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Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$ for $n\ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves?

I've never found such things in articles such like "What is a reciprocity law?", Ana Caraiani's "Higher-dimensional reciprocity laws" or "Reciprocity laws and Galois representations: recent breakthroughs".

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  • $\begingroup$ Do you mean an application of the conjecture or of known theorems toward the conjecture? $\endgroup$
    – Will Sawin
    Commented Apr 2 at 17:48
  • $\begingroup$ I mean application $\endgroup$
    – Cloudifold
    Commented Apr 2 at 17:55
  • $\begingroup$ For genus $2$ curves there is the recent work of Boxer--Calegari--Gee--Pilloni, see arxiv.org/abs/1812.09269. $\endgroup$
    – Pol van Hoften
    Commented Apr 2 at 18:49
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    $\begingroup$ What do you mean by "Langlands conjecture"? Langlands made a number of conjectures. Do you mean this: mathoverflow.net/q/298944 $\endgroup$
    – Kimball
    Commented Apr 3 at 0:08
  • $\begingroup$ I mean the Global Langlands for GL(n) in this article math.berkeley.edu/~sander/speaking/… $\endgroup$
    – Cloudifold
    Commented Apr 3 at 0:16

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