Related papers: Reductions of Crystalline Representations for Smal…

We determine the set of quaternionic Serre weights for generic two-dimensional mod $p$ representation of $\operatorname{Gal}(\overline{\mathbb Q}_p / K)$, where $K$ is a finite unramified extension of $\mathbb{Q}_p$.

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The…

One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real…

This article studies the first-order $p$-adic deformations of classical weight one newforms, relating their fourier coefficients to the $p$-adic logarithms of algebraic numbers in the field cut out by the associated projective Galois…

The aim of this article is to prove Zariski density of crystalline representations in the rigid analytic space associated to the universal deformation ring of a d-dimensional mod p representation of Gal(\bar{K}/K) for any d and for any…

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…

We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation rho_0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic…

We prove (under certain assumptions) the irreducibility of the limit $\sigma_2$ of a sequence of irreducible essentially self-dual Galois representations $\sigma_k: G_{\mathbf{Q}} \to \mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$…

We prove modularity for any irreducible crystalline $\ell$-adic odd 2-dimensional Galois representation (with finite ramification set) unramified at 3 verifying an "ordinarity at 3" easy to check condition, with Hodge-Tate weights $\{0, w…

Let $G$ be the group of points of a split reductive group over a finite extension of ${\mathbb Q}_p$. In this paper, we compute the dimensions of certain classes of locally analytic $G$-representations. This includes principal series…

We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of $\mathbb{Q}$ with Galois…

In previous work we described when a single geometric representation, valued in a linear algebraic group, of the Galois group of a number field lifts through a central torus quotient to a geometric representation. In this paper we prove a…

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…

The aim of this paper is to investigate the algebraicity behavior of reductions of $D$-finite power series modulo prime numbers. For many classes of D-finite functions, such as diagonals of multivariate algebraic series or hypergeometric…

Let p be an odd prime, K a finite extension of Q_p, G=Gal(\bar K/K) the Galois group and e=e(K/Q_p) the ramification index. Suppose T is a p^n torsion representation such that T is isomorphic to a quotient of two G-stable Z_p-lattices in a…

We study G-valued semi-stable Galois deformation rings where G is a reductive group. We develop a theory of Kisin modules with G-structure and use this to identify the connected components of crystalline deformation rings of minuscule…

Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$ which are unramified outside $\{p, \infty\}$…

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…

We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a hypergeometric…

Given a continuous, odd, reducible and semi-simple $2$-dimensional representation $\bar\rho_0$ of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$, we study the relation between the universal deformation ring of the…