Slope filtrations of $F$-isocrystals and logarithmic decay

Let $k$ be a perfect field of positive characteristic and let $X$ be a smooth irreducible quasi-compact scheme over $k$. The Drinfeld–Kedlaya theorem states that for an irreducible $F$-isocrystal on $X$, the gap between consecutive generic slopes is bounded by one. In this note we provide a new proof of this theorem. Our proof utilizes the theory of $F$‑isocrystals with $r$‑log decay. We first show that a rank one $F$‑isocrystal with $r$‑log decay is overconvergent if $r \lt 1$. Next, we establish a connection between slope gaps and the rate of log-decay of the slope filtration. The Drinfeld–Kedlaya theorem then follows from a patching argument.

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Received 26 February 2019

Accepted 15 August 2019

Published 24 May 2022