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

Abstract

Let $p\ge 5$ 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 $a_p$.

49 pages