Specializations of one-parameter families of polynomials
[Spécialisation des familles à un paramètre de polynômes]
Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 1127-1163.

Soient K un corps de nombres et λ(x,t)K[x,t] un polynôme irréductible sur K(t). À partir de la géométrie algébrique et de la théorie des groupes, nous donnons des conditions suffisantes pour que l’ensemble K-exceptionnel de λ, c’est-à-dire l’ensemble des éléments α de K tels que λ(x,α) est réductible sur K, soit fini. Nos méthodes nous permettent alors de développer trois applications. Tout d’abord, nous obtenons que pour tout entier n plus grand que 10, à l’exception d’un nombre fini de cas, la K-spécialisation du polynôme de Laguerre généralisé L n (t) (x) de degré n est K-irréductible et a pour groupe de Galois S n . Ensuite, nous étudions les spécialisations du polynôme modulaire Φ n (x,t) (celui-ci s’annule en les j-invariants des paires de courbes elliptiques reliées entre elles par une n-isogénie cyclique). Nous montrons que pour tout n53, à l’exception d’un nombre fini de cas, les K-specialisations de Φ n (x,t) sont K-irréductibles et ont un groupe de Galois contenant SL 2 (/n)/{±I}. Enfin, nous obtenons que pour un revêtement simple π:Y K 1 de degré n7 et de genre au moins 2, à l’exception d’un nombre fini de cas, les K-spécialisations de π sont K-irréductibles et ont pour groupe de Galois S n .

Let K be a number field, and suppose λ(x,t)K[x,t] is irreducible over K(t). Using algebraic geometry and group theory, we describe conditions under which the K-exceptional set of λ, i.e. the set of αK for which the specialized polynomial λ(x,α) is K-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed n10, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L n (t) (x) are K-irreducible and have Galois group S n . Second, we study specializations of the modular polynomial Φ n (x,t) (which vanishes on the j-invariants of pairs of elliptic curves related by a cyclic n-isogeny), and show that for any n53, all but finitely many of the K-specializations of Φ n (x,t) are K-irreducible and have Galois group containing SL 2 (/n)/{±I}. Third, for a simple branched cover π:Y K 1 of degree n7 and of genus at least 2, all but finitely many K-specializations are K-irreducible and have Galois group S n .

DOI : 10.5802/aif.2208
Classification : 12H25, 11C08, 11G15, 11R09, 14H25, 33C45
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
Hajir, Farshid 1 ; Wong, Siman 1

1 University of Massachusetts Department of Mathematics & Statistics Amherst, MA 01003-9318 (USA)
@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://www.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://www.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://www.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, Tome 56 (2006) no. 4, pp. 1127-1163. doi : 10.5802/aif.2208. http://www.numdam.org/articles/10.5802/aif.2208/

[1] Artin, E. The gamma function, Holt, Rinehart and Winston, 1964 | MR | Zbl

[2] Butler, G.; McKay, J. The transitive subgroups of degree up to eleven, Comm. in Algebra, Volume 11 (1983), pp. 863-911 | DOI | MR | Zbl

[3] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. Atlas of finite groups : maximal subgroups and ordinary characters for simple groups, Oxford, 1985 | MR | Zbl

[4] Cox, D. Primes of the form x 2 +ny 2 , Wiley, 1989 | MR | Zbl

[5] Cummins, C. J.; Pauli, S. Congruence subgroups of PSL(2,) of genus less than or equal to 24, Exper. Math., Volume 12 (2003), pp. 243-255 | MR | Zbl

[6] Dèbes, P.; Fried, M. D. Integral specialization of families of rational functions, Pacific J. Math., Volume 190 (1999), pp. 45-85 | DOI | MR | Zbl

[7] Dennin, Jr., J. B. The genus of subfields of K(n), Proc. AMS, Volume 51 (1975), pp. 282-288 | MR | Zbl

[8] Dixon, J. D.; Mortimer, B. Permutation groups, Springer-Verlag, 1996 | MR | Zbl

[9] Dummit, D. Solving solvable quintics, Math. Comp., Volume 57 (1991), pp. 387-401 | DOI | MR | Zbl

[10] Feit, W. A ˜ 5 and A ˜ 7 are Galois groups over number fields, J. Algebra, Volume 104 (1986), pp. 231-260 | DOI | MR | Zbl

[11] Filaseta, M.; Lam, T. Y. On the irreducibility of the generalized Laguerre polynomials, Acta Arith., Volume 105 (2002), pp. 177-182 | DOI | MR | Zbl

[12] Filaseta, M.; Trifonov, O. The irreducibility of the Bessel polynomials, J. reine angew. Math., Volume 550 (2002), pp. 125-140 | DOI | MR | Zbl

[13] Fried, M. On Hilbert’s irreducibility theorem, J. Number Theory, Volume 6 (1974), pp. 211-231 | DOI | MR | Zbl

[14] Fulton, W. Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. Math., Volume 90 (1969), pp. 542-575 | DOI | MR | Zbl

[15] Gow, R. Some generalized Laguerre polynomials whose Galois groups are the alternating groups, J. Number Theory, Volume 31 (1989), pp. 201-207 | DOI | MR | Zbl

[16] Hajir, F. Algebraic properties of a family of generalized Laguerre polynomials (Preprint, 17 p)

[17] Hajir, F. Some A ˜ n -extensions obtained from generalized Laguerre polynomials, J. Number Theory, Volume 50 (1995), pp. 206-212 | DOI | MR | Zbl

[18] Hall, M. The theory of groups, Macmillan, 1959 | MR | Zbl

[19] Huppert, N.; Blackburn, N. Finite groups III, Springer-Verlag, 1982 | MR | Zbl

[20] Knopp, M. I.; Newman, M. Congruence subgroups of positive genus of the modular group, Ill. J. Math., Volume 9 (1965), pp. 577-583 | MR | Zbl

[21] Lang, S. Fundamentals of Diophantine Geometry, Springer-Verlag, 1983 | MR | Zbl

[22] Lang, S. Elliptic functions, 2nd ed, Springer-Verlag, 1987 | MR | Zbl

[23] Liebeck, M. W.; Saxl, J. 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] Macbeath, A. M. Extensions of the rationals with Galois group PGL(2, n ), Bull. London Math. Soc., Volume 1 (1969), pp. 332-338 | DOI | MR | Zbl

[25] Müller, P. Finiteness results for Hilbert’s irreducibility theorem, Ann. Inst. Fourier, Volume 52 (2002), pp. 983-1015 | DOI | Numdam | MR | Zbl

[26] Rosen, M. Number theory in function fields, Springer-Verlag, 2002 | MR | Zbl

[27] Schur, I. Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen. II, Sitzungsberichte der Berliner Akademie (1929), pp. 370-391

[28] Schur, I. Gleichungen ohne Affekt, Sitzungsberichte der Berliner Akademie (1930), pp. 443-449

[29] Schur, I. Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome, J. reine angew. Math., Volume 165 (1931), pp. 52-58 | DOI | Zbl

[30] Sell, E. On a family of generalized Laguerre polynomials (To appear in J. Number Theory, 13 p) | MR | Zbl

[31] Serre, J.-P. Lectures on the Mordell-Weil theorem, 2nd ed, Vieweg, 1990 | MR | Zbl

[32] Serre, J.-P. Topics in Galois theory, Jones and Bartlett Publ., 1992 | MR | Zbl

[33] Silverman, J. H. The arithmetic of elliptic curves, Springer-Verlag, 1986 | MR | Zbl

[34] Volklein, H. Groups as Galois groups : an introduction, Cambridge University Press, 1996 | MR | Zbl

[35] Wielandt, H. Finite permutation groups, Academic Press, 1964 | MR | Zbl

Cité par Sources :