This was useful. Thanks for recommending it.

]]>Hmm. It would appear that the char-p Langlands is the work of Drinfel’d(according to wikipedia), and indeed the Geometric Langlands is due to him.

It would seem that reciprocity is captured in some representation theoretic statement a la Langlands, and this is supposed to be geometrical. I have heard representation theorists proposing in a broad sense that “everything is representation theory”. When I just remarked the “everything is geometry” statement to someone, I got the reply that from the abstract point of view of category theory, pretty much anything could be made into pretty much anything else.

Yeah, it is enough to use the e-mail address in the comments. However it would be nice if you don’t reveal it to others(so that I can continue to be an unidentified novice).

]]>I could send it to you by email. Can I use the gmail address you are giving when you post your comments? I could also send you more references about the reciprocity stuff if you are interested.

]]>That’s a very interesting question, I’m sorry about the delay in replying. I was meaning to write a longer reply, but I simply don’t have the time at the moment, so I will just give a brief answer.

If you only look at classical reciprocity laws (Gauss etc), which are unified in class field theory (CFT), the subject doesn’t look very “geometric”, although it is possible to use group cohomology to prove many of the main results of CFT (See e.g. Cassels and FrÃ¶hlich: Algebraic number theory).

However, there are several generalizations of reciprocity laws/CFT which in different ways are more geometric than the classical stuff. Here is a brief list of such generalizations, and there are probably others as well.

1. The Langlands program. There are many parts to this, including local Langlands (in zero and positive characteristic), global Langlands (in zero and positive characteristic), geometric Langlands, and p-adic Langlands in the sense of Breuil and others. Local and global Langlands are generalizations of local and global CFT. For example, for K a number field, global Langlands for the group GL(n) relates (conjecturally) -adic representations of the absolute Galois group of K (which are de Rham at and almost everywhere unramified) to so called algebraic automorphic representation of GL(n,A), where A is the adele ring of the number field. For n = 1 we recover the statements of global class field theory, and similarly for local Langlands and local CFT. A lot is known about local Langlands, and the proofs we have use a lot of algebraic geometry, see for example Harris and Taylor: The geometry and cohomology of some simple Shimura varieties. When it comes to global Langlands, Lafforgue got the Fields medal for GL(n) in the function field case (char p) while in the number field case (char zero) there is only very limited progress. Without knowing much about Lafforgue’s work, it seems reasonable to think that he was able to do the function field case precisely because things can be interpreted geometrically, in terms of curves over finite fields, while in the number field case, one of the reasons things are hard is that we don’t have any “geometry over the field with one element”, so the geometric arguments are not available.

2. It is is possible to formulate “reciprocity laws” for arithmetic schemes, see for example Wiesend: Class field theory for arithmetic schemes, and older work by Kato and Saito. Link: http://www.springerlink.com/content/p225753t146h2533/

3. Kato also has a notion of “explicit reciprocity laws”, see for example http://www.numdam.org/item?id=BSMF_1991__119_4_397_0 but I don’t know much about this.

4. There might be a connection between local CFT/local Langlands with homotopy theory, hinted at the last page of this survey of Morava: http://front.math.ucdavis.edu/0707.3216

All in all, I think there is good reason to say that there is a lot of geometry in the subject of “generalized” reciprocity laws. Hope this gives at least a partial answer to your question.

]]>Btw I would be very interested in having a look at your write-up on Tate’s thesis. The link in your homepage is not working.

]]>And there is the implication in one of your points that everything could be geometry. Is it possible to present a hardcore arithmetical topic such as the reciprocity investigations of Gauss, Eisenstein etc., as geometry? Since you’ve done etale cohomology etc., I hope you are qualified to answer.

]]>Unfortunately I am not in a position to answer the question about how important brave new rings are. My interest in homotopy theory is a relatively recent thing, so I lack a lot of background. You have to ask someone with more experience in the subject.

]]>I checked wikipedia on the said novel. The world described therein is certainly quirky and the phrase seems to have a jocular content as well. Due to my naivete at first I took the phrase to mean great optimism.

How important are brave new rings in homotopy theory? Since you are doing homotopy theory and you still haven’t fully looked into it yet, it certainly can’t be absolutely essential for the subject.

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