Ideal Classes of Order $2^n$ in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves
Year: 2026
Type: preprint
Abstract: For every prime $\ell\geq5$ and integer $n\geq2$, we construct a totally real cubic field in the family studied by Ennola with a specified prime ideal above $\ell$ whose class has exact order $2^n$. For parameters satisfying two square conditions and explicit congruences, the same squareclass obstructions give four independent rational points on the associated elliptic curves. A Pell equation describes all such parameters. A genus-one base change gives five independent sections that generate a subgroup saturated at two. It also yields infinitely many elliptic curves over $\mathbb Q$, pairwise nonisomorphic over $\overline{\mathbb Q}$, with rank at least five.
Source: arXiv (Cornell University)
Author: Lu, Junyu
Institution: Yang-En University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220666731