Related papers: A level raising result for modular Galois represen…

Ribet has proven remarkable results about non-optimal levels of residually reducible Galois representations. We focus on a non-optimal level $N$ that is the product of two distinct primes and where the Galois deformation ring is not…

We extend Colmez's functor defined for $\operatorname{GL}_2(\mathbf{Q}_p)$ to the category of finitely generated smooth admissible mod-$p$ representations of the two-fold metaplectic cover of $\operatorname{GL}_2(\mathbf{Q}_p)$. We compute…

Let $p$ be an odd prime, and $\mathbf{Q}_{p^f}$ the unramified extension of $\mathbf{Q}_p$ of degree $f$. In this paper, we reduce the problem of constructing strongly divisible modules for $2$-dimensional semi-stable non-crystalline…

Let $\rho$ be a mod $\ell$ Galois representation attached to a newform $f$. Explicit methods are sometimes able to determine the image of $\rho$, or even the number field cut out by $\rho$, provided that $\ell$ and the level $N$ of $f$ are…

We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an…

We establish Bernstein-centre type of results for the category of mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. We treat all the remaining open cases, which occur when $p$ is $2$ or $3$. Our arguments carry over for all primes…

The Iwasawa $\mu$-invariant of the Selmer group of a residually reducible Galois representation arising from a Hecke eigencuspform is studied. Furthermore, certain Iwasawa-invariants refining the $\mu$-invariant are defined and analyzed. As…

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's…

We show that an infinite family of odd complex 2-dimensional Galois representations ramified at 5 having nonsolvable projective image are modular, thereby verifying Artin's conjecture for a new case of examples. Such a family contains the…

Let $p>3$ be a prime number and let $G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$. In this paper, we find Galois stable lattices in the irreducible $3$-dimensional semi-stable and non-crystalline representations of…

We construct an Euler system for the adjoint Galois representation of a modular form, using motivic cohomology classes arising from Hilbert modular surfaces. We use this Euler system to give an upper bound for the Selmer group of the…

We formulate and prove the weight part of Serre's conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption in \cite{arXiv:1512.06380},…

Let p>3 be a prime, f a positive integer and Q_{p^f} the unramified extension of Q_p of degree f. After Breuil and Paskunas, to a generic semi-simple continue modulo p representation of the absolute Galois group of Q_{p^f}, we can associate…

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…

We consider the classical theta operator $\theta$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $\theta$ mod $p^m$ will map forms of weight $k$ to forms of weight…

Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $\rho_1, \rho_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ having residual image a $p$-group. We explain…

Let $E/F$ be a quadratic extension of p-adic fields. We prove that every smooth irreducible ladder representation of the group $GL_n(E)$ which is contragredient to its own Galois conjugate, possesses the expected distinction properties…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is a prime power and let $A:= \mathbb{F}_{q}[T]$. By~\cite{PR09}, the adelic image of the Galois representation attached to a rank $2$ Drinfeld $A$-module $\varphi$ is open,…

In this paper, we show that an odd Galois representation rhobar: Gal(Qbar/Q) --> GL_2(F_9) satisfying certain local conditions at 3 and 5 is modular. Our main tool is an idea of Taylor, which reduces the problem to that of exhibiting points…

We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show…