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