On the local constancy of certain mod p Galois representations

In this article we study local constancy of the mod p reduction of certain 2-dimensional crystalline representations of \(\text {Gal}\left( \bar{\mathbb {Q}}_p/\mathbb {Q}_p\right) \) $\text{Gal}\left({\overline{\mathbb{Q}}}_p,/,{\mathbb{Q}}_p\right)$ using the mod p local Langlands correspondence. We prove local constancy in the weight space by giving an explicit lower bound on the local constancy radius centered around weights going up to \((p-1)^{2} +3\) ${(p-1)}^2+3$ and the slope in \((0, \ p-1)\) $(0,p-1)$ satisfying certain constraints. We establish the lower bound by determining explicitly the mod p reductions at nearby weights and applying a local constancy result of Berger.