Points on elliptic curves parametrizing dynamical Galois groups

Wade Hindes. "Points on elliptic curves parametrizing dynamical Galois groups." Acta Arithmetica 159.2 (2013): 149-167. <http://eudml.org/doc/279622>.

@article{WadeHindes2013,
abstract = {We show how rational points on certain varieties parametrize phenomena arising in the Galois theory of iterates of quadratic polynomials. As an example, we characterize completely the set of quadratic polynomials x²+c whose third iterate has a "small" Galois group by determining the rational points on some elliptic curves. It follows as a corollary that the only integer value with this property is c=3, answering a question of Rafe Jones. Furthermore, using a result of Granville's on the rational points on quadratic twists of a hyperelliptic curve, we indicate how the ABC conjecture implies a finite index result, suggesting a geometric interpretation of this problem.},
author = {Wade Hindes},
journal = {Acta Arithmetica},
keywords = {arithmetic dynamics; iterated polynomials; Galois group},
language = {eng},
number = {2},
pages = {149-167},
title = {Points on elliptic curves parametrizing dynamical Galois groups},
url = {http://eudml.org/doc/279622},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Wade Hindes
TI - Points on elliptic curves parametrizing dynamical Galois groups
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 2
SP - 149
EP - 167
AB - We show how rational points on certain varieties parametrize phenomena arising in the Galois theory of iterates of quadratic polynomials. As an example, we characterize completely the set of quadratic polynomials x²+c whose third iterate has a "small" Galois group by determining the rational points on some elliptic curves. It follows as a corollary that the only integer value with this property is c=3, answering a question of Rafe Jones. Furthermore, using a result of Granville's on the rational points on quadratic twists of a hyperelliptic curve, we indicate how the ABC conjecture implies a finite index result, suggesting a geometric interpretation of this problem.
LA - eng
KW - arithmetic dynamics; iterated polynomials; Galois group
UR - http://eudml.org/doc/279622
ER -