Specializations of one-parameter families of polynomials

Hajir, Farshid; Wong, Siman

Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 1127-1163.

Soient $K$ un corps de nombres et $\lambda (x,t)\in K[x,t]$ un polynôme irréductible sur $K\left(t\right)$. 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 $L_n^{\left(t\right)}\left(x\right)$ 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 ${\mathrm{SL}}_2(\mathbb{Z}/n)/\{\pm I\}$. Enfin, nous obtenons que pour un revêtement simple $\pi :Y\to {\mathbb{P}}_K^1$ 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 $S_n$.

Let $K$ be a number field, and suppose $\lambda (x,t)\in K[x,t]$ is irreducible over $K\left(t\right)$. 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 $L_n^{\left(t\right)}\left(x\right)$ 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 ${\mathrm{SL}}_2(\mathbb{Z}/n)/\{\pm I\}$. Third, for a simple branched cover $\pi :Y\to {\mathbb{P}}_K^1$ of degree $n\ge 7$ and of genus at least $2$, all but finitely many $K$-specializations are $K$-irreducible and have Galois group $S_n$.

[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 $x^2+n{y}^2$, Wiley, 1989 | MR | Zbl

[5] Cummins, C. J.; Pauli, S. Congruence subgroups of $PSL(2,\mathbb{Z})$ 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 $K\left(n\right)$, 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. ${\tilde{A}}_5$ and ${\tilde{A}}_7$ 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 ${\tilde{A}}_n$-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 $PGL(2,{\mathbb{Z}}_n)$, 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