My question is motivated by a statement in Silverman's The Arithmetic of Elliptic Curves.
We observe that $H_1(\mathbb{C}/\Lambda_1, \mathbb{Z})$ is naturally isomorphic to the lattice $\Lambda_1$ via the map $\gamma \mapsto \int_\gamma dz$.
My understanding of integration on manifolds is rather weak, but I don't think that this $dz$ is a differential form as I've seen in my differential geometry course. Silverman also treats $E(\mathbb{C})$ (for $E$ an elliptic curve with a fixed Weierstrass equation) as a complex Lie group and works with the invariant differential $dx/y$ analytically; this also makes no sense to me, and he seems to manipulate these like they're standard differential forms when proving theorems. So, for $\gamma \in H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ and $\gamma' \in H_1(E(\mathbb{C}), \mathbb{Z})$, can someone please help with how I should be thinking about the two integrals
$$\int_\gamma dz$$ and
$$\int_{\gamma'} \frac{dx}{y}?$$