This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing ... read morethat the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve di ers from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion.read less
Klagsbrun, Zev, and Robert J. Lemke Oliver. "THE DISTRIBUTION OF 2-SELMER RANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION." Mathematika 62, no. 01 (May 4, 2015): 67-78. doi:10.1112/s0025579315000121.
