1.2. Elliptic functions 7

UJi +UJ2

Figure 1. A parallelogram of a lattice

Thus A is a discrete, free abelian subgroup of C of rank 2, i.e., A is a lattice

(see Figure 1).

A function / : C — C is elliptic with respect to A iff

- / is meromorphic on C

- / is periodic with periods in A,

(1.15) f(u + u) = f(u) for all wGA.

Clearly for the latter condition it suffices that / has two periods u)±, u2 which

generate A. An elliptic function can be regarded as a meromorphic function

on the torus C/A.

The elliptic functions for a lattice A form a field, say E(A). If / has no

poles then it is holomorphic and bounded on C, so by Liouville's theorem /

is a constant function.

Choose a fundamental parallelogram for the lattice A, say

P = ti(Ji +t2UJ2 + 11, 0 *i,*2 lj

with J U G C such that / has neither poles nor zeros on the boundary dP.

Integrating along the boundary, we get by Cauchy's theorem

(1.16) J2

r e s

/M

=

°

weP

because the integrals along the opposite sides cancel out by periodicity.

Hence there is no elliptic function having exactly one simple pole (mod A).

The order of / at w is the integer m = rrif(w) such that

f(u) = (u-w)mg(u)

where g(u) is meromorphic with g(w) ^= 0, oo. Since f(u) is A-periodic, so

is rrif(w). Notice that rrif(w) ^ 0 for at most a finite number of points in a