Congruence and product formula for local constants
(Un exposé de Seidai Yasuda au séminaire de théorie
des nombres de Chevaleret le 10 février 2003)
Résumé :
The theory of local $\epsilon$-factors is generalized in the following
way.
Let $K$ be a complete discrete valuation field whose residue field
$k$ is perfect of
positive characteristic $p$. Let $W_K$ be the Weil group of $K$.
For each triple $(R,(\rho,V),\psi)$, where $R$ is a noetherian local
ring
with algebraically closed residue field of characteristic $\neq p$
such that
the $p$-power map $R^{\times}\to R^{\times}$ is surjective,
$(\rho,V)$ is a continuous representations of $W_K$ on
a finitely generated free $R$-module $V$,
and $\psi:K\to R^{\times}$ is a non-trivial additive character
sheaf, I defined the local $\epsilon_0$-character
$\epsilon_{0,R}(V,\psi)$ of $(R,(\rho,V),\psi)$
and discussed its basic properties. As applications,
I generalized Deligne-Laumon's product formula
describing the determinants of cohomologies of
$l$-adic etale sheaves on curves over finite fields,
and Saito's formula for describing the determinants of cohomologies
of
tamely ramified $l$-adic etale sheaves on varieties over finite fields,
to the case of $\Lambda$-sheaves for pro-finite rings $\Lambda$.