Let $X$ be a normal variety over a finite field $F\_q$. Fix a prime number $l$ relatively prime to $q$. Let $\sigma$ be a irreducible lisse $l$-adic sheaf on $X$ whose determinant has finite order. It has been proved by L. Lafforgue that for every closed point $x\in |X|$, the roots of the polynomial $\mathrm{det}\_{\sigma}(\mathrm{Id}-yFrob\_{x}^{-1})$ are algebraic numbers of absolute value 1. Is there a proof of this statement independent of Langlands correspondence over function fields? 



Deligne conjecture without Langlands correspondence
I do not know any proof independent of the Langlands correspondence. 
Anna Cadoret gave recently a Bourbaki talk on related works:  

<http://www.bourbaki.ens.fr/TEXTES/Exp1156-Cadoret.pdf> 


