Gerd Faltings, mathematician who proved Mordell’s conjecture, wins Abel Prize at 71

Mordell’s conjecture, now known as Faltings’ theorem, concerns the number of special points on a curve.

By Joseph Howlett edited by Clara Moskowitz

This year’s Abel Prize, an annual lifetime achievement award in mathematics, is awarded by the Norwegian Academy of Sciences and Letters and was modeled on the Nobel Prize, was awarded to Gerd Faltings, a German mathematician who is best known for proving the influential Mordell conjecture in 1983. This conjecture has since been named “Faltings’ theorem” in his honor.

This award adds to the many accolades that Faltings, 71, has accumulated during his long career. This list includes the Fields Medal, the most coveted prize in mathematics, which Faltings won at the age of 32. “At the beginning of my career, I received the Fields Medal. And towards the end, I received the Abel Prize,” Faltings says. “It’s a beautiful duality.”

Faltings’ theorem concerns curves. Often these can be described by simple equations with two variables multiplied and added. Plot the solutions of such an equation on a coordinate grid, and they will form a line or an ellipse or a more complicated sinuous curve.

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Since the beginning of mathematics, people have looked for a rarefied subset of these solutions: “rational” points on the curve, where the coordinates are integers or fractions. These particular points maintain rich and complex relationships that belie a hidden order that mathematicians are trying to discover.

But there are infinitely many curves, and it seemed impossible to define all their rational points – until Faltings’ theorem, of course. He proved that if the equation of a curve has a variable raised to a power greater than 3, then it must have a finite number of such points. Only lines, quadratics (like circles) and cubic equations can have an infinite number.

The proof is considered the cornerstone of arithmetic geometry, the field that studies the curves and shapes represented by these types of equations.

“It’s absolutely fundamental,” says Noam Elkies, a mathematician at Harvard University, of Faltings’ proof. “The fact that Mordell’s conjecture is now a theorem and all the structures he developed have informed much work since then in neighboring fields.”

Mathematicians are still working out the consequences of the theorem, first conjectured by Louis Mordell in 1922. Just a few weeks ago, mathematicians announced that they had found a real limit to the number of curves of rational points can have.

The theorem that bears his name was just one of Faltings’ many mathematical achievements. These include an extended generalization of the curve theorem to multidimensional forms, which he proved in 1991, and major contributions to an important field known as “p-adic Hodge Theory,” which provides methods for studying such shapes and the equations that form them.

The five-member committee met to make the decision at the Institute for Advanced Study in Princeton, New Jersey, near the end of January, just as a winter storm blanketed the Northeast with a few feet of snow. “We had nothing to do but just sit around and talk about math,” said Helge Holden, chair of the committee, at the Abel Symposium, an event held the following week. “The hotel was short of provisions, so the bread was getting drier and drier. »

The choice is never easy, says Holden, whose four-year term as president ends this year. But their selection is difficult to dispute. “Gerd Faltings is a towering figure in arithmetic geometry,” says Holden. “His ideas and results reshaped the field. »

The field of mathematics has changed in many ways since Faltings made his major contributions. He doesn’t envy today’s mathematicians who rush to tackle the richest open problems, he says. “Now it seems like in anything interesting, there’s a huge group of people doing things,” he says. “I’m pretty happy I don’t have to compete with them.”

As for the excitement over this major achievement, Faltings betrays little, even by the stoic standards of German mathematicians. “I’m old and a lot has happened in my life, so I don’t jump around,” he says. “But it’s a very good thing.”