On a Galois property of fields generated by the torsion of an abelian variety

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

Authors

  • Sara Checcoli
  • Gabriel A. Dill

Research Organisations

External Research Organisations

  • University Grenoble-Alpes (UGA)
  • University of Bonn
View graph of relations

Details

Original languageEnglish
Pages (from-to)3530-3541
Number of pages12
JournalBulletin of the London Mathematical Society
Volume56
Issue number11
Publication statusPublished - 3 Nov 2024

Abstract

In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{tors}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height.

Keywords

    math.NT, 11J95, 11R32

Cite this

On a Galois property of fields generated by the torsion of an abelian variety. / Checcoli, Sara; Dill, Gabriel A.
In: Bulletin of the London Mathematical Society, Vol. 56, No. 11, 03.11.2024, p. 3530-3541.

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

Checcoli S, Dill GA. On a Galois property of fields generated by the torsion of an abelian variety. Bulletin of the London Mathematical Society. 2024 Nov 3;56(11):3530-3541. doi: 10.1112/blms.13149, 10.48550/arXiv.2306.12138
Checcoli, Sara ; Dill, Gabriel A. / On a Galois property of fields generated by the torsion of an abelian variety. In: Bulletin of the London Mathematical Society. 2024 ; Vol. 56, No. 11. pp. 3530-3541.
Download
@article{138c8a9df6c54b9e8ef35b396e2c2c60,
title = "On a Galois property of fields generated by the torsion of an abelian variety",
abstract = " In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{tors}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height. ",
keywords = "math.NT, 11J95, 11R32",
author = "Sara Checcoli and Dill, {Gabriel A.}",
year = "2024",
month = nov,
day = "3",
doi = "10.1112/blms.13149",
language = "English",
volume = "56",
pages = "3530--3541",
journal = "Bulletin of the London Mathematical Society",
issn = "0024-6093",
publisher = "John Wiley and Sons Ltd",
number = "11",

}

Download

TY - JOUR

T1 - On a Galois property of fields generated by the torsion of an abelian variety

AU - Checcoli, Sara

AU - Dill, Gabriel A.

PY - 2024/11/3

Y1 - 2024/11/3

N2 - In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{tors}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height.

AB - In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{tors}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height.

KW - math.NT

KW - 11J95, 11R32

UR - http://www.scopus.com/inward/record.url?scp=85204281083&partnerID=8YFLogxK

U2 - 10.1112/blms.13149

DO - 10.1112/blms.13149

M3 - Article

VL - 56

SP - 3530

EP - 3541

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 11

ER -