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The arithmetic of curves defined by iteration

Volume 169 / 2015

Wade Hindes Acta Arithmetica 169 (2015), 1-27 MSC: Primary 14G05; Secondary 37P55. DOI: 10.4064/aa169-1-1

Abstract

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 $C_n:y^2=f^n(x)$ and their quadratic twists (here $f^n$ denotes the $n$th 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 for the Galois groups of the fourth iterate of rational 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.

Authors

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

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