7
$\begingroup$

I am in the process of ordering some book on the Langlands program, and learning more about it. In the mean time, I have a question which is easy to the experts, but being a beginner, I am not a 100% sure, so I could use some confirmation (or the opposite!).

In a trace formula, the geometric side is the trace of the matrix, which is the sum of the diagonal entries, while the spectral side is the sum of the eigenvalues. That these 2 sums are equal is the content of the trace formula.

If one were to label one side of the Langlands program as geometric, and the other as spectral, which side would that be?

If I were to guess, I would say that perhaps the Galois side is the spectral side, while the automorphic side is the geometric side, because I think that the Galois side is the side containing the information that we would like to obtain, so it is the spectral side, while the automorphic side is supposedly easier to calculate. Am I right, or do I have the analogy backwards?

Edit: A paper by Gaitsgory confirms my suspicion: http://www.math.harvard.edu/~gaitsgde/GL/outline.pdf in 0.2.1. The "Galois" side is the "spectral" side, while the "automorphic" side is the "geometric" side, according to Gaitsgory.

$\endgroup$
5
  • $\begingroup$ The person you can ask is Ngo Bao Chau-he is the one who proved Langlands's fundamental lemma which is the biggest stumbling block in the program and made all the related theorems to be true. He is at the University of Chicago. He knows more of this than most. $\endgroup$
    – DeepSea
    Dec 14, 2017 at 12:07
  • $\begingroup$ @DeepSea, yes sure. But it is just a "soft" philosophical question. I do not want to bother big shots with an email, so I thought a small post here instead may be a better idea. $\endgroup$
    – Malkoun
    Dec 14, 2017 at 12:10
  • $\begingroup$ Ngo Bao Chau is considered the leader in Langlands program. You can ask him or Ngo Dac Tuan at Univ of Paris Caen. Both are at the frontier of the knowledge. I know both of them are expert in Langlands program. $\endgroup$
    – DeepSea
    Dec 14, 2017 at 12:12
  • $\begingroup$ Yes, I know that Ngo Bao Chau is one of the main figures in the Langlands program. Thank you for informing me about Ngo Dac Tuan also. I am just starting to read about the subject. I do not want to email the experts. I know that they know the answer, but I do not want to bother them. Thank you for your suggestion though. $\endgroup$
    – Malkoun
    Dec 14, 2017 at 12:17
  • 1
    $\begingroup$ This post is the top result on my choice of search engine for the words "spectral langlands" which seems to indicate that this is maybe not as obvious or applicable as you might think. If you want some expert opinions from researchers you could ask on Mathoverflow as well $\endgroup$
    – neptun
    Dec 14, 2017 at 12:20

1 Answer 1

0
$\begingroup$

I have decided to answer my own question, since I found a paper by Gaitsgory which confirms my suspicion:

http://www.math.harvard.edu/~gaitsgde/GL/outline.pdf in 0.2.1. The "Galois" side is the "spectral" side, while the "automorphic" side is the "geometric" side, according to Gaitsgory.

I am still very shaky about the Langlands program, which is why I am planning to order an introductory book. Actually, I only found one introductory book, called "An introduction to the Langlands Program", with various authors contributing to the book. If someone happens to know of some other introductory textbooks, could you please post it as a comment perhaps?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.