From the article:

“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.”

Since ancient Greece, researchers have tried to isolate special rational points on curves. Now they have the first ever formula that applies uniformly to all curves.

By Joseph Howlett edited by Clara Moskowitz.