Gerd Faltings

fl. 1954, Gelsenkirchen, West Germany

Arithmetic and number theoryAlgebraGeometryFields Medal 1986Shaw Prize in Mathematical Sciences 2015

German mathematician who proved the Mordell conjecture on finiteness of rational points and founded the quantitative tools — heights, arithmetic intersection theory, p-adic Hodge theory — of modern arithmetic geometry.

Gerd Faltings was born on 28 July 1954 in Gelsenkirchen, West Germany. He studied mathematics and physics at the University of Münster, earning his doctorate in 1978 under Hans-Joachim Nastold, and spent a postdoctoral year at Harvard. After professorships at Wuppertal and Princeton (1985–1994), he returned to Germany in 1995 as a professor at the University of Bonn and scientific member of the Max Planck Institute for Mathematics, whose director he later became.

His 1983 paper 'Endlichkeitssätze für abelsche Varietäten über Zahlkörpern' proved the Tate conjecture for abelian varieties over number fields, the Shafarevich finiteness conjecture and, as the crowning consequence, the Mordell conjecture — that a curve of genus at least two has only finitely many rational points. He later proved the Lang conjectures for abelian varieties and developed p-adic Hodge theory, arithmetic Riemann–Roch theorems and the theory of almost étale extensions, methods that underlie much of today's p-adic arithmetic geometry.

Faltings received the Fields Medal in 1986 at Berkeley and the Shaw Prize in Mathematical Sciences in 2015. His doctoral students include Peter Scholze, and his school in Bonn remains a center of arithmetic geometry.

Workplaces: University of Bonn / Max Planck Institute for Mathematics, Princeton University

Germany

Award-winning work

Fields Medal 1986For his proof of the Mordell conjecture. In 1983 Faltings proved that a curve of genus at least two over a number field has only finitely many rational points, resolving a conjecture of Mordell from 1922 and immediately implying that for

n \geq 4

the Fermat equation

x^n + y^n = z^n

has at most finitely many coprime integer solutions. The proof proceeded by first establishing the Tate conjecture on endomorphisms of abelian varieties and the Shafarevich conjecture on finiteness of isomorphism classes, introducing the 'Faltings height' to compare arithmetic complexity of abelian varieties — tools that founded modern arithmetic geometry as a quantitative theory.

Contributions, linked to the library

Library topicArithmetic geometryHis finiteness theorems and p-adic methods built the quantitative foundations of arithmetic geometry.Great problemTate conjectureProved the Tate conjecture for abelian varieties over number fields (1983) as a step toward Mordell.Great problemFermat's Last TheoremThe Mordell conjecture implied that each Fermat equation with exponent at least 4 has only finitely many primitive solutions.