Pure and Applied Mathematics Quarterly
Volume 13 (2017)
Special Issue in Honor of Yuri Manin: Part 1
Guest Editors: Lizhen Ji, Kefeng Liu, Yuri Tschinkel, and Shing-Tung Yau
Slopes of indecomposable $F$-isocrystals
Pages: 131 – 192
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 result of V. Lafforgue on the slopes of Frobenius of an $\ell$-adic local system.
We also prove similar statements for $G$-local systems (crystalline and $\ell$-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 $p \gt 0$.
$F$-isocrystal, local system, slope, Newton polygon, Frobenius, hypergeometric sheaf
2010 Mathematics Subject Classification
Primary 11F80, 14F30. Secondary 11F70, 14G15.
Received 26 January 2017
Published 14 September 2018