# What is the status of Arthur's book?

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reminded that Arthur also refers to some of his preprints ([A24]-[A27] in the quasi-split case), which are still unpublished (unfinished?)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.

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").