Hecke’s precursor to endoscopy

Ngô Bao Châu recently won a Fields Medal for his work on the Fundamental Lemma, conjectured by Langlands many years ago. But the relationship between the result proved by him and Langlands’ original conjecture might not be so clear to someone who has not followed the development closely. The original idea involvedwhat Langlands calls endoscopy—to explain harmonic analysis on reductive groups over local fields as well as on arithmetic quotients in terms of what Langlands calls stable analysis on the original group and certain other, smaller, groups associated to it. These smaller groups are called endoscopic. The subject is in rapid flux, and the present state of exposition is necessarily, in view of our present understanding, generally very technical. However, I believe that the basic idea of endoscopy can be explained by looking at what was in fact the very first place in which it showed up, in work of Erich Hecke.