Is there a comprehensive reference dealing with the last part(theory in the large!) of Tate's thesis? Why is the group of S units modulo the roots of unity a free abelian group of rank m?
The last part of Tate'sThesis
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9Something seems to have gone missing. – 2011-04-12
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0Even clicking on edit does not show the missing part (which sometimes happens if you use `<` and `>`). – 2011-04-12
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13I believe the last part is mostly a summary of various – 2011-04-12
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3Don't worry, you guys, I believe Jonathan will – 2011-04-13
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0Sorry for the missing part... I had another question which thought I could work a lil more... I think the last part is not a summary but contains actual computations of zeta funcitons and relate them to the functional equation of Hecke. – 2011-04-13
1 Answers
An adelic proof of the Unit Theorem for $S$-integer rings can be found as Theorem 8 in these notes. (You will find an attribution to Ramakrishnan-Valenza's Fourier Analysis on Number Fields. I have found this text to be readable and useful in general, but not completely reliable on the details. For instance, if memory serves I actually had to fix a false lemma in their text in order to carry out their proof, but having done that their proof is a very nice one.)
By the way, although I trust you that it appears in there somewhere, the Unit Theorem is not the meaty part of Tate's Thesis. It's the stuff about Hecke characters, L-functions, zeta integrals, analytic continuation, etc. which was then novel and continues to be of the utmost importance today.
By the way, the last chapter of the book of Ramakrishnan and Valenza contains a working through of Tate's thesis, so you might look there to see if it helps you.
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0Thanks Pete. I read Tate's thesis from Cassels and Frolich. I had some background with algebraic number theory and then I learnt Pontryagin duality from Sydney Morris' book. So I thought I could avoid reading Ramakrishnan. The expositions of Tate's thesis Lang seem to deal with construction of the function f (in the large) differently. And Ramakrishnan(I have read it only occasionally) immediately jumps to Hecke L-Functions after proving the analytic continuation of the global zeta function. – 2011-04-13
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0I found theory in the large section interesting only that there were a few jumps which I could not justify well enough... or am not sure if my arguments are right and hence was looking for some exposition of that section. Thanks again for the notes :-) – 2011-04-13