Learning Euler Factors of Elliptic Curves

TL;DR

This paper demonstrates that transformer and neural network models can accurately predict Frobenius traces of elliptic curves, including modular reductions, without relying on traditional number-theoretic methods.

Contribution

It introduces machine learning approaches to predict elliptic curve invariants, showing high accuracy and partial interpretability, advancing computational techniques in number theory.

Findings

High accuracy in predicting Frobenius traces from elliptic curves.

Effective prediction of traces modulo 2.

Models perform well without explicit number-theoretic tools.

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 $a_p \bmod 2$ from $a_q \bmod 2$, and cross-analysis such as $a_p \bmod 2$ 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.