Uniform irreducibility of Galois action on the $\ell$-primary part of Abelian $3$-folds of Picard type

TL;DR

This paper extends Manin's classical result on uniform bounds for cyclic isogenies from elliptic curves to specific families of abelian 3-folds with complex multiplication, revealing new uniformity properties.

Contribution

It generalizes the uniform irreducibility of Galois actions to abelian 3-folds of Picard type with complex multiplication, beyond elliptic curves.

Findings

Established uniform bounds for isogenies in new abelian 3-fold families

Extended classical results from elliptic curves to higher-dimensional abelian varieties

Demonstrated irreducibility properties of Galois representations in this context

Abstract

Half a century ago Manin showed that given a number field $k$ and a rational prime $\ell$, there exists a uniform bound for the order of cyclic $\ell$-power isogenies between two non-CM elliptic curves over $k$. We generalize this to certain $2$-dimensional families of abelian $3$-folds with multiplication by an imaginary quadratic field.