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- W4300008286 abstract "We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. 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 for the Galois groups of the fourth iterate of quadratic polynomials $x^2+c$, using techniques in the theory of rational points on curves. Moreover, we show that the Hall-Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when $f(x)=x^2+3$. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank." @default.
- W4300008286 created "2022-10-03" @default.
- W4300008286 creator A5028002831 @default.
- W4300008286 date "2013-05-01" @default.
- W4300008286 modified "2023-10-14" @default.
- W4300008286 title "The Arithmetic of Curves Defined by Iteration" @default.
- W4300008286 doi "https://doi.org/10.48550/arxiv.1305.0222" @default.
- W4300008286 hasPublicationYear "2013" @default.
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