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The induction hypotheses Application to endoscopic and stable expansions Cancellation of p-adic singularities Separation by infinitesimal character Elimination of restrictions on f Local trace formulas Local Theorem 1 Weak approximation Global Theorems 1 and 2 10. Concluding remarks Introduction This paper is the last of three articles designed to stabilize the trace formula. Our goal is to stabilize the global trace formula for a general connected group, subject to a condition on the fundamental lemma that has been established in some special cases. In the first article [I], we laid out the foundations of the process. . | Annals of Mathematics A stable trace formula III. Proof of the main theorems By James Arthur Annals of Mathematics 158 2003 769 873 A stable trace formula III. Proof of the main theorems By James Arthur Contents 1. The induction hypotheses 2. Application to endoscopic and stable expansions 3. Cancellation of p-adic singularities 4. Separation by infinitesimal character 5. Elimination of restrictions on f 6. Local trace formulas 7. Local Theorem 1 8. Weak approximation 9. Global Theorems 1 and 2 10. Concluding remarks Introduction This paper is the last of three articles designed to stabilize the trace formula. Our goal is to stabilize the global trace formula for a general connected group subject to a condition on the fundamental lemma that has been established in some special cases. In the first article I we laid out the foundations of the process. We also stated a series of local and global theorems which together amount to a stabilization of each of the terms in the trace formula. In the second paper II we established a key reduction in the proof of one of the global theorems. In this paper we shall complete the proof of the theorems. We shall combine the global reduction of II with the expansions that were established in Section 10 of I . We refer the reader to the introduction of I for a general discussion of the problem of stabilization. The introduction of II contains further discussion of the trace formula with emphasis on the elliptic coefficients a ji ỸS . These objects are basic ingredients of the geometric side of the trace formula. Supported in part by NSERC Operating Grant A3483. 770 JAMES ARTHUR However it is really the dual discrete coefficients Rdisc n that are the ultimate objects of study. These coefficients are basic ingredients of the spectral side of the trace formula. Any relationship among them can be regarded at least in theory as a reciprocity law for the arithmetic data that is encoded in automorphic representations. The relationships .
