[u, no head, "\|" description] & 1
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
& \operatorname{GL}_\Lambda(M) \arrow[dl] \arrow[dr] & \\
\mathfrak{S}_I \arrow[dr, "\mathrm{sg}"'] & & \mathfrak{D}_Q \arrow[dl, "\chi_C"] \\
& \{\pm 1\} &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& & & 1 \arrow[d] & & & \\
& 1 \arrow[dr] & & \mathfrak{S}_I^+ \times \mathbb{F}_2^{\underline{\omega}} \arrow[dl, "\mathrm{pr}_1"'] \arrow[dr, "\mathrm{pr}_2"] \arrow[dd] & & 1 \arrow[dl] & \\
& & \mathfrak{S}_I^+ \arrow[dl, Rightarrow] \arrow[dr] & & \mathbb{F}_2^{\underline{\omega}} \arrow[dl] \arrow[dr, Rightarrow] & & 1 \arrow[dl] \\
1 \arrow[r] & \mathfrak{S}_I^+ \arrow[dr] & & \operatorname{GL}_\Lambda(M) \arrow[dl] \arrow[dr] \arrow[dd, "\text{cf.\ (24)}"] & & \mathbb{F}_2^{\underline{\omega}} \simeq V_{S_Q} \arrow[dl] & \\
& & \mathfrak{S}_I \arrow[dl] \arrow[dr, "\mathrm{sg}"'] & & \mathfrak{D}_Q \arrow[dl, "\chi_C"] \arrow[dr] & & \\
& 1 & & \{\pm 1\} \arrow[dl] \arrow[d] \arrow[dr] & & 1 & \\
& & 1 & 1 & 1 & &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& & & 1 \arrow[d] & & & \\
& 1 \arrow[dr] & & \mathfrak{S}_I^+ \times \{1, -\mathrm{id}_M\} \arrow[dl, "\mathrm{pr}_1"'] \arrow[dr, "\mathrm{pr}_2"] \arrow[dd] & & 1 \arrow[dl] & \\
& & \mathfrak{S}_I^+ \arrow[dl, Rightarrow] \arrow[dr] & & \{1, -\mathrm{id}_M\} \arrow[dl] \arrow[dr, "\wr"] & & 1 \arrow[dl] \\
1 \arrow[r] & \mathfrak{S}_I^+ \arrow[dr] & & \operatorname{SL}_\Lambda(M) \arrow[dl] \arrow[dr] \arrow[dd, "\text{cf.\ (24)}"] & & \{1, \underline{a}_Q\} \simeq \mathbb{F}_2 \arrow[dl] & \\
& & \mathfrak{S}_I \arrow[dl] \arrow[dr, "\mathrm{sg}"'] & & \mathfrak{D}_Q^+ \arrow[dl] \arrow[dr] & & \\
& 1 & & \{\pm 1\} \arrow[dl] \arrow[d] \arrow[dr] & & 1 & \\
& & 1 & 1 & 1 & &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& & & \operatorname{GL}_\Lambda(M) \simeq [\mathfrak{S}_I \cdot \mathbb{F}_2^{\underline{\omega}}] & & \\
& & \mathfrak{S}_I^+ \times \mathbb{F}_2^{\underline{\omega}} \ [\mathfrak{S}_I \times \mathfrak{z}] \arrow[ur, "2"] & & \operatorname{SL}_\Lambda(M) \arrow[ul, "2"'] & \\
& M_0 \times \mathbb{F}_2^{\underline{\omega}} \simeq \mathfrak{gl}(M_0) \arrow[ur, "3"] & & \mathfrak{S}_I^+ \times \mathfrak{z} \arrow[ul, "2"'] \arrow[ur, "2"] \arrow[uu, dashed] & & \\
\mathbb{F}_2^{\underline{\omega}} \arrow[ur, "4"] & & M_0 \times \mathfrak{z} \simeq \mathfrak{sl}(M_0) \arrow[ul, "2"'] \arrow[ur, "3"] & & \mathfrak{S}_I^+ \arrow[ul, "2"'] & \\
& \mathfrak{z} \arrow[ul, "2"'] \arrow[ur, "4"] & & M_0 \arrow[ul, "2"'] \arrow[ur, "3"] & & \\
& & 1 \arrow[ul, "2"'] \arrow[ur, "4"] & & &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
1 \arrow[r] & \operatorname{End}(M_0)/\mathbb{F}_2 \cdot \mathrm{id}_{M_0} \arrow[r] \arrow[d, "\wr"] & \operatorname{GL}(M)' \arrow[r] \arrow[d, "\wr"] & \operatorname{GL}_{\mathbb{F}_2}(M_0) \simeq \mathfrak{S}_J \arrow[r] \arrow[d, "\wr"] & 1 \\
1 \arrow[r] & \mathbb{F}_2^J \simeq M_0 \times \mathbb{F}_2 \arrow[r] & \operatorname{Aut}_{\mathrm{oct}}(\Delta_M) \arrow[r] & \mathfrak{S}_J \arrow[r] & 1
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& \mathbb{F}_2^{\underline{\omega}} \arrow[dl] \arrow[dr, Rightarrow] & \\
\operatorname{GL}_\Lambda(M) \arrow[d] \arrow[dr] & & \mathbb{F}_2^{\underline{\omega}} \simeq V(S_Q) \arrow[dl] \\
\mathfrak{S}_I \arrow[dr] & \mathfrak{D}_Q \arrow[d, "\chi_C"] & \\
& \{\pm 1\} &
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \setminus \Delta_1 \arrow[r] \arrow[d, "\wr"'] & (X \setminus \Delta_1)/\{1, \sigma_1\} \arrow[d, "\wr"] \\
\mathbb{G}_{m\,S} \arrow[r, "\lambda \mapsto \lambda^2"] & \mathbb{G}_{m\,S}
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=large]
\underline{I} \subset E \arrow[r, "\text{étale}"] \arrow[d] & E_0 \ (\simeq E/\underline{M}_0) \arrow[d] \\
\underline{I} \hookrightarrow X \arrow[r] & \Sigma_{\underline{J}} \\
\Delta \arrow[u, hook] \arrow[r, dashed] & D_{\underline{J}} \arrow[u, hook]
\end{tikzcd}LaTeX source
\begin{tikzcd}
0 \arrow[r] & M_0 \arrow[r] & M \arrow[r, "\mathrm{can}"] & M_0 \arrow[r] & 0 \\
& M \arrow[u] \arrow[ur, "2\,\mathrm{id}_M"'] & & &
\end{tikzcd}LaTeX source
\begin{tikzcd}
T_0^{\otimes 2} \arrow[rr, "\kappa^{\otimes 2}", "\sim"'] \arrow[dr, "(12)"'] & & T^{\otimes 2} \arrow[dl, "(5)"] \\
& T_{\Sigma, t} &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
0 \arrow[r] & \mathcal{F}_{n-1} \arrow[r] \arrow[d, "\Theta_\delta"] &
\mathcal{F}_n \arrow[r] \arrow[d, "\Theta_\delta"] &
t_E^{\otimes n} \arrow[r] \arrow[d, "\mu_\delta"] & 0 \\
0 \arrow[r] & \mathcal{F}_n \arrow[r] & \mathcal{F}_{n+1} \arrow[r] &
t_E^{\otimes n+1} \arrow[r] & 0
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
0 \arrow[r] & t_E \otimes \mathcal{F}_{n-1} \arrow[r] \arrow[d, "\delta_{n-1}"] &
t_E \otimes \mathcal{F}_n \arrow[r] \arrow[d, "\delta_n"] &
t_E \otimes t_E^{\otimes n} \arrow[r] \arrow[d, "\wr\ \mathrm{can}"] & 0 \\
0 \arrow[r] & \mathcal{F}_n \arrow[r] & \mathcal{F}_{n+1} \arrow[r] &
t_E^{\otimes n+1} \arrow[r] & 0
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
0 \arrow[r] & F_t \arrow[r] \arrow[d, no head, "\wr"] &
V_J(k) \arrow[r] \arrow[d, no head, "\wr"] & \mathcal{O}_t(1) \arrow[r] & 0 \\
& \mathcal{O}_t(-1)(\underline{\omega}) &
\Gamma(\Sigma_J, \underline{\mathcal{O}}_{\Sigma_J}(1)) & &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
0 \arrow[r] & F_t \otimes \check F_t \arrow[r] \arrow[d, "\wr"] &
V_J(k) \otimes \check F_t \arrow[r] \arrow[d, "\wr"] &
\mathcal{O}_t(1) \otimes \check F_t \arrow[r] \arrow[d, "\wr"] & 0 \\
0 \arrow[r] & k \cdot 1_{\Sigma_J} \arrow[r] &
\Gamma(\Sigma_J, \underline{\mathcal{O}}_{\Sigma_J}\{t\}) \arrow[r] &
T_{\Sigma_J, t} \arrow[r] & 0
\end{tikzcd}LaTeX source
\begin{tikzcd}
\mathbb{C} \arrow[r, "\sim"] \arrow[d, "2"'] & t_E \arrow[d, no head, "\Vert" description] \\
\mathbb{C} \arrow[r] & t_E
\end{tikzcd}