Graduate Course in Mathematics Stieltjes Onderwijs week Algorithmen in de getaltheorie

December 3-5, 2001

Anabelian number theory and geometry

Organizers: F. Oort (Utrecht), P. Stevenhagen (Leiden)

Galois showed how the concept of a group lets one describe properties of field extensions. This tool, Galois theory, has been one of the most powerful ingredients of algebra and number theory since then. Independently geometers studied coverings of topological spaces. These are described by a group, the fundamental group. In 1961 Grothendieck showed that these are disguises of one and the same concept. The notion of the algebraic fundamental group is central in many aspects of arithmetic geometry.

For a number field, and for many algebraic varieties this group is anabelian , which means it is very non-commutative (it is not trivial and every finite index subgroup has trivial center). Grothendieck conjectured that ``anabelian objects'' are determined by their fundamental group. This has been proved in several cases. Uchida and Neukirch showed that an isomorphism between Galois groups of number fields implies the existence of an isomorphism between those number fields. For algebraic curves over finite fields, over number fields and over p-adic fields the Grothendieck anabelian conjecture also has been proved (Nakamura, Tamagawa, Mochizuki).

As we expect these ideas and theorems to play a role of increasing importance in number theory and arithmetic geometry, we will discuss them in our workshop. As in several previous instances (cf. the Shafarevich conjecture), one sees analogous structures play a role in number theory and geometry. This analogy is both aesthetically pleasing and technically useful. In the present case, the interplay between arithmetic and geometric structures is largely controlled by the action of the arithmetic part of the fundamental group on the geometric part, as encoded in the structure of the algebraic fundamental group. This is the central theme of our workshop.

Besides proving the main anabelian results on number fields and on algebraic curves, we hope to discuss several related topics and tools, such as: the anabelian conjecture for curves over local fields (Mochizuki), properties of good reduction of curves described by a Galois representation (Takayuki Oda), Hurwitz schemes, alterations (De Jong), and anabelian conjectures for function fields (Pop).

Maintained by Peter Stevenhagen (psh@math.leidenuniv.nl) Last modified: Friday, 29-Jun-2001 11:22:53 MEST