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

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.
 Comments: Banach Center Publ. (to appear). arXiv admin note: text overlap with arXiv:1611.07840, arXiv:1611.08181 Subjects: Number Theory (math.NT) Cite as: arXiv:1803.06932 [math.NT] (or arXiv:1803.06932v1 [math.NT] for this version)

Submission history

From: Lucjan Szymaszkiewicz [view email]
[v1] Thu, 15 Mar 2018 23:02:34 GMT (1070kb,D)