Existence of compatible systems of lisse sheaves on arithmetic schemes

TL;DR

This paper investigates the existence of compatible systems of lisse sheaves on arithmetic schemes, extending Deligne's conjecture beyond smooth varieties to regular flat schemes over Z, using recent advances in the field.

Contribution

It proves cases of the existence of compatible systems of lisse sheaves on arithmetic schemes, generalizing previous results from smooth varieties to regular flat schemes over Z.

Findings

Proved cases of the conjecture for regular flat schemes over Z

Extended the existence of compatible systems beyond smooth varieties

Utilized recent results by Lafforgue and others

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.