It is well known that all the compact orientable connected Hausdorff genus $1$ surfaces are homeomorphic, but they may have different complex structures.
In fact, consider the following connected region in $\mathbb{C}$. $$G=\{ z\in\mathbb{C}\colon |z|> 1, \Im(z)>0, |\Re(z)|<\frac{1}{2} \}.$$ Consider distinct points $w_1,w_2$ in the region $G$, we can form two lattices: $$\Gamma_1=\mathbb{Z}+w_1\mathbb{Z} \,\,\,\,\mbox{ and } \,\,\,\,\Gamma_2=\mathbb{Z}+w_2\mathbb{Z}.$$ Then $\mathbb{C}/\Gamma_1$ and $\mathbb{C}/\Gamma_2$ are non-equivalent compact Riemann surfaces of genus $1$. This is quite standard fact and its proof can be found in Ahlfors' classic text.
My question is about thinking on this fact in a little different way: can we find some property of Riemann surface $\mathbb{C}/\Gamma_1$ which is not satisfied by other one, so that they are not equivalent?
In algebra or topology, it is very common practice that proving two objects to be non-equivalent involves finding some property of one object which is not satisfied by other object. I am thinking in this way for the problem of non-equivalence of Riemann surfaces of genus $1$.
