Project overview

Developing powerful methods taken from geometry to study the arithmetical properties of algebraic equations

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.

Partnership

14 working groups from 7 countries with a wealth of expertise across all the fields

The project aims to study the relations between the different theories alluded to above by linking different working groups in several european countries, as well as some of their non european partners :

Coordinator :

M. BERTOLINI Tel : +39 02 50316125 Fax : +39 02 50316090 e-mail : Massimo.Bertolini@mat.unimi.it

Dipartimento di Matematica Università degli Studi di Milano Via Saldini 50 20133 Milano ITALIA

Work programme

The main themes of research of the network are:

Post- and pre-doctoral positions

The "Arithmetic Algebraic Geometry" network is sponsored by the European Commission's current programme for "Research Training Networks". In this framework, it is offering post- and pre-doctoral positions at each of its associated nodes, as part of its initiative to foster research in central subjects of this active domain of mathematical research.

Activities of the network

Previous network in arithmetic algebraic geometry

The present network replaces the network of the same name which was supported by the European Commission during the Fifth Framework Programme, under the programme "Improving Human Potential and the Socio-Economic Knowledge Base".