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."

Alain Connes-related material

A. Connes, "Noncommutative Geometry, the spectral standpoint" (preprint 10/2019)

[abstract:] "We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum—Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic cohomology, 5) The increasing role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology, 6) The understanding of the renormalization group as a motivic Galois group, 7) The development of quantum field theory on noncommutative spaces, 8) The discovery of a simple equation whose irreducible representations correspond to $4$-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity, 9) The discovery that very natural toposes such as the scaling site provide the missing algebro-geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of $L$-functions."

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*."

P. Cohen "Dedekind zeta functions and quantum statistical
mechanics"

[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
**Q**^{cycl} 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.

A. Connes
and M.
Marcolli, "From Physics to Number
Theory via Noncommutative Geometry. Part I: Quantum Statistical
Mechanics of **Q**-lattices" (preprint 04/04)

[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"

The "recent paper" was this one:

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

M. Marcolli and A. Connes, "From
physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann-Hilbert correspondence,
and motivic Galois theory", from *Frontiers in Number Theory,
Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems* (Springer, 2006)

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 AdS_{3} holography of
black holes. Moreover, in the case of Euclidean AdS_{2} 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."

M. Nardelli, "On the possible mathematical connections concerning
noncommutative minisuperspace cosmology, noncommutative quantum cosmology in low-energy string action,
noncommutative Kantowsky-Sachs quantum model, spectral action principle associated with a noncommutative space
and some aspects concerning the loop quantum gravity" (preprint, 2007)