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Ideal Classes of Order 2n in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves

  • Junyu Lu
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Abstract

For every prime ℓ≥5 and integer n≥2, we construct a totally real cubic field in the family studied by Ennola with a specified prime ideal above ℓ whose class has exact order 2n. 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 ℚ, pairwise nonisomorphic over ℚ̅, with rank at least five.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026ideal,
  title = {{Ideal Classes of Order $2^n$ in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves}},
  author = {Junyu Lu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04254},
  eprint = {2610.04254},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04254},
}
RIS
TY  - UNPB
TI  - Ideal Classes of Order 2^n in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves
AU  - Junyu Lu
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.04254
UR  - https://arxiv.org/abs/2610.04254
AB  - For every prime ℓ≥5 and integer n≥2, we construct a totally real cubic field in the family studied by Ennola with a specified prime ideal above ℓ 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 ℚ, pairwise nonisomorphic over ℚ̅, with rank at least five.
ER  -