An algorithm to compute Selmer groups via resolutions by permutations modules

Abstract

Given a number field with absolute Galois group $\mathcal{G}$, a finite Galois module $M$, and a Selmer system $\mathcal{L}$, this article gives a method to compute Sel${}_{\mathcal{L}}$, the Selmer group of $M$ attached to $\mathcal{L}$. First we describe an algorithm to obtain a resolution of $M$ where the morphisms are given by Hecke operators. Then we construct another group $H_S^1(\mathcal{G},M)$ and we prove, using the properties of Hecke operators, that $H_S^1(\mathcal{G},M)$ is a Selmer group containing Sel${}_{\mathcal{L}}$. Then, we discuss the time complexity of this method.