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Moments of one-level
densities in families of holomorphic cusp forms in the level
aspect
Peter Cohen, Justine Dell, Oscar E. González, Simran Khunger, Chung-Hang Kwan, Steven J. Miller, Alexander Shashkov, Alicia Reina Smith, Carsten Sprunger, Nicholas Triantafillou, Nhi Truong, Roger Van Peski and Stephen Willis |
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Vol. 20 (2026), No. 2, 237–298
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Abstract |
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We study the -th centered moments of the -level density for the low-lying zeros of -functions attached to holomorphic cuspidal newforms of large prime level and fixed weight. Assuming the generalized Riemann hypotheses, we compute this statistic for any and for all test functions whose Fourier transforms are supported in . This is believed to be the natural limit of the current technology. Our work significantly extends beyond the trivial range and surpasses the previous record of whenever . The Katz–Sarnak philosophy predicts that the aforementioned statistic can be modeled by the corresponding statistic for the eigenvalues of random orthogonal matrices. We prove that this is the case for test functions with Fourier support contained in . The main technical innovation is a tractable vantage to evaluate the combinatorial zoo of terms, similar to the work of Conrey, Snaith and Mason. As an application, our work provides better bounds on the order of vanishing at the central point for the -functions in our family. |
Keywords
low-lying zeros, optimal test function, $n$-level density,
Katz–Sarnak conjecture
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Mathematical Subject Classification
Primary: 11M26
Secondary: 11M41, 15A42
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Milestones
Received: 21 October 2022
Revised: 19 July 2024
Accepted: 3 March 2025
Published: 16 February 2026
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Authors |
| Peter Cohen | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 United States |
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| Justine Dell | |
| Department of Mathematics UCSD 9500 Gillman Drive San Diego, CA 92093 United States |
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| Oscar E. González | |
| Department of Mathematics University of Puerto Rico Box 70377 San Juan 00931 Puerto Rico |
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| Simran Khunger | |
| Department of Mathematics University of Michigan 530 Church St Ann Arbor, MI 48109 United States |
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| Chung-Hang Kwan | |
| Institute for Advanced Study Princeton, NJ United States |
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| Steven J. Miller | |
| Department of Mathematics and
Statistics Williams College 18 Hoxsey St Williamstown, MA 01267 United States |
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| Alexander Shashkov | |
| Department of Mathematics University of California, Berkeley Berkeley, CA United States |
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| Alicia Reina Smith | |
| Department of Mathematics Williams College Williamstown, MA United States |
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| Carsten Sprunger | |
| Department of Mathematics University of Michigan Ann Arbor, MI United States |
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| Nicholas Triantafillou | |
| IDA Center for Communications
Research Princeton, NJ United States |
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| Nhi Truong | |
| Roger Van Peski | |
| Department of Mathematics Columbia University New York, NY United States |
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| Stephen Willis | |
| Department of Mathematics Williams College Williamstown, MA United States |
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| © 2026 MSP (Mathematical Sciences Publishers). |
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