Fermat’s Last Theorem Superpowered

“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.”

“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.”

“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.”

“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.”

“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.”