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SUMMARY:The formalism of Shimura varieties - Richard Taylor\, Stanford Uni
 versity
DTSTART:20220503T133000Z
DTEND:20220503T143000Z
UID:TALK172220@talks.cam.ac.uk
CONTACT:Rong Zhou
DESCRIPTION:This is joint work with Jack Sempliner. In the 1970's Deligne 
 proposed a formalism for Shimura varieties and Langlands conjectured a for
 mula for the action of Galois on them. Deligne's formalism involves `Shimu
 ra data' which parametrizes a Shimura together with a preferred embedding 
 of its field of definition into the complex numbers. We propose a differen
 t formalism that directly parametrizes a Shimura variety over any field of
  characteristic 0. To this end we have to revisit the theory of conjugatio
 n of Shimura varieties conjectured by Langlands and established by Milne a
 nd others. We hope that our reformulation of this work also helps to clari
 fy the theory. A key tool is Kottwitz's definition of the groups B(G).\n\n
 In this talk I will sketch the formalism of Deligne and Langlands and poin
 t out what we see as its shortcomings. I will then introduce Kottwitz's co
 homology 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.
LOCATION:MR13
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