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Ana Caraiani

Publications and source records attributed to Ana Caraiani.

At least 19 recordsLinked to original sources
arXiv2026

Torsion-vanishing for Siegel modular varieties via supercuspidal congruences

We prove that the generic part of the cohomology of Siegel modular varieties with torsion coefficients is concentrated above the middle degree, under a suitable notion of genericity that is optimal in the unramified case. Our method relies on the semi-perversity of the relative cohomology of the Igusa stack and on a trace formula computation that is made possible by a congruence technique introduced by Scholze and Fintzen--Shin. Compared with the work of Yang--Zhu, which handles Shimura varieties of abelian type via categorical local Langlands, our method is adapted to Siegel modular varieties but handles coefficients of arbitrarily small characteristic.

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arXiv2026

Intersection Cohomology of Igusa Stacks

We study the intersection cohomology of minimally compactified Shimura varieties of PEL type AC using Igusa stacks and the work of Fargues-Scholze. More precisely, we construct a sheaf on the moduli stack of $G$-bundles on the Fargues-Fontaine curve, which recovers this intersection cohomology after applying a Hecke operator in the sense of geometric Langlands. We show that this sheaf has several desirable properties; for example, it is Verdier self-dual and perverse. This leads to several applications to intersection cohomology, including a version of the Mantovan product formula, as well as torsion-vanishing and Eichler-Shimura relations. Along the way, we investigate the interaction between Baily-Borel and Newton stratifications on minimally compactified Igusa stacks, and we study perverse t-structures on stratified v-stacks.

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arXiv2024

Components of moduli stacks of two-dimensional Galois representations

In a previous article we introduced various moduli stacks of two-dimensional tamely potentially Barsotti-Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil-Kisin modules with descent data. We study the irreducible components of these stacks, establishing in particular that the components of the former are naturally indexed by certain Serre weights.

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arXiv2023

On the generic part of the cohomology of non-compact unitary Shimura varieties

We prove that the generic part of the mod l cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending our previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Taylor and Thorne to get good control on these classes and deduce potential automorphy theorems without any self-duality hypothesis. Our main geometric result is a computation of the fibers of the Hodge-Tate period map on compactified Shimura varieties, in terms of similarly compactified Igusa varieties.

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arXiv2023

Recent progress on Langlands reciprocity for $\mathrm{GL}_n$: Shimura varieties and beyond

The goal of these lecture notes is to survey progress on the global Langlands reciprocity conjecture for $\mathrm{GL}_n$ over number fields from the last decade and a half. We highlight results and conjectures on Shimura varieties and more general locally symmetric spaces, with a view towards the Calegari-Geraghty method to prove modularity lifting theorems beyond the classical setting of Taylor-Wiles.

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arXiv2023

On the étale cohomology of Hilbert modular varieties with torsion coefficients

We study the étale cohomology of Hilbert modular varieties, building on the methods introduced for unitary Shimura varieties in [CS17, CS19]. We obtain the analogous vanishing theorem: in the "generic" case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet--Langlands functoriality established by Tian--Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ occurs in the completed homology of Hilbert modular varieties.

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arXiv2023

On the modularity of elliptic curves over imaginary quadratic fields

In this paper, we establish the modularity of every elliptic curve $E/F$, where $F$ runs over infinitely many imaginary quadratic fields, including $\mathbb{Q}(\sqrt{-d})$ for $d=1,2,3,5$. More precisely, let $F$ be imaginary quadratic and assume that the modular curve $X_0(15)$, which is an elliptic curve of rank $0$ over $\mathbb{Q}$, also has rank $0$ over $F$. Then we prove that all elliptic curves over $F$ are modular. More generally, when $F/\mathbb{Q}$ is an imaginary CM field that does not contain a primitive fifth root of unity, we prove the modularity of elliptic curves $E/F$ under a technical assumption on the image of the representation of $\mathrm{Gal}(\overline{F}/F)$ on $E[3]$ or $E[5]$. The key new technical ingredient we use is a local-global compatibility theorem for the $p$-adic Galois representations associated to torsion in the cohomology of the relevant locally symmetric spaces. We establish this result in the crystalline case, under some technical assumptions, but allowing arbitrary dimension, arbitrarily large regular Hodge--Tate weights, and allowing $p$ to be small and highly ramified in the imaginary CM field $F$.

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arXiv2022

Local geometry of moduli stacks of two-dimensional Galois representations

We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, as well as their resolutions by moduli stacks of two-dimensional Breuil-Kisin modules with tame descent data. We study the local geometry of these moduli stacks by comparing them with local models of Shimura varieties at hyperspecial and Iwahori level.

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arXiv2022

Potential automorphy over CM fields

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbf{A}_F)$.

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arXiv2019

Vanishing theorems for Shimura varieties at unipotent level

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite $\Gamma_1(p^\infty)$-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of the Shimura datum splits at $p$. This generalizes and strengthens the vanishing result proved in "Shimura varieties at level $\Gamma_1(p^\infty)$ and Galois representations". As an application of this vanishing theorem, we prove a result on the codimensions of ordinary completed homology for the same groups, analogous to conjectures of Calegari--Emerton for completed (Borel--Moore) homology.

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arXiv2019

Shimura varieties at level $Γ_1(p^\infty)$ and Galois representations

We show that the compactly supported cohomology of certain $\mathrm{U}(n,n)$ or $\mathrm{Sp}(2n)$-Shimura varieties with $Γ_1(p^\infty)$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\mathrm{GL}_n/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze and Newton-Thorne.

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arXiv2018

Kisin modules with descent data and parahoric local models

We construct a moduli space $Y^{μ, τ}$ of Kisin modules with tame descent datum $τ$ and with fixed $p$-adic Hodge type $\leq μ$, for some finite extension $K/\mathbb{Q}_p$. We show that this space is smoothly equivalent to the local model for $\mathrm{Res}_{K/\mathbb{Q}_p} \mathrm{GL}_n$, cocharacter $\{ μ\}$, and parahoric level structure. We use this to construct the analogue of Kottwitz-Rapoport strata on the special fiber $Y^{μ, τ}$ indexed by the $μ$-admissible set. We also relate $Y^{μ, τ}$ to potentially crystalline Galois deformation rings.

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arXiv2017

Patching and the p-adic Langlands program for GL(2, Q_p)

We present a new construction of the p-adic local Langlands correspondence for GL(2, Q_p) via the patching method of Taylor--Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and global aspects of the p-adic Langlands program; in particular, it gives a new proof of many cases of the second author's local-global compatibility theorem, and relaxes a hypothesis on the local mod p representation in that theorem.

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arXiv2016

Patching and the p-adic local Langlands correspondence

We use the patching method of Taylor--Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL_n(F), F a finite extension of Q_p. We use our construction to prove many new cases of the Breuil--Schneider conjecture.

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arXiv2015

p-adic q-expansion principles on unitary Shimura varieties

We formulate and prove certain vanishing theorems for p-adic automorphic forms on unitary groups of arbitrary signature. The p-adic q-expansion principle for p-adic modular forms on the Igusa tower says that if the coefficients of (sufficiently many of) the q-expansions of a p-adic modular form f are zero, then f vanishes everywhere on the Igusa tower. There is no p-adic q-expansion principle for unitary groups of arbitrary signature in the literature. By replacing q-expansions with Serre-Tate expansions (expansions in terms of Serre-Tate deformation coordinates) and replacing modular forms with automorphic forms on unitary groups of arbitrary signature, we prove an analogue of the p-adic q-expansion principle. More precisely, we show that if the coefficients of (sufficiently many of) the Serre-Tate expansions of a p-adic automorphic form f on the Igusa tower (over a unitary Shimura variety) are zero, then f vanishes identically on the Igusa tower. This paper also contains a substantial expository component. In particular, the expository component serves as a complement to Hida's extensive work on p-adic automorphic forms.

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arXiv2015

On the generic part of the cohomology of compact unitary Shimura varieties

The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties.

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