Rank distribution in cubic twist families of elliptic curves

Let a be an integer which is not of the form n² or -3 n² for n Z. Let Eₐ be the elliptic curve with rational 3-isogeny defined by Eₐ: y²=x³+a, and K: =Q (₃). Assume that the 3-Selmer group of Eₐ over K vanishes. It is shown that there is an explicit infinite set of cubefree integers m such that the 3-Selmer groups over K of E₌ℂ ₀ and E₌䃄 ₀ both vanish. In particular, the ranks of these cubic twists are seen to be 0 over K. Our results are proven by studying stability properties of 3-Selmer groups in cyclic cubic extensions of K, via local and global Galois cohomological techniques.