Quantum statistical mechanics over function fields

@article{Consani2006QuantumSM,
  title={Quantum statistical mechanics over function fields},
  author={Caterina Consani and Matilde Marcolli},
  journal={Journal of Number Theory},
  year={2006},
  volume={123},
  pages={487-528}
}
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TLDR
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References

SHOWING 1-10 OF 50 REFERENCES
From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices
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
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
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
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
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
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
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
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
...
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