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Murmurations of modular forms in the weight aspect

Jonathan Bober, Andrew R. Booker, Min Lee, David Lowry-Duda

TL;DR

This work proves murmurations in the weight aspect for level-1 holomorphic modular forms as weight $k\to\infty$, revealing a correlation between root numbers and Hecke eigenvalues at primes with $p/N$ in a fixed interval $E$, where $N$ is the analytic conductor $\mathcal N(k)$. The authors combine the Eichler–Selberg trace formula, a smooth partition of the weight sum, Poisson summation, and the circle method to isolate a main term governed by an explicit density $\nu(E)$, with error terms controlled under GRH. A key innovation is replacing sums over primes by an averaged Dirichlet-value $L$-function $L(1,\widetilde{\psi}_t)$ via a multiplicative-average mechanism, enabling a tractable circle-method analysis. The main term exhibits a non-absolutely continuous distribution with infinitely many point masses and a $1/\sqrt{N}$ scaling, providing a new archimedean analogue of murmurations and connecting to 1-level density phenomena. This extends prior fixed-weight murmurations to a family with varying archimedean parameter and suggests a broader joint-structure picture across L-function families.

Abstract

We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k\to\infty$, that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family.

Murmurations of modular forms in the weight aspect

TL;DR

This work proves murmurations in the weight aspect for level-1 holomorphic modular forms as weight , revealing a correlation between root numbers and Hecke eigenvalues at primes with in a fixed interval , where is the analytic conductor . The authors combine the Eichler–Selberg trace formula, a smooth partition of the weight sum, Poisson summation, and the circle method to isolate a main term governed by an explicit density , with error terms controlled under GRH. A key innovation is replacing sums over primes by an averaged Dirichlet-value -function via a multiplicative-average mechanism, enabling a tractable circle-method analysis. The main term exhibits a non-absolutely continuous distribution with infinitely many point masses and a scaling, providing a new archimedean analogue of murmurations and connecting to 1-level density phenomena. This extends prior fixed-weight murmurations to a family with varying archimedean parameter and suggests a broader joint-structure picture across L-function families.

Abstract

We prove the existence of "murmurations" in the family of holomorphic modular forms of level and weight , that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family.
Paper Structure (8 sections, 10 theorems, 193 equations, 3 figures)

This paper contains 8 sections, 10 theorems, 193 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The trace formula and overview of the proof
  3. Overview of the proof of Theorem \ref{['thm:main']}
  4. The sum over weights
  5. The sum over primes
  6. The exponential sum
  7. The Fourier series
  8. The circle method

Key Result

Theorem 1.1

Assume GRH for the $L$-functions of Dirichlet characters and modular forms. Fix $\varepsilon\in(0,\frac{1}{12})$, $\delta\in\{0,1\}$, and a compact interval $E\subset\mathbb{R}_{>0}$ with $|E|>0$. Let $K,H\in\mathbb{R}_{>0}$ with $K^{\frac{5}{6}+\varepsilon}<H<K^{1-\varepsilon}$, and set $N=\mathcal where and the $\ast$ indicates that terms occurring at the endpoints of $E$ are halved.

Figures (3)

  • Figure 1: top: Average of $a_p$ over isogeny classes of elliptic curves of conductor in $[7500,10000]$ and fixed rank (blue = rank 0, red = rank 1), for primes $p\in[2,7919]$ (reprinted from HLOP with permission from the authors). bottom: Starling shapes in the evening sky. cc-by-sa/2.0 -- © Walter Baxter -- https://www.geograph.org.uk/photo/1065181
  • Figure 2: A comparison of $M_2(y)$ for $y\in[0,2]$ (green) and the points $\bigl(p/2^{18},r(p)\bigr)$ for primes $p<2^{19}$ (purple), where $r(p)=\frac{\sum_{N\in I}\sum_{f\in H_2(N)}\epsilon_f\lambda_f(p)\sqrt{p}}{\sum_{N\in I}\sum_{f\in H_2(N)}1}$, and $I$ is the set of all squarefree integers in the range $[2^{18} \pm 2^{10}]$ (data computed by Andrew Sutherland).
  • Figure 3: A comparison of $(-1)^\delta\nu((0,t])$ and the left-hand side of \ref{['e:main_ratio']} scaled by $t\sqrt{N}$, for $K=3850$, $H=100$, and $t \in [0,2].$

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 8 more