Specializations of one-parameter families of polynomials

[Spécialisation des familles à un paramètre de polynômes]

Hajir, Farshid ; Wong, Siman

Annales de l'Institut Fourier, Tome 56 (2006), p. 1127-1163 / Harvested from Numdam

Résumé

Soient $K$ un corps de nombres et $ \lambda (x,t)\in 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 $\lambda $, c’est-à-dire l’ensemble des éléments $\alpha $ de $K$ tels que $\lambda (x,\alpha )$ 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 $\Phi _n(x,t)$ (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 $ n\ge 53 $, à l’exception d’un nombre fini de cas, les $K$-specialisations de $ \Phi _n(x, t) $ sont $K$-irréductibles et ont un groupe de Galois contenant ${\rm SL}_2(\mathbb{Z}/n)/\lbrace \pm I \rbrace $. Enfin, nous obtenons que pour un revêtement simple $\pi :Y\rightarrow \mathbb{P}_K^{1}$ de degré $n\ge 7$ et de genre au moins $2$, à l’exception d’un nombre fini de cas, les $K$-spécialisations de $\pi $ sont $K$-irréductibles et ont pour groupe de Galois $S_n$.

Abstract

Let $K$ be a number field, and suppose $ \lambda (x,t)\in K[x, t] $ is irreducible over $K(t)$. Using algebraic geometry and group theory, we describe conditions under which the $K$-exceptional set of $\lambda $, i.e. the set of $\alpha \in K$ for which the specialized polynomial $\lambda (x,\alpha )$ is $K$-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed $n\ge 10$, 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 $\Phi _n(x,t)$ (which vanishes on the $j$-invariants of pairs of elliptic curves related by a cyclic $n$-isogeny), and show that for any $ n\ge 53 $, all but finitely many of the $K$-specializations of $ \Phi _n(x, t) $ are $K$-irreducible and have Galois group containing ${\rm SL}_2(\mathbb{Z}/n)/\lbrace \pm I \rbrace $. Third, for a simple branched cover $\pi :Y\rightarrow \mathbb{P}_K^{1}$ of degree $n\ge 7$ and of genus at least $2$, all but finitely many $K$-specializations are $K$-irreducible and have Galois group $S_n$.

Détail

Publié le : 2006-01-01

MR 2266886 | Zbl 1160.12004

DOI : https://doi.org/10.5802/aif.2208
Classification:  12H25,  11C08,  11G15,  11R09,  14H25,  33C45
Mots clés: revêtement ramifié, multiplication complexe, théorème d’irréductibilité d’Hilbert, équation modulaire, polynômes orthogonaux, point rationnel, formule de Riemann-Hurwitz, revêtement simple, spécialisation

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