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

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 -