论文标题
$ gl_2（\ mathbb {q} _p）$的家庭中的agimimple mod mod $ p $ langlands通讯
A semisimple mod $p$ Langlands correspondence in families for $GL_2(\mathbb{Q}_p)$
论文作者
论文摘要
这是Arxiv：2007.01364v1的续集。令$ f $为任何本地字段，具有残留特征$ p> 0 $，而$ \ nathcal {h}^{（1）} _ {\ overline {\ mathbb {f}} _ p} _是mod $ p $ p $ p $ p $ -iwahori hecke yewahori hecke algebra of $ $ \_ mathbf} {gl_ f）在Arxiv：2007.01364v1中，我们已经构建了$ \ Mathcal {h}^{（1）} _ {\ overline {\ Mathbb {f}} _ p} $ - 通过某些模块的模块的参数化$ \ wideHat {\ MathBf {gl_2}}（\ edropline {\ Mathbb {f}} _ p）$ - SATAKE参数，以及$ \ Mathcal {H}^{（1）} _ {\ Matheres {\ MathBB} $ \ Mathcal {h}^{（1）} {\ p}的反；在这里，我们让$ f = \ mathbb {q} _p $（和$ p \ geq 5 $），然后从$ \ wideHat {\ mathbf {gl_2}}构建形态， $ \ wideHat {\ mathbf {gl_2}}}（\ overline {\ mathbb {f}} _ p）$ - langlands参数。结果，我们在Breuil的Semisimple Mod $ p $ p $ langlands的家族中获得了一个版本$ \ mathbf {gl_2}（\ Mathbb {q} _p）$具有中心字符。由于emerton-gee Moduli空间的emerton-gee $ \ wideHat {\ mathbf {gl_2}}}（\ edimelline {\ mathbb {f}} _ p）$ - GALOIS组$ {\ rm GAL} { \ Mathbb {q} _p）$。
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 arXiv:2007.01364v1 we have constructed a parametrization of the $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules by certain $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules. Here we let $F=\mathbb{Q}_p$ (and $p\geq 5$) and construct a morphism from $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters to $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuil's semisimple mod $p$ Langlands correspondence for $\mathbf{GL_2}(\mathbb{Q}_p)$ and of Paškūnas' parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $\mathbf{GL_2}(\mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-representations of the Galois group ${\rm Gal}(\overline{\mathbb{Q}}_p/ \mathbb{Q}_p)$.
