## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Shai Haran

### Non additive geometry

Abstract

We give a simple generalization of commutative rings. We show one can develope algebraic geometry a la Grothendieck using these generalized rings.The initial object is "the field with one element". The new geometry contains the old one fully faithfully, but also the "integers" and "residue field" at real and complex primes of a number field. The arithmetical plane $Z(x)Z$ is not reduced to its diagonal.We have the module of Kahler differentials satisfying the usual exact sequences, and the derivatives of integers are non trivial. We have a "zeta mechine" which associates a meromorfic function to a compact valuation generalized ring : for the $p-$adic integers it gives the $p-$ factor of zeta, for the "real integers" it gives the $\gamma$ factor.