Mathematicians just made a big leap forward on one of the field’s all-time favorite problems.
Curves—squiggly lines through space, such as a comet’s trajectory or a stock market trend—are some of math’s simplest objects. But even though they have been studied for thousands of years, mathematicians still have some basic questions about them left unanswered.
Number theorists have particularly sought special points on a curve with coordinates on an x–y grid that are either whole numbers or fractions. These rarified points are often interrelated in complicated and meaningful ways. “We’re mathematicians, and we care about structure,” says Barry Mazur, Gerhard Gade University Professor at Harvard University. That structure can sometimes be useful; the rational points on so-called elliptic curves gave birth to a whole branch of cryptography, for instance.
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But there’s a vast menagerie of curves out there, composed of numerous infinite families, and each has its own structure of rational points. Number theorists have dreamed of finding a concrete mathematical rule that applies to every curve. But such a unilateral formula has long eluded them.
That changed a few weeks ago. In a preprint paper posted on February 2, three Chinese mathematicians placed the first ever hard upper limit on the number of rational points any curve can have. The mathematical consequences are limitless.
“This really is an amazing result that sets a new standard for what to expect,” says Hector Pasten, a mathematician at the Pontifical Catholic University of Chile, who wasn’t involved in the work.
Finite or Infinite?
Curves are mathematically represented by simple equations called polynomials. They’re essentially a handful of variables multiplied and added together.
Think of the equation x2 + y2 = 1. If x and y are the two axes of a coordinate plane, this equation represents a circle. Every point on the circle corresponds to a different solution to this equation. For instance, the point x = 1 and y = 0, written as the coordinate pair (1, 0), is on the circle: if you put those values for x and y into the equation, you get 1 = 1, which is a valid solution.
Some solutions, including (1, 0) and (3⁄5, 4⁄5), are “rational,” meaning both x and y are either whole numbers or ratios of whole numbers. Other solutions, such as (1⁄√2, 1⁄√2), are “irrational.” Plug in these values for x and y, and you get a valid solution to the equation—the coordinates land right on the circle. But you can never express them in terms of whole numbers and their ratios.
Ancient Greek mathematicians were obsessed with finding rational points along curves. They wondered how many of these special points a given curve has. It’s one of the simplest questions in math, but it has vexed mathematicians for millennia. “These problems sit at the heart of number theory,” says Shenxuan Zhou, a mathematician at the Toulouse Mathematics Institute who co-authored the new result.
The circle—a particular kind of curve—has infinitely many rational points. The same is true for any other curve where neither x nor y is raised to a power bigger than 2. These “degree 2” equations always either have no rational points at all or infinitely many. The number of rational points on a curve that is one degree higher, degree 3, is sometimes infinite and sometimes finite.
But in 1922 Louis Mordell made a famous conjecture that indicated the situation sharply changes for higher-degree equations. It stated that when the degree of a curve is 4 or more, there will always be a finite number of rational points.
Sixty-one years later Gerd Faltings proved Mordell right; he was rewarded with a Fields Medal, math’s highest honor. But Mordell’s conjecture, now called Faltings’s theorem, says nothing about how many points these curves have.
Since then, mathematicians have sought a formula to answer this question. “We just know that there is a formula,” Pasten says. “It’s somewhere out there, and that’s good, but we want it.”
A Rule for Every Curve
That’s where the new proof comes in. Its authors present a formula that can be applied to any curve in the mathematical universe, whatever its degree. It doesn’t say precisely how many rational points that curve has, but it gives an upper limit on what that number can be.
Previous formulas of this kind either didn’t apply to all curves or depended on the specific equation used to define them. The new formula is something mathematicians have hoped for since Faltings’s proof, a “uniform” statement that applies to all curves without depending on the coefficients in their equations. “This one statement gives us a broad sweep of understanding,” Mazur says.
It depends on only two things. The first is the degree of the polynomial that defines the curve—the higher the degree is, the weaker the statement becomes. The second thing the formula depends on is called the “Jacobian variety,” a special surface that can be constructed from any curve. Jacobian varieties are interesting in their own right, and the formula offers a tantalizing path for studying them as well.
The new result is a first step toward knowing how many points curves have, not just whether or not they have an infinite number of points. “There are more questions on the horizon,” Pasten says. “We can get more ambitious now.”
Curves are also just a first foothold on the mathematical world of shapes carved out by equations. Polynomial equations with additional variables besides x and y can generate more complicated objects, such as surfaces or their higher-dimensional analogues, called “manifolds.” Manifolds are central to modern mathematics, as well as theoretical physics, where they’re used to map out space and time.
All these questions about rational points matter for those higher-dimensional objects, too. Pasten and mathematician Jerson Caro placed an upper bound on the number of rational points for certain surfaces in a 2023 paper, for example. The new result gives Pasten hope for further progress in this far broader quest.
This finding is one of several recent new results about rational points on curves. Taken together, the surge might signify a new chapter in this millennia-old saga.
“This is an exciting, fast-moving area,” Mazur says. “There’s something big happening right now.”