arXiv subjects
Publications and source records attributed to Matthew Emerton.
Let F be a totally real field, and let p be prime. Under standard Taylor--Wiles hypotheses, we show that an irreducible, 2-dimensional, totally odd mod p representation of the absolute Galois group of F admits lifts lying on arbitrary prescribed components of local deformation rings, allowing potentially semistable conditions with arbitrary regular Hodge--Tate weights.
Let p at least 5 be prime. We construct a fully faithful functor from the derived category of all smooth p-adic representations of GL_2(Q_p) (with a fixed central character) to a derived category of Ind-coherent sheaves on a stack of (phi,Gamma)-modules.
We prove that the Mazur-Tate elements of an eigenform $f$ sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic $\mathbb Z_p$-extension (up to scaling by a single constant). Our method begins with the construction of local cohomology classes built via the $p$-adic local Langlands correspondence. From these classes, we build algebraic analogues of the Mazur-Tate elements which we directly verify sit in the appropriate Fitting ideals. Using Kato's Euler system and explicit reciprocity laws, we prove that these algebraic elements divide the corresponding Mazur-Tate elements, implying our theorem.
We give an introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach" and "analytic" settings.
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.
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.
We establish a geometrisation of the Breuil-M\'ezard 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.
We show that the category of smooth representations of GL_2(Q_p) on p-power torsion modules localizes over a certain projective scheme, and give some applications.
This survey article is based on 10 lectures that we gave at the September 2019 Hausdorff School in Bonn on "the Emerton--Gee stack and related topics", and the sections correspond to the original lectures. We have retained the somewhat informal style of some of these lectures, but have filled them out with further details.
We construct stacks which algebraize Mazur's formal deformation rings of local Galois representations. More precisely, we construct Noetherian formal algebraic stacks over Spf Zp which parameterize \'etale (phi,Gamma)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. We use these stacks to show that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero, and indeed admit crystalline lifts. We also discuss the relationship between the geometry of our stacks and the Breuil-M\'ezard conjecture.
We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).
We study the Zariski closure of points in local deformation rings corresponding to potential semi-stable representations with certain prescribed $p$-adic Hodge theoretic properties. We show in favourable cases that the closure is equal to a union of irreducible components of the deformation space. We also study an analogous question for global Hecke algebras.
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".
We construct, over any CM field, compatible systems of l-adic Galois representations that appear in the cohomology of algebraic varieties and have (for all l) algebraic monodromy groups equal to the exceptional group of type E6.
We prove some basic results on the dimension theory of algebraic stacks, and on the multiplicities of their irreducible components, for which we do not know a reference.
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.
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.
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.