[Paper Review] Murmurations of elliptic curves

Yang‐Hui He, Kyu‐Hwan Lee+2 more|arXiv (Cornell University)|Apr 21, 2022

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper investigates the average behavior of the $a_p$-coefficients of elliptic curves over $\mathbb{Q}$, grouped by rank and conductor range, revealing unexpected oscillatory patterns in their distribution. Using logistic regression and principal component analysis on the first 10 $a_p$-values, the authors achieve over 95% accuracy in classifying curves of rank $\leq 1$ versus $\geq 2$, suggesting a novel data-driven approach to understanding rank distribution.

ABSTRACT

We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.

Motivation & Objective

• To investigate the average distribution of $a_p$-coefficients across elliptic curves of fixed rank and conductor range.
• To explore whether the oscillatory behavior in these averages can be leveraged for rank classification.
• To develop and test machine learning models—logistic regression and PCA—for distinguishing elliptic curves by rank using only finitely many $a_p$-coefficients.
• To provide a heuristic, data-driven framework for classifying elliptic curves based on their $a_p$-sequences, particularly for rank $\leq 1$ vs. $\geq 2$.
• To uncover new mathematical structures in the data, such as non-monotonic oscillations in $f_r(n)$, that may inform deeper number-theoretic insights.

Proposed method

• Define $f_r(n)$ as the average $a_{p_n}$-coefficient over all elliptic curves of rank $r$ and conductor in $[N_1, N_2]$.
• Use data from the LMFDB to compute $f_r(n)$ for $n \leq 1000$ (primes up to 7919) across conductor ranges $[7500,10000]$ and $[5000,10000]$.
• Apply logistic regression using the first 10 $a_p$-coefficients ($p = 2$ to $29$) as features to classify curves of rank $\leq 1$ vs. $\geq 2$.
• Perform principal component analysis (PCA) on $a_p$-vectors to visualize clustering patterns across ranks 0, 1, and 2.
• Fit parametric models to the oscillatory behavior of $f_r(n)$ to understand its functional form.
• Construct a heuristic logistic regression function $r(\mathbf{a}) = \frac{1}{1 + e^{-(\mathbf{w} \cdot \mathbf{a} + b)}}$ with learned weights $\mathbf{w}$ and bias $b$ for classification.

Experimental results

Research questions

• RQ1 Can the average $a_p$-coefficients of elliptic curves exhibit structured, non-monotonic oscillations when grouped by rank and conductor range?
• RQ2 To what extent can the first 10 $a_p$-coefficients distinguish elliptic curves of rank $\leq 1$ from those of rank $\geq 2$?
• RQ3 How do the principal components of $a_p$-vectors reflect underlying structural differences across ranks 0, 1, and 2?
• RQ4 Can a simple heuristic function based on early $a_p$-values achieve high classification accuracy for rank?
• RQ5 What mathematical implications arise from the observed oscillatory patterns in $f_r(n)$, especially in light of the Sato–Tate conjecture?

Key findings

• The average $a_p$-coefficients for elliptic curves of rank 0, 1, and 2 exhibit distinct, strikingly different oscillatory patterns when plotted against the index $n$ of the $n$-th prime.
• Logistic regression using only the first 10 $a_p$-coefficients ($p = 2$ to $29$) achieves approximately 95% accuracy in classifying curves of rank $\leq 1$ versus $\geq 2$ in the conductor range $[10000,20000]$.
• In the smaller conductor range $[1,10000]$, the same approach achieves approximately 97% accuracy, indicating improved performance with denser data.
• The learned logistic regression model for $[10000,20000]$ uses weights $\mathbf{w} = (-1.1198, -1.1273, \dots, -0.1939)$ and bias $b = -5.6277$, with a 0.5 threshold for classification.
• The model correctly classifies a rank 1 curve ($a_p = (1,1,1,1,-1,1,2,-4,-4,6)$) as rank $\leq 1$ ($r(\mathbf{a}) \approx 0.0001$) and a rank 2 curve ($a_p = (0,-2,-1,-4,0,1,0,-8,0,-1)$) as rank $\geq 2$ ($r(\mathbf{a}) \approx 0.9746$).
• PCA reveals clustering of curves by rank, with distinct loadings on the principal components that reflect the oscillatory behavior of $f_r(n)$, suggesting underlying geometric structure in the data.