Abstract
Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T) with group G such that E/k is regular, we produce some specializations of E/k(T) at points t0 ∈ P1(k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior — ramified or unramified — at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T).
Similar content being viewed by others
References
S. Beckmann, On extensions of number fields obtained by specializing branched coverings, Journal für die reine und angewandte Mathematik 419 (1991), 27–53.
S. Beckmann, Is every extension of Q the specialization of a branched covering?, Journal of Algebra 164 (1994), 430–451.
J. H. Conway et al., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray, Oxford University Press, Eynsham, 1985.
P. Dèbes, Galois covers with prescribed fibers: the Beckmann-Black problem, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie IV 28 (1999), 273–286.
P. Dèbes, Méthodes topologiques et analytiques en théorie inverse de Galois: théorème d'existence de Riemann, Arithmétique de revêtements algébriques (Saint-Étienne, 2000), Sémin. Congr., vol. 5, Société Mathématique de France, 2001, pp. 27–41.
P. Dèbes, Arithmétique des revêtements de la droite, Lecture notes, 2009, athttp://math.univ-lille1.fr/˜pde/ens.html.
P. Dèbes and M. D. Fried, Rigidity and real residue class fields, Acta Arithmetica 56 (1990), 291–323.
P. Dèbes and N. Ghazi, Specializations of Galois covers of the line, in Alexandru Myller Mathematical Seminar, AIP Conference Proceedings, Vol. 1329, AmericanInstitute of Physics, Melville, NY, 2011, pp. 98–108.
P. Dèbes and N. Ghazi, Galois covers and the Hilbert-Grunwald property, Annales de l'Institut Fourier 62 (2012), 989–1013.
P. Dèbes and F. Legrand, Twisted covers and specializations, in Galois-Teichmueller theory and Arithmetic Geometry, Proceedings for Conferences in Kyoto (October2010), H. Nakamura, F. Pop, L. Schneps, A. Tamagawa eds., Advanced Studies inPure Mathematics Vol. 63, 2012, pp. 141–162.
P. Dèbes and F. Legrand, Specialization results in Galois theory, Transactions of the American Mathematical Society 365 (2013), 5259–5275.
S. Flon, Mauvaises places ramifiées dans le corps des modules d'un revêtement, Ph.D. thesis, Université des Sciences et Technologies de Lille, France, 2002.
M. D. Fried, Introduction to modular towers: generalizing dihedral group–modular mcurve connections, in Recent developments in the inverse Galois problem (Seattle, WA, 1993), Contemporary Mathematics, Vol. 186, American Mathematical Society, Providence, RI, 1995, pp. 111–171.
W.-D. Geyer, Galois groups of intersections of local fields, Israel Journal of Mathematics 30 (1978), 382–396.
H. A. Heilbronn, Zeta-functions and L-functions, in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, DC, 1967, pp. 204–230.
C. U. Jensen, A. Ledet and N. Yui, Generic Polynomials, Constructive Aspects of the Inverse Galois Problem, Cambridge University Press, 2002.
C. Jordan, Recherches sur les substitutions, J. Liouville 17 (1872), 351–367.
J. Klüners and G. Malle, Counting nilpotent Galois extensions, Journal für die reine und angewandte Mathematik 572 (2004), 1–26.
S. Lang, Algebra, revised third ed., Graduate Texts in Mathematics, Vol. 211, Springer-Verlag, New York, 2002.
F. Legrand, Spécialisations de revêtements et théorie inverse de Galois, Ph.D. thesis, Université Lille 1, France, 2013, athttps://sites.google.com/site/francoislegranden/recherche.
F. Legrand, Parametric Galois extensions, Journal of Algebra 422 (2015), 187–222.
G. Malle and B. H. Matzat, Inverse Galois Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999.
J.-F. Mestre, Extensions régulières de Q (T) de groupe de Galois A˜n, Journal of Algebra 131 (1990), 483–495.
J. Neukirch, On solvable number fields, Inventiones Mathematicae 53 (1979), 135–164.
J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields, second ed., Grundlehren der mathematischen Wissenschaften, Vol. 323, Springer, Berlin, 2008.
B. Plans and N. Vila, Galois covers of P1 over Q with prescribed local or global behavior by specialization, Journal de Théorie des Nombres de Bordeaux 17 (2005), 271–282.
A. Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, Vol. 77, Cambridge University Press, Cambridge, 2000.
J.-P. Serre, Topics in Galois Theory, Research Notes in Mathematics, Vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992.
A. Travesa, Nombre d'extensions abéliennes sur Q, Sém. Théor. Nombres Bordeaux (2) 2 (1990), 413–423.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Legrand, F. Specialization results and ramification conditions. Isr. J. Math. 214, 621–650 (2016). https://doi.org/10.1007/s11856-016-1349-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-016-1349-y