Determining monogenity of pure cubic number fields using elliptic curves

2025/05/09 by Jordi Guàrdia, Guàrdia, Jordi, Francesc Pedret +1
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.06213

openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbF₃-orbit in the first cohomology group of the elliptic curve E^D: y² = 4x³ + D with respect to a certain 3-isogeny ϕ. Orbits corresponding to monogenic fields must lie in the soluble part of the Selmer group S^ϕ(E^D/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.

Citations

• The Scheme of Monogenic Generators I: Representability
• A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so
• Newton polygons of higher order in algebraic number theory

Related

• A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so
• Bounds for 2-Selmer ranks in terms of seminarrow class groups
• Explicit isogenies of prime degree over number fields
• Algebraic number fields generated by an infinite family of monogenic trinomials
• New isogenies of elliptic curves over number fields