Rational pullbacks of Galois covers

TL;DR

The paper characterizes finite groups for which all Galois covers of the projective line can be obtained by pullback from a small set of covers, highlighting a special property of subgroups of PGL(2,C).

Contribution

It proves that only finite subgroups of PGL(2,C) have the property that all Galois covers can be derived from a finite set via pullback, and explores implications for inverse Galois theory.

Findings

Finite subgroups of PGL(2,C) uniquely have the pullback property.

For these groups, a single cover with at most 3 branch points suffices.

Groups outside PGL(2,C) yield genuinely new Galois realizations as branch points grow.

Abstract

The finite subgroups of ${\rm PGL}_2(\mathbb{C})$ are shown to be the only finite groups $G$ with this property: for some integer $r_0$ (depending on $G$), 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 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 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 provides truly new Galois realizations $F/{\mathbb{C}}(T)$ of $G$. Another application is that the ``Beckmann--Black'' property that ``any two Galois covers of $\mathbb{P}^1_{\mathbb{C}}$ with the same group $G$ are always pullbacks of another Galois cover of group $G$'' only holds if $G\subset {\rm PGL}_2({\mathbb{C}})$.