Wydawnictwa / Banach Center Publications / Wszystkie tomy

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

Tom 118 / 2019

Banach Center Publications 118 (2019), 125-135 MSC: 11G05, 11G40, 11Y50. DOI: 10.4064/bc118-8

Streszczenie

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 (1987) and by Watkins (2007). 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.

Autorzy

• Andrzej DąbrowskiInstitute of Mathematics
University of Szczecin
Wielkopolska 15
70-451 Szczecin, Poland
e-mail
e-mail
• Lucjan SzymaszkiewiczInstitute of Mathematics
University of Szczecin
Wielkopolska 15
70-451 Szczecin, Poland
e-mail

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