Abstract: A rather notorious mistake occurs p. 217 in the proof of Lemma 6.3 of the book "Simple algebras, base change, and the advanced theory of the trace formula" (1989) by J. Arthur, L. Clozel. E. Lapid and J. Rogawski (1998) proposed a proof of this lemma based on what they called "Theorem A".

The lemma concerned descent of automorphic (cuspidal) representations for a cyclic extension E/F of number fields. C.S. Rajan later showed that one could, using "Theorem A", control descent in *solvable* extensions. Thanks to the mammoth work of Moeglin and Waldspurger on the stabilisation of the twisted trace formula ('endoscopy'), we are now able to prove "Theorem A" and therefore to complete the proof of solvable descent.

I will explain why the result should be true in view of the conjectural properties of the Langlands group, and how it reduces, using endoscopy, to the identity 0=0.