Cycles on Shimura varieties via geometric Satake
By Liang Xiao, Xinwen Zhu. 2017
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We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set of irreducible components of the affine Deligne-Lusztig variety associated to an unramified twisted conjugacy class.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b1a64226851c47f6d398a25295d51ead30917d26
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
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- Coherent sheaves on the stack of Langlands parameters (2020, methodology): “W;A W 0) !Hom Ztame H Kv (H I(W V);H(W 0 V)): Theorem 4.6.5. Such map exits if either K v is hyperspecial or Iwahori, and k= E. When K v is hyperspecial, this follows from our previous work with Xiao [XZ]. When K v is Iwahori, this will be established in the forthcoming work [HZ]. Now we move to the”