An application of "Selmer group Chabauty" to arithmetic dynamics

Abstract

We describe how one can use the "Selmer group Chabauty" method developed by the author to show that certain hyperelliptic curves of the form $$ C \colon y^2 = x^N + h(x)^2 \,, $$ where $N = 2g + 1$ is odd, $h \in \mathbb{Z}[x]$ with $\operatorname{deg} h \le g$ and $h(0)$ odd, have only the "obvious" rational points $\infty$ (the unique point at infinity on the smooth projective model of the curve) and $(0, \pm h(0))$. As an application of the method, we prove the following result. Let $c \in \mathbb{Q}$ and write $f_c(x) = x^2 + c$. We denote the iterates of $f_c$ by $f_c^{\circ n}$; i.e., we set $f_c^{\circ 0}(x) = x$ and $f_c^{\circ(n+1)}(x) = f_c(f_c^{\circ n}(x))$. If $f_c^{\circ 2}$ is irreducible, then $f_c^{\circ 6}$ is also irreducible. Assuming the Generalized Riemann Hypothesis (GRH), it also follows that $f_c^{\circ 10}$ is irreducible.