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

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.

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Published 1 January 2003