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Points on elliptic curves parametrizing dynamical Galois groups

Wade Hindes — 2013

Acta Arithmetica

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...

The arithmetic of curves defined by iteration

Wade Hindes — 2015

Acta Arithmetica

We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...

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