---
title: "A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so"
authors:
  - "Levent Alpoge"
  - "Manjul Bhargava"
  - "Ari Shnidman"
year: 2025
venue: "Mathematische Annalen"
arxiv: "2011.01186"
doi: "10.1007/s00208-024-03054-w"
source: "https://arxiv.org/abs/2011.01186"
topics:
  - "arithmetic-geometry"
  - "number-theory"
  - "elliptic-curves"
seed_rank: 1208
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 9
lineage: arithmetic-geometry
cites:
  - title: "A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so"
    url: "https://arxiv.org/abs/2107.05514"
    year: 2021
    arxiv: "2107.05514"
    doi: null
see:
  - "1207-a-positive-proportion-of-quartic-fields-are-not-monogenic-yet"
---

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

## One-sentence takeaway

Compare 2-descent and 3-descent on Mordell curves y^2=x^3+k: a positive proportion of cubic fields are non-monogenic with no local obstruction, and Sha[3] can be arbitrarily large on a positive-density set.

## Why it's lovely

Monogeneity as a Selmer comparison. By-product: for every r>=0, a positive proportion of E_k have dim_{F_3} Sha[3] >= r.

## Problem

Which cubic fields are monogenic? Local conditions are insufficient.

## Design

2- vs 3-descent on y^2=x^3+k, ordered by height.

## Evidence

arXiv:2011.01186. Math. Ann. 2025, DOI 10.1007/s00208-024-03054-w.

## Limitations

Cubic only; quartic is 1207.

## Implications for Broadside

The cubic monogenic paper; journal wins.

## Bottom line

Mint the Math. Ann. card; cite preprint.

## Links

- arXiv: [2011.01186](https://arxiv.org/abs/2011.01186)
- DOI: [10.1007/s00208-024-03054-w](https://doi.org/10.1007/s00208-024-03054-w)