On Some of the Mathematical Contributions of Gerd

To get a feeling for our level of ignorance in the face of such questions, consider that, before Faltings, there was not a single curve X (of genus > 1) for which we knew this statement to be true for all number fields K over which X is defined! Already in the twenties, Weil and Siegel made serious attempts to attack the problem. Siegel, influenced by Weil's thesis, used methods of diophantine approximation, to prove that the number of integral solutions to a polynomial equation f(Xi Y) = 0 (i.e., solutions in the ring of integers of a number field K) is finite, provided that / defines a curve over K of genus > 0, or a curve of genus 0 with at least three points at infinity. In his thesis, Weil generalized Mordell's theorem on the finite generation of the group of rational points on an elliptic curve, to abelian varieties of any dimension. Weil then hoped to use this finite generation result for the rational points on the jacobian of a curve to go on to show that when a curve of genus > 1 is imbedded in its jacobian, only a finite number of the rational points of the jacobian can lie on the curve. Not finding a way to do this, he decided to call his proof of finite generation (the "theorem of Mordell-Weil" ) a thesis, despite Hadamard's advice not to be satisfied with half a result! After this work of Weil and Siegel there was little progressibriJiirty years. It was in the sixties and early seventies that several new developments occurred in algebraic geometry and number theory which were to influence Faltings (work of