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A Bogomolov property for moduli spaces of polynomials over abelian extensions
Abstract
Let d≥2 be an integer, and let MPolyd be the moduli space of degree-d polynomials. For every number field K, we prove that there exists εK,d>0 such that {α∈MPolyd(Kab): hcrit(α)<εK,d}={α∈MPolyd(Kab): hcrit(α)=0} is finite, where hcrit is the critical height. In particular, only finitely many Kab-rational points of MPolyd are postcritically finite. The proof combines the correspondence method of Ji--Song--Xie with local ramification estimates and the uniform finiteness of intertwined classes in MPolyd. We also prove an analogous finiteness theorem for non-preperiodic marked points in non-isotrivial one-parameter polynomial families.
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BibTeX
@misc{zhang2026bogomolov,
title = {{A Bogomolov property for moduli spaces of polynomials over abelian extensions}},
author = {Geng-Rui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06382},
eprint = {2610.06382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06382},
}
RIS
TY - UNPB
TI - A Bogomolov property for moduli spaces of polynomials over abelian extensions
AU - Geng-Rui Zhang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.06382
UR - https://arxiv.org/abs/2610.06382
AB - Let d≥2 be an integer, and let MPoly^d be the moduli space of degree-d polynomials. For every number field K, we prove that there exists ε_K,d>0 such that {α∈MPoly^d(K^ab): h_crit(α)<ε_K,d}={α∈MPoly^d(K^ab): h_crit(α)=0} is finite, where h_crit is the critical height. In particular, only finitely many K^ab-rational points of MPoly^d are postcritically finite. The proof combines the correspondence method of Ji--Song--Xie with local ramification estimates and the uniform finiteness of intertwined classes in MPoly^d. We also prove an analogous finiteness theorem for non-preperiodic marked points in non-isotrivial one-parameter polynomial families.
ER -