If we observe the correspondence $$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$ we see the relationship between weight 2 and weight 3 modular forms, the coefficients of an elliptic curve, and a lattice $\Lambda$, i.e. a free $\mathbb{Z}$ module of rank 2.

Is there a higher dimensional correspondence between $\mathbb{C}/\Lambda$ and an abelian surface or abelian variety when $\Lambda$ is a free $\mathbb{Z}$-module of rank $d$? Does the equation(s) describing such an object include coefficients with higher weight modular forms like $g_{4}(\Lambda),g_{5}(\Lambda),...?$ Is there a way to embed odd-dimensional spaces in $\mathbb{C}^{2g}$, where $g$ is the genus, and then mod; I'm particularly interested in the case of free $\mathbb{Z}$-modules of rank $3$.