相关论文: Lifting and automorphy of reducible mod p Galois r…

If a $p$-adic Galois representation $\rho_{f,\nu}:\Gamma_{\mathbb Q} \to \GL_2(E_{f,\nu})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this…

Let $(\mathbb{T}_f,\mathfrak{m}_f)$ denote the mod $p$ local Hecke algebra attached to a normalised Hecke eigenform $f$, which is a commutative algebra over some finite field $\mathbb{F}_q$ of characteristic $p$ and with residue field…

We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the…

This brief note only contains a modest contribution: we just fix some inaccuracies in the proof of the prime level weight 2 case of Serre's conjecture given in Khare's preprint "On Serre's modularity conjecture for 2-dimensional mod p…

Given a totally real number field $F$ and a mod $p$ Galois representation $\rho\colon G_F\to \mathrm{GL}_2(\bar{\mathbf{F}}_p)$, we propose an explicit definition of the set of Serre weights $W(\rho)$ attached to $\rho$. We prove that our…

In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura…

We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.

Building on lifting results of Ramakrishna, Khare and Ramakrishna proved a purely Galois-theoretic level-raising theorem for two-dimensional odd representations of the Galois group of Q. In this paper, we generalize these techniques from…

We prove some new cases of weight part of Serre's conjectures for mod $p$ Galois representations associated to automorphic representations on unitary groups $U(d)$. The approach is a generalization of the work of Gee-Liu-Savitt, namely, we…

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,…

We show for all local fields $K/\mathbb{Q}_p$, with $p >3$, all representations $\bar\rho:G_K \to G_2(\bar{\mathbb{F}}_p)$ admit a crystalline lift $\rho: G_K\to G_2(\bar{\mathbb{Z}}_p)$, where $G_2$ is the exceptional Chevalley group of…

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual,…

We prove automorphy lifting theorems for 2-dimensional Galois representations of absolute Galois groups of totally real fields when the residual representation is of "exceptional" type. This exceptional case is when we are in characteristic…

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one…

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…

We construct a local deformation problem for residual Galois representations $\bar{\rho}$ valued in an arbitrary reductive group $\hat{G}$ which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles…

In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text{SL}_2(\mathbb Z)$. When the underlying modular…

A generalization of Serre's Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover it predicts the set of weights of such…

We studied framed deformations of two dimensional Galois representation of which the residue representation restrict to decomposition groups are scalars, and established a modular lifting theorem for certain cases. We then proved a family…

The global Langlands conjecture for $\text{GL}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $\pi$ of $\text{GL}_n(\mathbb{A}_F)$ and certain families $\{ \rho_{\pi,\ell} \}_\ell$…