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?
13
$\begingroup$
$\endgroup$
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/Exp1156Cadoret.pdf
New contributor

5$\begingroup$ welcome to mathoverflow, professor Lafforgue! $\endgroup$ – Venkataramana Feb 19 at 14:24
