Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces
Abstract
For a fixed prime p, let μp denote the p-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) μp-equidistribution of the p-th Hecke eigenvalues over the Atkin--Lehner subspaces Skσ(N) ⊆ Sk(N) and Sknew, σ(N) ⊆ Sknew(N). We then highlight five applications of this explicit equidistribution result. For the first application, we generalize Kim's vertical analog of the Atkin--Serre conjecture to the Atkin--Lehner setting. For the second application, we obtain explicit bounds on the number of newforms f ∈ Sknew, σ(N) for which p is extremal over Sknew, σ(N). For the third application, we prove explicit asymptotics for the number of 𝔽pr-points on the modular Jacobian J0(N), as well as on its factors J0new(N), J0σ(N), and J0new, σ(N). We also make explicit an asymptotic result of Serre concerning point counts of the modular curves X0(N). For the fourth application, we generalize lower bounds due to Murty and Sinha on the sizes of large ℚ-simple factors of J0(N) to analogous bounds for J0σ(N). Finally, for the fifth application (the details of which are given in a separate paper), we use our explicit equidistribution result to prove that only finitely many modular Jacobians are supersingular modulo any fixed prime.
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BibTeX
@misc{kumar2026effective,
title = {{Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces}},
author = {Aarya J. Kumar and Sargam Mondal and Erick Ross and Hui Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17806},
eprint = {2609.17806},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17806},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces
AU - Aarya J. Kumar
AU - Sargam Mondal
AU - Erick Ross
AU - Hui Xue
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17806
UR - https://arxiv.org/abs/2609.17806
AB - For a fixed prime p, let μ_p denote the p-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) μ_p-equidistribution of the p-th Hecke eigenvalues over the Atkin--Lehner subspaces S_k^σ(N) ⊆ S_k(N) and S_k^new, σ(N) ⊆ S_k^new(N). We then highlight five applications of this explicit equidistribution result. For the first application, we generalize Kim's vertical analog of the Atkin--Serre conjecture to the Atkin--Lehner setting. For the second application, we obtain explicit bounds on the number of newforms f ∈ S_k^new, σ(N) for which p is extremal over S_k^new, σ(N). For the third application, we prove explicit asymptotics for the number of 𝔽_p^r-points on the modular Jacobian J_0(N), as well as on its factors J_0^new(N), J_0^σ(N), and J_0^new, σ(N). We also make explicit an asymptotic result of Serre concerning point counts of the modular curves X_0(N). For the fourth application, we generalize lower bounds due to Murty and Sinha on the sizes of large ℚ-simple factors of J_0(N) to analogous bounds for J_0^σ(N). Finally, for the fifth application (the details of which are given in a separate paper), we use our explicit equidistribution result to prove that only finitely many modular Jacobians are supersingular modulo any fixed prime.
ER -