So they assumed the opposite: that there exists a noncongruence modular form with bounded denominators. By definition, it would live in the gap that Calegari, Dimitrov and Tang were trying to close. The three then showed that the existence of this noncongruence modular form automatically implied the existence of lots of other noncongruence modular forms with bounded denominators. It was as if a whole forest had grown from that single seed.

But they had already established the maximum size of the gap — and it was too small to fit that many noncongruence forms.

Which meant that even one such form couldn’t exist. They’d proved Atkin and Swinnerton-Dyer’s decades-old conjecture.

Mathematicians find the techniques used in the work even more intriguing than the result itself. “These ideas have never been used before in studying the arithmetic of modular forms,” Scholl said.

As Voight explains, though the study of modular forms started out as part of the field of complex analysis, current work has been the purview of number theory and algebraic geometry. The new paper, he said, marks a return to complex analysis: “It’s a refreshingly old perspective.”

**A Search for New Theories**

Mathematicians aren’t the only ones excited about the unbounded denominators conjecture. It also makes an appearance in theoretical physics.

In the 1970s, another story was unfolding in parallel to the one begun by Atkin and Swinnerton-Dyer. Mathematicians had noticed a strange connection between an object called the monster group and a modular form called the *j*-function. The coefficients of the *j*-function precisely reflected certain properties of the monster group.

Later research revealed that this connection was due to the fact that both the group and the modular form were related to an important model of particle interactions called a two-dimensional conformal field theory.

But the conformal field theory that linked the monster group to the *j*-function was just one example of an infinite number of conformal field theories. And while these theories don’t describe the universe we live in, understanding them can yield new insights into how more realistic quantum field theories might behave.

And so physicists have continued to study conformal field theories by looking at their associated modular forms. (In this context, physicists use a more general notion of a modular form, called a vector-valued modular form.)

To get a handle on what’s going on with a particular conformal field theory, you have to show that its modular form is congruence, said Michael Tuite, a mathematician and theoretical physicist at the University of Galway in Ireland. You can then start to describe conformal field theories, and even discover new ones you didn’t know to look for. This is particularly crucial for an ongoing effort to classify all conformal field theories — a project that physicists have dubbed the modular bootstrap.

“Once you know it’s a congruence modular form, that enables you to make enormous strides in this program,” Mason said.

Physicists developed a framework that allows them to assume this congruence property for the modular forms they’re studying. But that’s not the same as having a rigorous mathematical proof — and while other mathematicians were later able to provide such a proof, their argument only worked in certain settings. It also involved “a very tortuous, convoluted path” toward congruence, according to Mason, though he also pointed out that this convoluted path yielded important insights.

Calegari, Dimitrov and Tang’s proof of the unbounded denominators conjecture cuts through all that. That’s because, as it turns out, the modular forms associated to conformal field theories always have integer coefficients. By definition, integers have a denominator of 1, meaning their denominators are always bounded. And since the unbounded denominators conjecture states that bounded denominators are associated only with congruence modular forms, there’s no longer a need to make assumptions. “You don’t even need to know anything about [conformal field theories],” Tang said. The new proof automatically delivers congruence for all these cases — for free.

“It’s something that’s been in the air for decades,” Bost said. Now it’s finally resolved.

“It really is a miracle,” Mason said. “This just follows miraculously from the fact that these sequences are integers.”

He’s already started applying the result in his own work. “Ever since the day that paper appeared, I’ve been making use of it,” he said. “It provides a very welcome shortcut to results that I want to resolve. …It’s cutting out a huge amount of potential work that I couldn’t see my way through.”

It also puts the modular bootstrap program and other results on stronger mathematical footing. “This is going to allow mathematicians to re-prove [previous] results, or believe them,” Mason said.

“I think it’s really going to have an impact, especially on the mathematics side, just to really, really tie things down, to understand exactly what’s going on,” Tuite said.

**Mathematical Transcendence**

In the year since they posted their proof, Calegari, Dimitrov and Tang have continued their collaboration. They’ve now returned to the types of problems in transcendental number theory that originally sparked their interest in the conjecture. “We’re trying to finish what we started,” Tang said. In fact, they’ve already used their techniques to prove that several numbers of interest are irrational.

“They’re really pushing the [method] to the limit,” Fresán said. “I’m really very excited about this.”

These methods might also be applicable to other problems in number theory.

Techniques aside, the resolution of the unbounded denominators conjecture marks one of the first big milestones in the effort to gain a better understanding of noncongruence modular forms. “This is an amazing achievement, that we can make some progress on noncongruence forms in this way,” Franc said. “I’m excited for the next 10, 20 years, to see what happens.”

Li, Voight and others are already starting to look for patterns in the kinds of numbers that show up in the denominators of these mysterious modular forms. They hope that in doing so, they can find hints of deeper structure.

“This unbounded denominators conjecture was just the beginning,” Li said.