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arXiv · 1612.01164

The monodromy of $F$-isocrystals with log-decay

Joe Kramer-Miller
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Abstract

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 at each $x \in X-U$. 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 $F$-isocrystals in terms of asymptotic properties of higher ramification groups.

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Joe Kramer-Miller. 2016-12-04. The monodromy of $F$-isocrystals with log-decay. https://arxiv.org/abs/1612.01164

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