Quantum Statistical Mechanics of the Absolute Galois Group

Yuri I. Manin, Max Planck Institute for Mathematics, Germany, Matilde Marcolli, California Institute of Technology, USA
2020 Symmetry, Integrability and Geometry: Methods and Applications  
We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's dessins d'enfant, the embedding in the Grothendieck-Teichmüller group, and the Drinfeld-Ihara involution. An interested reader will find much more details and basic references in [21, Chapter 3, Section 2]. This construction of the Bost-Connes system was then generalized in [22, 36, 46, 63] , to the
more » ... of abelian extensions of arbitrary number fields. Given a number field K, there is an associated quantum statistical mechanical system BC K with the properties that its partition function is the Dedekind zeta function of K, and its symmetry group is the Galois group of the maximal abelian extension Gal K ab /K . Moreover, there is an arithmetic subalgebra A K with the property that evaluations of zero-temperature KMS states on elements of A K take values in K ab , and intertwine the action of Gal(K ab /K) by symmetries of the quantum statistical mechanical system with the Galois action on K ab . Later, it was shown in [24, 25] that the quantum statistical mechanical system BC K completely determines the number field K. Other generalizations of the Bost-Connes system were developed for the abelian varieties related to 2-lattices and the Shimura variety of GL 2 (see [20] ), and for more general Shimura varieties in [36] and abelian varieties in [62] . See also [21, Chapter 3] for an overview of these arithmetic quantum statistical mechanical models.
doi:10.3842/sigma.2020.038 fatcat:qbibph6e2nfb3f6d6nt7zk6344