On the essential torsion finiteness of abelian varieties over torsion fields

Abstract

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 both three, or when both have complex multiplication.

arXiv comment37 pages. Fix the gaps in the previous version, especially the proof of case 1 in Theorem 5.14