Automorphic representations, Shimura varieties, and motives. Ein Märchen

Abstract

1. Introduction. It had been my intention to survey the problems posed by the study of zetafunctions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representationtheoretic, but they both arise in the study of the zeta-function at the infinite places. The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too imprecise, and were not even in principle amenable to proof by Shimura’s methods of descent. The conjecture as it stands is the only statement I could discover that meets his criticism and is compatible with Shimura’s conjecture. The statement of the conjecture must be preceded by some constructions, which have implications that had escaped me. When combined with Deligne’s conception of Shimura varieties as parameter varieties for families of motives they suggest the introduction of a group, here called the Taniyama group, which may be of importance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed. With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of formulating one or two specific conjectures, but we begin to weave a tissue of surmise and hypothesis, and curiosity drives us on. Deligne’s ideas are reviewed in the fourth section, but to understand them one must be familiar at least with the elements of the formalism of tannakian categories underlying the conjectural theory of motives, say, with the main results of Chapter II of [40]. The present Summer Institute is predicated on the belief that there is a close relation between automorphic representations and motives. The relation is usually couched in terms of L-functions, * First appeared in Automorphic forms, representations, and L-functions, Proc. of Symp. in Pure