Let be a number field, and suppose is irreducible over . Using algebraic geometry and group theory, we describe conditions under which the -exceptional set of , i.e. the set of for which the specialized polynomial is -reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed , all but finitely many -specializations of the degree generalized Laguerre polynomial are -irreducible and have Galois group . Second, we study specializations of the modular polynomial (which vanishes on the -invariants of pairs of elliptic curves related by a cyclic -isogeny), and show that for any , all but finitely many of the -specializations of are -irreducible and have Galois group containing . Third, for a simple branched cover of degree and of genus at least , all but finitely many -specializations are -irreducible and have Galois group .
Soient un corps de nombres et un polynôme irréductible sur . À partir de la géométrie algébrique et de la théorie des groupes, nous donnons des conditions suffisantes pour que l’ensemble -exceptionnel de , c’est-à-dire l’ensemble des éléments de tels que est réductible sur , soit fini. Nos méthodes nous permettent alors de développer trois applications. Tout d’abord, nous obtenons que pour tout entier plus grand que , à l’exception d’un nombre fini de cas, la -spécialisation du polynôme de Laguerre généralisé de degré est -irréductible et a pour groupe de Galois . Ensuite, nous étudions les spécialisations du polynôme modulaire (celui-ci s’annule en les -invariants des paires de courbes elliptiques reliées entre elles par une -isogénie cyclique). Nous montrons que pour tout , à l’exception d’un nombre fini de cas, les -specialisations de sont -irréductibles et ont un groupe de Galois contenant . Enfin, nous obtenons que pour un revêtement simple de degré et de genre au moins , à l’exception d’un nombre fini de cas, les -spécialisations de sont -irréductibles et ont pour groupe de Galois .
Keywords: Branched cover, complex multiplication, Hilbert irreducibility, modular equation, orthogonal polynomial, rational point, Riemann-Hurwitz formula, simple cover, specialization
Mot clés : revêtement ramifié, multiplication complexe, théorème d’irréductibilité d’Hilbert, équation modulaire, polynômes orthogonaux, point rationnel, formule de Riemann-Hurwitz, revêtement simple, spécialisation
@article{AIF_2006__56_4_1127_0, author = {Hajir, Farshid and Wong, Siman}, title = {Specializations of one-parameter families of polynomials}, journal = {Annales de l'Institut Fourier}, pages = {1127--1163}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {4}, year = {2006}, doi = {10.5802/aif.2208}, zbl = {1160.12004}, mrnumber = {2266886}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2208/} }
TY - JOUR AU - Hajir, Farshid AU - Wong, Siman TI - Specializations of one-parameter families of polynomials JO - Annales de l'Institut Fourier PY - 2006 SP - 1127 EP - 1163 VL - 56 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2208/ DO - 10.5802/aif.2208 LA - en ID - AIF_2006__56_4_1127_0 ER -
%0 Journal Article %A Hajir, Farshid %A Wong, Siman %T Specializations of one-parameter families of polynomials %J Annales de l'Institut Fourier %D 2006 %P 1127-1163 %V 56 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2208/ %R 10.5802/aif.2208 %G en %F AIF_2006__56_4_1127_0
Hajir, Farshid; Wong, Siman. Specializations of one-parameter families of polynomials. Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 1127-1163. doi : 10.5802/aif.2208. http://archive.numdam.org/articles/10.5802/aif.2208/
[1] The gamma function, Holt, Rinehart and Winston, 1964 | MR | Zbl
[2] The transitive subgroups of degree up to eleven, Comm. in Algebra, Volume 11 (1983), pp. 863-911 | DOI | MR | Zbl
[3] Atlas of finite groups : maximal subgroups and ordinary characters for simple groups, Oxford, 1985 | MR | Zbl
[4] Primes of the form , Wiley, 1989 | MR | Zbl
[5] Congruence subgroups of of genus less than or equal to , Exper. Math., Volume 12 (2003), pp. 243-255 | MR | Zbl
[6] Integral specialization of families of rational functions, Pacific J. Math., Volume 190 (1999), pp. 45-85 | DOI | MR | Zbl
[7] The genus of subfields of , Proc. AMS, Volume 51 (1975), pp. 282-288 | MR | Zbl
[8] Permutation groups, Springer-Verlag, 1996 | MR | Zbl
[9] Solving solvable quintics, Math. Comp., Volume 57 (1991), pp. 387-401 | DOI | MR | Zbl
[10] and are Galois groups over number fields, J. Algebra, Volume 104 (1986), pp. 231-260 | DOI | MR | Zbl
[11] On the irreducibility of the generalized Laguerre polynomials, Acta Arith., Volume 105 (2002), pp. 177-182 | DOI | MR | Zbl
[12] The irreducibility of the Bessel polynomials, J. reine angew. Math., Volume 550 (2002), pp. 125-140 | DOI | MR | Zbl
[13] On Hilbert’s irreducibility theorem, J. Number Theory, Volume 6 (1974), pp. 211-231 | DOI | MR | Zbl
[14] Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. Math., Volume 90 (1969), pp. 542-575 | DOI | MR | Zbl
[15] Some generalized Laguerre polynomials whose Galois groups are the alternating groups, J. Number Theory, Volume 31 (1989), pp. 201-207 | DOI | MR | Zbl
[16] Algebraic properties of a family of generalized Laguerre polynomials (Preprint, 17 p)
[17] Some -extensions obtained from generalized Laguerre polynomials, J. Number Theory, Volume 50 (1995), pp. 206-212 | DOI | MR | Zbl
[18] The theory of groups, Macmillan, 1959 | MR | Zbl
[19] Finite groups III, Springer-Verlag, 1982 | MR | Zbl
[20] Congruence subgroups of positive genus of the modular group, Ill. J. Math., Volume 9 (1965), pp. 577-583 | MR | Zbl
[21] Fundamentals of Diophantine Geometry, Springer-Verlag, 1983 | MR | Zbl
[22] Elliptic functions, 2nd ed, Springer-Verlag, 1987 | MR | Zbl
[23] Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. London Math. Soc., Volume 63 (1991), pp. 266-314 | DOI | MR | Zbl
[24] Extensions of the rationals with Galois group , Bull. London Math. Soc., Volume 1 (1969), pp. 332-338 | DOI | MR | Zbl
[25] Finiteness results for Hilbert’s irreducibility theorem, Ann. Inst. Fourier, Volume 52 (2002), pp. 983-1015 | DOI | Numdam | MR | Zbl
[26] Number theory in function fields, Springer-Verlag, 2002 | MR | Zbl
[27] Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen. II, Sitzungsberichte der Berliner Akademie (1929), pp. 370-391
[28] Gleichungen ohne Affekt, Sitzungsberichte der Berliner Akademie (1930), pp. 443-449
[29] Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome, J. reine angew. Math., Volume 165 (1931), pp. 52-58 | DOI | Zbl
[30] On a family of generalized Laguerre polynomials (To appear in J. Number Theory, 13 p) | MR | Zbl
[31] Lectures on the Mordell-Weil theorem, 2nd ed, Vieweg, 1990 | MR | Zbl
[32] Topics in Galois theory, Jones and Bartlett Publ., 1992 | MR | Zbl
[33] The arithmetic of elliptic curves, Springer-Verlag, 1986 | MR | Zbl
[34] Groups as Galois groups : an introduction, Cambridge University Press, 1996 | MR | Zbl
[35] Finite permutation groups, Academic Press, 1964 | MR | Zbl
Cited by Sources: