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Points on elliptic curves parametrizing dynamical Galois groups

Volume 159 / 2013

Wade Hindes Acta Arithmetica 159 (2013), 149-167 MSC: Primary 14G05; Secondary 37P55. DOI: 10.4064/aa159-2-5


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.


  • Wade HindesDepartment of Mathematics
    Brown University
    Providence, RI 02912, U.S.A.

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