Moonshine doughnut maths proves pariahs are real

Number theorists have found a mysterious connection between seemingly unrelated areas of mathematics, writes Michael Lucy.

A doughnut.
Concrete visuals for objects like finite simple groups and elliptic curves are hard to find.
Travis Munson / EyeEm

In news that will excite number theorists but perhaps baffle the average reader, a trio of American mathematicians have traced a moonshine connection between the O’Nan pariah group and elliptic curves.

To understand this marvellous collection of words will require a detour through number theory, in particular the study of algebraic objects called finite simple groups.

Despite the name, finite simple groups are quite complicated, but it’s enough to know that all of them fit into one of 18 families, except for 26 that are known as sporadic groups.

Of these 26, 20 can be collected together as parts of the so-called Monster group (they are known as the Happy Family), and the lonely remaining 6 groups are known as the pariahs. (There is also an ambiguously classified group called the Tits group.)

Finite simple groups are fairly abstract, and their connections to anything outside group theory are not always clear.

This is where moonshine comes in: in 1979, Cambridge mathematicians John Conway and Simon Norton published a paper with the title ‘Monstrous moonshine’. The paper explained an unlikely connection between the Monster group and an object in number theory called the j-function. The connection – a relationship between the dimensions of the Monster group and the coefficients of the j-function – seemed so preposterous that when it was first proposed to Conway he dismissed it as “moonshine”.

Nonetheless, the connection existed and in the intervening decades moonshine theory – the study of mysterious connections between groups and functions – has blossomed, informing further mathematical investigations and ideas in physics including quantum gravity proposals and string theory.

Even so, the pariah groups have remained isolated. Until now.

“We’ve found a new form of moonshine,” says Ken Ono, a number theorist at Emory University in Atlanta in the United States and one of the authors of the new study. “And we’ve used this moonshine to show the mathematical usefulness of the O’Nan pariah group. It turns out that the O’Nan group knows deep information about elliptic curves.”

That’s good to hear, but what are elliptic curves? They’re a class of complex functions that describe collections of points on donut-shaped surfaces. They are also commonly used in cryptography, including day-to-day online security.

The O’Nan group, according to Ono’s co-author John Duncan, “organizes elliptic curves in a beautiful and systematic way”.

This is the first time any of the pariah groups – which were constructed to fill theoretical gaps in the catalogue of finite simple groups – have been connected to other parts of the mathematical universe.

“Our work proves that a pariah is real,” Ono says. “We found the O’Nan group living in nature.”

The paper, by Ono, Duncan and their colleague Michael Mertens, is published in Nature Communications.

Michael Lucy is features editor of Cosmos.
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