December 3-5, 2001
Anabelian number theory and geometry
Informal lecture notes for the workshop have been written by Frans Oort.
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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: