The Core of Fermat’s Last Theorem Just Got Superpowered
The new proof shows that two distant mathematical realms always match.
Nash Weerasekera for Quanta Magazine
Introduction
In 1994, an earthquake of a proof shook up the mathematical world. The mathematician Andrew Wiles had finally settled Fermat’s Last Theorem, a central problem in number theory that had remained open for over three centuries. The proof didn’t just enthrall mathematicians — it made the front page of The New York Times.
But to accomplish it, Wiles (with help from the mathematician Richard Taylor) first had to prove a more subtle intermediate statement — one with implications that extended beyond Fermat’s puzzle.
This intermediate proof involved showing that an important kind of equation called an elliptic curve can always be tied to a completely different mathematical object called a modular form. Wiles and Taylor had essentially unlocked a portal between disparate mathematical realms, revealing that each looks like a distorted mirror image of the other. If mathematicians want to understand something about an elliptic curve, Wiles and Taylor showed, they can move into the world of modular forms, find and study their object’s mirror image, then carry their conclusions back with them.
This connection between worlds, called “modularity,” didn’t just enable Wiles to prove Fermat’s Last Theorem. Mathematicians soon used it to make progress on all sorts of previously intractable problems.
Modularity also forms the foundation of the Langlands program, a sweeping set of conjectures aimed at developing a “grand unified theory” of mathematics. If the conjectures are true, then all sorts of equations beyond elliptic curves will be similarly tethered to objects in their mirror realm. Mathematicians will be able to jump between the worlds as they please to answer even more questions.
But proving the correspondence between elliptic curves and modular forms has been incredibly difficult. Many researchers thought that establishing some of these more complicated correspondences would be impossible.
Now, a team of four mathematicians has proved them wrong. In February, the quartet finally succeeded in extending the modularity connection from elliptic curves to more complicated equations called abelian surfaces. The team — Frank Calegari of the University of Chicago, George Boxer and Toby Gee of Imperial College London, and Vincent Pilloni of the French National Center for Scientific Research — proved that every abelian surface belonging to a certain major class can always be associated to a modular form.
From top: Toby Gee, Vincent Pilloni, Frank Calegari, and George Boxer (not pictured) spent nearly a decade on the proof.
From left: Toby Gee, Frank Calegari and Vincent Pilloni, along with George Boxer (not pictured), spent nearly a decade on the proof.
Courtesy of Toby Gee; Jayne Ion; MC
“We mostly believe that all the conjectures are true, but it’s so exciting to see it actually realized,” said Ana Caraiani, a mathematician at Imperial College London. “And in a case that you really thought was going to be out of reach.”
It’s just the beginning of a hunt that will take years — mathematicians ultimately want to show modularity for every abelian surface. But the result can already help answer many open questions, just as proving modularity for elliptic curves opened up all sorts of new research directions.
Through the Looking Glass
The elliptic curve is a particularly fundamental type of equation that uses just two variables — x and y. If you graph its solutions, you’ll see what appear to be simple curves. But these solutions are interrelated in rich and complicated ways, and they show up in many of number theory’s most important questions. The Birch and Swinnerton-Dyer conjecture, for instance — one of the toughest open problems in math, with a $1 million reward for whoever proves it first — is about the nature of solutions to elliptic curves.
Elliptic curves can be hard to study directly. So sometimes mathematicians prefer to approach them from a different angle.
That’s where modular forms come in. A modular form is a highly symmetric function that appears in an ostensibly separate area of mathematical study called analysis. Because they exhibit so many nice symmetries, modular forms can be easier to work with.
At first, these objects seem as though they shouldn’t be related. But Taylor and Wiles’ proof revealed that every elliptic curve corresponds to a specific modular form. They have certain properties in common — for instance, a set of numbers that describes the solutions to an elliptic curve will also crop up in its associated modular form. Mathematicians can therefore use modular forms to gain new insights into elliptic curves.
But mathematicians think Taylor and Wiles’ modularity theorem is just one instance of a universal fact. There’s a much more general class of objects beyond elliptic curves. And all of these objects should also have a partner in the broader world of symmetric functions like modular forms. This, in essence, is what the Langlands program is all about.
An elliptic curve has only two variables — x and y — so it can be graphed on a flat sheet of paper. But if you add another variable, z, you get a curvy surface that lives in three-dimensional space. This more complicated object is called an abelian surface, and as with elliptic curves, its solutions have an ornate structure that mathematicians want to understand.
It seemed natural that abelian surfaces should correspond to more complicated types of modular forms. But the extra variable makes them much harder to construct and their solutions much harder to find. Proving that they, too, satisfy a modularity theorem seemed completely out of reach. “It was a known problem not to think about, because people have thought about it and got stuck,” Gee said.
But Boxer, Calegari, Gee and Pilloni wanted to try.
Finding a Bridge
All four mathematicians were involved in research on the Langlands program, and they wanted to prove one of these conjectures for “an object that actually turns up in real life, rather than some weird thing,” Calegari said.
Not only do abelian surfaces show up in real life — the real life of a mathematician, that is — but proving a modularity theorem about them would open new mathematical doors. “There are lots of things you can do if you have this statement that you have no chance of doing otherwise,” Calegari said.
The mathematicians started working together in 2016, hoping to follow the same steps that Taylor and Wiles had in their proof about elliptic curves. But every one of those steps was much more complicated for abelian surfaces.
So they focused on a particular type of abelian surface, called an ordinary abelian surface, that was easier to work with. For any such surface, there’s a set of numbers that describes the structure of its solutions. If they could show that the same set of numbers could also be derived from a modular form, they’d be done. The numbers would serve as a unique tag, allowing them to pair each of their abelian surfaces with a modular form.
The problem was that while these numbers are straightforward to compute for a given abelian surface, mathematicians don’t know how to construct a modular form with the exact same tag. Modular forms are simply too difficult to build when the requirements are so constrained. “The objects you’re looking for, you don’t really know they exist,” Pilloni said.
Instead, the mathematicians showed that it would be enough to construct a modular form whose numbers matched those of the abelian surface in a weaker sense. The modular form’s numbers only had to be equivalent in the realm of what’s known as clock arithmetic.
Imagine a clock: If the hour hand starts at 10 and four hours pass, the clock will point to 2. But clock arithmetic can be done with any number, not just (as in the case of real-world clocks) the number 12.
Boxer, Calegari, Gee and Pilloni only needed to show that their two sets of numbers matched when they used a clock that goes up to 3. This meant that, for a given abelian surface, the mathematicians had more flexibility when it came to building the associated modular form.
But even this proved too difficult.
Then they stumbled on a trove of modular forms whose corresponding numbers were easy to calculate — so long as they defined their numbers according to a clock that goes up to 2. But the abelian surface needed one that goes up to 3.
The mathematicians had an idea of how to roughly bridge these two different clocks. But they didn’t know how to make the connection airtight so they could find a true match for the abelian surface in the world of modular forms. Then a new piece of mathematics appeared that turned out to be just what they needed.
Surprise Help
In 2020, a number theorist named Lue Pan posted a proof about modular forms that didn’t initially seem connected to the quartet’s problem. But they soon recognized that the techniques he’d developed were surprisingly relevant. “I didn’t expect that,” Pan said.
After years of regular meetings, mostly on Zoom, the mathematicians started to make progress adapting Pan’s techniques, but major hurdles remained. Then, in the summer of 2023, Boxer, Gee and Pilloni saw a conference in Bonn, Germany, as the perfect opportunity to come together. The only problem was that Calegari was supposed to travel to China at the same time to give a talk. But a difficult visit to the Chinese consulate in Chicago made him reconsider. “Eight hours later, my visa was rejected and my car was towed,” he said. He decided to scrap the China talk and join his collaborators in Germany.
Gee secured the team a room in the basement of the Hausdorff Research Institute, where they were unlikely to be disturbed by itinerant mathematicians. There, they spent an entire week working on Pan’s theorem, one 12-hour day after the next, only coming up to ground level occasionally for caffeine. “After a coffee, we would always joke that we had to go back to the mine,” Pilloni said.
The grind paid off. “There were many twists to come later,” Calegari said, “but at the end of that week I thought we more or less had it.”
It took another year and a half to turn Calegari’s conviction into a 230-page proof, which they posted online in February. Putting all the pieces together, they’d proved that any ordinary abelian surface has an associated modular form.
Their new portal could one day be as powerful as Taylor and Wiles’ result, revealing more about abelian surfaces than anyone thought possible. But first, the team will have to extend their result to non-ordinary abelian surfaces. They’ve teamed up with Pan to continue the hunt. “Ten years from now, I’d be surprised if we haven’t found almost all of them,” Gee said.
The work has also allowed mathematicians to formulate new conjectures — such as an analogue of the Birch and Swinnerton-Dyer conjecture that involves abelian surfaces instead of elliptic curves. “Now we at least know that the analogue makes sense” for these ordinary surfaces, said Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology. “Previously we did not know that.”
“Lots of things that I had dreamed we would be able to one day prove are now within reach because of this theorem,” he added. “It changes things.”
