Speaker
Description
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).
This talk discusses 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 the LMFDB dataset 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. The performance of the CNNs and the Mestre-Nagao sums is comparable on the custom dataset.
This is joint work with Domagoj Vlah.
Additionally, we will elaborate on an ongoing project with Zvonimir Bujanović, focusing on a detailed analysis of some aspects of Mestre-Nagao sums through the use of neural networks.
