Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields

Abstract

For integers $t\geq -1$, let $\theta_t$ be the largest real root of the Shanks polynomial $g_t(X)=X^3-tX^2-(t+3)X-1$ and put $K_t=\mathbb{Q}(\theta_t)$. We classify the points on $E_n:y^2=x^3-n^2x$ over $K_t$ with abscissa $\theta_t-1$ when $n$ is a positive integer and $E_n(\mathbb{Q})$ has rank zero. We determine the four pairs $(d,t)$ with $d>0$ rational for which $(\theta_t-1)/d^2$ is an abscissa on $E_3(K_t)$. We also 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 $n$.