Related papers: A Splitting Criterion for Galois Representations A…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

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 article investigates the splitting problem for finitely generated projective modules $P$ over affine algebras over algebraically closed fields and their polynomial extensions. We then address an open question due to M. Roitman on monic…

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…

It is known that any Galois representation $\rho : G_{\mathbb{Q}} \rightarrow \mathrm{GL}(2,\mathbb{F}_p)$ with determinant equal to the mod-$p$ cyclotomic character, arises from the $p$-torsion of an elliptic curve over $\mathbb{Q}$, if…

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…

In this paper we show that two dimensional (mod p) Galois representations satisfying mild hypotheses can be lifted to p-adic Galois representations ramified at infinitely many primes such that the characteristic polynomials of Frobenius at…

A powerful new perspective in the analysis of absolute Galois groups has recently emerged from the study of Galois modules related to classical parameterizing spaces of certain Galois extensions. The recurring trend in these decompositions…

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria…

Let $\bbk$ be an algebraically closed field of prime characteristic $p$. If $p$ does not divide $n$, irreducible modules over $\frak {sl}_n$ for regular and subregular nilpotent representations have already known(see \cite{Jan2} and…

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…

Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and…

We give a survey of recent work on the construction of differential operators on various types of modular forms (mod p). We also discuss a framework for determining the effect of such operators on the mod p Galois representations attached…

We study the rigid generic fiber $\mathcal{X}^\square_{\overline\rho}$ of the framed deformation space of the trivial representation $\overline\rho: G_K \to \text{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is…

Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf{F}$. Fix a representation $\overline{\rho}$ of the absolute Galois group of an unramified extension of $\mathbf{Q}_p$, valued in…

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 show that framed deformation rings of mod $p$ representations of the absolute Galois group of a $p$-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their…

In this expository article, we present a brief introduction to the theory of Hilbert modular forms and Galois representations, and describe what it means to attach a compatible system of Galois representations to a Hilbert modular form.

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types…

In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is totally inert in…