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BIMSA-YMSC Tsinghua Number Theory Seminar
The p-adic analog of the Hecke orbit conjecture and density theorems toward the p-adic monodromy
The p-adic analog of the Hecke orbit conjecture and density theorems toward the p-adic monodromy
Organizers
Hansheng Diao
, Heng Du
, Yueke Hu
, Bin Xu
, Yihang Zhu
Speaker
Yu Fu
Time
Monday, December 30, 2024 10:00 AM - 11:00 AM
Venue
Shuangqing-C654
Abstract
The Hecke orbit conjecture predicts that Hecke symmetries characterize the central foliation on Shimura varieties over an algebraically closed field k of characteristic p. The conjecture predicts that on the mod p reduction of a Shimura variety, any prime-to-p Hecke orbit is dense in the central leaf containing it, and was recently proved by a series of nice papers.
However, the behavior of Hecke correspondences induced by isogenies between abelian varieties in characteristic p and p-adically is significantly different from the behavior in characteristic zero and under the topology induced by Archimedean valuations. In this talk, we will formulate a p-adic analog of the Hecke orbit conjecture and investigate the p-adic monodromy of p-adic Galois representations attached to points of Shimura varieties of Hodge type. We prove a density theorem for the locus of formal neighborhood associated to the mod p points of the Shimura variety whose monodromy is large and use it to deduce the non-where density of Hecke orbits under certain circumstances.