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.