MA475 Example Sheet 4
10 March 2017
1. Show that the degree of the composition of two maps between compact
Riemann surfaces is the product of the degrees.
2. Show that
f (z) =
∞
X
1
(z − n)2
n=−∞
is holomorphic and well defined on C/Z.
3. Let ℘ be the Weierstrass ℘-function with respect to a lattice Λ ⊂ C.
Show that ℘ satisfies the differential equation ℘00 (z) = 6℘(z)2 + A for
some constant A. Show that there are at least three points and at most
five points in C/Λ at which ℘0 is not locally injective.
4. Show that the unit disc and the upper half plane are conformally equivalent by means of the linear fractional transformation f : U HP →
Disk
z−i
.
f (z) =
z+i
5. Show that the map f from the previous problem takes the metric ds2 =
|dz|2
4|dz|2
to the metric ds2 = (1−|z|
2 )2 .
|Im(z)|2
6. Let a1 , a2 , a3 be distinct points in CP1 . Show that there is a unique
Möbius transformation that takes 0, 1, ∞ to a1 , a2 , a3 (respecting the
order). Show that there is a Möbius transformation that takes any
ordered triple of distinct points to any other. Define the cross ratio of
a1 , a2 , a3 , a4 to be the image of a4 under the unique map that takes the
ordered triple a1 , a2 , a3 to the ordered triple 0, 1, ∞. Compute the cross
ratio. Show that two ordered quadruples are equivalent if and only if
they have the same cross ratio.
7. Show that no two of the following domains are conformally equivalent:
{1 < |z| < 2},
{0 < |z| < 1},
{0 < |z| < ∞}
8. (a) Let A and B be matrices that commute. Show that A takes eigenvectors of B to eigenvectors of B.
(b) Let α and β be elements of P SL(2, R) neither of which is the
identity. Show that α and β preserve the same geodesic, horocycle
or point in the U HP . Conclude that the group generated by α
and β cannot act discretely and freely on the U HP .
(c) Conclude that the torus cannot be given a hyperbolic structure.
It follows that the topology of a surface (not necessarily compact)
determines its “conformal type” with exactly two exceptions: the
disk and the annulus.
9. Show that the U HP is conformally equivalent to the strip S = {z ∈
C : 0 < Im(z) < π}. Let Γc be the group generated by z 7→ z + c.
Let Ac = S/Γc . Show that two annuli Ac and Ac0 are holomorphically
equivalent if only if c = c0 .