Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
-
Let $X$ be a smooth scheme over a finite field of characteristic $p$.Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adicWeil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell\neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$.Using the Langlands correspondence for global function fields in both the\'etale and crystalline settings (work of Lafforgue and Abe, respectively), onesees that on a curve, any coefficient object in one category has "companions"in the other categories with matching characteristic polynomials of Frobeniusat closed points. A similar statement is expected for general $X$; building onwork of Deligne, Drinfeld showed that any \'etale coefficient object has\'etale companions. We adapt Drinfeld's method to show that any crystallinecoefficient object has \'etale companions; this has been shown independently byAbe--Esnault. We also prove some auxiliary results relevant for theconstruction of crystalline companions of \'etale coefficient objects; thissubject will be pursued in a subsequent paper.more » « less
-
Anni, Samuele ; Karemaker, Valentijn ; Lorenzo García, Elisa (Ed.)