@battaglia01 - Lattices in ℂⁿ give complex tori, and the nice ones give 'abelian varieties': complex tori that are projective varieties. Only the latter really act like generalizations of elliptic curves. The moduli space of elliptic curves, and the theory of modular forms, generalizes to higher dimensions this way. But people discovered you must work with abelian varieties equipped with an extra structure, a 'polarization', to get this to work. When we do, we're led to study 'Siegel modular forms'. But unfortunately it seems the Eisenstein series trick for getting modular forms doesn't work in higher dimensions, because this sum over points ℓ in a lattice Λ⊂ℂⁿ:
[ \sum_{\ell \in \Lambda} \frac{1}{(z-\ell)^n} ]
only makes sense when (n = 1). You might hope that some trick would save us, but I haven't seen any analogue of Eisenstein series in higher dimensions. I'm just learning this theory, so I might have missed something, but I've looked around.