Langlands program
Description
The Langlands program, broadly speaking, seeks to unify algebraic geometry, number theory, and harmonic analysis. More precisely, it posits that there is a tight connection between representations arising from these three (ostensibly disparate) sources. Its development has been influenced by, and in turn influenced, several areas of arithmetic geometry, notably rigid geometry and p-adic Hodge theory.
Some key phrases
Shimura varieties (local and global), L-parameters, Arthur--Selberg trace formula, and endoscopy.
Annotated list of my relevant papers
Canonical Integral Models of Shimura Varieties of Abelian Type (joint with Patrick Daniels) || arXiv
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
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 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, 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
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 etale 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
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.
An approach to the characterization of the local Langlands correspondence (joint with Alexander Bertoloni Meli) || Representation Theory / arXiv
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
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.