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Murmurations in the depth aspect

Claire Burrin, Vivian Kuperberg, Min Lee, Catinca Mujdei, Hsin-Yi Yang

Abstract

We compute the murmuration density function for the family of Hecke forms of weight $k$ and prime power level $N=\ell^a$, with $\ell$ a fixed odd prime and $a\to \infty$.

Murmurations in the depth aspect

Abstract

We compute the murmuration density function for the family of Hecke forms of weight and prime power level , with a fixed odd prime and .

Paper Structure

This paper contains 7 sections, 8 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Determination of the murmuration density
  3. Sketch of proof of Theorem \ref{['thm:main']}
  4. Acknowledgments
  5. Eichler--Selberg trace formula
  6. Proof of Theorem \ref{['thm:main']}
  7. Bombieri--Vinogradov and necessary variants

Key Result

Theorem 1.1

As $a\to\infty$, we have where $C>0$ is any positive constant and $U_{k-2}$ is the Chebyshev polynomial given by

Figures (1)

  • Figure 1: Plots of the asymptotic murmuration density function in Theorem \ref{['thm:main']} for various values of $\ell,k$ and window $E=[1,t]$

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Theorem 4.1: MR3670199-baker, Theorem 1
  • ...and 4 more