He was awarded the Fields Medal at the ICM at Berkeley in 1986 for proving the Mordell conjecture, which states that any non-singular projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points. As a Fields Medallist he gave an ICM plenary talk Recent progress in arithmetic algebraic geometry. In 1994 as an ICM invited speaker in Zurich he gave a talk Mumford-Stabilität in der algebraischen Geometrie.