Reflex field of the reflex field (awaiting review)

Let $K$ be a CM number field and let $N$ a normal closure of $K$, let $\Phi \subset \mathrm{Hom}(K, \overline\mathbb{Q})$ be a CM type and $L$ its associated reflex field. Then $\Phi$ induces a CM type $\Phi_N \subset \mathrm{Hom}(N, \mathbb{C})$ by taking the maps that restrict to a map inside $\Phi$ on $K$. The maps in $\Phi_N$ are isomorphisms on the image $F$ of $N$ inside $\overline\mathbb{Q}$ and by inverting them, we obtain a CM type on $F$ with values in $N$. The reflex field of the reflex field is the reflex field of this CM type.

It can also be computed as follows. Consider the right action of $\mathrm{Gal}(N/K)$ on the set of CM types on $K$. Then the reflex field of the reflex field is the subfield corresponding to the subgroup stabilising $\Phi$.

The reflex field of the reflex field is also the smallest field of definition of the CM type $\Phi$, i.e. it is the largest subfield $M$ of $K$ such that $\Phi$ is induced from a CM type on $M$.