An application of "Selmer group Chabauty" to arithmetic dynamics

TL;DR

This paper applies the Selmer group Chabauty method to hyperelliptic curves to determine their rational points and explores implications for the irreducibility of polynomial iterates in arithmetic dynamics.

Contribution

It introduces a novel application of the Selmer group Chabauty technique to specific hyperelliptic curves and derives new irreducibility results for polynomial iterates.

Findings

Hyperelliptic curves of the form y^2 = x^N + h(x)^2 have only obvious rational points.

If f_c^{ ext{2}} is irreducible, then f_c^{ ext{6}} is also irreducible.

Under GRH, f_c^{ ext{10}} is irreducible.

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.