Do we have a finite coefficients Langlands correspondence for function fields? By which I mean a bijection between Galois representations $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)$$ and "mod $p$" automorphic forms, where $X$ is a smooth projective curve over a finite field (presumably there is a distinction to be made if $p$ is the characteristic of $X$ or not, but I'm interested in either case). I'm sure there is a nice survey on this question out there, if you could point in the right direction that'd be great!
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$\begingroup$ Good question, but as Langlands' geometric philosophy dictates, we should (perhaps?) ask it for locally constant residual $p$-adic sheaves instead of mod $p$ automorphic forms. $\endgroup$– Marsault ChabatOct 18, 2022 at 17:25
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$\begingroup$ @MarsaultChabat (perhaps) Indeed. I genuinely don't know what to expect. I just decided to say mod p automorphic forms because one would probably like to have some functoriality, in particular between p-adic and mod p coefficients $\endgroup$– curious math guyOct 18, 2022 at 18:49
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