Vol. 11, No. 1, 2017

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Existence of compatible systems of lisse sheaves on arithmetic schemes

Koji Shimizu

Vol. 11 (2017), No. 1, 181–211
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 $ℤ$ 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
Primary: 11G35
Secondary: 11F80