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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?

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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

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    $\begingroup$ welcome to mathoverflow, professor Lafforgue! $\endgroup$ – Venkataramana Feb 19 at 14:24
  • $\begingroup$ welcome et bienvenue $\endgroup$ – Abdelmalek Abdesselam 2 days ago

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