Determination of certain mod p Galois representations using local constancy

2024/06/21 by Abhik Ganguli, Suneel Kumar, Ganguli, Abhik +1
Engineering · #11F33 #11F70 #11F80 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.15600

openalex publication_date 2024/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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(ℚ_p/ℚ_p), where the slope ν(a_p) is constrained to be in (1, c) and non-integral. We use the mod p local Langlands correspondence for GL₂ (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.

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