GAAG seminar - Murmurations for elliptic curves ordered by height

While conducting a series of number-theoretic machine learning experiments in 2022, He-Lee-Oliver-Pozdnyakov noticed a curious oscillation in the averages of Frobenius traces of elliptic curves.  If one computes the average value of a_p(E) over all E/Q of a given rank with conductor constrained to a short interval, as p varies the averages oscillates with a decaying frequency that depends on the conductor range.  Similar oscillations have since found in many other families of L-functions, and while there are some known sources of bias (the influence of the rank on the distribution of Frobenius traces motivated both the BSD conjecture and these experiments), these oscillations do not appear to have been previously observed or predicted.  This may be due in part to the critical role played by the conductor; in arithmetic statistics it is common to order elliptic curves E/Q by height rather than conductor, but doing so obscures these oscillations.

In this talk I will present recent joint work with Will Sawin (arXiv:2503.12295) in which we obtain the first unconditional result for murmurations of elliptic curves.  We order elliptic curves by height and average against a smooth test function that accounts for the influence of the conductor in a way that allows us to obtain a rigorous result.  This leads to an explicit murmuration density function that we conjecture applies more generally and explains the murmuration phenomenon first observed by He-Lee-Oliver-Pozdnyakov.