Elliptic Curve 'Murmurations' Found with AI Take Flight (www.quantamagazine.org)

A collaborative research effort has unveiled unexpected patterns, dubbed “murmurations,” in elliptic curves using artificial intelligence (AI), marking a significant breakthrough in the intersection of mathematics and machine learning. Elliptic curves are crucial in number theory and cryptography, and understanding them could help resolve long-standing mathematical problems, including the Birch and Swinnerton-Dyer conjecture, which remains one of the "Millennium Prize Problems." The recent findings, first reported in 2022, have since attracted considerable attention, leading to the discovery of scale-invariant patterns across larger datasets of elliptic curves, suggesting these murmurations are a broader phenomenon related to L-functions. The significance of this discovery lies not only in the patterns themselves but also in how they were identified, utilizing machine learning techniques. Researchers trained algorithms on data from extensive elliptic curve databases, leading to predictions about their ranks and the identification of distinct visual patterns in the data. A breakthrough came when Zubrilina established an explicit formula for these murmurations, which is now considered an important advancement akin to well-known mathematical functions. This ongoing research continues to inspire new papers and investigations, underscoring the potential of AI to uncover deep mathematical truths and its transformative role in the field of mathematics.