On explicit units in a family of cubic fields and the arithmetic of curves arising from powers of these units

Abstract

For an integer $b\neq0,1$ 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 $q\ge1$ the curve $C_q$ is irreducible with $g(C_q)=\binom{q-1}{2}$, 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 $q\ge4$ of general type. In particular only finitely many $b$ have $q\mid n(b)$ for each $q\ge4$. A lower bound for linear forms in two logarithms gives $n(b)<6082138$ for every $b$, so $n(b)\in\{1,2\}$ for all but finitely many $b$, though not effectively. We add two computations: $n(b)$ is determined outright for $|b|\le10^{30}$, and the rational points of $C_4$ are determined by quadratic Chabauty, giving $4\mid n(b)\iff b\in\{-75,-3,5,96\}$. We state precisely which parts of the latter rest on certified arithmetic and which do not. Finally, we show that $K_b\cong K_{b'}$ with $b\neq b'$ 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 $q\ge5$. There are only finitely many such pairs.