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The hyperring of adèle classes

- A. Connes, C. Consani
- Mathematics
- 1 February 2011

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 K… Expand

Double Complexes and Euler L-Factors

- C. Consani
- Mathematics
- 1 May 1998

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 a… Expand

Schemes over 𝔽1 and zeta functions

- A. Connes, C. Consani
- Mathematics
- Compositio Mathematica
- 11 March 2009

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 Riemann… Expand

ARITHMETIC ON A QUINTIC THREEFOLD

- C. Consani, J. Scholten
- Mathematics
- 1 November 2001

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 action… Expand

The Arithmetic Site

- A. Connes, C. Consani
- Mathematics
- 18 May 2014

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… Expand

Characteristic 1 , entropy and the absolute point

- A. Connes, C. Consani
- Mathematics
- 1997

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… Expand

From monoids to hyperstructures: in search of an absolute arithmetic

- A. Connes, C. Consani
- Mathematics
- 24 June 2010

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 intersection… Expand

Quantum statistical mechanics over function fields

- C. Consani, M. Marcolli
- Mathematics
- 15 July 2006

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. It… Expand

Geometry of the Arithmetic Site

- A. Connes, C. Consani
- Mathematics
- 19 February 2015

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 of… Expand

On the arithmetic of the BC-system

- A. Connes, C. Consani
- Mathematics
- 24 March 2011

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-system… Expand

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