Slopes of indecomposable F-isocrystals

TL;DR

This paper establishes a bound on the gaps between slopes of indecomposable F-isocrystals over smooth varieties in characteristic p, with applications to Frobenius slopes, G-local systems, and automorphic representations.

Contribution

It proves a new bound on the slope gaps of indecomposable F-isocrystals and extends the results to G-local systems and automorphic forms in characteristic p.

Findings

Gap between consecutive slopes at the generic point is at most 1.

Refinement of Lafforgue's result on Frobenius slopes of l-adic local systems.

Properties of p-adic absolute values of Hecke eigenvalues for automorphic representations.

Abstract

We prove that for an indecomposable convergent or overconvergent F-isocrystal on a smooth irreducible variety over a perfect field of characteristic p, the gap between consecutive slopes at the generic point cannot exceed 1. (This may be thought of as a crystalline analogue of the following consequence of Griffiths transversality: for an indecomposable variation of complex Hodge structures, there cannot be a gap between nonzero Hodge numbers.) As an application, we deduce a refinement of a result of V.Lafforgue on the slopes of Frobenius of an l-adic local system. We also prove similar statements for G-local systems (crystalline and l-adic ones), where G is a reductive group. We translate our results on local systems into properties of the p-adic absolute values of the Hecke eigenvalues of a cuspidal automorphic representation of a reductive group over the adeles of a global field of characteristic p>0.