Extending the idea of Dabrowski [‘On the proportion of rank 0 twists of elliptic curves’, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 483–486] and using the 2-descent method, we provide three general families of elliptic curves over $\mathbb{Q}$ such that a positive proportion of prime-twists of such elliptic curves have rank zero simultaneously.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$ -torsion point over $\mathbb {Q}$ . This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$ . Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$ . It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $ .

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves En:y2=x3−n3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups for square-free positive integers n≤X is as X→∞. This is quite different from quadratic twists of elliptic curves with full 2-torsion points over ℚ [M. Xiong and A. Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553], where one Tate–Shafarevich group is almost always trivial while the other is much larger.