been studied by Manin and Marcolli [100] who proved a number of deep results
about "limiting" behavior of objects from the usual modular curve H/PSL2(Z)
that tend to points on the boundary.
On the other hand the space of orbits of P 1 under the action of PST^Z ) will
be one of the main examples of our theory in the spherical case; cf. the present
Introduction. It would be beautiful to understand if there is a connection between
the two approaches. Remark that from our perspective P
modulo PSL2(Z) is in a
"spherical situation" while in non-commutative geometry P
(R ) modulo PSZ/2(Z)
is at the boundary of a "hyperbolic commutative geometric object". Flat case. One of the first remarkable objects studied in non-com-
mutative geometry were the non-commutative tori of Connes and Rieffel [36], [40].
There are analogies between these objects and the objects relevant in the flat case
of our theory. Let us briefly explain this in the case of "complex dimension" one. A
complex torus of dimension one is an elliptic curve E over C. One can represent E
as the quotient ET (in the category of Riemann surfaces) of the complex plane C by
the action (by translation) of the subgroup Z + rZ, where r G H. Alternatively one
can describe ET as the quotient (in the category of Riemann surfaces) of C
by the
action of the subgroup (qT) where qT =
As r approaches a point 0 G R\Q ,
qT will approach q$ :=
which is on the unit circle S
c C , but not a root of
unity. In non-commutative geometry it is possible to define an object which can
be viewed as the limit of a family of elliptic curves ET when r 0. Here is the
construction. One considers the action
//a : Z x S 1 - + S 1 , ne(m,z) := q^1 z,
one attaches to this action a groupoid QQ and one defines the C*—algebra AQ :—
C*(Ge); the Morita equivalence class of AQ is interpreted as a non-commutative
elliptic curve which is the limit of Er as r 0. One of the crucial early discoveries
in non-commutative geometry was that for two irrational real numbers 0\ and
02 the C*—algebras AQ1 and AQ2 are Morita equivalent if and only if 0\ and 02
are P5Z,2(Z) conjugate. This justifies the claim in the previous example that
P 1 (R)/P5L2(Z) can be interpreted as a compactification of the modular curve
H/PSX2(Z). (Indeed P1(Q)/PSL2(Z) consists of one point, the classical cusp.)
Now let us note an analogy between the above example and what will be in-
volved in the flat case of our theory. The quotient of
by the action pe is a quotient
S1 /(qe) of the Lie group S1 by a cyclic subgroup (which acts "wildly"). Accordingly
one can propose to study (categorical) quotients of the form G/(gi, ...,gn) where G
is a commutative algebraic group and (gi,...,gn) is a finitely generated subgroup.
Such quotients have actually been studied in differential algebra [12], [13] in re-
lation to the Mordell conjecture over function fields and their study involves the
Manin maps discussed before. As already mentioned Manin maps have arithmetic
analogues that play a key role in this book. Hyperbolic case. Hecke correspondences play an important role in non-
commutative geometry; cf. [39], [37], [38], [101]. On the other hand Hecke cor-
respondences will be at the heart of the hyperbolic case of the present book. We
would like to compare the two theories in what follows. Let, as usual, A/ denote
the finite adeles of Q i.e.
/ : =
( l i m | , ) ® Q .
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