Talk:Iwasawa theory

"Γ is dual to the discrete group of all p-power roots of unity in the complex numbers." The p-power roots of unity in the complex numbers are not discrete w.r.t. the usual topology.

Urban-Skinner

I have made some minor changes in the history page.

I have put the sentences about Euler System methods in a separate paragraph before the announcement of Skinner-Urban, since it was ten years prior.

I have also removed the reference to Urban-Skinner's "note aux CRAS" which was a mistake, since this note has nothing to do with the main conjecture (it is a different announcement by the same guys).

In my opinion, the whole Urban-Skinner paragraph could as well be suppressed, since it does not meet Wikipedia's "verifiable source' requirement. —Preceding unsigned comment added by 160.39.47.33 (talk) 11:11, August 28, 2007 (UTC)

Does anyone knows if Skinner's and Urban's proof is valid or not? --85.76.80.222 (talk) 16:59, 9 December 2014 (UTC)

Iwasawa's originality

Ive always thought that the comment that "Iwasawas originality was to go off to infinity in a new direction" was completely without mathematical content, and not even useful as a vague explanation of what iwasawa theory is. In case anyone out there likes it, I wanted to leave a note to let you know that I was erasing it and give you a chance to change it back.

Jrdodge (talk) 03:49, 8 September 2009 (UTC)

Well, sorry about that. There is a general issue with heuristic remarks: they may have content that is not "mathematical content". As we know, mathematics can exist fine without any commentary at all. But do you really think that mathematics articles should only have such "mathematical content"? Over time (I mean five or six years) I have seen many people purge such remarks from Wikipedia. They tend not to add better heuristics of their own. In other words these edits tend to be purely negative. I wonder who else besides (pure) mathematicians think that no explanation at all its better than some vague explanation. It is not the kind of feedback we usually get, certainly. Charles Matthews (talk) 05:25, 8 September 2009 (UTC)

I think the comment about the new direction is a useful one. Perhaps it would help to find a source to support the claim. One certainly uses this kind of language in the field, speaking of the cyclotomic direction, the anticyclotomic direction, etc. I'll see if I come across something in the literature that makes such a claim about Iwasawa's originality. RobHar (talk) 08:47, 8 September 2009 (UTC)

The article could do with a whole section on the context. The remark was an old one: the Springer EoM link has some general comments about the analogy between finite field extensions and adding more roots of unity. There is probably something about the heuristics in Coates, J.: K-theory and Iwasawa's analogue of the Jacobian. In Algebraic K-theory II, p. 502-520. Lecture Notes in Mathematics 342. And other early papers of John Coates who (I guess) coined the term Iwasawa theory; certainly "main conjecture". There seem to have been many papers linking the main conjecture to the Bloch-Kato conjecture on Tamagawa numbers. I'm a bit out of touch, but I guess that that conjecture is not (equivalent to) the Bloch-Kato conjecture in K-theory that was announced completely proved earlier in the year. In other words, there is some pre-context and post-context to supply. Of course general expansion is required, also. Charles Matthews (talk) 18:22, 8 September 2009 (UTC)

Certainly some heuristic would be good, and it doesnt have to have mathematical content. I just think saying that Iwasawa went off to infinity in a new direction is unhelpful as an analogy or a heuristic because its completely unclear what the old direction to go off to infinity was. I dont think one should insist on having an unhelpful heuristic just so that non-mathematicians have SOMETHING to "picture".

Jrdodge (talk) 21:04, 9 September 2009 (UTC)

Formulation

I'm not a number theorist, so I might be missing some notation convention, but isn't it meant to be Z/p^nZ, not Z/p^n in the formulation paragraph?

Z/pⁿ is another notation for Z/pⁿZ. Z/(pⁿ) would be more proper, but I've certainly seen it both ways. RobHar (talk) 18:18, 12 August 2010 (UTC)

I'm not entirely sure the example given is correct. Perhaps one should swap n for n-1. For example take K=K_1=Q (the field of rational numbers). Then K contains the second roots of unity. The extension given by adjoining the fourth roots of unity is K_2=Q(i), and the Galois group of this over Q is Z/2Z, and not Z/4Z as is claimed in the article.

Moreover, Q adjoin the 2^nth roots of unity is a cyclotomic field extension whose Galois group is isomophic to the unit group of Z/2^nZ, and not isomophic to Z/2^nZ.

It looks like a series of edits over the years turned this example into a false one. I've fixed it. Thanks. RobHar (talk) 22:12, 11 October 2010 (UTC)