Cote n° 60 · pages 2–68
· 146 displayed formulas · Fonctions L / Équation fonctionnelle des fonctions L (cas géométrique) : notes manuscrites (s.d.).
Inventory dating : [à partir de 1962- à partir de 1971]
Édition de démonstration
\[P_f(t) = \det(1 - f t) = \prod_i (1 - \lambda_i t)
= \sum_i (-1)^i \bigl(\operatorname{Tr} \lambda^i(f)\bigr)\, t^i\]
LaTeX source
\[
P_f(t) = \det(1 - f t) = \prod_i (1 - \lambda_i t)
= \sum_i (-1)^i \bigl(\operatorname{Tr} \lambda^i(f)\bigr)\, t^i
\]\[Z_f(t) = \frac{1}{P_f(t)} = \prod_i \frac{1}{1 - \lambda_i t}
= \prod_i \sum_{\nu} \lambda_i^{\nu} t^{\nu}
= \sum_{\nu} t^{\nu} \sum_{\nu_1 + \cdots + \nu_n = \nu}
\lambda_1^{\nu_1} \cdots \lambda_n^{\nu_n}
= \sum_{\nu} \operatorname{Tr}\bigl(\mathrm{Sym}^{\nu}(f)\bigr)\, t^{\nu} .\]
LaTeX source
\[
Z_f(t) = \frac{1}{P_f(t)} = \prod_i \frac{1}{1 - \lambda_i t}
= \prod_i \sum_{\nu} \lambda_i^{\nu} t^{\nu}
= \sum_{\nu} t^{\nu} \sum_{\nu_1 + \cdots + \nu_n = \nu}
\lambda_1^{\nu_1} \cdots \lambda_n^{\nu_n}
= \sum_{\nu} \operatorname{Tr}\bigl(\mathrm{Sym}^{\nu}(f)\bigr)\, t^{\nu} .
\]\[(1 - a t) * (1 - b t) = 1 - a b t\]
LaTeX source
\[ (1 - a t) * (1 - b t) = 1 - a b t \]
\[\left\lbrace
\begin{array}{l}
P_{f \otimes f'}(t) = P_f(t) * P_{f'}(t) \\[4pt]
L_{f \otimes f'}(t) = \dfrac{1}{L_f(t) * L_{f'}(t)}
\end{array}
\right.
\qquad
L_{f_1 \otimes \cdots \otimes f_{\nu}}(t)
= \Bigl(\prod L_{f_i}(t)\Bigr)^{(-1)^{\nu+1}}\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
P_{f \otimes f'}(t) = P_f(t) * P_{f'}(t) \\[4pt]
L_{f \otimes f'}(t) = \dfrac{1}{L_f(t) * L_{f'}(t)}
\end{array}
\right.
\qquad
L_{f_1 \otimes \cdots \otimes f_{\nu}}(t)
= \Bigl(\prod L_{f_i}(t)\Bigr)^{(-1)^{\nu+1}}
\]\[\left\lbrace
\begin{array}{l}
P_{\lambda^i f}(t) = \lambda^i P_f(t) , \\[6pt]
\dfrac{1}{L_{\lambda^i f}(t)} = \lambda^i \dfrac{1}{L_f(t)}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
P_{\lambda^i f}(t) = \lambda^i P_f(t) , \\[6pt]
\dfrac{1}{L_{\lambda^i f}(t)} = \lambda^i \dfrac{1}{L_f(t)}
\end{array}
\right.
\]\[-\sum (-1)^i \bigl[L_{\lambda^i f}(t)\bigr] s^i
= \lambda^{*}\bigl(\bigl[-L_f(t)\bigr]\bigr)(-s)
= \frac{1}{\lambda^{*}\bigl[L_f(t)\bigr](-s)}
= \sum \sigma^i\bigl(L_f(t)\bigr) s^i\]
LaTeX source
\[
-\sum (-1)^i \bigl[L_{\lambda^i f}(t)\bigr] s^i
= \lambda^{*}\bigl(\bigl[-L_f(t)\bigr]\bigr)(-s)
= \frac{1}{\lambda^{*}\bigl[L_f(t)\bigr](-s)}
= \sum \sigma^i\bigl(L_f(t)\bigr) s^i
\]\[\left\lbrace
\begin{array}{l}
L_{\lambda^i(f)}(t) = \bigl(\sigma^i L_f(t)\bigr)^{(-1)^{i+1}} \\[4pt]
L_{\sigma^i(f)}(t) = \bigl(\lambda^i L_f(t)\bigr)^{(-1)^{i+1}}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
L_{\lambda^i(f)}(t) = \bigl(\sigma^i L_f(t)\bigr)^{(-1)^{i+1}} \\[4pt]
L_{\sigma^i(f)}(t) = \bigl(\lambda^i L_f(t)\bigr)^{(-1)^{i+1}}
\end{array}
\right.
\]\[P_{\check f}(t) = \prod (1 - \lambda_i^{-1} t)
= \prod_i (-\lambda_i^{-1} t)(1 - \lambda_i t^{-1})
= \frac{(-1)^n t^n}{\prod \lambda_i} \prod (1 - \lambda_i t^{-1})
= \frac{(-1)^n t^n}{\det f}\, P_f(t^{-1})\]
LaTeX source
\[
P_{\check f}(t) = \prod (1 - \lambda_i^{-1} t)
= \prod_i (-\lambda_i^{-1} t)(1 - \lambda_i t^{-1})
= \frac{(-1)^n t^n}{\prod \lambda_i} \prod (1 - \lambda_i t^{-1})
= \frac{(-1)^n t^n}{\det f}\, P_f(t^{-1})
\]\[\boxed{P_{\check f}(t) = (-t)^{\varepsilon(f)} \, (\det f)^{-1} \, P_f(t^{-1})}\]
LaTeX source
\[
\boxed{P_{\check f}(t) = (-t)^{\varepsilon(f)} \, (\det f)^{-1} \, P_f(t^{-1})}
\]\[\boxed{L_{\check f}(t) = (-t)^{-\varepsilon(f)} \, \det f \; L_f(t^{-1})}\]
LaTeX source
\[
\boxed{L_{\check f}(t) = (-t)^{-\varepsilon(f)} \, \det f \; L_f(t^{-1})}
\]\[L_{\check E}(t) = (-t)^{-n} \det f_E \; L_E(t^{-1})\]
LaTeX source
\[
L_{\check E}(t) = (-t)^{-n} \det f_E \; L_E(t^{-1})
\]\[\det f_E = \varepsilon(E)\, p^{\frac{n\rho}{2}}\]
LaTeX source
\[
\det f_E = \varepsilon(E)\, p^{\frac{n\rho}{2}}
\]\[L_{\check E}(t) = \varepsilon(E) \Bigl(-\frac{p^{\rho/2}}{t}\Bigr)^{n} L_E(t^{-1}) .\]
LaTeX source
\[
L_{\check E}(t) = \varepsilon(E) \Bigl(-\frac{p^{\rho/2}}{t}\Bigr)^{n} L_E(t^{-1}) .
\]\[\check E \simeq E(\rho)\]
LaTeX source
\[ \check E \simeq E(\rho) \]
\[L_{\check E}(t) = \bigl(L_E(t) * L_{\mathbb{Q}_\ell(\rho)}(t)\bigr)^{-1}
= \Bigl(L_E(t) * \frac{1}{1 - p^{-\rho} t}\Bigr)^{-1}
= L_E(t) * (1 - p^{-\rho} t)\]
LaTeX source
\[
L_{\check E}(t) = \bigl(L_E(t) * L_{\mathbb{Q}_\ell(\rho)}(t)\bigr)^{-1}
= \Bigl(L_E(t) * \frac{1}{1 - p^{-\rho} t}\Bigr)^{-1}
= L_E(t) * (1 - p^{-\rho} t)
\]\[L_{\check E}(t) = L_E\Bigl(\frac{t}{p^{\rho}}\Bigr)\]
LaTeX source
\[
L_{\check E}(t) = L_E\Bigl(\frac{t}{p^{\rho}}\Bigr)
\]\[\boxed{L_E\Bigl(\frac{1}{t p^{\rho}}\Bigr)
= \det f_E \, (-t)^{n} \, L_E(t)}\]
LaTeX source
\[
\boxed{L_E\Bigl(\frac{1}{t p^{\rho}}\Bigr)
= \det f_E \, (-t)^{n} \, L_E(t)}
\]\[\boxed{H^i_c(X, DE) \simeq D\bigl(H^{-i}(X, E)\bigr)}\]
LaTeX source
\[
\boxed{H^i_c(X, DE) \simeq D\bigl(H^{-i}(X, E)\bigr)}
\]\[R f_{*}(DE) = D\bigl(R f_{*}(E)\bigr)\]
LaTeX source
\[
R f_{*}(DE) = D\bigl(R f_{*}(E)\bigr)
\]\[(1)\quad
\left\lbrace
\begin{array}{l}
\boxed{L_{DE}(t) = (-t)^{-\chi(E)}\, \delta(E)\, L_E(t^{-1})} \\[6pt]
\chi(E) = \operatorname{rang} R f_{*} E = \sum (-1)^i \operatorname{rg} H^i(\bar X, \bar E) \\[6pt]
\delta(E) = \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^i}
= \varepsilon(E)\, p^{\sum_i (-1)^i \frac{i + \rho}{2} b_i}
\end{array}
\right.\]
LaTeX source
\[
(1)\quad
\left\lbrace
\begin{array}{l}
\boxed{L_{DE}(t) = (-t)^{-\chi(E)}\, \delta(E)\, L_E(t^{-1})} \\[6pt]
\chi(E) = \operatorname{rang} R f_{*} E = \sum (-1)^i \operatorname{rg} H^i(\bar X, \bar E) \\[6pt]
\delta(E) = \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^i}
= \varepsilon(E)\, p^{\sum_i (-1)^i \frac{i + \rho}{2} b_i}
\end{array}
\right.
\]\[\sum_i (-1)^i \frac{i + \rho}{2} b_i
= \frac{\rho}{2}\chi(E) + \frac{1}{2}\sum_{i=0}^{2n} (-1)^i i\, b_i\]
LaTeX source
\[
\sum_i (-1)^i \frac{i + \rho}{2} b_i
= \frac{\rho}{2}\chi(E) + \frac{1}{2}\sum_{i=0}^{2n} (-1)^i i\, b_i
\]\[\sum_{i=0}^{2n}(-1)^i i\, b_i
= \sum_{i=0}^{n-1}\bigl[(-1)^i i\, b_i + (-1)^{2n-i}(2n-i)\, b_{2n-i}\bigr]
+ (-1)^n n\, b_n\]
LaTeX source
\[
\sum_{i=0}^{2n}(-1)^i i\, b_i
= \sum_{i=0}^{n-1}\bigl[(-1)^i i\, b_i + (-1)^{2n-i}(2n-i)\, b_{2n-i}\bigr]
+ (-1)^n n\, b_n
\]\[= n\sum_{i=0}^{n-1} (-1)^i b_i + (-1)^n n\, b_n
= n \sum_{i=0}^{2n} (-1)^i b_i = n\,\chi(E)\]
LaTeX source
\[
= n\sum_{i=0}^{n-1} (-1)^i b_i + (-1)^n n\, b_n
= n \sum_{i=0}^{2n} (-1)^i b_i = n\,\chi(E)
\]\[\sum_i (-1)^i \frac{i + \rho}{2} b_i = \frac{n + \rho}{2}\,\chi(E)\]
LaTeX source
\[
\sum_i (-1)^i \frac{i + \rho}{2} b_i = \frac{n + \rho}{2}\,\chi(E)
\]\[\delta(E) = \varepsilon(E)\, p^{\frac{n+\rho}{2}\chi(E)} ,
\qquad \varepsilon(E) = \pm 1\]
LaTeX source
\[
\delta(E) = \varepsilon(E)\, p^{\frac{n+\rho}{2}\chi(E)} ,
\qquad \varepsilon(E) = \pm 1
\]\[(2)\quad
L_{DE}(t) = \varepsilon(E)\Bigl(-\frac{p^{\frac{n+\rho}{2}}}{t}\Bigr)^{-\chi(E)} L_E(t^{-1})\]
LaTeX source
\[
(2)\quad
L_{DE}(t) = \varepsilon(E)\Bigl(-\frac{p^{\frac{n+\rho}{2}}}{t}\Bigr)^{-\chi(E)} L_E(t^{-1})
\]\[\check E \simeq E(\rho)\]
LaTeX source
\[ \check E \simeq E(\rho) \]
\[D(E) \simeq \check E(n)[2n] \simeq E(\rho + n)[2n]\]
LaTeX source
\[ D(E) \simeq \check E(n)[2n] \simeq E(\rho + n)[2n] \]
\[R f_{*}(D(E)) \simeq R f_{*}(E)(\rho + n)[2n]\]
LaTeX source
\[
R f_{*}(D(E)) \simeq R f_{*}(E)(\rho + n)[2n]
\]\[L_{DE}(t) = L_{Rf_{*}(D(E))}(t) \simeq L_{Rf_{*}(E)}\bigl(p^{-(\rho+n)} t\bigr)\]
LaTeX source
\[
L_{DE}(t) = L_{Rf_{*}(D(E))}(t) \simeq L_{Rf_{*}(E)}\bigl(p^{-(\rho+n)} t\bigr)
\]\[L_{DE}(t) = L_E\bigl(p^{-(n+\rho)} t\bigr)\]
LaTeX source
\[
L_{DE}(t) = L_E\bigl(p^{-(n+\rho)} t\bigr)
\]\[(3)\quad
\boxed{L_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr)
= \varepsilon(E)\bigl(-p^{\frac{n+\rho}{2}} t\bigr)^{\chi(E)} L_E(t)}\]
LaTeX source
\[
(3)\quad
\boxed{L_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr)
= \varepsilon(E)\bigl(-p^{\frac{n+\rho}{2}} t\bigr)^{\chi(E)} L_E(t)}
\]\[\boxed{\xi_E(t) = t^{\chi(E)/2} L_E(t)}\]
LaTeX source
\[
\boxed{\xi_E(t) = t^{\chi(E)/2} L_E(t)}
\]\[\boxed{\xi\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr) = \varepsilon'(E)\, \xi(t)}\]
LaTeX source
\[
\boxed{\xi\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr) = \varepsilon'(E)\, \xi(t)}
\]\[\varepsilon'(E) = \pm 1 , \qquad
\varepsilon'(E) = (-1)^{\chi(E)} \prod_i \operatorname{signe} \det f_{H^i(\bar X, \bar F)}
= (-1)^{\chi(E)} \operatorname{signe} \det f_{H^{*}(\bar X, \bar F)}\]
LaTeX source
\[
\varepsilon'(E) = \pm 1 , \qquad
\varepsilon'(E) = (-1)^{\chi(E)} \prod_i \operatorname{signe} \det f_{H^i(\bar X, \bar F)}
= (-1)^{\chi(E)} \operatorname{signe} \det f_{H^{*}(\bar X, \bar F)}
\]\[\boxed{\varepsilon(E) = \operatorname{signe} \det f_{H^n(\bar X, \bar E)}} .\]
LaTeX source
\[
\boxed{\varepsilon(E) = \operatorname{signe} \det f_{H^n(\bar X, \bar E)}} .
\]\[\varepsilon'(E) = (-1)^{b_n(E)} \operatorname{signe} \det f_{H^n(X, E)}\]
LaTeX source
\[
\varepsilon'(E) = (-1)^{b_n(E)} \operatorname{signe} \det f_{H^n(X, E)}
\]\[(-1)^{b_n} = (-1)^{\varepsilon_1 + \varepsilon_{-1}} , \qquad
\varepsilon(E) = (-1)^{\varepsilon_{-1}} ,\]
LaTeX source
\[
(-1)^{b_n} = (-1)^{\varepsilon_1 + \varepsilon_{-1}} , \qquad
\varepsilon(E) = (-1)^{\varepsilon_{-1}} ,
\]\[\boxed{\varepsilon'(E) = (-1)^{\varepsilon_1}} .\]
LaTeX source
\[
\boxed{\varepsilon'(E) = (-1)^{\varepsilon_1}} .
\]\[D\, R f_{!}(E) = R f_{*}(DE)\]
LaTeX source
\[
D\, R f_{!}(E) = R f_{*}(DE)
\]\[DE = \check E(n)[2n] \simeq E(\rho + n)[2n]\]
LaTeX source
\[ DE = \check E(n)[2n] \simeq E(\rho + n)[2n] \]
\[D\, R f_{!}(E) \simeq \bigl(R f_{*}(E)\bigr)(\rho + n)[2n]\]
LaTeX source
\[
D\, R f_{!}(E) \simeq \bigl(R f_{*}(E)\bigr)(\rho + n)[2n]
\]\[\begin{aligned}
\operatorname{cl} D\bigl(R f_{!}(E)\bigr)
&= \bigl(\operatorname{cl} R f_{*}(E)\bigr)(\rho + n) \\
&= \operatorname{cl}\bigl(R f_{!}(E)\bigr)(\rho + n)
+ \operatorname{cl} R f_{\infty}(E)(\rho + n)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\operatorname{cl} D\bigl(R f_{!}(E)\bigr)
&= \bigl(\operatorname{cl} R f_{*}(E)\bigr)(\rho + n) \\
&= \operatorname{cl}\bigl(R f_{!}(E)\bigr)(\rho + n)
+ \operatorname{cl} R f_{\infty}(E)(\rho + n)
\end{aligned}
\]\[(-t)^{-\chi(E)}\, \delta(E)\, L_E\Bigl(\frac{1}{t}\Bigr)
= L_E\bigl(p^{-(\rho+n)} t\bigr)\, L^{\infty}_E\bigl(p^{-(\rho+n)} t\bigr)\]
LaTeX source
\[
(-t)^{-\chi(E)}\, \delta(E)\, L_E\Bigl(\frac{1}{t}\Bigr)
= L_E\bigl(p^{-(\rho+n)} t\bigr)\, L^{\infty}_E\bigl(p^{-(\rho+n)} t\bigr)
\]\[\boxed{L_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr)
= A(t)\, \delta(E)\, (-t)^{\chi(E)} L_E(t)}\]
LaTeX source
\[
\boxed{L_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr)
= A(t)\, \delta(E)\, (-t)^{\chi(E)} L_E(t)}
\]\[\left\lbrace
\begin{array}{l}
A(t) = L^{\infty}_E\Bigl(\dfrac{1}{p^{n+\rho}\, t}\Bigr)^{-1} \\[8pt]
\chi(E) = \sum (-1)^i \operatorname{rang} H^i_!(\bar X, \bar F) \\[6pt]
\delta(E) = \prod_i \bigl(\det f_{H^i_!(\bar X, \bar F)}\bigr)^{(-1)^i}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
A(t) = L^{\infty}_E\Bigl(\dfrac{1}{p^{n+\rho}\, t}\Bigr)^{-1} \\[8pt]
\chi(E) = \sum (-1)^i \operatorname{rang} H^i_!(\bar X, \bar F) \\[6pt]
\delta(E) = \prod_i \bigl(\det f_{H^i_!(\bar X, \bar F)}\bigr)^{(-1)^i}
\end{array}
\right.
\]\[\chi(E) = \varepsilon\bigl(R f_{!}(E)\bigr) , \qquad
\delta(E) = \delta\bigl(R f_{!}(E)\bigr)\]
LaTeX source
\[
\chi(E) = \varepsilon\bigl(R f_{!}(E)\bigr) , \qquad
\delta(E) = \delta\bigl(R f_{!}(E)\bigr)
\]\[\varepsilon\bigl(R f_{!}(E)\bigr) = \varepsilon\bigl(D R f_{!}(E)\bigr) ,\qquad
\delta\bigl(R f_{!}(E)\bigr) = \delta\bigl(D(R f_{!}(E))\bigr)^{-1}\]
LaTeX source
\[
\varepsilon\bigl(R f_{!}(E)\bigr) = \varepsilon\bigl(D R f_{!}(E)\bigr) ,\qquad
\delta\bigl(R f_{!}(E)\bigr) = \delta\bigl(D(R f_{!}(E))\bigr)^{-1}
\]\[D R f_{!}(E) = R f_{*}(DE) = \bigl(R f_{*}(E)\bigr)(\rho + n)[2n]\]
LaTeX source
\[
D R f_{!}(E) = R f_{*}(DE) = \bigl(R f_{*}(E)\bigr)(\rho + n)[2n]
\]\[\left\lbrace
\begin{array}{l}
\chi(E) = \sum (-1)^i \operatorname{rang} R^i f_{*}(E) \\[6pt]
\delta(E) = p^{+(\rho+n)\chi(E)}\, \delta\bigl(R f_{*}(E)\bigr)^{-1}
= p^{(n+\rho)\chi(E)} \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^{i+1}}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\chi(E) = \sum (-1)^i \operatorname{rang} R^i f_{*}(E) \\[6pt]
\delta(E) = p^{+(\rho+n)\chi(E)}\, \delta\bigl(R f_{*}(E)\bigr)^{-1}
= p^{(n+\rho)\chi(E)} \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^{i+1}}
\end{array}
\right.
\]\[\delta(E)^2 = p^{(n+\rho)\chi(E)}\]
LaTeX source
\[
\delta(E)^2 = p^{(n+\rho)\chi(E)}
\]\[\left\lbrace
\begin{array}{l}
\chi(E) = \sum (-1)^i \beta_i(E) \\[6pt]
\delta(E) = \prod_i \delta\bigl(h^i_{!}(E)\bigr)^{(-1)^i}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\chi(E) = \sum (-1)^i \beta_i(E) \\[6pt]
\delta(E) = \prod_i \delta\bigl(h^i_{!}(E)\bigr)^{(-1)^i}
\end{array}
\right.
\]\[\delta\bigl(h^i_{!}(E)\bigr) = \varepsilon_i\, p^{\frac{i \beta_i}{2}}\]
LaTeX source
\[
\delta\bigl(h^i_{!}(E)\bigr) = \varepsilon_i\, p^{\frac{i \beta_i}{2}}
\]\[\left\lbrace
\begin{array}{l}
\delta(E) = \varepsilon(E)\, p^{\frac{1}{2}\sum_i (-1)^i i\, \beta_i(E)} \\[6pt]
\varepsilon(E) = \prod_i \varepsilon_i(E)
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\delta(E) = \varepsilon(E)\, p^{\frac{1}{2}\sum_i (-1)^i i\, \beta_i(E)} \\[6pt]
\varepsilon(E) = \prod_i \varepsilon_i(E)
\end{array}
\right.
\]\[R_{\infty} f(E) = R f_{*}\, j^{*}\bigl(R i_{*}(E)\bigr)\]
LaTeX source
\[
R_{\infty} f(E) = R f_{*}\, j^{*}\bigl(R i_{*}(E)\bigr)
\]\[\begin{aligned}
D\, R_{\infty} f(E) &= D\, R f_{*}(\quad) \\
&= R f_{*}\, D(\quad) \\
&= R f_{*}\, R j^{!}\bigl(D\, R i_{*}(E)\bigr) \\
&= R f_{*}\Bigl(R j^{!}\bigl(R i_{!}(D E)\bigr)\Bigr)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
D\, R_{\infty} f(E) &= D\, R f_{*}(\quad) \\
&= R f_{*}\, D(\quad) \\
&= R f_{*}\, R j^{!}\bigl(D\, R i_{*}(E)\bigr) \\
&= R f_{*}\Bigl(R j^{!}\bigl(R i_{!}(D E)\bigr)\Bigr)
\end{aligned}
\]\[\boxed{D\, R^{*}_{\infty} f(E) = R^{*}_{\infty} f\bigl(D(E)\bigr)\, [1]}\]
LaTeX source
\[
\boxed{D\, R^{*}_{\infty} f(E) = R^{*}_{\infty} f\bigl(D(E)\bigr)\, [1]}
\]\[D(E) \simeq \check E(n) \simeq E(n + \rho)\]
LaTeX source
\[ D(E) \simeq \check E(n) \simeq E(n + \rho) \]
\[\boxed{D\, R^{*}_{\infty} f(E) \simeq R^{*}_{\infty} f(E)\,(n + \rho)\,[1]}\]
LaTeX source
\[
\boxed{D\, R^{*}_{\infty} f(E) \simeq R^{*}_{\infty} f(E)\,(n + \rho)\,[1]}
\]\[(-t)^{-\chi_{\infty}(E)}\, \delta_{\infty}(E)\, L^{\infty}_E(t^{-1})
= L^{\infty}_E\bigl(p^{-(n+\rho)} t\bigr)^{-1}\]
LaTeX source
\[
(-t)^{-\chi_{\infty}(E)}\, \delta_{\infty}(E)\, L^{\infty}_E(t^{-1})
= L^{\infty}_E\bigl(p^{-(n+\rho)} t\bigr)^{-1}
\]\[\boxed{L^{\infty}_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr) L^{\infty}_E(t)\,
(-t)^{\chi_{\infty}(E)}\, \delta_{\infty}(E) = 1}\]
LaTeX source
\[
\boxed{L^{\infty}_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr) L^{\infty}_E(t)\,
(-t)^{\chi_{\infty}(E)}\, \delta_{\infty}(E) = 1}
\]\[\chi_{\infty}(E) = \varepsilon\bigl(R f_{*}(E)\bigr) - \varepsilon\bigl(R f_{!}(E)\bigr)\]
LaTeX source
\[
\chi_{\infty}(E) = \varepsilon\bigl(R f_{*}(E)\bigr) - \varepsilon\bigl(R f_{!}(E)\bigr)
\]\[\delta_{\infty}(E) = \delta\bigl(R f_{*}(E)\bigr)\, \delta\bigl(R f_{!}(E)\bigr)^{-1}\]
LaTeX source
\[
\delta_{\infty}(E) = \delta\bigl(R f_{*}(E)\bigr)\, \delta\bigl(R f_{!}(E)\bigr)^{-1}
\]\[p^{(n+\rho)\chi(E)}\, \delta\bigl(R f_{*}(E)\bigr)^{-1} = \delta\bigl(R f_{!}(E)\bigr)\]
LaTeX source
\[
p^{(n+\rho)\chi(E)}\, \delta\bigl(R f_{*}(E)\bigr)^{-1} = \delta\bigl(R f_{!}(E)\bigr)
\]\[\left\lbrace
\begin{array}{l}
\delta_{\infty}(E) = p^{(n+\rho)\chi(E)}\, \delta(E)^{-2} = \varepsilon(E)^{-2} \\[6pt]
\varepsilon(E) = \delta(E)\, p^{-(n+\rho)\chi(E)/2}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\delta_{\infty}(E) = p^{(n+\rho)\chi(E)}\, \delta(E)^{-2} = \varepsilon(E)^{-2} \\[6pt]
\varepsilon(E) = \delta(E)\, p^{-(n+\rho)\chi(E)/2}
\end{array}
\right.
\]\[\boxed{L^{\infty}_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr) L^{\infty}_E(t)\,
\delta_{\infty}(E) = 1}\]
LaTeX source
\[
\boxed{L^{\infty}_E\Bigl(\frac{1}{p^{n+\rho}\, t}\Bigr) L^{\infty}_E(t)\,
\delta_{\infty}(E) = 1}
\]\[\Bigl(\delta_{\infty}(E) = p^{(n+\rho)\chi(E)}\, \delta(E)^{-2} = \varepsilon(E)^{-2}\Bigr)\]
LaTeX source
\[
\Bigl(\delta_{\infty}(E) = p^{(n+\rho)\chi(E)}\, \delta(E)^{-2} = \varepsilon(E)^{-2}\Bigr)
\]\[\delta(E) = \delta^{*}_{!}(E) , \quad
\delta'(E) = \delta^{*}_{*}(E) , \quad
\varepsilon(E) , \quad
\delta_{\infty}(E) ]\]
LaTeX source
\[
\delta(E) = \delta^{*}_{!}(E) , \quad
\delta'(E) = \delta^{*}_{*}(E) , \quad
\varepsilon(E) , \quad
\delta_{\infty}(E) ]
\]\[\boxed{\delta(E)\,\delta'(E) = q^{\chi(E)}}
\qquad
\boxed{\varepsilon(E) = \delta(E)\, q^{-\chi(E)/2} = \delta'(E)\, q^{-\chi(E)/2}}\]
LaTeX source
\[
\boxed{\delta(E)\,\delta'(E) = q^{\chi(E)}}
\qquad
\boxed{\varepsilon(E) = \delta(E)\, q^{-\chi(E)/2} = \delta'(E)\, q^{-\chi(E)/2}}
\]\[\boxed{\delta_{\infty}(E) = q^{\chi(E)}\, \delta(E)^{-2} = q^{-\chi(E)}\, \delta'(E)^{2}
= \varepsilon(E)^{-2}}\]
LaTeX source
\[
\boxed{\delta_{\infty}(E) = q^{\chi(E)}\, \delta(E)^{-2} = q^{-\chi(E)}\, \delta'(E)^{2}
= \varepsilon(E)^{-2}}
\]\[L\Bigl(\frac{1}{qt}\Bigr) = B(t)\, L(t)\]
LaTeX source
\[
L\Bigl(\frac{1}{qt}\Bigr) = B(t)\, L(t)
\]\[L(t) = B\Bigl(\frac{1}{qt}\Bigr) L\Bigl(\frac{1}{qt}\Bigr)
= B\Bigl(\frac{1}{qt}\Bigr) B(t)\, L(t)\]
LaTeX source
\[
L(t) = B\Bigl(\frac{1}{qt}\Bigr) L\Bigl(\frac{1}{qt}\Bigr)
= B\Bigl(\frac{1}{qt}\Bigr) B(t)\, L(t)
\]\[B(t)\, B\Bigl(\frac{1}{qt}\Bigr) = 1 .\]
LaTeX source
\[
B(t)\, B\Bigl(\frac{1}{qt}\Bigr) = 1 .
\]\[L^{\infty}_E\Bigl(\frac{1}{qt}\Bigr)^{-1} \delta(E)\, (-t)^{\chi(E)}\,
L^{\infty}_E(t)^{-1}\, \delta(E) \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)} = 1\]
LaTeX source
\[
L^{\infty}_E\Bigl(\frac{1}{qt}\Bigr)^{-1} \delta(E)\, (-t)^{\chi(E)}\,
L^{\infty}_E(t)^{-1}\, \delta(E) \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)} = 1
\]\[L^{\infty}_E(t)\, L^{\infty}_E\Bigl(\frac{1}{qt}\Bigr) = \delta(E)^2\, q^{-\chi(E)}\]
LaTeX source
\[
L^{\infty}_E(t)\, L^{\infty}_E\Bigl(\frac{1}{qt}\Bigr) = \delta(E)^2\, q^{-\chi(E)}
\]\[\xi_E(t) = t^{\frac{\chi(E)}{2}} L_E(t)\]
LaTeX source
\[
\xi_E(t) = t^{\frac{\chi(E)}{2}} L_E(t)
\]\[\begin{aligned}
\xi_E\Bigl(\frac{1}{qt}\Bigr)
&= \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)/2} L_E\Bigl(\frac{1}{qt}\Bigr)
= \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)/2} A(t)\, \delta(E)\, (-t)^{\chi(E)} L_E(t) \\
&= q^{-\chi(E)/2} A(t)\, \delta(E)\, (-1)^{\chi(E)}\; t^{\chi(E)/2} L_E(t)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\xi_E\Bigl(\frac{1}{qt}\Bigr)
&= \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)/2} L_E\Bigl(\frac{1}{qt}\Bigr)
= \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)/2} A(t)\, \delta(E)\, (-t)^{\chi(E)} L_E(t) \\
&= q^{-\chi(E)/2} A(t)\, \delta(E)\, (-1)^{\chi(E)}\; t^{\chi(E)/2} L_E(t)
\end{aligned}
\]\[\boxed{\xi_E\Bigl(\frac{1}{qt}\Bigr) = A(t)\, \varepsilon'(E)\, \xi_E(t)}
\qquad q = p^{n\rho}\]
LaTeX source
\[
\boxed{\xi_E\Bigl(\frac{1}{qt}\Bigr) = A(t)\, \varepsilon'(E)\, \xi_E(t)}
\qquad q = p^{n\rho}
\]\[\left\lbrace
\begin{array}{l}
A(t) = L^{\infty}_E\Bigl(\dfrac{1}{qt}\Bigr)^{-1} \\[8pt]
\varepsilon'(E) = \varepsilon(E)\, (-1)^{\chi(E)}
\end{array}
\right.
\qquad
\varepsilon(E) = \delta(E)\, q^{-\chi(E)/2}\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
A(t) = L^{\infty}_E\Bigl(\dfrac{1}{qt}\Bigr)^{-1} \\[8pt]
\varepsilon'(E) = \varepsilon(E)\, (-1)^{\chi(E)}
\end{array}
\right.
\qquad
\varepsilon(E) = \delta(E)\, q^{-\chi(E)/2}
\]\[D(E) = i_{*}(\check E_\eta)(1)\]
LaTeX source
\[
D(E) = i_{*}(\check E_\eta)(1)
\]\[L^{*}_{\check E_\eta}\Bigl(\frac{1}{pt}\Bigr)
= (-t)^{\chi^{*}(\check E_\eta)}\, \delta^{*}(E_\eta)\, L^{*}_{E_\eta}(t)\]
LaTeX source
\[
L^{*}_{\check E_\eta}\Bigl(\frac{1}{pt}\Bigr)
= (-t)^{\chi^{*}(\check E_\eta)}\, \delta^{*}(E_\eta)\, L^{*}_{E_\eta}(t)
\]\[\chi^{*}(E_\eta) = \chi\bigl(i_{*}(E_\eta)\bigr) , \qquad
\delta^{*}(E_\eta) = \delta\bigl(i_{*}(E_\eta)\bigr)\]
LaTeX source
\[
\chi^{*}(E_\eta) = \chi\bigl(i_{*}(E_\eta)\bigr) , \qquad
\delta^{*}(E_\eta) = \delta\bigl(i_{*}(E_\eta)\bigr)
\]\[\check E_\eta = E_\eta(\rho) \qquad \text{d'où} \qquad
i_{*}(\check E_\eta) = i_{*}(E_\eta)(\rho)\]
LaTeX source
\[
\check E_\eta = E_\eta(\rho) \qquad \text{d'où} \qquad
i_{*}(\check E_\eta) = i_{*}(E_\eta)(\rho)
\]\[\boxed{L^{*}_{E_\eta}\Bigl(\frac{1}{qt}\Bigr)
= (-t)^{\chi^{*}(E_\eta)}\, \delta^{*}(E_\eta)\, L^{*}_{E_\eta}(t)}
\qquad (q = p^{1+\rho}\]
LaTeX source
\[
\boxed{L^{*}_{E_\eta}\Bigl(\frac{1}{qt}\Bigr)
= (-t)^{\chi^{*}(E_\eta)}\, \delta^{*}(E_\eta)\, L^{*}_{E_\eta}(t)}
\qquad (q = p^{1+\rho}
\]\[\xi^{*}_{E_\eta}(t) = t^{\chi^{*}(E_\eta)/2} L^{*}_{E_\eta}(t)\]
LaTeX source
\[
\xi^{*}_{E_\eta}(t) = t^{\chi^{*}(E_\eta)/2} L^{*}_{E_\eta}(t)
\]\[\boxed{\xi^{*}_{E_\eta}\Bigl(\frac{1}{qt}\Bigr)
= \varepsilon'^{*}(E_\eta)\, \xi^{*}_{E_\eta}(t)}\]
LaTeX source
\[
\boxed{\xi^{*}_{E_\eta}\Bigl(\frac{1}{qt}\Bigr)
= \varepsilon'^{*}(E_\eta)\, \xi^{*}_{E_\eta}(t)}
\]\[\left\lbrace
\begin{array}{l}
\varepsilon'^{*}(E_\eta) = \delta^{*}(E_\eta)\, q^{-\chi^{*}(E_\eta)/2}\, (-1)^{\chi^{*}(E_\eta)} \\[6pt]
q = p^{1+\rho}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\varepsilon'^{*}(E_\eta) = \delta^{*}(E_\eta)\, q^{-\chi^{*}(E_\eta)/2}\, (-1)^{\chi^{*}(E_\eta)} \\[6pt]
q = p^{1+\rho}
\end{array}
\right.
\]\[0 \to E'_\eta \to E_\eta \to E''_\eta \to 0\]
LaTeX source
\[ 0 \to E'_\eta \to E_\eta \to E''_\eta \to 0 \]
\[0 \to i_{*}(E'_\eta) \to i_{*}(E_\eta) \to i_{*}(E''_\eta) \to Q \to 0\]
LaTeX source
\[
0 \to i_{*}(E'_\eta) \to i_{*}(E_\eta) \to i_{*}(E''_\eta) \to Q \to 0
\]\[Q \simeq \coprod_{x \text{ pt ramifié de } E''_\eta} j_x(\ill{})\]
LaTeX source
\[
Q \simeq \coprod_{x \text{ pt ramifié de } E''_\eta} j_x(\ill{})
\]\[\left\lbrace
\begin{array}{l}
L^{*}_{E_\eta}(t) = L^{*}_{E'_\eta}(t)\, L^{*}_{E''_\eta}(t)\, L_Q(t)^{-1}
= L^{*}_{E'_\eta}(t)\, L^{*}_{E''_\eta}(t) \prod_x P_{Q_x}(t) \\[8pt]
Q_x = \operatorname{Coker}(E_x \to E''_x)
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
L^{*}_{E_\eta}(t) = L^{*}_{E'_\eta}(t)\, L^{*}_{E''_\eta}(t)\, L_Q(t)^{-1}
= L^{*}_{E'_\eta}(t)\, L^{*}_{E''_\eta}(t) \prod_x P_{Q_x}(t) \\[8pt]
Q_x = \operatorname{Coker}(E_x \to E''_x)
\end{array}
\right.
\]\[E \mapsto E^{I} \quad \text{de} \quad
\mathrm{Mod}(D/U, k) \to \mathrm{Mod}(D/I, k)\]
LaTeX source
\[
E \mapsto E^{I} \quad \text{de} \quad
\mathrm{Mod}(D/U, k) \to \mathrm{Mod}(D/I, k)
\]\[\natural_U : R(D/U, k) \to R(D/I, k)\]
LaTeX source
\[ \natural_U : R(D/U, k) \to R(D/I, k) \]
\[R(D/U, k) \longrightarrow R(D, k)\]
LaTeX source
\[ R(D/U, k) \longrightarrow R(D, k) \]
\[\operatorname{Tr}_{E^{\natural}}(\varphi) = \sum_i \operatorname{Tr}_{E_i^{\natural}}(\varphi)\]
LaTeX source
\[
\operatorname{Tr}_{E^{\natural}}(\varphi) = \sum_i \operatorname{Tr}_{E_i^{\natural}}(\varphi)
\]\[\ill{}\ E^{\natural} = \operatorname{Im} \pi , \qquad
\pi = \frac{1}{N} \sum_{i \in I/U} e_i , \qquad N = \operatorname{Card} I/U\]
LaTeX source
\[
\ill{}\ E^{\natural} = \operatorname{Im} \pi , \qquad
\pi = \frac{1}{N} \sum_{i \in I/U} e_i , \qquad N = \operatorname{Card} I/U
\]\[\operatorname{Tr}_{E^{\natural}}(\varphi) = \operatorname{Tr}_E(\pi g)
\struck{= \frac{1}{N} \sum \operatorname{Tr}_E(\ill{})}\]
LaTeX source
\[
\operatorname{Tr}_{E^{\natural}}(\varphi) = \operatorname{Tr}_E(\pi g)
\struck{= \frac{1}{N} \sum \operatorname{Tr}_E(\ill{})}
\]\[\boxed{\operatorname{Tr}_{E^{\natural}}(\varphi) = \frac{1}{N}
\sum_{\substack{\psi_\alpha \in D/U \\ \psi_\alpha \mapsto \varphi \in D/I}}
\operatorname{Tr}_E \psi_\alpha}\]
LaTeX source
\[
\boxed{\operatorname{Tr}_{E^{\natural}}(\varphi) = \frac{1}{N}
\sum_{\substack{\psi_\alpha \in D/U \\ \psi_\alpha \mapsto \varphi \in D/I}}
\operatorname{Tr}_E \psi_\alpha}
\]\[E_\eta(x) = E_\eta^{\natural(x)} \quad \text{comme module sur } \pi_x .\]
LaTeX source
\[
E_\eta(x) = E_\eta^{\natural(x)} \quad \text{comme module sur } \pi_x .
\]\[E_\eta^{\natural} = i_{!}(E_U) + \sum_{x \in Y} j_{x*}\bigl(E_\eta^{\natural(x)}\bigr)\]
LaTeX source
\[
E_\eta^{\natural} = i_{!}(E_U) + \sum_{x \in Y} j_{x*}\bigl(E_\eta^{\natural(x)}\bigr)
\]\[\begin{aligned}
D(E_\eta^{\natural}) &= D\bigl(i_{!}(E_U)\bigr)
+ \sum_{x \in Y} D\bigl(j_{x*}(E_\eta^{\natural(x)})\bigr) \\
&= R i_{*}(D E_U) + \sum_{x \in Y} j_{x*}\bigl(D(E_\eta^{\natural(x)})\bigr)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
D(E_\eta^{\natural}) &= D\bigl(i_{!}(E_U)\bigr)
+ \sum_{x \in Y} D\bigl(j_{x*}(E_\eta^{\natural(x)})\bigr) \\
&= R i_{*}(D E_U) + \sum_{x \in Y} j_{x*}\bigl(D(E_\eta^{\natural(x)})\bigr)
\end{aligned}
\]\[D(E_U) = \check E_U(1)[2] \quad \struck{E_U(\rho+1)} , \quad \text{d'où}\]
LaTeX source
\[
D(E_U) = \check E_U(1)[2] \quad \struck{E_U(\rho+1)} , \quad \text{d'où}
\]\[\begin{aligned}
D(E_\eta^{\natural}) &= \bigl(R i_{*}(\check E_U)\bigr)(1)
+ \sum_{x \in Y} j_{x*}\bigl(\check E_\eta^{\natural(x)}\bigr) \\
&= R i_{!}(\check E_U)(1) + \sum_{x \in Y} j_{x*}\Bigl[\check E_\eta^{\natural(x)}
+ j_x^{*} R i_{*}(\check E_U)(1)\Bigr] \\
&= \check E_\eta^{\natural}(1) + \sum_{x \in Y} j_{x*}\bigl(\mu_x(E)\bigr)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
D(E_\eta^{\natural}) &= \bigl(R i_{*}(\check E_U)\bigr)(1)
+ \sum_{x \in Y} j_{x*}\bigl(\check E_\eta^{\natural(x)}\bigr) \\
&= R i_{!}(\check E_U)(1) + \sum_{x \in Y} j_{x*}\Bigl[\check E_\eta^{\natural(x)}
+ j_x^{*} R i_{*}(\check E_U)(1)\Bigr] \\
&= \check E_\eta^{\natural}(1) + \sum_{x \in Y} j_{x*}\bigl(\mu_x(E)\bigr)
\end{aligned}
\]\[\mu_x(E) = \check E_\eta^{\natural(x)} \otimes \bigl(\mathbb{Q}_\ell - \mathbb{Q}_\ell(1)\bigr)
+ j_x^{*} R i_{\eta}(\check E_\eta)(1)\]
LaTeX source
\[
\mu_x(E) = \check E_\eta^{\natural(x)} \otimes \bigl(\mathbb{Q}_\ell - \mathbb{Q}_\ell(1)\bigr)
+ j_x^{*} R i_{\eta}(\check E_\eta)(1)
\]\[j_x^{*}\bigl(R i_{\eta}(\check E_\eta)\bigr) = \check E_\eta(x) - \bigl(E_\eta(x)\bigr)^{\vee}(-1)\]
LaTeX source
\[
j_x^{*}\bigl(R i_{\eta}(\check E_\eta)\bigr) = \check E_\eta(x) - \bigl(E_\eta(x)\bigr)^{\vee}(-1)
\]\[\boxed{\alpha_x(E_\eta) = E_\eta^{\natural(x)} - E_\eta(x)}\]
LaTeX source
\[
\boxed{\alpha_x(E_\eta) = E_\eta^{\natural(x)} - E_\eta(x)}
\]\[\boxed{\mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(\check E_\eta)(1)}\]
LaTeX source
\[
\boxed{\mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(\check E_\eta)(1)}
\]\[\mu_x(\check E) = \alpha_x(\check E_\eta)^{\vee} - \alpha_x(E_\eta)(1)
= -\mu_x(E)^{\vee}(1) \quad \text{i.e.}\]
LaTeX source
\[
\mu_x(\check E) = \alpha_x(\check E_\eta)^{\vee} - \alpha_x(E_\eta)(1)
= -\mu_x(E)^{\vee}(1) \quad \text{i.e.}
\]\[\struck{\mu_x(\check E) = \mu_x}\]
LaTeX source
\[
\struck{\mu_x(\check E) = \mu_x}
\]\[\left\lbrace
\begin{array}{l}
\mu_x(\check E) = -\mu_x(E)^{\vee}(1) \\
\mu_x(E) = -\mu_x(\check E)^{\vee}(1)
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\mu_x(\check E) = -\mu_x(E)^{\vee}(1) \\
\mu_x(E) = -\mu_x(\check E)^{\vee}(1)
\end{array}
\right.
\]\[\boxed{
\left\lbrace
\begin{array}{l}
D(E_\eta^{\natural}) = \check E_\eta^{\natural}(1)
+ \displaystyle\sum_{x \in X^{(0)}} j_{x*}\bigl(\mu_x(E)\bigr) \\[10pt]
\mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(\check E_\eta)(1) \qquad
\alpha_x(E_\eta) = E_\eta^{\natural(x)} - E_\eta(x)
\end{array}
\right.}\]
LaTeX source
\[
\boxed{
\left\lbrace
\begin{array}{l}
D(E_\eta^{\natural}) = \check E_\eta^{\natural}(1)
+ \displaystyle\sum_{x \in X^{(0)}} j_{x*}\bigl(\mu_x(E)\bigr) \\[10pt]
\mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(\check E_\eta)(1) \qquad
\alpha_x(E_\eta) = E_\eta^{\natural(x)} - E_\eta(x)
\end{array}
\right.}
\]\[\boxed{
\begin{array}{l}
D(E_\eta^{\natural}) = E_\eta^{\natural}(\rho+1)
+ \displaystyle\sum_{x \in X^{(0)}} j_{x*}\bigl(\mu_x(E)\bigr) \\[10pt]
\mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(E_\eta)(\rho+1)
\end{array}}\]
LaTeX source
\[
\boxed{
\begin{array}{l}
D(E_\eta^{\natural}) = E_\eta^{\natural}(\rho+1)
+ \displaystyle\sum_{x \in X^{(0)}} j_{x*}\bigl(\mu_x(E)\bigr) \\[10pt]
\mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(E_\eta)(\rho+1)
\end{array}}
\]\[L_{\check E_\eta}(p^{-1} t) \prod_x \lambda_x(E)
= (-t)^{-\chi^{\natural}(E_\eta)}\, \delta^{\natural}(E_\eta)\, L_{E_\eta}(t^{-1})\]
LaTeX source
\[
L_{\check E_\eta}(p^{-1} t) \prod_x \lambda_x(E)
= (-t)^{-\chi^{\natural}(E_\eta)}\, \delta^{\natural}(E_\eta)\, L_{E_\eta}(t^{-1})
\]\[\left\lbrace
\begin{array}{l}
L_{E_\eta}\Bigl(\dfrac{1}{pt}\Bigr) = \prod_x \lambda_x(E)\,
(-pt)^{\chi^{\natural}(E_\eta)}\, \delta^{\natural}(E_\eta)^{-1}\, L_{\check E_\eta}(t) \\[10pt]
\chi^{\natural}(E_\eta) = \chi(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
\delta^{\natural}(E_\eta) = \delta(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
\lambda_x(E) = L_{\mu_x(E)}(t) = L_{\alpha_x(E_\eta)^{\vee}}(t) / L_{\alpha_x(\check E_\eta)}(p^{-1} t)
= (-t)^{\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
\dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(\check E_\eta)}(t)}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
L_{E_\eta}\Bigl(\dfrac{1}{pt}\Bigr) = \prod_x \lambda_x(E)\,
(-pt)^{\chi^{\natural}(E_\eta)}\, \delta^{\natural}(E_\eta)^{-1}\, L_{\check E_\eta}(t) \\[10pt]
\chi^{\natural}(E_\eta) = \chi(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
\delta^{\natural}(E_\eta) = \delta(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
\lambda_x(E) = L_{\mu_x(E)}(t) = L_{\alpha_x(E_\eta)^{\vee}}(t) / L_{\alpha_x(\check E_\eta)}(p^{-1} t)
= (-t)^{\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
\dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(\check E_\eta)}(t)}
\end{array}
\right.
\]\[\check E \simeq E(\rho) , \qquad L_{\check E}(t) = L_E(p^{-\rho} t) ,\]
LaTeX source
\[
\check E \simeq E(\rho) , \qquad L_{\check E}(t) = L_E(p^{-\rho} t) ,
\]\[\boxed{L_{E_\eta}\Bigl(\frac{1}{qt}\Bigr) = \Bigl(\prod_x \lambda_x(E)(t)\Bigr)\,
\delta^{\natural}(E_\eta)^{-1}\, (-qt)^{\chi^{\natural}(E_\eta)}\, L_E(t)}\]
LaTeX source
\[
\boxed{L_{E_\eta}\Bigl(\frac{1}{qt}\Bigr) = \Bigl(\prod_x \lambda_x(E)(t)\Bigr)\,
\delta^{\natural}(E_\eta)^{-1}\, (-qt)^{\chi^{\natural}(E_\eta)}\, L_E(t)}
\]\[\Bigl[\delta^{\natural}(E_\eta)^{-1} q^{\chi^{\natural}(E_\eta)}
\overset{?}{=} \struck{\varepsilon(E)}\ \delta^{\natural}(E_\eta)
\quad \text{i.e.} \quad
\delta^{\natural}(E_\eta)^2 = q^{\chi^{\natural}(E_\eta)} \ ??\Bigr]
\longleftarrow ??\]
LaTeX source
\[
\Bigl[\delta^{\natural}(E_\eta)^{-1} q^{\chi^{\natural}(E_\eta)}
\overset{?}{=} \struck{\varepsilon(E)}\ \delta^{\natural}(E_\eta)
\quad \text{i.e.} \quad
\delta^{\natural}(E_\eta)^2 = q^{\chi^{\natural}(E_\eta)} \ ??\Bigr]
\longleftarrow ??
\]\[\left\lbrace
\begin{array}{l}
q = p^{1+\rho} \\[4pt]
\lambda_x(E_\eta) = L_{\mu_x(E_\eta)}(t)
= (-t)^{-\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
\dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(E_\eta)}(p^{-\rho} t)} \\[10pt]
\chi_x(E_\eta) = \chi\bigl(\alpha_x(E_\eta)\bigr) \\[4pt]
\delta_x(E_\eta) = \delta\bigl(\alpha_x(E_\eta)\bigr)
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
q = p^{1+\rho} \\[4pt]
\lambda_x(E_\eta) = L_{\mu_x(E_\eta)}(t)
= (-t)^{-\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
\dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(E_\eta)}(p^{-\rho} t)} \\[10pt]
\chi_x(E_\eta) = \chi\bigl(\alpha_x(E_\eta)\bigr) \\[4pt]
\delta_x(E_\eta) = \delta\bigl(\alpha_x(E_\eta)\bigr)
\end{array}
\right.
\]\[L_{E_\eta}\Bigl(\frac{1}{qt}\Bigr) = A(t)\, L_{E_\eta}(t)\]
LaTeX source
\[
L_{E_\eta}\Bigl(\frac{1}{qt}\Bigr) = A(t)\, L_{E_\eta}(t)
\]\[A(t) = \prod_x \frac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(E_\eta)}(p^{-\rho} t)}\;
(-t)^{\chi^{\natural}(E_\eta) - \sum_x \chi_x(E_\eta)}\;
\frac{q^{\chi^{\natural}(E_\eta)}}{\delta^{\natural}(E_\eta) \prod_x \delta_x(E_\eta)^{-1}}\]
LaTeX source
\[
A(t) = \prod_x \frac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(E_\eta)}(p^{-\rho} t)}\;
(-t)^{\chi^{\natural}(E_\eta) - \sum_x \chi_x(E_\eta)}\;
\frac{q^{\chi^{\natural}(E_\eta)}}{\delta^{\natural}(E_\eta) \prod_x \delta_x(E_\eta)^{-1}}
\]\[L_{\alpha_x(E_\eta)}\Bigl(\frac{1}{p^{\rho} t}\Bigr) = A_x(t)\, L_{\alpha_x(E_\eta)}(t)\]
LaTeX source
\[
L_{\alpha_x(E_\eta)}\Bigl(\frac{1}{p^{\rho} t}\Bigr) = A_x(t)\, L_{\alpha_x(E_\eta)}(t)
\]\[\delta^{\natural}(E_\eta)^{-1} = \delta^{\natural}(E_\eta)\,
p^{-\chi^{\natural}(E)(\rho+1)} \prod_x \delta_x\bigl(\mu_x(E)\bigr)\]
LaTeX source
\[
\delta^{\natural}(E_\eta)^{-1} = \delta^{\natural}(E_\eta)\,
p^{-\chi^{\natural}(E)(\rho+1)} \prod_x \delta_x\bigl(\mu_x(E)\bigr)
\]\[\begin{aligned}
\delta\bigl(\mu_x(E)\bigr) &= \delta\bigl(\alpha_x(E_\eta)\bigr)^{-1}
\Bigl[\delta\bigl(\alpha_x(E_\eta)\bigr)\, p^{-\chi_x(E_\eta)(\rho+1)}\Bigr]^{-1} \\
&= \delta_x(E_\eta)^{-2}\, q^{+\chi_x(E_\eta)}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\delta\bigl(\mu_x(E)\bigr) &= \delta\bigl(\alpha_x(E_\eta)\bigr)^{-1}
\Bigl[\delta\bigl(\alpha_x(E_\eta)\bigr)\, p^{-\chi_x(E_\eta)(\rho+1)}\Bigr]^{-1} \\
&= \delta_x(E_\eta)^{-2}\, q^{+\chi_x(E_\eta)}
\end{aligned}
\]\[\Bigl\lbrace \prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}} \Bigr\rbrace
= \delta^{\natural}(E_\eta)^2\, q^{\sum \chi_x(E_\eta)}\]
LaTeX source
\[
\Bigl\lbrace \prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}} \Bigr\rbrace
= \delta^{\natural}(E_\eta)^2\, q^{\sum \chi_x(E_\eta)}
\]\[\left\lbrace
\begin{array}{l}
\dfrac{\delta^{\natural}(E_\eta)^2}{q^{\chi^{\natural}(E)}}
= \displaystyle\prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}} \\[14pt]
\qquad \parallel \\[2pt]
\varepsilon^{\natural}(E_\eta)^2
= \displaystyle\prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\dfrac{\delta^{\natural}(E_\eta)^2}{q^{\chi^{\natural}(E)}}
= \displaystyle\prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}} \\[14pt]
\qquad \parallel \\[2pt]
\varepsilon^{\natural}(E_\eta)^2
= \displaystyle\prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}}
\end{array}
\right.
\]\[\varepsilon^{\natural}(E_\eta)
= \pm \prod_x \frac{\delta_x(E_\eta)}{q^{\chi_x(E_\eta)/2}}
= \Bigl(\prod_x \frac{\delta_x(E_\eta)}{p^{\rho\chi_x(E_\eta)/2}}\Bigr) \frac{1}{p^{\chi/2}}\]
LaTeX source
\[
\varepsilon^{\natural}(E_\eta)
= \pm \prod_x \frac{\delta_x(E_\eta)}{q^{\chi_x(E_\eta)/2}}
= \Bigl(\prod_x \frac{\delta_x(E_\eta)}{p^{\rho\chi_x(E_\eta)/2}}\Bigr) \frac{1}{p^{\chi/2}}
\]\[\varepsilon^{\natural}(E_\eta) = \frac{\delta^{\natural}(E_\eta)}{q^{\chi^{\natural}(E_\eta)/2}}
\qquad\qquad
\chi = \sum_x \chi_x(E_\eta)\]
LaTeX source
\[
\varepsilon^{\natural}(E_\eta) = \frac{\delta^{\natural}(E_\eta)}{q^{\chi^{\natural}(E_\eta)/2}}
\qquad\qquad
\chi = \sum_x \chi_x(E_\eta)
\]\[\varepsilon^{\natural}(E_\eta) = \pm \Bigl(\prod_x \varepsilon_x(E_\eta)\Bigr) p^{-\chi/2}\]
LaTeX source
\[
\varepsilon^{\natural}(E_\eta) = \pm \Bigl(\prod_x \varepsilon_x(E_\eta)\Bigr) p^{-\chi/2}
\]\[\boxed{\cdots \to R^{i} f_{!}(F) \to R^{i} f_{*}(F) \to R^{i}_{\infty} f(F)
\to R^{i+1} f_{!}(F) \to \cdots}\]
LaTeX source
\[
\boxed{\cdots \to R^{i} f_{!}(F) \to R^{i} f_{*}(F) \to R^{i}_{\infty} f(F)
\to R^{i+1} f_{!}(F) \to \cdots}
\]\[R^{i}_{\infty} f(F) = R^{i} \bar f_{*}\bigl(R i_{*}(F)|Y\bigr)\]
LaTeX source
\[
R^{i}_{\infty} f(F) = R^{i} \bar f_{*}\bigl(R i_{*}(F)|Y\bigr)
\]\[L\Bigl(\frac{1}{qt}\Bigr) = A(t) L(t)\]
LaTeX source
\[
L\Bigl(\frac{1}{qt}\Bigr) = A(t) L(t)
\]\[L'(t) = \lambda(t) L(t)\]
LaTeX source
\[ L'(t) = \lambda(t) L(t) \]
\[L'\Bigl(\frac{1}{qt}\Bigr) = \lambda\Bigl(\frac{1}{qt}\Bigr) L\Bigl(\frac{1}{qt}\Bigr)
= \lambda\Bigl(\frac{1}{qt}\Bigr) A(t) L(t)
= \lambda\Bigl(\frac{1}{qt}\Bigr) \lambda(t)^{-1} A(t) L'(t)\]
LaTeX source
\[
L'\Bigl(\frac{1}{qt}\Bigr) = \lambda\Bigl(\frac{1}{qt}\Bigr) L\Bigl(\frac{1}{qt}\Bigr)
= \lambda\Bigl(\frac{1}{qt}\Bigr) A(t) L(t)
= \lambda\Bigl(\frac{1}{qt}\Bigr) \lambda(t)^{-1} A(t) L'(t)
\]\[\frac{\lambda(\frac{1}{qt})}{\lambda(t)} = c A(t)^{-1}\]
LaTeX source
\[
\frac{\lambda(\frac{1}{qt})}{\lambda(t)} = c A(t)^{-1}
\]\[\frac{\alpha(\frac{1}{qt})}{\alpha(t)} = \frac{c'}{c}
\qquad \text{i.e.} \qquad
\alpha\Bigl(\frac{1}{qt}\Bigr) = a\,\alpha(t)\]
LaTeX source
\[
\frac{\alpha(\frac{1}{qt})}{\alpha(t)} = \frac{c'}{c}
\qquad \text{i.e.} \qquad
\alpha\Bigl(\frac{1}{qt}\Bigr) = a\,\alpha(t)
\]\[\boxed{\underline{\lambda}^{g}\, \lambda^{-1} =}\]
LaTeX source
\[
\boxed{\underline{\lambda}^{g}\, \lambda^{-1} =}
\]\[\lambda = \frac{1}{L} \qquad \text{i.e.} \qquad L'(t) = 1 .\]
LaTeX source
\[
\lambda = \frac{1}{L} \qquad \text{i.e.} \qquad L'(t) = 1 .
\]\[L(t) = L_{\rho}(t) \cdots L_{\rho+n}(t)\,
L'_{\rho+n-1}(pt)\, L'_{\rho+n-2}(p^{2}t) \cdots L'_{\rho}(p^{n}t)\]
LaTeX source
\[
L(t) = L_{\rho}(t) \cdots L_{\rho+n}(t)\,
L'_{\rho+n-1}(pt)\, L'_{\rho+n-2}(p^{2}t) \cdots L'_{\rho}(p^{n}t)
\]\[p^{\frac{\rho+n-i}{2}}\, p^{i} = p^{\frac{\rho+n+i}{2}} .\]
LaTeX source
\[
p^{\frac{\rho+n-i}{2}}\, p^{i} = p^{\frac{\rho+n+i}{2}} .
\]\[\begin{array}{l}
\lambda_{\rho}(t) \cdots \lambda_{\rho+n-1}(t)\,
\lambda_{\rho+n-2}(pt)\, \lambda_{\rho+n-1}(pt)\, \lambda_{\rho+n-2}(p^{2}t) \\
\qquad\qquad \cdots \lambda_{\rho}(p^{n}t)
\end{array}\]
LaTeX source
\[
\begin{array}{l}
\lambda_{\rho}(t) \cdots \lambda_{\rho+n-1}(t)\,
\lambda_{\rho+n-2}(pt)\, \lambda_{\rho+n-1}(pt)\, \lambda_{\rho+n-2}(p^{2}t) \\
\qquad\qquad \cdots \lambda_{\rho}(p^{n}t)
\end{array}
\]\[\begin{array}{l}
L\Bigl(\frac{1}{p^{n+\rho}t}\Bigr)
= L'_{\rho}\Bigl(\frac{1}{p^{\rho}t}\Bigr) L'_{\rho+1}\Bigl(\frac{1}{p^{\rho+1}t}\Bigr)
\cdots L'_{\rho+n-1}\Bigl(\frac{1}{p^{n+\rho-1}t}\Bigr) \\[6pt]
\qquad\qquad L_{\rho+n}\Bigl(\frac{1}{p^{\rho+n}t}\Bigr)
L_{\rho+n-1}\Bigl(\frac{1}{p^{n+\rho}t}\Bigr) \cdots L_{\rho}\Bigl(\frac{1}{p^{n+\rho}t}\Bigr)
\end{array}\]
LaTeX source
\[
\begin{array}{l}
L\Bigl(\frac{1}{p^{n+\rho}t}\Bigr)
= L'_{\rho}\Bigl(\frac{1}{p^{\rho}t}\Bigr) L'_{\rho+1}\Bigl(\frac{1}{p^{\rho+1}t}\Bigr)
\cdots L'_{\rho+n-1}\Bigl(\frac{1}{p^{n+\rho-1}t}\Bigr) \\[6pt]
\qquad\qquad L_{\rho+n}\Bigl(\frac{1}{p^{\rho+n}t}\Bigr)
L_{\rho+n-1}\Bigl(\frac{1}{p^{n+\rho}t}\Bigr) \cdots L_{\rho}\Bigl(\frac{1}{p^{n+\rho}t}\Bigr)
\end{array}
\]\[\begin{array}{l}
= \delta'_{\rho}\, t^{\chi'_{\rho}} L'_{\rho}(t) \cdots
\delta'_{\rho+n-1}\, t^{\chi'_{\rho+n-1}} L'_{\rho+n-1}(t) \\[4pt]
\qquad \delta_{\rho+n}\, t^{\chi_{\rho}} L_{\rho+n}(t)\,
\delta_{\rho+n-1}\, t^{\chi_{\rho+n-1}} L_{\rho+n-1}(pt) \cdots
\delta_{\rho}\, t^{\chi_{\rho}} L_{\rho}(p^{n}t) \\[4pt]
= \delta\, t^{\chi} L'_{\rho}(t) \cdots L'_{\rho+n-1}(t)\,
L_{\rho+n}(t)\, L_{\rho+n-1}(pt) \cdots L_{\rho}(p^{n}t)
\end{array}\]
LaTeX source
\[
\begin{array}{l}
= \delta'_{\rho}\, t^{\chi'_{\rho}} L'_{\rho}(t) \cdots
\delta'_{\rho+n-1}\, t^{\chi'_{\rho+n-1}} L'_{\rho+n-1}(t) \\[4pt]
\qquad \delta_{\rho+n}\, t^{\chi_{\rho}} L_{\rho+n}(t)\,
\delta_{\rho+n-1}\, t^{\chi_{\rho+n-1}} L_{\rho+n-1}(pt) \cdots
\delta_{\rho}\, t^{\chi_{\rho}} L_{\rho}(p^{n}t) \\[4pt]
= \delta\, t^{\chi} L'_{\rho}(t) \cdots L'_{\rho+n-1}(t)\,
L_{\rho+n}(t)\, L_{\rho+n-1}(pt) \cdots L_{\rho}(p^{n}t)
\end{array}
\]\[\frac{L(\frac{1}{qt})}{L(t)}
= \delta\, t^{\chi}
\Bigl[\frac{L'_{\rho}(t)}{L'_{\rho}(p^{n}t)} \cdots
\frac{L'_{\rho+n-1}(t)}{L'_{\rho+n-1}(pt)}\Bigr]
\Bigl[\frac{L_{\rho+n-1}(pt)}{L_{\rho+n-1}(t)} \cdots
\frac{L_{\rho}(p^{n}t)}{L_{\rho}(t)}\Bigr]\]
LaTeX source
\[
\frac{L(\frac{1}{qt})}{L(t)}
= \delta\, t^{\chi}
\Bigl[\frac{L'_{\rho}(t)}{L'_{\rho}(p^{n}t)} \cdots
\frac{L'_{\rho+n-1}(t)}{L'_{\rho+n-1}(pt)}\Bigr]
\Bigl[\frac{L_{\rho+n-1}(pt)}{L_{\rho+n-1}(t)} \cdots
\frac{L_{\rho}(p^{n}t)}{L_{\rho}(t)}\Bigr]
\]\[A(t) = \delta\, t^{\chi}
\prod_i \frac{L'_i(t)}{L'_i(p^{\rho+n-i}t)}
\prod_i \frac{L_i(p^{\rho+n-i}t)}{L_i(t)}\]
LaTeX source
\[
A(t) = \delta\, t^{\chi}
\prod_i \frac{L'_i(t)}{L'_i(p^{\rho+n-i}t)}
\prod_i \frac{L_i(p^{\rho+n-i}t)}{L_i(t)}
\]\[\underbrace{L_{\rho}(t)}\, L_{\rho+1}(t)\, \underbrace{L'_{\rho}(pt)}\]
LaTeX source
\[
\underbrace{L_{\rho}(t)}\, L_{\rho+1}(t)\, \underbrace{L'_{\rho}(pt)}
\]\[L_{\rho}(t)\, L'_{\rho}(pt)\]
LaTeX source
\[
L_{\rho}(t)\, L'_{\rho}(pt)
\]