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arXiv · 1803.06932

Orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$

Andrzej Dąbrowski·Lucjan Szymaszkiewicz
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Abstract

This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz \cite{ZK} and by Watkins \cite{Wat}. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic formula for the class numbers of real quadratic fields.

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Andrzej Dąbrowski, Lucjan Szymaszkiewicz. 2018-03-15. Orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. https://arxiv.org/abs/1803.06932

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