PreprintCC BY 4.0arXiv
Exceptional maps and abelian points in backward orbits
Abstract
Consider a number field K, a polarized endomorphism f:X→ X on a normal projective variety, and a non-exceptional point α for f. We show that the extension K(f−∞(α))/K generated by the backward orbit of α is virtually abelian if and only if f is an exceptional map with CM and α is f-preperiodic. This answers a conjecture of Andrews and Petsche and extends it to higher dimensions. To this end, we develop the theory of exceptional maps on normal projective varieties over ℂ and prove that commuting polarized endomorphisms of multiplicatively independent degrees are exceptional.
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BibTeX
@misc{ji2026exceptional,
title = {{Exceptional maps and abelian points in backward orbits}},
author = {Zhuchao Ji and Jiarui Song and Junyi Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24701},
eprint = {2609.24701},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24701},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exceptional maps and abelian points in backward orbits
AU - Zhuchao Ji
AU - Jiarui Song
AU - Junyi Xie
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24701
UR - https://arxiv.org/abs/2609.24701
AB - Consider a number field K, a polarized endomorphism f:X→ X on a normal projective variety, and a non-exceptional point α for f. We show that the extension K(f^−∞(α))/K generated by the backward orbit of α is virtually abelian if and only if f is an exceptional map with CM and α is f-preperiodic. This answers a conjecture of Andrews and Petsche and extends it to higher dimensions. To this end, we develop the theory of exceptional maps on normal projective varieties over ℂ and prove that commuting polarized endomorphisms of multiplicatively independent degrees are exceptional.
ER -