2018/12/06 by Joe Kramer-Miller, Kramer-Miller, Joe
Mathematics · #11G20 #11R58 #14H25 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1812.02803
openalex publication_date 2018/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℓ-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 pn, 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.