arXiv subjects

Patrick Daniels

Publications and source records attributed to Patrick Daniels.

7 recordsLinked to original sources

arXiv2026

Igusa Stacks and the Cohomology of Shimura Varieties II

We construct Igusa stacks for all Shimura varieties of abelian type and derive consequences for the cohomology of these Shimura varieties. As an application, we prove that the Fargues--Scholze local Langlands correspondence agrees with the semi-simplification of the local Langlands correspondences constructed by Arthur, Mok and others, for all classical groups of type $A$, $B$ and $D$; this extends work of Hamann, Bertoloni Meli--Hamann--Nguyen and Peng.

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arXiv2024

Igusa Stacks and the Cohomology of Shimura Varieties

We construct functorial Igusa stacks for all Hodge-type Shimura varieties, proving a conjecture of Scholze and extending earlier results of the fourth-named author for PEL-type Shimura varieties. Using the Igusa stack, we construct a sheaf on $\mathrm{Bun}_G$ that controls the cohomology of the corresponding Shimura variety. We use this sheaf and the spectral action of Fargues-Scholze to prove a compatibility between the cohomology of Shimura varieties of Hodge type and the semisimple local Langlands correspondence of Fargues-Scholze, generalizing the Eichler-Shimura relation of Blasius-Rogawski to arbitrary level at $p$. When the given Shimura variety is proper, we show moreover that the sheaf is perverse, which allows us to prove new torsion vanishing results for the cohomology of Shimura varieties.

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arXiv2024

On a conjecture of Pappas and Rapoport

We prove a conjecture of Pappas and Rapoport about the existence of ''canonical'' integral models of Shimura varieties of Hodge type with quasi-parahoric level structure at a prime $p$. For these integral models, we moreover show uniformization of isogeny classes by integral local Shimura varieties, and prove a conjecture of Kisin and Pappas on local model diagrams.

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arXiv2024

Canonical Integral Models of Shimura Varieties of Abelian Type

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.

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arXiv2023

$G$-displays of Hodge type and formal $p$-divisible groups

Let $G$ be a reductive group scheme over the $p$-adic integers, and let $μ$ be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a functor from nilpotent $(G,μ)$-displays over $p$-nilpotent rings $R$ to formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors. When $R/pR$ has a $p$-basis étale locally, we show that this defines an equivalence between the two categories. The definition of the functor relies on the construction of a $G$-crystal associated with any adjoint nilpotent $(G,μ)$-display, which extends the construction of the Dieudonné crystal associated with a nilpotent Zink display. As an application, we obtain an explicit comparison between the Rapoport-Zink functors of Hodge type defined by Kim and by Bültel and Pappas.

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arXiv2022

Canonical integral models for Shimura varieties of toral type

We prove the Pappas-Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt-Scholze theory of prismatic $F$-crystals, that there is a fully faithful functor from $\mathcal{G}$-valued crystalline representations of Gal$(\bar{K}/K)$ to $\mathcal{G}$-shtukas over Spd$(\mathcal{O}_K)$, where $\mathcal{G}$ is a parahoric group scheme over $\mathbb{Z}_p$ and $\mathcal{O}_K$ is the ring of integers in a $p$-adic field $K$.

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arXiv2019

A Tannakian framework for $G$-displays and Rapoport-Zink spaces

We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport-Zink functors associated to triples $(G,\{μ\},[b])$, where $G$ is a flat affine group scheme over $\mathbb{Z}_p$ and $μ$ is a cocharacter of $G$ defined over a finite unramified extension of $\mathbb{Z}_p$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport-Zink spaces given by Bültel and Pappas. As an application, we prove a special case of a conjecture of Bültel and Pappas by showing their definition coincides with that of Rapoport and Zink in the case of unramified EL-type local Shimura data.

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