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The finite subgroups of PGL2(C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm{PGL}_2({\mathbb {C}})$$\end{document} are shown to be the only finite groups G with this property: for some integer r0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_0$$\end{document} (depending on G), all Galois covers X→PC1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X\rightarrow {\mathbb {P}}^1_{\mathbb {C}}$$\end{document} of group G can be obtained by pulling back those with at most r0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_0$$\end{document} branch points along non-constant rational maps PC1→PC1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {P}}^1_{\mathbb {C}}\rightarrow {\mathbb {P}}^1_{\mathbb {C}}$$\end{document}. For G⊂PGL2(C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G\subset \mathrm{PGL}_2({\mathbb {C}})$$\end{document}, 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⊄PGL2(C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G\not \subset \mathrm{PGL}_2({\mathbb {C}})$$\end{document}, letting the branch point number grow provides truly new Galois realizations F/C(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F/{\mathbb {C}}(T)$$\end{document} of G. Another application is that the “Beckmann–Black” property that “any two Galois covers of PC1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {P}}^1_{\mathbb {C}}$$\end{document} with the same group G are always pullbacks of another Galois cover of group G” only holds if G⊂PGL2(C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G\subset \mathrm{PGL}_2({\mathbb {C}})$$\end{document}.
Mathematische Zeitschrift – Springer Journals
Published: Dec 1, 2021
Keywords: Galois covers; Rational pullback; Inverse Galois theory; Primary 12F12; 11R58; 14E20; Secondary 14E22; 12E30; 11Gxx
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