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arXiv · 1511.03813

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

Zhangjie Wang
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Abstract

We study the distribution of a subclass congruent elliptic curve $E^{(n)}: y^2=x^3-n^2x$, where $n$ is congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. 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 $\big(\mathbb Z /2\mathbb Z\big)^2$. 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 $\big(\mathbb Z /2\mathbb Z\big)^{4}$.

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Zhangjie Wang. 2015-11-12. Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups: Distribution Part. https://arxiv.org/abs/1511.03813

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