Galois covers of 1 over with prescribed local or global behavior by specialization
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, p. 271-282

This paper considers some refined versions of the Inverse Galois Problem. We study the local or global behavior of rational specializations of some finite Galois covers of 1 .

On considère des versions raffinées du Problème Inverse de Galois. Nous étudions le comportement local et global des spécialisations rationnelles de quelques revêtements galoisiens finis de 1 .

@article{JTNB_2005__17_1_271_0,
     author = {Plans, Bernat and Vila, N\'uria},
     title = {Galois covers of $\mathbb{P}^1$ over $\mathbb{Q}$ with prescribed local or global behavior by specialization},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     pages = {271-282},
     doi = {10.5802/jtnb.490},
     mrnumber = {2152224},
     zbl = {1087.11068},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2005__17_1_271_0}
}
Plans, Bernat; Vila, Núria. Galois covers of $\mathbb{P}^1$ over $\mathbb{Q}$ with prescribed local or global behavior by specialization. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 271-282. doi : 10.5802/jtnb.490. http://www.numdam.org/item/JTNB_2005__17_1_271_0/

[1] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups. New York: Clarendon press, 1985. | MR 827219 | Zbl 0568.20001

[2] S. Beckmann, On extensions of number fields obtained by specializing branched coverings. J. Reine Angew. Math. 419 (1991), 27–53. | MR 1116916 | Zbl 0721.11052

[3] S. Beckmann, Is every extension of the specialization of a branched covering? J. Algebra 165 (1994), 430–451. | MR 1271246 | Zbl 0802.12003

[4] B. Birch, Noncongruence subgroups, Covers and Drawings. Leila Schneps, editor, The Grothendieck theory of dessins d’enfants. Cambridge Univ. Press (1994), 25–46. | MR 1305392 | Zbl 0930.11024

[5] E. Black, Deformations of dihedral 2-group extensions of fields. Trans. Amer. Math. Soc. 351 (1999), 3229–3241. | MR 1467461 | Zbl 0931.12005

[6] E. Black, On semidirect products and the arithmetic lifting property. J. London Math. Soc. (2) 60 (1999), 677–688. | MR 1753807 | Zbl 0944.12001

[7] J.-L. Colliot-Thélène, Rational connectedness and Galois covers of the projective line. Ann. of Math. 151 (2000), 359–373. | MR 1745009 | Zbl 0990.12003

[8] P. Dèbes, Some arithmetic properties of algebraic covers. H. Völklein, D. Harbater, P. Müller, and J. G. Thompson, editors, Aspects of Galois theory. London Math. Soc. LNS 256 (2). Cambridge Univ. Press (1999), 66–84. | MR 1708602 | Zbl 0977.14009

[9] P. Dèbes, Galois Covers with Prescribed Fibers: the Beckmann-Black Problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 273–286. | Numdam | MR 1736229 | Zbl 0954.12002

[10] P. Dèbes, Density results for Hilbert subsets. Indian J. pure appl. Math. 30 (1) (1999), 109–127. | MR 1677959 | Zbl 0923.12001

[11] C. U. Jensen, A. Ledet, N. Yui, Generic polynomials. Cambridge Univ. Press, Cambridge, 2002. | MR 1969648 | Zbl 1042.12001

[12] J. Klüners, G. Malle, A database for field extensions of the rationals. LMS J. Comput. Math. 4 (2001), 182–196. | MR 1901356 | Zbl 1067.11516

[13] J. Klüners, G. Malle, Counting nilpotent Galois extensions. J. reine angew. Math. 572 (2004), 1–26. | MR 2076117 | Zbl 1052.11075

[14] G. Malle, B. H. Matzat, Inverse Galois Theory. Springer, Berlin, 1999. | MR 1711577 | Zbl 0940.12001

[15] J.-F. Mestre, Extensions régulières de (T) de groupe de Galois A ˜ n . J. Algebra 131 (1990), 483–495. | MR 1058560 | Zbl 0714.11074

[16] J.-F. Mestre, Relèvement d’extensions de groupe de Galois PSL 2 (𝔽 7 ). Preprint (2004), arXiv:math.GR/0402187.

[17] J. Montes, E. Nart, On a Theorem of Ore. J. Algebra 146 (1992), 318–334. | MR 1152908 | Zbl 0762.11045

[18] L. Moret-Bailly, Construction de revêtements de courbes pointées. J. Algebra 240 (2001), 505–534. | MR 1841345 | Zbl 1047.14013

[19] Y. Morita, A Note on the Hilbert Irreducibility Theorem. Japan Acad. Ser. A Math. Sci. 66 (1990), 101–104. | MR 1065782 | Zbl 0725.12003

[20] Ö. Ore, Newtonsche Polygone in der Theorie der algebraischen Körper. Math. Ann. 99 (1928), 84–117. | JFM 54.0191.02 | MR 1512440

[21] B. Plans, Central embedding problems, the arithmetic lifting property and tame extensions of . Internat. Math. Res. Notices 2003 (23) (2003), 1249–1267. | MR 1967317 | Zbl 1044.12004

[22] B. Plans, N. Vila, Tame A n -extensions of . J. Algebra 266 (2003), 27–33. | MR 1994526 | Zbl 1057.12003

[23] B. Plans, N. Vila, Trinomial extensions of with ramification conditions. J. Number Theory 105 (2004), 387–400. | MR 2040165 | Zbl 1048.11086

[24] D. Saltman, Generic Galois extensions and problems in field theory. Adv. Math. 43 (1982), 250–283. | MR 648801 | Zbl 0484.12004

[25] J.-P. Serre, Groupes de Galois sur . Sém. Bourbaki 1987-1988, no 689. | Numdam | MR 992203 | Zbl 0684.12009

[26] J.-P. Serre, Topics in Galois theory. Jones and Bartlett, Boston, 1992. | MR 1162313 | Zbl 0746.12001

[27] R. Swan, Noether’s problem in Galois theory. J. D. Sally and B. Srinivasan, editors, Emmy Noether in Bryn Mawr. Springer (1983), 21–40. | MR 713790 | Zbl 0538.12012

[28] N. Vila, On central extensions of A n as Galois group over . Arch. Math. 44 (1985), 424–437. | MR 792366 | Zbl 0562.12011