# The monodromy of unit-root $F$-isocrystals with geometric origin

## TL;DR

This paper investigates the monodromy of geometric rank-one $p$-adic sheaves over curves, revealing exponential growth of ramification breaks and confirming predictions of polynomial behavior in special cases, advancing understanding of $F$-isocrystals.

## Contribution

It introduces a new approach using $F$-isocrystals with log-decay to analyze monodromy, proving exponential ramification growth and polynomial predictions in the ordinary case.

## Key findings

- Ramification breaks grow exponentially for geometric $p$-adic sheaves.
- In the ordinary case, ramification breaks follow polynomial patterns in $p^n$.
- Provides a new proof of the Drinfeld-Kedlaya theorem for irreducible $F$-isocrystals.

## Abstract

Let $C$ be a smooth curve over a finite field in characteristic $p$ and let $M$ be an overconvergent $F$-isocrystal over $C$. After replacing $C$ with a dense open subset $M$ obtains a slope filtration, whose steps interpolate the Frobenius eigenvalues of $M$ with bounded slope. 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. While they fit together to build an overconvergent $F$-isocrystal, which should have motivic origin, individually they are not motivic. In this article we study the monodromy of geometric lisse $p$-adic sheaves with rank one. We prove that the ramification breaks grow exponentially. In the case where $M$ is ordinary we prove that the ramification breaks are predicted by polynomials in $p^n$, which implies a variant of Wan's genus stability conjecture. The crux of the proof is the theory of $F$-isocrystals with log-decay. We prove a monodromy theorem for these $F$-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 $F$-isocrystals on curves.