Andrew Sutherland: Murmurations of arithmetic L-functions (NTWS 180)
Channel: Number Theory Web Seminar
Date: 2023-09-17

Abstract: While conducting a series of number-theoretic machine learning experiments last year, 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 short interval, as p increases the average oscillates with a decaying frequency determined by 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.  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 naive height rather than conductor, but doing so obscures these oscillations.
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 of 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 based on joint work with Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov.

Link to slides: https://drive.google.com/file/d/17VR0FPH8kNTL3nPXKrY4zu4m6C2Ml_9G

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, September 14, 2023 (9am PDT, 12pm EDT, 5pm BST, 6pm CEST, 7pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, September 15, 2023 (12am CST, 2am AEST, 4am NZST)