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

TL;DR

This paper demonstrates that convolutional neural networks can effectively predict the analytic rank of elliptic curves from Frobenius traces, revealing insights through saliency analysis and connecting murmurations with Mestre--Nagao sums.

Contribution

It introduces a novel application of CNNs to elliptic curve data and interprets their predictions via saliency, linking murmurations and Mestre--Nagao sums.

Findings

High accuracy in predicting analytic rank.

Saliency curves reveal interpretability.

Interplay between murmurations and Mestre--Nagao sums.

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, Bujanovi\'{c}--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.