Points on elliptic curves parametrizing dynamical Galois groups

Wade Hindes¹

¹ Department of Mathematics Brown University Providence, RI 02912, U.S.A.

Acta Arithmetica, Tome 159 (2013) no. 2, pp. 149-167.

Résumé

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 $x^2+c$ 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.

Détail

DOI : 10.4064/aa159-2-5

Keywords: rational points certain varieties parametrize phenomena arising galois theory iterates quadratic polynomials example characterize completely set quadratic polynomials whose third iterate has small galois group determining rational points elliptic curves follows corollary only integer value property answering question rafe jones furthermore using result granvilles rational points quadratic twists hyperelliptic curve indicate abc conjecture implies finite index result suggesting geometric interpretation problem