Deformations of dihedral 2-group extensions of fields

Abstract:

Given a $G$-Galois extension of number fields $L/K$ we ask whether it is a specialization of a regular $G$-Galois cover of $\mathbb {P}^{1}_{K}$. This is the “inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral $2$-groups under certain assumptions on the base field $k$. We also show that dihedral groups of order $8$ and $16$ have generic extensions over any base field $k$ with characteristic different from $2$.