Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 1

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

DOI: http://dx.doi.org/10.4310/PAMQ.2017.v13.n1.a5

Authors

Vladimir Drinfeld (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

Kiran S. Kedlaya (Department of Mathematics, University of California at San Diego)

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 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 $p \gt 0$.

Keywords

$F$-isocrystal, local system, slope, Newton polygon, Frobenius, hypergeometric sheaf

2010 Mathematics Subject Classification

Primary 11F80, 14F30. Secondary 11F70, 14G15.

Full Text (PDF format)

Received 26 January 2017

Published 14 September 2018