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 4 and weight 6 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 $d$? Do(es) 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 $3$.



Complex plane mod lattice to elliptic curve correspondence generalization
There is a higher dimensional correspondence that works for some full-rank lattices $\Lambda \subseteq \mathbb{C}^{g}$, but not all! The reference I know for this is Hindry and Silverman's book "Diophantine Geometry". Theorem A.5.0.1 states that the complex torus $\mathbb{C}^{g}/\Lambda$ is an abelian variety if and only if there exists a positive definite Hermitian form on $\mathbb{C}^{g} \times \mathbb{C}^{g}$ whose imaginary part takes integer values when restricted to $\Lambda \times \Lambda$. They give examples to show that this is sometimes true, and sometimes false. (In this construction, $\Lambda$ must be a lattice of full rank in $\mathbb{C}^{g}$, and hence $\Lambda$ must have even rank.)


There are some analogues in these higher-dimensional cases of the results for $g = 1$. There is a vector space of Riemann theta functions that gives an embedding of $\mathbb{C}^{g}/\Lambda$ into projective space. The coefficients of the resulting equation for the abelian surface will involve Siegel modular forms. This is rather painful to do explicitly. (In Cassels's and Flynn's book "Prolegomena to a middlebrow arithmetic of curves of genus 2", they embed abelian surfaces in $\mathbb{P}^{15}$, and the equations defining them are a system of 72 quadrics. In the paper "Defining equations of the universal abelian surfaces with level three structure" by Gunji (Manuscripta Math, 2005), abelian surfaces are embedded in $\mathbb{P}^{8}$.)


