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Algebra and Geometry Seminar

Monday, January 28, 2019
4:00pm to 5:00pm
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Linde Hall 387
Special Cycles on Unitary Groups, Superspecial Abelian Varieties and Bruhat--Tits Buildings
Dimitar Jetchev, EPFL,
We study a collection of special cycles on unitary Shimura varieties that are higher-dimensio​nal analogues of Heegner points on modular curves, and describe these cycles adelically. Using Bruhat--Tits theory for unitary groups, we prove that they satisfy certain rigidity relations (known as distribution relations), thus allowing to recover arithmetic information from cohomological data (cycle class images under p-adic Abel--Jacobi maps). We then state certain equidistributio​n conjectures for the specialization of these cycles to the special fiber of the (canonical) integral models of the ambient Shimura varieties related to non-vanishing of the cohomology classes (the latter formulated in terms of superspecial abelian varieties and mass formulas for unitary groups). As arithmetic applications of the distribution relations and the non-vanishing conjectures, we prove novel results towards the Bloch--Kato--Be​ilinson conjectures for ranks of groups of rational cycles in terms of analytic ranks of automorphic L-functions.
For more information, please contact Mathematics Dept. by phone at 626-395-4335 or by email at [email protected].