0.3. COMPARISON WITH OTHER THEORIES xxix

Then, following Shimura, one considers the space

(0.30) Sh = GL2(Q)\(GL2(Af) x H

±

) .

For any congruence subgroup V C SL(2, Z) consider the modular curve

Shr := H / r ,

where we wrote H / r instead of T\H in order to make notation agree with notation

in the present book. One can consider then the inverse limit

Sh° :=limShr

over all congruence subgroups T C SL2(Z). The space Sh° turns out to be a

connected component of Sh. All this is classical, i.e. commutative. Now, following

[38] one can embed the set Sh into the "much larger" set

S/inc

:= GL

2

(Q)\(M

2

(A

/

) x H

±

) .

The set

Sh^nc^

does not have a nice "commutative space" structure but one can

attach to it a C*—algebra A2 starting from a certain (alternative !) presentation

of

Sh^nc^

as a quotient by an action; cf. [101], p. 67. We refer to loc. cit. for

an overview of the beautiful and deep theory of A2 and also for an overview of the

Bost-Connes A\ that was its GL\ prototype.

Now turning to our theory let us consider a prime number /, the matrix

n : =

and the subgroup

I 0

0 1

r,r,)cGL2(Q)

(where T C SL2(Z) is any congruence subgroup). Define the set

S7irz) :=H/(r,r/) .

This set should be viewed, roughly, as the quotient of Shr by a Hecke correspon-

dence. Since the group (r,r/) does not act properly discontinuously on H the set

5/i(r,rz is n °t an object of analytic geometry anymore. Our theory can be viewed as

a way to put on the set Sh^^n) a geometric structure in 5—geometry. Summarizing

we have the following inclusions of sets

Sh° cShc

Sh{nc)

and the following projections of sets

Sh° — Shr — 5/i(r,n)-

The set S/i(nc) is replaced, in non-commutative geometry, by the C*—algebra A2

while the set Sh^^Tl) is replaced, in our theory, by an object of S—geometry. It is

plain from the above discussion that the non-commutative theory and our theory

"diverge" in this example. The set we are interested in, 5/i(r,rz ?ls a "wild" quotient

of the "commutative part" of Sh^nc\