Let $\wp(u) = \frac{1}{u^2} + \sum\limits\_{\omega \in L, \omega \neq 0} \left(\frac{1}{(u-\omega)^2} - \frac{1}{\omega^2}\right)$ be a Weierstrass pe function.


My question is, how can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$


If $αL=L$, the answer is easy, we can replace sum up over $L$ to sum up over $αL$, $\wp(αu)=1/α^2\wp(u)$.


So the problem is in the case of $αL\subsetneq L$. Are there some known results ?



How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$
$\alpha L$ is of finite index in $L$ so you can write $L = \cup\_{i=1}^n (\alpha L + \lambda\_i)$ for some $\lambda\_i, n$.


If you rewrite the sum defining $\wp(\alpha u)$ as a sum over $\cup\_{i=1}^n (\alpha L + \lambda\_i)$ and rearrange terms (without worrying too much about convergence) you should get something like


$$\wp(\alpha u) = (1/\alpha)^2 \sum\_{i=1}^n \wp(u - \lambda\_i/\alpha) + c$$


for some constant $c$. As Gro-Tsen suggested, you should think in terms of isogenies of elliptic curves and this boils down to Velú's formulas.


