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 $$E_k :y^2 = x^3 + k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> . As a by-product of our methods, we show that, for every $$r \ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , a positive proportion of curves $$E_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> have Tate–Shafarevich group with 3-rank at least r .