Hi, my name is Xinyu Zhou. I am a PhD student at Boston University. My main interest is arithmetic geometry. Recently, I am more focusing on some problems in p-adic geometry and the geometry of moduli spaces of shtukas. Currently, I am working on applications of p-adic cohomology of diamonds to chromatic homotopy theory.
My Email Address is xyz6 (at) bu.edu
My office is CDS 345.
On the Harris--Viehmann conjecture for Hodge-Newton reducible local Shimura data of abelian type(joint with Sandra Nair)ArXiv
We address a new case of the Harris-Viehmann conjecture, which establishes a parabolic induction formula on the cohomology groups associated to non-basic local Shimura data. It follows that all supercuspidal representations on a Shimura variety are concentrated along the basic locus, making the conjecture relevant to the Langlands program. Historically, many cases of the Harris-Viehmann conjecture have been approached with the additional condition of Hodge-Newton reducibility on the underlying local Shimura datum. Building on previous work by Mantovan (EL/PEL case) and Hong (Hodge case), we extend the proof of the conjecture to unramified non-basic local Shimura data of abelian type under the assumption of Hodge-Newton reducibility. We leverage Shen's construction of Rapoport-Zink spaces of abelian type at the hyperspecial level.
On the Langlands-Kottwitz Method for Drinfeld Modular Varieties at Bad Primes(In preparation)
We use Scholze's approach on Langlands-Kottwitz method for some Shimura varieties to determine the (semisimple) local factors of the Hasse-Weil zeta functions of Drinfeld modular varieties at bad primes. In this process, we overcome several difficuties that do not appear in the Shimura variety case. We also give a construction of the "canonical level structure" map for Drinfeld modular varieties, which gives reduced fibers at bad primes and also leads to a more general duality theory for Drinfeld modules.
Here are some notes I wrote. Intersection Theory. These notes give introductions to Chern classes and Segre classes on schemes. Brauer-Manin Obstructions. This is the note for the STAGE talk at MIT on the étale Brauer obstructions and insufficiencies. Formal Vanishing Cycles. These are the notes for my talk at a learning seminar on Scholze's proof of Local Langlands for \(GL_n\). I discussed some basic properties of formal vanishing cycles and deformation spaces of divisible modules. Verdier duality. Introduction to Verdier duality Ramanujan-Petersson Conjecture--Part 1. An introduction to the Ramanujan-Petersson conjecture. Introduction to divided power structures. Elementary. It has a detailed example of non-uniqueness of PD structures. So it may be helpful if you are looking for examples. Drinfeld's lemma in v-cohomology. Slides for (one of) my talk(s) on the Drinfeld's Lemma in quasicoherent v-cohomology of v-stacks.
