Learning Euler Factors of Elliptic Curves

TL;DR

This work investigates whether transformer-based and neural-network methods can infer Frobenius traces $a_p$ of elliptic curves from nearby traces $a_q$, leveraging datasets of isogeny classes with small conductors. It demonstrates that models can achieve high accuracy in predicting $a_p$ and, notably, reliably predict $a_p\bmod 2$, suggesting that parity and modular structure underpin the learned relationships. Through PCA visualizations and saliency analyses, the study provides evidence that the models internally encode modular information (especially parity and modulo-4 patterns) and often rely on predicting $a_p\bmod 2$ as an intermediate step to recovering $a_p$. These findings offer partial interpretability in a number-theoretic setting and point to modular patterns as a latent organizing principle learned by neural networks in arithmetic geometry tasks.

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.

Figures (13)

• Figure 1: We implement an encoder only transformer architecture using Int2Int.
• Figure 2: Predicting $a_{97}$. Proportion of cases where the model predicts a given value as a function of its true value.
• Figure 3: Confusion matrix heat map for predicting $a_2$ (left) and $a_3$ (right).
• Figure 4: The ratios of curves with $a_p \equiv 0 \pmod 2$ for various $p < 100$ when curves have good reduction at $p$. There is a consistent bias of $a_p \equiv 0 \mod 2$ when $p$ is not "small".
• Figure 5: The results of predicting $a_p \mod 2$ using $\{a_q\}_{q \ne p, q < 100}$ for curves with good reduction at $p$ show that, except for $p = 2, 3, 43$, the model achieves accuracies close to 0.94. The highest accuracy, 0.9472, occurs at $p = 83$, while the lowest, 0.8935, is observed at $p = 2$. Similarly, the model achieves MCC around 0.84, except for $p = 2, 79$. The highest MCC, 5, 0.8703, is recorded at $p = 5$, whereas the lowest, 0.7735, occurs at $p = 2$.