2
1. The modular group and elliptic function theory
1. Mobius transformations
The material in this book deals mostly with compact Riemann surfaces and
subgroups of PSL(2,Z). We start with a discussion of the more general
group: PSL(2, C). Good additional references for this introductory material
are [4, Ch. 4], [12, Ch. I] and [21, Ch. I].
An element A = \ G PSL(2,C) acts on the Riemann sphere
C = CU{oo} as the Mobius transformation
az + b
Z
H-
cz + a
We think of PSL(2,C) as a topological group with the following topology.
Consider SL(2,C) as the closed subset of complex euclidean 4-space defined
by the equation
(a, 6, c, d) G
C4;
ad be = 1,
and give PSL(2,C) = SL(2,C)/{±J} the quotient topology.
It is convenient to introduce a most important and useful classification
a b
c d
for Mobius transformations. Let A =
(recall that ad bc= 1)
A is parabolic 4=
tr2(A)
= (a +
d)2
= 4,
A is elliptic ^= 0
tr2(A)
4,
A is hyperbolic =^
tr2(^4)
4,
G PSL(2,C), A^L Then
and
A is loxodromic = tr (A) 0 [0,
We note that in our definition hyperbolic is a special case of loxodromic.
The (square of the) trace of an element of PSL(2,C) is a conjugacy class
invariant; that is,
tr(i4) - tr(B o i o
B~l)
for all A,B G SL(2,C).
Conversely, if A ^ / and B ^ I are two nontrivial Mobius
transformations6
with the same squared trace, then they are conjugate in PSL(2,C); if the
two motions are in PSL(2,R) and they have the same squared trace, then
we can only conclude that A is conjugate in PSL(2,E) to either B or
B~1,
6We denote the identity matrix and the identity Mobius transformation by the same symbol
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