\[ f\colon Y \longrightarrow S \]

\[ x \in H^m(Y, F_Y) . \]

\[ u(x) \in E_2^{1,m-1} = H^1(S, R^{m-1} f_{*}(F_Y)) . \]

\[ Z \in H^N(Y \times_S Y', G), \]

\[ x' \longmapsto Z(x')\colon H^{m'}(Y', F') \longrightarrow H^m(Y, F), \]

\[ u(Z(x')) = Z(u(x')) , \]

\[ a' \longmapsto Z(a')\ :\ R^{m'-1} f'_{*}(F_{Y'}) \longrightarrow R^{m-1}
f_{*}(F_Y) , \]

\[ x' \in H^2(C, \underline{\mathbf{Z}}_\ell(1)) \]

\[ Z \in H^{2n}(Y \times_S C, \underline{\mathbf{Z}}_\ell(n)) \]

\[ x \in H^{2n}(Y, \underline{\mathbf{Z}}_\ell(n)) \]

\[ z = Z(t_1) - Z(t_0) \]

\[ (*) \qquad Z\colon\ R^1 g_{*}(\underline{\mathbf{Z}}_\ell(1)) \longrightarrow
R^{2n-1} f_{*}(\underline{\mathbf{Z}}_\ell(n)) , \]

\[ (**) \qquad Z\colon\ H^1(S, R^1 g_{*}(\underline{\mathbf{Z}}_\ell(1)))
\longrightarrow H^1(S, R^{2n-1} f_{*}(\underline{\mathbf{Z}}_\ell(n))) , \]

\[ (*\ \mathrm{bis}) \qquad H^1(C_{\bar s}, \underline{\mathbf{Z}}_\ell(1))
\longrightarrow H^{2n-1}(Y_{\bar s}, \underline{\mathbf{Z}}_\ell(n)) \]

\[ i\colon Y \longrightarrow X \]

\[ (***) \qquad i^{*}\colon\ H^{2n-1}(X, \underline{\mathbf{Q}}_\ell(n))
\longrightarrow H^{2n-1}(Y, \underline{\mathbf{Q}}_\ell(n)) , \]

\[ Z\colon\ H^1(C, \underline{\mathbf{Q}}_\ell(1)) \longrightarrow H^{2n-1}(Y,
\underline{\mathbf{Q}}_\ell(n)) \]

\[ z' = Z'(t_1) - Z'(t_0), \qquad z'' = Z''(t_1) - Z''(t_0) . \]

\[ i\colon Y \longrightarrow X . \]

\[ u(z) \in H^1(S, R^{2n-1} f_{*}(\underline{\mathbf{Z}}_\ell(n))) \]

\[ \tilde f\colon \widetilde{X} \longrightarrow \mathbf{P}^1_k \]

\[ H^m(X) \to H^m(\widetilde{X}) \longrightarrow H^m(Y) \longrightarrow H^0(S, R^m
f_{*}(\underline{\mathbf{Q}}_\ell)) . \]

\[ P^m(X) \longrightarrow H^1(S, R^{m-1} f_{*}(\underline{\mathbf{Q}}_\ell)) \]

\[ i^{*}\colon\ H^1(S, R^{m-1} g_{*}(\underline{\mathbf{Q}}_\ell)) = H^1(S,
H^{m-1}(X, \underline{\mathbf{Q}}_\ell)) \longrightarrow H^1(S, R^{m-1}
f_{*}(\underline{\mathbf{Q}}_\ell)) . \]

\[ \operatorname{card}(Y_t(k')) \equiv a \mod p , \]

\[ f\colon \underline{Y} \longrightarrow S_d \]

\[ \operatorname{card} Y_t(k') \equiv c_t + (-1)^{n-1} \operatorname{Tr} f_t , \]

\[ c_t = \sum_{0 \leqslant i \leqslant n-1} (-1)^i\, \text{Trace de }
\mathrm{Frob}_{k'} \text{ opérant sur } H^i(X, \underline{\mathcal{O}}_X) , \]

\[ \sum_{0 \leqslant i \leqslant n-2} (-1)^i\, \text{Trace } \mathrm{Frob}_{H^i(X_{\bar
t}, \mathbf{Q}_\ell)} \bigl(1 + q^{\,2n \ldots}\bigr) + \ldots\, \text{Trace }
\mathrm{Frob}_{H^{n-1}(X_{\bar t}, \ldots)} \ldots \]

\[ \ldots + (-1)^{\ldots}\, \text{Trace } \mathrm{Frob}_{E^{n-1}(Y_s,
\underline{\mathbf{Q}}_\ell)} , \]

\[
(*) \qquad H^i(X, \underline{\mathcal{O}}_X(-d)) = 0
\quad \text{pour tout } d \geqslant d_0,\ i < n.
\]

\[
(**) \qquad H^i(X_{k'}, \underline{\mathcal{O}}_{X_{k'}}) \longrightarrow
H^i(Y_s, \underline{\mathcal{O}}_{Y_s})
\]

\[
(***) \qquad \operatorname{card} Y_s(k') \equiv c + (-1)^{n-1} \operatorname{Tr} f_s ,
\]

\[
N \geqslant r d_1^{\,n} .
\]

\[
\phi = \underline{\phi}(\phi_0, \ldots, \phi_n)
\]

\[
d'_0 = d_1 d_2 + (n+1) = (d_1 + 1)(n+1)
\]

\[
\begin{array}{cc}
B & B' \\
{\scriptstyle i}\uparrow & \uparrow{\scriptstyle i'} \\
M & M'
\end{array}
\qquad \text{biext.\ } \varepsilon \text{ vers } \mathbb{G}_m
\]

\[
B' \xrightarrow{\ \alpha'\ } \underline{\operatorname{Ext}}^1(B, \mathbb{G}_m) \to
\underline{\operatorname{Ext}}^1(M, \mathbb{G}_m) \ \text{nul}
\qquad \text{p.ex.\ } \operatorname{Hom}(B', \underline{\operatorname{Ext}}^1(M, \mathbb{G}_m)) = 0 \ ?
\]

\[
B \xrightarrow{\ \alpha\ } \underline{\operatorname{Ext}}^1(B', \mathbb{G}_m) \to
\underline{\operatorname{Ext}}^1(M', \mathbb{G}_m) \ \text{nul}
\qquad \text{p.ex.\ } \operatorname{Hom}(B, \underline{\operatorname{Ext}}^1(M', \mathbb{G}_m)) = 0 \ ?
\]

\[
0 \to DM \xrightarrow{\ j'\ } G' \xrightarrow{\ p'\ } B' \to 0, \qquad
0 \to DM' \xrightarrow{\ j\ } G \xrightarrow{\ p\ } B \to 0
\]

\[
\operatorname{Hom}(M', \underline{\operatorname{Ext}}^1(B, \mathbb{G}_m))
\rightleftarrows \operatorname{Biext}(B, M'; \mathbb{G}_m) \leftleftarrows
\operatorname{Ext}^1(B, D(M'))
\]

\[
M \to G \to B, \qquad M' \to G' \to B'
\]

\[
A = [M \to G], \qquad A' = [M' \to G']
\]

\[
\xi \in \operatorname{Ext}^1(A \overset{L}{\otimes} A', \mathbb{G}_m),
\]

\[
{}_nA \overset{\text{déf}}{=} \underline{\operatorname{Ext}}^0(\mathbb{Z}/n\mathbb{Z}, A),
\qquad
{}_nA' = \underline{\operatorname{Ext}}^0(\mathbb{Z}/n\mathbb{Z}, A')
\]

\[
\begin{cases}
{}_nM = 0 \\
{}_nG = G \quad \text{i.e.}\ nB = B,\ nT = T
\end{cases}
\]

\[
0 \to {}_nG \to {}_nA \to M_n \to 0
\]

\[
0 \to {}_nG' \to {}_nA' \to M'_n \to 0
\]

\[
\begin{cases}
0 \to T_\ell(G) \to T_\ell(A) \to M \otimes \mathbb{Z}_\ell \to 0 \\
0 \to T_\ell(G') \to T_\ell(A') \to M' \otimes \mathbb{Z}_\ell \to 0
\end{cases}
\]

\[
{}_nA \to A, \qquad {}_nA' \to A'
\]

\[
\varphi^\xi_n \colon {}_nA \otimes {}_nA' \to \mathbb{G}_m
\]

\[
T_\ell(A) \times T_\ell(A') \longrightarrow T_\ell(\mathbb{G}_m)
\overset{\text{déf}}{=} \mathbb{Z}_\ell(1).
\]

\[
{}_nA'/{}_nG' \times D(M'_n) \to \mathbb{G}_m, \qquad
D(M_n) \times {}_nA/{}_nG \to \mathbb{G}_m,
\]

\[
H^0(X, \mathfrak{z}_\ell) = U_\ell,
\]

\[
\begin{aligned}
H^2(X, \mathbb{Z}/\ell) &= H^2(X, \mathfrak{z}_\ell) \otimes_{\mathbb{Z}/\ell}
\operatorname{Hom}(U_\ell, \mathbb{Z}/\ell) \\
&= \mathbb{Z}/\ell \otimes_{\mathbb{Z}/\ell}
\operatorname{Hom}(U_\ell, \mathbb{Z}/\ell) \simeq \operatorname{Hom}(U_\ell, \mathbb{Z}/\ell) ;
\end{aligned}
\]

\[
H^1(X, \mathbb{Z}/\ell) = H^1(X, \mathfrak{z}_\ell) \otimes_{\mathbb{Z}/\ell}
\operatorname{Hom}(U_\ell, \mathbb{Z}/\ell)
= P^{(\ell)} \otimes_{\mathbb{Z}/\ell} \operatorname{Hom}(U_\ell, \mathbb{Z}/\ell) ;
\]

\[
H^2(X, \mathfrak{z}_\ell) \simeq \mathbb{Z}/\ell, \qquad
H^0(X, \mathbb{Z}/\ell) \simeq \mathbb{Z}/\ell .
\]

\[
H^0(X, \mathfrak{z}_p) = 0, \qquad
H^2(X, \mathbb{Z}/p) \simeq H^2(X, \mathbb{Z}/p) = 0 ;
\]

\[
H^2(X, \mathfrak{z}_p) \simeq \mathbb{Z}/p, \qquad H^0(X, \mathbb{Z}/p) \simeq
\mathbb{Z}/p .
\]

\[
H^0(X, i_p) = 0, \qquad H^1(X, i_p) \simeq {}_F H^1(X, \underline{\mathcal{O}}_X),
\qquad H^2(X, i_p) = H^1(X, \underline{\mathcal{O}}_X)_F .
\]

\[
H^1(X, \underline{\operatorname{Ext}}^1(f^{*}(B), \mathbb{G}_{m\,X}))
\]

\[
\operatorname{Ext}_X(f^{*}(B), \mathbb{G}_{m\,X}) \simeq
\operatorname{Ext}_S(B, f^{!}(\mathbb{G}_{m\,X}))
\]

\[
H^p(X, \underline{\operatorname{Ext}}^q_X(A, \mathbb{G}_{m\,X}))
\]

\[
\underline{\operatorname{Hom}}(\mathbb{G}_a, \mathbb{G}_m) \qquad
\mathbb{Z} \to \mathbb{Z}
\]

\[
H^1(X, A) = \varinjlim_n {}_nH^1(X, A) ;
\]

\[
0 \to \operatorname{Hom}(X, A)_n \to H^1(X, {}_nA) \to {}_nH^1(X, A) \to 0
\]

\[
0 \to \operatorname{Hom}(\operatorname{Alb}_X, A)_n \to
\operatorname{Hom}(D({}_nA), \operatorname{Pic}_X) \to {}_nH^1(X, A) \to 0
\]

\[
\operatorname{Hom}(\operatorname{Alb}_X, A) \simeq \operatorname{Hom}(A',
(\operatorname{Alb}_X)') = \operatorname{Hom}(A', \operatorname{Pic}^0_X),
\qquad D({}_nA) \simeq {}_nA'
\]

\[
(*) \qquad 0 \to \operatorname{Hom}(A', \operatorname{Pic}^0_X)_n \to
\operatorname{Hom}({}_nA', \operatorname{Pic}_X) \to {}_nH^1(X, A) \to 0
\]

\[
\boxed{0 \to \operatorname{Hom}(A', \underline{\operatorname{Pic}}^0_X)
\otimes_{\mathbb{Z}} \mathbb{Q} \to \operatorname{Hom}(T_{\cdot}(A'),
\underline{\operatorname{Pic}}_X) \to H^1(X, A) \to 0}
\]

\[
(**) \qquad 0 \to \operatorname{Hom}(A', \underline{\operatorname{Pic}}_X)_n \to
\operatorname{Hom}({}_nA', \operatorname{Pic}_X) \to {}_n\operatorname{Ext}^1(A',
\underline{\operatorname{Pic}}^0_X) \to 0
\]

\[
\boxed{0 \to \operatorname{Hom}(A', \operatorname{Pic}_X) \otimes_{\mathbb{Z}}
\mathbb{Q} \to \operatorname{Hom}(T_{\cdot}(A'), \operatorname{Pic}_X) \to
\operatorname{Ext}^1(A', \underline{\operatorname{Pic}}_X) \to 0}
\]

\[
\boxed{H^1(X, A) \simeq \operatorname{Ext}^1(A', \underline{\operatorname{Pic}}_X)}
\]

\[
\boxed{H^1(X, G) \simeq \operatorname{Hom}(D(G), \underline{\operatorname{Pic}}_X)}
\]

\[
\boxed{\underline{\mathcal{H}}^1(X/S; \underline{\operatorname{Hom}}_{S\text{-gr}}(\gamma,
G)) \simeq \underline{\operatorname{Hom}}_{S\text{-gr}}(\gamma,
\underline{\mathcal{H}}^1(X/S; G))}
\]

\[
\boxed{\underline{\mathcal{H}}^1(X/S; \underline{\operatorname{Ext}}^1_{S\text{-gr}}(\gamma,
G)) \simeq \underline{\operatorname{Ext}}^1_{S\text{-gr}}(\gamma,
\underline{\mathcal{H}}^1(X/S; G))}
\]

\[
\underline{\mathcal{H}}^1(X/S, \underline{\operatorname{Hom}}_{S\text{-gr}}(\gamma, G))
\to \underline{\operatorname{Hom}}(\gamma, \underline{\mathcal{H}}^1(X/S, G))
\]

\[
\underline{\mathcal{H}}^1(X, \operatorname{Hom}(A, B)) \simeq
\]

\[
\boxed{\begin{array}{c}
H^1(X/S; \underline{\operatorname{Hom}}_{S\text{-gr}}(\gamma, G)) \\
\downarrow \\
\underline{\operatorname{Hom}}_{S\text{-gr}}(\gamma, H^1(X/S, G))
\end{array}}
\]

\[
gf = i, \qquad fg = j
\]

\[
\mathcal{H}^1(X; A) \qquad \mathcal{E}xt^1(A, \underline{\operatorname{Pic}})
\]

\[
\boxed{\mathcal{H}^1(X, G) \simeq \operatorname{Hom}(D(G),
\underline{\operatorname{Pic}}_X)}
\]

\[
\mathcal{H}^i(X, D(G)) \simeq \operatorname{Ext}^i(G,
\underline{\operatorname{Pic}}_X)
\]

\[
\mathcal{E}xt^i_X(f^*(G), \mathbb{G}_m) \simeq \mathcal{E}xt^i_Y(G,
f_{\cdot}(\mathbb{G}_m))
\]

\[
\boxed{\underline{\mathcal{E}xt}^i_X(f^*(G), H^{\cdot}) \simeq
\underline{\mathcal{E}xt}^i_Y(G, f_!(H^{\cdot}))}
\qquad \text{avec } f\colon X \to Y
\]

\[
\underline{\mathcal{E}xt}^i_X(f^*(G), \mathbb{G}_m) \simeq
\underline{\mathcal{E}xt}^i_Y(G, \underline{\operatorname{Pic}}^{\bullet}_{X/Y})
\]

\[
\begin{array}{ll}
i = 0 & H^0(X, D(G)) = D(G) \\
i = 1 & H^1(X, D(G)) = \operatorname{Hom}(G, \underline{\operatorname{Pic}}^{(1)}) \\
i = 2 & H^2(X, D(G)) \neq \mathcal{E}xt^1(G, \underline{\operatorname{Pic}}^{(1)}_{X/Y})
+ \operatorname{Hom}(G, \underline{\operatorname{Pic}}^{(2)}_{X})
\end{array}
\]

\[
\mathcal{H}^1(X, \underline{\operatorname{Hom}}(\gamma, G)) \simeq
\mathcal{H}^1(X, \underline{\operatorname{Ext}}^1(\gamma, G))
\]

\[
X \to \underline{\operatorname{Hom}}_{S\text{-gr}}(\gamma, G), \qquad
X \times \gamma \to G, \qquad
\gamma \to \underline{\operatorname{Hom}}_S(X, G)
\]

\[
\prod^{\circ}_{X/S} \gamma_X = \underline{\operatorname{Hom}}_S(X, \gamma)
\doteq \gamma
\]

\[
e \to \gamma \to \underline{\operatorname{Hom}}_S(X, \gamma) \to
\operatorname{Hom}(X, a; \gamma) \to 0
\]

\[
H^1(A, \mathbb{Z}/p) \simeq H^1(A, \underline{O}_A)^{(p)}
\qquad \text{éléments de } H^1(A, \underline{O}_A) \text{ tels que } Fx = x
\]

\[
H^1(A, \mathbb{Z}/p^n) \simeq H^1(A, \underline{O}_A(W_n))^{(p)}
\qquad H^1(A, \underline{O}_A(W_n)) \text{ tels que } Fx = x
\]

\[
\boxed{\operatorname{Hom}(\pi_1^p(A), \mathbb{Z}_p) = H^1(A, \mathbb{Z}_p) \simeq
H^1(A, \mathbb{W})^{(p)}}
\]

\[
\boxed{\operatorname{Hom}_{\mathrm{cont}}(\pi_1^p(A), \Lambda) \simeq H^1(A,
\mathbb{W})_s}
\]

\[
\boxed{\pi_1^p(A) \simeq T_p(A)}
\]

\[
\boxed{T_p(A) \times H^1(A, \mathbb{W})^{(p)} \longrightarrow \mathbb{Z}_p}
\]

\[
\boxed{\operatorname{Hom}_{\mathrm{cont}}(T_p(A), W_\infty(k)) \simeq H^1(A,
\mathbb{W})_s}
\]

\[
\pi_1^{\neq p}(A) \simeq \varprojlim_{\substack{n \\ (n,p)=1}} A^{(n)}
\]

\[
\boxed{\pi_1^{\ell}(A) \simeq \varprojlim_{k} A^{(\ell^k)} \simeq T_\ell(A) =
\operatorname{Hom}(\mathbb{Q}_\ell/\mathbb{Z}_\ell, A)}
\qquad \text{module de Tate}
\]

\[
\operatorname{Hom}(\pi_1^{\ell}(A), U_\ell) \simeq
\text{sous-groupe de } \underline{\operatorname{Pic}}(A) = H^1(A,
\underline{O}_A^*)
\]

\[
\operatorname{Hom}(\pi_1^{\ell}(A), U_\ell) \simeq A^{*(\ell^\infty)}, \quad
\text{i.e.}
\]

\[
\boxed{T_\ell(A) \times T_\ell(A^*) \longrightarrow T_\ell(U_\ell)}
\]

\[
T_\ell(A)/\ell^k T_\ell(A) \times T_\ell(A^*)/\ell^k T_\ell(A^*) \longrightarrow
T_\ell(U_\ell)/\ell^k T_\ell(U_\ell)
\]

\[
A^{(\ell^k)} \times A^{*(\ell^k)} \longrightarrow U^{(\ell^k)}
\]

\[
0 \to P' \to P \to P'' \to 0 \qquad Q
\]

\[
{}_nP' \times {}_nQ \xrightarrow{\ E'(n)\ } G
\]

\[
{}_nP \times {}_nQ \xrightarrow{\ \varphi\ } G
\]

\[
\varphi|({}_nP' \times {}_nQ) = E'(n)
\]

\[
\varphi\colon T_{\mathbb{P}}(P) \times T_{\mathbb{P}}(Q) \longrightarrow
T_{\mathbb{P}}(G)
\]

\[
\varphi_{\mathbb{P}, E'}\colon T_{\mathbb{P}}(P') \times T_{\mathbb{P}}(Q)
\longrightarrow T_{\mathbb{P}}(G)
\]

\[
0 \to P' \to P \to P'' \to 0, \qquad 0 \to Q' \to Q \to Q'' \to 0
\]

\[
E \longmapsto (E', \varphi)
\]

\[
\varphi\colon T_{\mathbb{P}}(P) \times T_{\mathbb{P}}(Q) \longrightarrow
T_{\mathbb{P}}(G)
\]

\[
\varphi_{E';\mathbb{P}}\colon T_{\mathbb{P}}(P') \times T_{\mathbb{P}}(Q')
\longrightarrow T_{\mathbb{P}}(G)
\]

\[
\operatorname{Ext}^1(A, G) \xrightarrow{\ \sim\ } H^0(S,
\underline{\operatorname{Ext}}^1(A, G))
\]

\[
H^1(A, G_A) \simeq H^1(S, G) \times H^0(S, R^1f_*(G_A))
\]

\[
\begin{array}{c}
H^0(S, R^1f_*(G_A)) \\
\wr \\
\text{Classes d'isom.\ \uncertain{prises} des } G_A\text{-torseurs rigidifiés
\uncertain{rel.}\ à } e_A \\
\wr \\
\text{Classes de } G_A\text{-torseurs rigidifiables par } e_A.
\end{array}
\]

\[
0 \longrightarrow \operatorname{Ext}^1(A, G) \longrightarrow H^1(A, G_A) \quad );
\]

\[
\pi^*(L) \simeq \mathrm{pr}_1^*(L)\, \mathrm{pr}_2^*(L) \qquad \text{i.e.\ tels
que}
\]

\[
\pi^*(L)\, \mathrm{pr}_1^*(L)^{-1}\, \mathrm{pr}_2^*(L)^{-1} \simeq 0 .
\]

\[
A \times A \overset{\mathrm{pr}_1,\ \mathrm{pr}_2,\ \pi}{\longrightarrow} A
\]

\[
\operatorname{Biext}(A, B; G) \xrightarrow{\ \sim\ }
\underline{\operatorname{Hom}}_{\mathrm{gr}}(B, \underline{\operatorname{Ext}}^1(A,
G))
\]

\[
\operatorname{Biext}(A, B; G) = H^0(S, \underline{\operatorname{Biext}}(A, B; G))
\simeq \underline{\operatorname{Hom}}_{\mathrm{gr}}(B,
\underline{\operatorname{Ext}}^1(A, G))
\]

\[
\operatorname{Biext}(A, B; G) \hookrightarrow
\underline{\operatorname{Hom}}_{\mathrm{gr}}(B, R^1f_*(G_A)) \hookrightarrow
H^0(B, R^1f_{B*}(G_{A \times_S B}))
\]

\[
\begin{array}{c}
H^0(B, R^1f_{B*}(G_{A \times_S B})) \\
\Vert \\
\text{classes de } G_{A \times_S B}\text{-torseurs rigidifiés par : }
e_{A \times_S B} \\
\downarrow \\
H^1(A \times_S B, G_{A \times_S B})
\end{array}
\]

\[
\underline{\operatorname{BIEXT}}(A, B; G) \longrightarrow
\underline{\operatorname{TORS\,BIRIG}}_{e_A, e_B}(A, B; G) \longrightarrow
\underline{\operatorname{TORS\,RIG}}_{e_{A \times_S B}}(G_{A \times_S B})
\]

\[
R^1f_*(G_A) \xleftarrow{\ \Delta^*\ }
\underline{\operatorname{Tors}}\,[\underline{\operatorname{birig}}\
\underline{\operatorname{sym}}](A, A; G)
\]

\[
\Delta^* j(L) \simeq (2\,\mathrm{id}_A)^*(L)\, L^{-2}
\]

\[
\operatorname{Hom}(A, B) \longrightarrow \operatorname{Hom}(\underline{NS}(B),
\underline{NS}(A))
\]

\[
\varphi(u+v) - \varphi(u-v) = 2(\varphi(u) + \varphi(v)) .
\]

\[
\begin{array}{c}
\operatorname{Hom}(A, B) \longrightarrow
\operatorname{Hom}(\underline{\operatorname{Bitors}}(B, B; G),
\underline{\operatorname{Bitors}}(A, A; G))\ ! \\
\downarrow \ \text{linéaire} \\
\operatorname{Hom}(\underline{NS}(B), \underline{NS}(A))
\end{array}
\]

\[
\Delta^* j\colon \underline{NS}_{A/S} \longrightarrow \underline{P}_{A/S}
\]

\[
\begin{array}{ll}
(1) & X_0 = \sum_{i \in J} d_i D_i, \qquad \text{où } d_i \geqslant 1 , \\
(2) & d = \operatorname{pgcd}(d_i)_{i \in J}.
\end{array}
\]

\[
(3) \qquad \Delta \simeq \underline{\mathbf{Z}}^J
\]

\[
(4) \qquad \Gamma \simeq {}/\underline{\mathbf{Z}}.X_0 \hookrightarrow
\operatorname{Pic}(X)
\]

\[
(5) \qquad w\colon \Gamma \longrightarrow \mathrm{NS}(X_0)
\overset{\mathrm{dfn}}{=} \operatorname{Im}(\operatorname{Pic}(X_0) \to
\underline{\mathrm{NS}}_{X_0/k}(k)) ,
\]

\[
\Gamma \overset{i}{\hookrightarrow} \operatorname{Pic}(X) \longrightarrow
\operatorname{Pic}(X_0) \longrightarrow \mathrm{NS}(X_0) .
\]

\[
(6) \qquad 0 \to {}_n\Gamma \to H^1(X, \mu_n) \xrightarrow{\ u_n\ }
H^1(X_{\bar\eta}, \mu_n)^I \to \operatorname{Ker} w_n \longrightarrow 0 ,
\]

\[
w_n \colon \Gamma_n \longrightarrow \mathrm{NS}(X_0)_n
\]

\[
(7) \qquad 0 \to H^1(X, \underline{\mathbf{Z}}_\ell(1))
\xrightarrow{\ u(\ell)\ } H^1(X_{\bar\eta}, \underline{\mathbf{Z}}_\ell(1))^I
\to R \otimes \underline{\mathbf{Z}}_\ell \to 0 ,
\]

\[
(8) \qquad R \otimes \underline{\mathbf{Z}}_\ell \simeq
\begin{cases}
0 & \text{si } \ell \nmid d', \\
\underline{\mathbf{Z}}/\ell\underline{\mathbf{Z}} & \text{si } \ell \mid d' .
\end{cases}
\]

\[
(10) \qquad u(\ell) \colon H^1(X, \underline{\mathbf{Z}}_\ell(1)) \longrightarrow
H^1(X_{\bar\eta}, \underline{\mathbf{Z}}_\ell(1))^I
\]

\[
(11) \qquad 0 \to {}_n\Gamma \to H^1(X, \mu_n) \xrightarrow{\ u_n\ }
H^1(X_{\bar\eta}, \mu_n)^I \to Q(n) \to 0 ,
\]

\[
(12) \qquad 0 \to \operatorname{Ker} w_n \to Q(n) \to \Theta(n) \longrightarrow 0 ,
\]

\[
(13) \qquad \Theta(n) \hookrightarrow {}_n\operatorname{Ker}\bigl(\mathrm{Br}(K)
\to \mathrm{Br}(X_{\bar\eta})\bigr) \hookrightarrow {}_{(n,d^t)}\mathrm{Br}(K) ,
\]

\[
(14) \qquad p^m \Theta(n) = 0 \qquad \text{pour tout } n .
\]

\[
(16) \qquad d^t = p^r a , \qquad \text{avec } (a,p) = 1 .
\]

\[
u'_n \colon H^1(X', \mu_n) \longrightarrow H^1(X'_{\bar\eta}, \mu_n)^{I'}
\]

\[
u'(\ell) \colon H^1(X', \underline{\mathbf{Z}}_\ell) \longrightarrow
H^1(X'_{\bar\eta}, \underline{\mathbf{Z}}_\ell)^{I'}
\]

\[
\Delta \to \mathrm{Pic}(X) \to \mathrm{Pic}(X_0) \to
\underline{\mathrm{Pic}}_{X_0/k}(k) \to \underline{NS}_{X_0/k}(k) \to N
\]

\[
0 \to {}_n\Gamma \to H^1(X, \mu_n) \xrightarrow{\ u_n\ }
H^1(X_{\bar\eta}, \mu_n)^I \longrightarrow {}_n\delta''
\to \struck{\ill{}}\ {}_n\delta' \to \delta_n \to \delta''_n \to 0
\]

\[
H^1(X, \mathbf{Z}_\ell) \xrightarrow{\ \sim\ } H^1(X_{\bar\eta}, \mathbf{Z}_\ell)^I
\]

\[
(**) \qquad u' \colon {}_n\mathrm{Pic}(X) \to {}_n\mathrm{Pic}(X_\eta)
\]

\[
0 \to \Gamma \xrightarrow{\ i\ } \mathrm{Pic}(X) \xrightarrow{\ v\ }
\mathrm{Pic}(X_\eta) \to 0
\]

\[
\operatorname{Coker} u_n \simeq \operatorname{Coker} {}_nv \simeq
\operatorname{Ker} i_n , \quad \text{où } i_n \colon \Gamma_n \to
\mathrm{Pic}(X)_n .
\]

\[
j_X \colon \Gamma \to \mathrm{Pic}(X_0)
\]

\[
\operatorname{Ker} i_n \subset \operatorname{Ker} j_n .
\]

\[
\mathrm{Pic}(X_0) = \underline{\mathrm{Pic}}_{X_0/k}(k)
\]

\[
0 \to \mathrm{Pic}^0(X_0) \to \mathrm{Pic}(X_0) \to NS(X_0) \to 0 ,
\]

\[
\mathrm{Pic}(X_0)_n \hookrightarrow NS(X_0)_n
\]

\[
\operatorname{Coker} u_n \simeq \operatorname{Ker} k_n .
\]

\[
0 \to \Gamma \xrightarrow{\ k'\ } NS(X_0) \xrightarrow{\ \lambda\ } \delta \to 0
\]

\[
\operatorname{Ker} k_n = \operatorname{Ker} k'_n \simeq \operatorname{Coker} {}_n\lambda .
\]

\[
0 \to \delta' \to \delta \to \delta'' \to 0
\]

\[
0 \to {}_n\delta' \to {}_n\delta \to {}_n\delta'' \to \delta'_n \to \delta_n
\to \delta''_n
\]

\[
0 \to H^1(X, \mu_n) \to H^1(X_{\bar\eta}, \mu_n)^I \to {}_n\Gamma_n \to
NS(X_0)_n
\]

\[
0 \to H^1(X, \mathbf{Z}_\ell(1)) \to H^1(X_{\bar\eta}, \mathbf{Z}_\ell(1))^I
\to T_\ell(\Gamma \otimes \ldots) \to T_\ell(NS(X_0) \otimes \mathbf{Z}_\ell)
\]

\[
0 \to H^1(X, \mathbf{Z}_\ell(1)) \to H^1(X_{\bar\eta}, \mathbf{Z}_\ell(1))^I
\to R \otimes \mathbf{Z}_\ell \to 0
\]

\[
\to \operatorname{Pic}(X_\eta) \to P(K) \to \Theta \to 0 , \quad \text{où }
\Theta \subset \mathrm{Br}(K)
\]

\[
0 \to {}_n\Gamma \to H^1(X, \mu_n) \to H^1(X_{\bar\eta}, \mu_n)^I \to
\Gamma_n \to \mathrm{NS}(X_0)_n \to
\]

\[
0 \to H^1(X, \mathbf{Z}_\ell(1)) \to H^1(X_{\bar\eta}, \mathbf{Z}_\ell(1))^I
\to \Gamma \otimes \mathbf{Z}_\ell \to \mathrm{NS}(X_0) \otimes \mathbf{Z}_\ell
\]

\[
\hookrightarrow R \otimes \mathbf{Z}_\ell \to 0 , \qquad R = \operatorname{Ker}
\bigl(\Gamma \to \mathrm{NS}(X_0)\bigr)
\]

\[
\Gamma' = \Gamma/\struck{\ill{}}\,T, \qquad T = \operatorname{Tors} \Gamma
\]

\[
0 \to T_n \to \Gamma_n \to \Gamma'_n \to 0
\]

\[
\operatorname{Ker}\bigl(\Gamma_n \to \mathrm{NS}(X_0)_n\bigr) \simeq T_n +
\underbrace{\operatorname{Ker}\bigl(\Gamma'_n \to \mathrm{NS}(X_0)_n\bigr)}_{\operatorname{Ker}(\bar\Gamma_n
\to \mathrm{NS}(X_0)_n)}
\]

\[
0 \to \delta' \to \delta \to \delta'' \to 0
\]

\[
T_\ell(\tilde S) \subset T_\ell(G), \qquad T_\ell(\tilde T) \subset T_\ell(G'),
\]

\[
T_\ell(\tilde S)|U \subset T_\ell(A), \qquad T_\ell(\tilde T)|U \subset T_\ell(A') .
\]

\[
\varphi_\xi \colon \ldots \longrightarrow T_\ell(\mathbb{G}_{m,U}) =
T_\ell(\widetilde{\mathbb{G}_{m,Y}})|U
\]

\[
T_\ell(S) \times T_\ell(T) \to T_\ell(\mathbb{G}_{m,k}) .
\]

\[
T_\ell(S) \times T_\ell(T) \to T_\ell(\mathbb{G}_m)
\]

\[
T_\ell(S_0) \times T_\ell(T) \to T_\ell(\mathbb{G}_m)
\]

\[
V = \varinjlim A_i , \quad A_i \subset V , \quad A_i \ni \ldots \text{ de t.\ f.\ sur } \mathbf{Z} .
\]

\[
X = \varprojlim X_i , \quad Y = \varprojlim Y_i , \quad U = \varprojlim U_i .
\]

\[
T_\ell(\tilde T)|U \subset T_\ell(\tilde G_k)|U \subset T_\ell(A)
\]

\[
M(A)^I_0 \subset M(A)^I \subset M(A) = T_\ell(A(\bar K))
\]

\[
\varphi_\xi \colon M(A) \times M(A') \to T_\ell(\bar K^*) \struck{= \ldots} ,
\]

\[
\varphi_\xi \colon M(A) \times M(A) \to T_\ell(\bar K^*)
\]

\[
\varphi_\xi(M^I, M^I_0) = 0 .
\]

\[
{}_{\ell^\nu}A'(K) \times {}_{\ell^\nu}T(k) \to {}_{\ell^\nu}k^* \simeq {}_{\ell^\nu}K^*
\]

\[
\begin{aligned}
T_\ell(A(\bar K)) \supset T_\ell(A(\bar K))^I &= T_\ell\bigl(A(\bar K)^I\bigr)
= T_\ell(A(K)) = T_\ell(G(V)) \\
&= T_\ell(G(k)) = T_\ell(G^0(k)) \supset T_\ell(T(k))
\end{aligned}
\]

\[
\begin{gathered}
M \supset M^I \supset M^I_0 \\
M' \supset M'^I \supset M'^I_0
\end{gathered}
\qquad \text{de } \ldots
\]

\[
M \times M' \xrightarrow{\ \varphi\ } T_\ell(\mathbb{G}_m(\bar K)) =
T_\ell(\bar K^*) \quad \bigl[\ \overset{\text{non can.}}{\simeq}
\mathbf{Z}_\ell\ \bigr]
\]

\[
\begin{cases}
\varphi(M^I_0, M'^I) = 0 \\
\varphi(M^I, M'^I_0) = 0
\end{cases}
\]

\[
V = \varinjlim A_i , \quad A_i \subset V \text{ de t.f.\ sur } \mathbf{Z},
\ \text{\uncertain{avec} } t, \ \text{\uncertain{normaux}}.
\]

\[
X_i = \operatorname{Spec}(A_i), \quad Y_i = V(t_{X_i}), \quad U_i = X_i - Y_i
= X_{i,t} ,
\]

\[
{}_\ell G(k) = \varinjlim {}_\ell G_i(Y_i), \qquad {}_\ell G'(k) = \varinjlim
{}_\ell G'_i(Y_i)
\]

\[
{}_\ell G(k) \simeq {}_\ell G_i(Y_i), \qquad {}_\ell G'(k) \simeq {}_\ell
G'_i(Y_i) \qquad \Big|\ {}_\ell T
\]

\[
P' = \operatorname{Ker}\bigl(P \xrightarrow{\ \deg\ } \mathbb{Z}_S\bigr), \qquad
Q' = \operatorname{Ker}\bigl(Q \xrightarrow{\ \deg\ } \mathbb{Z}_S\bigr).
\]

\[
\Phi_{(s)} = \operatorname{Ker} u \simeq \mathbb{Z}/d\mathbb{Z}, \qquad
\Psi_{(s)} = \operatorname{Coker} u \qquad (d = \operatorname{pgcd}(d_i)),
\]

\[
u \colon E_{(s)} \longrightarrow E^*_{(s)}
\]

\[
(\beta u \alpha)\bigl((n_i)_i\bigr) = \Bigl(\sum_{1 \leqslant i \leqslant r}
n_i (C_i \cdot C_j)/\delta_j\Bigr)_{1 \leqslant j \leqslant r} .
\]

\[
Q'_\eta = P'_\eta = P^0_\eta = \underline{\mathrm{Pic}}^0_{X_\eta/k(\eta)} .
\]

\[
\bigl((n_i), (n'_i)\bigr) = \sum \delta_i n_i n'_i .
\]

\[
(\alpha(x), y^*) = (x, \beta y^*) \qquad x \in \mathbb{Z}^r,\ y^* \in E^*(s),
\]

\[
E(s) \times E^*(s) \longrightarrow \mathbb{Z} .
\]

\[
(x, u(y)) = (y, u(x)) \qquad \text{pour } x, y \in E(s).
\]

\[
\begin{aligned}
(\alpha(\underline{n}), u(y)) &= (\underline{n}, \beta u \alpha(\underline{n}')) \\
(\alpha(\underline{n}'), u(x)) &= (\underline{n}', \beta u \alpha(\underline{n})) .
\end{aligned}
\]

\[
(\underline{n}, \beta u \alpha(\underline{n}')) = \sum n_i n'_j (C_i \cdot C_j)
\]

\[
E^*(s) \simeq \operatorname{Hom}(E(s), \mathbb{Z})
\]