# Dynamical system problem studied by Bost--Connes

We state the problem solved by Bost and Connes in [BC] (J-B. Bost, A. Connes, Selecta Math. (New Series), 1, (1995) 411-457) and its analogue for number fields, considered in [HL](D.  Harari, E. Leichtnam, Selecta Mathematica, New Series 3 (1997), 205-243), [ALR](J. Arledge, M. Laca, I. Raeburn, Doc. Mathematica 2 (1997) 115-138) and [Coh](Paula B Cohen, J. Théorie des Nombres de Bordeaux, 11 (1999), 15-30). A (unital) -algebra is an (unital) algebra over the complex numbers with an adjoint , , that is, an anti-linear map with , , , and a norm with respect to which is complete and addition and multiplication are continuous operations. One requires in addition that for all . A dynamical system is a pair , where is a 1-parameter group of -automorphisms . A state on a -algebra is a positive linear functional on satisfying . The definition of Kubo-Martin-Schwinger (KMS) of an equilibrium state at inverse temperature is as follows.

Definition: Let be a dynamical system, and a state on . Then is an equilibrium state at inverse temperature , or -state, if for each there is a function , bounded and holomorphic in the band and continuous on its closure, such that for all ,

 (1)

A symmetry group of the dynamical system is a subgroup of commuting with :

Consider now a system with interaction. Then, guided by quantum statistical mechanics, we expect that at a critical temperature a phase transition occurs and the symmetry is broken. The symmetry group then permutes transitively a family of extremal - states generating the possible states of the system after phase transition: the -state is no longer unique. This phase transition phenomenon is known as spontaneous symmetry breaking at the critical inverse temperature . We state the problem related to the Riemann zeta function and solved by Bost and Connes.

Problem 1: Construct a dynamical system with partition function the zeta function of Riemann, where is the inverse temperature, having spontaneous symmetry breaking at the pole of the zeta function with respect to a natural symmetry group.

The symmetry group turns out to be the unit group of the ideles, given by where the product is over the primes and . We use here the normalisation . This is the same as the Galois group , where is the maximal abelian extension of the rational number field . The interaction detected in the phase transition comes about from the interaction between the primes coming from considering at once all the embeddings of the non-zero rational numbers into the completions of with respect to the prime valuations . The following natural generalisation of this problem to the number field case and the Dedekind zeta function was solved in [Coh] (see also [HL], [ALR]).

Problem 2: Given a number field , construct a dynamical system with partition function the Dedekind zeta function , where is the inverse temperature, having spontaneous symmetry breaking at the pole of the Dedekind function with respect to a natural symmetry group.

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