Sparsity of rational points on torsion level covers of Hilbert modular varieties
Abstract
Let F be a totally real field of degree n and discriminant ΔF, and let X1(η) be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by 𝒪F, together with a torsion point having annihilator η. Let L=KX̅1(η)+D be the log-canonical bundle on a smooth toroidal compactification, and let HL be an associated multiplicative height. We prove that rational points on X1(η) become sparser as |Nm(η)|→∞, with (η,ΔF)=1. More precisely, for every number field K, set Nη,K(B)=#{x∈ X1(η)(K):HL(x)≤ B}. If |Nm(η)|≥ 5n and (η,ΔF)=1, we prove limsupB→∞logmax{1,Nη,K(B)}/(log B) ≤δη,K,n, δη,K,n≪[K:ℚ],n|Nm(η)|−1/(2n). In particular, δη,K,n→0 uniformly when n and [K:ℚ] are bounded and |Nm(η)|→∞. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of X1(η). Combining this estimate with recent progress derived from determinant-method, due to Ellenberg--Lawrence--Venkatesh and Brunebarbe--Maculan, we obtain the sparsity result above. We also prove that, for sufficiently large |Nm(η)|, every subvariety of X1(η) is of general type, and establish a higher-dimensional generalization of the geometric torsion theorem of Bakker--Tsimerman. Namely, for a family of abelian varieties with real multiplication over a quasi-projective base of arbitrary dimension, we bound the torsion subgroup of its Mordell--Weil group in terms of the canonical volume of the base, uniformly in the totally real multiplication field of fixed degree.
Related
- Completed cohomology of Hilbert modular varieties below middle degree
- Plectic Lie Algebra Action on Hilbert modular varieties
- Uniformity in rational torsion and small points on abelian varieties
- Rational curves on toroidal compactifications of Picard modular varieties
- Asymptotics of holomorphic torsion on orbifolds and an equivariant arithmetic Hilbert-Samuel theorem
BibTeX
@misc{memariansorkhabi2026sparsity,
title = {{Sparsity of rational points on torsion level covers of Hilbert modular varieties}},
author = {Soheil Memariansorkhabi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30033},
eprint = {2609.30033},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30033},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sparsity of rational points on torsion level covers of Hilbert modular varieties
AU - Soheil Memariansorkhabi
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30033
UR - https://arxiv.org/abs/2609.30033
AB - Let F be a totally real field of degree n and discriminant Δ_F, and let X_1(η) be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by 𝒪_F, together with a torsion point having annihilator η. Let L=K_X̅_1(η)+D be the log-canonical bundle on a smooth toroidal compactification, and let H_L be an associated multiplicative height. We prove that rational points on X_1(η) become sparser as |Nm(η)|→∞, with (η,Δ_F)=1. More precisely, for every number field K, set N_η,K(B)=#{x∈ X_1(η)(K):H_L(x)≤ B}. If |Nm(η)|≥ 5^n and (η,Δ_F)=1, we prove limsup_B→∞logmax{1,N_η,K(B)}/(log B) ≤δ_η,K,n, δ_η,K,n≪_[K:ℚ],n|Nm(η)|^−1/(2n). In particular, δ_η,K,n→0 uniformly when n and [K:ℚ] are bounded and |Nm(η)|→∞. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of X_1(η). Combining this estimate with recent progress derived from determinant-method, due to Ellenberg--Lawrence--Venkatesh and Brunebarbe--Maculan, we obtain the sparsity result above. We also prove that, for sufficiently large |Nm(η)|, every subvariety of X_1(η) is of general type, and establish a higher-dimensional generalization of the geometric torsion theorem of Bakker--Tsimerman. Namely, for a family of abelian varieties with real multiplication over a quasi-projective base of arbitrary dimension, we bound the torsion subgroup of its Mordell--Weil group in terms of the canonical volume of the base, uniformly in the totally real multiplication field of fixed degree.
ER -