Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups

Abstract

We study a subclass of congruent elliptic curves $E^{(n)}:y^2=x^3-n^{2}x$, where $n$ is a positive integer congruent to $1(\mathrm{mod}8)$ with all prime factors congruent to $1(\mathrm{mod}4)$. We characterize such $E^{(n)}$ with Mordell-Weil rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^2$. We also discuss such $E^{(n)}$ with 2-primary part of Shafarevich-Tate group isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^{2k}$ with $k\ge 2$.

Cited by in corpus (3)

• Non-trivial Shafarevich-Tate Groups of Elliptic Curves
• On non-congruent numbers with $8a\pm 1$ type odd prime factors and tame kernels
• Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups: Distribution Part