$\mathbb{Z}_p$-folded surfaces and the $\mathbb{Z}_p$-Thurston norm

$\mathbb{Z}_p$-folded surfaces are a class of oriented folded surfaces with a fundamental $H_2(\mathbb{Z}/p\mathbb{Z})$ class. In this paper we study the $\mathbb{Z}_p$-folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the $\mathbb{Z}_p$-Thurston norm