arithmetische-geometrie.txt

=== Arithmetischer Situs

Alain Connes, Caterina Consani

http://arxiv.org/abs/1405.4527

We show that the non-commutative geometric approach to the Riemann zeta
function has an algebraic geometric incarnation: the "Arithmetic Site". This
site involves the tropical semiring viewed as a sheaf on the topos which is the
dual of the multiplicative semigroup of positive integers. We prove that the
set of points of the arithmetic site over the maximal compact subring of the
tropical semifield is the non-commutative space quotient of the adele class
space of Q by the action of the maximal compact subgroup of the idele class
group. We realize the Frobenius correspondences in the square of the
"Arithmetic Site" and compute their composition. This note provides the
algebraic geometric space underlying the non-commutative approach to RH.


=== Körper mit einem Element

Yuri I. Manin
Cyclotomy and analytic geometry over F_1

http://arxiv.org/abs/0809.1564

Geometry over non--existent "field with one element" F1 conceived by Jacques
Tits [Ti] half a century ago recently found an incarnation, in at least two
related but different guises. In this paper I analyze the crucial role of roots
of unity in this geometry and propose a version of the notion of "analytic
functions" over F1. The paper combines a focused survey with some new
constructions.
In new version, several local additions and changes are made, references added.