A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of 2-descent and 3-descent in a certain family of Mordell curves \(E_k :y^2 = x^3 + k\) $E_k:y^2=x^3+k$ . As a by-product of our methods, we show that, for every \(r \ge 0\) $r\ge 0$ , a positive proportion of curves \(E_k\) $E_k$ have Tate–Shafarevich group with 3-rank at least r.