1822 CHAPTER 58. ELLIPTIC FUNCTIONS

This proves the theorem.Hille says this relation is due to Liouville. There is also a simple corollary which

follows from the above theorem applied to the elliptic function, f (z)− c.

Corollary 58.1.9 Let f be a non constant elliptic function and suppose the function, f (z)−c has poles p1, · · · , pm and zeros, z1, · · · ,zm on Pα , listed according to multiplicity where∂Pα contains no poles or zeros of f (z)− c. Then ∑

mk=1 zk−∑

mk=1 pk ∈ M, the module of

periods.

58.1.1 The Unimodular TransformationsDefinition 58.1.10 Suppose f is a nonconstant elliptic function and the module of periodsis of the form {aw1 +bw2} where a,b are integers and w1/w2 is not real. Then by analogywith linear algebra, {w1,w2} is referred to as a basis. The unimodular transformationswill refer to matrices of the form (

a bc d

)where all entries are integers and

ad−bc =±1.

These linear transformations are also called the modular group.

The following is an interesting lemma which ties matrices with the fractional lineartransformations.

Lemma 58.1.11 Define

φ

((a bc d

))≡ az+b

cz+d.

Thenφ (AB) = φ (A)◦φ (B) , (58.1.3)

φ (A)(z) = z if and only ifA = cI

where I is the identity matrix and c ̸= 0. Also if f (z) = az+bcz+d , then f−1 (z) exists if and only

if ad− cb ̸= 0. Furthermore it is easy to find f−1.

Proof: The equation in 58.1.3 is just a simple computation. Now suppose φ (A)(z) = z.

Then letting A =

(a bc d

), this requires

az+b = z(cz+d)

and so az+b = cz2 +dz. Since this is to hold for all z it follows c = 0 = b and a = d. Theother direction is obvious.