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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-2-5,
author = {Wade Hindes},
title = {Points on elliptic curves parametrizing dynamical Galois groups},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {149-167},
zbl = {1296.14017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-2-5}
}
Wade Hindes. Points on elliptic curves parametrizing dynamical Galois groups. Acta Arithmetica, Tome 161 (2013) pp. 149-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-2-5/