Holmstrom Field with one element II

http://mathoverflow.net/questions/2300/what-is-the-field-with-one-element

http://mathoverflow.net/questions/1628/kf-1-sphere-spectrum

http://ncatlab.org/nlab/show/field+with+one+element

http://www.noncommutative.org/index.php/the-smirnov-letters.html

http://www.noncommutative.org/index.php/prep-notes-dump.html

angst seminar

http://www.noncommutative.org/index.php/mathbbf_1-and-noncommutative-geometry.html

http://matrix.cmi.ua.ac.be/fun/index.php/connesconsani2011.html

Nonadditive prolegomena

See all arxiv articles of Connes, Consani, maybe Marcolli

arXiv:0911.3537 Characteristic one, entropy and the absolute point from arXiv Front: math.NT by Alain Connes, Caterina Consani We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point. After introducing the notion of “perfect” semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in positive characteristic to the limit case of characteristic one. This construction unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F_1-scheme and extend our earlier computations of the zeta function to cover the case of F_1-schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map from monoids to sets which comes close to fulfill the requirements for the hypothetical curve compactifying Spec Z over the absolute point. Finally, we test the computation of the zeta function on elliptic curves over the rational numbers.

arXiv:0910.3879 Meromorphicity of some deformed multivariable zeta functions for $F_1$-schemes from arXiv Front: math.NT by Norihiko Minami Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani

Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani

We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity

When restricted to the log derivative of the modified Soulé type zeta functions, we find our invariant $\mu(A)$ for a finite abelian group $A$, introduced in ArXiv-0907.0918v2, plays an extremely important role in the Soulé type zeta functions.

nLab page on Field with one element II