Let U be a smooth geometrically connected affine curve over F p with compactification X. Following Dwork and Katz, a p-adic representation ρ of π 1 (U ) corresponds to an F -isocrystal. By work of Tsuzuki and Crew an F -isocrystal is overconvergent precisely when ρ has finite monodromy at each x ∈ X − U . However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log growth Fisocrystals in terms of asymptotic properties of higher ramification groups.
Let $C$ be a smooth curve over a finite field of characteristic $p$ and let $M$ be an overconvergent $\mathbf {F}$ -isocrystal over $C$ . After replacing $C$ with a dense open subset, $M$ obtains a slope filtration. This is a purely $p$ -adic phenomenon; there is no counterpart in the theory of lisse $\ell$ -adic sheaves. The graded pieces of this slope filtration correspond to lisse $p$ -adic sheaves, which we call geometric. Geometric lisse $p$ -adic sheaves are mysterious, as there is no $\ell$ -adic analogue. In this article, we study the monodromy of geometric lisse $p$ -adic sheaves with rank one. More precisely, we prove exponential bounds on their ramification breaks. When the generic slopes of $M$ are integers, we show that the local ramification breaks satisfy a certain type of periodicity. The crux of the proof is the theory of $\mathbf {F}$ -isocrystals with log-decay. We prove a monodromy theorem for these $\mathbf {F}$ -isocrystals, as well as a theorem relating the slopes of $M$ to the rate of log-decay of the slope filtration. As a consequence of these methods, we provide a new proof of the Drinfeld–Kedlaya theorem for irreducible $\mathbf {F}$ -isocrystals on curves.
In this article we study the behavior of Newton polygons along Z p -towers of curves. Fix an ordinary curve X over a finite field F q of characteristic p. By a Z p -tower X ∞ /X we mean a tower of coversWe show that if the ramification along the tower is sufficiently moderate, then the slopes of the Newton polygon of X n are equidistributed in the interval [0, 1] as n tends to ∞. Under a stronger congruence assumption on the ramification invariants, we completely determine the slopes of the Newton polygon of each curve. This is the first result towards 'regularity' in Newton polygon behavior for Z p -towers over higher genus curves. We also obtain similar results for Z p -towers twisted by a generic tame character.
Let k be a perfect field of positive characteristic and let X be a smooth irreducible quasicompact scheme over k. The Drinfeld-Kedlaya theorem states that for an irreducible Fisocrystal 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 < 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 simple patching argument.
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