Arithmetic Algebraic Geometry (AAG)

Grunddaten zu diesem Projekt

Beschreibung

Algebraic equations and their arithmetical properties have interested mankind since antiquity. One has only to think of the works of Pythagoras and Diophantus, which were a milestone in their time. For many centuries such problems have fascinated both serious mathematicians (Fermat, Gauss, ...) and amateurs alike. However, developments in recent years have transformed the subject into one of the central areas of mathematical research, which has relations with, or applications to, virtually every mathematical field, as well as an impact to contemporary everyday life (for example, the use of prime numbers and factorisation for encoding "smart" cards). The classical treatment of equations by analysis and geometry in the realm of complex numbers in this century has found a counterpart, in the similar theories over finite and p-adic fields, which have particular significance for arithmetic questions. The study of certain functions encoding arithmetic information and generalising the Riemann zeta-function (L-functions) has produced unexpected phenomena and links to groups constructed in a geometric way (K-groups). These functions are in turn related in a mysterious way to particular objects of representation theory (automorphic representations). The combination of integer arithmetic and complex analysis has found an interpretation which is motivated by the classical theory of Riemannian manifolds (Arakelov theory). Through the interaction of arithmetic and geometry, these different theories have led to a complex and far-reaching web of conjectures proposing a deep explanation for the observed phenomena. At the same time, this interaction and the combination of the new, powerful methods have enabled the solution of some of these conjectures as well as of some long-standing ones (Fermat's Last Theorem). It has turned out that only the combined effort of specialists from different areas made true success possible.