Abstract

This is joint work with Jack Sempliner. In the 1970’s Deligne proposed a formalism for Shimura varieties and Langlands conjectured a formula for the action of Galois on them. Deligne’s formalism involves `Shimura data’ which parametrizes a Shimura together with a preferred embedding of its field of definition into the complex numbers. We propose a different formalism that directly parametrizes a Shimura variety over any field of characteristic 0. To this end we have to revisit the theory of conjugation of Shimura varieties conjectured by Langlands and established by Milne and others. We hope that our reformulation of this work also helps to clarify the theory. A key tool is Kottwitz’s definition of the groups B(G).

In this talk I will sketch the formalism of Deligne and Langlands and point out what we see as its shortcomings. I will then introduce Kottwitz’s cohomology theory for extensions of Galois groups, and explain how to make cocycles a canonical object, not just cohomology classes. Finally I shall explain a reformulation of the theory of conjugation of Deligne’s Shimura varieties and propose an alternative formalism for Shimura varieties.