This paper gives an overview of the various approaches towards F1-geometry. In the first
part, we review all known theories in literature so far, which are: Deitmar's F1-schemes …

We develop basic notions and methods of algebraic geometry over the algebraic objects
called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an …

We determine the real counting function N (q)(q∈[1,∞)) for the hypothetical 'curve'over 𝔽1,
whose corresponding zeta function is the complete Riemann zeta function. We show that …

TEXT: We show that the theory of hyperrings, due to M. Krasner, supplies a perfect
framework to understand the algebraic structure of the adèle class space HK= AK/K× of a …

We show that the trace formula interpretation of the explicit formulas expresses the counting
function Nq/of the hypothetical curve C associated to the Riemann zeta function, as an …

In this paper, we introduce the category of blueprints, which is a category of algebraic
objects that include both commutative (semi) rings and commutative monoids. This …

We give conditions for the Mayer–Vietoris property to hold for the algebraic K-theory of blow-
up squares of toric varieties and schemes, using the theory of monoid schemes. These …

The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an
enrichment of the theory of commutative rings. In the same way, we can enrich usual …

TEXT: We show that the algebra and the endomotive of the quantum statistical mechanical
system of Bost–Connes naturally arises by extension of scalars from the “field with one …

We show that the mathematical meaning of working in characteristic 1 is directly connected
to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea …