Determining monogenity of pure cubic number fields using elliptic curves

TL;DR

This paper investigates the monogenity of pure cubic number fields by analyzing Selmer groups of associated elliptic curves, providing criteria to identify monogenic fields and bounds on their quantity, with some results assuming GRH.

Contribution

It introduces a novel criterion using elliptic curve Selmer groups to determine monogenity of pure cubic fields and derives bounds based on elliptic curve ranks.

Findings

Criteria to discard monogenic fields based on Selmer group analysis

Bounds on the number of monogenic cubic fields related to elliptic curve rank

Determination of monogenity for many pure cubic fields assuming GRH

Abstract

We study monogenity of pure cubic number fields by means of Selmer groups of certain elliptic curves. A cubic number field with discriminant $D$ determines a unique nontrivial $\mathbb{F}_3$-orbit in the first cohomology group of the elliptic curve $E^D: y^2 = 4x^3 + D$ with respect to a certain 3-isogeny $\phi$. Orbits corresponding to monogenic fields must lie in the soluble part of the Selmer group $S^{\phi}(E^D/\mathbb{Q})$, and this gives a criterion to discard monogenity. From this, we can derive bounds on the number of monogenic cubic fields in terms of the rank of the elliptic curve. We can also determine the monogenity of many concrete pure cubic fields assuming GRH.