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arXiv · 1511.03810

Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups

Zhangjie Wang
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Abstract

We study a subclass of congruent elliptic curves $E^{(n)}: y^2=x^3-n^2x$, where $n$ is a positive integer congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We characterize such $E^{(n)}$ with Mordell-Weil rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^2$. We also discuss such $E^{(n)}$ with 2-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^{2k}$ with $k\ge2$.

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Zhangjie Wang. 2016-06-11. Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups. https://doi.org/10.1007/s11425-015-0741-3

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