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Specialization results in Galois theory


Authors: Pierre Dèbes and François Legrand
Journal: Trans. Amer. Math. Soc. 365 (2013), 5259-5275
MSC (2010): Primary 11R58, 12E30, 12E25, 14G05, 14H30; Secondary 12Fxx, 14Gxx, 14H10
DOI: https://doi.org/10.1090/S0002-9947-2013-05800-X
Published electronically: March 25, 2013
MathSciNet review: 3074373
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Abstract: The central topic of this paper is this question: is a given $ k$-étale algebra $ \prod _lE_l/k$ the specialization of a given $ k$-cover $ f:X\rightarrow B$ of the same degree at some unramified point $ t_0\in B(k)$? We reduce it to finding $ k$-rational points on a certain $ k$-variety, which we then study over various fields $ k$ of diophantine interest: finite fields, local fields, number fields, etc. We have three main applications. The first one is the following Hilbert-Grunwald statement. If $ f:X\rightarrow \mathbb{P}^1$ is a degree $ n$ $ \mathbb{Q}$-cover with monodromy group $ S_n$ over $ \overline {\mathbb{Q}}$, and finitely many suitably large primes $ p$ are given with partitions $ \{d_{p,1}, \ldots , d_{p,s_p}\}$ of $ n$, there exist infinitely many specializations of $ f$ at points $ t_0\in \mathbb{Q}$ that are degree $ n$ field extensions with residue degrees $ d_{p,1}, \ldots , d_{p,s_p}$ at each prescribed prime $ p$. The second one provides a description of the separable closure of a PAC field $ k$ of characteristic $ p\not =2$: it is generated by all elements $ y$ such that $ y^m-y\in k$ for some $ m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers.


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  • [Ang07] Julien Angeli, Trinômes irréductibles résolubles sur un corps de nombres, Acta Arith. 127 (2007), no. 2, 169-178. MR 2289982 (2007m:11155)
  • [Ang09] -, Trinômes à petits groupes de Galois, Thèse de doctorat, Université de Limoges, 2009.
  • [Bec91] Sybilla Beckmann, On extensions of number fields obtained by specializing branched coverings, J. Reine Angew. Math. 419 (1991), 27-53. MR 1116916 (93a:11095)
  • [BS09] Lior Bary-Soroker, Dirichlet's theorem for polynomial rings, Proc. Amer. Math. Soc. 137 (2009), 73-83. MR 2439427 (2009h:12007)
  • [BS10] -, Irreducible values of polynomials, Adv. Math. 229 (2012), no. 2, 854-874. MR 2855080 (2012j:11184)
  • [Coh80] Stephan D. Cohen, The Galois group of a polynomial with two indeterminate coefficients, Pacific J. Math. 90 (1980), no. 1, 63-76. MR 599320 (83j:12020a)
  • [Coh81] -, Corrections to [Coh80], Pacific J. Math. 97 (1981), no. 2, 483-486. MR 641176 (83j:12020b)
  • [DD97] Pierre Dèbes and Jean-Claude Douai, Algebraic covers: Field of moduli versus field of definition, Annales Sci. E.N.S. 30 (1997), 303-338. MR 1443489 (98k:11081)
  • [Dèb99a] Pierre Dèbes, Arithmétique et espaces de modules de revêtements, Number Theory in Progress, (K. Gyory, H. Iwaniec and J. Urbanowicz, eds.), Walter de Gruyter, 1999, pp. 75-102. MR 1689500 (2000c:14029)
  • [Dèb99b] -, Density results for Hilbert subsets, Indian J. Pure and Applied Math. 30 (1999), no. 1, 109-127. MR 1677959 (2000c:12004)
  • [Dèb99c] -, Galois covers with prescribed fibers: The Beckmann-Black problem, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4) 28 (1999), 273-286. MR 1736229 (2000m:12006)
  • [Dèb09] -, Arithmétique des revêtements de la droite, at http://math.univ-lille1.fr/˜pde/ens.html.
  • [DG11] Pierre Dèbes and Nour Ghazi, Galois covers and the Hilbert-Grunwald property, Ann. Inst. Fourier 62 (2012), no. 3, 989-1013. MR 3013814
  • [DH98] Pierre Dèbes and David Harbater, Fields of definition of $ p$-adic covers, J. Reine Angew. Math. 498 (1998), 223-236. MR 1629870 (99e:12006)
  • [DL] Pierre Dèbes and François Legrand, Twisted covers and specializations, Galois-Teichmueller theory and Arithmetic Geometry, Proceedings for Conferences in Kyoto (October 2010), H. Nakamura, F. Pop, L. Schneps, A. Tamagawa eds., Advanced Studies in Pure Mathematics 63, 2012, pages 141-163.
  • [DW08] Pierre Dèbes and Yann Walkowiak, Bounds for Hilbert's irreducibility theorem, Pure & Applied Math. Quarterly 4/4 (2008), 1059-1083. MR 2441693 (2010e:12003)
  • [FJ04] Michael D. Fried and Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 11, Springer-Verlag, Berlin, 2004 (first edition 1986). MR 868860 (89b:12010)
  • [Sch00] Andrzej Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, 2000. MR 1770638 (2001h:11135)
  • [Ser92] Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics, Jones and Bartlett Publishers, 1992. MR 1162313 (94d:12006)
  • [Uch70] Koji Uchida, Galois group of an equation $ x^n-ax+b=0$, Tohoku Math. Journ. 22 (1970), 670-678. MR 0277505 (43:3238)
  • [Völ96] Helmut Völklein, Groups as Galois groups, Cambridge Studies in Advanced Mathematics, vol. 53, Cambridge University Press, 1996. MR 1405612 (98b:12003)

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Additional Information

Pierre Dèbes
Affiliation: Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Email: Pierre.Debes@math.univ-lille1.fr

François Legrand
Affiliation: Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Email: Francois.Legrand@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9947-2013-05800-X
Keywords: Specialization, algebraic covers, twisting lemma, Hilbert's irreducibility theorem, Grunwald's problem, PAC fields, local fields, global fields, Hurwitz spaces
Received by editor(s): September 4, 2011
Received by editor(s) in revised form: January 13, 2012, and January 26, 2012
Published electronically: March 25, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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