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The hyperring of adèle classes
Abstract Text We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adele class space H K = A K / K × of a global field KExpand
Double Complexes and Euler L-Factors
In this paper we take into consideration a conjecture of Bloch equating in a suitable range and under some standard conjectures, the rank of the motivic cohomology of the special fiber of aExpand
Schemes over 𝔽1 and zeta functions
Abstract We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, whose corresponding zeta function is the complete RiemannExpand
This paper investigates some aspects of the arithmetic of a quintic threefold in Pr4 with double points singularities. Particular emphasis is given to the study of the L-function of the Galois actionExpand
The Arithmetic Site
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 aExpand
Characteristic 1 , entropy and the absolute point
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 notionExpand
From monoids to hyperstructures: in search of an absolute arithmetic
We show that the trace formula interpretation of the explicit formulas expresses the counting functionN.q/ of the hypothetical curveC associated to the Riemann zeta function, as an intersectionExpand
Quantum statistical mechanics over function fields
In this paper we construct a noncommutative space of "pointed Drinfeld modules" that generalizes to the case of function fields the noncommutative spaces of commensurability classes of Q-lattices. ItExpand
Geometry of the Arithmetic Site
We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient ofExpand
On the arithmetic of the BC-system
For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-systemExpand