ON THE ESSENTIAL TORSION FINITENESS OF ABELIAN VARIETIES OVER TORSION FIELDS

The classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$ by a result of Ribet. In this article,...

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Bibliographic Details
Published in:Nagoya mathematical journal 2024-03, Vol.253, p.91-127
Main Authors: ACHTER, JEFFREY D., DUAN, LIAN, WANG, XIYUAN
Format: Article
Language:English
Subjects:
Algebra
Multiplication
Number theory
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Summary:The classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety A over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety B. Assuming the Mumford–Tate conjecture, and up to a finite extension of the base field K, we give a necessary and sufficient condition for the finiteness of $A(K_B)_{\mathrm tors}$ in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension at most 3, or when both have complex multiplication.
ISSN:0027-7630
2152-6842
DOI:10.1017/nmj.2023.19