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

2026-03-18

AI Summary

This study investigates the prediction of analytic ranks of elliptic curves over the rational numbers, exploring their deep connection with Frobenius traces. We propose a one-dimensional convolutional neural network model that leverages sequences of Frobenius traces to predict analytic ranks and employs saliency analysis to interpret the model’s decisions. Our research uncovers novel patterns linking murmurations phenomena with Mestre–Nagao sums, revealing how these relationships vary with conductor and predicted rank. Across multiple conductor ranges, the model achieves high predictive accuracy while offering mathematically interpretable insights, thereby opening new avenues at the intersection of arithmetic geometry and machine learning.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models
• Cognitive Modeling & Cognitive Systems: Analogy

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web search models and ranking

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.

Problem

• elliptic curves
• analytic rank
• Frobenius traces
• murmurations
• Mestre--Nagao sums

Innovation

• convolutional neural networks
• elliptic curves
• Frobenius traces
• murmurations
• Mestre–Nagao sums