Timeline for Is this a subcase of the fundamental lemma?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2011 at 10:45 | comment | added | Kevin Buzzard | Ok I finally remembered to look. It seems to me that Laumon is just doing what Arthur-Clozel do (but perhaps in char $p$ rather than char 0): proving the base change fundamental lemma. In particular he is proving that an orbital integral downstairs is equal to a twisted orbital integral upstairs. You do not seem to have the twist in your question, so it seems to me that Laumon's calculations are not directly relevant. | |
| Nov 26, 2011 at 10:52 | comment | added | Kevin Buzzard | ...it's OK for the identity function. So in some simple situations the problem of transferring functions is equivalent to the fundamental lemma. The moment one moves away from these simple situations, I know nothing and I'm afraid I'm not the person to ask. I'll try to remember to take a look at Laumon's book on Monday when I'm back at work, but I am a little worried that it will still mean very little to me on anything other than a superficial level. Ask me the same questions again in 5 years and we'll see if I've learnt any more! I'm still struggling through Labesse-Langlands! | |
| Nov 26, 2011 at 10:49 | comment | added | Kevin Buzzard | I am not an expert so I can only offer you a superficial overview rather than practical help with details. My general impression is that in certain situations (e.g. an unramified endoscopic situation), the transfer map is staring at you in the face -- it's coming from the theory of the Satake isomorphism. The problem is proving that the "obvious" transfer does the job you want, i.e. that one orbital integral equals another. I think that if you can check it for the identity functions then you can check it for all the functions, and the fundamental lemma is the assertion that... | |
| Nov 26, 2011 at 10:26 | vote | accept | Marc Palm | ||
| Nov 26, 2011 at 10:25 | comment | added | Marc Palm | so I will have to reprove most of the chapter 4 in Laumon. Also my field extensions will be ramified in general. Are there some standard tools, which allow unramified extensions to be treated as ramified ones? | |
| Nov 26, 2011 at 10:19 | comment | added | Marc Palm | Thanks. You are only adressing in the last part of your answer my original question. The rest of your ellaboration actually adresses a question, which was implicit or even only in the title;) So thanks for reading inbetween the lines. After your explanation, I think the answer is contained in Laumon "Cohomology of Drinfeld modules" Part 1, Proposition 4.7.1, page 114, at least for unramfied extensions. So you claim it is sufficient to prove every thing for the unit? This uses invariance here? I probably have to also include weights in my orbital integrals.... | |
| Nov 26, 2011 at 8:39 | history | answered | Kevin Buzzard | CC BY-SA 3.0 |