Product formula forp-adic epsilon factors

Abstract

AbstractLet$X$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.