Related papers: A deformation problem for Galois representations o…

We extend the lifting methods of our previous paper to lift reducible odd representations $\bar{\rho}:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when…

In this paper we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum we construct…

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\bar{\rho}:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow…

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…

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…

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 study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised reductive group schemes, such as $L$-groups and $C$-groups. We show that the corresponding deformation rings are complete…

We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.

We study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised tori, such as $L$-groups of (possibly non-split) tori. We show that the corresponding deformation rings are formally smooth over a…

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…

We prove automorphy lifting results for geometric representations $\rho:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime,…

We prove an integral R = T theorem for odd two dimensional p-adic representations of the absolute Galois group which are unramified at p, extending results of [CG] to the non-minimal case. We prove, for any p, the existence of Katz modular…

Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with residue…

We prove an almost minimal R=T theorem for self-dual Galois representations with coefficients in a finite field satisfying a property called rigid. We also prove the rigidity property for a large family of residual Galois representations…

This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet's Lemma for $\mathrm{GL}_2$ in the residually indistinguishable case. We suppose we are given a Galois representation taking values…

We prove that every odd semisimple reducible (2-dimensional) mod l Galois representation arises from a cuspidal eigenform. In addition, we investigate the possible different types (level, weight, character) of such a modular form. When the…

This is the second in a pair of papers about residually reducible Galois deformation rings with non-optimal level. In the first paper, we proved a Galois-theoretic criterion for the deformation ring to be as small as possible. This paper…

We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discrete valuation rings. In order to prove these results we…

For any simple algebraic group $G$ of exceptional type, we construct geometric $\ell$-adic Galois representations with algebraic monodromy group equal to $G$, in particular producing the first such examples in types $\mathrm{F}_4$ and…

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular…