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Non-Abelian Fundamental Groups in Arithmetic Geometry

19th July 2009 to 18th December 2009

Organisers: Professor M Kim (University College London), Professor J Coates (Cambridge), Professor F Pop (Pennsylvania), Dr M Saidi (Exeter) and Professor P Schneider (Münster)

Programme Theme

In the 1980's Grothendieck formulated his anabelian conjectures that brought to an hitherto-unexplored depth the interaction between topology and arithmetic. This suggested that the study of non-abelian fundamental groups could lead to a new understanding of deep arithmetic phenomena, including the arithmetic theory of moduli and Diophantine finiteness on hyperbolic curves. A certain amount of work in recent years linking fundamental groups to Diophantine geometry intimates deep and mysterious connections to the theory of motives and Iwasawa theory, with their links with arithmetic problems on special values of L-functions such as the conjecture of Birch and Swinnerton-Dyer. In fact, the work thus far suggests that the still-unresolved section conjecture of Grothendieck, whereby maps from Galois groups of number fields to fundamental groups of arithmetic curves are all proposed to be of geometric origin, is exactly the sort of key problem that touches the core of all these areas of number theory and more.

The goal of this programme is to investigate the ideas and problems of anabelian geometry within the global context of mainstream arithmetic geometry. By bringing together leading experts in number theory and arithmetic geometry we hope to shed more light on the inter-connections between anabelian geometry and more classical Diophantine problems, and hopefully make progress towards the solution of the section conjecture.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons