So for the equation x2 + y2 = 1, one rational solution is x = 3/5 and y = 4/5. The problem Kim is wrestling with dates all the way back to Diophantus of Alexandria, who studied such “Diophantine equations” in the third century A.D. The most significant recent result on the topic provided an important but blunt reframing of the problem: In 1986, Gerd Faltings won the Fields Medal, math’s highest honor, primarily for proving that certain classes of Diophantine equations have only finitely many rational solutions.
Faltings’ proof was what mathematicians call “Ineffective,” meaning that it didn’t actually count the number of rational solutions, let alone identify them.
This kind of proof arises especially often when mathematicians argue by contradiction: If there are infinitely many rational solutions, then you get a logical contradiction; therefore, there must be only finitely many rational solutions.
Since Faltings’ result, mathematicians have looked for “Effective” methods for finding rational solutions to Diophantine equations, and Kim has one of the most promising new ideas.
Mathematicians have made steady progress on Serre’s question over the last 40-plus years, but it involves an equation they just couldn’t handle-the cursed curve.
In 2002 the mathematician Steven Galbraith identified seven rational solutions to the cursed curve, but a harder and more important task remained: to prove that those seven are the only ones.
They constructed a specific geometric object that intersects the graph of the cursed curve at exactly the points associated to rational solutions.