The 3-Selmer ranks of the elliptic curves y^2=x^3+a(x-b)^2 over Q(zeta_3) are controlled by ideal class group 3-ranks, yielding large-rank families and a positive proportion of rank-one curves.
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$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks
The 3-Selmer ranks of the elliptic curves y^2=x^3+a(x-b)^2 over Q(zeta_3) are controlled by ideal class group 3-ranks, yielding large-rank families and a positive proportion of rank-one curves.