# Arithmetic Algebraic Geometry

### Gerd Faltings

Max-Planck-Institut für Mathematik, Bonn, Germany### Günter Harder

Universität Bonn, Germany### Nicholas M. Katz

Princeton University, USA

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## Abstract

At this workshop various aspects of Arithmetic Algebraic Geometry have been discussed. It was organized by G. Faltings, G. Harder (Max-Planck-Institute for Mathematics Bonn) and N. Katz (Princeton university) The main goal of this field is, to obtain information on the solution of diophantine problems by applying the tools provided by algebraic geometry. The workshop was attended by 42 participants and we had the total number of 18 talks. A very interesting diophantine problem, which has a very geometric flavour, is the investigation of the structure of the Brauer group of a scheme or more in geometric terms of a complex algebraic variety. This aspect has been discussed in the talks by Gabber, who reported on some recent progress in direction of the purity conjecture. It has also been discussed in the talks of Lieblich and de Jong, in which some more geometric questions have been discussed. Another tool to obtain information on diophantine problems is provided by $p$-adic methods. Here certain analogies between classical analytic theory over $\mathbf{C}$ and $p$-adic analytic theory have to be developed. We have to understand the meaning of local systems. This topic was discussed in the talks of Deninger and Ramero. Minhyong Kim outlined a program how to used a $p$-adic unipotent Albanese map to prove finiteness in diophantine geometry, for instance the classical theorem of Siegel. Shimura varieties are certainly interesting objects in Arithmetic Algebraic Geometry, they provide interesting examples of algebraic varieties. Ben Moonen's talk was on the borderline between $p$-adic methods and Shimura varieties. Laumon reported on the fundamental Lemma for $U(n)$. Various talks discussed the Galois representations attached to automorphic forms and abelan varieties (B\"ultel, Edixhoven,Harris, Diamond). In the talks of Pink and Ullmo Wildeshaus, Rapoport, M. Kings and Ullmo some other aspects of this field were discussed. Goncharov discussed some interesting aspects of higher Teichm\"uller theory.