Journal of the Ramanujan Mathematical Society
Volume 41, Issue 1, March 2026 pp. 19–34.
Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of
elliptic curves
Authors:
Andrzej Dąbrowski and Lucjan Szymaszkiewicz
Author institution:Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.
Summary:
We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists
of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve E over ℚ and positive integer
k, an asymptotic formula for the number of quadratic twists Ed, d positive square-free integers less than X, with
finite group E{d}(ℚ) and |Ш(E{d})| = k{2}. This paper continues the authors previous investigations concerning orders of
Tate-Shafarevich groups in quadratic twists of the curve X{0}(49).
Contents
Full-Text PDF