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Product formula for p -adic epsilon factors
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 14 / Issue 2 / April 2015
- Published online by Cambridge University Press:
- 08 May 2014, pp. 275-377
- Print publication:
- April 2015
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- Article
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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.
