Ranks of elliptic curves and deep neural networks

TL;DR

This paper introduces deep convolutional neural network heuristics to classify the rank of elliptic curves using conductor and a_p sequences, outperforming traditional Mestre-Nagao sum methods on large datasets.

Contribution

It develops CNN-based heuristics for elliptic curve rank prediction, demonstrating improved accuracy over classical methods on extensive datasets.

Findings

CNN outperforms Mestre-Nagao sums on LMFDB dataset

Neural network with all sums performs best among tested models

Comparable performance of CNN and Mestre-Nagao sums on custom dataset

Abstract

Determining the rank of an elliptic curve E/Q is a hard problem, and in some applications (e.g. when searching for curves of high rank) one has to rely on heuristics aimed at estimating the analytic rank (which is equal to the rank under the Birch and Swinnerton-Dyer conjecture). In this paper, we develop rank classification heuristics modeled by deep convolutional neural networks (CNN). Similarly to widely used Mestre-Nagao sums, it takes as an input the conductor of E and a sequence of normalized a_p-s (where a_p=p+1-#E(F_p) if p is a prime of good reduction) in some range (p<10^k for k=3,4,5), and tries to predict rank (or detect curves of ``high'' rank). The model has been trained and tested on two datasets: the LMFDB and a custom dataset consisting of elliptic curves with trivial torsion, conductor up to 10^30, and rank up to 10. For comparison, eight simple neural network models of Mestre-Nagao sums have also been developed. Experiments showed that CNN performed better than Mestre-Nagao sums on the LMFDB dataset (interestingly neural network that took as an input all Mestre-Nagao sums performed much better than each sum individually), while they were roughly equal on custom made dataset.