| This thesis investigates the 2-Selmer rank in quadratic-twist families of elliptic curves defined over number fields, presenting new results in this area for curves having E(K)[2] = 0 and E(K)[2] ≃ Z/2Z . In particular, we show that all elliptic curves with E( K)[2] = 0 have twists with 2-Selmer rank equal to r for every r ≥ 0 subject to the condition of constant 2-Selmer parity, and give a lower bound on the number of such twists as a function of the conductor. We do the same for all elliptic curves with E( K)[2] ≃ Z/2Z that do not have a cyclic 4-isogeny defined over K( E[2]). Lastly, we present an infinite family of elliptic curves with coefficients in Q such that if 2 splits completely in K, then the 2-Selmer rank of EF/K is bounded below by r2(K) for every quadratic F/K. |
