1. TH E NC G DICTIONAR Y 3

many proof s hav e no t bee n include d i n th e text , th e reade r wil l find

references to the relevant literature, where complete proofs are provided

(in particular [33] , [46], [40], and [88]) .

More explicitly, th e text i s organized a s follows:

• W e start b y recalling a few preliminary notion s of noneommu-

tative geometr y (followin g [22]) .

• Th e secon d chapte r describe s ho w variou s arithmeti c proper -

ties of modular curve s can be see n by their "noncommutativ e

boundary". Thi s part i s based on the joint work of Yuri Manin

and th e author . Th e mai n reference s ar e [88] , [89] , [90].

• Th e thir d chapte r include s a n accoun t o f the wor k o f Conne s

and the autho r [33 ] on the noncommutative geometr y o f com-

mensurability classes of Q-lattices. I t also includes a discussion

of the relatio n o f th e noncommutativ e spac e o f eommensura -

bility classe s o f Q-lattice s t o th e Hilber t 12th proble m o f ex -

plicit clas s field theory an d a section on the results of Connes,

Ramachandran an d th e autho r [40 ] o n th e constructio n o f a

quantum statistica l mechnica l syste m tha t full y recover s th e

explicit clas s field theor y o f imaginar y quadrati c fields. W e

also included a brief discussio n o f Manin's rea l multiplicatio n

program [79] , [80] and th e proble m o f real quadratic fields.

• Th e noncommutativ e geometr y o f th e fibers a t "arithmeti c

infinity" o f varietie s ove r numbe r fields i s th e conten t o f th e

remaining chapter , base d o n join t wor k o f Consan i an d th e

author, fo r whic h the references ar e [46] , [47], [48], [49], [50].

This chapte r als o contains a detaile d accoun t o f Manin' s for -

mula fo r th e Gree n functio n o f Arakelo v geometr y fo r arith -

metic surfaces , base d o n [83] , and a propose d physica l inter -

pretation o f this formula , a s in [87] .

1. Th e NC G dictionar y

There i s a dictionary ^cf. [22j y relatin g concept s of ordinary geom -

etry t o th e correspondin g counterpart s i n noncommutativ e geometry .

The entries can be arranged according to the finer structures considere d

on the underlyin g space , roughl y accordin g to the followin g table .

measure theor y

topology

smooth structure s

Riemannian geometr y

von Neuman n algebra s

C*-algebras

smooth subalgebra s

spectral triple s