Ranks of elliptic curves and deep neural networks

Abstract

Determining the rank of an elliptic curve $E/\mathbb {Q}$ is a difficult problem. In applications such as the search for curves of high rank, one often relies on heuristics to estimate the analytic rank (which is equal to the rank under the Birch and Swinnerton-Dyer conjecture). In this paper, we propose a novel rank classification method based on deep convolutional neural networks (CNNs). The method takes as input the conductor of E and a sequence of normalized Frobenius traces $a_p$ for primes p in a certain range ($p<10^k$ for $k=3,4,5$), and aims to predict the rank or detect curves of “high” rank. We compare our method with eight simple neural network models of the Mestre–Nagao sums, which are widely used heuristics for estimating the rank of elliptic curves. We evaluate our method 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. Our experiments demonstrate that the CNNs outperform the Mestre–Nagao sums on the LMFDB dataset (remarkably, the neural network that took as an input all Mestre–Nagao sums performed much better than each sum individually). On the custom dataset, the performance of the CNNs and the Mestre–Nagao sums is comparable.