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I'm very new to the Langlands program and I was going through the Gauss reciprocity law, Hilbert's 9th problem, Artin's reciprocity law which allowed him to identify the Artin's L-functions with the Hecke L-functions, all the way to the Shimura-Tayinama conjecture and the corresponding reciprocity law.

Could someone provide a modern example of a reciprocity law and how it fits in the Langlands picture? These laws all seem to hinge on the identification of certain eigenvalues with cardinalities of solution sets of equations, which is quite suprising. References are also very welcome. Thank you.

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    $\begingroup$ the generalizations of the reciprocity law would not be about the identification of certain eigenvalues with cardinalities of solution sets of equations. That is the Grothendieck trace formula (which does not have to be mentioned in the formulation of the local or global Langlands conjectures; though for instance L. Lafforgue used it to prove a particular result in the Langlands program but that is logically a separate thing because it involved stacks). $\endgroup$
    – user158636
    Commented Jul 21, 2020 at 19:52
  • $\begingroup$ Thank you crispr, I'm gonna look up the Grothendieck trace formula. $\endgroup$
    – Rachid Atmai
    Commented Jul 21, 2020 at 20:53
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    $\begingroup$ Gelbart and Weinstein have survey articles about reciprocity/Langlands program, both in the Bulletin of the AMS if I recall. Have you tried reading either of those? $\endgroup$
    – Kimball
    Commented Jul 21, 2020 at 21:32
  • $\begingroup$ Thanks for the references Kimball, I wasn't aware of these survey articles $\endgroup$
    – Rachid Atmai
    Commented Jul 21, 2020 at 21:57
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    $\begingroup$ Artin did not "identity the Artin's $L$-functions with the Hecke $L$-functions": neither collection of $L$-functions is a subset of the other. Artin was able to identity the $L$-functions of finite-order Hecke characters with the 1-dimensional Artin $L$-functions. $\endgroup$
    – KConrad
    Commented Jul 21, 2020 at 22:41

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