Rational pullbacks of Galois covers

Pierre Dèbes, Joachim Koenig, François Legrand, Danny Neftin

Published 2018-07-05Version 1

The finite subgroups of ${\rm{PGL}}_2(\mathbb{C})$ are shown to be the only finite groups $G$ with the following property: for some bound $r_0$, all Galois covers $X\rightarrow \mathbb{P}^1_\mathbb{C}$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along all non-constant rational maps $\mathbb{P}^1_\mathbb{C} \rightarrow \mathbb{P}^1_\mathbb{C}$. For $G\subset {\rm PGL}_2(\mathbb{C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A worthwhile consequence of the converse for inverse Galois theory is that, for $G\not \subset {\rm PGL}_2(\mathbb{C})$, letting the branch point number grow always provides truly new realizations $F/\mathbb{C}(T)$ of $G$. Our approach also leads to some improvements of results of Buhler-Reichstein about generic polynomials with one parameter. For example, if $G$ is neither cyclic nor odd dihedral, no polynomial $P \in \mathbb{C}[T,Y]$ of Galois group $G$ can {parametrize}, via specialization of $T$, all Galois extensions $E/k$ of group $G$; here $k$ is some specific base field, not depending on $G$, viz. $k=\mathbb{C}((V))(U)$. A final application, related to an old problem of Schinzel, provides, subject to the Birch and Swinnerton-Dyer conjecture, a $1$-parameter family of affine curves over some number field, all with a rational point, but with no rational generic point.