Profinite version of the Beckmann-Black problem

Abstract

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point $t_0\in {\mathbb{P}}^1(K)$ of some finite regular Galois extension $F/K(T)$ with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the $\mathcal{l}$-universal Frattini cover of an arbitrary finite group.

KEYWORDS:

• Algebraic covers
• inverse Galois theory

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 12F12
• Secondary: 12E25
• 14H30

Acknowledgments

I wish to thank P. Dèbes for very helpful comments and many valuable suggestions. Also, I am indebted to an anonymous reviewer for insightful comments and a kind help and his interest in my paper.

Notes

1 Unramified point $t_0\in {\mathbb{P}}^1(K)$ in the extension $F/K(T)$ here means that the ideal $〈T-t_0〉$ is unramified in $F/K(T).$

2 A field K is said to be an ample field if each geometrically irreducible smooth curve defined over K with at least one K-rational point has infinitely many such points (for more detail, see [3, 8, 14, 18]).

3 This means that t is globally invariant under the action of G_K and that each coordinate is separable.

4 A field K is said to be a real closed field if K is an ordered field for which no non-trivial algebraic extension can be ordered. The field of real numbers $\mathbb{R}$ is a real closed field.

5 The viewpoints of cover and field extensions are equivalent, thus allowing to define a regular realization in terms of cover.

6 The branch point set of a cover $f:X\to {\mathbb{P}}^1$ is the finite set of points $t\in {\mathbb{P}}^1$ such that the associated discrete valuations are ramified in the corresponding function field extension $\overline{K}(X)/\overline{K}(T).$

7 We can drop the hypothesis “K is an uncountable field” in proportion 3.1 of [13] because we assume that the tower of G-extensions has an unramified point $t_0\in K.$

8 In similar proof as in Corollary 4.2, we have the same result for K is real closed field. e.g. $K=\mathbb{R},$ because it is a regular ψ-free field (see [9, theorem 2.3]) and for $K={\mathbb{Q}}^{\text{ab}}((x))$ is a field of formal Laurent series with coefficients in ${\mathbb{Q}}^{\text{ab}},$ because it is a regular ψ-free field (see [9, Section 3.2]).