Wojciech J. Gajda

Determining the algebraic rank of an elliptic curve E/Q is challenging, often relying on heuristics to estimate the analytic rank, which is conjecturally equal to the algebraic rank under the Birch and Swinnerton-Dyer conjecture. This talk presents a novel rank classification method utilizing deep convolutional neural networks (CNNs). The method takes the conductor of E and a sequence of Frobenius traces a_p as input to predict rank or detect "high" rank curves. Our method and eight simple neural network models utilizing Mestre-Nagao sums, commonly employed heuristics are compared. Results from evaluating both methods on the LMFDB dataset and a custom dataset show that CNNs outperform Mestre-Nagao sums on the LMFDB dataset while demonstrating comparable performance on the custom dataset. This is joint work with Domagoj Vlah. Additionally, we will elaborate on a recent project with Zvonimir Bujanović and Lukas Novak. We'll explain how the recently observed phenomenon of murmurations of elliptic curves can help us understand fluctuations in the averages of Mestre-Nagao sums, leading to improved classification quality.