Related papers: The semi-stable Local Langlands Correspondence

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 extend a comparison theorem of Anandavardhanan-Borisagar between the quotient of the induction of a mod $p$ character by the image of an Iwahori-Hecke operator and compact induction of a weight to the case of the trivial character. This…

We determine the mod $p$ reductions of the semi-stable representations $V_{k, \mathcal{L}}$ of weight $k \in [p + 5, 2p]\cup[2p + 6, 3p + 1]$ and $v_p(\mathcal{L}) < 1-k/2$ for primes $p \geq 5$. In particular, this shows that 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 take some initial steps towards illuminating the (hypothetical) $p$-adic local Langlands functoriality principle relating Galois representations of a $p$-adic field $L$ and admissible unitary Banach space representations of $G(L)$ when…

Let $\rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2\ne 0$ on $D_{\mathrm{st}}(\rho_p)$. When…

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 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 establish a "matrix simultaneous diagonalization theorem" for disconnected reductive groups which relaxes both the semisimplicity condition and the commutativity condition. As an application, we prove the following basic results…

This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $\mathbf{GL_2}(F)$. In…

We survey results related to our geometrization of a part of the $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbf Q}_p)$.

The p-adic local Langlands correspondence for GL_2(Q_p) is given by an exact functor from unitary Banach representations of GL_2(Q_p) to representations of the absolute Galois group G_{Q_p} of Q_p. We prove, using characteristic 0 methods,…

We reprove the Local Langlands Correspondence for $\GL_n$ over $p$-adic fields as well as the existence of $\ell$-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for…

We give a categorical formulation of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$,as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on…

Let $\ell$ and $p$ be distinct primes, $n$ a positive integer, $F_\ell$ an $\ell$-adic local field of characteristic $0,$ and let $W(k)$ denote the ring of Witt vectors over an algebraically closed field of characteristic $p$. Work of…

Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the…

We develop a general strategy for constructing the explicit Local Langlands Correspondences for $p$-adic reductive groups via reduction to LLC for supercuspidal representations of proper Levi subgroups, using Hecke algebra techniques. As an…

We develop the $p$-adic representation theory of $p$-adic Lie groups on solid vector spaces over a complete non-archimedean extension of $\mathbb{Q}_p$. More precisely, we define and study categories of solid, solid locally analytic and…

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…

We prove that the reduction mod \ell of the local Langlands correspondence between supercuspidal representations of GL_n(F), where F is a finite extension of Q_p, and representations of the Galois group of F is well-defined. The results and…