Comparing Galois representations in the residually reducible case

Abstract

Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $ρ_1, ρ_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ 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 $ρ_1$ and $ρ_2$ are isomorphic. As an application, we show that an irreducible representation $ρ: G_{\mathbb{Q}(\sqrt{-3})} \to \operatorname{GL}_2(\mathbb{Z}_3)$ 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.