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Points on Elliptic Curves Parametrizing Dynamical Galois Groups

Authors
  • Hindes, Wade
Type
Preprint
Publication Date
Sep 27, 2012
Submission Date
Aug 30, 2012
Identifiers
arXiv ID: 1208.6220
Source
arXiv
License
Yellow
External links

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 $x^2+c$ 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 such 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.

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