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Facteurs epsilon p-adiques

Published online by Cambridge University Press:  01 March 2008

Adriano Marmora*
Affiliation:
LAGA, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France (email: [email protected])
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Abstract

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We develop and study the epsilon factor of a ‘local system’ of p-adic coefficients over the spectrum of a complete discrete valuation field K with finite residue field of characteristic p>0. In the equal characteristic case, we define the epsilon factor of an overconvergent F-isocrystal over Spec(K), using the p-adic monodromy theorem. We conjecture a global formula, the p-adic product formula, analogous to Deligne’s formula for étale -adic sheaves proved by Laumon, which explains the importance of this local invariant. Namely, for an overconvergent F-isocrystal over an open subset of a projective smooth curve X, the constant of the functional equation of the L-series is expressed as a product of the local epsilon factors at the points of X. We prove the conjecture for rank-one overconvergent F-isocrystals and for finite unit-root overconvergent F-isocrystals. In the mixed characteristic case, we study the behavior of the epsilon factor by deformation to the field of norms.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2008

References

Part of this work has been supported by JSPS postdoctoral fellowship number PE06005.