zeros of $\wp (u)-\wp (w)$. (Weierstrass elliptic function)

Question

In Iwaniec's Topics in Classical Automorphic Forms, he says (chapter 1.2 equation 1.21)

$\wp(u)-\wp(w)$ has a double zero at $u=w$ exactly when $w\equiv-w \pmod{\Lambda}$.

Can anyone show me how to prove this both ways?

Answer 1

Fix $w$ and write $f(u)=\wp(u)-\wp(w)$. As $f$ has two poles in each fundamental domain, it also has two zeros, which must be simple and at $w$ and $-w$, as long as $w$ and $-w$ are distinct modulo $\Lambda$.

But if $w\equiv -w\pmod\Lambda$, $f(w)=0$ and $f'(z)=\wp'(z)$ also has a zero at $w$. This is because $f'(u+2w)=f'(u)=-f'(-u)$, so putting $u=-w$ gives $f'(w)=-f'(w)$. So $f$ has a double zero at $w$. This double zero accounts for all zeroes in the fundamental domain.