Determination of certain mod $p$ Galois representations using local constancy

Abstract

Let $p \geq 5$ be a prime. Let $k = b + c(p-1)$ be an integer in $[2p+2, p^2 - p +3]$, where $b \in [2,p]$ and $c \in [2, p-1]$. We prove local constancy in the weight space of the mod $p$ reduction of certain two-dimensional crystalline representations of $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, where the slope $\nu(a_p)$ is constrained to be in $(1, c)$ and non-integral. We use the mod $p$ local Langlands correspondence for $\text{GL}_{2} (\mathbb{Q}_{p})$ to compute the mod $p$ reductions explicitly, thereby also giving a lower bound on the radius of constancy around the weights $k$ in the above range and under additional conditions on the slope. As an application of local constancy, we obtain explicit mod $p$ reductions at many new values of $k$ and $a_p$.