## 20 Citations

QUANTUM STATISTICAL MECHANICS, L-SERIES AND ANABELIAN GEOMETRY

- Mathematics
- 2010

It is known that two number fields with the same Dedekind zeta f unction are not neces- sarily isomorphic. The zeta function of a number field can be i nterpreted as the partition function of an…

The Weil Proof and the Geometry of the Adèles Class Space

- Mathematics
- 2009

This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adeles class space, which is the noncommutative space underlying Connes'…

Cyclotomy and endomotives

- Mathematics
- 2009

AbstractWe compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the…

Bost-Connes systems associated with function fields

- Mathematics
- 2011

With a global function field K with constant field F_q, a finite set S of primes in K and an abelian extension L of K, finite or infinite, we associate a C*-dynamical system. The systems, or at least…

Noncommutative Geometry and Arithmetic

- Mathematics
- 2010

This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the…

Curves, dynamical systems, and weighted point counting

- MathematicsProceedings of the National Academy of Sciences
- 2013

The method of proof is to show that this is equivalent to the respective class field theories of the curves being isomorphic as dynamical systems, in a sense that the result solves the analogue of the isospectrality problem for curves over finite fields.

Bost-Connes type dynamics and arithmetic of function fields

- Mathematics
- 2019

The work of this thesis falls within two main themes: the analogy between number fields and function fields of an algebraic curve over a finite field; and the recovery of arithmetic information from…

A walk in the noncommutative garden

- Mathematics
- 2006

This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a…

## References

SHOWING 1-10 OF 50 REFERENCES

From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices

- Mathematics
- 2004

Several recent results reveal a surprising connection between modular forms and noncommutative geometry. The first occurrence came from the classification of noncommutative three spheres,…

KMS states and complex multiplication

- Mathematics
- 2005

Abstract.We construct a quantum statistical mechanical system which generalizes the Bost–Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explicit…

From Physics to Number theory via Noncommutative Geometry

- Mathematics
- 2004

We give here a comprehensive treatment of the mathematical theory of per-turbative renormalization (in the minimal subtraction scheme with dimensional regularization), in the framework of the…

Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory

- Mathematics
- 1995

In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann ζ function. Our construction is general and associates to an inclusion of rings (under a suitable…

On the equilibrium states in quantum statistical mechanics

- Mathematics
- 1967

AbstractRepresentations of theC*-algebra
$$\mathfrak{A}$$
of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential μ are studied. Both for finite…

Bost–Connes type systems for function fields

- Mathematics
- 2006

We describe a construction which associates to any function field k and any place 1 of k a C*-dynamical system .Ck;1;� t/ that is analogous to the Bost-Connes system associated to Q and its…

Trace formula in noncommutative geometry and the zeros of the Riemann zeta function

- Mathematics
- 1998

Abstract. We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric…

Bost-Connes-Marcolli systems for Shimura varieties. Part I. Definitions and formal analytic properties

- Mathematics
- 2005

We construct a quantum statistical mechanical system $(A,s)$ analogous to the systems constructed by Bost-Connes and Connes-Marcolli in the case of Shimura varieties. Along the way, we define a new…