PreprintCC BY 4.0arXiv
Non-Archimedean Rigidity and the Uniform Dynamical Manin--Mumford for (ℙ1)n
Abstract
We prove a uniform form of the dynamical Manin-Mumford conjecture for split endomorphisms Φ= (f1,…,fn): (ℙ1)n → (ℙ1)n over ℂ, where each fi has degree d ≥ 2 and at most one fi is exceptional. Specifically, we bound the number of irreducible components of Prep(Φ)∩ V(ℂ)̅Zar uniformly, depending only on (n,d,deg V). We also obtain uniform statements for the dynamical Bogomolov conjecture. Our approach combines a degeneration argument via ultrafilters with a new non-archimedean rigidity theorem, along with the quantitative dynamical Zhang's inequality by Mavraki and the second author.
Related
BibTeX
@misc{yap2026nonarchimedean,
title = {{Non-Archimedean Rigidity and the Uniform Dynamical Manin--Mumford for $\left(\mathbbP^1\right)^n$}},
author = {Jit Wu Yap and Zhuchao Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40152},
eprint = {2610.40152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40152},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-Archimedean Rigidity and the Uniform Dynamical Manin--Mumford for (ℙ^1)^n
AU - Jit Wu Yap
AU - Zhuchao Ji
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40152
UR - https://arxiv.org/abs/2610.40152
AB - We prove a uniform form of the dynamical Manin-Mumford conjecture for split endomorphisms Φ= (f_1,…,f_n): (ℙ^1)^n → (ℙ^1)^n over ℂ, where each f_i has degree d ≥ 2 and at most one f_i is exceptional. Specifically, we bound the number of irreducible components of Prep(Φ)∩ V(ℂ)̅^Zar uniformly, depending only on (n,d,deg V). We also obtain uniform statements for the dynamical Bogomolov conjecture. Our approach combines a degeneration argument via ultrafilters with a new non-archimedean rigidity theorem, along with the quantitative dynamical Zhang's inequality by Mavraki and the second author.
ER -