Extending the unconditional support in an Iwaniec–Luo–Sarnak family

Devin, Lucile; Fiorilli, Daniel; Södergren, Anders

doi:10.2140/ant.2025.19.1621

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Extending the unconditional support in an Iwaniec–Luo–Sarnak family

Lucile Devin, Daniel Fiorilli and Anders Södergren

Vol. 19 (2025), No. 8, 1621–1635

DOI: 10.2140/ant.2025.19.1621

Abstract

We study the harmonically weighted one-level density of low-lying zeros of $L -$ functions in the family of holomorphic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz–Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(- 3 ∕ 2, 3 ∕ 2)$ . This result was improved by Ricotta–Royer, who increased the admissible support for $k \geq 4$ in a way that is asymptotically as good as the best known GRH result. We extend the admissible support for all $k \geq 2$ to $(- Θ_{k}, Θ_{k})$ , where $Θ_{2} = 1.866 \dots$ and $Θ_{k}$ tends monotonically to $2$ asymptotically five times faster than what was previously known. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$ -functions.

Keywords

low-lying zeros, Katz–Sarnak heuristics, zero-density estimates, holomorphic modular forms

Mathematical Subject Classification

Primary: 11F11, 11M26, 11M41

Secondary: 11M50

Milestones

Received: 26 July 2023

Revised: 5 March 2024

Accepted: 3 September 2024

Published: 12 June 2025

Authors

Lucile Devin
Université du Littoral Côte d’Opale, UR 2597 LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville Calais France

Daniel Fiorilli
Université Paris-Saclay CNRS, Laboratoire de mathématiques d’Orsay Orsay France

Anders Södergren
Department of Mathematical Sciences Chalmers University of Technology and the University of Gothenburg Gothenburg Sweden

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Vol. 19, No. 8, 2025