2
1. OUVERTUR E
of Conne s an d Moscovic i [43] , [44 ] o n th e modula r Heck e algebras .
This show s tha t th e Rankin-Cohe n brackets , a n importan t algebrai c
structure o n modular form s [115] , have a natural interpretatio n i n th e
language of noncommutative geometry , in terms of the Hop f algebr a of
the transvers e geometr y o f codimensio n on e foliations . Th e modula r
Hecke algebras, which naturally combin e products an d actio n of Hecke
operators o n modular forms , ca n be viewed a s the "holomorphi c part 55
of the algebr a o f coordinates o n the spac e of commensurability classe s
of 2-dimensiona l Q-lattice s constructe d i n join t wor k o f Conne s an d
the autho r [33] ,
Cases of occurrences o f interesting numbe r theor y withi n noncom -
mutative geometr y ca n b e foun d i n th e classificatio n o f noncommuta -
tive three-sphere s b y Conne s an d Dubois-Violett e [29] , [30]. Her e th e
corresponding moduli space has a ramified cove r by a noncommutativ e
nilmanifold, wher e the noncommutative analo g of the Jacobia n o f thi s
covering map is expressed naturall y i n terms of the ninth powe r o f th e
Dedekind et a function . Anothe r suc h cas e occur s i n Connes
5
calcula -
tion [26 ] of the explicit cyclic cohomology Chern character of a spectral
triple o n SU q(2) define d b y Chakrabort y an d Pa l [14].
Other instances of noncommutative space s that aris e in the contex t
of numbe r theor y an d arithmeti c geometr y ca n b e foun d i n th e non -
commutative compactificatio n o f modula r curve s o f [28] , [88] . Thi s
noncommutative spac e i s again relate d t o th e noncommutativ e geom -
etry o f Q-lattices . I n fact , i t ca n b e see n a s a stratu m i n th e com -
pactification o f the spac e of commensurability classe s of 2-dimensiona l
Q-lattices (cf. [33]) .
Another context in which noncommutative geometry provides a use-
ful too l fo r arithmeti c geometr y i s i n th e descriptio n o f th e totall y
degenerate fibers a t "arithmeti c infinity 55 o f arithmeti c varietie s ove r
number fields, analyzed in joint work of the author with Katia Consan i
([46], [47] , [48] , [49]).
The present tex t i s based on a series of lectures given by the autho r
at Vanderbil t Universit y i n Ma y 2004 , a s well as on previou s serie s of
lectures given at th e Field s Institut e i n Toronto (2002) , at th e Univer -
sity o f Nottingham (2003) , and a t CIR M i n Lumin y (2004) .
The mai n focu s o f th e lecture s i s th e noncommutativ e geometr y
of modula r curve s (followin g [88] ) an d o f th e archimedea n fibers o f
arithmetic varieties (followin g [46]) . A chapter on the noncommutativ e
space o f commensurabilit y classe s o f 2-dimensiona l Q-lattice s i s als o
included (followin g [33]) . Th e tex t reflect s ver y closel y th e styl e o f
the lectures . I n particular , w e have trie d mor e t o conve y th e genera l
picture tha n th e detail s o f th e proof s o f th e specifi c results . Thoug h
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