# The last part of Tate'sThesis

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?

-
Something seems to have gone missing. –  joriki Apr 12 '11 at 19:05
Even clicking on edit does not show the missing part (which sometimes happens if you use < and >). –  Aryabhata Apr 12 '11 at 19:15
I believe the last part is mostly a summary of various –  user641 Apr 12 '11 at 19:44
Don't worry, you guys, I believe Jonathan will –  Guess who it is. Apr 13 '11 at 1:21
Sorry 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. –  Jonathan Apr 13 '11 at 5:03

## 1 Answer

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.

-
Thanks 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. –  Jonathan Apr 13 '11 at 4:51
I 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 :-) –  Jonathan Apr 13 '11 at 4:53