paper with Consani The Hyperring of Adele Classes:

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Since the last activity here, there have been some interesting new Fun-developments, Connes and Consani posted at least two papers, Marcolli at least one, Manin put a new version of his cyclotomy paper online and there was even a ‘survey’-paper relating all different notions of Fun-geometry. So, imho, there’s plenty of new material to cover to keep the site going for a while. Interested in helping out?

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 the category of finite abelian groups by the category of monoids… where have I heard about that before?

During his lecture, he went on following the Connes-Consani paper and their construction of the gadget associated to Chevalley groups, that gives a variety defined over the quadratic extension .

• Lecture V: Chevalley schemes are defined over

Tomorrow, Peter Arndt from the University of Göttingen will give us an introduction to Durov’s approach to geometry. This will be the last talk at the F_un study seminar before Christmas. Hopefully Bram Mesland and Oliver Lorscheid will keep it running in January.

I will try to convince you that Grothendieck’s ‘dessins d’enfant’ form an example of a noncommutative manifold over the mythical field with one element (in the sense of Soule and Connes-Consani).

However, dessins only appear at the final slide. The main part of the talk consisted in explaining one sentence in Manin’s recent paper (page 4, line 3):

Soule’s definition of an -scheme involves besides , a -algebra , and each cyclotomic point of coming from must assign ‘values’ to the elements of . His choice of for the multiplicative group is that of continuous functions on the unit circle in … We suggest to consider the ring of Habiro’s analytic functions…

I promised Jacques to do a proper write-up of the talk (and include some more details on the final slide) so I might as well do a couple of posts on it, later.

After a very brief historical intro, I’ll focus on this basic question : what geometric object is considered over the ‘absolute point’ ? In particular, what is its ‘dimension’ and what geometric object corresponds to prime numbers? I’ll follow the answers and motivations given by Yuri Manin in The notion of dimension in geometry and algebra. There are (at least) three different answers to these questions :

dim = 1 : This is the classical approach we know from global field theory, is analogous to the affine line over a finite field, and, more generally prime ideals in number fields correspond to points on curves (over finite fields). Applications are to the Riemann hypothesis on zeta functions using the concept of ‘absolute motives’ as in Manin’s 1995 paper and recent work of Kurokawa.

dim = 3 : This is based on the Artin-Verdier-Mazur duality in etale topology suggesting that might be considered as the three-sphere with prime ideals corresponding to knots. Applications include the interpretation of power residue symbols and reciprocity laws as (higher) linking numbers as in the Kapranov-Smirnov paper and supported by recent work of Morishita.

dim = : This is supported by the fact that we are unable to realize as an affine -variety. Still, in his recent paper, Manin suggests that a Soule-version of Witt vectors, by restricting the values of the ‘ghost variables’ to cyclotomic numbers, might furnish a formal -approximation to the elusive arithmetic line. In this set-up, primes correspond to factors of these Witt-functors as exemplified by the decomposition of the profinite numbers.

My prep-notes are far from ideal and are only meant to prevent me from getting too lost in these ghost-worlds. Anyway, here they are. Comments are very wellcome!

I hope to turn these notes into a series of more readable posts in the upcoming days and weeks…

• Lecture III: The Bost-Connes system and the Riemann zeta function.
• Lecture IV: The BC system on “Fun with F_un”.

Be aware that these notes contain only the information I could take or the points I considered interesting, and not everything that Bram said on the lecture. All omissions or mistakes are solely my own fault.

For people interested on a more comprehensive introduction to the BC system, this post series by Lieven is a good starting point.

Next Tuesday at the F_un study seminar, Pierre-Emmanuel Chaput will tell us about the construction of the geometries associated to Chevalley group schemes, and how they are defined over [Unparseable or potentially dangerous latex formula. Error 6 ].

It is a bit odd hearing a talk without seeing any slides or blackboard images, but then, here is the streaming audio (you need to have Real Player installed).

Consani F_un talk at Fields Institute