Specializations of one-parameter families of polynomials
Annales de l'Institut Fourier, Volume 56 (2006) no. 4, p. 1127-1163
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 .
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 .
DOI : https://doi.org/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
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {4},
     year = {2006},
     pages = {1127-1163},
     doi = {10.5802/aif.2208},
     zbl = {1160.12004},
     mrnumber = {2266886},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2006__56_4_1127_0}
}
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. https://aif.centre-mersenne.org/item/AIF_2006__56_4_1127_0/

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