Related papers: A zig-zag conjecture and local constancy for Galoi…

The zig-zag conjecture says that the reductions of two-dimensional crystalline representations of the Galois group of ${\mathbb {Q}}_p$ of large exceptional weights and half-integral slopes up to $\frac{p-1}{2}$ vary through an alternating…

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 prove a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of $G_{{\mathbb{Q}}_p}$ of slope $\frac{3}{2}$ and exceptional weights. This along with previous works completes the…

Let $p \geq 5$ be a prime. Let $k = b + c(p-1)$ be an integer in $[2p+2, p^2 - p +3]$, where $b \in [2,p]$ and $c \in [2, p-1]$. We prove local constancy in the weight space of the mod $p$ reduction of certain two-dimensional crystalline…

We study the mod $p$ reduction of crystalline local Galois representations of dimension 2 under certain conditions on its weight and slope. Berger showed that for a fixed non-zero trace of the Frobenius, the reduction process is locally…

We prove the existence of local constancy phenomena for reductions in a general prime power setting of two-dimensional irreducible crystalline representations. Up to twist, these representations depend on two parameters: a trace $a_p$ and a…

Gouv\^ea-Mazur [GM] made a conjecture on the local constancy of slopes of modular forms when the weight varies $p$-adically. Since one may decompose the space of modular forms according to associated residual Galois representations, the…

We use the p-adic local Langlands correspondence for GL_2(Q_p) to find the reduction modulo p of certain two-dimensional crystalline Galois representations. In particular, we resolve a conjecture of Breuil, Buzzard, and Emerton in the case…

The question of computing the reductions modulo $p$ of two-dimensional crystalline $p$-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small…

A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle…

Irreducible crystalline representations of dimension 2 of Gal(Qpbar/Qp) depend up to twist on two parameters, the weight k and the trace of frobenius a_p. We show that the reduction modulo p of such a representation is a locally constant…

We construct an explicit sequence $V_{k_n,a_n}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation $V_{k,{\mathcal{L}}}$ of…

A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle…

We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbf{Q}_p}$ of slope 1, for primes $p \geq 5$, and all weights. We describe the semisimplification of the reductions completely. In particular,…

We describe the semi-simplification of the mod $p$ reduction of certain crystalline two dimensional local Galois representations of slopes in the interval $(1,2)$ and all weights. The proof uses the mod $p$ Local Langlands Correspondence…

We determine rational Kisin modules associated with two-dimensional, irreducible, crystalline representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ of Hodge-Tate weights $0, k-1$. If the slope is larger than $\lfloor…

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 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…

We establish a geometrisation of the Breuil-M\'ezard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute…

We determine reductions of 2-dimensional, irreducible, semi-stable, and non-crystalline representations of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ with Hodge--Tate weights $0 < k-1$ and with $\mathcal L$-invariant whose $p$-adic…