COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

372 Harvey Cohn
We also consider GZ the subgroup (of matrices or transformations) for which
S = .E (mod 2)
where E is the unit matrix.
(2.4)
The fundamental domain F for G is classically given by the region t;
shown in Fig. 1. Thus F is determined by the inequalities
!Rez(ci
(2.5)
1+-l I
with boundary identified by making 00 A coincide with 00 C according to
zo = z+ 1 while AB coincides with CB according to z. = - l/z. (We have
compactified, of course, by adjoining -.) (See 131, pp. 84, 127.)
A @ C
FIG. 1. Fundamental domain for G and G,. We see F with floor ABC projected
onto segment AC on the left. We see F, assembled from six replicas of Fl on the
right so as to form a 2-sphere.
In a one-dimensional world, we would see the floor of the region P or
arc ABC projected as segment ABC (see interval in Fig. 1). Also, the walls
of the region Fare trivial by comparison. They are merely the boundaries
of the fundamental region for G” (the subgroup of G which Ieaves -
unchanged). Here Gm is simply
z. = zfn (n integral). (2.6)
To visualize the fundamental domain F2 for Gz we would note that Gz
is a subgroup of G of index 6 with cosets determined by
s1= (:, y), s, = (A i), s, = (; --A),
s4 = (y I;), s, = (; -3, s, = (f 0).
I
(2.7)
Algebraic topology on bicomplex manifolds
Each right coset G&,, in G relocates F in a well-defined manner (to
within equivalences under Gz). Thus Fz consists of six replicas shown
at the left of Fig. 1. (Naturally FS has no floor since it touches the real
axis at 0 and 1.) It is easy to see, from the diagram on the right of Fig. 1,
how the fundamental domain Fz becomes a sphere under “trivial boundary
identifications”. The trivial boundary identifications are possible only
because the floor of F, namely IzI = 1, is mapped into itself under
zo= -l/z, the transformation mapping F into the region (OABC)
immediately below it.
A deceptively simple intermediate stage is provided by the Picard modular
group (like Klein’s except that in (2.2), a, b, c, dare Gaussian integers).
Here the three-dimensional representation makes for a simple analogy
to Fig. 1 and indeed the analog of G and the analog of Gz are 3-spheres
(see [61).
We know that going to four dimensions, the domain of definition of
an algebraic function field in two complex variables cannot be a 4-sphere.
Therefore, we know some degree of dif%culty must be encountered in extending
the construction of FZ to two complex variables!
3. Hilbert modular group. We summarize the construction of the fundamental
domain, here, only in sufficient detail to define necessary terms
and symbols. The justification appears in earlier work {[l], [2]).
The theory is restricted to the quadratic field Q (2Z). We deal with
three closely related groups,
r* = (ordinary) Hilbert modular group,
r = symmetrized Hilbert modular group,
rZ = subgroup of r z E (mod 2’) (principal congruence subgroup).
Here we have the Cartesian product UX U of two upper half planes
written as “formal” conjugates z, z’
Im z z 0, Im 2’ z 0. (3.1)
We define I’* as the group of linear transformations (sometimes called
“hyperabelian”),
zo = Z(z) = (az+/q/(yz+Q, z; = Z’(z’) = (a’z’+,Y)/(y’z’+ 6’) (3.2)
where a, b,. . . , a’, ,5’, . . . are conjugate algebraic integers in Q ( 2; ) and
ad--fly = ef, a’#-16’7’ = (&$“’ (3.3)
where e. = 1+2+ is the fundamental unit
an integer. The corresponding matrices are
and likewise for the conjugate.
( ES =
0
373
3+2.2; 1 , and t is
(3.4)