##
**A stable trace formula. III: Proof of the main theorems.**
*(English)*
Zbl 1051.11027

The stable trace formula is very important for applications to the functorial principle in the Langlands program. In a series of three papers, of which this is the last one, the author stabilizes the Arthur-Selberg trace formula for a general connected group under the assumption that the fundamental lemma holds.

Relations between these three parts are described in the introduction of this paper as follows: “In the first article [I: A stable trace formula. I: General expansions, J. Inst. Math. Jussieu 1, 175–277 (2002; Zbl 1040.11038)], we laid out the foundations of the process. We also stated a series of local and global theorems, which together amounts to a stabilization of each of the terms in the trace formula. In the second part [II: A stable trace formula. II: Global descent, Invent. Math. 143, 157–220 (2001; Zbl 0978.11025)], we established a key reduction in the proof of the theorems. We shall combine the global reduction of [II] with the expansions that were established in [I]”.

Relations between these three parts are described in the introduction of this paper as follows: “In the first article [I: A stable trace formula. I: General expansions, J. Inst. Math. Jussieu 1, 175–277 (2002; Zbl 1040.11038)], we laid out the foundations of the process. We also stated a series of local and global theorems, which together amounts to a stabilization of each of the terms in the trace formula. In the second part [II: A stable trace formula. II: Global descent, Invent. Math. 143, 157–220 (2001; Zbl 0978.11025)], we established a key reduction in the proof of the theorems. We shall combine the global reduction of [II] with the expansions that were established in [I]”.

Reviewer: Lizhen Ji (Ann Arbor)