## noncommutative geometry and number theory

Paul Smith's excellent "Noncommutative Geometry and Algebra" page

M. Marcolli, "Lectures on arithmetic noncommutative geometry"

[abstract:] "This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory."

C. Consani and M. Marcolli (eds.), Noncommutative Geometry and Number Theory: Where Arithmetic meets Geometry and Physics (Vieweg Verlag, 2006)

[AMS website description:] "In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and $K$-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive $p$-adic groups, Shimura varieties, the local $L$-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect."

A. Connes, "Noncommutative geometry and the Riemann zeta function" from Mathematics: Frontiers and Perspectives 2000, V. Arnold et.al., eds. (2000) 35-55.

A. Connes, "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function", Selecta Math. (N.S.) 5 (1999) 29-106.

A. Connes, "Noncommutative geometry year 2000", GAFA special volume 2000 (2001)

[Abstract:] "We describe basic concepts of noncommutative geometry and a general construction extending the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. Basic tools of the theory, K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic index theorems, Hopf algebra symmetry are reviewed. They cover the global aspects of noncommutative spaces, such as the transformation $\theta \to 1/\theta$ for the NC torus $\Tb_{\theta}^2$, unseen in perturbative expansions in $\theta$ such as star or Moyal products. We discuss the foundational problem of "what is a manifold in NCG" and explain the role of Poincare duality in K-homology which is the basic reason for the spectral point of view. When specializing to 4-geometries this leads to the universal "Instanton algebra". We describe our work with G. Landi which gives NC-spheres $S_{\theta}^4$ from representations of the Instanton algebra. We show that any compact Riemannian spin manifold whose isometry group has rank $r \geq 2$ admits isospectral deformations to noncommutative geometries. We give a survey of our work with H. Moscovici on the transverse geometry of foliations which yields a diffeomorphism invariant geometry on the bundle of metrics on a manifold and a natural extension of cyclic cohomology to Hopf algebras. Then, our work with D. Kreimer on renormalization and the Riemann-Hilbert problem. Finally we describe the spectral realization of zeros of zeta and L-functions from the noncommutative space of Adele classes on a global field and its relation with the Arthur-Selberg trace formula in the Langlands program. We end with a tentalizing connection between the renormalization group and the missing Galois theory at Archimedian places."

A. Connes, "Formule de trace en geometrie non commutative et hypothese de Riemann", C.R.Sci. Paris, t.323, Serie 1 (Analyse) (1996) 1231-1235.;

[Abstract:] "We reduce the Riemann hypothesis for L-functions on a global field k to the validity (not rigorously justified) of a trace formula for the action of the idele class group on the noncommutative space quotient of the adeles of k by the multiplicative group of k."

[excerpt:] "[We] construct a quantum dynamical system with partition function the Riemann zeta function, or the Dedekind zeta function in the general number field case. In order for the quantum dynamical system to reflect the arithmetic of the primes it must capture also some sort of interaction between them. This last feature translates in the statistical mechanical language into the phenomenon of spontaneous symmetry breaking at critical temperature with respect to a natural symmetry group. In the region of high temperature, there is a unique equilibrium state as the system is in disorder and symmetric with respect to a natural symmetry group. In the region of low temperature, a phase transition occurs and the symmetry is broken. This symmetry group acts transitively on a family of possible extremal equilibrium states. The construction of a quantum dynamical system with partition function the Riemann zeta function zeta(beta) and spontaneous symmetry breaking or phase transition at its pole beta = 1 with respect to a natural symmetry group was achieved by Bost and Connes in [BC].

A different construction of the basic algebra using crossed products was proposed by Laca and Raeburn and extended to the number field case by them with Arledge in [ALR].

An extension of the work of Bost and Connes to general global fields was done by Harari and Leichtnam in [HL]. The generalisation proposed by Harari and Leichtnam in [HL] fails to capture the Dedekind zeta function as partition function in the case of a number field with class number greater than 1. Their partition function in that case is the Dedekind zeta function with a finite number of non-canonically chosen Euler factors removed. This prompted the author's paper [Coh1] where the full Dedekind zeta function is recovered as partition function. This is achieved by recasting the original construction of Bost and Connes more completely in terms of adeles and ideles."

[BC] J.-B. Bost and A. Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457.

"In this paper, we construct a natural C*-dynamical system whose partition function is the Riemann zeta function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax + b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings Z provides the desired C*- dynamical system, which admits the zeta function as partition function and the Galois group Gal(Q cycl/ Q) of the cyclotomic extension Qcycl of Q as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature beta = 1. The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe])."

[excerpt from p.413:] "We shall now describe (the precise motivation will be explained below) a C* dynamical system intimately related to the distribution of prime numbers and exhibiting the above behaviour of spontaneous symmetry breaking."

[ALR] J. Arledge, M. Laca, I. Raeburn, "Semigroup crossed products and Hecke algebras arising from number fields", Doc. Mathematica 2 (1997) 115-138.

[HL] D. Harari and E. Leichtnam "Extension du phenomene de brisure spontanee de symetrie de Bost-Connes au cas des corps global quelconques"

[Coh1] P.B. Cohen, "A C*-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking", soumis aux Actes des Journees Arithmetiques de Limoges, 1997. Preprint de l'IRMA de l'UST de Lille.

[abstract:] "This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field."

M. Planat explains:

"[This paper] is quite remarkable: it still generalizes the 1995 Bost and Connes paper to a more general Hamiltonian (proposition 1.17), which is the logarithmic determinant of 2 by 2 matrices associated to an integer lattice. Instead of the Riemann zeta function at temperature \beta, the partition function becomes a product of 2 Riemann zeta functions at \beta and \beta -1. This product appears also as a Mellin transform of the logarithm for the number of unrestricted partitions p(n), a function I used in a recent paper, which generalizes Planck theory of radiation"

M. Planat, "Quantum 1/f noise in equilibrium: from Planck to Ramanujan", Physica A 318 (2003) 371

M. Marcolli, "Number Theory in Physics" (survey article, 07/05)

A. Connes and M. Marcolli, "A walk in the noncommutative garden" (preprint 01/06)

[abstract:] "This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the "Tehran program") of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions."

J. Baez, This Week's Finds in Mathematical Physics week 218 contains an illuminating discussion of the work of Connes, Marcolli, Haran, also framing certain issues concerning the bewildering array of zeta and L-functions in terms of category theory.

E. Ha and F. Paugam, "Bost-Connes-Marcolli systems for Shimura varieties" (preprint 03/05)

[abstract:] "We construct a Quantum Statistical Mechanical system $(A,\sigma_t)$ analogous to the Bost-Connes-Marcolli system...in the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the "correct" symmetries and "correct" partition function. We give a formalism that applies to general Shimura data (G,X). The object of this series of papers is to show that these systems have phase transitions and spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature."   [additional background information]

Yu. Manin and M. Marcolli, "Holography principle and arithmetic of algebraic curves", Adv. Theor. Math. Phys. 5 (2001), no. 3, 617–650.

[abstract:] "According to the holography principle (due to G. 't Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS3 holography of black holes. Moreover, in the case of Euclidean AdS2 holography, we present some results on bulk/boundary correspondence where the boundary is a non-commutative space."

M. Planat, P. Solé, S. Omar, "Riemann hypothesis and quantum mechanics" (preprint 12/2010)

[abstract:] "In their 1995 paper, Jean-Beno\^{i}t Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function $\zeta(\beta)$, where $\beta$ is an inverse temperature. We formulate Riemann hypothesis (RH) as a property of the low temperature Kubo-Martin-Schwinger (KMS) states of this theory. More precisely, the expectation value of the BC phase operator can be written as $$\phi_{\beta}(q)=N_{q-1}^{\beta-1} \psi_{\beta-1}(N_q),$$ where $N_q=\prod_{k=1}^qp_k$ is the primorial number of order $q$ and $\psi_b$ a generalized Dedekind $\psi$ function depending on one real parameter $b$ as $$\psi_b (q)=q \prod_{p \in \mathcal{P,}p | q}\frac{1-1/p^b}{1-1/p}.$$ Fix a large inverse temperature $\beta >2.$ The Riemann hypothesis is then shown to be equivalent to the inequality $$\phi_\beta (N_q)\zeta(\beta-1) >e^\gamma \log \log N_q,$$ for $q$ large enough. Under RH, extra formulas for high temperatures KMS states ($1.5< \beta <2$) are derived."

M. Lapidus, In Search of the Riemann Zeros (AMS, 2008)

[from publisher's description:] "In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated modui space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true.

Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical and mathematical or theoretical physics."

[more relevant work by Lapidus listed here]

V. Gayral, B. Iochum and D.V. Vassilevich, "Heat kernel and number theory on NC-torus" (preprint 07/2006)

[abstract:] "The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the number-theoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find non-local counterterms in the NC $\phi^4$ theory on $\T^4$, we show that this theory can be made renormalizable at least at one loop, and may be even beyond."

I. Fesenko, "Several nonstandard remarks" (preprint, 2003)

[abstract:] "This text aims to present and discuss a number of situations in analysis, geometry, number theory and mathematical physics which can profit from developing their nonstandard description or interpretation and then using it to prove standard results and/or establish standard theories."

Section 8 concerns "Nonstandard interpretations of interactions between noncommutative differential geometry and number theory."

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