arXiv subjects

Alex Youcis

Publications and source records attributed to Alex Youcis.

13 recordsLinked to original sources

arXiv2026

On canonicity for integral models of Shimura varieties with hyperspecial level

We give a new definition -- and in some cases, a new construction -- of integral canonical models of Shimura varieties that uses the notion of an aperture appearing in work of Gardner--Madapusi on some conjectures of Drinfeld. This applies to Shimura varieties of pre-abelian type at odd primes of hyperspecial level, recovering and extending previous work of Kisin, Kim--Madapusi and Imai--Kato--Youcis, but also to exceptional Shimura varieties for large enough primes. The characterization in the exceptional case is \emph{a priori} different from the one recently shown by Bakker--Shankar--Tsimerman, and recovers many of their results, such as the existence of prime-to-$p$ Hecke operators, the non-emptiness of the $\mu$-ordinary stratum and the theory of the canonical lift. In fact, we give a uniform proof of the non-emptiness of \emph{all} possible Newton strata, and of the non-emptiness of Ekedahl--Oort strata and central leaves as well. An important ingredient in the proofs is a generalization of Tate's full faithfulness theorem for $p$-divisible groups to the context of apertures. This leads to a mapping property for the integral canonical model that characterizes maps into it from all normal, flat and excellent schemes over $\mathbb{Z}_{(p)}$.

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arXiv2025

An integral comparison of crystalline and de Rham cohomology

Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X/\mathcal{O}_K$ there exists a comparison between the de Rham cohomology groups $\mathrm{H}^i_\mathrm{dR}(X/\mathcal{O}_K)$ and the crystalline cohomology groups $\mathrm{H}^i_\mathrm{crys}(X_k/W(k))$ of the special fibre, after tensoring with $K$. In this article, we use the stacky perspective on prismatic cohomology, due to Drinfeld and Bhatt--Lurie, to give a version of this comparison result with coefficients in a perfect complex of prismatic $F$-crystals on $X$. Our method is of an integral nature and suggests new tools to understand the relationship between torsion in de Rham and crystalline cohomology.

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arXiv2025

An integral analogue of Fontaine's crystalline functor

For a smooth formal scheme $\mathfrak{X}$ over the Witt vectors $W$ of a perfect field $k$, we construct a functor $\mathbb{D}_\mathrm{crys}$ from the category of prismatic $F$-crystals $(\mathcal{E},\varphi_\mathcal{E})$ (or prismatic $F$-gauges) on $\mathfrak{X}$ to the category of filtered $F$-crystals on $\mathfrak{X}$. We show that $\mathbb{D}_\mathrm{crys}(\mathcal{E},\varphi_\mathcal{E})$ enjoys strong properties when $(\mathcal{E},\varphi_\mathcal{E})$ is what we call locally filtered free (lff). Most significantly, we show that $\mathbb{D}_\mathrm{crys}$ actually induces an equivalence between the category of prismatic $F$-gauges on $\mathfrak{X}$ with Hodge--Tate weights in $[0,p-2]$ and the category of Fontaine--Laffaille modules on $\mathfrak{X}$. Finally, we use our functor $\mathbb{D}_\mathrm{crys}$ to enhance the study of prismatic Dieduonn\'e theory of $p$-divisible groups (as initiated by Ansch\"{u}tz--Le Bras) allowing one to recover the filtered crystalline Dieudonn\'e crystal from the prismatic Dieudonn\'e crystal. This in turn allows us to clarify the relationship between prismatic Dieudonn\'e theory and the work of Kim on classifying $p$-divisible groups using Breuil--Kisin modules.

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arXiv2024

Beauville-Laszlo gluing of algebraic spaces

For a complete discrete valuation field $K$, we show that one may always glue a separated formal algebraic space $\mathfrak{X}$ over $\mathcal{O}_K$ to a separated algebraic space $U$ over $K$ along an open immersion of rigid spaces $j\colon \mathfrak{X}^{\rm rig}\to U^{\rm an}$, producing a separated algebraic space $X$ over $\mathcal{O}_K$. This process gives rise to an equivalence between such `gluing triples' $(U,\mathfrak{X},j)$ and separated algebraic spaces $X$ over $\mathcal{O}_K$, which one might interpret as a version of the Beauville--Laszlo theorem for algebraic spaces rather than coherent sheaves. Moreover, an analogous equivalence exists over any excellent base. Examples due to Matsumoto imply that the result of such a gluing might be a genuine algebraic space (not a scheme) even if $U$ and the special fiber of $\mathfrak{X}$ are projective. The proof is a combination of Nagata compactification theorem for algebraic spaces and of Artin's contraction theorem. We give multiple examples and applications of this idea.

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arXiv2024

A Tannakian framework for prismatic $F$-crystals

We develop the Tannakian theory of (analytic) prismatic $F$-crystals on a smooth formal scheme $\mathfrak{X}$ over the ring of integers of a discretely valued field with perfect residue field. Our main result gives an equivalence between the $\mathcal{G}$-objects of prismatic $F$-crystals on $\mathfrak{X}$ and $\mathcal{G}$-objects on a newly-defined category of $\mathbb{Z}_p$-local systems on $\mathfrak{X}_\eta$: those of prismatically good reduction. Additionally, we develop a shtuka realization functor for (analytic) prismatic $F$-crystals on $p$-adic (formal) schemes and show it satisfies several compatibilities with previous work on the Tannakian theory of shtukas over such objects.

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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

The prismatic realization functor for Shimura varieties of abelian type

For the integral canonical model $\mathscr{S}_{\mathsf{K}^p}$ of a Shimura variety $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$ of abelian type at hyperspecial level $K_0=\mathcal{G}(\mathbb{Z}_p)$, we construct a prismatic $F$-gauge model for the `universal' $\mathcal{G}(\mathbb{Z}_p)$-local system on $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$. We use this to obtain several new results about the $p$-adic geometry of Shimura varieties, notably an abelian-type analogue of the Serre--Tate deformation theorem (realizing an expectation of Drinfeld in the abelian-type case) and a prismatic characterization of these models at individual level.

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arXiv2023

The analytic topology suffices for the $B_{\mathrm{dR}}^+$-Grassmannian

The $B_{\mathrm{dR}}^+$-affine Grassmannian was introduced by Scholze in the context of the geometric local Langlands program in mixed characteristic and is the Fargues-Fontaine curve analogue of the equal characteristic Beilinson-Drinfeld affine Grassmannian. For a reductive group $G$, it is defined as the \'{e}tale (equivalently, $v$-) sheafification of the presheaf quotient $LG/L^+G$ of the $B_{\mathrm{dR}}$-loop group $LG$ by the $B_{\mathrm{dR}}^+$-loop subgroup $L^+G$. We combine algebraization and approximation techniques with known cases of the Grothendieck-Serre conjecture to show that the analytic topology suffices for this sheafification, more precisely, that the $B_{\mathrm{dR}}^+$-affine Grassmannian agrees with the analytic sheafification of the aforementioned presheaf quotient $LG/L^+G$.

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arXiv2022

Variants of the de Jong fundamental group

For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ of coverings which are locally in the admissible topology on $X$ the disjoint union of finite etale coverings: we show that this class is different from the one used by de Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of $\mathbf{Cov}^\mathrm{et}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze.

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arXiv2022

The Jacobson--Morozov morphism for Langlands parameters in the relative setting

We construct a moduli space $\mathsf{LP}_G$ of $\mathrm{SL}_2$-parameters over $\mathbb{Q}$, and show that it has good geometric properties (e.g. explicitly parametrized geometric connected components and smoothness). We construct a Jacobson--Morozov morphism $\mathsf{JM}\colon \mathsf{LP}_G\to\mathsf{WDP}_G$ (where $\mathsf{WDP}_G$ is the moduli space of Weil--Deligne parameters considered by several other authors). We show that $\mathsf{JM}$ is an isomorphism over a dense open of $\mathsf{WDP}_G$, that it induces an isomorphism between the discrete loci $\mathsf{LP}^{\mathrm{disc}}_G\to\mathsf{WDP}_G^{\mathrm{disc}}$, and that for any $\mathbb{Q}$-algebra $A$ it induces a bijection between Frobenius semi-simple equivalence classes in $\mathsf{LP}_G(A)$ and Frobenius semi-simple equivalence classes in $\mathsf{WDP}_G(A)$ with constant (up to conjugacy) monodromy operator.

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arXiv2021

Specialization for the pro-\'etale fundamental group

For a formal scheme $\mathfrak{X}$ of finite type over a complete rank one valuation ring, we construct a specialization morphism \[ \pi^{\rm dJ}_1(\mathfrak{X}_\eta) \to \pi^{\rm proet}_1(\mathfrak{X}_k) \] from the de Jong fundamental group of the rigid generic fiber to the Bhatt-Scholze pro-\'etale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of $\mathfrak{X}$ and normalizations of the irreducible components of $\mathfrak{X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy \'etale descent.

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arXiv2021

Geometric arcs and fundamental groups of rigid spaces

We develop the notion of a geometric covering of a rigid space X, which yields a much larger class of covering spaces than that studied previously by de Jong. Geometric coverings of X are closed under disjoint unions and are \'etale local on X. If X is connected, its geometric coverings form a tame infinite Galois category, and hence are classified by a topological group. The definition is based on the property of lifting of "geometric arcs," making it similar to geometric coverings of schemes studied by Bhatt and Scholze as well as semicoverings of topological spaces introduced by Brazas.

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arXiv2020

An Approach to the Characterization of the Local Langlands Correspondence

In this paper, we give a method for characterizing the local Langlands conjectures in the vein of Scholze's alternate proof of the local Langlands conjecture for $\mathrm{GL}_n$. More specifically, we show that if a local Langlands correspondence satisfies a Scholze--Shin equation, as in the paper of Scholze and Shin, in addition to the usual desiderata expected of such a correspondence then these properties uniquely characterize the correspondence.

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