1.2. Elliptic functions 7
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
r e s
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