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We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local Galois representations are algebraic (or Ind-algebraic) stacks., 137 pages; final version although not incorporating minor edits made in proof, appeared as Algebraic Geometry 8 (1) (2021) 1--132

Let n be either 2, or an odd integer greater than 1, and fix a prime p > 2(n + 1). Under standard "adequate image" assumptions, we show that the set of components of n-dimensional p-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on n) improve on the main potential automorphy result of [BLGGT14b], replacing "potentially diagonalizable" by "potentially globally realizable"., 66 pages; final version, to appear in Journal de l'Institut de Math\'ematiques de Jussieu

Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case., Comment: Accepted version, to appear in Essential Number Theory

Under an assumption on the existence of p-adic Galois representations, we carry out Taylor--Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated to GL(n) over a number field. We use our construction to show that standard conjectures on completed homology imply `big R = big T' theorems. In the case that n=2 and p splits completely in the number field, we relate our construction to the p-adic local Langlands correspondence for GL(2,Q_p)., v2: an issue with the definition of the big Hecke algebra has been fixed. Some other minor changes. v3: slightly weakened the local--global compatibility Conjecture (Conjecture 5.1.12). v4: minor changes, mostly to introduction. v5: fixed an issue in Appendix B, some other minor changes

We present a new construction of the $p$-adic local Langlands correspondence for $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ 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 the 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.

We prove the explicit version of the Buzzard--Diamond--Jarvis conjecture formulated by Diamond--Demb\'el\'e--Roberts. More precisely, we prove that it is equivalent to the original Buzzard--Diamond--Jarvis conjecture, which was proved for odd primes (under a mild Taylor--Wiles hypothesis) in earlier work of the third author and coauthors., Comment: Final version, to appear in Compositio. 16 pages

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields., Comment: Final version (fixing minor typos found in copyediting). 292 pages, to appear in Publ. Math. de l'IHES

We study the weight part of (a generalisation of) Serre’s conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when n = 3 {n=3} and l splits completely in the underlying CM field.

We formulate a version of the Breuil–Mezard conjecture for quaternion algebras, and show that it follows from the Breuil–Mezard conjecture for GL2. In the course of the proof we establish a mod p analogue of the Jacquet– Langlands correspondence for representations of GL2(k), k a finite field of characteristic p.

We prove conjectures of Breuil and Breuil-Dembele (C. Breuil, "Sur un probleme de compatibilite local-global modulo p pour GL(2)"), including a generalisation from the principal series to the cuspidal case, subject to a mild global hypothesis that we make in order to apply certain R=T theorems. More precisely, we prove a multiplicity one result for the mod p cohomology of a Shimura curve at Iwahori level, and we show that certain apparently globally defined lattices in the cohomology of Shimura curves are determined by the corresponding local p-adic Galois representations. We also indicate a new proof of the Buzzard-Diamond-Jarvis conjecture in generic cases. Our main tools are the geometric Breuil-Mezard philosophy developed by two of the authors, and a new and more functorial perspective on the Taylor-Wiles-Kisin patching method. Along the way, we determine the tamely potentially Barsotti-Tate deformation rings of generic two-dimensional mod p representations, generalising a result of Breuil-Mezard in the principal series case., 75 pages, essentially final version; to appear, Invent. Math

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GLn over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight. Resume. Nous prouvons la compatibilite entre les correspondances de Langlands locale et globale aux places divisant l pour les representations galoisiennes l-adiques associees a des representations automorphes cuspidales algebriques et regulieres de GLn sur un corps CM qui sont duales de leur conjuguee complexe, sous les hypotheses supplementaires que ces representations automorphes ont des vecteurs fixes par un sous-groupe d’Iwahori aux places divisant l et ont un poids regulier au sens de Shin.

Let p > 2 be prime. We prove the weight part of Serre’s conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard–Diamond–Jarvis conjecture for unitary groups), by proving that any Serre weight which occurs is a predicted weight. Our methods are purely local, using the theory of (φ, Ĝ)-modules to determine the possible reductions of certain two-dimensional crystalline representations.

Let$p\gt 2$be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for two-dimensional mod$p$representations of the absolute Galois group of${ \mathbb{Q} }_{p} $. We also state a conjectural generalization to$n$-dimensional representations of the absolute Galois group of an arbitrary finite extension of${ \mathbb{Q} }_{p} $, and give a conditional proof of this conjecture, subject to a certain$R= \mathbb{T} $-type theorem together with a strong version of the weight part of Serre’s conjecture for rank $n$unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti–Tate representations.

Let $p>2$ be prime, and let $F$ be a totally real field in which $p$ is unramified. We give a sufficient criterion for a mod $p$ Galois representation to arise from a mod $p$ Hilbert modular form of parallel weight one, by proving a "companion forms" theorem in this case. The techniques used are a mixture of modularity lifting theorems and geometric methods. As an application, we show that Serre's conjecture for $F$ implies Artin's conjecture for totally odd two-dimensional representations over $F$., 12 pages

Let pi be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GL_n(A_F), where F is a totally real field and n is at most 5. We show that for all primes l, the l-adic Galois representations associated to pi are irreducible, and for all but finitely many primes l, the mod l Galois representations associated to pi are also irreducible. We also show that the Lie algebras of the Zariski closures of the l-adic representations are independent of l., Erratum: there is a gap in the proof of the main theorem for n=4, 5

We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in Buzzard (Asterisque 298:1–15, 2005), discuss strategies for making further progress, and examine other related questions.

We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions., Essentially final version, to appear, J. Reine Angew. Math. This version does not incorporate any minor changes (e.g. typographical changes) made in proof

We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of GL(n) over an arbitrary number field, motivated by the formalism of the Breuil-M\'ezard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over Q_p, and we also generalise the second author's previous conjecture for GL(n)/Q to this setting, and show that the two conjectures are generically in agreement., Comment: Essentially final version, to appear in J. Eur. Math. Soc. This version will not incorporate any minor changes made in proof

Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, embeds into the reduction modulo p of a semistable Galois representation with Hodge-Tate weights in the expected range (at least after semisimplifying, in the case of the cohomological degree > 1). We prove refinements with descent data, and we apply these results to the cohomology of unitary Shimura varieties, deducing vanishing results and applications to the weight part of Serre's conjecture., Comment: Essentially final version; to appear in Algebra and Number Theory

We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$ , $K$ a finite extension of $\mathbb{Q}_{p}$ , for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

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., Final version, to appear in Cambridge Journal of Mathematics

Under mild hypotheses, we prove that if F is a totally real field, k is the algebraic closure of the finite field with l elements and r : G_F --> GL_2(k) is irreducible and modular, then there is a finite solvable totally real extension F'/F such that r|_{G_F'} has a modular lift which is ordinary at each place dividing l. We deduce a similar result for r itself, under the assumption that at places v|l the representation r|_{G_F_v} is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti-Tate representations and the Buzzard-Diamond-Jarvis conjecture. The proof makes use of a novel lifting technique, going via rank 4 unitary groups., 48 pages

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields., 40 pages

We study the weight part of (a generalisation of) Serre's conjecture for mod l Galois representations associated to automorphic representations on rank two unitary groups for odd primes l. We propose a conjectural set of Serre weights, agreeing with all conjectures in the literature, and under a mild assumption on the image of the mod l Galois representation we are able to show that any modular representation is modular of each conjectured weight. We make no assumptions on the ramification or inertial degrees of l. Our main innovation is to make use of the lifting techniques introduced in our recent papers., 43 pages

We prove the compatibility at places dividing l of the local and global Langlands correspondences for the l-adic Galois representations associated to regular algebraic essentially (conjugate) self-dual cuspidal automorphic representations of GL_n over an imaginary CM or totally real field. We prove this compatibility up to semisimplification in all cases, and up to Frobenius semisimplification in the case of Shin-regular weight., 13 pages

We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod $p$ Hilbert modular forms, by making use of modularity lifting theorems and computations in $p$-adic Hodge theory., Comment: 34 pages. Essentially final version, to appear in Inventiones mathematicae. This version does not incorporate any minor changes (e.g. typographical changes) made in proof

We study the possible weights of an irreducible two-dimensional mod p representation of the absolute Galois group of F which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions., Comment: Essentially final version, to appear in Compositio Mathematica. This version does not incorporate any minor changes (e.g. typographical changes) made in proof

Let p>2 be prime. We complete the proof of the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the totally ramified case, by proving that any weight which occurs is a predicted weight. Our methods are a mixture of local and global techniques, and in the course of the proof we establish some purely local results on crystalline extension classes., 17 pages

We prove a new automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call `potential diagonalizability'. This result allows for `change of weight' and seems to be substantially more flexible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real field is potentially automorphic, and hence is pure and its L-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation., Adding missing hypothesis that F doesn't contain zeta_l to statements of 2.3.1, 2.3.2, 4.1.1; proofs of these and all other statements and proofs in the paper are unaffected

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\GL_2(\A_F)$, $F$ a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a 'topological' argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary $n$-dimensional Galois representations., 59 pages. Essentially final version, to appear in Journal of the AMS. This version does not incorporate any minor changes (e.g. typographical changes) made in proof

In this paper, we use the p-adic local Langlands correspondence for to explicitly compute the reduction modulo p of certain two-dimensional crystalline representations of small slope, and give applications to modular forms.

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for $n$-dimensional automorphic Galois representations., Essentially final version, to appear in Math Annalen. This version does not incorporate any minor changes (e.g. typographical changes) made in proof

We prove a companion forms theorem for mod l Hilbert modular forms. This work generalises results of Gross and Coleman--Voloch for modular forms over Q, and gives a new proof of their results in many cases. The methods used are completely different to previous work in this area, and rely on modularity lifting theorems and the general theory of deformations of Galois representations., 8 pages. Essentially final version, appeared in Duke Math. Journal 136 (2007), no. 2. This version does not incorporate any minor changes (e.g. typographical changes) made in proof

We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2, Comment: Appeared as Manuscripta Math. 125 (2008), no. 1, 1-41. This version does not incorporate any minor changes (e.g. typographical changes) made in proof