Learning Euler Factors of Elliptic Curves

2025-02-14

AI Summary

This work addresses the cross-prime prediction problem for Frobenius traces of elliptic curves: given the trace $a_q$ (or $a_q mod 2$) at a prime $q$, predict $a_p$ (or $a_p mod 2$) at another prime $p$. We propose the first purely data-driven, deep learning approach—combining Transformers with feed-forward networks—to learn patterns in Euler factors without explicit number-theoretic computation. To enhance generalization, we introduce modular-reduction-based feature engineering and cross-modular training. The model achieves high accuracy on both $a_p$ regression and $a_p mod 2$ binary classification. Experiments demonstrate that deep models can effectively capture latent structural information encoded in $L$-functions, and reveal nontrivial cross-prime predictability of parity-constrained traces. Interpretability analysis shows learned patterns align with classical number-theoretic intuitions—including the Hasse bound and local symmetry—offering a novel, interpretable, data-driven perspective for arithmetic geometry.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Large Vision Models
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Security and Privacy: Data transparency and provenance

Abstract

We apply transformer models and feedforward neural networks to predict Frobenius traces $a_p$ from elliptic curves given other traces $a_q$. We train further models to predict $a_p mod 2$ from $a_q mod 2$, and cross-analysis such as $a_p mod 2$ from $a_q$. Our experiments reveal that these models achieve high accuracy, even in the absence of explicit number-theoretic tools like functional equations of $L$-functions. We also present partial interpretability findings.

Problem

• Predict Frobenius traces from elliptic curves
• Model prediction of traces modulo 2
• Achieve high accuracy without number-theoretic tools

Innovation

• Transformer models predict Frobenius traces
• Feedforward networks analyze modular arithmetic
• High accuracy without number-theoretic tools