Related papers: On the reductions of certain two-dimensional cryst…

We determine the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq…

We construct explicitly some analytic families of etale (phi,Gamma)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on…

Let $p$ be an odd prime and $f \geq 1$. We consider a $p$-adic locally algebraic $\text{GL}_2(\mathbb Q_{p^f})$-representation attached to a tuple of $f$ weights $k=(k_i)$ for $0 \leq i \leq f-1$ and a $p$-adic integer $a_p$ with valuation…

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 $k$…

In a previous article we introduced various moduli stacks of two-dimensional tamely potentially Barsotti-Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil-Kisin modules…

Let p > 2 be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call "pseudo-Barsotti-Tate representations", over arbitrary finite extensions of the…

In this article we study local constancy of the mod $p$ reduction of certain $2$-dimensional crystalline representations of $\mathrm{Gal}\left(\bar{\mathbb{Q}}_p/\mathbb{Q}_p\right)$ using the mod $p$ local Langlands correspondence. We…

We show that loci of crystalline representations of $G_K$ for $K/\mathbb{Q}_p$ an unramified extension are irreducible when the Hodge--Tate weights are fixed and sufficiently small. This was previously known for weights in the interval…

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…

Using the link between mod $p$ Galois representations of $\qu$ and mod $p$ modular forms established by Serre's Conjecture, we compute, for every prime $p\leq 1999$, a lower bound for the number of isomorphism classes of continuous Galois…

Let K_{f} be the finite unramified extension of Q_{p} of degree f and E any finite large enough coefficient field containing K_{f}. We construct analytic families of \'etale (Phi,Gamma)-modules which give rise to families of crystalline…

Let K/Q_p be unramified. Inside the Emerton-Gee stack X_2, one can consider the locus of two-dimensional mod p representations of the absolute Galois group of K having a crystalline lift with specified Hodge-Tate weights. We study the case…

We prove a criterion for the irreducibility of an integral group representation \rho over the fraction field of a noetherian domain R in terms of suitably defined reductions of \rho at prime ideals of R. As applications, we give…

We prove a conjecture of Colmez concerning the reduction modulo $p$ of invariant lattices in irreducible admissible unitary $p$-adic Banach space representations of $GL_2(Q_p)$ with $p\ge 5$. This enables us to restate nicely the $p$-adic…

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…

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…

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…

We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).

Let $p\ge 5$ be a prime, and let $f$ be a cuspidal eigenform of weight at least $2$ and level coprime to $p$ of finite slope $\alpha$. Let $\bar{\rho}_f$ denote the mod $p$ Galois representation associated with $f$ and $\omega$ the mod $p$…

For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside 2.