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Fourier transforms and $p$-adic ‘Weil II’

Published online by Cambridge University Press:  24 November 2006

Kiran S. Kedlaya
Kiran S. Kedlaya
Affiliation:
Department of Mathematics, Room 2-165, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, [email protected]
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Abstract

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We give a purity theorem in the manner of Deligne's ‘Weil II’ theorem for rigid cohomology with coefficients in an overconvergent $F$-isocrystal; the proof mostly follows Laumon's Fourier-theoretic approach, transposed into the setting of arithmetic $\mathcal{D}$-modules. This yields in particular a complete, purely $p$-adic proof of the Weil conjectures when combined with recent results on $p$-adic differential equations by André, Christol, Crew, Kedlaya, Matsuda, Mebkhout and Tsuzuki.

Keywords

rigid cohomologyFourier transformWeil conjectures14F3014G10
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 6 , November 2006 , pp. 1426 - 1450
Copyright
Foundation Compositio Mathematica 2006