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Murmurations of Hecke $L$-Functions of Imaginary Quadratic Fields

Zeyu Wang

TL;DR

The paper establishes a Murmurations framework for the family of Hecke $L$-functions attached to imaginary quadratic fields, showing a universal murmuration density compatible with Katz-Sarnak 1-level density while revealing an almost periodic main term in $p$-dependent averages. By carefully degenerating the problem through orthogonality of characters, truncation of $L$-functions, and a square-free sieve, it derives explicit main-term contributions involving $c(p)$, $\delta_y(p)$, and $\vartheta(y)$, and proves that averaging over primes yields a density $M(\\Xi)$ that, when integrated against smooth weights $\\Phi$, reproduces the predicted limiting behavior $M_\\Phi(\\Xi) \to -\tfrac{1}{2}$ as $\\Xi \to \infty$. The results highlight an almost periodic dependence on the prime parameter, offering a refined description of murmuration without heavy prime averaging and illustrating a connection between murmuration densities and 1-level densities in symplectic symmetry. Numerics corroborate the analytic formulas, and the framework provides a blueprint for analyzing murmuration phenomena in other $L$-function families.

Abstract

We calculate the murmuration density for the family of Hecke $L$-functions of imaginary quadratic fields associated to non-trivial characters. This density exhibits a universality property like Zubrilina's density for the murmurations of holomorphic modular forms. We show all murmuration functions obtained by averaging over the family with a compactly supported smooth weight function has asymptotics compatible with the 1-level density conjecture of Katz and Sarnak. The novelty of the murmurations of this family of $L$-functions is its pronounced almost periodic feature, which allows one to describe this murmuration without averaging over primes, and which is non-existent or previously unnoticed for other families.

Murmurations of Hecke $L$-Functions of Imaginary Quadratic Fields

TL;DR

The paper establishes a Murmurations framework for the family of Hecke -functions attached to imaginary quadratic fields, showing a universal murmuration density compatible with Katz-Sarnak 1-level density while revealing an almost periodic main term in -dependent averages. By carefully degenerating the problem through orthogonality of characters, truncation of -functions, and a square-free sieve, it derives explicit main-term contributions involving , , and , and proves that averaging over primes yields a density that, when integrated against smooth weights , reproduces the predicted limiting behavior as . The results highlight an almost periodic dependence on the prime parameter, offering a refined description of murmuration without heavy prime averaging and illustrating a connection between murmuration densities and 1-level densities in symplectic symmetry. Numerics corroborate the analytic formulas, and the framework provides a blueprint for analyzing murmuration phenomena in other -function families.

Abstract

We calculate the murmuration density for the family of Hecke -functions of imaginary quadratic fields associated to non-trivial characters. This density exhibits a universality property like Zubrilina's density for the murmurations of holomorphic modular forms. We show all murmuration functions obtained by averaging over the family with a compactly supported smooth weight function has asymptotics compatible with the 1-level density conjecture of Katz and Sarnak. The novelty of the murmurations of this family of -functions is its pronounced almost periodic feature, which allows one to describe this murmuration without averaging over primes, and which is non-existent or previously unnoticed for other families.

Paper Structure

This paper contains 30 sections, 16 theorems, 167 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Murmurations of Elliptic Curves and Other Families
  3. General Formulation of Murmurations
  4. Properties of the Family $\mathscr{F}$
  5. Results: Murmuration Density for the Family $\mathscr{F}$
  6. Easy Evaluations
  7. Orthogonality of Characters
  8. Evaluation of $G_{\mathop{\mathrm{denom}}\nolimits}$ and $G_{\mathop{\mathrm{num}}\nolimits}^-$
  9. Evaluation of $G_{\mathop{\mathrm{num}}\nolimits}^+$: $y=1$ Terms
  10. Overview of Computation
  11. Small $a$, Small $n$
  12. Evaluation and Summation of $C_{8na^2,p,a}$
  13. Evaluation of $G_{\mathop{\mathrm{num}}\nolimits,y}^+$
  14. Overview of Computation
  15. New Error Bounds
  16. ...and 15 more sections

Key Result

Theorem 1

Let $p,X,Y$ be parameters going to $\infty$ with $Y\sim X^{1-\delta_Y}$ and $p\ll X^{1+\delta_p}$ prime and $\delta_Y,\delta_p>0$, and assume that $\xi=p/X$ is bounded away from squares of half-integers as $p,X,Y\to\infty$, in the sense that there exists $\delta_\xi>0$ such that $\xi-y^2/4\geq\delta where In particular, for any $\delta_p<\frac{13-2\sqrt{37}}{108}\approx0.0077$, one can find some

Figures (13)

  • Figure 1: Oscillatory pattern observed in HLOP and the murmurations of starlings.
  • Figure 2: Frobenius trace averages of newforms of fixed weight with level in a particular bounded interval.
  • Figure 3: $G_{\text{avg}}(P,N)$ (from numerics, with rolling average over primes in intervals of size $H=P^{0.55}$) vs. murmuration function $M_\Phi(P/N)$ for $\Phi=\chi_{[1,2]}$ (integrated from the density $M(\Xi)$ in \ref{['eqn:murmuration_density']}), shown for $N=2^{19},\Xi\in[0,8]$.
  • Figure 3: $G(p,N)$ vs $\xi=p/N$ up to $\xi=9/4$ for $N=2^{19}$, with $p$ colored according to different moduli.
  • Figure 4: Asymptotic behavior of the murmuration function $M_\Phi(\Xi)$ for $\Phi=\chi_{[1,2]}$.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 16 more