Slopes of indecomposable $F$-isocrystals

Vladimir Drinfeld, Kiran S. Kedlaya
2017 Pure and Applied Mathematics Quarterly  
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 non-zero Hodge numbers.) As an application, we deduce a refinement of a
more » ... ult of V. Lafforgue on the slopes of Frobenius of an -adic local system. We also prove similar statements for G-local systems (crystalline and -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.
doi:10.4310/pamq.2017.v13.n1.a5 fatcat:s27hvghmkrfkjks3ii2wlg5d6m