Let $E$ be an elliptic curve over complex field. To integral holomorphic differential form $ω＝dx/y$ on $E$, we cut riemann surface and glue them together to avoid multiplicity of integrand, and gain a torus.
We integrate $ω$ on torus, but I have question.
Integral on torus may be different from original path on $E($$\mathbb{C}$$)$. For example, if we take path $α$ on $E($$\mathbb{C}$$)$ and corresponding path $α'$ on torus, there is no need of conincidence of two integral $∮ω$ on $α$ and $∮ω$ on $α'$. The integral change if we change the domain.
Nevertheless, why (or in what kind of perspective) can we calculate the integral $∮ω$ on $E$ by studying integral on torus?