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Variations on murmurations

Kimball Martin

TL;DR

The note extends murmurations from root-number–coefficient correlations to a suite of variants including unweighted averages, Möbius-twisted traces, local Atkin–Lehner decompositions, short class-number sums, twisted L-values, and representations by quadratic forms. It leverages trace formulas (e.g., Eichler–Selberg, Jacquet's relative trace formula) to produce scale-invariant averages in the regime $\frac{p}{X}$ across modular forms and elliptic curves, with several settings showing predictive murmurations and others highlighting delicate cancellations. The work unifies these diverse perspectives under a common probabilistic/arithmetic-statistics framework, illustrating how different arithmetic objects exhibit similar scale-invariant fluctuations and proposing further avenues via local root numbers and relative trace formulas. At a broader level, these results deepen connections between Fourier coefficients, class numbers, L-values, and representation counts, with potential implications for understanding fluctuations in arithmetic statistics and the distribution of arithmetic invariants.

Abstract

We explore several variations on the recently discovered phenomena of murmurations for elliptic curves and modular forms.

Variations on murmurations

TL;DR

The note extends murmurations from root-number–coefficient correlations to a suite of variants including unweighted averages, Möbius-twisted traces, local Atkin–Lehner decompositions, short class-number sums, twisted L-values, and representations by quadratic forms. It leverages trace formulas (e.g., Eichler–Selberg, Jacquet's relative trace formula) to produce scale-invariant averages in the regime across modular forms and elliptic curves, with several settings showing predictive murmurations and others highlighting delicate cancellations. The work unifies these diverse perspectives under a common probabilistic/arithmetic-statistics framework, illustrating how different arithmetic objects exhibit similar scale-invariant fluctuations and proposing further avenues via local root numbers and relative trace formulas. At a broader level, these results deepen connections between Fourier coefficients, class numbers, L-values, and representation counts, with potential implications for understanding fluctuations in arithmetic statistics and the distribution of arithmetic invariants.

Abstract

We explore several variations on the recently discovered phenomena of murmurations for elliptic curves and modular forms.
Paper Structure (8 sections, 21 equations, 25 figures)

This paper contains 8 sections, 21 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Review of murmurations
  3. No root numbers
  4. Möbius sums
  5. Local root numbers
  6. Class number sums
  7. $L$-values
  8. Representations by quadratic forms

Figures (25)

  • Figure 1: Murmurations for weight 2 modular forms of squarefree level $1000 \le N \le 2000$
  • Figure 2: Murmurations for weight 2 modular forms of squarefree levels $2000 \le N \le 4000$
  • Figure 3: Weight 2 murmurations without root number for $X=2000$
  • Figure 4: Weight 2 murmurations without root number for $X=4000$
  • Figure 5: Elliptic curve $\sqrt N a_p$ averages without root number for $X=2000$
  • ...and 20 more figures