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We show that a positive proportion of cubic fields are not monogenic, despitehaving no local obstruction to being monogenic. Our proof involves thecomparison of $2$-descent and $3$-descent in a certain family of Mordell curves$E_k \colon y^2 = x^3 + k$. As a by-product of our methods, we show that, forevery $r \geq 0$, a positive proportion of curves $E_k$ have Tate--Shafarevichgroup with $3$-rank at least $r$.

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