Specializations of one-parameter families of polynomials

Farshid Hajir[1]; Siman Wong[1]

  • [1] University of Massachusetts Department of Mathematics & Statistics Amherst, MA 01003-9318 (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 4, page 1127-1163
  • ISSN: 0373-0956

Abstract

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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 α ∈ K 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 n ≥ 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 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 n ≥ 53 , all but finitely many of the K -specializations of Φ n ( x , t ) are K -irreducible and have Galois group containing SL 2 ( ℤ / n ) / { ± I } . Third, for a simple branched cover π : Y → ℙ K 1 of degree n ≥ 7 and of genus at least 2 , all but finitely many K -specializations are  K -irreducible and have Galois group S n .

How to cite

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Hajir, Farshid, and Wong, Siman. "Specializations of one-parameter families of polynomials." Annales de l’institut Fourier 56.4 (2006): 1127-1163. <http://eudml.org/doc/10168>.

@article{Hajir2006,
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 $S_n$.},
affiliation = {University of Massachusetts Department of Mathematics & Statistics Amherst, MA 01003-9318 (USA); University of Massachusetts Department of Mathematics & Statistics Amherst, MA 01003-9318 (USA)},
author = {Hajir, Farshid, Wong, Siman},
journal = {Annales de l’institut Fourier},
keywords = {Branched cover; complex multiplication; Hilbert irreducibility; modular equation; orthogonal polynomial; rational point; Riemann-Hurwitz formula; simple cover; specialization; branched cover},
language = {eng},
number = {4},
pages = {1127-1163},
publisher = {Association des Annales de l’institut Fourier},
title = {Specializations of one-parameter families of polynomials},
url = {http://eudml.org/doc/10168},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Hajir, Farshid
AU - Wong, Siman
TI - Specializations of one-parameter families of polynomials
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 4
SP - 1127
EP - 1163
AB - 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 $S_n$.
LA - eng
KW - Branched cover; complex multiplication; Hilbert irreducibility; modular equation; orthogonal polynomial; rational point; Riemann-Hurwitz formula; simple cover; specialization; branched cover
UR - http://eudml.org/doc/10168
ER -

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