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Points on elliptic curves parametrizing dynamical Galois groups. (English) Zbl 1296.14017
The Galois groups of the iterates of a polynomial are, typically, as large as they can be. For the quadratic family \(f(x)=x^2+c\) over the integers, congruence relations are known, which ensure that this will be the case. Little is known for rational parameter values.
The author considers the characterization of polynomials having iterates with “small” Galois groups. The symplest non-trivial case is the third iterate of a two-parameter quadratic family over the rationals (the second parameter being the position of the critical point). The author shows that the parameter values corresponding to a small Galois group can be obtained by determining the rational points on some elliptic surfaces, analysing in detail the case of the critical point at the origin. He also shows that for all but finitely many rational critical points, the set of rational values of \(c\) for which the Galois group of the third iterate is small is infinite.
Regarding the case of large Galois groups, the author indicates (using a result of Granville’s on the rational points on quadratic twists of a hyperelliptic curve) how, in the case of the quadratic family, the ABC conjecture implies a finite index result for the Galois groups of iterates of arbitrary order.

14G05 Rational points
37P55 Arithmetic dynamics on general algebraic varieties
Magma; SageMath
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