2 Consider the unordered triple T 0 l oo of points in C Det

2. Consider the unordered triple T = {0, l, oo} of points in C, Determine all the Mobius t ransofmations m satisfying [12] m(T) = T.

Solution

Let f:P1P1 be a Möbius transformation z(az+b)/(cz+d) sending {0,1,} to {0,1,} with adbc=1

A non-trivial example is f(z)=1/(1z). It sends 0 to 1, 1 to and to 0. Note thatf(z)=z+1 is not an example, because adbc=1 in this case complex-analysis complex-numbers projective-space

If 00, then b=0. This means that d=1a.

If f(1)=1 and f()=, then we must have a=c+d and c=0, so ad=1, a=1. Hence a=d=±1, b=c=0 The only transformation is the identity.

If f(1)= and f()=1, then we must have a=c and c+d=0; since d=1a=c=a, we have a2=1, so a=c=±i, d=i

If 01, then b=d, so (ac)d=1.

If 0, then d=0, so bc=1.