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Rational pullbacks of Galois covers

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Abstract

The finite subgroups of \(\mathrm{PGL}_2({\mathbb {C}})\) 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 \(G\subset \mathrm{PGL}_2({\mathbb {C}})\), 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 \(G\not \subset \mathrm{PGL}_2({\mathbb {C}})\), 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 \(G\subset \mathrm{PGL}_2({\mathbb {C}})\).

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Notes

  1. The term “regularly” will be fully justified with the general definition of “k-regular parametricity” for which the base field k is not necessarily algebraically closed (see Definition 2.1).

  2. Above and throughout the paper, the condition “\(G \subset \mathrm{PGL}_2({\mathbb {C}})\)” (resp., “\(G \not \subset \mathrm{PGL}_2({\mathbb {C}})\)”) really means that G is isomorphic (resp., is not isomorphic) to a subgroup of \(\mathrm{PGL}_2({\mathbb {C}})\).

  3. The case \(N=1\) is particularly significant as it supports Hilbert’s strategy to solve the Inverse Galois Problem by first producing a \({\mathbb {Q}}\)-regular Galois cover \(f:X\rightarrow {\mathbb {P}}^1_{\mathbb {Q}}\) of given group. It is known to hold for some groups: abelian, \(S_n\), \(A_n\), dihedral of order 2n with \(n>1\) odd, etc.

  4. which is the function field extension associated with \(f \otimes _k {\overline{k}} : X \otimes _k {\overline{k}} \rightarrow {\mathbb {P}}^1_{{\overline{k}}}\).

  5. in the sense that they correspond to \(e^{2i\pi /e_i}\) in the canonical isomorphism \(I_{{\mathfrak {P}}} \rightarrow \mu _{e_i} =\langle e^{2i\pi /e_i} \rangle \).

  6. Due to our definition of the categories \({{\textsf {H}}}_{G,r}(k)\) and \({{\textsf {H}}}_{G,r}(\mathbf{C})(k)\), it is the so-called inner version of Hurwitz spaces that we shall be working with.

  7. In particular, the degree of \(T_0\in k(U)\) (the maximum of numerator degree and denominator degree in coprime notation) is the same as the degree of the associated map \({\mathbb {P}}^1_k \rightarrow {\mathbb {P}}^1_k\).

  8. Here, a subset of a topological space is called constructible if it is a finite union of locally closed sets.

  9. In fact, \(|G|\in \{2,3,4,6\}\), since G is then a group of automorphisms of some elliptic curve, see [28, Chapter III, Theorem 10.1].

  10. Note here that equivalent covers have the same defining equations by definition, so that the term “defining equation for an element of \({\mathcal {H}}_{G,r}(\mathbf{C})(k)\)” is indeed well-defined.

  11. There may be a priori two different \(j_i\in \{1,\ldots ,r\}\) such that \({{\mathcal {C}}}_i\in \mathbf{C}_{f,T_0,j_i}\).

  12. Definition of “ample field” is recalled in Remark 1.4(a).

  13. In fact, the assumption on H to be non-solvable can be removed with a bit of extra effort. Moreover, if H is not a regular Galois group over k, then G is not either. Hence, Theorem 1.1(b) trivially fails.

  14. We note that, for \(R\ge 6\), this claim also follows for any choice of y from [7, Theorem 3.1(b-2)], since the latter implies that every pullback of a k-cover in \({{\textsf {H}}}_{G,R}({\mathbf{C}})\) has at least \(R+2\) branch points, and hence is not in \({{\textsf {H}}}_{G,R+1}(\mathbf{D}_y)(k).\)

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Acknowledgements

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and ISF grants No. 577/15 and No. 696/13.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Paul Painlevé, Université de Lille, 59655, Villeneuve d’Ascq Cedex, France

    Pierre Dèbes

  2. Department of Mathematics Education, Korea National University of Education, 28173, Cheongju, South Korea

    Joachim König

  3. Institut für Algebra, Fachrichtung Mathematik, TU Dresden, 01062, Dresden, Germany

    François Legrand

  4. Department of Mathematics, Technion, Israel Institute of Technology, Haifa, 32000, Israel

    Danny Neftin

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Dèbes, P., König, J., Legrand, F. et al. Rational pullbacks of Galois covers. Math. Z. 299, 1507–1531 (2021). https://doi.org/10.1007/s00209-021-02703-z

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