Bounds on torsion of CM abelian varieties over a $p$-adic field with values in a field of $p$-power roots

TL;DR

This paper establishes a uniform bound on the torsion subgroup size of CM abelian varieties over a specific $p$-adic extension field, depending only on the base field and dimension.

Contribution

It proves the existence of a universal bound for torsion points of CM abelian varieties over a field obtained by adjoining all $p$-power roots, depending solely on the base field and dimension.

Findings

Existence of a constant $C$ bounding torsion subgroup size.

Bound depends only on the base $p$-adic field and the dimension.

Bound applies uniformly to all CM abelian varieties of fixed dimension.

Abstract

Let $p$ be a prime number and $M$ the extension field of a $p$-adic field $K$ obtained by adjoining all $p$-power roots of all elements of $K$. In this paper, we show that there exists a constant $C$, depending only on $K$ and an integer $g>0$, which satisfies the following property:If $A_{/K}$ is a $g$-dimensional CM abelian variety, then the order of the torsion subgroup of $A(M)$ is bounded by $C$.