Let $U$ be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification $X$. Following Dwork and Katz, a $p$-adic representation $\rho$ of $\pi_1(U)$ corresponds to an etale F-isocrystal. By work of Tsuzuki and Crew an F-isocrystal is overconvergent precisely when $\rho$ has finite monodromy. However, in practice most F-isocrystals arising geometrically are not overconvergent and instead have logarithmic decay at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log decay F-isocrystals in terms of asymptotic properties of higher ramification groups. We apply this theory to a conjecture of Daqing Wan.
