Determination of certain mod p Galois representations using local constancy

Let p ≥ 5 be a prime. Let $k = b + c(p-1)$ be an integer in [2p+2, p² - p +3], where b ∈ [2,p] and c ∈ [2, p-1]. We prove local constancy in the weight space of the mod p reduction of certain two-dimensional crystalline representations of Gal(Q̄ₚ/Qₚ), where the slope ν(aₚ) is constrained to be in $(1, c)$ and non-integral. We use the mod p local Langlands correspondence for GL₂ (Qₚ) 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 aₚ.