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

TL;DR

This paper investigates the stability of mod p reductions of certain 2-dimensional crystalline Galois representations over Q_p, providing explicit bounds on when these reductions remain constant in weight space.

Contribution

It establishes explicit lower bounds for local constancy of mod p Galois representations in weight space, expanding understanding of their behavior under the mod p local Langlands correspondence.

Findings

Proves local constancy in weight space for specific Galois representations.

Provides explicit lower bounds on the radius of local constancy.

Determines mod p reductions at nearby weights explicitly.

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.