xxviii INTRODUCTION

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

1

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".

0.3.5.2. 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

x

by the

action of the subgroup (qT) where qT =

e2nir.

As r approaches a point 0 G R\Q ,

qT will approach q$ :=

e2nie

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

S1

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.

0.3.5.3. 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.

A

/ : =

( l i m | , ) ® Q .