Lubin-Tate moduli space of semisimple mod p Galois representations for GL_2 and Hecke modules

Let $p$ be an odd prime. Let $F$ be a non-archimedean local field of residue characteristic $p$, and let ${\mathbb{F}}_q$ be its residue field. Let ${\mathcal{H}}_{{\mathbb{F}}_q}^{(1)}$ be the pro-$p$-Iwahori-Hecke algebra of the $p$-adic group $\text{G}{L}_2(F)$ with coefficients in ${\mathbb{F}}_q$, and let $Z({\mathcal{H}}_{{\mathbb{F}}_q}^{(1)})$ be its center. We define a scheme $X(q{)}_{{\mathbb{F}}_q}$ whose geometric points parametrize the semisimple two-dimensional Galois representations of $\text{G}al(\bar{F}\mathrm{/}F)$ over ${\bar{\mathbb{F}}}_q$. Then we construct a morphism from the spectrum of $Z({\mathcal{H}}_{{\mathbb{F}}_q}^{(1)})$ to $X(q{)}_{{\mathbb{F}}_q}$ generalizing the morphism appearing in \cite{PS2} for $F={\mathbb{Q}}_p$. In the case $F\mathrm{/}{\mathbb{Q}}_p$, we show that the induced map from Hecke modules to Galois representations, when restricted to supersingular modules, coincides with Grosse-Klönne’s bijection \cite{GK18}. For this, we determine the Lubin-Tate $(\phi ,\mathrm{\Gamma })$-modules associated to absolutely irreducible Galois representations.