Let $\Lambda = w_1\mathbb Z+w_2\mathbb Z$ be a lattice and $$\wp(z)=\frac1{z^2}+\sum_{w\in\Lambda - 0}\frac1{(z-w)^2}-\frac1{w^2}$$ its associated Weierstrass $\wp$-function. Let $n>1$ be an integer. Denote by $(\mathbb C/\Lambda)[n]$ its $n$-torsion points. Show that for every $n$, there exist polynomials $P_n(X)\in\mathbb C[X]$ such that:
(a) If $n$ is odd, $$(P_n(\wp(z)))^2=n^2\prod_{0\neq u\in (\mathbb C/\Lambda)[n] }(\wp(z)-\wp(u))$$
(b) If $n$ is even, $$\wp'(z)^2(P_n(\wp(z)))^2=n^2\prod_{0\neq u\in (\mathbb C/\Lambda)[n] }(\wp(z)-\wp(u))$$
This was an exam question so it's hinted at in the previous question, but I'm still completely stumped.
Let $u\in(\mathbb C/\Lambda)[n]$. Then $u\equiv-u\mod \Lambda $ if and only if $u\equiv 0\mod\Lambda $ if $n$ is odd, or $u\equiv 0,\frac{w_1}2,\frac{w_2}2\frac{w_1+w_2}2\mod\Lambda $ if $n$ is even.
As is usual with any time the second I post a question here, a burst of inspiration appeared, but feel free to check my work:
In (a), the left hand side is clearly a polynomial in $\wp$ since there are only finitely many $n$-torsion points. Since $u\equiv -u\mod\Lambda$ implies $u\in\Lambda$ and $\wp$ is an even function, then there are two of every term in the product, hence giving us that $f(z)=P(\wp(z))^2$ for some polynomial $P$. Hence assign half of the torsion points to a set $A$, such that $-A$ consists of the other half. We can now take $$P_n^+(X)=n\prod_{u\in A} (X-\wp(u)), \;\; P_n^-(X)=n\prod_{u\in -A} (X-\wp(u))$$ Hence $P_n(\wp)^2=P_n^+(\wp)P_n^-(\wp)$ is indeed the right hand side of (a) by the previous discussion. The second case follows analogously after realizing that the three other terms $w_1/2,\cdots$ correspond to the three zeroes of $\wp'$, and since both $u$ and $-u$ gives us a factor, we obtain $\wp'^2$ in the decomposition.
Let $$H_n = \{ w_1 \frac{a}{n}+w_2\frac{b}{n}, a,b \in 0 \ldots n-1, 0 < a+b\}$$ And for $n$ odd $$h_n = \{ w_1 \frac{a}{n}+w_2\frac{b}{n}, a,b \in 0 \ldots n-1,0 < a+b< n\}$$ Since $\wp(u) = \wp(-u)$ and $H_n = h_n \cup -h_n \bmod \Lambda$ (disjoint union) we have $$ \prod_{u \in H_n} (\wp(z)-\wp(u))=(\prod_{u \in h_n} (\wp(z)-\wp(u)))^2 =P_n(\wp(z))^2 $$