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We can view the Langlands program in each of its versions (local/global, arithmetic/geometric) as giving a description of the finite-dimensional representations of the étale fundamental group of a scheme $X$ over a field $k$ into a connected reductive group $G$ (the Galois side) in terms of certain functions/sheaves on a space built from the dual group of $G$ and $X$ (the automorphic side.)

In each version the scheme $X$ considered is of the lowest dimension in its context — a number ring, a $p$-adic ring, a curve over a finite field, or a curve over complex numbers.

Can we expect that this kind of description is possible for representations of fundamental groups of higher-dimensional schemes as well? What would the automorphic side look like in that case?

Is there any proposal for a description of the finite-dimensional representations of $\pi_1^\text{ét}(X)$ where $X$ is complex algebraic surface, or a curve over $\mathbb{Z}$, or a curve over $\mathbb{\overline{Q}}_p$ for that matter which in a sense is a counterpart to the curve over $\mathbb{C}$ case (geometric Langlands)?

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    $\begingroup$ This is tangentially related at best but in case you are not familiar with it, you may be interested in anabelian geometry, which does manage to cover a lot of such higher dimensional "arithmetic" objects, although it is less concerned with representations of these groups but rather what information the groups themselves carry. $\endgroup$
    – Wojowu
    Commented Oct 5 at 14:06
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    $\begingroup$ I am not an expert, but I know that there is a lot of work on higher dimensional class field theory. Since class field theory is concerned with the abelianisation of the fundamental group/Galois group, this is classically considered the $\operatorname{GL}_1$-case of the Langlands programme. But this is clearly a lot easier, and the tools often look very different from the representation-theoretic and category-theoretic techniques in the Langlands programme. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 5 at 15:34
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    $\begingroup$ There are lots of higher dimensional abelian duality theorems. However for G nonabelian the basic problem is you can only have low (H^0 and H^1) degree cohomology with coefficients in G, while the duality will try to pair it with a higher cohomology group which doesn't make sense in any obvious way. There have however been various attempts, most notably by Kapranov and Kontsevich, to make sense of higher dimensional Langlands conjectures. $\endgroup$
    – David Ben-Zvi
    Commented Oct 6 at 3:13

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