01 · A life in mathematics
The person behind the work
Gerd Faltings was born in Gelsenkirchen-Buer in what was then West Germany and was a Princeton professor when he received the 1986 Fields Medal. His proof of the Mordell conjecture established that a smooth projective curve of genus greater than one over a number field has only finitely many rational points over that field. The proof connected curves with their Jacobian abelian varieties and deep finiteness theorems in arithmetic geometry. It transformed a question about solving equations into one about the organisation of geometric objects. Faltings’ result proves finiteness; it does not, by itself, provide a general algorithm that lists every point.
The contribution, explained
A rational point on a curve is a solution whose coordinates lie in the chosen number field. Curves behave differently according to their genus: some have infinitely many rational points, while sufficiently complicated ones do not. Faltings proved that every smooth projective curve of genus greater than one over a number field has only finitely many such points. His approach studies the curve’s Jacobian and finiteness properties of abelian varieties, bringing geometry and arithmetic together. The theorem rules out an endless supply of solutions without necessarily showing where the last one lies or giving an effective search bound.
Why it matters
The proof resolved the Mordell conjecture and established arithmetic-geometric methods as powerful tools for studying rational solutions, influencing later finiteness questions far beyond individual equations.