On explicit units in a family of cubic fields and the arithmetic of curves arising from powers of these units
Abstract
For an integer b≠0,1 let Kb be the cubic field generated by the unique real root of x3−3bx−b3, let η0 be its fundamental unit, and let ηb be the explicit unit studied in the companion paper (arXiv:2609.37389). Write ηb=η0n(b). Deciding whether n(b) is divisible by q amounts to finding the integral points of a plane curve Cq of degree 2q. We prove that for every integer q≥1 the curve Cq is irreducible with g(Cq)=binomq−12, the genus of a smooth plane curve of degree q rather than of degree 2q. The proof is not a genus computation but a structural one: Cq is the inverse image, under the q-power map of a two-dimensional torus, of the curve traced by the minimal polynomial of ηb as b varies, and the formula falls out of two applications of Riemann-Hurwitz in the resulting tower. As a consequence the problem splits into exactly three regimes: q=2 rational, q=3 elliptic, and q≥4 of general type. In particular only finitely many b have q| n(b) for each q≥4. A lower bound for linear forms in two logarithms gives n(b)<6082138 for every b, so n(b)∈{1,2} for all but finitely many b, though not effectively. We add two computations: n(b) is determined outright for |b|≤1030, and the rational points of C4 are determined by quadratic Chabauty, giving 4| n(b)⟺ b∈{−75,−3,5,96}. We state precisely which parts of the latter rest on certified arithmetic and which do not. Finally, we show that Kb≅ Kb' with b≠ b' holds for {b,b'}={−1,5}, and otherwise only if one of the two parameters, c say, satisfies |c|>1030 and n(c) is divisible by a prime q≥5. There are only finitely many such pairs.
Related
- Fermat curves and explicit class field theory for real quadratic number fields
- Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
- Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields
- Ideal Classes of Order 2n in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves
- On an infinite family of cubic fields with explicit fundamental units
BibTeX
@misc{kimura2026explicit,
title = {{On explicit units in a family of cubic fields and the arithmetic of curves arising from powers of these units}},
author = {Iwao Kimura and Hikaru Umemoto},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07333},
eprint = {2611.07333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07333},
}
RIS
TY - UNPB
TI - On explicit units in a family of cubic fields and the arithmetic of curves arising from powers of these units
AU - Iwao Kimura
AU - Hikaru Umemoto
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07333
UR - https://arxiv.org/abs/2611.07333
AB - For an integer b≠0,1 let K_b be the cubic field generated by the unique real root of x^3−3bx−b^3, let η_0 be its fundamental unit, and let η_b be the explicit unit studied in the companion paper (arXiv:2609.37389). Write η_b=η_0^n(b). Deciding whether n(b) is divisible by q amounts to finding the integral points of a plane curve C_q of degree 2q. We prove that for every integer q≥1 the curve C_q is irreducible with g(C_q)=binomq−12, the genus of a smooth plane curve of degree q rather than of degree 2q. The proof is not a genus computation but a structural one: C_q is the inverse image, under the q-power map of a two-dimensional torus, of the curve traced by the minimal polynomial of η_b as b varies, and the formula falls out of two applications of Riemann-Hurwitz in the resulting tower. As a consequence the problem splits into exactly three regimes: q=2 rational, q=3 elliptic, and q≥4 of general type. In particular only finitely many b have q| n(b) for each q≥4. A lower bound for linear forms in two logarithms gives n(b)<6082138 for every b, so n(b)∈{1,2} for all but finitely many b, though not effectively. We add two computations: n(b) is determined outright for |b|≤10^30, and the rational points of C_4 are determined by quadratic Chabauty, giving 4| n(b)⟺ b∈{−75,−3,5,96}. We state precisely which parts of the latter rest on certified arithmetic and which do not. Finally, we show that K_b≅ K_b' with b≠ b' holds for {b,b'}={−1,5}, and otherwise only if one of the two parameters, c say, satisfies |c|>10^30 and n(c) is divisible by a prime q≥5. There are only finitely many such pairs.
ER -