The image of Colmez's Montreal functor

A mod-$p$ metaplectic Montréal functor

We extend Colmez's functor defined for $\operatorname{GL}_2(\mathbf{Q}_p)$ to the category of finitely generated smooth admissible mod-$p$ representations of the two-fold metaplectic cover of $\operatorname{GL}_2(\mathbf{Q}_p)$. We compute the images of the absolutely irreducible genuine objects and obtain a bijection between the genuine supersingular representations and four-dimensional irreducible Galois representations invariant under twist by all characters of order two. Restricted to genuine objects, the extended functor naturally takes values in the category of what we call metaplectic G…

广义Montréal函子的有限性性质及其在GLₙ(ℚₚ)模p表示中的应用

Finiteness properties of generalized Montréal functors with applications to mod $p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$

本文研究广义Montréal函子的有限性性质，证明其将GLₙ(ℚₚ)的光滑p幂挠表示对应到伽罗瓦侧有限维表示，确定函子核并刻画抛物诱导表示，还提出关于本质像的猜想及部分结果。

First steps towards $p$-adic Langlands functoriality

By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the …

Locally algebraic automorphisms of the ${\rm PGL}_2(F)$-tree and ${\mathfrak o}$-torsion representations

For a local field $F$ and an Artinian local coefficient ring $Λ$ with the same positive residue characteristic $p$ we define, for any $e\in{\mathbb N}$, a category ${\mathfrak C}^{(e)}(Λ)$ of ${\rm GL}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There is an obvious functor from the category of ${\rm GL}_2(F)$-representations over $Λ$ to ${\mathfrak C}^{(e)}(Λ)$. If $F={\mathbb Q}_p$ then ${\mathfrak C}^{(1)}(Λ)$ is equivalent to the category of smooth ${\rm GL}_2({\mathbb Q}_p)$-representations over $Λ$ generated by their invariants under a pro-$p$-Iw…