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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 that the distribution of 2‐Selmer ranks in the quadratic twist family of such an elliptic curve differs 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.

THE DISTRIBUTION OF 2‐SELMER RANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO‐TORSION