Deligne conjectured that a single -adic lisse sheaf on a normal variety over a finite field can be embedded into a compatible system of -adic lisse sheaves with various . 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 and prove some cases using Lafforgue’s result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor.
"Existence of compatible systems of lisse sheaves on arithmetic schemes." Algebra Number Theory 11 (1) 181 - 211, 2017. https://doi.org/10.2140/ant.2017.11.181