Comparing Galois representations in the residually reducible case

Abstract

Let $n\ge 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $\ddot{I}{}_1,\ddot{I}{}_2:\mathrm{Gal}(\overline{K}/K)\to {\mathrm{GL}}_n({\mathbb{Z}}_p)$ 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 $\ddot{I}{}_1$ and $\ddot{I}{}_2$ are isomorphic. As an application, we show that an irreducible representation $\ddot{I}:G_{\mathbb{Q}(\sqrt{-3})}\to {\mathrm{GL}}_2({\mathbb{Z}}_3)$ 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.

17 pages, the associated code can be found in https://github.com/IgnasiSanchez/ComparingReducibleReps