Comparing Galois representations in the residually reducible case

2025/10/14 by Nuno Freitas, Freitas, Nuno, Ignasi Sánchez-Rodríguez +1
Computer Science · Engineering · #11F80 #11Y40 (Primary) 11G05 (Secondary) #Advanced Numerical Analysis Techniques #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2510.12956

openalex publication_date 2025/10/14 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

Let n ≥ 2 and p be a prime. Let K be a number field and consider two Galois representations ρ₁, ρ₂ : Gal(K / K) → GL_n(ℤ_p) having residual image a p-group. We explain and implement an algorithm that makes effective a result of Loïc Grenié to decide wether the semisimplifications of ρ₁ and ρ₂ are isomorphic. As an application, we show that an irreducible representation ρ: G_ℚ(√(-3)) → GL₂(ℤ₃) unramified outside 3 is determined by the characteristic polynomials of Frobenius elements at five primes of small norm. As an additional check, we apply it to a 2-adic example studied by Grenié, recovering Grenié's result in a fully automated way.

Citations

• On the modularity of elliptic curves over imaginary quadratic fields
• Proving modularity for a given elliptic curve over an imaginary quadratic field

Related

• Lifting N-dimensional Galois representations to characteristic zero
• Lifting residual Galois representations with the same semi-simplification
• A Theorem on GL(n) a la Tchebotarev
• On the images of certain G₂-valued automorphic Galois representations
• Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations