Local acyclicity in $p$-adic cohomology

TL;DR

This paper establishes a $p$-adic analogue of the Deligne--Laumon theorem, characterizing when the cohomology sheaves of an overconvergent $F$-isocrystal on a curve are overconvergent, based on Swan conductor conditions.

Contribution

It proves a new criterion linking overconvergence of cohomology sheaves to constant Swan conductors for $p$-adic coefficients on curves.

Findings

Cohomology sheaves are overconvergent if and only if Swan conductor is constant at infinity.

Provides a $p$-adic analogue of a classical theorem for curves.

Characterizes local acyclicity in the context of $p$-adic cohomology.

Abstract

We prove an analogue for $p$-adic coefficients of the Deligne--Laumon theorem on local acyclicity for curves. That is, for an overconvergent $F$-isocrystal $E$ on a relative curve $f:U\rightarrow S$ admitting a good compactification, we show that the cohomology sheaves of $\mathbf{R}f_!E$ are overconvergent isocrystals if and only if $E$ has constant Swan conductor at infinity.