Determining monogenity of pure cubic number fields using elliptic curves

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 $\varphi$. Orbits corresponding to monogenic fields must lie in the soluble part of the Selmer group $S^{\varphi }(E^D/\mathbb{Q})$, 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.

Added the hypothesis that ${\mathcal{O}}_L$ is free over ${\mathcal{O}}_K$ to the main theorem, as this condition is necessary for our methods to apply. Since our primary focus is the case $K=\mathbb{Q}$, this correction does not impact our other results or computations