---
title: "Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field"
authors:
  - "Levent Alpoge"
  - "Manjul Bhargava"
  - "Wei Ho"
  - "Ari Shnidman"
year: 2026
venue: "Inventiones mathematicae"
arxiv: "2501.18774"
doi: "10.1007/s00222-025-01392-3"
source: "https://arxiv.org/abs/2501.18774"
topics:
  - "arithmetic-geometry"
  - "number-theory"
  - "elliptic-curves"
seed_rank: 1199
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 10
lineage: arithmetic-geometry
cites:
  - title: "Ranks of twists of elliptic curves and Hilbert's tenth problem"
    url: "https://doi.org/10.1007/s00222-010-0248-9"
    year: 2010
    arxiv: null
    doi: "10.1007/s00222-010-0248-9"
  - title: "Hilbert's tenth problem via additive combinatorics"
    url: "https://arxiv.org/abs/2412.01768"
    year: 2025
    arxiv: "2412.01768"
    doi: null
see:
  - "1197-the-3-4-infinity-modular-family-of-2-tori-completed-at-its-thr"
  - "1198-more-than-two-thirds-of-the-zeros-of-the-riemann-zeta-function"
---

# Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

## One-sentence takeaway

For every quadratic extension of number fields K/F there is an abelian variety over F of positive rank that does not grow to K — hence Hilbert's tenth problem is undecidable over the ring of integers of every number field.

## Why it's lovely

Hilbert 10 over Z is Matiyasevich 1970. Over O_K the missing piece, after Poonen / Shlapentokh / Mazur-Rubin-Shlapentokh, was rank-stable positive-rank abelian varieties in every quadratic extension. Alpoge-Bhargava-Ho-Shnidman supply them unconditionally. Koymans-Pagano independently, by a different method.

## Problem

Shlapentokh: if every quadratic K/F admits A/F with rank A(F)=rank A(K)>0, then Z is Diophantine in every O_K, so H10(O_K) is undecidable. Mazur-Rubin had this under Sha-finiteness.

## Design

Construct, in each quadratic extension, an abelian variety whose rank is stable under the base change — arithmetic-statistics / twist-family input rather than a Sha hypothesis.

## Evidence

Invent. math. 243 (2026), 1129-1139. DOI 10.1007/s00222-025-01392-3. arXiv:2501.18774 (30 Jan 2025). Homepage: "the most recent paper is a solution of Hilbert's tenth problem over all number fields."

## Limitations

Independent of Koymans-Pagano 2412.01768 (additive combinatorics); cite them, do not remint here. Short Inventiones note, not a survey of H10.

## Implications for Broadside

Alpoge's headline number-theory theorem. See 1200 for the conditional algorithm that would list rational points if Hodge/Tate/FM hold.

## Bottom line

Mint the Inventiones journal version; cite the preprint.

## Links

- arXiv: [2501.18774](https://arxiv.org/abs/2501.18774)
- DOI: [10.1007/s00222-025-01392-3](https://doi.org/10.1007/s00222-025-01392-3)