ARIVAF

The main topics of the project under submission is how properties of arithmetic nature vary in families of varieties. The notion of family has to be understood either in its geometric classical meaning of a flat morphism of schemes or in its categorical one of algebraic stack. The researchers involved in the project under submission are working on special aspects of this fundamental question using approaches of different nature in the general frame of arithmetic geometry, such as deformation theory, general theory of stacks, rational points on moduli spaces (modular curves, Shimura varieties, Hurwitz spaces etc.), étale fundamental groups of schemes and stacks, l-adic representations, diophantine geometry etc., all of which are very active areas of investigation worldwide. Modern arithmetic geometry is often considered to be born in France, under the impulse of A. Grothendieck in the 60's, who elaborated, together with the French school, its fundations in the EGA and SGA. The theory of stacks, Galois categories, l-adic cohomology etc. were formalized during this period and have been developped since then, being fundamental theoretic tools to understand the objects of arithmetic geometry. Around the same period, diophantine geometry, which focusses on 'varieties', that is reduced schemes of finite type and separated over fields and stems from problem of more effective nature, started to develop powerful technical tools such as heights to investigate questions related to rational points on varieties. The theoretical tools of Grothendieck's formalism and the more technical ones of diophantine geometry are now intimately mixed and the conjugation of both has maintained the impressive development of modern arithmetic geometry for the past 50 years. The purpose of this project is to understand better some of the main problems at stake in the study of variation of arithmetic properties in families as well as the different approaches and technics that were developped to try and tackle them. Our basic strategy is to organize series of short workshops (averagely three days workshops, twice or three times a year) around common themes of interest. Half of the talks will be given by researchers involved in the project, as in a working group, whereas the other half will be given by specialists of the topics. Hopefully, this sharing of knowledges should bring new insights to the specific problems studied by each of the researchers and help develop collaborations between them. Due to its transversal dimension and its learning purposes, we hope that our research project will interact constructively with other more specialized projects in arithmetic or diophantine geometry.r