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

Abstract

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 $E_k:y^2=x^3+k$. As a by-product of our methods, we show that, for every $r\ge 0$, a positive proportion of curves $E_k$ have Tate--Shafarevich group with $3$-rank at least $r$.

Tweaked Definition 3, other minor changes