Murmurations for elliptic curves ordered by height

While conducting machine learning experiments in 2022, He-Lee-Oliver-Pozdnyakov noticed a curious oscillation (murmuration) in averages of Frobenius traces of elliptic curves over Q of particular ranks in prescribed conductor ranges. Similar oscillations have since been observed in many other families of L-functions. For L-functions of Hecke eigenforms with trivial character, Zubrilina used the Eichler-Selberg trace formula to derive a density function that completely explains the murmuration phenomenon in this setting. Zubrilina’s methods have since been applied in other settings where a suitable trace formula is available, but an explanation for the murmurations originally observed in elliptic curves has remained elusive.

In this talk I will present joint work with Will Sawin (arXiv:2504.12295) in which we use the Voronoi summation formula to analyze murmurations in the elliptic curve setting. We order elliptic curves by height and average against a smooth test function, which allows us to obtain an unconditional result. This leads to an explicit murmuration density function that we conjecture applies more generally and explains the original murmuration phenomenon observed by He-Lee-Oliver-Pozdnyakov, in which elliptic curves are ordered by conductor rather than height.