Introduction

Abstract

At the moment fascinating developments are taking place in arithmetic algebraic geometry. Very prominent among these is that of Arakelov geometry. This is a way of “completing” a variety over the ring of integers of a number field by adding fibres over the archimedean places. In this way the analogy between algebraic number fields and function fields of algebraic curves is extended to a more precise analogy between arithmetic varieties and varieties fibered over a complete curve. Thus a completely new tool for attacking arithmetic problems has emerged from the Russian school of arithmetic algebraic geometry. The importance of this development lies not only in its direct results (like the proof of the Mordell Conjecture), but also in the link it establishes between number theory and complex analytic geometry.

Es zeigt sich hier einmal mehr, dass die Zahlentheorie zwar mit recht die Königin der Mathematik genannt wird, sie aber ihren Glanz, wie auch Königinnen selbst, nicht so sehr aus sich selbst als vielmehr aus den Kräften ihrer Untertanen zieht.

G. Faltings (1984)