Rational points of exact period three for the quadratic family

Abstract

For the quadratic family $f_c(x) = x^2 + c$ over $\Q$, we isolate a general $S$-unit and ABC template for rational periodic cycles: under the ABC conjecture, any rational quantity built from the consecutive-point sums of a rational period-$n$ cycle whose value and shift-by-$1$ are both $S$-units (with $S$ the support of the denominator of $c$) has logarithmic height at most $(1+\eps)\sum_{p \in S} \log p + O_\eps(1)$. Specializing to period $3$ in the change of coordinates $u = x_0 + x_1$ and $v = x_0 + x_2$, the period-$3$ condition reduces to $v = -1/(1+u)$, forcing the parameter $t = u$ to satisfy $t, t+1 \in \Z[S^{-1}]^\times$; the template then yields, under ABC, $\max\{|a|,|b|,|a+b|\} \le C_\eps \rad(d)^{1+\eps}$ for $t = a/b$ in lowest terms. We also characterize the period-$3$ locus on the $(u,v)$-plane intrinsically, isolating the ghost curve. Extending the height bound to higher periods remains an open structural question.