Torsion of Abelian varieties over solvable extensions of number fields

TL;DR

The paper addresses the finiteness of torsion for Abelian varieties over solvable extensions of a number field $K$, under the assumption that $A_{ar K}$ is isogenous to a product of non-CM simple varieties. It proves that $A$ has finitely many torsion points over $K^{n\textnormal{-}\mathrm{solv}}$ for all $n$ and finitely many torsion points of prime order over $K^{\mathrm{solv}}$, using reductions to absolutely simple factors with connected Zariski closures and leveraging deep results on mod-$\ell$ Galois representations (Serre, Larsen–Pink, Wintenberger) together with Zarhin’s theorems; an ultraproduct variant is also presented. The work sharpens Zarhin's finiteness results and clarifies torsion behavior in prosolvable towers, highlighting the role of open-image phenomena in arithmetic geometry. These results have implications for understanding how torsion points can propagate in large solvable extensions and connect Galois representations with the arithmetic of abelian varieties.

Abstract

Let $K$ be a number field, and let $A$ be an Abelian variety over $K$ which has no CM 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 $K$.

Theorems & Definitions (8)

• Theorem 1.1
• Remark 1.2
• proof : Proof of (a$'$)
• Remark 2.1
• proof : Proof of (b)
• Remark 2.2
• Lemma 3.1
• proof : Proof sketch