ON SELMER GROUPS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH A TWO‐TORSION OVER

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 . 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 and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves . Mathematika 58 (2012), 236–274.] for the special elliptic curves . It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is for square-free positive integers as .