Related papers: Adapted bases of Kisin modules and Serre weights

We show for all local fields $K/\mathbb{Q}_p$, with $p >3$, all representations $\bar\rho:G_K \to G_2(\bar{\mathbb{F}}_p)$ admit a crystalline lift $\rho: G_K\to G_2(\bar{\mathbb{Z}}_p)$, where $G_2$ is the exceptional Chevalley group of…

Let K_0 be a finite unramified extension of Q_p. We show that all crystalline representations of G_{K_0} (the absolute Galois group of K_0) with Hodge-Tate weights in {0, ..., p-1} are potentially diagonalizable.

We prove that, in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$, every Kirillov--Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion…

Let $K$ be a complete discrete valuation field of characteristic $(0, p)$ with perfect residue field, and let $T$ be an integral $\mathbb{Z}_p$-representation of $\mathrm{Gal}(\overline{K}/K)$. A theorem of T. Liu says that if $T/p^n T$ is…

Let $\mathcal{T}$ be an $\mathcal{O}_K$-linear idempotent-complete, small smooth proper stable $\infty$-category, where $K$ is a finite extension of $\mathbb{Q}_p$. We give a Breuil-Kisin module structure on the topological negative cyclic…

Let V be a p-adic representation of the absolute Galois group G of Q_p that becomes crystalline over a finite tame extension, and assume p odd. We provide necessary and sufficient conditions for V to be isomorphic to the Tate module V_p(A)…

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such…

We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil…

We compute explicit reductions of crystalline representations of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_{p^f})$ with labeled Hodge-Tate weights in the range $p+2\le k_{0}\le 2p-4$ and $2\le k_i\le p-3$ for…

Let $p$ be an odd prime number, $K_{f}$ the finite unramified extension of $\mathbb{Q}_{p}$ of degree $f$, and $G_{K_{f}}$ its absolute Galois group. We construct analytic families of \'etale $(\varphi,\Gamma)$-modules which give rise to…

We study the weight part of (a generalisation of) Serre's conjecture for mod l Galois representations associated to automorphic representations on rank two unitary groups for odd primes l. We propose a conjectural set of Serre weights,…

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…

Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}_p$. Let $\rho_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the…

Let $N/K$ be a finite Galois extension of $p$-adic number fields and let $\rho^\mathrm{nr} : G_K \to \mathrm{Gl}_r(\mathbb Z_p)$ be an $r$-dimensional unramified representation of the absolute Galois group $G_K$ which is the restriction of…

Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre's conjecture for continuous, totally odd, two-dimensional mod p representations rhobar of the absolute Galois group of F that are…

Let $K$ be a mixed characteristic complete discrete valuation field with residue field admitting a finite $p$-basis, and let $G_K$ be the Galois group. We first classify semi-stable representations of $G_K$ by weakly admissible filtered…

We obtain character formulas of minimal affinizations of representations of quantum groups when the underlying simple Lie algebra is orthogonal and the support of the highest weight is contained in the first three nodes of the Dynkin…

We prove Breuil's lattice conjecture for higher Hodge-Tate weights in the case of $\mathrm{GL}_2(K)$ where $K$ is an unramified extension of $\mathbb{Q}_p$. More precisely, under some genericity conditions, we show that the lattice inside a…

Motivated by the weight part of Serre's conjecture we consider the following question. Let $K/\mathbb{Q}_p$ be a finite extension and suppose $\overline{\rho} \colon G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{F}}_p)$ admits a…

Let $p>5$ be a prime integer and $K/\mathbb{Q}_p$ a finite ramified extension with ring of integers $\mathcal{O}$ and uniformizer $\pi$. Let $n>1$ be a positive integer and $\rho_n:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O}/\pi^n)$ be a…