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 Andrew Wiles’ celebrated proof of Fermat’s Last Theorem in the 1990s. They are key tools in modern cryptography. And in 2000, the Clay Mathematics Institute named a conjecture about the statistics of elliptic curves one of seven “Millennium Prize Problems,” each of which carries a $1 million prize for its solution. That conjecture, first ventured by Bryan Birch and Peter Swinnerton-Dyer in the 1960s, still hasn’t been proved.
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. “It was just a matter of time before machine learning landed on our front doorstep with something interesting,” said Peter Sarnak, a mathematician at the Institute for Advanced Study and Princeton University. Initially, nobody could explain why the newly discovered patterns exist. Since then, in a series of recent papers, mathematicians have begun to unlock the reasons behind the patterns, dubbed “murmurations” for their resemblance to the fluid shapes of flocking starlings, and have started to prove that they must occur not only in the particular examples examined in 2022, but in elliptic curves more generally.

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