Mathematicians just took a giant leap forward on one of the field’s all-time favorite problems.
Curves – sinuous lines through space, such as the path of a comet or a stock market trend – are some of the simplest objects in mathematics. But even though they have been studied for thousands of years, mathematicians still have some fundamental questions about them unanswered.
Number theorists have particularly sought special points on a curve with coordinates of one x–y grids that are either whole numbers or fractions. These rarified points are often connected in complicated and meaningful ways. “We are 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, for example, gave birth to a whole branch of cryptography.
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But there is a vast menagerie of curves out there, composed of many infinite families, each having its own structure of rational points. Number theorists have dreamed of finding a concrete mathematical rule that applies to every curve. But such a one-sided formula has long eluded them.
That changed a few weeks ago. In a preprint paper posted on February 2, three Chinese mathematicians set the first ever hard upper bound on the number of rational points a curve can have. The mathematical consequences are unlimited.
“This is really an amazing result that sets a new standard for what you can expect,” said Hector Pasten, a mathematician at the Pontifical Catholic University of Chile, who was not involved in the work.
Finite or infinite?
Curves are mathematically represented by simple equations called polynomials. They are essentially a handful of variables multiplied and added together.
Consider the equation x2 + y2 = 1. If x and y are the two axes of a coordinate plane, this equation represents a circle. Each point on the circle corresponds to a different solution to this equation. For example, the point x = 1 and y = 0, written as the coordinate pair (1, 0), is on the circle: if you set these 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 yand 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 ratio.
Ancient Greek mathematicians were obsessed with finding rational points along curves. They wondered how many of these special points a given curve has. It is one of the simplest questions in mathematics, but it has vexed mathematicians for millennia. “These problems sit at the heart of number theory,” said Shengxuan Zhou, a mathematician at the Toulouse Mathematics Institute who co-authored the new result.
The circle – a special type of curve – has infinitely many rational points. The same applies to all other curves where none of them x neither y is raised to a power greater than 2. These “degree 2” equations always have either no rational points at all or infinitely many. The number of rational points on a curve one degree higher, degree 3, is sometimes infinite and sometimes finite.
But in 1922 Louis Mordell came up with a famous conjecture indicating that the situation changed greatly for equations of higher order. It stated that when the degree of a curve is 4 or more, there will always be a finite number of rational points.
61 years later, Gerd Faltings proved that Mordell was right; he was awarded a Fields Medal, mathematics’ highest honor. But Mordell’s conjecture, now called Falting’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 it there is a formula,” Pasten says. “It’s somewhere out there, and it’s good, but we want it.”
A rule for each curve
This is where the new evidence comes in. The authors present a formula that can be applied to any curve in the mathematical universe, regardless of degree. It doesn’t say exactly how many rational points that curve has, but it gives an upper bound on what the number can be.
Previous formulas of this type either did not apply to all curves or depended on the specific equation used to define them. The new formula is something mathematicians have been hoping for since Falting’s proof, a “uniform” statement that applies to all curves without depending on the coefficients in their equations. “This one statement gives us a wide range of understanding,” says Mazur.
It just depends on two things. The first is the degree of the polynomial that defines the curve – the higher the degree, the weaker the statement. The other thing the formula depends on is called the “Jacobian variant”, a special surface that can be constructed from any curve. Jacobian varieties are interesting in their own right, and the formula offers a tempting avenue for studying them as well.
The new result is a first step towards knowing how many points curves have, not just whether they have an infinite number of points or not. “There are more questions on the horizon,” says Pasten. – We can become more ambitious now.
Curves are also only a first foothold in the mathematical world of shapes carved out of equations. Polynomial equations with additional variables in addition 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 are used to map space and time.
All these questions about rational points also mean for the higher dimensional objects. For example, Pasten and mathematician Jerson Caro set an upper limit for the number of rational points for certain surfaces in a paper from 2023. The new result gives Pasten hope for further progress in this much broader search.
This finding is one of several recent results about rational points on curves. Altogether, the wave could mean a new chapter in this thousand-year-old saga.
“This is an exciting, fast-moving area,” says Mazur. – Something big is happening right now.
