Murmurations of Mestre–Nagao sums

Abstract

This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre–Nagao sum [Formula: see text] where [Formula: see text] is defined as [Formula: see text] for an elliptic curve [Formula: see text] with good reduction at prime [Formula: see text]. This approach is inspired by the Birch and Swinnerton-Dyer (BSD) conjecture. Our observations reveal an oscillatory behavior in the sums, closely associated with the recently discovered phenomena of murmurations of elliptic curves [Y.-H. He, K.-H. Lee, T. Oliver and A. Pozdnyakov, Murmurations of elliptic curves, preprint (2022), arXiv:2204.10140.]. Surprisingly, this suggests that in some cases, opting for a smaller value of [Formula: see text] yields a more accurate classification than choosing a larger one. For instance, when considering elliptic curves with conductors within the range of [Formula: see text], the rank classification based on [Formula: see text]’s with [Formula: see text] produces better results compared to using [Formula: see text]. This phenomenon finds partial explanation in the recent work of Zubrilina [Murmurations (2023)].