Torsion of Abelian varieties over solvable extensions of number fields

TL;DR

The paper proves finiteness results for torsion points on Abelian varieties over certain solvable extensions of number fields, extending understanding of torsion behavior in these infinite extensions.

Contribution

It establishes that Abelian varieties without CM factors have finitely many torsion points over maximal solvable extensions, a new result in the arithmetic of Abelian varieties.

Findings

Finitely many torsion points over maximal n-step-solvable extensions.

Finitely many prime order torsion points over maximal prosolvable extensions.

Results apply to Abelian varieties with no CM isogeny-factors.

Abstract

Let $K$ be a number field, and let $A$ be an Abelian variety over $K$ which has no CM isogeny-factors over $\overline{K}$. We prove that $A$ has only finitely many torsion points over the maximal $n$-step-solvable extension of $K$ for any $n$ and only finitely many torsion points of prime order over the maximal prosolvable extension of $K$.