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Formule du conducteur pour un caractère l-adique

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  01 May 2009

Isabelle Vidal
Isabelle Vidal*
Affiliation:
LAGA-Institut Galilée, Université Paris 13, 99 av. J.-B. Clément, 93430 Villetaneuse, France (email: [email protected])
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Abstract

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Let K be a local field of equal characteristic p>2, let XK/K be a smooth proper relative curve, and let ℱ be a rank 1 smooth l-adic sheaf (l≠p) on a dense open subset UK⊂XK. In this paper, under some assumptions on the wild ramification of ℱ, we prove a conductor formula that computes the Swan conductor of the etale cohomology of the vanishing cycles of ℱ. Our conductor formula is a generalization of the conductor formula of Bloch, but for non-constant coefficients.

Résumé

Soit K un corps local d’égale caractéristique p>2, XK/K une courbe relative propre et lisse, ℱ un caractère l-adique (avec l≠p) lisse sur un ouvert dense UK⊂XK. Dans cet article, sous certaines hypothèses sur la ramification sauvage de ℱ, on prouve une formule qui calcule le conducteur de Swan de la cohomologie des cycles évanescents de ℱ. Notre formule du conducteur est une généralisation, pour des coefficients non constants, de la formule du conducteur de Bloch.

Keywords

ramificationvanishing cyclesSwan conductor

MSC classification

Secondary: 14G20: Local ground fields 11G25: Varieties over finite and local fields 11S15: Ramification and extension theory
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 687 - 717
Copyright
Copyright © Foundation Compositio Mathematica 2009

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