October 9, 2024
4 min read
A Century-Old Question Is Still Revealing Answers in Fundamental Math
Mathematicians have made lots of recent progress on a question called the Mordell conjecture, which was posed a century ago
A boy at a blackboard working on a conceptual representation of math.
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After German mathematician Gerd Faltings proved the Mordell conjecture in 1983, he was awarded the Fields Medal, often described as the “Nobel Prize of Mathematics.” The conjecture describes the set of conditions under which a polynomial equation in two variables (such as x2 + y4 = 4) is guaranteed to have only a finite number of solutions that can be written as a fraction.
Faltings’s proof answered a question that had been open since the early 1900s. Furthermore, it opened new mathematical doors to other unanswered questions, many of which researchers are still exploring today. In recent years mathematicians have made tantalizing progress in understanding these offshoots and their implications for even fundamental mathematics.
The proof of the Mordell conjecture concerns the following situation: Suppose that a polynomial equation in two variables defines a curved line. The question at the heart of the Mordell conjecture is: What’s the relationship between the genus of the curve and the number of rational solutions that exist for the polynomial equation that defines it? The genus is a property related to the highest exponent in the polynomial equation describing the curve. It is an invariant property, meaning that it remains the same even when certain operations or transformations are applied to the curve.
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The answer to the Mordell conjecture’s central question, it turns out, is that if an algebraic curve is of genus two or greater, there will be a finite number of rational solutions to the polynomial equation. (This number excludes solutions that are just multiples of other solutions.) For genus zero or genus one curves, there can be infinitely many rational solutions.
“Just over 100 years ago, Mordell conjectured that this genus controlled the finiteness or infiniteness of rational points on one of these curves,” says Holly Krieger, a mathematician at the University of Cambridge. Consider a point (x, y). If both x and y are numbers that can be written as fractions, then (x, y) is a rational point. For instance, (1⁄3, 3) is a rational point, but (√2, 3) isn’t. Mordell’s idea meant that “if your genus was sufficiently large, your curve is somehow geometrically complicated,” Krieger says. She gave an invited lecture at the 2024 Joint Mathematics Meetings about the about the history of the Mordell conjecture and some of the work that has followed it.
Faulting’s proof ignited new possibilities for exploring questions that expand on the Mordell conjecture. One such exciting question—the Uniform Mordell-Lang conjecture—was posed in 1986, the same year that Faltings was awarded the Fields Medal.
The Uniform Mordell-Lang conjecture, which was formalized by Barry Mazur of Harvard University, was “proved in a series of papers culminating in 2021,” Krieger says. The work of four mathematicians—Vesselin Dimitrov of the California Institute of Technology, Ziyang Gao of the University of California, Los Angeles, and Philipp Habegger of the University of Basel in Switzerland, who were collaborators, and Lars Kühne of University College Dublin, who worked individually—led to proving that conjecture.
For the Uniform Mordell-Lang conjecture, mathematicians have been asking: What happens if you broaden the mathematical discussion to include higher-dimensional objects? What, then, can be said about the relationship between the genus of a mathematical object and the number of associated rational points? The answer, it turns out, is that the upper bound—meaning highest possible number—of rational points associated with a curve or higher-dimensional object such as a surface depends only on the genus of that object. For surfaces, the genus corresponds to the number of holes in the surface.
There’s an important caveat, however, according to Dimitrov, Gao and Habegger. “The geometric objects (curves, surfaces, threefolds etc.) [must] be contained inside a very special kind of ambient space, a so-called abelian variety,” they wrote in an e-mail to Scientific American. “An abelian variety is itself also ultimately defined by polynomial equations, but it comes equipped with a group structure. Abelian varieties have many surprising properties and it is somewhat of a miracle that they even exist.”
The proof of the Uniform Mordell-Lang conjecture “is not only the resolution of a problem that’s been open for 40 years,” Krieger says. “It touches at the heart of the most basic questions in mathematics.” Those questions are focused on finding rational solutions—ones that can be written as a fraction—to polynomial equations. Such questions are often called Diophantine problems.
The Mordell conjecture “is kind of an instance of what it means for geometry to determine arithmetic,” Habegger says. The team’s contribution to proving the Uniform Mordell-Lang conjecture showed “that the number of [rational] points is essentially bounded by the geometry,” he says. Therefore, having proved Uniform Mordell-Lang doesn’t give mathematicians an exact number on how many rational solutions there will be for a given genus. But it does tell them the maximum possible number of solutions.
The 2021 proof certainly isn’t the final chapter on problems that are offshoots from the Mordell conjecture. “The beauty of Mordell’s original conjecture is that it opens up a world of further questions,” Mazur says. According to Habegger, “the main open question is proving Effective Mordell”—an offshoot of the original conjecture. Solving that problem would mean entering another mathematical realm in which it’s possible to identify exactly how many rational solutions exist for a given scenario.
There’s a significant gap to bridge between the information given by having proved the Uniform Mordell-Lang conjecture and actually solving the Effective Mordell problem. Knowing the bound on how many rational solutions there are for a given situation “doesn’t really help you” pin down what those solutions are, Habegger says.
“Let’s say you know that the number of solutions is at most a million. And if you only find two solutions, you’ll never know if there are more,” he says. If mathematicians can solve Effective Mordell, that will put them tremendously closer to being able to use a computer algorithm to quickly find all rational solutions rather than having to tediously search for them one by one.
