Alain Connes and Katia Consani have updated their paper On the notion of geometry over F_un In the revised version, they include a definition for the notion of varieties that are not affine and go a bit further adding a new section with proposals for a new notion of scheme over F_un that involves replacing [...]

Matilde Marcolli: Cyclotomy and endomotives; arXived on January 20th, 2009: arXiv:0901.3167v1.

Lecture notes (in PDF) from Pierre-Emmanuel Chaput’s talk at the MPI F_un seminar

Handwritten notes from the third and fourth lectures at the MPI F_un seminar.

Handwritten notes from the lectures at the MPI F_un seminar

Grothendieck’s anabelian geometry is an example of noncommutative F_un geometry. Javier starts this series with ramblings on how the folklore about F_un can be used to relate linear and permutation representations of finite groups.

Inspired by the equivalent definition of a scheme as a particular kind of functor, Soulé gives his definition of a gadget over F_un as a couple consisting on a functor, that takes the rôle of the functor of points, and an algebra, which encodes a ‘topology at infinity’ and gives us the extra information that is missing in the functor.

Nobushige Kurokawa: Zeta functions over Proc. Japan Acad., 81, Ser. A (2005), 180-184 Click on the picture below to get a PDF-version of the paper.

Any possible definition of a variey over F_un should have a nice extension of scalars to the integers. In particular, we might expect to be able to control the behavior of such extension for the easiest family of rings defined over F_un: its finite abelian extensions F_un^n.