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 .
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 .
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://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/
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