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 talk, we discuss 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. On the custom dataset, the performance of the CNNs and the Mestre-Nagao sums is comparable. This is joint work with Domagoj Vlah.

machine learningmathematical physicsalgebraic geometryalgebraic topology

Audience: researchers in the topic

DANGER3: Data, Numbers, and Geometry