Existence of compatible systems of lisse sheaves on arithmetic schemes

Shimizu, Koji

doi:10.2140/ant.2017.11.181

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Existence of compatible systems of lisse sheaves on arithmetic schemes

Koji Shimizu

Vol. 11 (2017), No. 1, 181–211

DOI: 10.2140/ant.2017.11.181

Abstract

Deligne conjectured that a single $ℓ$ -adic lisse sheaf on a normal variety over a finite field can be embedded into a compatible system of $ℓ^{'}$ -adic lisse sheaves with various $ℓ^{'}$ . Drinfeld used Lafforgue’s result as an input and proved this conjecture when the variety is smooth. We consider an analogous existence problem for a regular flat scheme over $ℤ$ and prove some cases using Lafforgue’s result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor.

Keywords

arithmetic geometry, lisse sheaves, compatible system

Mathematical Subject Classification 2010

Primary: 11G35

Secondary: 11F80

Milestones

Received: 22 February 2016

Revised: 19 October 2016

Accepted: 17 November 2016

Published: 23 January 2017

Authors

Koji Shimizu
Department of Mathematics Harvard University Cambridge, MA 02138 United States

Vol. 11, No. 1, 2017