Comparing Galois representations in the residually reducible case

Nuno Freitas; Ignasi Sánchez-Rodríguez

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arXiv · 2510.12956

Comparing Galois representations in the residually reducible case

Nuno Freitas·Ignasi Sánchez-Rodríguez

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Abstract

Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $\rho_1, \rho_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ having residual image a $p$-group. We explain and implement an algorithm that makes effective a result of Lo\"ic Greni\'e to decide wether the semisimplifications of $\rho_1$ and $\rho_2$ are isomorphic. As an application, we show that an irreducible representation $\rho : G_{\mathbb{Q}(\sqrt{-3})} \to \operatorname{GL}_2(\mathbb{Z}_3)$ 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 Greni\'e, recovering Greni\'e's result in a fully automated way.

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Nuno Freitas, Ignasi Sánchez-Rodríguez. 2025-10-14. Comparing Galois representations in the residually reducible case. https://arxiv.org/abs/2510.12956

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