From: jr@redmink.demon.co.uk (John R Ramsden)
Subject: Re: endoscopy?
Date: Sat, 14 Aug 1999 18:49:43 GMT
Newsgroups: sci.math,sci.math.research
Keywords: what is the Langland's program?

On 13 Aug 1999 19:43:55 -0400, lrudolph@panix.com (Lee Rudolph) wrote:

>mal11@my-deja.com writes:
>
>>I've seen recently the term endoscopy used on some mathematics
>>web sites.
>>
>>Could someone explain how they are using this term and why?
>>
>>And what does it have to do with the Langlands program?
>>
>>
>>In medicine, endoscopy is the visualization of the cavities
>>inside the human body with a camera, normally the GI tract.
>>You can go down the throat or up the rectum.
>
>A friend and former colleague of mine who I will not name 
>explicitly (but he is, I'm pretty sure, on the path that gives 
>me my Erdos number--whatever it may equal) claims to have been 
>the one who suggested the term to whoever it was who introduced 
>it into the mathematical literature (Langlands himself? Shelstad 
>or Jacquet, also his and my former colleagues at about the right 
>time? I dunno), with full awareness of the medical meaning on both 
>their parts.
>
>>p.s. I very well may be the first person to use the word rectum in
>>sci.math.....hmmmmm
>
>By no means.  A few years ago, Robin Chapman even was kind enough
>to define "latus rectum" for a questioner here.
>
>Oh--as to your mathematical questions, I have no idea of the
>answers whatsoever.  The entire Langlands Program is a black hole
>to me.
>
>Lee Rudolph

As I understand it, "classical" class field theory is the study
of correspondences between cyclotomic extensions of algebraic
fields and Abelian groups.

Class Field Theory was developed between about the '20s and the
'50s by many people, including among others D Hilbert, E Artin,
and A Weil. It allowed many old problems to be solved, including
general reciprocity laws for cyclotomic fields. But it obstinately
refused to generalize to non-cyclotomic extensions (for which the
correspondence would require Galois groups that are non-Abelian).

The Langlands Program, started about thirty years ago, is an
attempt to make this leap to general extensions. It requires a
close study of endomorphisms of something or other, and I presume
the term "endoscope" is related to this aspect. There was a survey
article on the Langlands Program in the Bulletin of the AMS about
fifteen years ago.

Well hopefully I've made enough of a pig's ass of this description
to entice a few experts into giving the _real_ story, including the
current state of play. (I've also ventured to add sci.math.research
to the list of groups.)


Cheers

---

John R Ramsden             #   "No one who has not shared a submarine
                           #    with a camel can claim to have plumbed
 (jr@redmink.demon.co.uk)  #    the depths of human misery."
                           #
                           #      Ritter von Haske
                           #        "Adventures of a U-boat Commander".

[duplicate sig deleted -- djr]