Author

Jun-Yu Lu

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Preprint Sep 2026

Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields

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...

Jun-Yu Lu · 0 citations

Preprint Aug 2026

Unit Indices of Shanks Orders

For an integer $t\geq-1$, let $\theta_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and set $R_t=\mathbb{Z}[\theta_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(\theta_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We determine $\varepsilon_t$ when $N_t$ is squar...

Jun-Yu Lu · 0 citations