Existence of compatible systems of lisse sheaves on arithmetic schemes

Abstract

Deligne conjectured that a single $\ell$-adic lisse sheaf on a normal variety over a finite field can be embedded into a compatible system of ${\ell }^{\prime }$-adic lisse sheaves with various ${\ell }^{\prime }$. 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 $\mathbb{Z}$ and prove some cases using Lafforgue’s result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor.