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# Mathematical Research Letters

## Volume 10 (2003)

### Number 2

### Semistable reduction for overconvergent $F$-isocrystals on a curve

Pages: 151 – 159

DOI: http://dx.doi.org/10.4310/MRL.2003.v10.n2.a2

#### Author

#### Abstract

Let $X$ be a smooth affine curve over a field $k$ of characteristic $p >0$ and $\calE$ an overconvergent $F^a$-isocrystal on $X$ for some positive integer $a$. We prove that after replacing $k$ by some finite purely inseparable extension, there exists a finite separable morphism $X' \to X$, the pullback of $\mathcal E$ along which extends to a log-$F^a$-isocrystal on a smooth compactification of $X'$. This resolves a weak form of the global version of a conjecture of Crew; the proof uses the local version of the conjecture, established (separately) by André, Mebkhout and the author.