Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields

Abstract

Let $K_t=\mathbb{Q}(\theta_t)$, where $\theta_t$ is the largest root of $X^3-tX^2-(t+3)X-1$ and $t\geq-1$ is an integer. We classify the points in $E_n(K_t)$ with abscissa $\theta_t-1$, for $E_n:y^2=x^3-n^2x$, when $n$ is a positive integer and $E_n(\mathbb{Q})$ has rank zero. The main step excludes every nonzero two-torsion value of the group trace. The classification reduces to $v^2=2n^2-9$, and the conjugates of every resulting point generate a subgroup of rank two. A classical quartic equation then gives exactly four pairs $(d,t)$ with $d>0$ rational for which $(\theta_t-1)/d^2$ is an abscissa on $E_3$. Without a rank assumption, we exclude the abscissa $\theta_t-1$ on $E_5(K_t)$ and $E_6(K_t)$ and prove that only finitely many parameters $t$ admit this abscissa for each fixed positive integer $n$. For an integral shift $\theta_t-r$, we obtain a simultaneous-square criterion for trace zero. We use it to construct points on $E_3$ over infinitely many pairwise nonisomorphic simplest cubic fields.