On measure-preserving ${\mathbb{F}}_p^{\omega }$-systems of order $k$

Building on previous work in the nilspace-theoretic approach to the study of Host-Kra factors of measure-preserving systems, we prove that every ergodic ${\mathbb{F}}_p^{\omega }$-system of order $k$ is a factor of an Abramov ${\mathbb{F}}_p^{\omega }$-system of order $k$. This answers a question of Jamneshan, Shalom and Tao.