I want to show that there exist two elliptic curves over $\Bbb C$ that are non-isomorphic.
Let us write $\Gamma_1=\Bbb{Z}\oplus \Bbb{Z}\tau_1$ and $\Gamma_2=\Bbb{Z}\oplus\Bbb{Z}\tau_2$ where $\tau_1,\tau_2\in\Bbb C\backslash\Bbb R$ and $\tau_2\not\in\Bbb{R}\{\tau_1\}$, and consider two elliptic curves $X=\Bbb{C}/\Gamma_1$ and $Y=\Bbb{C}/\Gamma_2$.
(Or in words: Let $\Gamma_1,\Gamma_2$ be non-equivalent rank two $\Bbb{Z}$-lattices in $\Bbb C$. In particular let it be so that $\operatorname{SL}_2(\Bbb Z)\cdot \Gamma_1\not\ni\Gamma_2$.)
Clearly $X,Y$ are topologically equivalent to a torus (and hence one-another).
I want to distinguish the two of these as complex manifolds. Could someone tell me how to do this by means of `periods'? Apparently period integrals can take us back to the lattice, and this distinguishes them, but I don't understand this theory yet.