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 Ek:y2=x3+k. As a by-product of our methods, we show that, for every r≥0, a positive proportion of curves Ek have Tate–Shafarevich group with 3-rank at least r.
Original language | English |
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Journal | Mathematische Annalen |
DOIs | |
State | Accepted/In press - 2024 |
All Science Journal Classification (ASJC) codes
- General Mathematics