Liouville's theorems on elliptic functions

Remarks

1. An elliptic function without poles is continuous on the closure of $P$, hence bounded on $\C$ by periodicity, hence constant by Liouville's theorem.
2. By the residue theorem, the sum of the residues is $\frac{1}{2\pi i}\int_{\partial P} f(z)\,dz$, and the integrals over opposite sides cancel by periodicity.
3. Apply part 2 to the elliptic function $f'/(f - c)$ and use the argument principle.

The common number in part 3 is the order of $f$. Part 2 shows that the order is never $1$: a single simple pole would have a nonzero residue. So a nonconstant elliptic function has order at least $2$; the Weierstrass function $\wp$ has order $2$, with one double pole in $P$. A further result of the same kind says that the sum of the zeros minus the sum of the poles in $P$, with multiplicity, lies in $\Lambda$. Each point of $\C$ is congruent modulo $\Lambda$ to exactly one point of $P$, so the counts in parts 2 and 3 do not depend on $a$. The contour arguments use a choice of $a$ for which $f$ has no zeros, poles or $c$-points on the boundary of $P$.

Going deeper

For a student

All three proofs use one idea: integrate around the boundary of a period parallelogram, where opposite sides cancel by periodicity. The results show that the smallest possible order of a nonconstant elliptic function is $2$. This is why the Weierstrass function has a double pole and the Jacobi functions have two simple poles per cell. On the torus $\C/\Lambda$, they are the first instances of general facts on compact Riemann surfaces. There a holomorphic function is constant, and a meromorphic function has as many zeros as poles.