Some remarks on the companions conjecture for normal varieties

Abstract

Drinfeld in 2010 proved the companions conjecture for smooth varieties over a finite field, generalizing L. Lafforgue’s result for smooth curves. We study the obstruction to prove the conjecture for arbitrary normal varieties. To do this, we introduce a new property of morphisms. We verify this property in some cases, showing thereby the companions conjecture for some singular normal varieties.

Keywords

• $\ell $-adic representation
• independence of $\ell $
• étale fundamental group