Improving elliptic curve rank classification using multi-value and learned Mestre–Nagao sums

Determining the rank of an elliptic curve [Formula: see text] remains a central challenge in number theory. Heuristics such as Mestre–Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre–Nagao sums. First, we propose a “multi-value” approach that simultaneously uses two distinct sums, [Formula: see text] and [Formula: see text], evaluated over multiple ranges. This multi-sum perspective significantly improves classification accuracy over traditional single-sum heuristics. Second, we employ machine learning — specifically deep neural networks — to learn optimal, potentially conductor-dependent weightings for Mestre–Nagao sums directly from data. Our results indicate that adaptively weighted sums offer a slight edge in rank classification over traditional methods.