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Galois covers with prescribed fibers: The Beckmann-Black problem. (English) Zbl 0954.12002
The Beckmann-Black problem asks whether any Galois extension \(E/K\) is the specialization of a Galois branched cover of \({\mathbb P}^1\) defined over \(K\) with the same Galois group. It is conjectured by E. V. Black [J. Lond. Math. Soc., II. Ser. 60, No. 3, 677-688 (1999; Zbl 0944.12001)] that this is always true. The author proves three results about this conjecture.
The first one is that its validity would imply the Regular Inverse Galois Problem (this has been observed independently by A. Tamagawa, as the author himself remarks). The second one deals with a variation of the original Beckmann-Black problem, where one asks that the realizing Galois cover defined over \(\overline K\) be only a ‘mere’ cover, i.e. without the Galois action, and be Galois over \(\bar{K}\). It is shown that this problem has an affermative answer if \(K\) contains an ample field (a field \(k\) is called ample if every smooth \(k\)-curve has infinitely many \(k\)-rational points provided that there is at least one). The third one is the proof of the original Beckmann-Black problem in the case when \(K\) is a PAC (pseudo-algebraically closed) field.

MSC:
12F12 Inverse Galois theory
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
14H30 Coverings of curves, fundamental group
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