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 chromatic splitting conjecture in coheight 1(joint with Tobias Barthel, Lucas Mann, Rin Ray, Andy Senger, Tomer Schlank, and Jared Weinstein.)Draft
This article confirms predictions made by Hopkins' chromatic splitting conjecture in stable homotopy theory. We use methods from the theory of p-divisible groups and p-adic geometry. Along the way we prove an entirely new kind of two-tower isomorphism, in analogy with Faltings' isomorphism between the Lubin--Tate and Drinfeld towers, which parametrizes p-divisible groups over characteristic-p perfectoid spaces with fixed generic fiber in terms of negative Banach--Colmez spaces. Using the two-tower isomorphism, we gain control over the continuous cohomology of the Morava stabilizer group \(\mathbb{G}_h\) with coefficients in \(\pi_*L_{K(h-1)}E_h\).
To study the continuous action of \(\mathbb{G}_h\) on \(L_{K(h-1)} E_h\), it was necessary to develop the theory of solid Lubin--Tate spectra. Adapting Lurie's constructions, we attach a solid Lubin-Tate spectrum to p-divisible groups over a broad class of solid rings in characteristic p.
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.
