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

authored by

Sara Checcoli, Gabriel A. Dill

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

University Grenoble-Alpes (UGA)
University of Bonn

Type

Article

Journal

Bulletin of the London Mathematical Society

Volume

56

Pages

3530-3541

ISSN

0024-6093

Publication date

03.11.2024

Publication status

Published

Peer reviewed

Yes

Electronic version(s)

https://doi.org/10.1112/blms.13149 (Access: Closed)
https://doi.org/10.48550/arXiv.2306.12138 (Access: Open)