# Soft Question: in the Langlands program, which side is "geometric", which side is "spectral"?

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.

• 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. Dec 14, 2017 at 12:07
• @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. Dec 14, 2017 at 12:10
• 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. Dec 14, 2017 at 12:12
• 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. Dec 14, 2017 at 12:17
• 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 Dec 14, 2017 at 12:20