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\
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