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
smooth structure s
Riemannian geometr y
von Neuman n algebra s
smooth subalgebra s
spectral triple s
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