TY - JOUR
T1 - A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so
AU - Alpöge, Levent
AU - Bhargava, Manjul
AU - Shnidman, Ari
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85212097961&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85212097961&partnerID=8YFLogxK
U2 - 10.1007/s00208-024-03054-w
DO - 10.1007/s00208-024-03054-w
M3 - Article
AN - SCOPUS:85212097961
SN - 0025-5831
JO - Mathematische Annalen
JF - Mathematische Annalen
ER -