##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Thomas Grubb

#### UCSD

## 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.

### October 21, 2021

### 2:00 PM

APM 6402 and Zoom; see https://www.math.ucsd.edu/$\sim$nts/

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