A cut-by-curves criterion for overconvergence of $F$-isocrystals

Abstract: Let $X$ be a smooth, geometrically irreducible scheme over a finite field of characteristic $p > 0$. With respect to rigid cohomology, $p$-adic coefficient objects on $X$ come in two types: convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals. Overconvergent isocrystals are related to $\ell$-adic etale objects ($\ell\neq p$) via companions theory, and as such it is desirable to understand when an isocrystal is overconvergent. We show (under a geometric tameness hypothesis) that a convergent $F$-isocrystal $E$ is overconvergent if and only if its restriction to all smooth curves on $X$ is. The technique reduces to an algebraic setting where we use skeleton sheaves and crystalline companions to compare $E$ to an isocrystal which is patently overconvergent. Joint with Kiran Kedlaya and James Upton.

number theory

Audience: researchers in the topic

Comments: pre-talk at 1:30

UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.