Trace-zero property for prescribed-abscissa points at positive rational rank

Determine whether every point on the congruent-number curve E_n: y^2=x^3-n^2x over a simplest cubic field K_t, with abscissa x=\theta_t-1 and positive rational rank rank E_n(\mathbb{Q})>0, has group trace zero.

Background

The paper studies points P=(\theta_t-1,y) on the congruent-number curves E_n over the simplest cubic fields K_t=\mathbb{Q}(\theta_t), where \theta_t is the largest root of the Shanks polynomial X^3-tX^2-(t+3)X-1. When rank E_n(\mathbb{Q})=0, the authors prove that every such point has trace zero, classify all of them through a Pell-type equation, and show that the conjugates generate a rank-two subgroup.

Without the rank-zero hypothesis, the authors prove that nonzero two-torsion cannot occur as the group trace. They further show that a point with nonzero trace must have a quadratic ordinate in z=\theta_t-1; its fourth intersection with E_n is a rational nontorsion point, and the point yields an integral solution of the associated norm equation. The unresolved issue is whether such quadratic-ordinate points actually exist when rank E_n(\mathbb{Q})>0.