The math of doughnuts: 'Moonshine' sheds light on elliptic curves

By Carol Clark | eScienceCommons | Sept. 26, 2017

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In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says. Emory Photo/Video

Mathematicians have opened a new chapter in the theory of moonshine, one which begins to harness the power of the pariahs – sporadic simple groups that previously had no known application.

“We’ve found a new form of moonshine, which in math refers to an idea so farfetched as to sound like lunacy,” says Ken Ono, a number theorist at Emory University. “And we’ve used this moonshine to show the mathematical usefulness of the O’Nan pariah group in a way that moves it from theory to reality. It turns out that the O’Nan group knows deep information about elliptic curves.”

Nature Communications published the representation theory for the O’Nan group developed by Ono, John Duncan (also a number theorist at Emory) and Michael Mertens (a former post-doctoral fellow at Emory who is now at the University of Cologne).

“We’ve shown that the O’Nan group, a very large pariah group, actually organizes elliptic curves in a beautiful and systematic way,” Duncan says. “And not only does it organize them, it allows us to see some of their deepest properties. It sees infinitely many curves, which allows us to then use our moonshine to make predictions about their general behavior. That’s important, because these objects underlie some of the hardest questions at the very horizon of number theory.”

Elliptic curves may sound esoteric, but they are part of our day-to-day lives. They are used in cryptography – the creation of codes that are difficult to break. An elliptic curve is not an ellipse, rather it is a complex torus, or doughnut shape.

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