Halloween-talk : prep-notes

Posted by on Oct 31, 2008 in KapranovSmirnov, Kurokawa2005, Manin1995, Manin2006, Manin2008, newsNo comments

This afternoon I’ll give the first in a series of talks on F_un-geometry in our Art-seminar. In the following sessions I will give a detailed account of the construction of commutative and non-commutative algebraic geometry over \mathbb{F}_1, but as it is Halloween today, I’ll start off with a couple of ghost-stories on conjectural applications of F_un to other fields.

After a very brief historical intro, I’ll focus on this basic question : what geometric object is \wis{spec}(\Z) considered over the ‘absolute point’ \wis{spec}(\mathbb{F}_1)? 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, \wis{spec}(\Z) 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 \wis{spec}(\Z) might be considered as the three-sphere S^3 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 = \infty : This is supported by the fact that we are unable to realize \wis{spec}(\Z) as an affine \mathbb{F}_1-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 \mathbb{F}_1-approximation to the elusive arithmetic line. In this set-up, primes correspond to factors of these Witt-functors as exemplified by the decomposition \hat{\Z} = \prod_p \hat{\Z}_p 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…

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