Class Field Theory states the correspondence between abelian extensions of k and congruence divisor class. In idelic language, there is a surjective map from $J_k/k^*$ to $Gal(k^{ab}/k)$ with its kernel unkonwn.

Tate's Thesis proved some functional equations and analytic continuity(with a finite character of $J_k/k^*$).

Question: Why Tate's thesis contributed to class field theory?

  • 1
    $\begingroup$ Why do you think it did? It reproved results of Hecke. $\endgroup$ – Charles Matthews Jun 22 '10 at 17:11
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    $\begingroup$ Hindsight is 20-20: in retrospect, Tate's work provides input into the Langlands program for ${\rm{GL}}_1$, in terms of established analytic properties of the $L$-functions for objects given on the adelic side. But class field theory is needed to connect those with the Galois side, so one could say that class field theory plus Tate's thesis tells us analytic facts about $L$-functions attached to characters on Galois groups or Weil groups. In that sense they are closely related, but neither logically depends on the other. (See Tate's own description of necessary background in his thesis!) $\endgroup$ – Boyarsky Jun 22 '10 at 17:22
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    $\begingroup$ Yes, good mathematics, bad history. $\endgroup$ – Charles Matthews Jun 22 '10 at 18:05

Boyarsky has already answered the question in the comments section. These are a couple of expositions which put John Tate's contribution in perspective.

  1. Stephen Kudla's chapter on Tate's thesis
  2. Stephen Gelbart's article on Elementary introduction to Langlands' program. See the section beginning page 194

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