The Arithmetic of Curves Defined by Iteration

TL;DR

This paper explores the connection between the Galois groups of iterated quadratic polynomials and rational points on certain algebraic curves, providing new insights and applications in arithmetic dynamics.

Contribution

It introduces a geometric framework linking polynomial iteration to rational points on curves, establishing new theorems and conjectural results in dynamical Galois theory.

Findings

Maximality theorem for the Galois group of the fourth iterate of quadratic polynomials

Connection between the Hall-Lang conjecture and finite index results in dynamical Galois groups

Construction of a family of curves with explicitly determined rational points and special Jacobian properties

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.