Preprints and publications

The Langlands program

Title	Collaborators	Journal/Status	Links	Description
Canonical Integral Models of Shimura Varieties of Abelian Type	Patrick Daniels	Submitted	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. In particular, 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 for Shimura Varieties of Abelian Type	Naoki Imai and Hiroki Kato	Submitted	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 \(D_{\mathrm{crys}}\), and relate it to Fontaine--Lafaille and Dieudonné theory.
The analytic topology is enough for the \(B_\mathrm{dR}^+\)-Grassmannian	Kęstutis Česnavičius	Submitted	arXiv	In this paper we show that the \(B_\mathrm{dR}^+\)-affine Grassmannian can be computed as the sheafification of the presheaf quotient \(LG/L^+G\) for the analytic open topology (opposed to the étale topology) for a reductive group \(G\).
The Jacobson--Morozov morphism for Langlands parameters in the relative setting	Alexander Bertoloni Meli and Naoki Imai	International Mathematics Research Notices	Journal/ arXiv	In this paper we define a moduli space of \(L\)-parameters over \(\mathbb{Q}\), 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	Alexander Bertoloni Meli	Representation Theory	Journal/ arXiv	In this paper we show that, under suitable hypotheses, the local Langlands conjecture may be characterized by the so-called Scholze--Shin equations.
The Scholze--Shin conjecture for unramified unitary groups	Alexander Bertoloni Meli	Draft	pdf	In this paper we prove that the Scholze--Shin conjectures hold true for unramified unitary groups.
The Langlands--Kottwitz--Scholze method for deformation spaces of abelian type	N/A	In preparation, available on request	N/A	In this paper I, following previous work of Scholze, define certain overconvergent open subsets of local Shimura varieties of abelian type, and show that their cohomology allows one to give a formula for the trace of the Hecke/Galois action on the cohomology of Shimura varieties of abelian type.

Rigid geometry

Title	Collaborators	Journal/Status	Links	Description
Specialization for the pro-etale fundamental group	Piotr Achinger and Marcin Lara	Compositio Mathematica	Journal/ arXiv	For a formal scheme \(\mathfrak{X}\) over a height \(1\) valuation ring \(\mathcal{O}\), we construct a continuous specialization homomorphism \(\pi_1^\mathrm{dJ}(\mathfrak{X}_\eta)\to \pi_1^\mathrm{proet}(\mathfrak{X}_s)\) from the de Jong fundamental group of the generic fiber to the Bhatt--Scholze pro-etale fundamental group of the special fiber. We show that it is surjective in good cases. We also show that the notion of a de Jong covering space is etale local on the target for a smooth quasi-compact rigid space over an equicharacteristic \(0\) non-archimedean field \(K\).
Geometric arcs and fundamental groups of rigid spaces	Piotr Achinger and Marcin Lara	Journal für die reine und angewandte Mathematik	Journal/ arXiv	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--Scholze, but 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	Piotr Achinger and Marcin Lara	Submitted	arXiv	In this paper, which is a companion paper to Geometric arcs and fundamental groups of rigid spaces, we use the theory of geometric coverings to study some previously studied objects. For a (Grothendieck) topology \(\tau\) on a rigid space \(X\) write \(\mathbf{Cov}^\tau_X\) for the category of morphisms \(Y\to X\) which split into a disjoint union of finite etale pieces \(\tau\)-locally. We answer two questions of de Jong by showing that in generality his notion of covering space is not admissible (and thus certainly not etale) local on the target, but that the resulting enlarged categories \(\mathbf{Cov}^\mathrm{adm}_X\) and \(\mathbf{Cov}^\mathrm{et}_X\) are still (essentially) tame infinite Galois categories. We also show that the category \(\mathbf{Loc}(X_\mathrm{proet})\) of pro-etale local systems on \(X\) is equivalent to the category \(\mathbf{Cov}^\mathrm{et}_X\) and so the former is (essentially) a tame infinite Galois category.
Fundamental groups and specialization in rigid geometry	N/A	数理解析研究所講究録 (Kōkyūroku)	pdf	This is a survery article for the paper "Specialization for the pro-etale fundamental group''.

Misc.

Title	Collaborators	Journal/Status	Links	Description
Bijective projections on parabolic quotients of affine Weyl groups	E. Beazley, M. Nichols, M. Park, and X. Shi	Journal of Algebraic Combinatorics	Journal / arXiv	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.