Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields

Abstract

For integers $t\ge -1$, let ${\theta }_t$ be the largest real root of the Shanks polynomial $g_t(X)=X^3-t{X}^2-(t+3)X-1$ and put $K_t=\mathbb{Q}({\theta }_t)$. We classify the points on $E_n:y^2=x^3-n^{2}x$ 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 $n$.