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A uniform effective André--Oort result
Abstract
We prove the André--Oort conjecture for hypersurfaces V ⊂ Y(1)n ≅ 𝔸nC defined by an equation a1 x1m + … + an xnm = b, where a1, …, an, b ∈ ℚ̅ and m ∈ ℤ>0. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients a1, …, an, b. This is the first effective proof of a uniform André--Oort statement for a class of subvarieties with arbitrary dimension and non-empty special locus. We also prove an analogous result for hypersurfaces V ⊂ Y(1)n × 𝔾ml.
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BibTeX
@misc{fowler2026uniform,
title = {{A uniform effective André--Oort result}},
author = {Guy Fowler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18934},
eprint = {2609.18934},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18934},
}
RIS
TY - UNPB
TI - A uniform effective André--Oort result
AU - Guy Fowler
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18934
UR - https://arxiv.org/abs/2609.18934
AB - We prove the André--Oort conjecture for hypersurfaces V ⊂ Y(1)^n ≅ 𝔸^n_C defined by an equation a_1 x_1^m + … + a_n x_n^m = b, where a_1, …, a_n, b ∈ ℚ̅ and m ∈ ℤ_>0. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients a_1, …, a_n, b. This is the first effective proof of a uniform André--Oort statement for a class of subvarieties with arbitrary dimension and non-empty special locus. We also prove an analogous result for hypersurfaces V ⊂ Y(1)^n × 𝔾_m^l.
ER -