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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".

MO link: https://mathoverflow.net/questions/468233

Cloudifold
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  • Yeah, "higher reciprocity laws" (e.g., local Langlands conjectures for GL(n)) are not elementary. "The modularity theorem" (Wiles-Taylor and Breuil-Conrad-et-al), formerly the Taniyama-Shimura conjecture, although it can be used to prove Fermat's Last Theorem, is not at all obviously useful for anything elementary (if we didn't know about Frey curves...) – paul garrett Apr 01 '24 at 20:35
  • Sato-Tate sounds like an obvious answer here. But you really should have asked this on MO instead. – Aphelli Apr 01 '24 at 22:33
  • But... Sato-Tate is for elliptic curves? It might be generalized to other varieties and be deduced from Langlands, but I don't quite know about them. And thank you for your advice, I'll ask on MO. – Cloudifold Apr 02 '24 at 17:20
  • @Cloudifold Sato-Tate is also for Fourier coefficients of modular forms, Satake parameters of Maass forms, and so on. – Jakob Streipel Apr 02 '24 at 19:29

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