Related papers: Modularity Lifting beyond the Taylor-Wiles Method

Let $p>5$ be a prime integer and $K/\mathbb{Q}_p$ a finite ramified extension with ring of integers $\mathcal{O}$ and uniformizer $\pi$. Let $n>1$ be a positive integer and $\rho_n:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O}/\pi^n)$ be a…

Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p. For a…

We prove the existence of a potentially diagonalizable lift of a given automorphic mod $p$ Galois representation $\overline{\rho}:{\rm Gal}(\overline{F}/F)\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ for any totally real field $F$…

In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local…

We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called `Taylor--Wiles hypothesis'. We apply this to the problem of the…

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1…

This work is motivated by the search for an "explicit" proof of the Bloch-Kato conjecture in Galois cohomology, proved by Voevodsky. Our concern here is to lay the foundation for a theory that, we believe, will lead to such a proof- and to…

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 introduce a new partial resolution of crystalline spaces of Galois representations when the gaps in Hodge--Tate weights are smaller than $p$, with no bound on ramification. Furthermore, when $n =3$ in the case of minimal regular weight,…

Typos in the abstract have been corrected. Let $\rho_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $GL_2(\mathcal O/\pi^n)$. Here $\mathcal O$ is a ramified DVR with uniformiser $\pi$. If $\rho_n$…

We generalize the main result of arXiv:1206.6631 [math.NT] to all totally real fields. In other words, for $p>2$ prime, we prove (under a mild Taylor-Wiles hypothesis) that if a modular representation is unramified and $p$-distinguished at…

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms.…

We develop local cohomology techniques to study the finite slope part of the coherent cohomology of Shimura varieties. The local cohomology groups we consider are a generalization of overconvergent modular forms, and they are defined by…

We prove that over totally real fields, the $p$-adic Galois representations attached to non-self-dual regular algebraic cuspidal automorphic representations of $\mathrm{GL}(4)$ are irreducible. We then develop the theory of extra-twists in…

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all…

We show that the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\CF$ ,…

Let \rho be a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that \rho has large image and admits a low weight crystalline modular deformation we show that any low weight…

In this paper we study deformations of mod $p$ Galois representations $\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\tau_1$ and $\tau_2$. As opposed to our…

Let $F$ be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations $Gal(\bar{F}/F) \to PGL_n(C)$ lift to $GL_n(C)$. We take…

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…