1820 CHAPTER 58. ELLIPTIC FUNCTIONS

Proof: Since f has no poles it is analytic. Now consider the parallelograms determinedby the vertices, mw1 + nw2 for m,n ∈ Z. By periodicity of f it must be bounded becauseits values are identical on each of these parallelograms. Therefore, it equals a constant byLiouville’s theorem.

Definition 58.1.5 Define Pa to be the parallelogram determined by the points

a+mw1 +nw2,a+(m+1)w1 +nw2,a+mw1 +(n+1)w2,

a+(m+1)w1 +(n+1)w2

Such Pa will be referred to as a period parallelogram. The sum of the orders of the poles ina period parallelogram which contains no poles or zeros of f on its boundary is called theorder of the function. This is well defined because of the periodic property of f .

Theorem 58.1.6 The sum of the residues of any elliptic function, f equals zero on every Paif a is chosen so that there are no poles on ∂Pa.

Proof: Choose a such that there are no poles of f on the boundary of Pa. By periodicity,∫∂Pa

f (z)dz = 0

because the integrals over opposite sides of the parallelogram cancel out because the valuesof f are the same on these sides and the orientations are opposite. It follows from theresidue theorem that the sum of the residues in Pa equals 0.

Theorem 58.1.7 Let Pa be a period parallelogram for a nonconstant elliptic function, fwhich has order equal to m. Then f assumes every value in f (Pa) exactly m times.

Proof: Let c ∈ f (Pa) and consider Pa′ such that f−1 (c)∩Pa′ = f−1 (c)∩Pa and Pa′

contains the same poles and zeros of f − c as Pa but Pa′ has no zeros of f (z)− c or polesof f on its boundary. Thus f ′ (z)/ ( f (z)− c) is also an elliptic function and so Theorem58.1.6 applies. Consider

12πi

∫∂Pa′

f ′ (z)f (z)− c

dz.

By the argument principle, this equals Nz−Np where Nz equals the number of zeros off (z)− c and Np equals the number of the poles of f (z). From Theorem 58.1.6 this mustequal zero because it is the sum of the residues of f ′/( f − c) and so Nz = Np. Now Npequals the number of poles in Pa counted according to multiplicity.

There is an even better theorem than this one.

Theorem 58.1.8 Let f be a non constant elliptic function with poles p1, · · · , pm and zeros,z1, · · · ,zm in Pα , listed according to multiplicity where ∂Pα contains no poles or zeros of f .Then ∑

mk=1 zk−∑

mk=1 pk ∈M, the module of periods.