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

Levent Alpöge; Manjul Bhargava; Ari Shnidman

doi:10.1007/s00208-024-03054-w

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

Levent Alpöge, Manjul Bhargava, Ari Shnidman

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

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
Pages (from-to)	5535-5551
Number of pages	17
Journal	Mathematische Annalen
Volume	391
Issue number	4
DOIs	https://doi.org/10.1007/s00208-024-03054-w
State	Published - Apr 2025

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

10.1007/s00208-024-03054-w

Cite this

@article{74b2d79e3c484bb3a72d3f521ec61598,

title = "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 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.",

author = "Levent Alp{\"o}ge and Manjul Bhargava and Ari Shnidman",

note = "Publisher Copyright: {\textcopyright} The Author(s) 2024.",

year = "2025",

month = apr,

doi = "10.1007/s00208-024-03054-w",

language = "الإنجليزيّة",

volume = "391",

pages = "5535--5551",

journal = "Mathematische Annalen",

issn = "0025-5831",

publisher = "Springer New York",

number = "4",

}

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

PY - 2025/4

Y1 - 2025/4

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=105001580975&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/record.url?scp=85212097961&partnerID=8YFLogxK

U2 - 10.1007/s00208-024-03054-w

DO - 10.1007/s00208-024-03054-w

M3 - مقالة

SN - 0025-5831

VL - 391

SP - 5535

EP - 5551

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 4

ER -

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

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this