An algorithm to compute Selmer groups via resolutions by permutations modules
Fabrice Etienne
TL;DR
The paper addresses the computational problem of determining Selmer groups attached to a finite Galois module over a number field. It introduces an algorithm that builds a resolution of the module using permutation modules with morphisms defined by sums of Hecke operators, then constructs a group $H^1_S(\mathcal{G},M)$ of $S$-restricted cohomology that contains the desired Selmer group. By establishing that $H^1_S(\mathcal{G},M)$ is a Selmer group under suitable choices of $S$, the method reduces the computation to analyzing $S$-units in étale-algebra components and the action of Hecke operators, followed by matching local conditions to isolate $\mathrm{Sel}_{\mathcal{L}}$. The framework generalizes to arbitrary finite Galois modules and provides a complexity-based roadmap under reasonable computational subroutines for subfields, unit groups, and class groups. This approach thus offers a broad, modular route to finitely generated Selmer groups with potential practical impact in arithmetic geometry and number theory computations.
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 $H^1_S(\mathcal{G}, M)$ and we prove, using the properties of Hecke operators, that $H^1_S(\mathcal{G}, M)$ is a Selmer group containing Sel$_\mathcal{L}$. Then, we discuss the time complexity of this method.