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(μ₃). 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 Em² a and Em⁴ a 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.