Component ideal (awaiting review)

The adelic Hilbert modular surface may be geometrically reducible, and its components are in bijection with the narrow class group of the base field $F$. The component corresponding to a given narrow ideal class $[\mathfrak{b}]$ parametrizes abelian surfaces with endomorphism algebra isomorphic to $F$ whose polarization module is isomorphic to $\mathbb{Z}_F \oplus \mathfrak{b}$. In particular, the surfaces parametrized are principally polarized if and only if the component ideal is the inverse different of the base field $F$.