\[
  D_a(A_K) \times Z_0^{*}(A)_K
\]

\[
  v(X, \underline{a}) = \sum_{x_0 \in A_0} i(\mathrm{div}_{g}(\bar{f}), \bar{\underline{a}} ; x_0)
\]

\[
  \boxed{(X, \underline{a})_v = v(X, \underline{a}) + \sum_{x_0 \in A_0} i(\bar{X}, \bar{\underline{a}}, x_0)}
\]

\[
  v(X_{\underline{a}}, \underline{b}) = v(X^{-}_{\underline{b}}, \underline{a})
  \qquad (= v(X_{\underline{b}^{-}}, \underline{a}^{-}))
\]

\[
  D_a^{*}(A_K) \times Z_0^{*}(A_K) \longrightarrow \mathbb{R}
\]

\[
  (X, \underline{a}) = \sum_v (X, \underline{a})_v
\]

\[
  A^{*}(K) \times A^{*}(K) \longrightarrow \mathbb{R} .
\]

\[
  \begin{array}{l}
    \mathrm{Pic}(X^{*})' \longrightarrow J^{0}(\tilde{X}_{\bar{\eta}}) \\
    \mathrm{Pic}(\tilde{X})' \longrightarrow J^{0}(\tilde{X}_{\bar{\eta}})
  \end{array}
\]

\[
  \tilde{X}_{y_0} = \tilde{Y}_0 \simeq Y_0 .
\]

\[
  \varphi^{*}(\xi_0) - \sum_{z \in Z} \tilde{z} ,
\]

\[
  \mathrm{Pic}(X)' \longrightarrow \mathrm{Pic}(\tilde{X}_{\bar{\eta}}) = \mathrm{Pic}(Y_{\bar{\eta}})
\]

\[
  \boxed{\rho(\tilde{X}) - 2 =}\ \boxed{\rho(X) + z - 2 = \rho(J')}
\]

\[
  \boxed{Q(i(\xi), i(\eta)) = -\, \xi \cdot \eta} \qquad (\xi, \eta \in NS(X)')
\]

\[
  \langle C_\eta, D_\eta \rangle_v
\]

\[
  \left\{
  \begin{array}{l}
    c_1 + \dots + c_k = n + 1 \\
    \text{[deux parmi les termes]}\ Z_{i\eta}\ \text{alg. équiv. à } 0 \\
    Z_1 \cdot \ldots \cdot Z_k \ \text{défini}
  \end{array}
  \right.
\]

\[
  i(Z_1 \cdot \ldots \cdot Z_k) = (Z_{1\eta}, \dots, Z_{k\eta})_v
\]

\[
  (Z_{1\eta}, \dots, Z_{k\eta}) = v(f_\eta(Z_{1\eta} \cdot \ldots \cdot Z_{k-1,\eta} \cdot Z'_{k\eta}))
\]

\[
  -\langle D_\eta, C_\eta \rangle = D.C
\]

\[
  -\langle D_\eta, C_\eta \rangle
  = \sum_v \langle D_\eta, C_\eta \rangle_v = \sum_v (D.C)_v
\]

\[
  \boxed{\ \langle C_\eta, D_\eta \rangle_v = i(C.D)\ }
\]

\[
  \langle C_\eta, \operatorname{div} f_\eta \rangle_v
  = i(C.\operatorname{div} f)
  = v\bigl(f_\eta(C_\eta)\bigr)
\]

\[
  \alpha^*(D'_\eta) = D_\eta + \operatorname{div}_{X_\eta}(f)
\]

\[
  D = \alpha^*(D')
\]

\[
  C' = \alpha_*(C) .
\]

\[
  C'.D' = \alpha_*(C).D' = \alpha_*\bigl(C.\alpha^*(D')\bigr)
  = \alpha_*(C.D)
\]

\[
  i(C.D) = i\bigl(\alpha_*(C.D)\bigr) = i(C'.D')
\]

\[
  \boxed{\ i\bigl(\overline{C'_\eta}.\overline{D'_\eta}\bigr)
  = \langle C'_\eta, D'_\eta \rangle_v\ }
\]

\[
  D_\eta \sim \alpha^*(D'_\eta) ,
\]

\[
  D'_\eta \not\supset \alpha_\eta(X_\eta) \quad\text{et}\quad
  \overline{D'_\eta} \not\supset \alpha_0(X_0)
\]

\[
  D = \alpha^*(D') + \operatorname{div}(f) ,
\]

\[
  \boxed{\ \langle C_\eta, \operatorname{div} f_\eta \rangle_v
  = i(C.\operatorname{div} f)\ }
  = v\bigl(f_\eta(C_\eta)\bigr)
\]

\[
  M_{\overline{K}} / \Gamma \xrightarrow{\ \sim\ } M_K
\]

\[
  \varphi : X(\overline{K}) \times M_{\overline{K}} \to \mathbf{R}
\]

\[
  \varphi(\sigma P, \sigma v) = \varphi(P, v)
\]

\[
  \varphi(\sigma P, \sigma v) = \varphi(P, v) \quad \text{pour tt } v, \sigma ,
\]

\[
  X(K^{p^{-\infty}}) \times M_K \to \mathbf{R}
\]

\[
  \varphi_L : X(L) \times M_L \to \mathbf{R} ,
\]

\[
  \varphi_L(P, v|L) \simeq \varphi_{L'}(P, v)
  \qquad \left| \begin{array}{l} P \in X(L) \\ v \in M_{L'} \end{array} \right.
\]

\[
  \widehat{\Phi}(X)
\]

\[
  \widehat{\Phi}(X/K) \to \widehat{\Phi}(X_{K'}/K')
\]

\[
  \widehat{\Phi}(X/K) \simeq \widehat{\Phi}(X_{K'}/K')^{\pi} .
\]

\[
  |\varphi(\xi)| \leqslant c(\pi(\xi))
\]

\[
  \varphi(\xi) = 0 \quad \text{si } \pi(\xi)|K \notin S
\]

\[
  E(U; x_1, \ldots, x_n; c) ,
\]

\[
  v(\underline{x}(P)) = \operatorname*{Inf}_{1 \leqslant i \leqslant n}
  v(x_i(P)) \geqslant c(v)
\]

\[
  \{P\} \times M_{\overline{K}} , \quad \text{où } P \in X(\overline{K}) ,
\]

\[
  \widetilde{\varphi}(P) = \bigl(v \mapsto \varphi(P, v)\bigr) :
  M_{\overline{K}} \to \mathbf{R}
\]

\[
  M_{\overline{K}} \to M_{K'} \to \mathbf{R} ,
\]

\[
  \operatorname{Div}(M_{\overline{K}})
  = \varinjlim_{K'/K} \operatorname{Div}(M_{K'})
\]

\[
  \widetilde{\varphi} : X(\overline{K}) \to \operatorname{Div}(M_{\overline{K}})
\]

\[
  \varphi_f(P, v) = v(f(P)) \qquad \text{si } P \in X(\overline{K}),\
  v \in M_{\overline{K}} .
\]

\[
  \operatorname*{Inf}_i v(x_i(P)) \geqslant c(v) \Longrightarrow
  |v(f(P))| \leqslant c'(v) .
\]

\[
  \varphi_f(x, v) = v(x)
\]

\[
  \varphi : X(\overline{K}) \times M_{\overline{K}} \to \mathbf{R}
\]

\[
  \underline{\mathcal{O}}^*_X \xrightarrow{\ f \mapsto \varphi_f\ } \underline{\Phi}_X
\]

\[
  \underline{\mathcal{O}}^*_X \hookrightarrow \underline{R}^*_X
\]

\[
  \underline{QF}_X = \underline{\Phi}_X \amalg_{\underline{\mathcal{O}}^*_X} \underline{R}^*_X
  \quad \text{(somme amalgamée)}
\]

\[
  \simeq \underline{\Phi}_X \times \underline{R}^*_X / \theta(\underline{\mathcal{O}}^*_X)
\]

\[
  \theta : \underline{\mathcal{O}}^*_X \longrightarrow \underline{\Phi}_X \times \underline{R}^*_X
\]

\[
  \theta(f) = (-\varphi_f, f^{\ill{}})
\]

\[
  \underline{\Phi}_X \longrightarrow \underline{QF}_X \xrightarrow{\ \mathrm{div}\ } \underline{\mathrm{Div}}_X \longrightarrow 0
  \qquad \bigl(\underline{\mathrm{Div}}_X \simeq \underline{R}^*_X / \underline{\mathcal{O}}^*_X \text{, diviseurs de Cartier}\bigr)
\]

\[
  \underline{R}^*_X \xrightarrow{\ f \mapsto \{f\}\ } \underline{QF}_X \longrightarrow \underline{\Phi}_X / \mathrm{Im}\, \underline{\mathcal{O}}^*_X \longrightarrow 0
\]

\[
  0 \to \underline{\Phi}_X \longrightarrow \underline{QF}_X \longrightarrow \underline{\mathrm{Div}}_X \longrightarrow 0
\]

\[
  0 \to \underline{R}^*_X / \underline{U}_X \longrightarrow \underline{QF}_X \longrightarrow \underline{\Phi}_X / \mathrm{Im}\, \underline{\mathcal{O}}^*_X \longrightarrow 0
\]

\[
  \varphi : X(K) \times M_K \longrightarrow \mathbb{R}
\]

\[
  \tilde\varphi : X(K) \longrightarrow \mathrm{Div}(M_K)
\]

\[
  \Gamma(X, \underline{\Phi}_X / (\underline{\mathcal{O}}^*_X / \underline{U}_X)) \longrightarrow \mathrm{Pic}(X)
\]

\[
  \varphi : X(K') \times M_{K'} \longrightarrow \mathbb{R}
\]

\[
  \varphi : X(K) \times M_K \longrightarrow \mathbb{R}
\]