For $\tau\in \mathbb H=\{ x+iy\in \mathbb C \lvert x,y\in \mathbb R, \, y>0\}$ and $z\in \mathbb C$, let us define $$ \wp=\wp(\cdot,\tau): \mathbb C \rightarrow \mathbb P^1\, , \quad z\mapsto \frac{1}{z^2}+\sum_{(m,n)}' \frac{1}{(z+m+n\tau)^2}-\frac{1}{(m+n\tau)^2} $$ where $\sum_{(m,n)}'$ stands for the summation over the non-zero pairs $(m,n)\in \mathbb Z^2$.
Question: for $\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in SL_2(\mathbb Z)$ and $z$ arbitrary, is there a way to express $\wp(z,\frac{a\tau+b}{c\tau+d})$ in terms of $\wp(\tilde z,\tau)$, for a certain explicit $\tilde z$ (depending on $a,b,c,d$ and $z$)?
Thanks!
Sure, the formula is $$\wp(\frac{z}{c\tau+d},\frac{a \tau+b}{c \tau+d})=(c \tau+d)^2 \wp(z,\tau).$$