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

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 ov...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Checcoli, Sara, Dill, Gabriel Andreas
Format: Article
Language:English
Subjects:
Number theory
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2306.12138