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PreprintCC BY 4.0Zenodo

Murmurations in L-Functions: Statistical Correlations in Elliptic Curves of Conductor 24 — E8 Intelligence Research

  • Andrew Stewart Caldin1
  1. 1Independent Researcher, United Kingdom
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Abstract

FINDING: Murmurations in arithmetic L-functions reveal unexpected statistical correlations between elliptic curve coefficients and modular form data, tied to conductor 24 families. | MATH: The murmuration phenomenon involves averaging Fourier coefficients ap(E) of elliptic curves over families, revealing oscillatory patterns in the distribution of ap as a function of conductor N. Key objects: L(E,s) = ∑n≥1 an n−s, conductor N (with N=24 as a notable small case), and the modular form correspondence fE(q) = ∑ an qn. The observed murmurations are statistical — not exact equalities — but show coherent phase alignment in coefficient averages across thousands of curves, suggesting hidden structure in the Sato–Tate distribution beyond the predicted sin2θ law. | CONNECTION: Conductor 24 factors as 24 = 23 · 3, and the associated elliptic curves have complex multiplication or CM-like structure in some cases. The number 24 app

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Cite

BibTeXRIS
BibTeX
@misc{caldin2026murmurations,
  title = {{Murmurations in L-Functions: Statistical Correlations in Elliptic Curves of Conductor 24 — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23635506},
  url = {https://zenodo.org/records/23635506},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Murmurations in L-Functions: Statistical Correlations in Elliptic Curves of Conductor 24 — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/25
PB  - Zenodo
DO  - 10.5281/zenodo.23635506
UR  - https://zenodo.org/records/23635506
AB  - FINDING: Murmurations in arithmetic L-functions reveal unexpected statistical correlations between elliptic curve coefficients and modular form data, tied to conductor 24 families. | MATH: The murmuration phenomenon involves averaging Fourier coefficients a_p(E) of elliptic curves over families, revealing oscillatory patterns in the distribution of a_p as a function of conductor N. Key objects: L(E,s) = ∑_n≥1 a_n n^−s, conductor N (with N=24 as a notable small case), and the modular form correspondence f_E(q) = ∑ a_n q^n. The observed murmurations are statistical — not exact equalities — but show coherent phase alignment in coefficient averages across thousands of curves, suggesting hidden structure in the Sato–Tate distribution beyond the predicted sin^2θ law. | CONNECTION: Conductor 24 factors as 24 = 2^3 · 3, and the associated elliptic curves have complex multiplication or CM-like structure in some cases. The number 24 app  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -