Semisimple Langlands for GL2(Qp) and mod p Hecke modules

Abstract

Let $p\ge 5$ and let $Z\left(H_{{\bar{F}}_p}^{\left(1\right)}\right)$ be the centre of the mod p pro-p-Iwahori Hecke algebra of ${\mathrm{GL}}_2\left(Q_p\right)$. Let X be the projective curve parametrizing 2-dimensional mod p semi-simple representations of the absolute Galois group $\mathrm{Gal}({\bar{Q}}_p/Q_p)$. We construct a quotient morphism of schemes $L:\mathrm{Spec}Z\left(H_{{\bar{F}}_p}^{\left(1\right)}\right)\to X$. We then show that the correspondence between the specialization $M_{{\bar{F}}_p,z}^{\left(1\right)}$ of the spherical $H_{{\bar{F}}_p}^{\left(1\right)}$-module $M_{{\bar{F}}_p}^{\left(1\right)}$ from [PS] in closed points $z\in \mathrm{Spec}Z\left(H_{{\bar{F}}_p}^{\left(1\right)}\right)$ and the Galois representation ${\rho }_{L\left(z\right)}$ is the semi-simple mod p local Langlands correspondence for the group ${\mathrm{GL}}_2\left(Q_p\right)$.