Beauville—GLuing of algebraic spaces (joint with Piotr Achinger) || arXiv
Tags: Rigid analytic geometry
Beauville—GLuing of algebraic spaces (joint with Piotr Achinger) || arXiv
Tags: Rigid analytic geometry
We show that one may always glue a separated formal algebraic space XX over Zp to a separated algebraic space X over Qp via an open embedding of XXη into the analytification of X, resulting in algebraic space. In fact, we show that this gluing procedure gives rise to an equivalence between such gluing triples (X,XX,j) and the category of separated algebraic spaces over Zp. Moreover, one may replace Zp by an essentially arbitrary base. This is a sort of Beauville--Laszlo gluing theorem for (algebraic) spaces, opposed to coherent sheaves. We apply this to better understand some well-documented phenomena in arithmetic geometry.
A Tannakian Framework for Prismatic F-crystals (joint with Naoki Imai and Hiroki Kato) || arXiv
Tags: p-adic Hodge theory, prismatic F-crystals, p-adic shtukas, Tannakian formalism
We study some Tannakian aspects of the theory of prismatic F-crystals, a recent notion in integral p-adic Hodge theory developed by Bhatt--Scholze and others. Additionally, we explain how to connect this Tannakian theory to the prervious Tannakian theory of shtukas studied by Scholze et al.
Canonical Integral Models of Shimura Varieties of Abelian Type (joint with Patrick Daniels) || arXiv
Tags: Shimura varieties, integral models, p-adic Hodge theory, p-adic shtukas
We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when p > 3 by showing that the Kisin–-Pappas–Zhou integral models of Shimura varieties of abelian type are canonical. Inparticular, this shows that these models of are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data.
The Prismatic Realization Functor on Shimura Varieties of Abelian type (joint with Naoki Imai and Hiroki Kato) || arXiv
Tags: Shimura varieties, integral models, p-adic Hodge theory, prismatic F-gauges, Fontaine--Laffaille theory
In this paper we construct an object on the integral canonical models of Shimura varieties of abelian type (and hyperspecial level) which should be thought of as the prismatic F-gauge realization of the universal G-motive on such an object. We use this to obtain new p-adic Hodge-theoretic information about such Shimura varieties, notably an analogue of the Serre--Tate theorem, as well as characterizing them (even at finite levels). Along the way we develop an integral analogue of the functor D_crys, and relate it to Fontaine--Lafaille and Dieudonné theory.
The analytic topology suffices for the BdR^+-Grassmannian (joint with Kęstutis Česnavičius) || arXiv
Tags: p-adic Hodge theory, torsors, Grothendieck--Serre conjecture
In this article we show that Scholze's BdR^+-affine Grassmannian can be defined using sheafification with respect to the analytic (i.e., usual topological open cover Grothendieck topology) opposed to étale or v topologies.
The Jacobson--Morozov morphism for Langlands parameters in the relative setting (joint with Alexander Bertoloni Meli and Naoki Imai) || International Mathematics Research Notices (IMRN) / arXiv
Tags: Langlands conjecture, L-parameters, moduli theory
In this paper we define a moduli space of L-parameters over the rational numbers, show it has good geometric properties (i.e. is smooth with explicitly parameterized geometric connected components), and show that there is a morphism from this moduli space to the moduli space of Weil--Deligne parameters which is an isomorphism over a dense open of the target.
Specialization for the pro-etale fundamental group (joint with Piotr Achinger and Marcin Lara) || Compositio Mathematica / arXiv
Tags: Rigid analytic geometry, covering space theory, de Jong fundamental group, proétale fundamental group
In previous work of Bhatt and Scholze, there is defined the notion of a pro-étale fundamental group of a scheme to account for covers missed by the étale fundmanetal group of singular schemes. On the other hand, in previous work of de Jong, there is defined a fundamental group (that we call the de Jong fundamental group) of a rigid space, meant to capture richer analytic covering spaces missed by the étale fundmanetal group. In this article we show that these, ostensibly unrelated groups, are connected by construction a specialization morphism from the de Jong fundamental group to the pro-étale fundamental group.
Geometric arcs and fundamental groups of rigid spaces (joint with Piotr Achinger and Marcin Lara) || Journal für die reine und angewandte Mathematik (Crelle's journal) / arXiv
Tags: Rigid analytic geometry, covering space theory, geometric arcs, de Jong fundamental group
In this paper we develop a new notion of covering spaces, called geometric coverings, in rigid geometry. Our definition is modeled after the notion of geometric coverings developed by Bhatt and Scholze in their work on the pro-étale topology for schemes. This definition must be modified to account for the more subtle topology of rigid spaces. We show that geometric coverings are closed under composition, disjoint union, and are etale local on the target. We also show that the category of geometric coverings is a tame infinite Galois category, and so supports a notion of fundamental group.
Variants of the de Jong fundamental group (joint with Piotr Achinger and Marcin Lara) || (to appear in American Journal of Mathematics) arXiv
Tags: Rigid analytic geometry, covering space theory, de Jong fundamental group, proétale topology
In this paper we give an example showing that de Jong covering spaces of a rigid space, developed in previous work of de Jong (where it is called an étale covering space), cannot be glued together in the analytic (i.e., normal topological open cover) topology. We then use our work in the article Geometric arcs and fundamental groups of rigid spaces to produce enlargements of de Jong's category fixing this issue, and show how they are related to the pro-étale topology of rigid spaces as developed in work of Scholze.
An approach to the characterization of the local Langlands correspondence (joint with Alexander Bertoloni Meli) || Representation Theory / arXiv
Tags: The Langlands program, representation theory, endoscopy
In previous work Scholze and Shin conjectured certain equations (the Scholze--Shin equations) should hold for the local Langlands correspondence for any group. In this article we showed that the Scholze--Shin equations (in fact only those for the trivial endoscopic groups) are enough (in addition to the usual desiderata) to characterize the local Langlands correspondence for many groups.
The Scholze--Shin conjecture for unramified unitary groups (joint with Alexander Bertoloni Meli) || PDF
Tags: Shimura varieties, The Langlands program, representation theory, endoscopy, Arthur--Selberg trace formula
In previous work Scholze and Shin conjectured certain equations (the Scholze--Shin equations) should hold for the local Langlands correspondence for any group. In this article we showed that the Scholze--Shin equations hold for the local Langlands correspondence for unramified unitary groups as developed by work of Mok.
The Langlands--Kottwitz--Scholze method for deformation spaces of abelian type (PhD Thesis) || ProQuest
Tags: Shimura varieties, The Langlands program, rigid analytic geometry
In this paper I, following previous work of Scholze, define certain overconvergent open subsets of local Shimura varieties of abelian type and use them to define test functions for such local Shimura data.
Bijective projections on parabolic quotients of affine Weyl groups (joint with Elizabeth Milićević (Beazley), Margaret Nichols, Min Hae Park, and XaoLin Shi) || Journal of Algebraic Combinatorics / arXiv
Tags: Algebraic combinatorics, root theory, alcoves
In previous work of Berg--Jones--Vazirani there was introduced a bijection between two sets of certain combinatorial objects of representation theoretic signifiance. Namely, there was a bijection between n-cores with first part equal to k and (n-1)-cores with first part less than or equal to k. In this article we develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometrically as a projection of alcoves onto the hyperplane containing their coroot lattice points.