Sarnak Quanta — Alex Ramsay, Alamy Stock Photo

Patterns formed by thousands of elliptic curves bear a striking similarity to murmurations of starlings.

Elliptic Curve ‘Murmurations’ Found With AI Take Flight

"Elliptic curves are among the more beguiling objects in modern mathematics. They don’t seem complicated, but they form an expressway between the math that many people learn in high school and research mathematics at its most abstruse. They were central to [frequent Member and Visitor in the School of Mathematics] Andrew Wiles’ celebrated proof of Fermat’s Last Theorem in the 1990s. They are key tools in modern cryptography."

"Understanding elliptic curves is a high-stakes endeavor that has been central to math. So in 2022, when a transatlantic collaboration used statistical techniques and artificial intelligence to discover completely unexpected patterns in elliptic curves, it was a welcome, if unexpected, contribution."

Peter Sarnak, Gopal Prasad Professor in the School of Mathematics, spoke with Quanta about the impact of this work. He engaged in correspondence with one of the key contributors, Andrew Sutherland (MIT), who discovered that elliptic curves stayed the same shape even as he looked at the curves over larger and larger prime numbers. Sarnak's graduate student at Princeton University, Nina Zubrilina, also made a major impact. In a paper posted in 2023, she proved that a specific type of murmuration follows an explicit formula that she discovered.

"Though Zubrilina’s formula was the first, others have followed. 'Every week now, there’s a new paper out,' Sarnak said, 'mainly using Zubrilina’s tools, explaining other aspects of murmurations.'"