Lorentz Center December 3-5, 2001 Anabelian number theory and geometry

F. Oort (Utrecht) and P. Stevenhagen (Leiden)

Speakers:

Frits Beukers
Gunther Cornelissen
Bart de Smit
Richard Groenewegen
Ben Moonen
Frans Oort
Peter Stevenhagen
Jaap Top
Marius van der Put
Torsten Wedhorn

Informal lecture notes for the workshop have been written by Frans Oort.

Please register on-line if you want to attend. This will put you on the list of participants.
Registered participants are entitled to free coffee and tea. There is no registration fee.

Theme

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.

The following topics will be discussed:

• anabelian results for number fields (the Neukirch-Uchida theorem);
• the Grothendieck conjecture for affine curves (Nakamura-Tamagawa)
• the Grothendieck conjecture for curves over local fields (Mochizuki)

Maintained by Peter Stevenhagen (psh@math.leidenuniv.nl)
Last modified: Thursday, 22-Nov-2001 15:54:42 CET