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Product formula for p -adic epsilon factors

Part of: Algebraic number theory: local and $p$-adic fields (Co)homology theory

Published online by Cambridge University Press:  08 May 2014

Tomoyuki Abe  and
Adriano Marmora
Tomoyuki Abe
Affiliation:
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan ([email protected])
Adriano Marmora
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France ([email protected])
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Abstract

Let $X$ be a smooth proper curve over a finite field of characteristic $p$. We prove a product formula for $p$-adic epsilon factors of arithmetic $\mathscr{D}$-modules on $X$. In particular we deduce the analogous formula for overconvergent $F$-isocrystals, which was conjectured previously. The $p$-adic product formula is a counterpart in rigid cohomology of the Deligne–Laumon formula for epsilon factors in $\ell$-adic étale cohomology (for $\ell \neq p$). One of the main tools in the proof of this $p$-adic formula is a theorem of regular stationary phase for arithmetic $\mathscr{D}$-modules that we prove by microlocal techniques.

Keywords

rigid cohomologyp-adic differential equationsarithmetic D-modulesp-adic Fourier transformepsilon factors

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 11S40: Zeta functions and $L$-functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 2 , April 2015 , pp. 275 - 377
Copyright
© Cambridge University Press 2014 

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