As far as I know, Tate's thesis has been successfully applied in two fronts:

  • Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid)

  • Automorphic L-functions, by Jacquet, Shalika, Shapiro etc.

Of course the first approach works for everything below Hecke in the L-function food chain (Dedekind, Dirichlet, Riemann).

My question is, given that the same ideas work for those extremes, abelian on the one side and... well, supposedly all of them on the other, why hasn't been any progress using harmonic analysis to study Artin L-functions?

For example, Tate's thesis for 1-dimensional Artin representations is just... Tate's thesis again (this is, using $W_k$ instead of Galois). This detail is obvious enough, but I haven't seen the details ever worked out in any detailed (I don't think I've seen it mentioned at all).

If the analysis goes through for general automorphic objects, I can't imagine the problem being in dealing with noncommutative locally compact groups. So, where is the problem exactly?

  • $\begingroup$ If you like my answer, please accept it officially (so that it turns green). Thanks in advance! $\endgroup$ – GH from MO Aug 19 '18 at 23:23

I think there is a misunderstanding on your side. Tate's thesis is not about special Artin $L$-functions, but about special automorphic $L$-functions. It is a reformulation of Hecke's ideas in a uniform adelic language. Of course the automorphic $L$-functions occurring in Tate's thesis (namely those associated to automorphic forms on ${\rm GL}_1$) agree with the $L$-functions of one-dimensional Galois representations, but this is not due to Tate but to class field theory established before his thesis.

The short answer to your question "where is the problem exactly" is to show that Artin's $L$-functions are special automorphic $L$-functions. This is part of a bigger program formulated by Langlands.

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    $\begingroup$ Yes, perhaps to further clarify, "work" might mean "give L-functions with analytic continuation and functional equations". This sense is guaranteed for many automorphic L-functions, especially the "standard" ones attached to cuspforms on $GL(n)$, etc. On the other hand, Artin L-functions (and Hasse-Weil zetas of varieties) and other L-functions coming from Galois repns do not have any obvious general analytic continuation (for example), ... so are proven to have such by proving that they are, in fact, automorphic. Classfield theory does the abelian case. Further cases ... [cont'd] $\endgroup$ – paul garrett Dec 23 '14 at 21:19
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    $\begingroup$ ... are highly non-trivial, such as Wiles' argument that, in effect, L-functions of (nice...) elliptic curves are automorphic, to prove Fermat. It was Langlands' epiphany in the 1960s that non-abelian Artin L-functions (Galois objects) should be automorphic. $\endgroup$ – paul garrett Dec 23 '14 at 21:20
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    $\begingroup$ @Myshkin: Once again, Tate's thesis is not about Artin $L$-functions. There is no Galois representation in his thesis. Instead, he works with idele class group characters, i.e. very special automorphic forms. It was generalized to more general automorphic forms, that is all. $\endgroup$ – GH from MO Dec 23 '14 at 22:33
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    $\begingroup$ @Myshkin: A Galois representation is a global object, just as an automorphic form. But we don't know how to define the $L$-function as directly in terms of a Galois representation as in terms of an automorphic form. So the analytic properties of an Artin $L$-function are much less accessible than those of automorphic $L$-functions. Tate's thesis is completely analytic, it lives on the automorphic side. $\endgroup$ – GH from MO Dec 23 '14 at 23:11
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    $\begingroup$ @Myshkin: I am not familiar with Fesenko's work. Of course you don't need to know beforehand that the underlying object is automorphic. It is possible that Artin's conjecture will be proven without automorphic forms. At any rate, it seems your original question is too vague. $\endgroup$ – GH from MO Dec 24 '14 at 0:33

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