Rational points of exact period three for the quadratic family
Abstract
For the quadratic family fc(x) = x2 + 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)∑p ∈ S log p + Oeps(1). Specializing to period $3$ in the change of coordinates u = x0 + x1 and v = x0 + x2, the period-$3$ condition reduces to v = −1/(1+u), forcing the parameter t = u to satisfy t, t+1 ∈ Z[S−1]×; the template then yields, under ABC, max{|a|,|b|,|a+b|} ≤ Ceps 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.
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BibTeX
@misc{kaenmaki2026rational,
title = {{Rational points of exact period three for the quadratic family}},
author = {Antti Käenmäki and Pekka Kosunen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39820},
eprint = {2610.39820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39820},
}
RIS
TY - UNPB
TI - Rational points of exact period three for the quadratic family
AU - Antti Käenmäki
AU - Pekka Kosunen
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39820
UR - https://arxiv.org/abs/2610.39820
AB - 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)∑_p ∈ 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 ∈ Z[S^−1]^×; the template then yields, under ABC, max{|a|,|b|,|a+b|} ≤ 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.
ER -