Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion

Abstract: This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results about the minimum number of twists of E with rank 0, and subject to standard conjectures, the number of twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K.