Log-growth filtration and Frobenius slope filtration of \(F\)-isocrystals at the generic and special points
type: Item
instance of: scholarly article
title: Log-growth filtration and Frobenius slope filtration of \(F\)-isocrystals at the generic and special points
zbMATH Open document ID: 1242.12006
published in: Documenta Mathematica
publication date: 2011-05-05T00:00:00Z
full work available at URL: 02.html
review text: The goal of this paper is to study the relation between the Frobenius slope filtration and the log-growth filtration on a \(\varphi\)-\(\nabla\)-module. The authors prove that, under certain conditions, the two filtrations agree at both ``the generic point'' and ``the special point''. From this, they deduce the truth of a conjecture of Dwork in the special case in which the generic log-growth polygon lies above the special log-growth polygon. This follows as a corollary of the corresponding statement for Frobenius slope polygons. To be more precise, let \(K\) be a complete discrete valuation field of mixed characteristic, equipped with an action \(\varphi\) lifting the Frobenius action on the residue field. Let \(K\lfloor x \rfloor_0\) denote the ring of formal power series with bounded coefficients in \(K\) and \(\mathcal E\) the \(p\)-adic completion of the fraction field of \(K\lfloor x \rfloor_0\). For \(\lambda \in \mathbb R_{\geq 0}\), let \(K \lfloor x\rfloor_\lambda\) denote the ring of power series of log-growth \(\lambda\). Let \(\mathcal A_K(0,1^-)\) denote the open unit disc. For a \(\varphi\)-\(\nabla\)-module \(M\) over \(K\lfloor x\rfloor_0\), let \(\mathrm{Sol}_\lambda(M)\) denote the \(K\)-vector space of solutions \(\mathrm{Hom}_{K\lfloor x\rfloor_0[\nabla]}(M, K\lfloor x\rfloor_\lambda)\). On one hand, one may define a decreasing filtration \(V^\bullet(M)\) on the set of horizontal sections \(V(M)\) of \(M\) over \(\mathcal A_K(0,1^-)\), by taking the orthogonal space of \(\mathrm{Sol}_\bullet(M)\) under the natural pairing. The special log-growth polygon is formed by taking the ranks of the successive quotients of the decreasing filtration \(V^\bullet(M)\). On the other hand, the \(\varphi\)-action on \(V(M)\) induces an increasing slope filtration \(S_\bullet(V(M))\). If \(M'\) is a \(\varphi\)-\(\nabla\)-module over \(\mathcal E\), then pulling it back to the open unit disc around a generic point \(t\) of radius \(1\) defines a \(\varphi\)-\(\n…
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 5884792
zbMATH Keywords: p-adic differential equations
zbMATH Keywords: logarithmic growth
zbMATH Keywords: Frobenius slopes
zbMATH Keywords: Newton polygon
author: Bruno Chiarellotto
author: Nobuo Tsuzuki
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5884792