Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups: Distribution Part

Abstract

We study the distribution of a subclass congruent elliptic curve $E^{(n)}:y^2=x^3-n^{2}x$, where $n$ is congruent to $1(\mathrm{mod}8)$ with all prime factors congruent to $1(\mathrm{mod}4)$. We prove an independence of residue symbol property. Consequently we get the distribution of rank zero such $E^{(n)}$ with $2$-primary part of Shafarevich-Tate group isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^2$. We also obtain a lower bound of the number of such $E^{(n)}$ with rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^4$.

References in corpus (1)

• Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups