40
$\begingroup$

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted:

  1. The stabilization of the twisted trace formula for GL($N$) and SO($2n$).

  2. Orthogonality relations for elliptic tempered characters.

  3. Weak spectral transfer of tempered $p$-adic characters

  4. A twisted version of Shelstad's archimedean theory of endoscopy

Assumption 1, and implicitly 2 and 3, are in Arthur's Hypothesis 3.2.1. Assumption 4 is made in the proof of Arthur's Proposition 2.1.1 (see also Remark 5 after Theorem 2.2.1). (I think these are all of Arthur's working hypotheses, but let me know if I missed something.)

Q1: Recently, in a series of 10-11 preprints, Waldspurger and Moeglin finished the stabilization of the twisted trace formula. This seems to verify Assumption 1--does it also take care of implicit Assumptions 2 and 3? If yes, where are 2 and 3 proved?

Arthur said that Assumption 4 should be taken care of by work in progress of Shelstad and Mezo.

Q2: Is Assumption 4 verified yet? Mezo and Shelstad have some recent preprints, but I don't know if they prove exactly what's needed. If not, any explanation of what is still needed would be appreciated.

If not all of these assumptions are proven yet, are any of the specific main results of Arthur (beyond, say, SO(3)) now unconditional?

Edit: I was recently reminded that Arthur also refers to some of his preprints ([A24]-[A27] in the quasi-split case), which are still unpublished (unfinished?) as of mid 2018, for some proofs, but based on Arthur's earlier verbal confirmation to me, I will keep my answer as accepted unless other information comes to light.

$\endgroup$
17
$\begingroup$

Now I think the answer is yes, Arthur's work is now unconditional for quasi-split special orthogonal and symplectic groups. Wee Teck Gan kindly directed me to the relevant papers of Waldspurger and Moeglin addressing the assumptions 2-4 above, though I was not able to verify with certainty that their stated results precisely cover what Arthur requires. Here are the relevant papers:

  1. The stabilization is completed in Moeglin and Waldspurger's Stabilisation X paper.
  2. The orthogonality for elliptic tempered characters appears to be completed in Waldspurger's twisted local trace formulas paper, generalizing Arthur's 1993 Acta paper. See Theorem 7.3.
  3. I believe the $p$-adic weak spectral transfer that Arthur requires, specifically twisted analogues of Theorems 6.1 and 6.2 from his 1996 Selecta paper, is done by Moeglin in her Représentations elliptiques... paper (which is unofficially called "Stabilisation XI").
  4. Twisted archimedean endoscopy seems to be done in Waldspurger's Stabilisation IV paper, which uses Shelstad's 2012 Annals paper. This is independent from the treatment in Mezo's Tempered spectral transfer... paper, but I cannot tell if Mezo's results are unconditionally sufficient for Arthur's needs.

Edit: Arthur confirmed that the results in his book (except the last chapter on non-quasi-split forms) are now unconditional.

$\endgroup$

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.