From Physics to Number Theory via Noncommutative Geometry

Connes, Alain; Marcolli, Matilde

doi:10.1007/3-540-31347-8_8

Cited by 23 publications

(24 citation statements)

References 125 publications

(68 reference statements)

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“…The existence of an algebraic model of A Q was already shown in [3]. Ten years later Connes, Marcolli and Ramachandran [9] constructed in a beautiful way arithmetic models of A K in the case of K imaginary quadratic by drawing the connection to the theory of Complex Multiplication on the modular curve by using the GL 2 -system of [7]. A first approach towards the construction of (partial) arithmetic models of more general BC-systems A K was undertaken in [20] where the theory of Complex Multiplication on general Siegel modular varieties and the GSp 2n -systems of [12] were used to construct partial arithmetic models of A K in the case of K containing a CM field.…”

Section: Introductionmentioning

confidence: 93%

“…In [12] property (i) and (ii) were shown to hold for all BC-systems A K . The difficult problem of classifying the KM S β -states of BC-systems was solved by Laca, Larsen and Neshveyev [16] by building upon earlier work of [3], Connes and Marcolli [7], Laca [14] and Neshveyev [18], thus proving property (iii) and (iv) for all A K . From an arithmetic view point the most interesting property of BC-systems is the existence of arithmetic models.…”

Section: Introductionmentioning

confidence: 99%

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On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

Yalkinoglu

2012

Invent. math.

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This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes [3]. In particular our construction shows how the class field theory of an arbitrary number field can be realized through the dynamics of a certain operator algebra. This is achieved by working in the framework of Endomotives, introduced by Connes, Marcolli and Consani [5], and using a classification result of Borger and de Smit [1] for certain Λ-rings in terms of the Deligne-Ribet monoid. Moreover the uniqueness of the arithmetic model is shown by Sergey Neshveyev in an appendix. In the second part of the paper we introduce a base-change functor for a class of algebraic endomotives and construct in this way an algebraic refinement of a functor from the category of number fields to the category of Bost-Connes systems, constructed recently by Laca, Neshveyev and Trifkovic [13].

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

Yalkinoglu

2012

Invent. math.

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“…We recall some definitions and refer to [8], [12], and Chapter 3 of [13] for more information and for some physics background. After that, we introduce isomorphism of QSM-systems, and prove it preserves KMS-states (cf.…”

Section: Isomorphism Of Qsm Systemsmentioning

confidence: 99%

“…Bost and Connes ( [7]) introduced a QSM-system for the field of rational numbers, and [14], [15] did so for imaginary quadratic fields. More general QSM-systems associated to arbitrary number fields were constructed by Ha and Paugam in [27] as a special case of their more general class of systems for Shimura varieties, which in turn generalize the GL(2)-system of [12]. We recall here briefly the construction of the systems for number fields in an equivalent formulation (cf.…”

Section: A Qsm-system For Number Fieldsmentioning

confidence: 99%

Quantum Statistical Mechanics, L-Series and Anabelian Geometry I: Partition Functions

Cornelissen

Marcolli

2014

Trends in Contemporary Mathematics

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Abstract. It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C * -algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck's "anabelian" program, much like the Neukirch-Uchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups.In the second part of the paper, we use these systems to prove the following. If there is a group isomorphism ψ :L between the character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding L-series s) (not just the zeta function), then the number fields are isomorphic. This is also equivalent to the purely algebraic statement that there exists an isomorphism ψ as a above and a norm-preserving group isomorphism between the ideals of K and L that is compatible with the Artin maps via ψ.

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“…[19], [20] and §3 of [21]). We will analyze more closely the relation to the wild fundamental group in [11].…”

Section: The Universal Singular Framementioning

confidence: 99%

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Connes¹,

Marcolli²

2004

Internat. Math. Res. Notices

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We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain " motivic Galois group" U * , which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup of U * . The group U * arises through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U * is a semi-direct product by the multiplicative group Gm of a pro-unipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over Q, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.

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From Physics to Number Theory via Noncommutative Geometry

Cited by 23 publications

References 125 publications

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

Quantum Statistical Mechanics, L-Series and Anabelian Geometry I: Partition Functions

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