On the local constancy of certain mod $p$ Galois representations

Abstract

In this article we study local constancy of the mod $p$ reduction of certain $2$-dimensional crystalline representations of $\mathrm{Gal}\left({\bar{\mathbb{Q}}}_p/{\mathbb{Q}}_p\right)$ 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 $(p-1{)}^2+3$ and the slope fixed in $(0,p-1)$ 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.

An updated version, 47 pages