Points of small order on three-dimensional abelian varieties; with an appendix by Yuri Zarhin

Let A be a three-dimensional abelian variety defined over a number field K, and let ℓ∈{3,5}. We classify the images of the mod ℓ representations of those three-dimensional abelian varieties which possess an ℓ-torsion point modulo p for almost all primes p of K, but for which there does not exist a K-isogenous A′ with a rational point of order ℓ.