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On the 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 Smith Reina, Carsten Sprunger, Nicholas Triantafillou, Nhi Truong, Roger Van Peski, Stephen Willis

TL;DR

The paper proves that the $n$-th centered moment of the $1$-level density for holomorphic cusp forms in the level aspect, under GRH, matches the corresponding moment from the eigenvalue statistics of random orthogonal matrices for test functions with Fourier support in $(-\tfrac{2}{n}, \tfrac{2}{n})$. It achieves this by combining the explicit formula with the Petersson trace formula, and by a detailed combinatorial analysis that isolates the main-term contributions while showing most non-main terms vanish in the limit. The authors provide explicit integral representations and correction terms $\mathcal{R}(n,a;\phi)$, enabling precise comparison with random matrix theory and yielding sharper bounds on the order of vanishing at the central point for the associated $L$-functions. The work extends prior results that were limited to smaller supports, and furnishes new techniques for handling the combinatorics of higher moments in orthogonal families, with potential applications to vanishing bounds and related statistics in automorphic $L$-functions. Overall, the paper confirms Katz–Sarnak predictions in a nontrivial setting and strengthens the connection between number theory and random matrix theory in the level aspect.

Abstract

We study the $n^{\rm th}$ centered moments of the $1$-level density for the low-lying zeros of $L$-functions attached to holomorphic cuspidal newforms of large prime level and fixed weight. Assuming the Generalized Riemann Hypotheses, we compute this statistic for any $n\ge 1$ and for all test functions whose Fourier transforms are supported in $\left(-2/n, \, 2/n\right)$. This is believed to be the natural limit of the current technology. Our work significantly extends beyond the trivial range $(-1/n, \, 1/n)$ and surpasses the previous record of $(-1/(n-1),\, 1/(n-1))$ whenever $n>2$. 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 $(-2/n,\, 2/n)$. The main technical innovation is a tractable vantage to evaluate the combinatorial zoo of terms, similar to the work of Conrey-Snaith and Mason-Snaith. As an application, our work provides better bounds on the order of vanishing at the central point for the $L$-functions in our family.

On the moments of one-level densities in families of holomorphic cusp forms in the level aspect

TL;DR

The paper proves that the -th centered moment of the -level density for holomorphic cusp forms in the level aspect, under GRH, matches the corresponding moment from the eigenvalue statistics of random orthogonal matrices for test functions with Fourier support in . It achieves this by combining the explicit formula with the Petersson trace formula, and by a detailed combinatorial analysis that isolates the main-term contributions while showing most non-main terms vanish in the limit. The authors provide explicit integral representations and correction terms , enabling precise comparison with random matrix theory and yielding sharper bounds on the order of vanishing at the central point for the associated -functions. The work extends prior results that were limited to smaller supports, and furnishes new techniques for handling the combinatorics of higher moments in orthogonal families, with potential applications to vanishing bounds and related statistics in automorphic -functions. Overall, the paper confirms Katz–Sarnak predictions in a nontrivial setting and strengthens the connection between number theory and random matrix theory in the level aspect.

Abstract

We study the 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-Snaith. As an application, our work provides better bounds on the order of vanishing at the central point for the -functions in our family.
Paper Structure (38 sections, 57 theorems, 302 equations)

This paper contains 38 sections, 57 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Historical Perspectives
  3. Number theory setup
  4. Random matrix theory setup
  5. Main Results
  6. Applications
  7. Proof sketch and structure of the paper
  8. Preliminaries
  9. Notations and Conventions
  10. Analytic Preliminaries
  11. Automorphic Preliminaries
  12. Density and moment sums
  13. Proof of Theorem \ref{['thm:lfnsn-2']} assuming Propositions \ref{['lem:nonmaintermcalc']} and \ref{['lem:maintermcalc']}
  14. Combinatorial expansion and cleaning
  15. Cleaning with Weil's bound and Prime Number Theorem (PNT)
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $k, n\ge 2$ be positive integers with $k$ even. Assume GRH for $L(s, f)$ for all $f \in H_k^*(N)$, where $N$ is any prime or $N = 1$. Assume also RH for $\zeta(s)$ and GRH for all primitive Dirichlet $L$-functions. Then for $\phi \in \mathcal{S}_{ec}(\mathbb{R})$ with $\mathop{\mathrm{supp}}\nol The moments $\mathcal{D}^{\pm}_{n}(N; \phi)$ and $\mathcal{Z}_{n}(M; \phi)$ are defined in (defcent

Theorems & Definitions (120)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 110 more