Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curves

Abstract

We apply one-dimensional convolutional neural networks to the Frobenius traces of elliptic curves over $\mathbb{Q}$ and evaluate and interpret their predictive capacity. In keeping with similar experiments by Kazalicki--Vlah, Bujanović--Kazalicki--Novak, and Pozdnyakov, we observe high accuracy predictions for the analytic rank across a range of conductors. We interpret the prediction using saliency curves and explore the interesting interplay between murmurations and Mestre--Nagao sums, the details of which vary with the conductor and the (predicted) rank.

Figures (10)

• Figure 2.1: Distribution of conductors in XECQ.
• Figure 2.2: Learning the rank in XECQ[A,B]: percentage accuracy against epoch. Each colour corresponds to one of the $4$ conductor intervals specified.
• Figure 2.3: Learning the rank in XECQ[A,B]: (Left) test accuracy against number of primes used in training for different choices of $[A,B]$. (Right) test accuracy against the logarithm of the number of primes used in training for different choices of $[A,B]$.
• Figure 3.1: Comparison of the saliency scores $w_p$ with the Mestre--Nagao weighting. In this figure, the saliency scores are computed after $100$ epochs.
• Figure 3.2: Comparison the normalized saliency scores $\widetilde{w}_p$ with the Mestre--Nagao weighting. In this figure, the normalised saliency scores are computed after $100$ epochs.