Murmurations, Mestre–Nagao sums, and convolutional neural networks for elliptic curves

Abstract

Abstract We apply one-dimensional convolutional neural networks to the Frobenius traces of elliptic curves over $$\mathbb {Q}$$ 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 saliency, murmurations and Mestre–Nagao sums.