Drew Sutherland (MIT)

While conducting a series of number-theoretic machine learning experiments, He, Lee, Oliver, and Pozdnyakov noticed a curious oscillation in the averages of Frobenius traces of elliptic curves over Q.  If one computes the average value of a_p(E) for E/Q of fixed rank with conductor in a fixed interval, as p increases the average of a_p(E) oscillates with a decaying frequency determined by the average size of the conductor.  That the rank influences the distribution of Frobenius traces has long been known (indeed, this was the impetus for the experiments that led to the conjecture of Birch and Swinnerton-Dyer), but these oscillations do not appear to have been noticed previously.

I will present results from an ongoing investigation of this phenomenon, which is remarkably robust and not specific to elliptic curves. One finds similar oscillations in the averages of Dirichlet coefficients for many types of L-functions when organized by conductor and root number, including those associated to modular forms and abelian varieties.  The source of these murmurations in the case of weight-2 newforms with trivial nebentypus is now understood, thanks to recent work of Zubrilina, but all other cases remain open.

This is joint work with Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov.