A Century-Old Question Is Still Revealing Answers in Fundamental Math

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October 9, 2024, 11:00 AM

• •The Mordell conjecture describes the set of conditions under which a polynomial equation in two variables is guaranteed to have only a finite number of solutions that can be written as a fraction.
• •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.
• •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.”

The Mordell conjecture is a question in number theory that was posed over a century ago. It asks about the number of rational solutions to a certain type of polynomial equation. In 1983, German mathematician Gerd Faltings proved the conjecture, which was a major breakthrough in number theory. In recent years, mathematicians have made progress on related questions, such as the Uniform Mordell-Lang conjecture. This conjecture was proved in 2021, and it has implications for even more fundamental questions in mathematics.

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