## Abstract

The zeta function of a number field can be interpreted as the partition

function of an associated quantum statistical mechanical (QSM) system, built from

abelian class field theory.

We introduce a general notion of isomorphism of QSM-systems and prove that it

preserves (extremal) KMS equilibrium states.

We prove that two number fields with isomorphic quantum statistical mechanical

systems are arithmetically equivalent, i.e., have the same zeta function. If one of

the fields is normal over Q, this implies that the fields are isomorphic. Thus, in this

case, isomorphism of QSM-systems is the same as isomorphism of number fields,

and the noncommutative space built from the abelianized Galois group can replace

the anabelian absolute Galois group from the theorem of Neukirch and Uchida.

function of an associated quantum statistical mechanical (QSM) system, built from

abelian class field theory.

We introduce a general notion of isomorphism of QSM-systems and prove that it

preserves (extremal) KMS equilibrium states.

We prove that two number fields with isomorphic quantum statistical mechanical

systems are arithmetically equivalent, i.e., have the same zeta function. If one of

the fields is normal over Q, this implies that the fields are isomorphic. Thus, in this

case, isomorphism of QSM-systems is the same as isomorphism of number fields,

and the noncommutative space built from the abelianized Galois group can replace

the anabelian absolute Galois group from the theorem of Neukirch and Uchida.

Original language | Undefined/Unknown |
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Title of host publication | Trends in Contemporary Mathematics |

Editors | Vincenzo Ancona, Elisabetta Strickland |

Publisher | Springer |

Pages | 47-57 |

Number of pages | 11 |

ISBN (Print) | 978-3-319-05254-0 |

DOIs | |

Publication status | Published - 2014 |

### Publication series

Name | Springer INdAM Series |
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Publisher | Springer International Publishing |