Ana Caraiani's papers

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Publications and preprints (click on title to view abstract):

Recent progress on Langlands reciprocity for GL_n: Shimura varieties and beyond (with Sug Woo Shin), to appear in Proceedings of the 2022 IHES summer school on the Langlands program.	PDF	arXiv
The goal of these lecture notes is to survey progress on the global Langlands reciprocity conjecture for 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.
On the modularity of elliptic curves over imaginary quadratic fields (with James Newton)	PDF	arXiv
In this paper, we establish the modularity of every elliptic curve E/F, where F runs over infinitely many imaginary quadratic fields, including ℚ(√(-d)) for d=1,2,3,5. More precisely, let F be imaginary quadratic and assume that the modular curve X₀(15), which is an elliptic curve of rank 0 over ℚ, also has rank 0 over F. Then we prove that all elliptic curves over F are modular. More generally, when F/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 Gal(F̅/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.
The cohomology of Shimura varieties with torsion coefficients, ICM–International Congress of Mathematicians. Vol. 3. Sections 1–4, 1744–1766.	PDF		Book	Video
In this article, we survey recent work on some vanishing conjectures for the cohomology of Shimura varieties with torsion coefficients, under both local and global conditions. We discuss the p-adic geometry of Shimura varieties and of the associated Hodge–Tate period morphism, and explain how this can be used to make progress on these conjectures. Finally, we describe some applications of these results, in particular to the proof of the Sato–Tate conjecture for elliptic curves over CM fields.
The geometric Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate Galois representations (with Matthew Emerton, Toby Gee, and David Savitt), to appear in Algebra Number Theory.	PDF	arXiv
We establish a geometrisation of the Breuil–Mézard conjecture for potentially Barsotti–Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field.
Components of moduli stacks of two-dimensional Galois representations (with Matthew Emerton, Toby Gee, and David Savitt), Forum Math. Sigma 12 (2024), e31, 1–62.	PDF	arXiv	Journal
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.
Local geometry of moduli stacks of two-dimensional Galois representations (with Matthew Emerton, Toby Gee, and David Savitt), to appear in Proceedings of the International Colloquium on 'Arithmetic Geometry', TIFR Mumbai, Jan. 6–10, 2020.	PDF	arXiv
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.
On the étale cohomology of Hilbert modular varieties with torsion coefficients (with Matteo Tamiozzo), Compositio Math. 159 (2023), no. 11, 2279–2325.	PDF	arXiv	Journal
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 GL₂(ℚ_p) occurs in the completed homology of Hilbert modular varieties.
New frontiers in Langlands reciprocity, EMS Magazine (2021), no. 119.	PDF		Journal
In this survey, I discuss some recent developments at the crossroads of arithmetic geometry and the Langlands programme. The emphasis is on recent progress on the Ramanujan–Petersson and Sato–Tate conjectures. This relies on new results about Shimura varieties and torsion in the cohomology of locally symmetric spaces.
Vanishing theorems for Shimura varieties at unipotent level (with Daniel R. Gulotta and Christian Johansson), J. Eur. Math. Soc. 25 (2023), no. 3, 869–911.	PDF	arXiv	Journal
We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite Γ₁(p^∞)-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 Γ₁(p^∞) 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.
On the generic part of the cohomology of non-compact unitary Shimura varieties (with Peter Scholze), Annals of Math. 199 (2024), no. 2, 483–590.	PDF	arXiv	Journal
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 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 [ACC+18] 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.
Potential automorphy over CM fields (with Patrick B. Allen, Frank Calegari, Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack A. Thorne), Annals of Math. 197 (2023), no. 3, 897–1113.	PDF	arXiv	Journal
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 GL₂(𝔸_F).
Perfectoid Shimura varieties, in Perfectoid spaces: Lectures from the 2017 Arizona Winter School.	PDF		Book	Video
This is an expanded version of the lecture notes for the minicourse I gave at the 2017 Arizona Winter School. In these notes, I discuss Scholze's construction of Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n, with a focus on his proof that Shimura varieties of Hodge type with infinite level at p acquire the structure of perfectoid spaces. I also briefly discuss some recent vanishing results for the cohomology of Shimura varieties with infinite level at p.
Shimura varieties at level Γ₁(p^∞) and Galois representations (with Daniel R. Gulotta, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih), Compositio Math. 156 (2020), no. 6, 1152–1230.	PDF	arXiv	Journal	Video
We show that the compactly supported cohomology of certain U(n,n) or Sp(2n)-Shimura varieties with Γ₁(p^∞)-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 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.
Patching and the p-adic Langlands program for GL(2,ℚ_p) (with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paškūnas, and Sug Woo Shin), Compositio Math. 154 (2018), no. 3, 503–548.	PDF	arXiv	Journal
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.
Kisin modules with descent data and parahoric local models (with Brandon Levin), Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 1, 181–213.	PDF	arXiv	Journal
We construct a moduli space Y^μ,τ of Kisin modules with tame descent datum τ and with fixed p-adic Hodge type μ, for some finite extension K/ℚ_p. We show that this space is smoothly equivalent to the local model for Res_{K/ℚ_p}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.
On the generic part of the cohomology of compact unitary Shimura varieties (with Peter Scholze), Annals of Math. 186 (2017), no. 3, 649–766.	PDF	arXiv	Journal
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.
p-adic q-expansion principles on unitary Shimura varieties (with Ellen Eischen, Jessica Fintzen, Elena Mantovan, and Ila Varma), Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop. Springer International Publishing (2016), 197–243.	PDF	arXiv	Book
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.
On the image of complex conjugation in certain Galois representations (with Bao V. Le Hung), Compositio Math. 152 (2016), no. 7, 1476–1488.	PDF	arXiv	Journal
We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for GL_n over a totally real field F.
Patching and the p-adic local Langlands correspondence (with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paškūnas, and Sug Woo Shin), Cambridge Journal of Math. 4 (2016), no. 2, 197–287.	PDF	arXiv	Journal	Video
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.
Monodromy and local-global compatibility for l=p, Algebra Number Theory 8 (2014), no. 7, 1597–1646.	PDF	arXiv	Journal
We strengthen the compatibility between local and global Langlands correspondences for GL_n when n is even and l=p. Let L be a CM field and Π a cuspidal automorphic representation of GL_n(A_L) which is conjugate self-dual and regular algebraic. In this case, there is an l-adic Galois representation associated to Π, which is known to be compatible with local Langlands in almost all cases when l=p by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless Π has Shin-regular weight. We extend the compatibility to Frobenius semisimplification in all cases by identifying the monodromy operator on the global side. To achieve this, we derive a generalization of Mokrane's weight spectral sequence for log crystalline cohomology.
Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J. 161 (2012), no. 12, 2311–2413.	PDF	arXiv	Journal	Video
We strengthen the local-global compatibility of Langlands correspondences for GL_n in the case when n is even and l≠p. Let L be a CM field and Π be a cuspidal automorphic representation of GL_n(A_L) which is conjugate self-dual. Assume that Π_∞ is cohomological and not "slightly regular", as defined by Shin. In this case, Chenevier and Harris constructed an l-adic Galois representation R_l(Π) and proved the local-global compatibility up to semisimplification at primes v not dividing l. We extend this compatibility by showing that the Frobenius semisimplification of the restriction of R_l(Π) to the decomposition group at v corresponds to the image of Π_v via the local Langlands correspondence. We follow the strategy of Taylor-Yoshida, where it was assumed that Π is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator N on the complex of nearby cycles on a scheme which is locally etale over a product of semistable schemes and derive a generalization of the weight-spectral sequence in this case. We also prove the Ramanujan–Petersson conjecture for Π as above.

Oberwolfach reports:

Patching and p-adic local Langlands, in Algebraische Zahlentheorie, joint with M. Emerton, T. Gee, D. Geraghty, V. Paškūnas, and S.W. Shin, Oberwolfach Report No. 32 (2014), 31–33.

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Undergraduate papers:

Multiplicative semigroups related to the 3x+1 problem, Adv. Appl. Math. 45 (2010), no. 3, 373–389.

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